Optics and Lens Design

7 

Optics and Lens Design MORLEY FORD and JANET STONE, revised by RONALD RABBETTS

CHAPTER CONTENTS

Practical Effects of Optical Differences Between Contact Lenses and Spectacles, 132 Optical Considerations of Contact Lenses on the Eye,  145 Astigmatism of the Front Surface of the Cornea and the Effect of the Liquid Lens, 146

Some aspects of contact lens design and optics are rarely applied nowadays, as they are less relevant to modern practice, but readers may still need to refer to them. They are available at available at: https://expertconsult.inkling.com/, whereas the more relevant topics are included here. The appendix, available at available at: https://expertconsult. inkling.com/, also describes various equations that are used in contact lens work, while the online resources include programs relating to both the optics and to lens design. There are two main aspects to be considered when dealing with the optics of contact lenses: the effects on the wearer of the optical differences from spectacles and the necessity for the practitioner to understand the components which affect the back vertex power (BVP) of the contact-lens/liquid*-lens system. There is some overlap of these two aspects, but for the sake of convenience they are discussed separately in the first two sections of the chapter. In the second section a set of approximate rules is included, the use of which should permit practitioners to make quick and reasonably accurate estimates of changes in power caused by altering certain lens parameters. The Cartesian sign convention2 is used throughout. For further understanding of the basic principles involved, readers are referred to the works of Bennett (1985), Tunnacliffe (1993), Freeman and Hull (2003), Douthwaite (2006) and Rabbetts (2007).

Practical Effects of Optical Differences Between Contact Lenses and Spectacles SPECTACLE AND OCULAR REFRACTION: EFFECTIVITY

Toric Contact Lenses,  148 Power Variations as a Sequel to Other Changes, 150 Bifocal Contact Lenses,  153

differ significantly from the spectacle lens power, especially in higher powers. To correct fully an eye’s refractive error, the BVP of the contact lens† must equal the ocular refraction. The symbol Ko is used rather than the simple letter K that is generally used in texts on visual optics to avoid possible confusion with the keratometer reading, also often denoted as ‘K’. The principle of effectivity is shown in Fig. 7.1, which illustrates a hypermetropic eye viewing a distant object and corrected by a spectacle lens of power Fsp and hence of focal length f ′sp, giving the position of the far point plane M′R. The distance from the corneal vertex to this plane, k, is the distance f ′sp reduced by the vertex distance d, so the required power of the contact lens and the ocular refraction Ko are given by: k = f ′ sp − d , hence Ko =

Fsp 1 = f ′ sp − d 1 − dFsp

or approximately ≈ Fsp (1 + dFsp ) = Fsp + dFsp2 KEY POINT

Since dFsp2 is always positive, the ocular refraction and hence the contact lens power for a hypermetrope is always greater than the spectacle refraction, whereas it is numerically less for a myope.

Because the spectacle lens is positioned 10 mm or more in front of the cornea, the effective power of the lens at the cornea may

Conversely, given Ko, the spectacle refraction is given by: Fsp =

*The ‘liquid lens’ between a contact lens and the cornea is also known as the ‘tears lens’. 2 In the Cartesian sign convention, light moves from left to right. Distances to the right of the lens are positive, and to the left of the lens are negative. Distances above the optic axis are positive and below the optic axis are negative.

132

†

Ko 1 + dK o

Strictly speaking, for rigid lenses, the power of the contact-lens/liquid-lens system.

133

7  •  Optics and Lens Design F ′s

Table 7.2  Comparison of Accommodation With Spectacles and Contact Lenses

L

Ocular Accommodation (D) at 0.33 Metre From Spectacle Plane Spectacle Refraction (D)

d fs ′

l

Fig. 7.1  Effective power of spectacle lens at contact lens is L = 1/l = Fs′/ (1 − dFs′) (see Appendix A and Formula I on the CD-ROM).

Ocular refraction and Difference (ocular - spectacle) (D)

30

Ocular Difference

25 20 15

0.25 Metre From Spectacle Plane

Spectacles

Contact Lenses

Spectacles

Contact Lenses

−20

1.89

2.90

2.50

3.82

−15

2.09

2.90

2.76

3.82

−10

2.32

2.90

3.06

3.82

−5

2.58

2.90

3.40

3.82

0

2.90

2.90

3.82

3.82

+5

3.27

2.90

4.31

3.82

+10

3.72

2.90

4.89

3.82

+15

4.25

2.90

5.61

3.82

10 5 0

KEY POINT

–5

An observer sees the wearer’s eyes looking their normal size in contact lenses – smaller than with spectacles for a hypermetrope and larger for a myope.

–10 –15 –20

–20

–15

–10

–5 0 5 10 Spectacle lens power (D)

15

20

Fig. 7.2  Ocular refraction and the difference between the ocular refraction and the spectacle refraction, both as a function of the spectacle refraction.

Disturbing appearances due to the prismatic effects and surface reflections of spectacle lenses are also removed.

ACCOMMODATION KEY POINT

Fig. 7.2 shows the ocular refraction as a function of the spectacle correction and the difference between the two values. For astigmatic prescriptions, the calculation must be made individually for the two principal powers since the cylindrical power also changes, as shown in the example below. If the prescription had been for a hypermetrope, then the ocular cylinder would have been greater than the spectacle cylinder, as predicted by the steeper curve in Fig. 7.2. Vertex distance tables using this calculation are given in Appendix A at the end of this book, while the spherical rigid gas permeable (RGP) lens calculator available at: https://expertconsult.inkling.com/ includes this conversion (Table 7.1). Table 7.1  Spectacle and Ocular Refraction Spectacle refraction

−6.00/−2.00 × 180

Back vertex distance

12.5 mm

Spectacle refraction in crossed cylinder form

−6.00 × 90/−8.00 × 180

Ocular refraction in crossed cylinder form after allowing for vertex distance

−5.58 × 90/−7.27 × 180

Ocular refraction

−5.58/−1.69 × 180

COSMETIC APPEARANCE Besides the change in appearance achieved by removing the need for spectacles, the magnification or minification effect of the spectacle lenses is eliminated.

More accommodation is required by myopes and less by hypermetropes when they transfer from spectacles to contact lenses (Table 7.2 and Fig. 7.3). To calculate the ocular accommodation (A) (the demand on accommodation to be exerted by the eye), it is necessary to determine the ocular refraction (Ko) and the distance (b) from the eye at which the near object of regard is imaged by the spectacle lens of power Fsp. For example, Fig. 7.4a shows a myope wearing a spectacle lens of −8.00D, who reads at a distance (l) 25 cm from the spectacle plane. Thus the demand on spectacle accommodation (Asp) is 4D. Now if the spectacle lens is worn 12 mm from the eye, the ocular refraction (Ko) is −7.30D. In Fig. 7.4a the near object, O, is imaged by the spectacle lens at O′. Now l = −250 mm Therefore, L = −4 D Fsp = −8 D Therefore, L ′ = −12 D Thus, l ′ = −83.3 mm But b = l ′ − d = −83.3 − 12 = −95.3 mm 1 b (in metres) = −10.5 D

B=

134

SECTION 3  •  Instrumentation and Lens Design

But −7.30D of this corrects the ocular refraction. The remaining −3.20D must be overcome by the use of the myope’s accommodation, i.e. A = Ko − B. This demonstrates the effectivity of the spectacle lens in permitting such a myope to need only 3.2D of accommodation, whereas if a contact lens were worn, the same near object would be 262 mm from the eye, necessitating 3.82D of accommodation. Fig. 7.4b shows a similar situation for a hypermetrope. Fsp = +8.00 D, l = −250 mm and d = 12 mm. Thus Ko = +8.85D.

6

Now L = −4 D, Fsp = +8 D Therefore, L ′ = +4 D and l ′ = +250 mm But b = l ′ − d = +250 − 12 = +238 mm Therefore, B = +4.2 D Since A = K o − B, A = +8.85 − 4.2 = +4.65 D. This demonstrates how the ocular accommodation of a hypermetrope wearing spectacles is greater than that required when contact lenses are worn. In this example, 4.65D of accommodation is required compared with 3.82D in contact lenses.

PRISMATIC EFFECTS

Contact lenses

cle

s

1. Those of spectacle lenses during convergence. 2. Those due to an anisometropic spectacle correction when the eyes make version movements.

cta

5

Spe

Accommodation (D)

There are two oculomotor imbalance effects caused by spectacles:

Convergence Spectacles optically centred for distance vision but which are used for all distances of gaze exert a prismatic effect during off-axis viewing. Contact lenses which move with the eyes remain centred (or nearly so) for all distances, and positions of gaze cause no such prismatic effect. The resulting effect is that:

4

KEY POINT 3

During near vision:

Contact lenses

• a spectacle-wearing myope experiences a base-in prismatic effect • a spectacle-wearing hypermetrope experiences a base-out effect (see Fig. 7.5).

les

tac

c Spe

-20

-15

Therefore for a given object distance:

2

-10

-5 0 +5 Spectacle refraction (D)

+10

+15

Fig. 7.3  Accommodation with spectacles and contact lenses. Dotted lines at 0.25 metre; continuous lines at 0.33 metre using the figures from Table 7.2 (compare this with Fig. 7.6).

(a)

• a contact-lens-wearing myope has to exert more convergence than with spectacles • a contact-lens-wearing hypermetrope has to exert less convergence than with spectacles. Table 7.3 gives the amount of convergence in prism dioptres (Δ) exerted by both eyes in various degrees of ametropia,

(b)

Fig. 7.4  Near vision through spectacle lenses. The near object of regard, O, is imaged at O′, which is at a distance b from the eye. The object, image and vertex distances from the spectacle lens are l, l′ and d respectively. (a) myopia; (b) hypermetropia.

135

7  •  Optics and Lens Design O

Table 7.3  Comparison of Binocular Convergence with Spectacles and Contact Lenses Convergence (prism dioptres, Δ) at 0.33 Metre From Spectacle Plane

O R' Myope: base-in effect

O L'

Spectacle Refraction (D)

O R'

O L'

0.25 Metre From Spectacle Plane

Spectacles

Contact Lenses

Spectacles

Contact Lenses

−20

11.11

16.66

14.56

21.66

−15

12.11

16.66

15.87

21.66

−10

13.33

16.66

17.56

21.66

−5

14.80

16.66

19.31

21.66

0

16.66

16.66

21.66

21.66

+5

19.03

16.66

24.67

21.66

+10

22.19

16.66

28.63

21.66

+15

26.64

16.66

34.14

21.66

O 35

Hypermetrope: base-out effect

les ctac

31

Spe

Convergence (∇)

33

29 27 25 23

Fig. 7.5  Spectacles centred for distance vision give prismatic effects when the eyes converge.

Contact lenses 21

assuming spectacles centred for a distance CD of 60 mm and worn 27 mm in front of the eyes’ centres of rotation*, and also assuming contact lenses giving an equivalent power, remaining centred for all distances of gaze and worn 15 mm in front of the eyes’ centres of rotation. The method of calculation of the convergence in prism dioptres (Δ) is as described below for the calculation of prismatic effects in anisometropia. Table 7.3 is used as a basis for the graph in Fig. 7.6. The significance of this difference in convergence must be considered in association with changes in accommodation (see ‘Accommodation’, p. 137), where it is shown that the ratio between accommodation and convergence remains the same with spectacles and contact lenses. The effect of the change in convergence alone, when transferring *For convenience, the centre of rotation of the eye is taken as 15 mm behind the anterior pole of the cornea, and the spectacle back vertex distance is 12 mm.

19 Contact lenses

17 15

s

cle

cta

e Sp

13 11

-20

-15

-10

-5

0

+5

+10

+15

Spectacle refraction (D) Fig. 7.6  Convergence with spectacles and contact lenses. Dotted lines at 0.25 metre; continuous lines at 0.33 metre using the figures from Table 7.3 (compare this with Fig. 7.3).

136

SECTION 3  •  Instrumentation and Lens Design

from spectacles to contact lenses, is most likely to prove difficult in a myope whose near point of convergence is abnormally remote, when the removal of the base-in prism may be sufficient to disrupt binocular vision at near; the change may also prove difficult in wearers with unusually high or low AC/A (accommodative convergence: convergence) ratio. KEY POINT

The general effect of transferring from spectacles to contact lenses for near vision is as if the myope had brought the near task a little closer, since more convergence and accommodation are required, whereas for the hypermetrope the reverse applies and it is as if the working distance had been increased.

Anisometropia Since contact lenses move with the eyes, the visual axes always pass through their optical centres or very close to them. Thus differential prismatic effects which can create difficulties for anisometropic spectacle wearers are essentially removed. (The effects of contact lens movement on the eyes are considered in ‘Incorporation of prism’ (p. 137).) An example will serve to illustrate this: Spectacle correction: R − 4.00DS L + 1.00DS The prismatic effect when looking down at an object 10 cm below the horizontal and 25 cm in front of the spectacle plane (assumed to be 27 mm in front of the eyes’ centres of rotation) is: R 3.55Δ, base-down L 1.00Δ, base-up.

As shown two paragraphs below, this is found by calculating the distance that the eyes have to turn to view the images formed by the lens, rather than simply by calculating using Prentice’s rule. The difference between the two eyes in the vertical meridian is nearly 5Δ, which is too great for the patient to obtain comfortable binocular single vision. This spectacle correction would therefore necessitate vertical head movements rather than eye movements. Fig. 7.7 illustrates the difference in vertical eye rotation that would be required with this spectacle correction, as well as the difference in magnification (see ‘Relative Spectacle Magnification’, p. 141). To calculate the prismatic effect, the positions and sizes of the images OR′ and OL′ formed by the spectacle lenses are first found (Fig. 7.7). Thus, for the right eye, l = −25 cm Therefore, L = −4 D Now Fsp = −4 D, and since L ′ = L + Fsp, L ′ = −8 D Therefore, l ′ = −12.5 cm Since,

h′ L −4 = , h′ R = × 10 cm = 5 cm h L′ −8

For the left eye, l again = −25 cm. Therefore, L = −4 D Now Fsp = +1 D, and therefore L ′ = −3 D Therefore, l ′ = −33.33 cm −4 and h ′ L = × 10 cm = 13.33 cm −3 Points T, at which the two visual axes intersect the spectacle lenses, must then be found. Using the similar triangles for each eye – that is, with apex at C and bases at O′ and ST, and assuming the distance z (SC) to be 27 mm, the following equations apply:

Myopic eye: base-down effect

Z

Hypermetropic eye: base-up effect

Z

Fig. 7.7  Anisometropia: During near vision when wearing spectacles, the visual axis of the hypermetropic eye is depressed more than that of the myopic eye. The image seen by the hypermetropic eye is also larger than that seen by the myopic eye. O, object; OR′ and OL′, images of O formed by the spectacle lenses; OR′′ and OL′′, retinal images.

137

7  •  Optics and Lens Design

For the right eye, and ST For the left eye, and ST

ST z = h′ R z − l ′ 2 .7 = 5× = 0.888 cm 15.2 2 .7 ST = = h′ L 36.03 2 .7 = 13.33 × = 0.999 cm 36.03

From Prentice’s law (see p. 138), the prismatic effect of the right lens is −4 × 0.888 = 3.55 Δ base-down, and for the left lens it is +1 × 0.999 = 1.00 Δ base-up. This gives 4.55 Δ difference between the two eyes. An alternative way of looking at this is to calculate the actual angles through which each eye rotates downwards and then find the difference. The right eye rotates downwards by an angle of θR given by: tan θ R =

h′ R z − l′

Since the angle θ in prism dioptres is given by the value 100 × tan θ,

θ R = 100 ×

5 5 = 100 × = 32.89 ∆ 2.7 − ( −12.5) 15.2

The left eye rotates downwards by an angle of

θ L = 100 ×

13.33 13.33 = 100 × = 37.00 ∆ 2.7 − ( −33.33) 36.03

The difference between the rotation required of the two eyes is thus 4.10 Δ, which differs a little from the value measured in the spectacle plane, which was 4.55 Δ. The latter method of determining the angles through which each eye rotates is the way in which angular values for convergence are also calculated. An object located on the midline between the two eyes (see Fig. 7.5) is then considered as an object of height (h) equal to one-half the interpupillary distance, because this is its distance from the optical axis of the spectacle lens. Because of the larger fusional reserves, horizontal prism differences are more easily tolerated than vertical differences. During version movements of the eyes, the anisometropic spectacle wearer learns to make allowance for the increasing prismatic difference as the visual axes intersect points at increasing distances from the optical centres. This habit of allowing for the prismatic difference shows as a noncomitant heterophoria which may persist for some time after changing from spectacles to contact lens wear. From habit, one eye moves more than the other, and objects tend to be seen double until a new extraocular muscle balance is achieved.

ACCOMMODATION AND CONVERGENCE If a comparison is made of the graphs of accommodation in Fig. 7.3 and the graphs of convergence in Fig. 7.6, it will be noted that the slopes showing convergence and

Table 7.4  Ratio Between Accommodation and Convergence With Spectacles and Contact Lenses Ratio of Accommodation (D) to Convergence (δ) at 0.33 Metre From Spectacle Plane Spectacle Refraction (D)

0.25 Metre From Spectacle Plane

Spectacles

Contact Lenses

Spectacles

Contact Lenses

−20

0.170

0.174

0.172

0.176

−15

0.173

0.174

0.174

0.176

−10

0.174

0.174

0.174

0.176

−5

0.174

0.174

0.176

0.176

0

0.174

0.174

0.176

0.176

+5

0.172

0.174

0.175

0.176

+10

0.168

0.174

0.171

0.176

+15

0.160

0.174

0.164

0.176

accommodation with spectacles are the same. They are also the same with contact lenses. As Westheimer (1962) stated, this implies that the accommodation/convergence ratio (A/C ratio) is the same with contact lenses as it is with spectacles. Stone (1967) also showed that if contact lenses remain centred for all working distances and a comparison is made with spectacles centred for distance vision, the A/C ratio remains approximately the same with both forms of correction (Table 7.4). The values for the basis of this table are derived from Tables 7.2 and 7.3 and show that: A C ratio with spectacles ≈ 1 − Fsp ( z − 2d) A C ratio with contact lenses where Fsp = spectacle lens power in dioptres z = distance from spectacle plane to centre of rotation of eye, in metres d = back vertex distance of spectacle lens, in metres. Now, if d = z/2, it can be seen that the A/C ratio is the same with both spectacles and contact lenses. In Table 7.4, d was taken as 12 mm and z as 27 mm, which accounts for the slight discrepancies between the values found for the two forms of correction. But as d is always approximately z/2, the ratios are always approximately the same. The interpupillary distance and spectacle centration distance were taken as 60 mm. Changes in accommodation should therefore cause difficulty only in the presbyopic or prepresbyopic myope, who may have trouble in exerting the extra accommodation (and convergence) when transferring to contact lenses from spectacles.

INCORPORATION OF PRISM Most manufacturers prefer not to incorporate more than 3 Δ into a contact lens because the thickness difference makes more than this amount impracticable with such steeply curved surfaces (except in higher powers).

138

SECTION 3  •  Instrumentation and Lens Design

Because the prism base always rotates down and slightly in, it is impossible to prescribe a horizontal prism satisfactorily, and a vertical prism is therefore also limited to 3 Δ, as it can be prescribed in only one lens. Thus most prisms must be incorporated in spectacles to be worn in addition to contact lenses. It may be possible, at least in theory, to incorporate a low-power horizontal prism in just the optic zone of a soft lens, relying on a different method of peripheral stabilisation to give the required alignment (see Chapter 11). Full-size scleral lenses can be made with a horizontal prism. Sometimes the expected prism is not required in contact lenses, as they provide a better standard of binocular vision than spectacles. This is where tolerance trials in contact lenses are useful, as are fixation disparity tests for uncompensated heterophoria. With contact lenses, some unwanted prismatic effect occurs due to movement of the lenses on the eyes. One of the aims of correct fitting is to ensure that this movement is similar for both lenses so that little prismatic difference between the two eyes is experienced. The prismatic effect due to such movement is given by: P = c × F (Prentices’s law) where P is the prismatic effect in Δ F is the BVP, in dioptres, of the contact-lens/liquid-lens system c is the displacement or movement in cm. If the powers of the two lenses are the same and the movement is similar, then no prismatic difference occurs. If the two lens powers are not the same, the amount of movement enables the prismatic difference to be calculated. It is worth noting that if a person wears a negative-powered contact lens in one eye and a positive-powered lens in the other, to counteract the prismatic effects due to vertical lens movement, the negative lens should move up (prism base-down) as the positive lens moves down (also prism base-down). With corneal lenses, because of the position of the centre of gravity and the action of the lids during blinking (see Chapter 9), this desirable opposite movement of positive and negative lenses occasionally occurs. If a lens tilts due to pressure of the upper lid on a corneal lens, e.g. on a ‘with-the-rule’ cornea, then a certain amount of extra prism base-down is introduced due to the liquid lens. Again, this tilt may be ignored as long as it is similar in the two eyes. Where a contact lens has prism worked on it, then the power of the prism on the eye is the same as that in air. The liquid lens (unless it contains a prism element of its own due to tilt of the contact lens) has no effect on the prismatic element of the contact lens.

ASTIGMATIC EFFECT INTRODUCED BY PRISM, DECENTRATION AND TILT It has been suggested that a prism in a contact lens introduces a significant cylindrical element; this is incorrect. Although the refracted pencil through the front surface is slightly oblique, the amount of cylinder introduced is negligible. For example, taking a lens of BVP +10.00D, back optic zone

radius (BOZR) 7.80 mm and centre thickness 0.40 mm, calculation using the Coddington differential equations shows that the cylindrical element introduced is only 0.28D for a prism of 3 Δ, and it drops to 0.12D for a 2 Δ prism. Expressed as a positive cylinder, the power is along the prism base–apex line and the axis perpendicular to it. Thus in the example given, if the prism is base-down along 90, there will be +0.28DC × 180. (In this calculation, it is assumed that the eye has rotated to view the object, so that the chief ray and visual axis still pass normally through the back surface, unlike with a spectacle lens incorporating a prism. The astigmatic effect is therefore caused solely by the tilt in the front surface; in this case the angle of incidence is just over 5° and the surface power nearly +71.5D.) Lenses can also tilt by decentring on the eye. If the decentration occurs as a rotation about the centre of curvature (C2) of the back surface, then C2 will not be displaced, but C1, the centre of curvature of the front surface, is displaced. This introduces a small amount of prism and hence, as shown in the previous example, astigmatism. The axis of the induced positive cylinder is perpendicular to the direction of decentration. In the case of negative lenses of similar numerical power, the astigmatism induced is less than for positive-powered lenses because of the flatter front surface. In the normal range, the BOZR has little influence on the result. Clearly, the induced astigmatism may be increased if the lens actually tilts on the eye as well as decentring. When the whole lens tilts through a small angle, the resulting astigmatism can be approximated by the equation: Cylinder = F tan2 θ where θ is the angle of tilt and F is the BVP of the lens. The cylinder axis is perpendicular to the direction of tilt. Thus if F = +10.00D and θ = 5°, the induced cylinder is +0.0765D. If the direction of tilt is vertical, i.e. about a horizontal axis, then the plus cylinder axis is horizontal. There is also a very small change in the spherical element given approximately by the equation: Sphere = F(1 + 1 3 sin2 θ) Thus in the same example, the sphere is increased by 0.025D to +10.025D, which is of no significance. A tilt of 5° is extremely unlikely – a corneal lens of BOZR of 8.00 mm fitted to the flat meridian of a cornea with a steep meridian of 7.60 mm, and aligning 2 mm above the centre will tilt by only about 1°. Sarver (1963) studied the effect of contact lens tilt on residual astigmatism, and his experimental observations confirmed the above theoretical findings.

FIELDS OF VIEW OR FIXATION The macular field of view, i.e. the angular extent that can be fixated through an appliance with the moving eye, of someone wearing centred contact lenses equals the field of fixation, and it is limited only by the extent to which the eyes can move. This normally gives a clear field of view of about 100°. By comparison, the clear field of view of the spectacle wearer is limited by the size and vertex distance of the

7  •  Optics and Lens Design

139

The sizes of the real and apparent fields of view through the spectacle lens are easily calculated. The angular subtense of the spectacle lens at the eye’s centre of rotation, C, gives the apparent macular field of view, B. For example, if the centre of rotation distance, z, from the spectacle lens to C is 25 mm and the size of spectacle lens is 50 mm, then its semidiameter is 25 mm. A

B

Therefore 1 2 B = arctan (25 25) = 45° and thus B = 90°,

C

where arctan = inverse tan, atan or tan −1.

Fig. 7.8  Field of view of a myope through a spectacle lens. A, actual macular field of view; B, apparent macular field of view; A > B; C, centre of rotation of eye. Hatched area is seen double due to prismatic effect (doubling is minimised by the spectacle frame, if present).

To obtain the size of the real macular field of view, A, requires that the position of the image of C, as formed by the spectacle lens, has to be found. A is then the angular subtense of the spectacle lens at that point. Using the same example as above and reversing the path of the light rays shown in Fig. 7.8, if the lens has a power of −10.00D and making use of the usual nomenclature for object and image distances, then l = −25 mm, and therefore L = −40.00D; Fsp = −10.00D. Thus L ′ = L + Fsp = −50.00 D l ′ = 1000 L ′ = −20 mm C is thus imaged 20 mm from the spectacle lens, on the same side as C. Therefore 1 2 A = arctan (25 20) = 51.3°. And thus A = 102.7°.

B

C

A

In addition, the prismatic effects of the spectacle lenses cause blind areas in the peripheral visual field of the hypermetrope and areas of doubled vision for the myope, as illustrated in Figs 7.9 and 7.8, respectively. The blind area experienced by a hypermetrope is enlarged due to the thickness of the spectacle frame. This prismatic effect and the blind area are particularly troublesome to aphakes owing to the high power of the spectacle lenses. Contact lenses afford great relief.

MAGNIFICATION

Fig. 7.9  Field of view of a hypermetrope through a spectacle lens. A, actual macular field of view; B, apparent macular field of view; A < B; C, centre of rotation of eye. Blind area due to prismatic effect (and spectacle frame when present) is shown hatched.

spectacle lens and is restricted to an apparent field of about 80° (although blurred vision is possible beyond the limits of the spectacle lens or frame as far as the eyes can rotate). Fig. 7.8 shows that, in fact, the myopic spectacle wearer has a larger real field of view than this, depending on the power of the spectacle lens. However, the aberrations from the periphery of the spectacle lenses, especially in higher powers, also reduce the usability of the outer field. The hypermetropic spectacle wearer (Fig. 7.9) has a real field of view smaller than 80°. This means that, on transferring to contact lenses, the myope must move the eyes more to see the same area of the visual field as seen with spectacles. The reverse applies to the hypermetrope.

Any correction, be it a spectacle lens or a contact lens, alters the size of the basic retinal image. (The basic retinal image is taken to be the size in the uncorrected eye assuming blur circles of zero diameter, i.e. ‘pinpoint’ pupils.) This is known as ‘spectacle’ magnification, even when it is the magnification due to contact lenses. To compare spectacles and contact lenses, the differences in magnification for both spherical and toric correcting lenses must be considered. This is affected by the form and thickness of the lens.

Spherical Lenses KEY POINT

Positive spectacle lenses magnify images, and negative lenses minify, and the magnification increases with the vertex distance. Only if a corrective lens is worn in the plane of the eye’s entrance pupil is unit magnification of the basic retinal image achieved. Thus a contact lens worn on the cornea approaches unit magnification. (An intraocular implant is fitted even closer to the entrance pupil plane.)

140

SECTION 3  •  Instrumentation and Lens Design

The size of the retinal image is proportional to the angular subtense of the object at the entrance pupil, as shown in Fig. 7.10. The angular subtense is w′ when the spectacle lens, assumed to be infinitely thin, is present and w when it is not (a distant object is assumed). Now, w ′ =

h′ h′ and w = (1 F ) (1 F ) − a

Thus the component of spectacle magnification due to the power of the lens, the Power Factor, PF, =

(1 F ) w′ 1 = = ≈ 1 + aF w (1 ) − a 1 − aF F

Note that with a contact lens, a is about 3 mm (the approximate distance of the entrance pupil plane from the cornea); with a spectacle lens, a equals the vertex distance plus the approximate distance of the entrance pupil plane from the cornea, i.e. 12 mm plus 3 mm. Therefore a is approximately 15 mm, and the approximate expression gives 1.5% per dioptre. Fig. 7.11 illustrates these results for theoretically infinitely thin spectacle lenses and shows mean values for typical aligning corneal lenses and monthly replacement hydrogel soft lenses.* The mean results are shown because there is negligible difference between the results for these two types. The soft lens was assumed to have no tears lens between it and the cornea, whereas the distance a (see Fig. 7.10) was increased by 0.008 mm for the corneal lens, but being divided by the refractive index of tears, taken to be 1.336. Fig. 7.12 also illustrates these results, but the contact lens magnification is expressed as a relative increase or decrease compared with the thin lens spectacle magnification. The powers of the spectacle lenses were those correcting the ocular refraction, but are illustrated lined up with the ocular refraction. For further explanation on Figs. 7.11 and 7.12 see Section 9, Addendum, available at: https://expertconsult.inkling.com/.

*Despite their greater corneal clearance and slightly greater thickness, the results for semiscleral lenses differ minimally from those of the RGP and soft lenses.

Magnification

Fig. 7.10  Spectacle magnification is w′/w. A distant off-axis object, subtending an angle w at the spectacle lens, is imaged by the lens of power F, in the far-point plane M suffix R. The image is of size h′ and subtends an angle w′ at the centre of the entrance pupil, which is situated at a distance of a metres from the spectacle lens. The left-hand diagram shows the situation in hypermetropia, the right-hand diagram in myopia.

1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7

"Thin" spectacle lens Typical contact lens "Thin" contact lens

-20 -15 -10 -5 0 5 10 15 Contact lens power (D)

20

Fig. 7.11  Power factor magnification produced by an imaginary “thin” spectacle lens, an imaginary “thin” contact lens rigid corneal and the total spectacle magnification (including the shape factor) of a typical contact lens and thin hydrogel soft lens. Data averaged for corneal and soft lenses. This graph compares the spectacle magnification for a spectacle lens (The blue line) and a contact lens (the green line) based only on their power and vertex distance (to the entrance pupil), ignoring magnification due to lens thickness, with that of a typical contact lens including the magnification due to lens thickness and front surface power.

KEY POINT

It can be seen that myopes who change from spectacles to contact lenses see objects larger than before, and hypermetropes see objects smaller than before. Myopes might expect increased acuity in contact lenses, but they may experience some disorientation when they first wear them, owing to the apparent increase in the size of objects. Conversely, hypermetropes may have poorer acuity with contact lenses, but since the difference in image size is of real significance only in the higher powers (except for those with poor visual acuity), it is only the high hypermetropes and aphakes who are affected (see Chapter 21). The latter are able to see objects reduced to only slightly larger than their normal size again. More realistic values are also shown that include the effect of lens thickness, again as compared with infinitely thin spectacle lenses.

Shape Factor The magnification of two lenses having the same BVP is affected by their front surface power and thickness. Shape factor (a term which is unfortunately also applied to the asphericity e2 = (1 − p) of conic sections, see ‘Corneal Shape’, Chapter 9) is the allowance that must be made for the increase

Percentage difference in rentinal image size

7  •  Optics and Lens Design

141

A more direct method of calculation was used for the total spectacle magnification for typical contact lenses shown in Figs 7.11 and 7.12. If a ray is traced from the entrance pupil back through the tears lens and contact lens, then the shape factor can be shown to be (Rabbetts 2007):

40 30 20 10

Rigid corneal lens

0

Soft hydrogel lens

-10

"Thin" contact lens

-20 -30 -20 -15 -10 -5 0 5 10 15 20 Contact lens power (D)

Fig. 7.12  Percentage difference in retinal image size: comparison of thin and real contact lenses with a thin spectacle lens.

in magnification due to the form and thickness of the lens. It is given as: SF =

1 1 − t n F1

which is the ratio between BVP and equivalent power, where n = refractive index t = centre thickness in metres F1 = front surface power in dioptres.

L ′1 × L ′2 × L ′3 L1 × L2 × L3

where L1, L2 and L3 are the vergences incident on the front surface of the lens, the lens-liquid interface and the back surface of the liquid lens, and where the dashed items are the vergences after refraction. The shape factor for contact lenses varies from about 1.003 (0.3%) to about 1.03 (3.0%), and it is illustrated by the closeness of the curves for the power factor magnification and the total magnification in Figs 7.11 and 7.12. Although the front surface power F1 is high, the lenses are relatively thin. Although not included in the figures, the shape factor for spectacle lenses is zero for plane front surface high minus power lenses, rising to nearly 15% for a +20.00 D aspherical front surface lenticular lens because of its thickness and front surface power.

CHANGING FROM SPECTACLES TO CONTACT LENSES This can be difficult because of the alteration in the perceived image due to the following:

To obtain the spectacle magnification, the values for the power factor magnification of the ‘thin’ lenses shown in Figs 7.11 and 7.12 should therefore be multiplied by the shape factor to give: SM =

SF =

1 1 × 1 − aF 1 − t F1 n

This expression is easily applied to spectacle lenses, but a contact lens system comprises a plastic lens and a tears lens in combination. The expression for the shape factor is correspondingly more involved. Bennett (1985) derived an approximate simplified expression for the shape factor of a contact lens system, based on values which are normally known. This is: SF = 1 + t(K o + K ) − (t1 n )F2 1

where t (total reduced thickness in metres of contact lens and liquid lens) = t1/n1 + t2/n2 Ko = ocular refraction in dioptres K = keratometer reading in dioptres (assuming refractive index of calibration equals that of tears, i.e. n2 = 1.336) t1 = thickness of plastic contact lens in metres, t2 = tear lens thickness n1 = refractive index of contact lens F2 = interface power in dioptres at the back optic zone surface of the contact lens.

Relative Spectacle Magnification This is defined as the ratio between the retinal image sizes in a corrected ametropic eye and a standard emmetropic eye. Various formulae have been given to calculate this, depending on whether the difference between the two eyes is axial or refractive (see below and Table 7.5). Its main use is in determining whether a particular type of correction is likely to improve or disrupt binocular vision, by comparing the two retinal image sizes. Bennett (1985) pointed out that such formulae can be misleading because the human emmetropic eye has a large range of powers and axial lengths. Therefore a much simpler approach is used in this chapter, based on ‘reduced eye’ data and values for ‘spectacle’ magnification to determine retinal image sizes. Where a person has different ocular refractions in the two eyes, the different magnification given by the two spectacle lenses (or contact lenses) may result in poor binocular vision. This is usually due to fusion difficulties resulting from unequal retinal image sizes. (Similar spacing between the retinal receptors in the two eyes is assumed, although this may be a false assumption.) It is common to think of anisometropia as being either axial or refractive. Sorsby et al. (1962) showed that most naturally occurring anisometropia is predominantly axial, but this is often accompanied by a smaller refractive component. By contrast, one obvious example of refractive anisometropia is unilateral aphakia where contact lenses provide greater similarity in retinal image sizes than do spectacle lenses. Fig. 7.13 illustrates how two eyes of the same length and corneal power, one aphakic and the other phakic, have similar basic retinal image sizes. Contact lenses give rise to a minimal change in this basic retinal image size, thereby permitting a good

142

SECTION 3  •  Instrumentation and Lens Design

w

w

Equal basic retinal image sizes w

w

Fig. 7.13  Refractive anisometropia: corneal and soft lenses cause minimal change in the basic retinal image size. w, angular subtense of distant object.

chance of binocular vision. However, retinal image size differences of as little as 1% may give rise to binocular problems in some patients. Only an intraocular implant for the aphakic eye can achieve ‘equality’ of retinal image sizes. Fig. 7.14 shows how two eyes of unequal length have unequal basic retinal image sizes. In this case, contact lenses – which scarcely affect this basic size – are theoretically unsatisfactory if fusion is to be achieved, assuming equally spaced retinal receptors in the two eyes. Spectacle lenses are worn in a position close to the eye’s anterior focal plane. Because the ray shown dashed passes through the eye’s anterior focus, it is refracted in the eye parallel to the eye’s axis, and hence it gives constant image height irrespective of axial length, making the retinal images in the two eyes similar in size. Typical of such a case is unilateral myopia. To compare or calculate retinal image sizes, it is simplest to assume a reduced eye as shown in Fig. 7.14 with a refractive index of 1.336 and a single spherical refracting surface of radius 5.6 mm, giving it a power of +60.00D. When the refractive error (Ko) is known to be axial, then the power of the reduced eye (Fe′) is assumed as +60.00D, and its length is k′. Now, k ′ =

n′ , where K ′ = K o + Fe K′

Thus if the ocular refraction Ko = −10.00D, then K ′ = +50 D and k ′ =

13666 × 1000 = 26.72 mm 60

A standard emmetropic eye has an axial length fe′ =

n ′ 1.336 × 1000 = 22.27 mm = Fe′ 60

When the error is known to be refractive (as in aphakia), then the power of the eye (F′e) is determined from its length

w

Unequal basic retinal image sizes

Fig. 7.14  Axial anisometropia: a spectacle lens suitably placed before the ametropic eye can give equality of retinal image sizes, but this may not be desirable (see text). w, angular subtense of distant object.

(k′) and its ocular refraction (Ko). For example, Ko = +12.00D, k′ = 22.27 mm. 1.336 × 1000 = +60.00 D 22.27 and F ′ e = K ′ − K o = 60 − 12 = +48.00 D Thus K ′ =

As can be seen from Fig. 7.14, the principal ray determining the basic retinal image size undergoes refraction according to Snell’s law* at the principal point of the eye. Considering this principal ray, prior to refraction the angle subtended at the eye’s principal point is w, and after refraction the angle subtended by the basic retinal image is thus w/n′. (All angles are small, and the sine, tangent and angle in radians then all become equal.) But

w Basic retinal image size = n′ k′

Thus basic retinal image size = k′(w/n′) = w/K′ (in metres, if w is in radians). Note that the principal ray may already have undergone refraction at a spectacle lens or contact lens, so that w is then equivalent to the w′ of Fig. 7.10. Thus the spectacle magnification is taken into account in determining the angular subtense at the principal point prior to refraction by the eye. The final retinal image size then becomes w 1 × (in metres) 1− aF K ′ where F is the power of the spectacle lens or contact-lens/ tears-lens combination. In the standard emmetropic eye the retinal image size is thus (w ÷ 60) metres.

*Snell’s Law. The angle of refraction at a boundary of two materials depends on the angle of incidence of the light and on the indices of refraction of the two materials.

7  •  Optics and Lens Design

Table 7.5 illustrates the differences between axial and refractive anisometropia. The vertex distance is assumed to be 12 mm and the distance from cornea to entrance pupil to be 3 mm. Shape factor has not been taken into account. In summary: Refractive anisometropia is demonstrated by a unilateral aphakic with equal unaided retinal image sizes because both eyes are similar in length. Thus the spectacle magnification afforded by both spectacles and contact lenses has a direct effect on the retinal image sizes. With spectacles, the difference in magnification between the two eyes is large; therefore so is the percentage difference between retinal image sizes. With contact lenses, it is small, providing a greater chance of binocular vision. ■ Axial anisometropia is demonstrated by unilateral axial myopia in which the power of both eyes is assumed to be 60.00D. Thus the basic retinal image sizes are proportional to the axial lengths (or inversely proportional to the dioptric lengths, K′L:K′R, as shown in Table 7.5; one must remember to calculate these on the basis of ocular refraction). These basic image sizes are affected by the spectacle magnifications. With spectacles, where the difference in magnification is large, there is only a small difference in the retinal image sizes. With contact lenses, the spectacle magnifications are almost the same, but the retinal images are very different in size. Although theoretically spectacles provide the better chance for binocular vision, this is usually not the case in practice because the retinal receptor distribution may be different in the two eyes, being more widely spaced in the bigger myopic eye. This reduces the enlargement caused by the longer axial length (Table 7.5). In practice it is found that axially anisometropic patients achieve better binocular fusion in contact lenses than in spectacles (Winn et al. 1986) (see Chapter 21). Thus all types of anisometropia and antimetropia are better corrected by contact lenses than spectacles if optimum binocular vision is to be achieved. ■

The extremes of purely refractive or purely axial anisometropia, as shown in the examples in Table 7.5, are rare. In most anisometropes, contact lenses afford other advantages such as absence of differential prism. Ford and Stone (1997) suggested that the perceptual process which allows fusion of different-sized images

143

was more readily adaptable than the extraocular musculature which has to cope with dissimilar prismatic effects. Large, stable lenses, whether soft or rigid, are better to use for anisometropia than small mobile corneal lenses.

Toric Lenses A toric spectacle lens gives different magnification in different meridians, which produces distortion of the retinal image. This is particularly noticeable where the principal meridians are at oblique axes (Bennett 1985). A square object seen through a toric spectacle lens may look rectangular if the principal meridians are horizontal and vertical, or may look diagonal like a parallelogram if the principal meridians are oblique (Fig. 7.15). This distortion of shape is minimised with a contact lens because the meridional difference in magnification is reduced (see Fig. 7.11). Difficulty may arise when a toric spectacle correction has been worn for many years and a perceptual allowance has been made for the distortion. On transferring to contact lenses which give a less distorted retinal image, the perceived image

Retinal image

Perceived image

With spectacles

Retinal image

Perceived image

With contact lenses when first worn Fig. 7.15  An exaggerated image of perceptual compensation for retinal image distortion: This is acquired during spectacle wear and continues when contact lenses are first worn.

Table 7.5  Differences Between Axial and Refractive Anisometropia (Upper Row: Refractive Anisometropia; Lower Row: Axial Ametropia)

Spectacle Correction (D)

With Spectacle Correction

With Contact Lenses

Spectacle Magnification

Spectacle Magnification

R

L

Ratio of Basic Retinal Image Sizes in Uncorrected Eyes (R : L)

(1)

+2

+12

60.00 : 60.00

1.03

1.22

(2)

−1

−10

51.07 : 59.01

0.99

0.87

R

L

Difference in Retinal Damage Sizes (%)

R

L

Difference in Retinal Image Sizes (%)

18.29

1.01

1.04

3.75

1.98

1.00

0.97

12.87

144

SECTION 3  •  Instrumentation and Lens Design

may appear distorted to the wearer (see Fig. 7.15) until the brain has had a chance to adapt.

A'B

B'B

Oblique Aberrations.  Even best-form spectacle lenses allow objects viewed through their periphery to suffer from the effects of oblique aberrations: oblique astigmatism distortion ■ transverse chromatic aberration ■ curvature of field and the closely related mean oblique (power) error ■ ■

Contact lenses remain almost centred in all directions of gaze, and distortions are therefore kept to a minimum. The higher the spectacle prescription, the greater the amount of aberration. The relief afforded by contact lenses is then considerable; conversely, returning to spectacles from contact lenses can give rise to disorientation and nausea and may require modification of the spectacle prescription to keep the power as low as possible, yet consistent with adequate visual acuity. Any cylindrical correction needs to be kept to an absolute minimum and possibly removed altogether, provided visual acuity is not unduly compromised. Spectacle lenses should be kept as small as possible to avoid some of the peripheral distortion. The Effective Binocular Object.  Although each eye views the image formed by the spectacle or contact lens, the eyes as a pair align on the ‘effective binocular object’. Bennett, cited in Rabbetts (2007), showed that for a myope wearing spectacles that are centred for distance vision, the effective binocular object is positioned further away. Fig. 7.16, which is grossly out of scale, shows two eyes viewing an object AB in a plane parallel to the line joining the two eyes, and in which A and B are directly in front of the left and right eyes, respectively. For the left eye, the image, B′L, of B formed by the lens lies on the line joining B to the optical centre OL of the spectacle lens. Since B lies on the optical axis of the right lens, so does its image, B′R. If the eyes look at B, then the left eye rotates about its centre of rotation ZL so that B′L lies on the visual axis; the two visual axes converge to meet at the effective binocular object, B′B, which is situated behind B. By analogy, A′B is similarly positioned further away than A. This could be predicted from the base-in prismatic effect shown in Fig. 7.5. Conversely, for a hypermetrope, the effective binocular object lies closer than the real object. Because contact lenses stay centred on the visual axis with eye rotation, this effect does not occur when wearing them. Because it allows for the distortion produced by spectacle lenses, accurate ray tracing shows a slightly more complicated situation, in that the effective binocular object is curved, bulging slightly towards a myope and giving the appearance of being in a bowl for a hypermetrope. Fig. 7.16 illustrates the horizontal plane, but the effective object is also curved in the vertical plane, though to a lesser extent. These effects will be very much more pronounced for high prescriptions and make occasional spectacle wear disconcerting. A similar effect probably occurs stereoscopically, though the reference here will be the pupil centre with a stationary eye rather than the centre of rotation.

A

A'L

B

B'L

A'R

B'R

OL

OR

ZL

ZR

Fig. 7.16  Effective binocular object. The myopic spectacle lens wearer sees the object AB as the line A′BB′B. Simple theory gives the straight line, but this is curved when spectacle lens distortion is taken into account.

Moreover, if the refractive errors in the two eyes differ significantly, then the differing distances to the respective image planes from ZR and ZL result in the effective binocular object being at a slope, closer to the more ametropic eye. The effects are likely to be even more complicated for a person with a significant astigmatic error, particularly if at oblique axes, since the images formed by the spectacle lenses will show scissors distortion as shown in Fig. 7.15. Swim on Head Movements.  If the spectacle wearer rotates the head, then the image formed by the spectacle lens will also move. The brain has developed compensatory processes to neutralise this, but on reverting to spectacles from contact lenses, the world appears to swim, causing a ‘with the head’ movement for minus-powered lenses and an ‘against the head’ movement for plus-powered lenses.

7  •  Optics and Lens Design

Optical Considerations of Contact Lenses on the Eye (Refer to https://expertconsult.inkling. com/ Appropriate Formulae in Bold Roman numerals in this Section) To understand why a contact lens correction often differs considerably from a spectacle correction, the significance of the following points must be fully understood: effectivity – the difference between spectacle and ocular refraction ■ the contribution made by the liquid (tears) lens for rigid lenses ■ the effects of radius changes on the BVP of the contactlens/liquid-lens system ■ the differences between total and corneal astigmatism. ■

This section is intended as a practical guide in determining soft and rigid corneal lens powers, and employs the method of specifying BOZR in millimetres rather than in terms of a keratometer reading in dioptres. Different aspects of refraction with contact lenses will now be considered.

OCULAR REFRACTION The difference between the spectacle and ocular refractions was discussed on pp. 132–133. To correct fully an eye’s refractive error, the BVP of the contact-lens/liquid-lens system must equal the ocular refraction (Ko). As shown in Fig. 7.17, a simplification is to imagine an infinitely thin air layer separating the cornea from the liquid lens, and another layer between the liquid lens and the back surface of the contact lens. This enables all power calculations to be made as if the surfaces were in air, particularly for changes in BOZR. In the following considerations,

Air

145

it will be assumed that surface powers are additive, provided that their separations are small. This leads to some approximations. It will also be assumed that the BVP of a contact lens can be directly added to the vergence of the light reaching it – although this, again, is an approximation. BVP of final contact lens in air + BVP of liquid lens in air (liquid lens assumed thin) = Ocular refraction Approximate methods of calculation are useful to crosscheck that formulae and computer programs have been used correctly. However: KEY POINT

A contact lens having a very different BVP from that required should not be used for refraction. In addition: KEY POINT

A combination of positive spectacle lens and negative contact lens, which constitutes a Galilean telescope system, should be avoided wherever possible since it gives a higher magnification than that obtained with the final contact lens. This gives a false assessment of visual acuity, leading to disappointment when the final contact lens gives poorer vision. However, it can occasionally prove a useful method of improving visual acuity.

Refraction With a Contact Lens of Incorrect BOZR When a refraction is carried out using a rigid lens of incorrect BOZR, for example if a trial lens is fitted that is slightly different from the lens to be ordered, then: Liquid lens power in air + trial contact lens BVP in air + effective power at the contact lens of the additional spectaclle lens = Ocular refraction If the BOZR is flatter than that to be ordered, the liquid lens is more negative than it will be with the final contact lens (Fig. 7.17). The vergence of the light reaching the front surface of the liquid lens must therefore be adjusted to allow for this. Positive power must be added to counteract the extra negative power of the final liquid lens. KEY POINT: APPROXIMATE RULE [1]

Trial contact lens Extra positive liquid lens with final back optic zone radius

Cornea Liquid lens

Fig. 7.17  Refraction with a trial contact lens: The trial contact lens has too flat a back optic zone radius. The diagram also illustrates the imaginary air layer separating the components.

When fitting rigid corneal lenses: (i) If the BOZR of the lens to be supplied needs to be steeper than the trial lens, for each 0.20 mm that the final BOZR is steepened, add –1.00D to the BVP. (ii) If the BOZR of the lens to be supplied needs to be flatter than the trial lens, for each 0.20 mm of that the final BOZR is flattened, add +1.00D to the BVP.

146

SECTION 3  •  Instrumentation and Lens Design

(Some authorities suggest remembering that a difference of 0.05 mm, a typical step in RGP lens BOZR, is equivalent to 0.25 D, but the present writer prefers to avoid decimal values on both sides of the rule.) These rules do not apply when fitting thin soft lenses, because they conform to the central corneal contour. However, if the soft lens is thick and there is a large difference in radius between the lens and the cornea, there may be a liquid lens, but this is unlikely with modern lenses and in any case its power would be unpredictable. Examples.  A typical corneal lens problem is shown in Table 7.6. This shows that the error of the approximate method is sufficiently small to be ignored. Fig. 7.18 shows the power difference corresponding to a 0.2  mm radius difference (calculated for each radius ± 0.1 mm) for a refractive index of 1.336. This rule is approximately correct at about 8.1 mm. The values for radii that are largely different from average (for example in keratoconus) should be calculated or evaluated using radius/power tables for a refractive index of 1.336. The power and radius scales on keratometers may also be used (see Chapter 8). Although keratometers are usually calibrated for an index of 1.3375, this difference will be insignificant – see ‘Keratometry and Corneal Astigmatism’ p. 147. Table 7.6  Difference between the accurate and approximate methods for BVP:BOZR compensation Accurate Method BOZR used (mm)

8.10

BOZR ordered (mm)

8.00

Approximate Rule [1] Method:

From:

+41.48 D

To:

+42.00 D

Change:

+0.52 D

0.10 mm

Contact lens BVP change (D) to counteract this

−0.52

−0.50

Error of approximate method (D)

+0.02

Change in power (D) / radius (mm) of liquid lens front surface in air

OCULAR ASTIGMATISM Ocular astigmatism is the cylindrical component of the ocular refraction, and is given by the following combination: Front surface corneal astigmatism + back surface corneal astigmatism + crystalline lens astigmatism (referred to the corneal plane) = Total ocular astigmatism KEY POINT

The front surface of the cornea usually has greater positive power in the near vertical meridian, i.e. ‘with-the-rule’ astigmatism, whereas the back surface of the cornea and the crystalline lens normally have ‘against-the-rule’ astigmatism. The total effect is usually with-the-rule, although this decreases and may reverse with age.

Astigmatism of the Front Surface of the Cornea and the Effect of the Liquid Lens (see also Chapter 11) Refractive index of tears, nt = 1.336 Refractive index of cornea, nc = 1.376 When a rigid contact lens with spherical back surface is placed on the eye, the front surface of the liquid lens is spherical because it is formed by the back surface of the contact lens. If the front surface of the cornea is toroidal, then the back surface of the liquid lens is also toroidal, with radii (r) in mm equal to that of the cornea, but having negative power. The powers in air of the back surface of the liquid lens are given by: F=

and the powers of the front surface of the cornea are given by:

Power difference (D) (n = 1.336)

1.6 1.5 1.4

F=

1.3 1.2 1.1 1.0 0.9 0.8

(1 − 1.336) 1000 r

6.5 6.7 6.9 7.1 7.3 7.5 7.7 7.9 8.1 8.3 8.5 8.7 8.9 Radius (mm)

Fig. 7.18  The power difference between two surfaces differing by 0.2 mm in radius for a refractive index of 1.336 as a function of radius.

(1.376 − 1)1000 r

This means that the front surface astigmatism of the cornea is partly neutralised by the back surface astigmatism of the liquid lens. The amount neutralised thus is 336/376, which is almost 90%. This is of importance with toroidal corneas because it is likely that the back surface of the cornea itself will neutralise the remaining 10% of its front surface astigmatism. Thus with a rigid, nonflexing, spherical contact lens on the eye, any residual astigmatism found is almost entirely due to the crystalline lens since practically all the corneal astigmatism is corrected, i.e.

7  •  Optics and Lens Design

Back surface astigmatism of liquid lens + front surface astigmatism of cornea + back surface astigmatism of cornea = Zero (approximately)

KERATOMETRY AND CORNEAL ASTIGMATISM (see also Chapters 8 and 11) Keratometers (and topographers’ simulated K readings) measure front surface corneal radii but give total corneal power on the assumption, given above, that the back surface of the cornea has −10% of the power of the front surface. The true refractive index of the cornea (1.376) is therefore not used to calibrate keratometers. Instead, an index of 1.3375 is usually used (nk). This allows the instrument to read total corneal power (or approximately 90% of the front surface power). However, nk and nt are almost the same (1.3375 and 1.336). Indeed, some keratometers are calibrated for an index of 1.336 or even 1.332. Therefore the astigmatism measured by the keratometer is almost the same as that corrected by the back surface of the liquid lens. In fact, the use of nk instead of nt gives a power value which is slightly too high: For a radius of 8 mm: nk gives Fk = 42.19D and nt gives Ft = 42.00D. Astigmatism is the difference between the two principal powers; therefore the error due to the slight difference in the refractive indices is reduced to an insignificant amount. This is illustrated in the following example of a highly astigmatic cornea (Table 7.7). Even in such an extreme example, it can be seen that the amount of total corneal astigmatism uncorrected by the liquid lens is insignificant (+6.02 − 6.00 = +0.02D). It is therefore valid to state that all the astigmatism measured by keratometry is corrected by the back surface of the liquid lens. Although this large difference of 1 mm between the two meridians gave 6D of corneal astigmatism, a difference of 0.2 mm at the average corneal radius of 7.8 mm results in approximately 1.00D of corneal astigmatism, the same relation as for changes in BOZR of a rigid corneal lens as given by Approximate rule [1] – see Fig. 7.18.

147

Since the amount of astigmatism corrected by the back surface of the liquid lens can be measured by keratometry (as shown above), the amount of residual astigmatism with a spherical contact lens may be predicted in advance, although this assumes that the lens is reasonably thick and does not flex (see p. 148). If this is so: Total ocular astigmatism − astigmatism measured by keratometry = Residual astigmatism When determining the total ocular astigmatism, the effective change in power of both the principal meridians must be calculated from the spectacle refraction and the vertex distance, as shown on page 133. Prediction of residual astigmatism allows the effect of this amount of astigmatism to be simulated by the use of a trial cylinder in front of the patient’s usual spectacle correction. The sphere power may then be adjusted to obtain the best visual acuity. If this is inadequate, it is obvious that the contact lens must incorporate a cylinder for the correction of the residual astigmatism to obtain satisfactory visual acuity. A suitable lens design may then be selected at the outset of the fitting. Because soft lenses drape over the cornea, they replicate the corneal astigmatism; hence the residual astigmatism with a spherical soft lens is usually almost the same as the ocular astigmatism; if this is 1D or more, a toric soft lens or a spherical rigid lens may be necessary to obtain adequate visual acuity. RGP materials vary in the amount of lens flexure that takes place on a toroidal cornea (see ‘Flexure’, p. 151). In addition, a thin spherical RGP lens that corrects all the corneal astigmatism when first worn may, after several weeks of wear, alter shape, thereby increasing the amount of residual astigmatism. KEY POINT: APPROXIMATE RULE [2]

RESIDUAL ASTIGMATISM (see also Chapter 11)

When rigid nonflexing spherical lenses are to be fitted, if the corneal astigmatism and total ocular astigmatism are both with-the-rule or against-the-rule and the difference between them is less than 0.75D, this cylinder (which represents the expected residual astigmatism) may be ignored. When spherical soft or RGP lenses are to be fitted, ocular astigmatism of 0.75D or less may usually be ignored, though for RGP lenses this assumes that the corneal astigmatism is also low.

Residual astigmatism is defined as that remaining when a nonflexing rigid contact lens is placed on an eye.

Consider a patient with ocular refraction: −2.00 DS / −0.50 DC × 180 and K readings of:

Table 7.7  Corneal and liquid lens astigmatism. Keratometry (nk = 1.3375)

8 mm (+42.19D) along 180 7 mm (+48.21D) along 90

Total corneal astigmatism

+ 6.02DC × 180

Liquid lens back surface powers (nt = 1.336)

8 mm (−42.00D) along 180 7 mm (−48.00D) along 90

Liquid lens back surface astigmatism

−6.00DC × 180

8.00 along 180 × 7.80 along 90. Approximate rule [1] suggests that the corneal astigmatism (in terms of the lens needed to correct the astigmatism) is −1.00 × 180. Hence the ocular astigmatism is less than the corneal astigmatism, predicting a residual astigmatism of −0.50 × 90. If, however, the patient had been fitted with a soft lens, then the ocular astigmatism would remain, i.e. −0.50 × 180.

148

SECTION 3  •  Instrumentation and Lens Design

Table 7.8  Scheme for Selecting the Type of Contact Lens to Be Fitted, Depending Upon the Ocular Refraction and Corneal Astigmatism Refractive Error

Spherical Cornea

Astigmatic Cornea

Spherical

Either soft or RGP

Soft

Low to moderate cylindrical power

Toric soft

Spherical RGP if the refractive astigmatism matches the corneal astigmatism

High cylindrical power (>2.50 D)

Toric soft

Table 7.10  Calculation of the BVP when fitting a back surface toric lens. Flat Meridian

Steep Meridian

Keratometry (mm)

8.00

7.60

Corneal powers (D)

42.00

44.21

Spectacle refraction

−3.25

−5.75

Ocular refraction

−3.13

−5.38

−2.25 × 180

In those rare cases where the ocular astigmatism is low but corneal astigmatism is significant, i.e. the corneal astigmatism is neutralised by the crystalline lens, a spherical soft lens is the lens of choice, because the corneal astigmatism is transferred through the lens to the front surface. This is summarised in Table 7.8, which gives a scheme for selecting, on optical grounds, the type of contact lens to be fitted.

Liquid lens

+0.53

−1.68

−2.21 × 180

BVP required = ocular refraction − liquid lens power

−3.66

−3.70

negligible

Toric lens: radii 8.00 × 7.60 Lens back surface power (n = 1.450)

−56.25

−59.21

−2.96

Liquid lens

0.00

0.00

0.00

BVP required

−3.13

−5.38

−2.25

But if the lens has power of −3.13 in the flat meridian, power in steep meridian =

Toric Contact Lenses (see Chapter 11)

Hence induced astigmatism =

CALCULATION METHOD – INDUCED ASTIGMATISM AND TORIC BACK OPTIC ZONE RIGID LENSES Induced astigmatism is that astigmatism generated when a rigid toric lens is placed on an eye. It is simplest to take an example (Table 7.9). Using a notional nk of 1.336, the same as tears, the corneal astigmatism is 2.21D with-the-rule, as shown in Table 7.10.

K readings

8.00 along 180 7.60 along 90

Refractive error

−3.25/−2.50 × 180 at 12 mm

−6.09 (−3.13 − 2.96) +0.71 {−5.38 −(−6.09)}

When a spherical lens is placed on the eye, for example one of BOZR 7.90 which lies between the two keratometry readings, the liquid lens in one meridian is positive since the lens is steeper than the cornea, while in the other it is negative since the lens is flatter. In this example, the residual astigmatism is negligible (about 0.04 D). When a toric lens exactly matching the keratometry readings is placed on the eye, the liquid lens is of zero power in both meridians. The back surface of the toric lens has, however, −2.96 D of astigmatic power, whereas the eye needs only −2.25D. Hence the eye is now overcorrected by the difference, +0.71 D × 180. KEY POINT: APPROXIMATE RULE [3]

If the cornea has 2D of astigmatism, then an aligning rigid contact lens will show approximately 3D of astigmatism in air, whereas on the eye there is an overcorrection of approximately −1D, the negative sign signifying that the sign of the induced refractive error is opposite to the original, or that its axis has rotated through 90°. This is a simple 3 : 2:−1 rule. Several points arise: It is often simplest to do the power calculation in the two meridians separately. It is only necessary at the end to consider astigmatic powers, i.e. the difference between the meridional powers needed on the lens and the astigmatic power of the back surface of the contact lens. ■ If all the ocular astigmatism is in the cornea, then fitting a toric contact lens will always induce some astigmatic error, though a partial match will reduce the induced ■

Table 7.9  Example of astigmatic eye used in calculations in Table 7.10

−2.21 × 180

Spherical lens − radius 7.90

Back surface toric or bitoric RGP or toric silicone hydrogel

Both optical and fitting considerations of toric polymethylmethacrylate (PMMA) corneal lenses have been dealt with in detail by Capelli (1964), Stone (1966) and Westerhout (1969). The principles will apply equally to RGP lenses. In summary, it may be said that if the back optic zone of a rigid contact lens is to be made toroidal, the BVP required should ideally be found by refraction over a lens having the correct toroidal BOZR and appropriate power. Many laboratories will manufacture a diagnostic lens if given the patient’s spectacle prescription, vertex distance and keratometry readings. Alternatively, the computer program available at: https:// expertconsult.inkling.com/, or Rabbetts’ (1992) spreadsheet program can be used. Alternatively, an over-refraction may be carried out with a spherical lens having a BOZR equal to the flatter meridian of the toric lens to be ordered. The calculation method will be taken first.

Astigmatism

7  •  Optics and Lens Design

error: in the example above, a lens of 7.95 × 7.65 BOZR reduces the induced error to +0.52 D. The ideal patient for a toric fitting is the rare individual with greater ocular than corneal astigmatism. ■ The rule, which is more accurate for PMMA lenses, comes from the relative refractive effect of tears or the cornea and of the contact lens. Since the refractive index of lens material is around 1.45, its refractive effect in air is proportional to (1.5 − 1.0 ≈ 1/2); the refractive effect of tears or the cornea is proportional to (1.336 − 1.0 ≈ 1/3); whilst the refractive effect of the lens on the eye is proportional to (1.5 − 1.336 ≈ 1/6). With a modern lens material having an index of, say, 1.45, the ratios are nearer 4 : 3:−1, but the approximate rule gives a feeling for what will occur in practice.

TORIC POWER MEASUREMENT FROM REFRACTING THROUGH A SPHERICAL LENS In this case, which is not as straightforward as refracting through a lens with the correct toric back optic surface, some calculation is necessary to determine the BVP of the final toric lens to be ordered. Since the BOZR of one meridian is to be steepened by a known amount when ordering, the calculation is the same as that for a spherical lens where refraction has been carried out with a trial lens of incorrect BOZR, as outlined on p. 145 and summarised in Approximate rule [1]. It is a simple matter of allowing for the fact that the liquid lens power in one meridian will be different, with the final lens in place, from the value with the spherical trial lens in place. An allowance for this difference must therefore be made on the contact lens itself. Example.  A spherical lens might be tried on a cornea with a 0.5 mm difference in K readings, only to be followed with the decision that because of a poor fit, a toric lens should be used (Table 7.11). When the radius change is as large as this, it is more accurate to work out the change in power of the liquid lens, remembering that this is a change of the front surface of the liquid lens in air (see p. 146 and Formulae III and V on the website) for a refractive index difference of 1.336 − 1. This example gives a change of −2.73, i.e. 0.23 D more than

Table 7.11  Calculation of the BVP Needed in the Steep Meridian

149

the value given by Approximate rule [1]. Thus the BVP of the final contact lens along 90 should be −6.48 D. It can now be established whether or not a toroidal front surface will also be necessary on the final toroidal back surface lens. This depends on whether or not the cylindrical power of the back surface in air is the same as the required cylindrical element of the BVP of the lens in air (Table 7.12). The induced astigmatism in this example is 1.42 –0.50 (over-refraction with the spherical lens) = –0.92 × 90, increased for the ordered cylinder by the over-refraction at the same orientation. Rabbetts (1992) gives a spreadsheet listing for calculating the required powers when refracting over a toric or spherical lens, as in this example. The computer program available at: https://expertconsult.inkling.com/ may also be used to determine the back surface cylinder power in air and, if each meridian is treated separately, Formula III may be used. The above example is an obvious case where a front surface cylinder is necessary to give good visual acuity. Frequently the front surface cylinder calculated in this way is quite small, and the practitioner may prefer to order a lens with a spherical front surface and risk leaving the patient with a small amount of uncorrected (or, more commonly, overcorrected) astigmatism. The order for the final lens may state that it is to have a back toroidal surface only, and may give the BVP along the flat meridian, i.e. the maximum positive or least negative power. Although in theory the contact lens powers can be obtained this way, the instability of the spherical lens on a toroidal cornea in the example given above would result in a poorly fitting lens and might cancel out the notional advantages. It may be preferable to calculate a trial toric lens specification as in the first example, or to send the refraction and keratometry details to the laboratory and let them do the calculations.

COMPENSATED BITORIC OR SPHERICAL POWER–EQUIVALENT BITORIC LENSES These are rigid lenses in which the front toroidal surface in air has an equal and opposite cylindrical power to the back toroidal surface in tears. It thus acts like a spherical lens on the eye, both in terms of power and in that rotation of the lens on the eye does not induce any cylindrical effect (see Chapter 11). Such a lens may be required for reasons of comfort or fit, or if a spherical lens flexes on a toroidal cornea.

BOZR of lens to be ordered

8.10 along 180 7.60 along 90

Table 7.12  Calculation of the Front Surface Astigmatism

BOZR of spherical trial lens for refraction

8.10 mm

Required BVP of final lens (in air)

BVP of spherical trial lens for refraction

−2.75D

Over-refraction

−1.00/−0.50 × 90

When expressed in normal sphero-cylindrical form

which, when converted to crossed cylinder form, gives

−1.50 D along 180/−1.00 D along 90

For an RGP lens back surface, powers (in air) (from Formula III for surface power (1.450 − 1))

−59.21 along 90 and −55.56 along 180

BVP of final lens along 180

−2.75 + (−1.50) = −4.25

Back surface cylinder in air is thus

−59.51 –(–55.56) = −3.65 × 180

Front surface cylinder required (ignoring thickness)

(−2.23− [−3.65]) × 180 = +1.42 × 180

BVP of final lens along 90 is approximately −2.75 + (−1.00) + allowance of −2.50 D for radius change (see Approximate rule [1]) = −6.25 D in total

−4.25 along 180 −6.48 along 90 = −4.25/−2.23 × 180

150

SECTION 3  •  Instrumentation and Lens Design

If, for example, a rigid spherical RGP lens corrects the astigmatism of an eye but for physiological (and/or fitting) reasons, an RGP lens with a toroidal back surface to match the corneal radii may be chosen. This prevents flexure but introduces astigmatism at the interface of the lens back surface with the liquid lens. The amount of this induced astigmatism is given by: 1000(n′ − n) 1000(n ′ − n) − rF rS where

RESULTANT OF TWO CYLINDRICAL POWERS Any resultant cylindrical effect from misalignment of the two astigmatic surfaces or of the lens on the eye can be found graphically by Stokes’ construction (Jalie 1984/2016, Rabbetts 2007), calculated by ‘astigmatic decomposition’ (Bennett 1984), Fourier decomposition (Thibos et al. 1997, Rabbetts 2007) or by using the online programs.

Power Variations as a Sequel to Other Changes

n′ = refractive index of contact lens n = refractive index of tears = 1.336 rF = BOZR along flat meridian of lens back surface rS = BOZR along steep meridian of lens back surface.

POWER AND SPHERICAL ABERRATION (see ‘Aberrations of Contact Lenses’, page 153 and also Section 9, Addendum, available at: https://expertconsult.inkling.com/)

To compensate for this astigmatism introduced at the back surface of the lens in situ, a front toroidal surface with principal meridians parallel to those of the back surface must be made which, since it is in air, will not be as toroidal as the back surface. The ratio between front and back surface toricity depends on the refractive index, n′, of the lens material and is given by:

The steep curvature of contact lenses may result in significant spherical aberration, whether over the area of the wearer’s pupil or the aperture of the lens support of a focimeter used to measure the power. Because of their shallow curvature, most spectacle lenses do not suffer from significant spherical aberration, so that paraxial optical calculations may be used to describe their BVP. Although paraxial BVP may also be used for contact lenses, some manufacturers may use a nominal value based on an average over the area of the pupil or focimeter support, or the similar value calculated for a diameter approximately 0.7 times the full diameter (Campbell 2009). Manufacturers of RGP lenses may use the paraxial values when calculating the lens parameters, but the ability to modify the lens or exchange it obviates any problem. Some manufacturers make a tiny compensatory adjustment to the value of the refractive index in the lens calculations to give the label power rather than the paraxial power. The 2017 revision of the ISO contact lens vocabulary standard ISO 18369-1 introduced the term ‘label back vertex power’ to describe the optimal focus over the optical zone of a lens. Studies by Young et al. (1999), Belda-Salmerón et al. (2013) and Wagner et al. (2015) showed that frequentreplacement soft lenses from various manufacturers had powers that differed for the same labelled power. Hence a wearer may need slightly different powers for different brands of lenses, even ignoring the need to check for fitting and physiological differences. Wagner et al. (2015) postulated that this may be caused by slight errors in the refractive index of the lens – finding that some lenses differed by 0.01 from the nominal index. Although this would give a significant error in the power of the front surface, for a lens as a whole, the effect would be negligible. If the formula for the power of a surface is differentiated,

1 (n ′ − 1) n ′ − 1.336 = 1 (n ′ − 1.336) n′ − 1 For an RGP material of refractive index 1.45 it is 1.45 − 1.336 = 0.253 or approximately 1 4 1.45 − 1 Thus to provide a spherical effect on the eye, a compensated parallel bitoric lens has a front surface cylinder which, depending upon the refractive index of the lens material, counteracts about a quarter of the back surface cylinder in air. This information is useful when checking the lens on a focimeter. Its total cylindrical power is thus about three-quarters of the back surface cylinder in air, the latter being easily obtained from radiuscope readings (see Chapter 18) and then by radius/power conversion (see Formulae III–V). If such lenses are not manufactured accurately with the principal meridians absolutely parallel on the front and back surfaces, then not only is the cylindrical power of the lens in air different from that expected, but the lens will also not provide the correct ‘spherical equivalent’ effect on the eye (Douthwaite 1988, 2006). Similarly, if the power of the cylinder on the front surface does not have the correct ratio to that of the back surface, then again it will not be the equivalent of a spherical lens on the eye. This may not be problematical unless the lens rotates, when the effect of the swinging cylinder may give rise to a reduction and variation in visual acuity. These erroneous effects are compounded if the front surface cylinder is both incorrect in power and not parallel to that of the back surface – which may explain the reluctance of some manufacturers to supply such lenses, as they are extremely difficult to manufacture accurately.

dF 1 F = = dn r n − 1 So, if the effects of lens thickness on BVP are ignored, the sum of the front and back surface powers of the lens may be regarded as the BVP, F′V, of the lens, and the formula above applied to both surfaces; hence, assuming a refractive index of 1.4:

7  •  Optics and Lens Design

∆n 0.01 ≈ × F ′ V ≈ 0.025 × F ′ V n − 1 0 .4

After the lenses are allowed to settle on the eye, an overrefraction through disposable trial lenses similarly avoids the problem of an unexpected residual error when a lens is ordered on the basis of the ocular refraction. It also overcomes the next three potential effects.

0 –.05

Bennett Bibby 1.75

–.10 Power error (D)

∆F ′ V = F ′ V ×

151

–.15 –.20 –.25 –.30 –.35 –.40

POWER CHANGES OF SOFT LENSES

–.45

Before being placed on an eye, a soft contact lens is normally in a fully hydrated state in physiological saline solution and the refractive index is at its lowest value. The lens is also at room temperature, and its curvature (i.e. its BOZR, or back optic radius if the back surface is aspherical) and power should be as specified by the manufacturer. After being placed on the eye, the following changes to the soft lens occur, all of which affect the power of the lens. Ford (1976) termed this altered state of the lens on the eye as ‘the equilibrated state’, which takes into account changes due to:

–.50 –20

flexure temperature ■ evaporation.

–15

–10

–5

Flexure The BOZR of modern soft lenses is considerably flatter than the central cornea (see Chapter 10), and so when placed on an eye, the centre of the back surface of the lens alters to take up the same curvature as the central cornea, or almost so. This change in curvature is commonly referred to as flexure or draping. The amount of the resultant power change due to flexure, be it spherical or toroidal, is small for thin lenses but becomes significant for lenses of high positive power (Fatt & Chaston 1981). Plainis and Charman (1998) reviewed some of the many formulae for predicting power changes due to this flexure, and took an experimental survey of various lenses fitted. Their results fitted both Bennett’s (1976) analysis and an equation based on a constant sagittal change for the two surfaces. Bennett based his argument on the following: The volume of the lens remains constant even though its curvature changes. ■ There was no redistribution of lens thickness. ■ The centre thickness remains unchanged. ■ The front surface of the lens remains spherical if the cornea is spherical. ■

Bearing these factors in mind, he calculated that, for the same central thickness, both positive and negative lenses change power with flexure by the same amount that concentric lenses (i.e. lenses that have a common centre of curvature for back and front surface radii) change power when they are bent. As *

For simplicity, since this discussion considers only the optical zones of the lens, the subscripts 1 and 2 referring to the front and back surfaces of the lens identifying the radius r are used rather than the more complicated subscripts referred to in ISO 18369-1.

10

15

20

Fig. 7.19  Power changes due to flexing to drape onto a cornea of radius 7.8 mm. Averaged results for two types of contact lenses using Bennett’s and the Bibby beam-bending models (see text).

an approximation, Bennett derived a simplified equation for the BVP of a lens with concentric surfaces:

■ ■

0 5 Lens power

F′V = −

1000 (n − 1) t × 2 n r2

where F′V is BVP, n is refractive index, t is centre thickness and r2 is the BOZR*, where both of these are in millimetres. For a value of n of 1.40, this gives: F′v = −

286t r22

Thus the change in power on flexing the lens is given by: 1  1 ∆F ′ v = −286t  − 2 2  (r2 ′) r2  where r2′ is the radius to which r2 changes after flexure. Using the method and more detailed assumptions by Bennett that are described in the online website available at: https:// expertconsult.inkling.com/, Fig. 7.19 illustrates the averaged results for monthly replacement hydrogel and a siliconehydrogel lenses (except for the +20.00 D lens which was available only in the hydrogel material) with indices around 1.39 and 1.40, respectively, when flexing from an initial radius of 8.60 mm to match a cornea of 7.80 mm. The error is negligible for the negative-powered lenses (less than 0.01 D), rising to just over 0.25 D for a +5.00 D lens and 0.50 D for a +15.00 D lens because of their greater thickness. It should be noted that the power change is always in the negative direction. Although it could be argued that Bennett’s assumptions are not absolutely correct, mathematical considerations show that errors introduced by their acceptance are of an insignificant order (Bennett 1976). Bibby (1980) likened the flexing of a contact lens to the bending of a beam or girder under load. He assumed that in an unflexed contact lens, an arc can be found such that the

152

SECTION 3  •  Instrumentation and Lens Design

radial distances (perpendicular to the arc) from the arc to the front and back surfaces of the lens are equal. On draping onto the cornea, the outer surface is assumed to stretch and the inner surface to compress, so that: 1. The surface containing the arc remains without stress; hence, the length of the arc remains constant. 2. At any point on the arc, the radial distances to the front and back surfaces are the same as before flexing. The results of calculations using this method are also shown in Fig. 7.19. Although they are less regular than those from Bennett’s hypothesis, the trend for positive lenses to have increased power loss with increasing power is the same. There is, however, a slight tendency for more negativepowered lenses to be more negative on the eye compared with Bennett’s approach. This approach cannot be simplified to an equation. Unfortunately, these theoretical power changes are not always those found in practice, and some of the differences may be caused by temperature and evaporation changes (see below).

Temperature Effects As the temperature of the cornea is around 36°C and room temperature is about 20°C, there is a change in temperature of the lens when it is put on the eye (Purslow et al. 2005). Even when the lens is saturated, there may be a loss of water content, leading to a tiny increase in the refractive index of the material; however, this makes no significant difference to lens powers. Evaporation Effects The increase in temperature and the exposure of the front surface of the soft lens to air leads to evaporation. This causes the water content of the lens to decrease slightly on the eye, leading to a small increase in refractive index – for a typical hydrogel lens, the equilibrium water content can be calculated from the refractive index by the formula, e.g. Lira et al. (2008): EWC = 952.85 × n2 − 3193.9 × n + 2664 Solving this for n shows that a change in water content from 60% to 57% leads to an increase in index of about 0.0058. Assuming that this change in index is uniform throughout the lens, i.e. not restricted to the front surface, the result would be an increase in power of the lens of approximately 1.5%, an insignificant amount on standard lenses. If the back surface maintained hydration and index, and therefore negative power, with only the front surface refraction altering because of the dehydration, then there would be a positive power gain of almost 1.0 D at +20 D, falling to 0.34 D at −20 D. Unfortunately, the amount of change in the lens due to evaporation depends on the tear output of the wearer (Ford 1974) and on the lens material (Purslow et al. 2005, Lira et al. 2008). Thus greater changes may occur in lenses worn in dry eyes than in those with normal or excessive tear output. It is important therefore to allow soft lenses adequate time to settle before over-refracting and to use diagnostic lenses of approximately the correct power, so that the effects on the eye will be similar. However, the discussion may explain why the final power needed may not agree with the ocular refraction.

THE EFFECTS ON ASTIGMATISM OF POWER CHANGES DUE TO SOFT LENS FLEXURE AND EQUILIBRATION Spherical soft lenses flex to match the corneal contour (see ‘Flexure’ page 151). They therefore replicate the front surface corneal toricity on their own front surface. The lens thickness may reduce the amount of toricity transferred, but as almost all soft lenses have a refractive index higher than that of the cornea, the amount of astigmatism transferred to the soft lens front surface is usually slightly higher than that of the corneal front surface. This, however, is neutralised by the toricity of the back surface, so that the lens maintains an essentially spherical power on the eye. Hence a spherical soft lens should make negligible difference to the cylindrical component of the ocular refraction. The discussion above showed that as a soft lens with back optic radius flatter than the cornea flexes to match the steeper cornea, the power becomes more negative. This is theoretically more so along the steeper meridian, but the amount of correction of corneal astigmatism will be negligible for negative-powered lenses because of their thinness (around 0.03 D for a 0.4 mm difference in K reading); the correction is slightly more for positive-powered lenses (around 0.10 D at +5.00 D).

THE EFFECTS OF FLEXURE ON RIGID CORNEAL LENSES A spherical corneal lens may be chosen to fit a mildly astigmatic cornea since the liquid lens will neutralise the corneal astigmatism. In the same way that a soft lens flexes to conform to the corneal contour, there is a tendency for rigid lenses to flex on toroidal corneas, with the front surface partially replicating the corneal astigmatism. The back surface of the lens will flex similarly, so the power of the lens on the eye will remain essentially spherical, leaving the astigmatism produced by the front surface of the liquid lens to become manifest. The flexing on the eye may be demonstrated objectively by keratometry. Measurement of the BOZR with a radiuscope may also give a toric reading, since although initially the lens may recover its spherical form on removal from the eye, it may warp permanently with time. Thus a lens that corrected the corneal astigmatism on issue may result in residual astigmatism after a few weeks’ wear. Since both front and back surfaces will have changed by similar amounts, there will be no significant cylindrical power visible on the focimeter in air. Unfortunately, flexure may vary, as the lens moves on the eye and is best assessed by refractive techniques (both objective and subjective) with the lens in situ. Collins et al. (2001) found that with both PMMA and RGP (Boston XO) lenses, the flexure increased as the centre thickness decreased. When fitted in approximate alignment to the flattest meridian of corneas with 1.00–2.00 D of astigmatism, the front surface of −3 D lenses with central thickness of 0.05 mm showed regular astigmatism close to that of the cornea, whereas lenses of 0.15 mm thickness showed less than 0.5 D of astigmatism. Surprisingly, the very thin lenses showed more astigmatic deformation than the cornea on near-spherical corneas, presumably caused by lid pressure during blinking.

7  •  Optics and Lens Design

Lens flexure can be minimised by fitting the BOZR as flat as possible (Pole 1983, Stone & Collins 1984) and by keeping the back optic zone diameter (BOZD) as small as possible (Brown et al. 1984). Although the flexure is not as great with an RGP lens as with a soft lens, it may necessitate fitting a compensated bitoric lens (Douthwaite 1988 and see ‘Compensated Bitoric or Spherical Power-Equivalent Bitoric Lenses’, page 149). Such flexure will be beneficial optically in those rare cases where some of the residual against-the-rule astigmatism from the crystalline lens will be neutralised by lens flexure on a cornea having with-the-rule astigmatism.

ABERRATIONS OF CONTACT LENSES (see Section 9, Addendum, available at: https://expertconsult.inkling.com/)

2.5

produced by a soft lens differs between off-eye and on-eye conditions (see Fig. 7.20). Assuming an average corneal asphericity, soft lens manufacturers could use aspherical surfaces to avoid the lens affecting the eye’s aberration, while another approach is to design the lens surfaces to correct the average ocular spherical aberration. Lindskoog Pettersson et al. (2008) found that, by measuring the spherical aberration of the lens-wearing eye, on average a standard lens gave less residual spherical aberration in myopic wearers than lenses designed to correct the lens and eye aberration. These results were repeated when the same spherical lens was compared with two different aspherical designs (Lindskoog Pettersson et al. 2011), but measurement of high- and low-contrast visual acuity at distance and of high-contrast visual acuity and contrast sensitivity at near showed no significant difference. Spherical aberration differs between eyes; therefore it may be better to compare the subjective response to differing designs during the lens-fitting procedures. Although reducing any positive aberration may be beneficial for distance vision, with the usual lag of accommodation, it might have the opposite effect at near. Because spherical aberration increases approximately with the square of the pupil diameter (see Fig. 7.21), large pupils may be more affected than small pupils. Aberrations are more complicated in rigid lenses with aspheric back surfaces, particularly since these move on the eye. The extra aberrations and astigmatism induced by this surface may require the front surface to be also aspheric to cancel them out (Hammer & Holden 1994, El-Nashar 1999, Hong et al. 2001). Aspheric front surfaces are also used to create an ‘extended’ focus or progressive power (see ‘Progressive Power Contact Lenses’ below and Chapter 13).

Bifocal Contact Lenses Chapter 13 deals with the various designs and methods of fitting all types of bifocal contact lenses. An appreciation of the optical principles of concentric solid bifocals with the distance or near portion in the centre permits a general understanding of all other designs of bifocal contact lenses.

In Air On-Eye

2.0 1.5 1.0

Spherical aberration (D)

Spherical Aberration @ 6mm Pupil (D)

Because of their steeply curved surfaces, the cornea and contact lenses can produce spherical aberration. The cornea’s potential contribution to the overall spherical aberration of the eye is reduced by its peripheral flattening, but the population studies by Porter et al. (2001) of the eye’s aberrations show that the relaxed young eye has some undercorrected or positive aberration, i.e. the peripheral rays through the pupil focus closer to the cornea than the paraxial rays. Because of their rotational symmetry, contact lenses may affect the ocular spherical aberration and, if decentred, coma. When a rigid lens is placed on the eye, the overall aberrations are likely to increase, particularly for positivepowered lenses with their steeper curves. Cox (1990) suggested that this was true for lenses of power more positive than −3.00D. With soft lenses, it is uncertain how much corneal asphericity is transferred to the front surface of the lens during flexing, and what effect the different anterior radius of curvature has on the p-value. Cox’s (1990) calculations using the beam-bending model suggest that soft lenses would increase ocular aberrations for powers outside the range −6.00 to +3.00D. Using the beam-bending model (see Fig. 7.19), Cox (2000) also showed that the calculated aberration

.50 –.50 0.0 –1.0 –1.5 –2.0 –2.5

153

–12

–9

–6

–3

0 3 BVP (D)

6

9

12

Fig. 7.20  Theoretical spherical aberration for low-water soft lenses ‘in air’ and ‘on eye’ (conformed to a corneal shape 7.8 mm radius, p = 0.7). (With permission from Cox, I.G., 2000. The why and wherefore of soft lens visual performance. Cont. Lens Anterior Eye 23, 3–9.)

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

0.5

1 1.5 Distance from axis (mm)

2

2.5

Fig. 7.21  Variation of spherical aberration with height of incident ray for a contact lens (n = 1.490) of +12.00 D BVP, BOZR of 7.80 mm and tc of 0.35 mm. Parallel incident light is assumed.

154

SECTION 3  •  Instrumentation and Lens Design

Similarly, for the near portion, F1NP

F1DP

since L1 = 0.00 D and F1NP = +61.00 D L ′1 = +61.00 D.

F2NP F2DP

(a)

(b)

Fig. 7.22  Concentric solid bifocal corneal lenses: (a) front surface addition; (b) back surface addition.

CONCENTRIC AND FLAT-TOP SOLID BIFOCALS Rigid lenses are available with the addition worked on either the front or back optic surface (Fig. 7.22) (and, of course, a combination of back and front surface additions can also be used), soft lenses only on the front. When the addition is on the front surface (a plastic/air interface), the front optic zone has two radii worked on it, the steeper corresponding to the near portion of the lens. Then, provided that the lens is assumed to be infinitely thin, the near addition is equal to the difference between the two front surface powers. For example, if the near addition to be incorporated in an RGP lens is +3.00D, where F1DP and F1NP are the front surface powers of the distance and near portions, respectively,* then F1NP = F1DP + 3.00. Since this is a plastic/air interface, the appropriate front optic zone radii may be obtained from Formulae III–V taking n2 = 1.47 and using n1 = 1.00. If, in a particular case, F1DP is calculated to be +58.00D, this gives a radius, r1DP, of

The 0.22 mm thickness adds 0.56D to the reduced vergence reaching the back surface. The difference between 0.46D and 0.56D is 0.10D, which is small enough to be ignored, but it indicates that F1NP should be reduced by this amount, from +61.00D to 60.90D, giving r1NP as 7.72 mm instead of 7.70 mm. In practice this small radius change is not worth making. There is a tendency for a small negative-powered liquid lens to collect in front of the upper and lower portions of any corneal lens due to the tears prism along the eyelid margins. The configuration of the front surface of a solid bifocal with front surface addition (Fig. 7.22a) is such that this tear lens may slightly reduce the front surface positive power at the periphery. It is wise, therefore, to err on the positive side (by as much as +1.00D) to allow for this negative tear lens, although it varies depending on tear output and evaporation. Executive style segments have been made as solid front surface bifocals, with a junction ridge (see Chapter 13). The optical theory is the same as for concentric bifocals. When the addition is on the back surface (Fig. 7.21b), no allowance for the effect of thickness need be considered; however, the major consideration here is that it is a plastic/ tears interface, rather than a plastic/air interface. In air the power of the RGP surface depends on 1 − 1.470 r whereas in tears it depends on

(1.470 − 1)1000 = 8.10 mm 58.00

1.336 − 1.470 r

For an addition of +3.00D, F1NP must therefore be +61.00D, giving a radius of 7.70 mm. If thickness is to be taken into account, reference to Formula VI should be made. In the example just given, if the centre thickness of the distance portion were 0.20 mm and that of the near portion 0.22 mm,

This is a factor of 0.470/0.134, or approximately 3.51, for RGP lenses of this index. (For a refractive index of 1.45, this factor becomes 3.95; for soft lenses of refractive index of 1.4, the factor becomes over 60, so this design is impractical and, moreover, the draping effect would transfer any back surface curvature changes onto the front.) The back surface radii must therefore provide approximately three to four times the addition on the back surface (when measured in air) than is really required, due to the neutralising effect of the tears. Usually this type of bifocal is fitted with a steep BOZR (r2DP)* and a small BOZD, with the back peripheral optic zone radius (r2NP) providing the near addition and fitted so as to align or be just flatter than the cornea. For example:

since L1 = 0.00 D and F1DP = +58.00 D L ′1 = +58.00 D for the distance portion. It can be seen that the 0.20 mm thickness adds 0.46D to this power. The reduced vergence reaching the back surface is thus +58.46D.

If the BPOZR (r2NP) is 8.50 mm, reference to Formula III Available at: https://expertconsult.inkling.com/ for 1.336–1.470 shows F2NP to be −15.764D. (It is negative in power because the medium of higher refractive index has a concave surface.) ■ To give a +3.00D addition requires that F2DP be −18.764D; thus r2DP (the BOZR) is 7.14 mm (from Formula IV). ■

*For simplicity, since this discussion considers only the optical zones of the lens, the subscripts 1 and 2 referring to the front and back surfaces of the lens identifying the radius r are used rather than the more complicated subscripts referred to in ISO 18369-1.

7  •  Optics and Lens Design

155

A Addition

B C D E

B 3

2 1 0 1 2 Distance from centre of lens (mm)

3

A

C Fig. 7.23  (a) Power profile of a centre-near, front surface, progressive power lens. (b) Schematic view (grossly out of scale) of the image formation by the front surface. The dotted line indicates a spherical surface having the same radius of curvature as the apical radius of the lens. (c) Illustration of the bundle of rays forming a blur disc when viewing a distant object.

If this lens is measured in air on a focimeter, the radii of 8.50 mm and 7.14 mm would have surface powers, for 1–1.470, of −55.29D and −65.82D, respectively (see Formula III).

■

Thus the near addition measured in air is +10.52D, i.e. the near addition in tears × 3.51 (approximately) as stated above, depending on the refractive index of the material used.

PROGRESSIVE POWER CONTACT LENSES* (see Section 9, Addendum, available at: https://expertconsult.inkling.com/) Rather like progressive-power† spectacle lenses, contact lenses may be made with a steadily flattening or steepening surface to give a continuous variation of power to provide a longer range of focus. In contact lens form, these lenses are made with radial symmetry. Soft lenses of this type will, like soft bifocal lenses, be of simultaneous vision design. A possible power profile for a centre near design is shown in *Despite the terms and definitions in ISO 18369-1, contact lens literature frequently uses the word ‘multifocal’ lens to describe lenses with a near addition, whether bifocal, blended bifocal or progressive power. † The ISO spectacle lens vocabulary hyphenates progressive-power, the contact lens one does not.

Fig. 7.23a, while a ray trace, grossly out of scale, is shown in Fig. 7.23b. Provided the lens is reasonably well centred on the eye, rays from a distant object passing through A and E will be brought to a focus on the retina, while those from a nearer object passing through the more steeply curved central part at B and D will similarly be in focus. If the lens decentres, however, the behaviour will be more like that of the rigid lens described below. When the eye views a distant, intermediate or near object, the sharp image formed by the appropriate zone of the lens has superimposed on it the blur disc formed by the remainder of the lens, as indicated in Fig. 7.23c. With a centre-near design, the pupil miosis in near vision should reduce the blur, but pupil constriction in bright light out of doors conversely means that the majority of the pupil is viewing through the near zone. The retina’s ability to enhance contrast and the Stiles Crawford effect, or the greater importance of rays passing through the centre of the pupil, may also have an effect on vision. Aspheric curves can also be used in rigid lenses. A centredistance design is used so that as the lens moves up on the cornea in down-gaze for near vision, the peripheral zone slides in front of the pupil, thus giving a partly alternating design. Simplistically, a single aspheric curve could be used, as with a soft lens. Fig. 7.24 shows, again grossly exaggerated, the front surface of a steepening ellipse (p >1), having decentred on the eye. To provide the near addition, the radius of curvature of the element TT of the lens surface in the plane of

156

SECTION 3  •  Instrumentation and Lens Design

References

A Cs T S Ct P

Co

T S

Fig. 7.24  The astigmatic effect when viewing through a peripheral zone of an aspheric surface. The lens is assumed to have moved up on the wearer’s cornea, while the visual axis of the wearer lies along the line PCS. The dotted line indicates a spherical surface having the same radius of curvature as the apical radius of the lens.

the diagram at point P, given by PCT, has to be shorter than the apical radius of curvature AC0. The sagittal radius of curvature, PCS, corresponding to an element of the surface SS at right angles to the plane of the diagram, will differ again. If the wearer has a small pupil, the bundle of light forming the image is that from a small zone surrounding P. Light passing through this single peripheral zone may be significantly astigmatic, whereas with the soft lens, the useful image is that provided by rays passing symmetrically on either side of the apex of the lens. Just as in videokeratoscopy or keratometry, it is the sagittal or axial radius of curvature that is important with the soft lens. It is instructive to consider the astigmatism for a rigid lens. For example, a front surface progressive lens with an addition of +2.00D at 3 mm from the apex in a material of refractive index 1.45 would have a p-value of 1.164. Table 7.13 shows the addition at various heights from the pole or apex, and the astigmatism is calculated from the difference between the sagittal and tangential powers. However, in practice oblique refraction at the surfaces will slightly alter the value entering the eye. Not only is the induced astigmatism clinically significant, but the variation in power in the tangential plane across the pupil is also large, giving rise to coma. The resulting beam pattern reaching the retina will be very irregular. To access the accompanying appendix to this chapter, please visit https://expertconsult.inkling.com/. Table 7.13  The Addition and Surface Astigmatism of a Centre Distance Aspheric Surface Height From Apex (mm)

Addition (D)

Surface Astigmatism (D)

1

0.22

0.14

2

0.87

0.58

3

2.00

1.34

4

3.64

2.45

Belda-Salmerón, L., Madrid-Costa, D., Ferrer-Blasco, T., et al., 2013. In vitro power profiles of daily disposable contact lenses. Cont. Lens Anterior Eye 36, 247–252. Bennett, A.G., 1976. Power changes in soft contact lenses due to bending. Ophthal. Optician 16, 939–945. Bennett, A.G., 1984. A new approach to the statistical analysis of ocular astigmatism and astigmatic prescriptions. In: Charman, W.N. (Ed.), Transactions of the First International Congress: The Frontiers of Optometry, vol. 2. British College of Ophthalmic Opticians (Optometrists), London, pp. 35–42. Bennett, A.G., 1985. Optics of Contact Lenses, 5th ed. Association of Dispensing Opticians, London. Bibby, M.M., 1980. A model for lens flexure – validation and predictions. Int. Cont. Lens Clin. 7, 124–138. Brown, S., Baldwin, M., Pole, J., 1984. Effect of the optic zone diameter on lens flexure and residual astigmatism. Int. Cont. Lens Clin. 11, 759–766. BS EN ISO 18369, 2017. Ophthalmic optics – Contact lenses – Part 1: Vocabulary, classification system and recommendations for labelling specifications. British Standards Institution, London. Campbell, C.E., 2009. Spherical aberration of a hydrogel contact lens when measured in a wet cell. Optom. Vis. Sci. 86 (7), 900–903. Capelli, Q.A., 1964. Determining final power of bitoric lenses. Br. J. Physiol. Opt. 21, 256–263. Collins, M.J., Ranklin, R., Carney, L.G., et al., 2001. Flexure of thin rigid contact lenses. Cont. Lens Anterior Eye 24, 59–64. Cox, I., 1990. Theoretical calculations of the longitudinal spherical aberration of rigid and soft contact lenses. Optom. Vis. Sci. 67, 277–282. Cox, I.G., 2000. The why and wherefore of soft lens visual performance. Cont. Lens Anterior Eye 23, 3–9. Douthwaite, W.A., 1988. Technical note: compensated toric rigid contact lenses. J. Br. Cont. Lens Assoc. 11 (2), 35–38. Douthwaite, W.A., 2006. Contact Lens Optics and Lens Design, 3rd ed. Butterworth-Heinemann, Oxford. El-Nashar, N.F., 1999. Longitudinal spherical aberration and mass of biaspheric aphakic rigid contact lenses. Ophthalmic Physiol. Opt. 19, 441–445. Fatt, I., Chaston, J., 1981. The response of vertex power to changes in dimensions of hydrogel contact lenses. Int. Contact Lens Clin. 8 (1), 22–28. Ford, M.W., 1974. Changes in hydrophilic lenses when placed on an eye. Paper read at the joint International Congress of The Contact Lens Society and The National Eye Research Foundation, Montreux, Switzerland. Ford, M.W., 1976. Computation of the back vertex powers of hydrophilic lenses. Paper read at the Interdisciplinary Conference on Contact Lenses, Department of Ophthalmic Optics and Visual Science, The City University, London. Ford, M.W., Stone, J., 1997. Practical optics and computer design of contact lenses. In: Phillips, A.J., Speedwell, L. (Eds.), Contact Lenses, 4th ed. Butterworth-Heinemann, Oxford, pp. 154–231. Freeman, M.H., Hull, C.G., 2003. Optics, 11th ed. Butterworth-Heinemann, Oxford. Hammer, R.M., Holden, B.A., 1994. Spherical aberration of aspheric contact lenses on eye. Optom. Vis. Sci. 71, 522–528. Hong, X., Himebaugh, N., Thibos, L.N., 2001. On-eye evaluation of optical performance of rigid and soft contact lenses. Optom. Vis. Sci. 78, 872–880. Jalie, M., 1984, 2016. The principles of ophthalmic lenses, 3rd, 4th ed. abdo, London. Lindskoog Pettersson, A., Jarkö, C., Alvin, Å., et al., 2008. Spherical aberration in contact lens wearers. Cont. Lens Anterior Eye 31, 189–193. Lindskoog Pettersson, A., Mårtensson, L., Salkic, J., et al., 2011. Spherical aberration in relation to visual performance in contact lens wear. Cont. Lens Anterior Eye 34, 12–16. Lira, M., Santos, L., Azeredo, J., et al., 2008. The effect of lens wear on refractive index of conventional hydrogel and silicone-hydrogel contact lenses: A comparative study. Cont. Lens Anterior Eye 31, 89–94. Plainis, S., Charman, W.N., 1998. On-eye power characteristics of soft contact lenses. Optom. Vis. Sci. 75, 44–54. Pole, J.J., 1983. The effect of the base curve on the flexure of Polycon lenses. Int. Contact Lens Clin. 10 (1), 49–52. Porter, J., Guirao, A., Cox, I.G., et al., 2001. Monochromatic aberrations of the human eye in a large population. J. Opt. Soc. Am. A. Opt Image Sci Vis. 18 (8), 1793–1803. Purslow, C., Wolffsohn, J.S., Santodomingo-Rubido, J., 2005. The effect of contact lens wear on dynamic ocular surface temperature. Cont. Lens Anterior Eye 28, 29–36.

7  •  Optics and Lens Design Rabbetts, R.B., 1992. Spreadsheet power calculation for toric lenses. J. Br. Cont. Lens Assoc. 15, 75–76. (1993), 16, 41. Rabbetts, R.B., 2007. Bennett and Rabbetts’ Clinical Visual Optics, 4th ed. Butterworths, Oxford. Sarver, M.D., 1963. The effect of contact lens tilt upon residual astigmatism. Am J Optom Arch Am Acad Optom. 40, 730–744. Sorsby, A., Leary, G.A., Richards, M.J., 1962. The optical components in anisometropia. Vision Res. 3, 43–51. Stone, J., 1966. The use of contact lenses in the correction of astigmatism. Optica Int. 3, 6–23. Stone, J., 1967. Near vision difficulties in non-presbyopic corneal lens wearers. Cont. Lens 1 (2), 14–25. Stone, J., Collins, C., 1984. Flexure of gas permeable lenses on toroidal corneas. Optician 188 (4951), 8–10. Thibos, L.N., Wheeler, W., Horner, D., 1997. Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error. Optom. Vis. Sci. 74, 367–375.

157

Tunnacliffe, A., 1993. Introduction to Visual Optics, 4th ed. Association of British Dispensing Opticians, London. Wagner, S., Conrad, F., Bakaraju, R.C., et al., 2015. Power profiles of single vision and multifocal soft contact lenses. Cont. Lens Anterior Eye 38, 2–14. Westerhout, D., 1969. Clinical observations in fitting bitoric and toric forms of corneal lenses. Contact Lens 2 (3), 5–21, 36. Westheimer, G., 1962. The visual world of the new contact lens wearer. J. Am. Optom. Assoc. 34, 135–140. Winn, B., Ackerley, R.G., Brown, C.A., et al., 1986. The superiority of contact lenses in the correction of all anisometropia. Trans. BCLA Conference 95–100. Young, G., Lewis, Y., Coleman, S., et al., 1999. Process capability measurement of frequent replacement spherical soft contact lenses. Cont. Lens Anterior Eye 22 (4), 127–135.

Appendix Aspects of Contact Lens Design The main purpose of this appendix is to give some guidelines to those contact lens practitioners who wish to design their own lenses. It is hoped that they can then avoid the pitfall of ordering a lens of such thickness that it is impossible to manufacture. While intended primarily for the design of rigid corneal lenses since the practitioner has the option to specify the detailed design, the principles outlined in this section may be applied to any type of contact lens. Readers are referred to the book by Douthwaite (2006) and papers by Campbell (1987) on the calculation of tear volume between lens and cornea, and Young (1988) who gives an overview of computer-assisted contact lens design. The formulae given should allow users of computers and programmable scientific calculators to carry out the following tedious calculations extremely quickly. The formulae may easily be programmed into spreadsheets enabling calculations to be performed quickly and easily. It is advisable, however, to consult Figs 7.25–7.31 and the accompanying explanatory text first before working through some examples with a calculator in order to provide a firm understanding of the optical or geometrical principles involved. The use of a computer or programmable scientific calculator permits much more complex lens forms to be considered than the multicurve spherical design considered in the text here. It is thus possible to cope with aspherical lens surfaces in conjunction with ellipsoidal corneal surfaces of specific asphericity, and to design lenses to give a required central tear layer thickness and a specified corneal clearance at the lens edge. Practitioners are recommended to study the definitive work of Bennett (1988) on the subject of aspheric and continuous curve contact lenses. In this revision, the author has mostly continued the original authors’ number of decimal places in the answers to calculations in order to show the numerical differences in, for example, tear layer thickness when lens specifications are rounded to the nearest 0.05 mm. Whether such differences are significant in terms of fit may be questionable.

sagitta 3.1.5.9) only in relation to scleral lenses, where it has a different meaning. It should be obvious from Fig. 7.25 that, for both positive and negative lenses, the sagitta or sag value, sap, of the front surface of a lens plus the axial edge thickness, tEA, must equal the primary sag (of the back surface), sp, plus the centre thickness, tC. Thus sap + tEA = sp + tC. For a positive lens there is a danger of ordering the centre thickness too small to permit adequate edge thickness, and for a negative lens the attempt to keep the edge thickness reasonably small may result in the centre of the lens becoming excessively thin. The values for sp and sap may be found as follows.

Primary Sag (sp) (see Formula XXXII) From Fig. 7.26 it can be seen that the primary sag of the back surface of a tricurve corneal lens is x0 + x1 + x2 where x0 is the sag of the BOZR at the BOZD = sØ0r0. x1 and x2 may be determined by studying Fig. 7.26b,c. Thus, x1 = s Ø1r 1 − s Ø0r 1 where s Ø1r 1 is the sag of BPR1 at BPD1 and s Ø0r 1 is the sag of BPR1 at BOZD. x2 is determined in exactly the same manner as x1 (see Fig. 7.26c). Thus, x2 = s ØTr2 − s Ø1r2 where s ØTr2 is the sag of BPR2 at TD (or BPD2 if the lens has more than three back surface curves) and s Ø1r2 is the sag of BPR2 at BPD1. It can be seen, then, that the primary sag of a C3 lens is given by s p = x0 + (s Ø1r 1 − s Ø0r 1 ) + (s ØTr 2 − s Ø1r 2 ) and for a C4 lens s p = x0 + (s Ø1r 1 − s Ø0r 1 ) + (s Ø2r2 − s Ø1r2 ) + (s ØTr3 − s Ø2r3 ) . All the individual sag values may be obtained directly from Formula XI (or XII for ellipsoidal surfaces). For example the tri-curve corneal lens C3 8.00:7.50/8.90:9.60/10.00:9.60 has a primary sag, sp, which can be calculated as follows:‡ x0 = sag of BOZR of 8.00 mm at BOZD of 7.5 mm = 0.933 mm x1 = s Ø1r 1 − s Ø0r 1 Now s Ø1r 1 = sag of 8.90 mm (BPR1 ) at 8.6 mm ( BPD1 ) = 1.108 mm s Ø0r 1 = sag of 8.90 mm (BPR1 ) at 7.5 mm ( BOZD) = 0.829 mm

SAGITTA OF FRONT AND BACK SURFACES (See Formulae XI and XII† for spherical and ellipsoidal surfaces, respectively.) The following sections refer to the ‘primary sag’ of the back surface. This is defined as the total sag of the back surface of the contact lens, including all peripheral curves, but excluding the effects from rounding the lens edge. BS EN ISO 18369-1: (2017) uses this term (in the form: primary

The sags, radii, thicknesses, etc. have been calculated to an accuracy of ten decimal places where appropriate. This explanation is given in case a student is puzzled by the slightly different results obtained if the rounded figures are entered into the various formulae.

‡ †

All figures in bold Roman numerals refer to the numbers of the appropriate formulae in the Formulae section.

The values shown above have been rounded to three decimal places for clarity on the printed page. However, calculators and computers operate in floating decimal mode, even when the number of decimal places displayed is deliberately limited.

157.e1

157.e2

SECTION 3  •  Instrumentation and Lens Design

(a) tC sap

sp

tEA (b)

tC sp

sap tEA

Fig. 7.25  Sap+ tEA = Sp + tC (a) Positive lens; (b) negative lens. (Note: sap = sa0 when the front surface of the lens has one single curve of radius ra0)

x0

ø0

BOZD BPD1 TD

x1 x2

søTr0

sp

ø1 øT

IEA2

IEA1 IEA

(a)

X1

sø0r1

BOZD BPD1

sø1r1

(b)

x2

BPD1

sø1r2

TD

søTr2

(c) Fig. 7.26  Back surface of a lens. (a) sp = x0 + x1 + x2; lEA = sØT r0 − sp ; lEA = lEA1 + lEA2 ; (b) x1 = sØ1r 1 − sØ0r 1 ; (c) x 2 = sØT r2 − sØ1r2 . (Note: BOZD = Ø0, BPD1 = Ø1, TD = ØT)

Thus x1 = 1.108 − 0.829 = 0.279 mm x2 = s ØTr2 − s Ø1r2 Now s ØTr2 = sag of 10.00 mm (BPR2 ) at 9.6 mm (TD) = 1.227 mm and s Ø1r2 = sag of 10.00 mm (BPR2 ) at 8.6 mm (BPD1 ) = 0.972 mm So x2 = 1.227 − 0.972 = 0.2 256 mm and primary sag, s p = x 0 + x1 + x2 = 0.933 + 0.279 + 0.256 = 1.468 mm (Note: Small discrepancies may occur in the third place of decimals due to rounding.)

Axial Edge Thickness (tEA) When using sagitta-involved methods of designing lenses it is necessary to measure thickness values parallel to the primary axis of the lens, although this involves using axial junction thickness, a term which is not used in International Standard BS EN ISO 18369:1 (2017). The abbreviations tEA and tER are used in the Standard to denote axial and radial edge thickness, and have been augmented here by tJA and tJR to denote axial and radial junction thickness. (Radial thicknesses are measured perpendicular to the back surface.) In order to determine the centre thickness and therefore front surface details, tEA may be chosen as any desired value. It will therefore be taken as 0.15 mm for the purposes of this Appendix. Front Surface Sag (sap = sa0 for Lenses With a Single Front Surface Curve, ra0) Now suppose the lens with the above primary sag value of 1.468 mm is to be made up with F′v = +2.00D, using a material having a refractive index (n) = 1.43. First choose an appropriate axial thickness for the edge of the lens (tEA), and as indicated above, take the value 0.15 mm. The sagitta of the required front surface (sa0) can be calculated using Formula XXXVII (see Formulae section), where q, y and b are ‘sub’ values required for this formula, and other symbols are as above. (The lens is assumed to be infinitely thin at this stage.) (In the following calculations the symbol * represents a multiplication sign as now used by computers and calculators.) Back surface power = F2 = (1.00 − 1.43) 0.008 m (or 1000 ∗ (1.00 − 1.43) 8 mm ) = −53.75 D (from III ) Front surface power = F1 ‘thin’ = F ′ v − F2 = +2.00 − −53.75 = +55.75 D q = s p − tEA = 1.468 − 0.15 = 1.318 mm y = Ø T 2 = 9.6 2 = 4.8 mm b = [(n − 1) (2 − n)] ∗ [1000n F ′1‘thin’ − q ] = [0.43 0.57] ∗ [1430 55.75 − 1.318] = 0.887 ∗ 22.880 = 18.356 Enter the values b and y into Formula XXXVII to calculate sa0.

sa 0 = b − b2 − y 2 ∗ n ( 2 − n ) Sa 0 = 18.356 − 18.3562 − 4.82 ∗ 1.43 ( 2 − 1.43) = 18.356 − 336.937 − 23.04 ∗ 1.43 0.57 = 18.356 − 336.937 − 57.802 = 18.356 − 279.135 = 18.356 − 16.707 = 1.649 mm (from XXXVII )

7  •  Optics and Lens Design

157.e3

The values for Formula XXXVII become:

Anterior Optic Radius (ra0) ra 0 = ( y2 + sa 02 ) (2∗ sa 0 ) = (23.04 + 2.718) 3.297 = 25.758 3.297 = 7.812 mm m (from XVII )

y now = Øa 0 2 = 7.00 2 = 3.75 mm q now = s Øa 0 p − t JA As Øa 0 = Ø0, s Øa 0 p = s Ø0r 1 , the sag of the back central optic surface = 0.933 mm (from above).

Centre Thickness (tc) tC = sa 0 + tEA − s p (see Fig. 7.25) = 1.649 + 0.15 − 1.468 = 0.330 mm

t JA = tEA = 0.15 mm, so q now = 0.933 − 0.15 = 0.783 mm

Lenticular Lenses If, from the previous calculations, it is felt that the tC of 0.33 mm is too great, but the tEA of 0.15 mm is desirable, then a lenticular front surface must be designed. It is convenient, although not essential, to make the central lenticular diameter (Øa0) the same as the back optic zone diameter (Ø0), so let Øa0 = Ø0 = 7.50 mm and, so that the lens will have a parallel carrier zone, let the junction thickness (tJA) = tEA = 0.15 mm. Calculation of sa0 (now sØa0ra0 for the lenticular front surface; Fig. 7.27), ra0 and tC for the central lenticular surface is the same as for the anterior surface of the full aperture lens above except that tJA is used instead of tEA (in this example they are the same) and the primary sag of the back surface must be calculated at Øa0 instead of ØT.

n − 1  1000n  0.43  1430 b =  ∗ − 0.783 = 18.759 ∗ − q =   2 − n   F ′1‘thin’  0.57  55.75 sa 0 = sφa 0ra 0 = b − b2 −

y2 ∗ n − − − − XXXVII 2− n

3.752 ∗1.43 0.57 = 0.965 mm (from XXXVII ) = s Øa 0ra 0 is used to calculate the lenticular ra 0

sa 0 = sφa 0ra 0 = 18.759 − 18.7592 − sa 0

ra 0 = ( y2 + s Øa 0ra 0 2 ) (2 ∗ s Øa 0ra 0 ) − − − − XVII ra 0 = (14.06 + 0.965) 1.930 = 7.768 mm (from XVII )

∅a0 1

11 9

4 2

14 15

3

x1

6 5

x

10

13

16

12

7

x2

8

1 2 3 4

tC s∅Tr0 sp s∅0r0

5 6 7 8

(x1+x2)= tJA tEA lEA

∅0 ∅l ∅T 9 10 11 12

s∅a0ra1 s∅TrA1 s∅a0ra0 tEA

13 14 15 16

xa1 sap tC sp

Fig. 7.27  Positive lens with reduced optic zone and ‘parallel’ carrier zone (i.e. axial edge and junction thicknesses are the same) and having the same back and front optic zone diameters. The heavy lines on the left-hand side show the tricurve back surface with related sag values and thicknesses. On the right-hand side, the heavy lines show the bicurve front surface and related sag values

157.e4

SECTION 3  •  Instrumentation and Lens Design

and

sap = sØa 0ra 0 + (s ØTra1 − s Øa 0ra1 ) (see Fig. 7.27) = 0.962 + (1.312 − 0.7 777) = 1.496 mm

tC = s Øa 0ra 0 + t JA − s Ø0r0 (see Fig. 7.27) = 0.965 + 0.15 − 0.933 = 0.18 mm Thus lenticulation has reduced tC by about 43% from 0.33 to 0.18 mm.

Calculation of Front Peripheral Radius (ra1) The lens is to have a ‘parallel’ carrier, so tEA must = tJA. Therefore the anterior peripheral radius, ra1, must provide the same sag value (xa1) between Øa0 and ØT as the sag provided by the peripheral geometry of the back surface from Øa0 out to ØT. In this example, this back surface sag is x which = (x1 + x2) (see Figs 7.27 and 7.33 and Formulae section). This statement is true for any form of back surface. The data from the back surface may be used to design the front peripheral surface. There are two methods of determining the sag, x, of the back peripheral surface (which determines the value for xa1). In the example being used (for a lens of back surface specification C3 8.00:7.50/8.90:8.6/10.00:9.6) x = (x1 + x2 ) = (s Ø1r 1 − s Ø0r 1 ) + (s ØTr2 − s Ø1r2 ) (see Example 26 and Formu ulae XXVIII and XXXVIII ) = (1.1077 − 0.8286) + (1.2273 − 0.9717)(sags from XI ) = 0.5347 mm = xa1 Alternatively (see Formula XXIX), x may be determined if the axial edge lift, lEA, is known. Now l EA = s ØTr0 − s p (see Fig. 7.27) and s ØTr0 = 1.600 mm (from XI ) – s p = 1.468 mm (see ‘Primary sag’, above) Thus l EA = 0.132 mm Using Formula XXIX , xN = s ØTr0 − s Ø0r0 − DlEA (where DlEA = Desired l EA ) Thus xN = x = xa = 1.600 − 0.933 − 0.132 (sags from XI) = 0.535 mm (as above) This enables the anterior peripheral radius to be calculated as follows: ra1 = ((( y∅ T − y02 − xN2 ) ( z*xN ))2 + y∅T 2 ) − − − − XXVIII = (((4.82 − 3.752 − 0.5352 ) (2*0.535))2 + 4.82 ) = 9.439 mm Now that ra1 is known, confirmation that the carrier zone is indeed parallel surfaced may be established by showing that tEA = tJA (which was chosen to be 0.15 mm). This necessitates determining the primary sag, sap, of the anterior surface.1 1

See earlier footnote.

Figs 7.25 and 7.27 show that tEA = s p + tC − sap Thus tEA = 1.468 + 0.178 − 1.496 = 0.15 mm = tJA For negative and positive carrier zones and for lenses in which Øa0 differs from Ø0, see the Formulae section.

Form of the Carrier Zone The form of the carrier zone of a lenticular lens affects the position which the lens takes up on the eye. A lens is said to have a negative carrier zone when its edge thickness is greater than the junction thickness and a positive carrier zone when the reverse applies, the thickness relationship being similar to that of negative and positive lenses. It is desirable for the edge thickness to be equal to or greater than the junction thickness in order to provide a parallel surfaced or negative zone, respectively (see Chapter 9). Negative Lenses As these lenses do not suffer from excessive centre thickness problems, the approach to determining front surface radii is somewhat different from positive lenses. For full aperture lenses, see Formulae XXXV–XXXVII. If excessive edge thickness becomes a problem, a reduced optic design can be employed (Fig. 7.28). A front peripheral radius (ra1) can be calculated for the carrier zone which will give the desired edge thickness (see ‘Calculation of front peripheral radius’ for the +2.00D lens above and the Formulae section). With negative lenses of high power, a desirable edge thickness frequently results in the lens having a positive carrier zone because the edge thickness is less than the junction thickness. This positive carrier zone encourages the lens to drop whereas it may be more desirable for the lens to attach to the upper lid. Since ‘lid attachment’ is more likely to be achieved with a negative or parallel surfaced carrier zone, some way must be found of reducing the junction thickness. By creating a steep intermediate peripheral curve or Front Junction Radius (which becomes ra1), the junction thickness may be reduced so that it is equal to or less than the edge thickness, the carrier zone thereby becoming parallel surfaced or negative, respectively. Such an FJR (now ra1) and front peripheral radius (now ra2) are shown in Fig. 7.29. Occasionally, with very high-powered negative lenses, a front junction radius (ra1) is necessary anyway to join ra0 to ra2. From Fig. 7.29 it can be seen that the front junction radius (ra1) is a shorter radius than either ra0 or ra2. Example (Using the Same Back Surface Specification).  The C3 lens 8.00:7.50/8.90:8.60/10.00:9.60 is to be made with F′v = –10.00D, tC = 0.1 mm (= 0.0001 m), and, in this example, with Øa0 equal to Ø0. The power of the front surface (F1) can be found by tracing light backwards through the lens.

7  •  Optics and Lens Design

157.e5

The sagitta, sap, of the front surface at ØT = 9.86 − (9.862 − 4.82 ) = 1.247 mm (from XI) tEA = s p + tC − s ap (see Fig. 6.28). If the lens is made in full aperture form tEA = 1.468 + 0.1 − 1.247 = 0.321 mm which is thicker than is desirable. Axial thickness at Ø0, tØ0A FP R

= s Ø0r0 + tC − s Ø0ra 0

FO Z

R

= 0.933 + 0.1 − (9.859 − (9.8592 − 3.752 )) = 0.933 + 0.1 1 − 0.741 = 0.292 mm

Fig. 7.28  Negative lenticular lens

(tEA is similar to tØ0A because of the substance removed to create the back surface peripheral curves). To maintain a parallel or negative carrier requires a front junction radius to be calculated, and this is simplified by introducing the concept of front surface edge drop D, as distinct from back surface edge lift. In Formula XXIX, xN = (s ØGr N−1 − s ØG−1r N−1 − Dl EAN ) . This can be modified to xN = (s ØGr N−1 − s ØG−1r N−1 + Dl EAN ) - - - - XXIX(D), to produce a value xN which, when used with Formula XXVIII, will yield a Front Junction Radius to create a stipulated Axial (Edge) drop (dEA), as opposed to Axial (Edge) lift (lEA) at a Given diameter. In this example, let the Given diameter be a chosen anterior Junction diameter (ØaJ), say 8.20 mm, and ØG–1 = Øa0 = 7.50 mm. As this axial drop is not specifically at the edge of the lens in this calculation, the symbol Dd ØaJA will be used: Dd ØaJ A = tØaJ A − DtØaJ A (see Fig. 7.29)

n − 1 (1.43 − 1)*103 = r0 −8.00 430 = −53.75 D = −8 (from III )

The power of the back surface (F2 ) =

(r0 has a –ve sign because it is measured from surface to centre of curvature which is now in the opposite direction to that in which the light is travelling). L ′1 = F2 − F ′ v = −53.75 D − −10.00 D = −43.75 D L2 = L ′1 (1 − tC n ∗ L ′1 ) = −43.75 (1 − 0.1 1480 ∗ −43.75) = −43.62 D (from XXII ) Light must emerge parallel from F1 therefore F1 = +43.62 D ra 0 =

(1.43 − 1) ∗10 430 = = 9.86 mm (from IV ) F1 43.62 3

[Although the present writer (RR) prefers to do the calculations step by step, an alternative approach combining Formulae IV and a modified version of XXIII is: ra 0 = 103 ∗ (1 − n) ∗ (1 ((1 − n) ∗103 r0 − F ′ v ) − tC n 103 ) − − − − XXXV (r0 and tC in mm)

Where Dd ØaJA = Desired Axial (Edge) drop at the Junction diameter t1ØaJA = Axial thickness at the proposed anterior Junction diameter before the junction radius is incorporated. t1ØaJ A = s ØaJ p + tC − s ØaJ ra 0 s ØaJp = Primary sag at the proposed anterior Junction diameter. s ØaJ ra0 = sag of ra0 at the proposed anterior Junction diameter. DtØaJA = Desired Axial thickness at the proposed anterior Junction diameter. tC = centre thickness. s ØaJ p = s Ø0 r0 + s ØaJ r1 − s Ø0 r1 = 0.933 + 0.975 − 0.829 = 1.080 mm (sags from XI) + tC = 0.100 mm Total sag at ØaJ = s T ØaJ – s ØaJra 0

= 1.180 mm = 0.870 mm (from XI )

Thus, t1ØaJA – DtØaJA

= 0.309 mm = 0.150 mm

Thus, D dØaJA

= 0.159 mm

which is the Desired Axial (Edge) drop at the Junction diameter. [For calculators:

157.e6

SECTION 3  •  Instrumentation and Lens Design

11

2

1

12 3

5

13

10

6

14

15 16

4 8

7

∅0 ∅a0 ∅aJ

1 2 3 4

s∅a0r s∅aJra1 s∅aJP+tJA sØaJA

5

a

6 7 8

BOZD FOZD FJD

9

s∅a0r1 = s∅0r1 s∅aJra1 ra0 ra1

9 10 11 12

ra2 Dd∅aJA tC s∅a0ra0

13 14 15 16

s∅aJaP s∅aJP sØaJra0 + tØaJA Dt∅aJA

Fig. 7.29  Negative lenticular lens with front junction radius (ra1). The diagram shows all the parameters needed for calculation of ra1 to give a desired axial junction thickness at the anterior junction diameter. (Axial junction thickness symbols are tØaJA or tJA and DtØaJA – the desired axial junction thickness. Note: DtØaJA , the desired axial junction thickness, equals the original axial junction thickness, t1ØaJA , less the desired axial drop, DdØaJA , as shown on the right-hand side of the diagram)

Dd ∅aJ A = r0 − (r02 − y02 ) + (r1 − (r12 − yaJ2 )) − (r1 − (r12 − y02 )) + tC − (ra 0 − (ra 02 − yaJ2 )) − Dt∅aJ A where y0 = back optic zone diameter/2. yaJ = anterior Junction diameter/2 Other symbols as above. DdØaJ A = 0.933 + 0.975 − 0.829 + 0.1 − 0.870 − 0.15 = 0.159 mm (sags from XII )] Now, to determine front junction radius, ral:

x = s ØaJra 0 − s Øa 0ra 0 + DdØaJ A = 0.870 − 0.741 + 0.159 = 0.289 mm (from XXIX X (D ) above) ra1 = ((( yaJ2 − ya 02 − x2 ) (2* x))2 + yaJ2 ) (from XXVIII ) Thus, ra1 = (((4.052 − 3.752 − 0.2892 ) (2*0.289))2 + 4.052 ) ra1 = 5.630 mm 5.630 mm is considered to be fairly steep, although acceptable, for ra1. If the calculated value for ra1 is less than 5.5 mm, ra1 should be recalculated, increasing the proposed ØaJ in 0.1 mm steps to obtain a more reasonable value.

7  •  Optics and Lens Design

Thus x = s∅Tr0 − s∅aJr0 − Dl EA (from XXIX )

As a general guide, there is little advantage in using a junction radius greater than 8.5 mm unless ØT (TD) is greater than 10.00 mm. Confirmation of the correct axial junction thickness required is as follows: tØaJA = Axial thickness at anterior Junction diameter. tØaJA = s ØaJp + tC − s ØaJa p = s TØaJ − s ØaJa p Where s ØaJap = primary sag of anterior surface at anterior s ØaJap

Junction diameter. = s Øa 0ra 0 + s ØaJra1 − s Ø0ra1 = 0.741 + 1.719 − 1.431 = 1.030 mm (sags from XI ; other symbols as above).

157.e7

= 1.600 − 1.101 − 0.111 x = 0.388 mm (from XXIX ) Thus xa 2 = x = 0.388 mm| and ra 2 = ((( y∅aJ 2 − y∅aJ 2 − x2 ) 2 ∗ x))2 + y∅T 2 ra 2 =

(((4.82 − 4.052 − 0.3882 )

(from XXVIII )

(2 ∗ 0.388)) + 4.8 ) ra 2 = 9.63 mm (from XXVIII ) 2

2

To confirm that this radius gives the required edge thickness may be shown as follows: Establish the primary sag of the anterior surface (sap) sap = s Øa 0ra 0 + s ØaJra1 − s Øa 0ra1 + s ØTra 2 − s ØaJ ra 2

Now, s ØaJp + tC = s TØaJ = 1.180 mm (from above) − s ØaJap = 1.030 mm (from above) = tØaJA = 0.150 mm

= 0.741 + 1.719 − 1.431 + 1.281 − 0.893 (sags from XI ) = 1.418 mm Then tEA = s p + tC − sap = 1.468 + 0.1 − 1.418 = 0.150 mm = tØaJ A

[For calculators: t∅aJ A = r0 − (r02 − y02 )

( ) − (r − (r − y ) ) + t  (r − (rr − y ) )     −  + (r − (r − y ) )     − (r − (r − y ) ) + r1 − (r12 − yaJ2 ) 2 1

1

0

a0

a0

2

2

C

a0

2

a1

2 a1

aJ

2

a1

2 a1

a0

2

(Symbols as above) tØaJA = 0.806 + 0.836 − 0.704 + 0.1 − (0.659 + 1.353 − 1.124) = 0.150 mm ] To create a parallel surfaced carrier, ra2 must provide the same sag, xa2, as provided by the back surface peripheral curves from ØaJ out to ØT (sag x). The value for the back surface sag from ØaJ to ØT, in this case = s Ø1r 1 − s ØaJr 1 + s ØTr2 − s Ø1r2 = 0.388 mm Alternatively it may be found by determining the edge lift between ØaJ and ØT, DlEA, and using this in Formula XXIX as follows: Dl EA = l EA from ∅aJ out to ∅T = Total l EA − l EA at ∅aJ Dl EA = 0.132{from p.e.4} − (s∅aJr0 − s∅aJ p ) Dl EA = 0.132 − ((8.0 − (8.02 − 4.052 )) − 1.078 {from above} Dl EA = 0.132 − (1.101 − 1.078) = 0.132 − 0.021 = 0.111 mm

TEARS LAYER THICKNESS The BOZD and BOZR of a lens may be chosen to give a required central clearance between the lens and cornea. This is known as the apical tears layer thickness or TLT. It is the difference in sag value between the primary sag of the lens at the contact diameter, usually the BOZD, and the sag of the cornea at the contact diameter. In contact lens designs, the BOZD values used are too large to assume that the corneal cross-section is an arc of a circle at this diameter, so that the corneal contour must either be measured (see Chapter 7) or assumed. More detail is given in Chapter 9. Formulae XI–XVII show how the sagitta of spherical (XI) and ellipsoidal (XII) surfaces are derived; how the apical radius of the cornea can be obtained for a specific keratometer reading and known asphericity (XIII and XV) taking into account the mire separation of the keratometer used (XIV). Formula XVI then makes use of this information to obtain TLT and Formula XVII shows how the BOZR of a lens may be changed to give a required alteration to TLT. Values for TLT range from 0.01 to 0.02 mm and are often specified in micrometres (0.001 mm).

AXIAL EDGE LIFT AND AXIAL EDGE CLEARANCE Another aspect of lens design is to create peripheral curves which will give a desired axial edge lift (see Chapter 9). If a specific axial edge clearance (AEC) is required, the peripheral corneal contour must be assumed or measured (see Chapters 6 and 9) as for TLT (above). Then a lens of suitable axial edge lift is designed to give the necessary clearance at the total diameter of the lens. Suitable use of sag formulae is required. Axial edge lift, lEA, is then calculated in the following manner.

157.e8

SECTION 3  •  Instrumentation and Lens Design

and

(a) x0

BOZD BPD1

søTr1– sø0r1

IEA2 søTr0

(b) x0

søTr0 søTr1– søTr0

x1

IEA

IEA1

TD

Having determined xN, BPR1 may be obtained from Formula XXVIII where yØG = 4.75 mm ( = TD 2) yØN−1 = 3.75 mm ( = BOZD 2) xN = 0.562 mm ( = s ØTr1 − s Ø0r1 )

ø0

søTr1

ø1 øT

IEA1 øT ø1 ø0

søTr2

sø0r1

sø1r2

= 0.562 mm = xN in Formula a XXIX

sø0r1

øT

(c)

s ØTr 1 = s ØTr0 − x0 – l EA, = 1.563 − 0.933 − 0.0675

sp

søTr1

sø1r1

Thus rN = r1 = 8.69 mm2, rounded to 8.70 mm (= BPR1)

BPR2 In the same way, a value for BPR2 may be calculated which will yield a value of 0.0675 mm for l EA2. From Fig. 7.30c, l EA2 = s ØTr 1 − s Ø1r 1 − x2 where s ØTr 1 is the sag of BPR1 at TD, s Ø1r1 is the sag of BPR1 at BPD1 and x2 is now s ØTr2 − s Ø1r2 (see Fig. 7.30c). s ØTr2 is the sag of BPR2 at TD, and s Ø1r2 is the sag of BPR2 at BPD1.

X2

/EA2 Fig. 7.30  (a) Axial edge lift lEA = lEA1 + lEA 2 = sØT r0 − sp ; (b) lEA1 = sØT r0 − ( x 0 + sØT r1 − sØ0r 1; (c) lEA2 = sØT r 1 − sØ1r1 − x 2 and x 2 = sØT r2 − sØ1r2. (Note: BOZD = Ø0, BPD1 = Ø1, TD = ØT)

Thus x2 = s ØTr 1 − s Ø1r 1 − l EA2 Again, using Formulae XI and XXIX for the determination of the various sag values, for this portion of the peripheral zone, xN in Formula XXIX is x2. Thus s ØTr 1 = sag of 8.70 mm ( BPR1 ) at 9.50 mm (TD)

From Fig. 7.30a it is seen that, for a tricurve lens, the axial edge lift l EA = s ØTr0 − s p where sp is the primary sag and equals x0 + x1 + x2 (see Fig. 7.26a). If the two peripheral curves are to contribute equally to the edge lift, then l EA1 must be equal to l EA2 where l EA1 + l EA2 = l EA . For example, a lens has a BOZR of 8.00 mm. The BOZD is 7.50 mm, BPD1 is 8.50 mm and TD is 9.50 mm. An axial edge lift of 0.135 mm is required, of which 0.0675 mm is to be contributed by BPR1, and 0.0675 mm by BPR2. The values for BPR1 and BPR2 to give the necessary edge lift are determined as follows.

BPR1 From Fig. 7.30b it can be seen that the lens is treated first as if it were a bicurve lens with BPR1 extending out to the TD. Then l EA1 = s ØTr0 − (x0 + s ØTr 1 − s Ø0r 1 ) From above, s ØTr0 = l EA1 + x0 + s ØTr 1 − s Ø0r 1 and so s ØTr 1 − s Ø0r 1 = s ØTr0 − x0 − l EA1 Then, using Formula XI for the determination of sag values, and Formula XXIX to determine the sag value, xN, of the peripheral curve BPR1 between BOZD and TD, which is sØTr1 – sØ0r1, gives: s ØTr0 = sag of 8.00 mm (BOZR) at 9.50 mm (TD) = 1.563 mm x0 = sag of 8.00 mm (BOZR) at 7.50 mm ( BOZD) = 0.933 mm

= 1.411 mm sØ1r1 = sag of 8.70 mm (BPR1 ) at 8.50 mm ( BPD1 ) = 1.109 mm l EA2 = 0.0675 mm Since x2 = s ØTr 1 − s Ø1r 1 − l EA2 then x2 = xN = 1.411 − 1.109 − 0.0675 = 0.235 mm Using Formula XXVIII again allows BPR2 to be obtained where yØG = 4.75 mm ( = TD 2) yN −1 = 4.25 mm ( = BPD1 2) xN = 0.235 mm (= x2 ) Thus rN = r2 = 10.59 mm, rounded to 10.60 mm ( = BPR2 ) Therefore the two peripheral curves necessary to give a total axial edge lift of 0.12 mm are (to the nearest 0.05 mm) 8.70 mm for BPR1 and 10.60 mm for BPR2. The detailed calculations and the effect on axial edge lift of rounding the values for the radii to the nearest 0.01 mm (or indeed any value, such as 0.05 mm) are dealt with in the Formulae section. Formula XXX shows how the desired axial edge lift to be achieved from any one peripheral curve may be determined to arrive at a well-balanced peripheral zone. The total axial edge lift required must be known as well as the TD of the lens, and the diameter (Øc) at which the lens contacts the cornea – usually Øc = BOZD = Ø0. 2

See earlier footnote

7  •  Optics and Lens Design

Formula XXXI is a sag difference formula to give axial edge lift when peripheral radii have been given ‘rounded’ values. This enables the practitioner to decide if the error in axial edge lift introduced by ‘rounding’ is acceptable.

AXIAL EDGE LIFT CALCULATION OF EXISTING LENSES Formula XXXII may be used for this. It is another sag difference formula to allow calculation of the axial (edge) lift value of each peripheral curve at its own outer diameter, rather than at the edge of the lens. The totals of these axial lift values are equal to the total axial edge lift of the lens. Any lens of known back surface specification may thus have its axial edge lift determined by this method. Alternatively, the spreadsheets developed by Rabbetts (1993) may be used, either to calculate the edge lift of existing lenses or to calculate the required peripheral curves, given an edge lift or the tear layer thickness and edge clearance. Details of fitting sets with constant axial edge lift are given in Chapter 9.

Modification of Peripheral Radii to Alter Edge Lift Formula XXXII can also be used with XXVIIIA and XXIXA (adaptations of Formulae XXVIII and XXIX) to determine new peripheral radii in order to alter the axial edge lift of an existing lens. For those practitioners who modify their own lenses, this can be very useful in deciding on the curvature of the polishing tools to use. It is also helpful if a new lens is to be ordered to give increased or reduced edge clearance.

REVERSE GEOMETRY LENSES – LENSES FOR ORTHOKERATOLOGY This type of lens, in which the first peripheral curve is steeper than the BOZR and hence is frequently termed the reverse curve, may be used to try to provide a better fit than a conventional rigid lens on corneas that have undergone refractive surgery, or for use in orthokeratology (see Chapter 19). While the post-refractive surgery patient will have to be fitted by trial and error, even though based on corneal topography (video-keratoscope) data, in orthokeratology the back surface design may be calculated theoretically. The back surface profile will be similar to that of the front surface shown in Fig. 7.29, though the calculations will be similar to those for calculating axial edge lift and clearance. Example.  A cornea of apical radius r0 = 8.00 mm and p-value = 0.7 is to be fitted with a tetracurve orthokeratology lens to correct myopia of –3.00D. A central TLT of 0.005 mm and axial edge clearance of 0.05 mm is required. From Formula III, the corneal power assuming a notional corneal index of 1.336 is +42.00D. Allowing for a Jessen factor of –0.75D, the new corneal power required is +42.00 – 3.00 – 0.75 = +38.25D. From Formula IV, the new corneal radius required is 8.7843 mm, so the value 8.78 is chosen to be the BOZR. A BOZD of 6.0 mm, BPD1 of 7.5 mm, BPD2 of 9.5 mm and TD of 10.5 mm are to be used. Then the sag, x0, of the BOZR over the BOZD, using Formula XI is 0.52815 mm,

157.e9

while Formula XII gives the sag of the unaltered cornea, c0, over the same diameter as 0.57707 mm. The axial corneal clearance at the edge of the optic zone is given by the equation: –x0 + TLT + c0 = –0.52815 + 0.005 + 0.57707 = 0.05392 mm. The sag, c1, of the cornea over 7.5 mm, the BPD1, is given, also by Formula XII, as 0.91558 mm. Hence the increase in sag, c1 – c0, from the edge of the optic zone to the edge of the reverse curve is 0.33851 mm. The reverse curve is then designed to join the edge of the BOZR to the aligning zone of the third curve, so over the band width from 6.0 to 7.5 mm, the sag of the reverse curve, x1, is 0.05392 + 0.33851 = 0.39243 mm. Formula XXVII may now be used to calculate the radius, r1, of the reverse curve. Using yØG = 3.75, yØN−1 = 3.00 and xN = 0.39243 mm, we obtain 7.29 mm. So that the second peripheral curve aligns the cornea, its radius, r2, is chosen to give the same sag difference between 7.5 and 9.5 as that of the cornea. Again using Formula XII, the corneal sag at 9.5 mm is 1.50990 mm, so the sag difference is 1.50990 – 0.91558 = 0.59432 mm. Formula XXVII now gives the aligning second peripheral curve radius as 8.34 mm. The third peripheral curve radius, r3, is similarly obtained except that the sag difference is reduced by the required edge clearance. Hence the corneal sag at the TD of 10.50 mm is 1.87675 mm, giving a corneal sag difference of 0.36686 mm between that at 9.50 and 10.50 mm, which, in turn, gives the required sag difference for r3 between these two diameters of 0.36686 – 0.05 = 0.31386 mm. Again using Formula XXVII, r3 = 9.35 mm. The final back surface specification is therefore: C4 8.78:6.00/7.29:7.50/8.34:9.50/9.35:10.50.

DRAWING LENSES TO SCALE Another way to design lenses is to draw them to scale at ×40 full size as recommended by Mackie (1973), who described the method in detail. Graph paper, 56 × 38 cm, a drawing board and beam compass (preferably 50 cm long) are essential, as well as a contact lens slide rule or tables, or an electronic calculator. Some manufacturers have employed this method but computer programs and the formulae now make such drawing unnecessary. Fig. 7.31 shows how computer graphics may be used to draw lenses to scale, with the scale being magnified vertically to show the thickness and clearances more obviously, with extra magnification of the central tears layer.

Optical Formulae for Contact Lens Work With Examples to Illustrate Their Use It is hoped that the formulae incorporated here will help to simplify and to speed up optical calculations that occur in the contact lens field. The principles involved are well covered elsewhere (Bennett 1985, Douthwaite (2006), Jalie (2016)), and are therefore not dealt with in detail here.

157.e10

SECTION 3  •  Instrumentation and Lens Design

søTrok, tC, tJA, tEA and CEA (mm (TLT*10)

TLT*10=0.11

1.6 SøTr0k=1.642 1.4

2 tC=0.3026 1.8

ra0 =7.12 r0=7.98

TLT*10=0.11

BVP =+8.00 D

rK=8.0

SøTr0k, tC, tJA, tEA and CEA (mm (TLT*10)

2 tC=0.3026 1.8

n=1.47

r0 K =7.97

1.2 p=0.8

1

tJA =0.15 øa0 = 7.8

0.8

ra1=10.00

Bearing point

0.6 0.4 Parallel carrier

0.2 0

tEA=0.15

0

1 Cornea;

cEA=0.15

2 3 Distance from centre of lens (mm) Back surface ;

TLT;

4

5

Front surface

1.6 SøTr0k=1.642

ra0=7.12 r0=7.98

BVP =+8.00 D

rK=8.0

1.4

n=1.47

r0K=7.97

1.2 p=0.8

1

tJA =0.15 øa0=7.8

0.8

ra1=11.97 0.6

Bearing zone

0.4

0

tEA=0.24

Negative carrier

0.2 0

1

cEA=0.15

2 3 Distance from centre of lens (mm)

Cornea;

Back surface ;

TLT;

4

5

Front surface

Fig. 7.31  (a) Cross-section of half a C3 spherical lenticular lens on a cornea having a spherical keratometer radius of 8.00 mm and a p-value of 0.8. The lens back surface is of orthodox design giving 0.011 mm apical clearance (TLT) and 0.15 mm axial edge clearance (CEA). The lens contacts the cornea at the transition between r0 and r1. BVP = +8.00 D and the lens is depicted in lenticular form with Øa0 = Ø0 + 0.3 mm and the carrier is parallel surfaced, i.e. tEA = tJA. The horizontal axis shows distance from the lens centre and the vertical axis shows sags and thickness values: tC, TLT (×10), tJA, tEA, CEA and sØT r0 K . Values for ra0 of 7.12 mm, r0 of 7.98 mm and rK of 8.00 mm (r0K = 7.97 mm) are shown. (b) Here r1 and r2 have been changed so that the surface of r1 is parallel to the cornea, creating a bearing zone, and r2 is increased to give the same CEA (0.15 mm). The lens has a negatively surfaced carrier, tEA >tJA. Primary sag (sp) and all other parameters are the same as in (a). sp = 1.5012 mm for both lenses. In the ‘y’ scale of the graphs (0–2.0 mm) the TLT (0.011 mm) would be barely thicker than a line so the central tears layer only is shown expanded by a factor of 10 for clarity, and does not therefore show the true relationship between the back optic surface and the cornea. All other parameters and measurements are to scale, including CEA

All formulae are presented both in orthodox notation and using the arithmetic hierarchy recognized by most scientific calculators and computer spreadsheets. The use of calculators and computers enables results to be presented to much greater precision than that to which manufacturing equipment can be set or lenses manufactured. In general, precision should be maintained throughout a calculation with only the final result being rounded. Appendix A has been retained at the end of the book and is a table showing the effective power of spectacle lenses at various distances from the back surface of the spectacle lens. Thus, it can be seen from the table that a lens of back vertex power +7.00D has an effective power of +7.76D, in a plane 13 mm from the back surface of the lens, while the effective power of a –7.00D lens at the same vertex distance is –6.41D. The effective power of a trial spectacle lens at the eye can be obtained very quickly from this table. A comprehensive table of this type can be useful in other ways, such as comparing spectacle and ocular refraction, in assessing the correction of astigmatism in near vision and in the determination of spectacle and ocular accommodation. A system for ‘constructing’ the symbols used to indicate, define or describe values relating to a contact lens and/or the cornea is employed throughout this section. The system is based on (and is an extension of) that used in Table 1 of the British and International Standard BS EN ISO 18369-1: (2017). A symbol normally has three parts which represent, in sequence, the answers to the questions: What is it?, Where is it?, and Which is it? For example, ra0 is the symbol used to indicate the radius of the anterior optic surface. As with all good rules there is an exception, which is also taken from the ‘Standard’ system. The most frequently used contact lens specifications are those of the back surface so the ‘middle’ part of the symbol, Where is it?, is omitted

when referring to a back surface zone. r0 and r2 for example represent respectively the back central optic radius (or apical radius for an aspheric surface) and the second back peripheral radius. Back surface data other than zones require a full symbol. For example, sØ0r1 indicates the sagitta at the back central optic diameter (Ø0) of the first back peripheral radius (r1). Other examples are s ØT ra1 which is the sagitta at Total diameter (ØT) of the first anterior peripheral radius (ra1) (i.e. the anterior radius of the carrier portion of a lenticular lens); lEA is the lift at the Edge of a lens and it is Axial (as opposed to Radial – lER). Note: upper case subscript A = axial; lower case subscript a = anterior. Some of the symbols appear strange at first but become familiar quite quickly with use and the system has the advantage that if the significance of a symbol is forgotten a knowledge of the rules enables the meaning to be construed or a new symbol to be constructed. All the formulae used for spherical lenses can be used for toric lenses if each principal meridian is treated separately as though it were spherical. In all cases powers (F) are in dioptres (D). All distances are in metres (m), millimetres (mm = m/103), or micrometres (µm = m/106) as indicated.

SPECTACLE AND OCULAR REFRACTION Formulae I and II Ocular refraction = K o = Spectacle Rx = Fsp =

Fsp − − − −I 1 − dFsp Ko − − − − II 1 + dK o

7  •  Optics and Lens Design

where Fsp = BVP of spectacle lens (D), Ko = Ocular Rx(D), d = BVD (m). Formula I is the basic form of the ‘Effectivity’ equation, see Formula XXII. For calculations: K o = Fsp (1 − d ∗ Fsp ) − − − − I Fsp = K o (1 + d ∗ K o ) − − − − II Example 1.  What is the ocular refraction corresponding to a spectacle refraction of +8.00 DS at a vertex distance of 12 mm? To preserve scale when using Formula I (or II), the 12 mm BVD must be divided by 1000. Thus 12 mm/103 = 0.012 m. K o = 8.00 (1 − 12 103 ∗ 8.00) = +8.85 D (from I ) Other examples (Examples 2 and 3) are: For Fsp of − 12.50 D at 9 mm BVD : K o = −12.5 (1 − 0.009 ∗ −12.5) = −11.24 D (from I) For K o of + 18.00 D : Fsp at 10 mm BVD = 18 (1 + 0.01 ∗18) = +15.25 D (from II)

RADIUS AND SURFACE POWER

Lens − liquid interface power = (1.336 − 1.45) ∗103 7.90 = −114 7.90 = –14.436 D (from III ) Alternatively, if we had been given the front surface power: Front surface radius = (1.45 − 1.00) ∗103 +56.25 = 450 +56.25 = +8.00 mm (from IV )

CHANGE IN SURFACE POWER (δF) WHEN RADIUS (r1) IS CHANGED TO NEW RADIUS (r2) FOR GIVEN REFRACTIVE INDEX DIFFERENCES Formula V This calculation is based on Formula III (used twice) Change in surface power = δ F n −n n −n = 2 1 − 2 1 − − − −V r2 r1 (radii in metres) For calculations:

δ F = (n2 − n1 ) (r2 (mm ) 103 )

Formulae III and IV

n2 − n1 − − − − III r n −n Radius = r = 2 1 − − − − IV F

Surface power = F =

Where r = surface radius in metres; n1 = refractive index of first medium; n2 = refractive index of second medium. Because radius is normally in mm, r can either be divided by 103 to convert it to metres or the numerator multiplied by 103 so that r can remain in mm. For calculations: F = (n2 − n1 ) (r(mm ) 103 ) OR, since it is easier to multiply the refractive index by a thousand than divide the radius – indeed, for a surface in air, the refractive effect is obtained by dropping the unit before the decimal point, and writing three figures for the part after the decimal point, e.g. (1.487 – 1.000) becomes simply 487. F = 103 ∗ (n2 − n1 ) r(mm ) − − − − III r(mm ) = (n2 − n1 ) ∗103 F − − − − IV Examples 4, 5, 6 and 7.  A lens has FOZR = ra0 = 8.00 mm, BOZR = r0 = 7.90 mm, refractive index = n = 1.45 Front surface power in air = (1.45 − 1.00) (8.00 103 ) = 0.45 0.008 = +56.25 D (from III ) Back surface power in air = (1.00 − 1.45) (7.90 103 ) = 0.45 0.0079 = 56.92 D (from III )

157.e11

(n2 − n1 ) (r1 (mm ) 103 ) − − − −V Examples 8, 9 and 10.  A lens has FOZR = ra0 = 8.20 mm, BOZR = r0 = 7.95 mm, refractive index = 1.442 If ra0 is changed from 8.20 (r1) to 7.70 (r2):

δ F = (1.442 − 1.00) (7.70 103 ) (1.442 − 1.00) ∗103 8.20 = +3.50 D (from V ) If r0 is changed from 7.95 (r1) to 7.80 (r2):

δ F in air = (1.00 − 1.442) (7.80 103 ) − (1.00 − 1.442) (7.95 103 ) = 1.07 D (from V )

δ F in tears = (1.336 − 1.442) ∗103 7.80 − (1.336 − 1.442) ∗ 103 7.95 = −0.26 D (from V )

CHANGE IN REDUCED VERGENCE (δRV) DUE TO THICKNESS FOR A GIVEN INITIAL VERGENCE AND REFRACTIVE INDEX Formula VI δ RV = L2 − L1′ L1′ = − L1′ − − − −VI t 103 1− C L1′ n Where L′1 = initial vergence of light (after refraction at the first surface); L2 = the changed vergence of light after

157.e12

SECTION 3  •  Instrumentation and Lens Design

travelling through a given thickness; tC = centre thickness (in mm); n = refractive index. This type of calculation is based on the concept of reduced thickness and equivalent air distances.

r1

r2

n1 = 1.490

Example 11.  L1′ = 62.00D, tC = 0.4 mm, refractive index = 1.45

δ RV = 62.00 ∗{(1 − 0.4 ∗ 62.00 1.45 103 ) 1 − 1} = +1.08 D (from VI ) The change in vergence due to thickness, or an increase in thickness, is always +ve (or less –ve) and a decrease in thickness will always produce a –ve (or less +ve) change in vergence except when L′1 is plano, in which case there is no change in vergence due to thickness. Care must be taken when light is traced ‘backwards’ through a lens (i.e. if the lens is reversed), as the radius of the back (concave) surface is treated as a –ve value. Rather than use formulae, it may be simpler to use the “step-along” method introduced by Bennett, converting image vergence L1’ to either image distance or reduced image distance, subtracting the reduced thickness or actual thickness respectively, and converting back to the new object vergence L2. Using reduced distances and working in mm, this example gives: l1 ′ = 1000 62.00 = 16.129 (mm ), subtract tc n gives 15.853, hence L2 = 63.08 D.

DIFFERENCE IN SURFACE POWER (dF) OF A FUSED BIFOCAL LENS WHEN, FOR A GIVEN RADIUS THE REFRACTIVE INDEX CHANGES FROM THE MAIN LENS TO THE NEAR SEGMENT This type of lens no longer appears to be made, but the mathematical treatment is retained here for reference. These are very similar to fused bifocal spectacle lenses except that most corneal lenses have the segment on the back surface, and the RI of the fused segment when the main lens was made from PMMA was usually 1.56. The optical theory is easily understood if reference is made to Fig. 7.32, and more detail is given at the end of this appendix. Most fused bifocal corneal lenses have the segment on the back surface. As r0 is a concave surface common to both distance and near portions and the near segment has a higher refractive index than the distance portion, the power of the back surface of the near segment is more –ve than that of the distance portion. The change in surface power is proportional to the difference in refractive indices and dependent on r0. Note: For equation VII only, the sign of the radius (denominator) departs from the normal sign convention. If the bifocal surface is concave, the radius is entered as a negative value and for a front surface bifocal the radius is regarded as a positive value. Alternatively, the sign of the result for dF can be obtained by inspection. The use of this device enables equation VII to be used in the calculation of surface power differences for both back and front surface fused bifocals.

n2 = 1.560 r3 Contact surface

r2

Fig. 7.32  Fused bifocal corneal lens: r1, r2 and r3 are the radii of the front, back and contact surfaces, respectively; n1 and n2 are the refractive indices of the main lens and the near segment.

Formula VII Surface power of segment differs from that of the main lens by dF =

nN − nD − − − −VII r

Where nN = refractive index of near segment, nD = index of distance portion, r = surface radius in metres. Hence the external surface power of the segment differs (dF) from that of the main lens (dBF back surface seg, dFF front surface seg) by: dF = (nN − n D ) ∗103 ±r0 (mm ) OR dF = (nN − n D ) ( ±r0 (mm ) 103 ) − − − −VII Example 12.  A bifocal corneal lens has a segment fused into the back surface, the refractive index of the main lens = 1.49, the index of the segment = 1.56 and r0 = 8.00 mm. dB F = (1.56 − 1.49) ∗103 −8.00 = 0.07 ∗103 −8.00 = 70 −8.00 = −8.75 D (from VII) Although this has been worked out directly from the difference in refractive indices of the main and segment materials, the same result occurs if the main and segment surface powers are calculated in air or in tears, and the difference in powers calculated. Example 13.  For a front surface segment using the same refractive indices and FOZR = ra0 = 8.20 mm. dF F = (1.56 − 1.49) (8.2 103 ) = 0.07 0.0082 = +8.54 D (from VII )

7  •  Optics and Lens Design

SAGITTA CALCULATIONS

CONTACT SURFACE POWERS AND RADII FOR FUSED BIFOCAL LENSES OF VARIOUS OPTIC ZONE RADII AND NEAR ADDITIONS

Formulae XI and XII For a spherical surface or a principal meridian of a toroidal surface

Formulae VIII, IX and X

s = r − r2 − y2 − − − − XI

FCS = Add − (d BF or d FF ) − − − −VIII (dF is obtained from VII ) To differentiate between back and front surface segments the contact surface is denoted BCS and FCS appropriately. rBCS =

(n2 − n1 ) (n − n ) ∗103 − − − − IX rBCS (in mm) = N D FBCS FBCS

rFCS =

(n2 − n1 ) − − − −X FFCS

rFCS (in mm) =

157.e13

(n D − nN ) ∗103 FFCS

where CS = contact surface, F = power, dF = difference in power, r = radius, Add = near addition, n1 and n2 are the refractive indices of the 1st and 2nd media. Thus for a back surface segment, n1 = nD and n2 = nN, and vice versa for a front surface segment (see VII). Hence: FCS = Add – (d BF or d FF) − − − −VIII (dF is obtained from VII ) rBCS = (n N – n D ) ∗103 FCS − − − − IX (rBCS in mm) rFCS = (n D – n N ) ∗103 FCS − − − − X (rFCS in mm) If, in Example 12 (a back surface fused bifocal), the required near addition (Add) is +3.00D, the contact surface between the lens and the segment must have enough +ve power to provide the required Add AND to neutralize the unwanted –ve power dBF at the back surface of the seg (= –8.75D, from VII). To achieve this the contact surface (CS) must be convex for the higher index and its power, FBCS, must be +3.00D– –8.75D = +11.75D (from VIII). (1.56 − 1.49) ∗103 70 = +11.75 +77.75 = 5.96 mm (from IX )

rBCS =

where rBCS = the contact surface radius (back seg) and FBCS = its power. In Example 13 (a front surface fused bifocal), dFF = +8.54D (from VII). If the required near Add is +2.00D, the contact surface must be concave for the higher index and its –ve power must neutralize the unwanted part of the +ve power on the front surface of the seg. FFCS = +2.00 − 8.54 = −6.54 (from VII ) −70 (1.49 − 1.56) ∗103 rFCS = = −6.54 −6.54 = 10.70 mm (from X) Where rFCS = the contact surface radius (front seg) and FFCS = its power.

= r − (r2 − (Ø 2)2 ) − − − − XI Where s = sagitta, r = radius, Ø = diameter, y = Ø/2 Primary sag, s p = s Ø0r0 + s Ø0r 1 + ……s ØN rN − s ØN−1rN ; which involves XI used repeatedly (also see Figs 7.25, 7.26). For the prolate (flattening) curve of an ellipse: sK =

r0 K − r0 K 2 − py2 − − − − − XII p

(

= r0 K − r0 K 2 − p ∗ y2

)

p − − − − − XII

where p = 1 – e2 (e = eccentricity); sK and r0K are used respectively to differentiate between the sagitta and apical radius of an ‘ellipsoidal’ cornea (or lens) and those of a spherical lens or cornea when s and r0 and r0Ks are used; other symbols as above.

CALCULATION OF APICAL RADIUS OF CORNEA (r0K) FROM ‘K’ READING (rK) IN MM (FOR THE PROLATE (FLATTENING) CURVE OF AN ELLIPSE); DETERMINATION OF KERATOMETER MIRE IMAGE DIAMETER (ØMI) AND REFLECTING POWER (FKR) OF THE CORNEA Formulae XIII, XIV and XV r0K = rK 2 − (1 − p)yMI 2 − − − − XIII or r0K = rK 2 − (1 − p) ∗ yMI 2 − − − − XIII (r0 K, rK and yMI in mm) where rK = keratometer radius; yMI = Ø/2 of mire image; other symbols as before. The diameter of the mire image (ØMI) is taken to be the Ø of the annulus around the corneal apex (or more correctly around the visual axis) which is measured by the keratometer. In keratometers with fixed mires and hence variable doubling, ØMI increases in proportion to corneal radius, but with moveable mire keratometers, there is a slight decrease (Lehman 1967). There is a significant difference in ØMI between different instruments, varying from 2.0 to 3.5 mm for corneal radii of 7.00 mm, and 2.6 to 3.8 mm for radii of 9.25 mm. ØMI created by any keratometer can be calculated from rK 2 ∗ Ø M − − − − XIV d ≈ (rK 2 ∗ Ø M ) d − − − − XIV

ØMI ≈

where ØM = Ø of a fixed mire, or the separation between movable mires: d = distance between the mire plane and the mire image (ØM and d in mm).

157.e14

SECTION 3  •  Instrumentation and Lens Design

The mire image, formed at distance d from the mire plane, is slightly in front of the focal point of the cornea, but negligible error is introduced by accepting these two distances as being the same taking d ≈ mire plane to corneal apex + rK/2. The reflecting power of the cornea = FKR =

2000 − − − − XV (−rK in mm) (−rK )

or FKR = 2000/(−rK) - - - - XV Average FKR = 2000/(−7.9) ≈ −253D (from XV) The mire image is minified by a factor of approximately 22 to 1; therefore, in the absence of any data concerning the mire size or distance, the use of equation XIII with an assumed average ØMI of 3.00 mm will allow a more precise calculation of TLT than is obtained by using rK as a basis for determining corneal sag: 3.10 mm is said to be the ØMI created in an average cornea by a Bausch & Lomb keratometer.

CALCULATION OF TEAR LAYER THICKNESS (TLT) Formula XVI Example 14.  A spherical corneal lens of r0 = 7.80 mm and Ø0 = 8.00 mm is placed on a cornea having a ‘p’ value of 0.80 for which an unknown (but impeccable) keratometer has indicated a radius of 7.85 mm and no astigmatism. What will be the TLT? The mire image diameter, ØMI, is assumed to be 3.00 mm. (1)  Calculate sagitta of lens at Ø0;

Where: s ØCp = primary sag of lens at the Ø at which the back surface of the lens contacts the cornea (ØC). (This is usually Ø0 if there is apical clearance, but see Example 23.) and: s ØCr 0 K = sag of cornea at the contact diameter (Ø0 in the present case). TLT = 1.104 mm – 1.083 mm = 0.021mm (from XVI ) (0.021 mm = 21 micrometres (µm). If the above calculation gives a –ve value for TLT it indicates that there is central touch and the TLT represents Axial ‘Edge’ Clearance (at Ø0 in this case). Blending the junction between r0 and r1 spreads the contact between lens and cornea slightly, and although the effective contact diameter Ø0 is probably unaltered, it will reduce the TLT by a tiny amount. To calculate the TLT created by a lens when the junction between r0 and r1 is not the bearing surface, or for an aspheric lens, see Example 23. The sag of the cornea at Ø0 using rK (7.85) instead of r0K (7.82) in Example 14 is 1.078 mm, which gives the TLT as 26 µm, an error of ≈ 24%. A central TLT of 21 µm is considered excessive today, about half that value (when calculated using r0K) being preferred for rigid gas permeable lenses.

CALCULATION OF RADIUS (r) FROM SAGITTA (s) AND SEMI-DIAMETER (y), AND NEW BOZR (r0N) REQUIRED TO CHANGE TLT BY A SPECIFIED AMOUNT (δ) Formula XVII y2 + s 2 − − − − XVII 2s = ( y2 + s 2 ) (2 ∗ s) − − − − XVII

s Ø0r0 = 7.80 − (7.80)2 − (8 2)

2

r=

= 7.80 − (60.84 − 16) = 7.80 − (44.84) = 7.80 − 6.696 = 1.104 mm (from XI ) (2)  Calculate apical radius of cornea; r0 K = (7.852 − (1 − 0.8) ∗1.52 = (61.6225 − 0.2 ∗ 2.25)

Formula XVII, which is a rearrangement of the ‘sag’ Formula XI, will yield the radius of any spherical surface, given the sagitta (s) and the semi-diameter (y). Example 15.  The BOZD, Ø0, of a contact lens is 7.50 mm (y = 3.75 mm) and its sagitta (s) is 0.94 mm. What is its radius? r0 = (3.752 + 0.942 ) (2 ∗ 0.94) = (14.0625 + 0.8836) 1.88 = 7.95 mm (from XV VII)

= (61.6225 − 0.45) = (61.1725) = 7.82 mm (from XIII ) (3)  Calculate sagitta of cornea at Ø0; s Ø0r0 K = {7.82 − 7.822 − 0.8 * 42 } 0.8 = {7.82 − 61.1524 − 12.8} 0.8 = {7.82 − 48.352} 0.8 = {7.82 − 6.954} 0.8 = 1.083 mm (from XII ) TLT = s ØCp − s ØCr 0 K − − − − XVI

The value of s is modified in XVII below to give the radius required to change the TLT. r0N = ((Ø0 2)2 + (s ± δ)2 ) (2 ∗ (s ± δ)) − − − − XVII where r0N = New BOZR (r0) required to alter TLT by a specified amount; Ø0 = diameter of back optic zone (BOZ); y = Ø0/2; s = sag of BOZ at the present radius; δ = the desired change in TLT.

7  •  Optics and Lens Design

Example 16.  We require the TLT of 0.021 mm in Example 14 to be reduced by 10 µm, thus δ = –0.01 mm, and the desired TLT is: 0.021 mm − 0.01 mm = 0.011 mm = 11 µm s Ø0 = 1.104 mm (from XI) – from Example 14(1) r0N = ((8.00 2)2 + (1.104 – 0.01)2 ) (2 ∗ (1.104 – 0.01)) = (42 + 1.0942 ) (2 ∗1.094) = (16 + 1.1968) 2.188 = 17.1968 2.188 = 7.86 mm (from XVII) Checking, the sag of 7.86 mm radius at Ø8.00 mm = 1.0939 (from XI) less the corneal sag: −s Ø0r0 K = 1.083 (from XII ) (see Example 14(3)) TLT = 0.0109 mm = 10.9 µm ≈ 11 µm as required.

BACK AND FRONT VERTEX POWERS FOR VARIOUS CENTRE THICKNESSES AND BOZ RADII Formulae XVIII, XIX, XX, XXI, XXII, XXIII, XXIV and XXV The BOZR (r0), refractive index (n) and centre thickness (tC) must be known in order to find the back vertex power (F′V) given the front vertex power (FV) or to find FV given F′V. The relationship between FV and F′V can be seen from the formulae: 1 − (tC 1 − (tC 1 − (tC Fv = Fv′ 1 − (tC

FV′ = Fv

n)F2 − − − − XVIII n)F1 n)F1 − − − − XIX n)F2

where tC = centre thickness in metres, n = refractive index, F1 = the power of the convex surface, F2 = the power of the concave surface. In either case it is necessary to find the powers of both surfaces. For the power of F1 see Formula XXIII and Step 1 below. The power of F2 is found from Formula III. L1¢ = L1 + F1 when determining BVP (F ′ v ) L1¢ = L1 + F2 when determining FVP (Fv ) where L1 = vergence of incident light = zero when calculating vertex powers. Thus, for determining BVP, L1′ = F1 and for determining FVP, L1′ = F2. L2 = the vergence of light incident on the second surface (calculated from XXIV). Once the power of the convex surface (F1) is known (see XXIII below): Fv′ = L2 + F2 =

L1′ + F2 − − − − XX (Also see XXVII ) 1 − (tC n)L1′

or: F ¢ v = L1′ (1 − tC n ∗ L1′) + F2 − − − − XX

157.e15

(Note: F1 may be substituted for L1′ in Formula XX, see above) Fv = L2 + F1 =

L1′ + F1 − − − − XXI (Also see XXVI ) 1 − (tC n)L1′

or: Fv = L1′ (1 − tC n ∗ L1′) + F1 − − − − XXI (Note: F2 may be substituted for L1′ in Formula XXI, see above.) (tC in metres for XX and XXI.) Formula XX will yield F′v if calculated on the basis that parallel light is incident on the convex (first) surface F1. (F2 is the concave (second) surface.) Formula XXI yields Fv if parallel light is incident on the concave surface. The calculation is simplified if reduced thickness is used, allowing the use of reciprocals and, especially if using a calculator, a further simplification is gained by expressing r0 and tC in metres (mm/1000). Example 17: to Find Front Vertex Power (Fv).  A lens has r0 = 7.90 mm (0.0079 m), n = 1.45, tC = 0.4 mm (0.0004 m), F′v = +6.00D. What is the front vertex power F V? Step 1 is to find the power of the convex surface. The ‘step-along’ method can be employed, but using reduced thickness (tC/n) and effectivity (see XXII below). L1 = Vergence of incident light (the lens is reversed, so L1 = –6.00D and the first surface (concave) is F2; the second surface (convex) is F1). F2 = power of first surface = (1.45 − 1.00) (−7.90 103 ) = −56.96 D (from III) (When the lens is reversed, r0 is given a –ve sign because it is measured from the surface to the centre of curvature, i.e. in the opposite direction to which the light is now travelling.) Lens is reversed so L1 = FV′; L′1 = F2 − FV′ = −56.96 − 6.00 = -62.96 D L2 =

L′1

− − − − XXII 1 − (tC n)L′1 −62.96 L2 = 1 − (0.0004 1.45) ∗ −62.96 = -61.89 D (from XXII ) In XXII, Formula I has been modified to use tC/n (in metres) instead of d, but it is essentially the same ‘effectivity’ equation. Hence: L2 = L1′ (1 − tC n ∗ L1′) − − − − XXII L2 = −62.96 (1 − 0.0004 1.45 ∗ −62.96) = −62.96 1.0174 = −61.89 D (from XXII )

157.e16

SECTION 3  •  Instrumentation and Lens Design

L2 = 1 (((1.45 − 1.00) −0.0079) − 0.0004 1.45) −1

Light (L2′) emerging from the second surface is parallel so the vergence of L2′ = 0D.

= −56.08 D (from XXIV ) Fv = L2′ = L2 + F1 1 + F1 − − − − XXV Fv = 1 tC − F2 n = 1 (((n2 − n1 ) −r0 )−1 − tC n) + F1 − − − − XXV

L2′ = L2 + F1, therefore: F1 = L2′ − L2 = 0 − L2 = 0 − −61.89 = +61.89 D The individual steps above can be combined into one formula, XXIII, which is a mathematical expression of the ‘step-along’ method:

= 1 (−r0 (n2 − n1 ) − tC n) + F1 − − − − XXV (r0 and tC in metres)

L1¢ = F2 – F ′ v Power, F1 = 0 − L2 = 0 −

1 − − − − XXIII 1 tC − L1′ n

= 0 − 1 ((n2 − n1 ) −r0 − F ′ v )−1 − tC n) − − − − XXIII (r0 and tC in metres) F1 = 0 − 1 ((1.45 − 1.00) −0.0079 − 6..00)−1 − 0.0004 1.45) = +61.89 D (from XXIII ) Step 2. Now that F1 is known (from XXIII), if parallel light is incident on the back surface, L1 = 0D L1′ = F2 = –56.96 D (from III )

Example 17 Fv = 1 (−0.0079 (1.45 − 1.00) − 0.0004 1.45) + 61.89 = +5.81 D (from XXV ) Example 18.  To find the back vertex power (F′v) given r0 = 8.00 mm (0.008 m), tC = 0.20 mm (0.0002 m), n = 1.448 and FV = –9.86D. Step 1. To find the power of the front surface: parallel light is incident on the ‘reversed’ lens as in Step 2, Example 17, above. F2 is the power of the back surface and r0 is –ve. F1 is the power of the front surface. L1′ = L1 + F2, L1 = 0 therefore L1′ = F2 0.448 = −56.00 D (fro om III ) L1′ = F2 = −0.008 −56.00 L2 = 1 − (0.0002 1.448) ∗ −56.00 = -55.57 D (from XXII )

L2 is calculated from XXII and Fv = L2′ = L2 + F1 −56.96 = -56.08 D (from XXII ) 1 − (0.0004 1.45) ∗ −56.96 FV = L2′ = L2 + F1 = −56.08 + 61.09 (from XXII and XXIII ) = +5.81 D (using floating point calculations)

L2 =

Alternatively, using equation XXI: Fv = F2 (1 – tC n ∗ F2 ) + F1 − − − − XXI (Where L1 = 0 then L1′ = F2 ) Fv = –56.96 (1 – 0.0004 1.45 ∗ –56.96) + 61.89 = –56.08 + 61.89 = +5.81 D (from XXI ) Discrepancies may occur between calculations where the many decimal places are kept during the calculation, or where values are re-entered to only two decimal places. L2 may also be found by the ‘step-along’ method. If parallel light is incident on the concave surface as in Step 2 above, then L1′ = F2 = –59.49D (from III).

L2 should emerge (L2′) after refraction at F1 with vergence = Fv = –9.86D. L2¢ = Fv = L2 + F1, therefore F1 = Fv − L2 F1 = −9.86 − (−55.57) = +45.71 D Step 2. Now that the power of the front surface (F1) is known, F′v can be found if a pencil of parallel incident rays is traced ‘forwards’ through the lens. L1¢ = L1 + F1, L1 = 0 therefore L1¢ = F1 = +45.71 D L2 is calculated from XXII L2 =

45.71 = +46.00 D (from XXII ) 1 − (0.0002 1.448) ∗ 45.71

and F¢¢ v = L2 + F2 = 46.00 + −56.00 = -10.00 D

L2 = Reduced vergence of light incident on second surface 1 − − − − XXIV = 1 tC − F2 n = 1 ((n2 − n1 ) −r0 )−1 − tC n) − − − − XXIV (r0 and tC in metres)

RAPID CALCULATION OF FV AND F′V HAVING ESTABLISHED THREE VALUES Formulae XXVI and XXVII Most scientific and all programmable calculators have many memories. The use of three memories can simplify the above

7  •  Optics and Lens Design

calculations significantly. For spreadsheets, m1, m2 and m3 refer to the cell identification. Example 17: Find Fv given r0 = 7.90 mm, n = 1.45, tC = 0.4 mm, F′v = +6.00D (1) Calculate F2 = (n2 – n1)/r0(m) = (1.00 – 1.45)/(0.0079) = –56.96 D (from III). Store F2 in memory no. 1 (named m1). (2) Calculate reduced tC = tC(m)/n = 0.0004/1.45 = 0.00027586. Store tC/n in m2. (3) Calculate F1 = 0 – L2 = 0 – (m1– F′v)/(1 – m2 * (m1 – F′v)) = +61.89 D (L2 from XXII). Store F1 in m3. Fv = m1 (1 − m2 ∗ m1 ) + m3 − − − − XXVI (equivalent to XXI ) = +5.81 D Similarly, Formula XIX may be expressed as Fv = F ′ v ∗ (1 − m2 ∗ m3 ) (1 − m2 ∗ m1 ) Example 18: To find F′v given r0 = 8.00 mm, n = 1.448, tC = 0.2 mm, Fv = –9.86D (1) Calculate F2 = (n2 – n1)/r0(m) = (1.00 – 1.473)/0.008 = –56.00 D (from III). Store F2 in memory no. 1 (named m1). (2) Calculate reduced tC = tC(m)/n = 0.0002/1.448 = 0.0001381. Store tC/n in m2. (3) Calculate F1 = Fv – L2 = –9.86 – m1/(1 – m2 * m1) = +45.71 D (L2 from XXII). Store F1 in m3. F ¢ v = m3 (1 − m2 ∗ m3 ) + m1 − − − − XXVII (equivalent to XX ) = -10.00 D Similarly, Formula XVIII may be expressed as F¢¢ v = Fv ∗ (1 − m2 ∗ m1 ) (1 − m2 ∗ m3 )

CALCULATION OF PERIPHERAL RADII TO CREATE A STIPULATED AXIAL EDGE LIFT (lEA) Formulae XXVIII, XXIX, XXX, XXXI and XXXII (Note: The term ‘axial edge lift’ appears to imply that the measurement is made at the edge of the lens, but it may be made at the edge of any specified zone.) The radius (r) required so that any spherical peripheral surface (N), of sag value xN, will create a Desired Axial Edge lift (DlEA) at a Given diameter (ØG) can be calculated from 2

 y 2 − yØN−1 2 − xN 2  2 rN =  ØG  + yØG − − − − XXVIII  2 ∗ xN Given φG inner Ø of surface N ; yN-1 = 2 2 (where Ø = diameter), and

where yØG =

xN = (s ØGrN−1 − s ØN−1rN−1 − DlEAN ) − − − − XXIX (sags from XI ) (ØG must have the same value in both XXVIII and XXIX and must be ≥ØN.)

157.e17

where s ØGrN−1 = sagitta at ØG of rN–1 = radius of surface inside (central to) rN; ØN–1 = inner Ø of surface N; DlEAN = Axial Edge Lift Desired from rN. xN is the sag value of the portion of surface (N) involved. The formula may be derived as follows, using s2 to denote the sag at semi-diameter y2 of the required radius r, and s1 and y1 to denote the sag and semi-diameter at the inner edge of the zone, i.e. the values with suffix 2 are greater than those with suffix 1, and x equals the desired drop or difference in sags. s2 = r − r 2 − y22 s1 = r − r 2 − y12 s2 − s1 = x = r 2 − y12 − r 2 − y22 Squaring: x 2 = (r 2 − y12 ) + (r 2 − y22 ) − 2 (r 2 − y12 )(r 2 − y22 ) This can be rearranged: 2 r 4 − r 2 ( y12 + y22 ) + y12 y22 = 2r 2 − ( y12 + y22 ) − x2 Squaring: 4r 4 − 4r 2 ( y12 + y22 ) + 4 y12 y22 = 4r 4 + ( y12 + y22 )2 + x4 − 4r 2 ( y12 + y22 ) − 4r( y12 + y22 )2 x2 + 2( y12 + y22 )x2 4r 2x2 = −4 y12 y22 + ( y12 + y22 )2 + 2( y12 + y22 )x2 + x4 This can be rearranged: 4r 2x2 = ( y22 − y12 )2 + x4 − 2( y22 − y12 )x2 + 4x2 y22 r2 =

y22 − y12 − x2 + y 22 4x2

Taking the square root, and using the symbols used for contact lens work gives formula XXVIII. Example 19. The C3 lens described above (8.00:7.50/ ?.??:8.60/?.??:9.60) is to have total lEA = 0.132 mm (see also Fig. 7.30). Most modern rigid multicurve corneal lenses are designed so that the edge lift increases progressively from the diameter at which the back surface of the lens contacts the cornea (ØC) to the total diameter (ØT). For a given increase in peripheral diameter, the sag increases less as the peripheral radius is increased, so a fairly well-balanced lens is created if the total edge lift is apportioned in ratio to the peripheral band widths between ØC and ØT. The back peripheral radii so produced are usually reasonable but can, of course, be modified if this is thought to be necessary, particularly so that the increments between successive radii increase. The lEA share for any peripheral diameter (ØN) can be stated mathematically: Total l EA ∗ (Ø N − Ø N −1 ) DlEAN = − − − − XXX (all values in mm) ØT − ØC Hence: DlEAN = Total l EA ∗ (Ø N − Ø N −1 ) (Ø T − ØC ) − − − − XXX In Example 19, ØC = Ø0 and the diameter difference, 1.1 and 1.0 mm (peripheral band widths, 0.55 and 0.50 mm), are almost the same for r1 and r2.

157.e18

SECTION 3  •  Instrumentation and Lens Design

DlEA1 = 0.132 ∗ (8.6 − 7.5) (9.6 − 7.5) = 0.132 ∗1.1 2.1 = 0.1452 2.1 = 0.0691 mm (from XXX ) DlEA2 = 0.132 ∗ (9.6 − 8.6) (9.6 − 7.5) = 0.132 ∗1.0 2.1 = 0.132 2.1 = 0.0629 mm (from XXX )

Where y1 = (inner Ø of r2)/2 = (outer Ø of r1)/2 r2 = (((4.82 − 4.32 − 0.245272 ) (2 ∗ 0.24527))2 + 4.82 ) r2 = (((23.04 − 18.49 − 0.0616) 0.49055)2 + 23.04) which resolves to 10.33496 mm (from XXVIII) If r2 is Rounded to 10.35 mm (r2R) lEA2 = s ØTr 1R – s Ø1r 1R – (s ØTr2R – s Ø1r2R )

Total lEA

For calculations: rN =

((( yT 2 – yN – xN 2 ) (2 ∗ xN ))2 + yT 2 ) − − − − XXVIII

r1 = ((( yT 2 – y02 – x12 ) (2 ∗ x1 ))2 + yT 2 ) where x1 = (s ØTr0 − sØ0r0 − Dl EAN ) = (1.6 − 0.93335 − 0.06914) = 0.5975 (from XXIX ) r1 = (((4.82 – 3.752 – 0.59752 ) (2 ∗ 0.5975))2 + 4.82 ) = (((23.04 – 14.0625 – 0.35701) 1.19501) + 23.04) 2

The final specification for Example 19 is 8.00:7.50/ 8.65:8.60/10.35:9.60. In Example 19 the peripheral radii were calculated on the basis that they were produced to ØT (ØT = ØG for XXVIII and XXIX in this case) and that at ØT the lEA created by each peripheral radius would be about 0.066 mm relative to the preceding radius produced to ØT. In the finished lens, r1 extends only from Ø0 to Ø1 and lEA is normally stated relative to r0. The lEA contributed by r1 and r2 individually at their respective outer diameters is given by: lEANi = s ØN r0 – s ØN –1r0 – (s ØNrN – s ØN –1 rN ) − − − − XXXII

= ((8.62049 1.19501) + 23.04) 2

(sags from XI ) lEANi = the individual l EA contribution of rN

r1 = (7.213752 + 23.04) = 52.03816 + 23.04 = 75.07816 r1 = 8.66477 mm (frrom XXVIII) If the calculated value of rN is ‘Rounded’ (rNR) to a standard value (e.g. 0.05 mm), the axial edge lift produced by rNR, l EAN , (relative to rN–1) can be found from: l EAN = s ØTrN –1 – s ØN –1rN –1 – (s ØTrNR – s ØN –1rNR ) − − − − XXXI (sags from XI) The difference between Dl EAN and l EAN can then be added algebraically to Dl EAN+1 (the next peripheral surface out from N), when calculating the value of xN+1. If r1 is rounded to 8.65 mm (r1R): l EA1 = 1.6 − 0.93335 − (1.45399 − 0.85513) = 0.06778 mm (from XXXI ) the error = Dl EA1 – l EA1 = 0.0691 – 0.06778 = 0.00136 mm a very small error, but for the sake of the example, Dl EA2 (0.06 mm) is Modified to Dl EA2M = 0.06422 mm to compensate for the error introduced by rounding r1. x2 = (s ØTr 1R − s Ø1r 1R − Dl EA2M , where DlEA2M = Dl EA2 + (Dl EA 1 − l EA1 ) to take account of the error due to the rounding of r1. x2 = (1.45399 – 1.14450 – 0.06422) = 0.24527 mm (from XXIX ) r2 = (((yT 2 – y12 – x22 ) (2 ∗ x2 ))2 + yT2 )

= 1.45399 – 1.14450 – (1.180349 − 0.935516) = 0.0647 mm (from XXXI ) = l EA1 + l EA2 = 0.06778 + 0.0647 = 0.13244 mm. Error = +0.00044 mm (0.44 µm )

(relative to r0 ) in the final lens and lEA1i = s Ø1r0 – s Ø0r0 – (s Ø1r 1R – s Ø0r 1R ) = 1.25389 − 0.93335 − (1.14450 − 0.85513) = 0.031165 mm lEA2i = s ØTr0 – s Ø1r0 – (s ØTr2R – s Ø1r2R ) = 1.6 − 1.25389 − (1.18035 − 0.93552) = 0.101277 mm Total lEA (as above) = 0.13244 mm (from XXXII) Note the very different values for the two components depending upon the method of calculation, i.e. approximately 0.061 each for the first method compared with 0.031 and 0.101 for the second method. For a lens of this diameter, a C4 construction would probably be better.

CALCULATION OF NEW PERIPHERAL RADIUS TO ALTER AXIAL EDGE LIFT BY A GIVEN AMOUNT Formulae XXVIIIA, XXIXA and XXXII If the edge lift of an existing lens is to be increased by modification, then it is necessary to flatten the outermost curve and also the inner peripheral ones if necessary, taking care not to reduce the BOZD. If only an intermediate curve is flattened, then the position of the edge of the lens, as shown in Fig. 7.26, relative to the BOZR would not be altered, only the relative contributions to the edge lift provided by the different peripheral zones. In a C3 lens, flattening the first peripheral curve, r1, would increase the contribution from

7  •  Optics and Lens Design

this zone whilst reducing the contribution from the outer curve. The diameter Ø1, would be increased from its original value of 8.60 in this example, but could be restored back to its original value by reworking the outer peripheral zone, either with the same 10.35 radius or flatter. If a new lens is to be ordered, then calculations can be made as follows. Formulae XXVIII and XXIX can be modified so that they yield a spherical peripheral radius (rN) which will create a new DlEAN relative to r0 at ØN. This is achieved by using ØN as ØG in both formulae and, because the lEA created by rN is to be calculated relative to r0, r0 is used instead of rN–1 in XXIXA. In Example 20 below, r0 is rN–1 since the first peripheral curve is being modified, but r0 would be used instead of rN–1 in XXIXA when calculating a new radius for any other peripheral band. rN =

yØN 2 − yØN−1 2 − X N 2 + yØN 2 − − − − XXVIIIA (2 ∗ xN )2

rN = ((( yØN 2 – yØN –1 2 – xN 2 ) (2 ∗ xN ))2 + yØN 2 ) − − − − XXVIIIA xN = (s ØNrN – s ØN –1r0 – Dl EAN ) − − − − XXIXA (sags from XI ) (symbols as before) Example 20.  If it was decided to flatten r1R of the lens in Example 19 to increase its lEA1i contribution from 0.0311 to say 0.035 mm, the outer and inner diameters of the peripheral surface r1 (Ø1 and Ø0, respectively) would be used for yØN and yØN–1 in XXVIIIA and for ØN and ØN–1 in XXIXA. Also in Formula XXIXA, DlEA = 0.035 is used (where 0.06 was used previously) because the formula is now dealing with r1 produced to Ø1 (as opposed to ØT). New r1 = ((( yØ1 2 − yØ0 2 − x12 ) (2 ∗ x1 ))2 + yØ1 2 ) Where x1 = s Ø1r0 − s Ø0r0 − Dl EA1 x1 = 1.25389 – 0.93335 – 0.035 = 0.28554 mm (from XXIXA) (((4.32 – 3.752 – 0.285542 ) (2 ∗ 0.28554))2 + 4.32 ) = 8.741mm (from XXVIIIIA)

New r1 =

If the new r1 is rounded to 8.75 mm, l EA1i would be 0.03537 mm (from XXXII) which, with l EA2i (above), gives Total lEA = 0.1367 mm. The quantities in the above example are quite small because the proposed modification is relatively small. The increase in total lEA due to changing r1 from 8.65 to 8.75 mm is 3%. The values become more significant for greater modifications but the principle is demonstrated.

CALCULATION OF PERIPHERAL RADII TO CREATE A STIPULATED AXIAL EDGE CLEARANCE (CEA) Formulae XXVIII, XXIX, XXX, XXXIII and XXXIV [see also the spreadsheets in Rabbetts (1992)]. In order to calculate peripheral radii which will create a stipulated Axial Edge Clearance (CEA) either the facility to measure the contour of the cornea must be available or a corneal model must be adopted. It is widely accepted that

157.e19

the average cornea approximates most closely to the surface created when the prolate curve of an ellipse is rotated about its major axis and an analysis of the findings of most researchers indicates that the average p-value is in the order of 0.8, with a ‘normal range’ varying from 0.9 (less flat periphery) to 0.7 (flatter periphery). Cases have been reported (rarely, if those that are subject to ortho-keratology or have undergone refractive surgery for myopia are excluded) of corneas having a periphery which is steeper than the central region, with p-values greater than 1.0 (or rotation of the oblate curve of an ellipse about the minor axis). The following formulae are equally valid for either type of cornea provided that the p-value is known. To create a stipulated CEA at a specified diameter, ØN, the lEA of the lens over the peripheral zone between the contact diameter (ØC) and ØN must be equal to the sagitta of the cornea over the same peripheral zone + the Desired Axial Edge Clearance (DCEA). Dl EA = s ØNr0 K – s ØCr0 K + DCEA (at Ø N ) − − − − XXXIII The procedure for calculating peripheral radii which will create a desired CEA is similar to that used with Formulae XXVIII and XXIX, differing only in the derivation of the value ‘x’ for the first peripheral surface outside (wider than) ØC. Formula XXIX is modified to xN = s ØTr0 K – s ØN –1r0 K – DCEAN − − − − XXIXB because axial Clearance is relative to the sagitta created by the radius of the cornea (which changes with diameter depending on the p-value), not to that created by the adjacent back surface radius central to rN. xN represents an element of ‘corneal sag’ reduced by an element of CEA. Any subsequent peripheral radii are calculated using Formulae XXVIII and XXIX unmodified, (see Example 21 below). Formula XXX, which was used previously to apportion the share of the total lEA for each peripheral band, can be used to apportion the total CEA. The ‘CEA share’ for each band can be used in Formula XXIX instead of the ‘lEA share’. Example 21.  C5 ?.??:7.50/?.??:8.10/?.??:8.70/?.??:9.30/?. ??:9.80 A cornea having a p-value of 0.8 for which a Bausch & Lomb keratometer has indicated a spherical radius of 8.00 mm (rK) is to be fitted with a C5 RGP corneal lens designed to create a tear layer 0.011 mm deep at the corneal apex and a total axial edge clearance (CEA) of 100 micrometres (µm) = 0.1 mm. In this example (and most commonly), Ø0 is chosen to be the contact diameter (ØC). The C5 lens specification with unknown radii has diameters as shown above. Ø0 and ØT have been chosen to take account of the patient’s pupil diameter (4.5 mm in average indoor lighting), horizontal visible iris diameter (HVID) 13.50 mm, lower lid margin at the lower limbus and vertical palpebral aperture 9.0 mm between lids of normal tension. First convert the ‘K’ reading (rK) to corneal apical radius (r0K) using Formula XIII. ØMi is assumed to be 3.10 mm for the Bausch & Lomb keratometer and an average cornea. r0K = (82 − (1 − 0.8) ∗1.552 ) = (64 − 0.2 ∗ 2.4025) = (63.5195) = 7.9699 mm (from XIII)

157.e20

SECTION 3  •  Instrumentation and Lens Design

To create a tears layer (TLT) 0.011 mm thick, s ØC r0 must = s ØC r0 K + 0.011 mm

r3 = ((( yØT 2 − yØ2 2 − x32 ) (2 ∗ x3 ))2 + yØT 2 )

Thus s ØC r0 K = (7.9699 – (7.9699) – 0.8 × 3.75 ) 0.8 2

2

= 0.9252 mm (from XII ) Thus s ØC r0 = 0.9252 + 0.011 = 0.9362 mm

r3 = (((4.92 − 4.352 − 0.31712 ) (2 * 0.3171))2 + 4.92 ) = 9.2651§ mm (from XXVIII ) x4 = s ØTr3 – s Ø3r3 – DC EA4 = 1.4017 – 1.2513 – 0.0217 = 0.1287 mm (from XXIX ) r4 = (( yØT 2 – yØ3 2 – x42 ) (2 ∗ x4 ))2 + yØT 2 )

r0 = (0.93622 + 3.752 ) (2∗ 0.9362) = 14.9389 1.8224 = 7.9786 mm (from XVII) Total CEA ∗ (Ø1 − Ø0 ) ØT − ØC 0.100 ∗ (8.1 − 7.5) = 9 .8 − 7 . 5 0.100 ∗ 0.6 0.06 = = 2 .3 2 .3 DCEA1 = 0.0261 mm (from XXX )

DCEA share for r1 =

Since Ø1– Ø0, Ø2 – Ø1 and Ø3– Ø2 all = 0.6 mm, DCEA2 and DCEA3 = 0.0261mm (as DCEA1 ). 0.100 ∗ (9 9.8 − 9.3) DCEA share for r4 = 9.8 − 7.5 DCEA4 = 0.0217 mm (from XXX ) The sag formula for an ellipse (XII) must be used for corneal sag values involving r0K, i.e. for calculating x1,whereas Formula XI is used for the remaining sag values which all relate to spherical surfaces. See Fig. 7.29. x1 = s ØTr0 K – s Ø0r0 K – DC EA1 = 1.6415 – 0.9252 – 0.0261 = 0.6902§ mm (from XXIXB)

r4 = (((4.92 – 4.652 – 0.12872 ) (2 ∗ 0.1287))2 + 4.92) = 10.4349 mm (from XXVIII ) It is proposed that radii should be rounded to the nearest 0.01 mm, so therefore the final specification is: C5 7.98:7.5/8.43:8.1/8.74:8.7/9.27:9.3/10.43:9.8 With radii rounded to 0.01 mm and using Douthwaite’s (2006) formula, CEA may be found as follows: CEA = s ØTr0 K + TLT – s p − − − − XXXIV When r0 is rounded from 7.9786 to 7.98 mm, TLT is reduced from 0.011 to 0.0108 mm, so: CEA = 1.6415 + 0.0108 − 1.5524 = 0.0999 mm The small discrepancy (0.1 µm) is due to rounding the radii to 0.01 mm and limiting the calculation to 4 decimal places. Although the axial edge clearance was designed to be 0.1 mm, the axial edge lift is significantly larger at 0.129 mm The radii in Example 21 have been rounded to the nearest 0.01 mm rather than the traditional 0.05 mm because the tolerances contained in BS EN ISO 18369-2: (2006) (± 0.05 mm for r0 and ± 0.1 mm for peripheral radii) allow significant departure from the intended lens parameters. Rounding all the radii to the nearest 0.05 mm gives the specification:

r1 = ((( yØT 2 – yØ0 2 – x12 ) (2 ∗ x1 ))2 + yØT 2 )

C5 8.00:7.5/8.45:8.1/8.75:8.7/9.25:9.3/10.45:9.8

r1 = (((4.92 – 3.752 – 0.69022 ) (2 ∗ 0.6902))2 + 4.92 )

TLT is reduced from 10.8 to 8.17 µm (from XVI). r2 and r4 are each increased by 0.01 mm. r1 is increased by 0.02 mm but r3 is decreased by 0.02 mm and these therefore tend to cancel each other out. Thus the net flattening of the peripheral radii due to rounding in this case is in the order of 0.02 mm which increases CEA minimally from 99.9 to 100.5 µm (from XXXIV)§. However ‘Standard’ tolerances allow a manufacturer to supply the above lens as:

r1 = (((24.01 – 14.0625 – 0.4764) (1.3804))2 + 24.01) r1 = ((9.4711 1.3804)2 + 24.01) = (6.86112 + 24.01) r1 = (47.0751 + 24.01) = (71.0785) = 8.4308 mm (from XXVIII ) x2 = s ØTr1 – s Ø1r1 – DCEA2 = 1.5701 – 1.0364 – 0.0261 = 0.5076§ mm (from XXIX )

C5 8.05:7.5/8.55:8.1/8.85:8.7/9.35:9.3/10.55:9.8

r2 = ((( yØT 2 – yØ1 2 – x22 ) (2 ∗ x2 ))2 + yØT 2 ) r2 = (((4.92 – 4.052 – 0.50762 ) (2 ∗ 0.5076)) + 4.92 ) 2

= 8.7425§ mm (from XXVIII ) x3 = s ØTr2 – s Ø2r2 – DC EA3 = 1.5022 – 1.1590 – 0.0261 = 0.3171 mm (from XXIX )

§Very small discrepancies may occur in the calculated values here and in the following pages, depending on the number of decimal places used. This applies even when ‘floating decimal mode’ is used in a calculator or computer, as instruments vary in the number of decimal places they employ.

which reduces TLT from 8.17 to 1.61 µm but increases CEA from 100.5 to 109.06 µm. Example 22.  A C4 lens will now be designed to give the same DCEA (100 µm) on the same eye. r0, Ø0 and ØT are the same: C4 7.98:7.5/?.??:8.0/?.??:8.8/?.??:9.8 0.1∗ (8.0 − 7.5) 0.1∗ 0.5 0.05 = DC EA1 = = 9 .8 − 7 .5 2 .3 2 .3 = 0.0217 mm (from XXX) 0.1 ∗ (8.8 − 8.0) 0.1∗ 0.8 0.8 DC EA2 = = = 9 .8 − 7 .5 2 .3 2 .3 = 0.0348 mm m (from XXX )

7  •  Optics and Lens Design

0.1∗ (9.8 − 8.8) 0.1∗1.0 0.1 = = 9 .8 − 7 .5 2 .3 2 .3 = 0.0435 mm (from XXX) x1 = s ØTr0 K − s Ø0r0 K − DC EA1 = 1.6415 − 0.9252 − 0.0217 = 0.6946 mm (from XXIXB)

DC EA3 =

r1 = r1 =

(((y − y − x ) (2∗ x )) + y ) (((4.9 − 3.75 − 0.6946 ) (2∗ 0.6946)) + 4.9 ) ØT

2

Ø0

2

2

2 1

2

2

1

ØT

2

2

2

2

= 8.3919 mm (from XXVIII ) x2 = s ØTr 1 − s Ø1r 1 − DC EA1 = 1.5791 − 1.0146 − 0.0348 = 0.5297 mm (from XXIX ) r2 = (((yØT 2 − yØ1 2 − x22 ) (2∗ x2 ))2 + yØT 2 ) r2 = (((4.92 − 4.002 − 0.52972 ) (2∗ 0.5297))2 + 4.92 ) = 8.7890 mm (from XXVIII ) x3 = s ØTr2 − s Ø2r2 − DC EA3 = 1.4927 − 1.1807 − 0.0435 = 0.2685 mm (from XXIX ) r3 = r3 =

(((y − y − x ) (2∗ x )) + y ) (((4.9 − 4.40 − 0.2685 ) (2∗ 0.2685)) + 4.9 ) ØT

2

2

Ø2

2

2 3

2

2

3

ØT

2

2

2

2

= 9.8333 mm (from XXVIII ) The specification of the lens in Example 22 (with radii rounded to 0.01 mm) is: C4 7.98:7.5/8.39:8.0/8.79:8.8/9.83:9.8 Then CEA = s ØTr0 K + TLT – s p = 1.6415 + 0.0108 – 1.5524 (TLT from XVI ) = 0.0999 mm (≈ 0.1 mm) (from XXXIV )§ ‘Standard’ tolerances allow the above lens to be supplied in the form C4 8.03:7.5/8.49:8.0/8.89:8.8/9.93:9.8. This would reduce TLT to 0.0042 mm, an error of 61.8%, from the desired 0.011 mm. If all the calculated radii are rounded to 0.05 mm, the lens becomes C4 8.00:7.5/8.40:8.0/8.80:8.8/9.85:9.8, which allows the lens to be supplied as C4 8.05:7.5/8.50:8.0/8.90:8.8/9.95:9.8 TLT is then reduced to 0.0016 mm (error of 85.4%) and CEA = 0.1097 mm (error = 9.7% from the desired 0.1 mm). The change in TLT indicated is the ‘worst case’ result due to rounding and permitted tolerance for r0. The error in CEA could be significantly greater than that indicated above in the ‘worst case’ – if the calculated result for all three peripheral radii needed to be increased by 0.024 mm to round them to 0.05 mm. A further degradation of the intended specification may be introduced due to the tolerances permitted for diameters: ± 0.2 mm for optic and peripheral zones and ± 0.1 mm for total diameter. C4 8.05:7.3/8.50:7.8/8.90:8.6/9.95:9.7 could be supplied within tolerance! TLT virtually disappears being reduced to 0.00089 mm (error 92%) and CEA is increased to 0.1144 mm (error of 14.4%). The ‘worst case’ error

157.e21

for CEA could be more than this for the reasons explained above. Collectively these sources of error probably account for why two lenses, supposedly of the same specification, can look and behave quite differently on the eye. Example 23.  The C4 lens above will now be redesigned (for the same eye) so that r1 is parallel to the cornea and the desired CEA (0.1 mm) is provided by r2 and r3. CEA, r0, Ø0 and ØT are the same. The band width for r1 is increased to provide a greater bearing surface area although this, as with all other diameters, is optional. C4 7.98:7.5/???:8.5/???:9.3/???:9.8 shows the predetermined and unknown parameters. If r1 is to align with the cornea, the sag created by the surface of r1 must equal the sag of the cornea over the band Ø0 to Ø1, which is the value of xN required for Formula XXVIIIA (used instead of Formula XXVIII in this case as we are dealing with a single peripheral band). Formula XXIXB is modified to XXIXC which calculates a single band of corneal sag and the DCEA term is omitted as no CEA is required from this surface. xN = s Ø1r0 K – s Ø0 r0 K = 1.2062 – 0.9252 = 0.281mm (from XXIXC) rN = ((( yØN 2 – yØN –1 2 – xN 2 ) (2 ∗ xN ))2 + yØN 2 ) r1 = (((4.252 – 3.752 – 0.2812 ) (2 ∗ 0.281))2 + 4.252 ) = 8.1694 mm (from XXVIIIA). Round to 8.17 mm. Because r1 is in alignment with the cornea, Ø1 is the ‘effective’ contact diameter (ØC), i.e. the diameter from which edge clearance begins. The desired CEA (0.1 mm) is shared between r2 and r3 using Formula XXX. 0.1∗ (9.3 − 8.5) 0.1∗ 0.8 0.08 = = 9.8 − 8.5 1.3 1.3 = 0.0615 mm (from XXX ) 0.1 ∗ (9.8 − 9.3) 0.1 ∗ 0.5 0.05 DCEA3 = = = 9.8 − 8.5 1.3 1.3 = 0.0385 mm (from XXX ) x2 = s ØTr0 K − s Ø1r0 K − DC EA2 = 1.6415 − 1.2062 − 0.0615 = 0.3738 mm (from XXIXB)

DCEA2 =

r2 = ((( yØT 2 − yØ1 2 − x22 ) (2 ∗ x2 ))2 + yØT 2 ) r2 = (((4.92 − 4.252 − 0.37382 ) (2 ∗ 0.3738))2 + 4.92 ) = 9.1840 mm (from XXVIII)). Round to 9.18 mm. x3 = s ØTr2 − s Ø2r2 − DC EA3 = 1.4164 − 1.2642 − 0.0385 = 0.1137 mm (from XXIX ) r3 = ((( yØT 2 − yØ2 2 − x32 ) (2 ∗ x3 ))2 + yØT 2 ) r3 = (((4.92 − 4.652 − 0.11372 ) (2 ∗ 0.1137))2 + 4.92 ) = 11.5355 mm (from XXV VIII). Round to 11.54 mm. Thus the lens to order (radii rounded to 0.01 mm) is: C4 7.98:7.5/8.17:8.5/9.18:9.3/11.54:9.8

157.e22

SECTION 3  •  Instrumentation and Lens Design

The following calculation will confirm that the desired edge clearance (0.10 mm) is provided by r2 and r3. In this case the effective ØC is Ø1 so the edge clearance created between ØC and ØT = the sagitta of the cornea from Ø1 to ØT minus the sagitta of the back surface of the lens from Ø1 to ØT. Sagitta of cornea (Ø1 to Ø T ) = s ØTr0 K − s Ø1r0 K = 1.6415 − 1.2062 = 0.435 53 mm (sags from XII ) Sagitta of back surface of lens (Ø1 to ØT) = s Ø2r2 – s Ø1r2 + s ØTr3 – s Ø2r3 = 1.2648 – 1.0431 + 1.0920 – 0.9783 = 0.3354 mm (sags from XI ) C EA = 0.4353 – 0.3355 = 0.0999 mm The small discrepancy (0.1 µm) is due to the rounding of calculated values. The contact diameter (ØC) of any contact lens on a given cornea is the widest back surface diameter at which the primary sag of the lens (s ØWp) is ≥ the sag of the cornea at the same diameter. In Example 23, r1 was calculated to produce a surface parallel to the cornea, or as nearly parallel as a spherical surface with its radius rounded to the nearest 0.01 mm can be to an ellipsoidal surface. Although Ø1 was taken to be the ‘effective’ ØC for the purpose of calculating r2 and r3, either Ø0 or Ø1 can be used in Formula XVI to calculate TLT because the surface of r1 is parallel to the cornea (or very nearly so). Using r0 = 7.98 mm, r1 = 8.17 mm, r0K = 7.97 mm (all radii have been rounded to 0.01 mm): TLT for Ø0 = s Ø0r0 – s Ø0r0 K = 0.9360 – 0.9252 = 0.0108 mm (from XVI , XI and XII) TLT for Ø1 = s Ø1p – s Ø1r0 K = s Ø0r0 + s Ø1r 1 – s Ø0rs1 – s Ø1r0 K = 0.9360 + 1.1924 – 0.9115 – 1.2062 = 0.0107 mm (from XVI , XI and XII ) The similarity of these two results§ (within 0.1 µm) indicates that r1 aligns fairly closely with the cornea. As with surface power and radius, the same numerical change in radius will produce a greater change in sag with steep curves rather than flat curves. Hence the differences between r0 and r1, r1 and r2 increase as the BOZR, r0, flattens in order to produce the same lAE.

CALCULATION OF ANTERIOR OPTIC RADIUS (ra0) WHEN CENTRE THICKNESS (tC) IS KNOWN (OR CHOSEN), GIVEN SPHERICAL r0, REFRACTIVE INDEX (n) AND BVP (F′V) Formula XXXV An appropriate centre thickness (tC) is usually chosen when designing minus lenses. tC can be chosen empirically for plus lenses but may have to be modified by trial and error to achieve an acceptable tEA (or tJA for lenticular lenses), but see Formula XXXVII below. It is assumed that the lens is in air. ra0 (in m) = (from modified IV ) = (1 − n) (− L2 ) in air The lens is reversed to calculate L2 and, unless F′v is plano, XXIV is modified so that L1′ = F2 – F′v because the light incident on F2 is not parallel. L2 (vergence of light incident on F1 ) =

1 1 t − C F2 − F ′v n

(from modified XXIV) (tC in m) Combining modified expressions IV and XXIV: (1 − n) −1 t   1 − C  F2 − F ′ v n  t   1 = (1 − n) ∗  − C − − − − XXXV (tC in m)  F2 − F ′ v n 

ra0 =

The lens is reversed to calculate ra0, and r0 is given a –ve sign. [Care must be taken with (n2 – n1). Bearing in mind that the lens is reversed, the modified Formula IV and the first part of the ‘calculator’ version of XXXI are concerned with light which will emerge from F1 therefore (n2 – n1) = (1 – n); where n = the refractive index of the lens. ra 0 (in m ) = (1 − n) ∗ [((n − 1)/{−r0 } − Fv′ )−1 − tC n ] − − − − XXXV The middle part of this form of Formula XXXV deals with light entering F2 so (n2 – n1) = (n – 1)]. The ‘curly’ brackets enclosing {−r0} have no effect on the working of Formula XXXV and are simply to highlight the fact that r0 is given a –ve sign because it is measured in the opposite direction to the incident light. Example 24. A C3 lens having the back surface 8.00:7.70/8.85:8.20/10.30:9.20 and refractive index (n) 1.45, is to have BVP –5.00D. If tC is made 0.15 mm, what will be the radius of the front surface (ra0)? r0 = 8.00 mm. ra 0 (in mm) = (1.00 − 1.45) ∗ 1000 ∗ (((1.45 − 1.00) ∗ 1000 −8

§See footnote on p. 157.e20.

− −5.00)−1 − 0.15 1.45 ∗ 1000)

7  •  Optics and Lens Design

ra 0 (in mm) = −450 ∗ ((450 −8 − −5.00)−1 − 0.15 1450) = −450 ∗ ( −51.25−1 − 0.000103448) = −450 ∗ ( −0.019512195 − 0.000103448) = −450 ∗ ( −0.019615643) = 8.827 mm (from XXXV ) For Example 17, tC = 0.4 mm was chosen empirically; r0 = 7.9 mm; n = 1.45; Fv′ = +6.00 D ra0 (in mm) = (−0.45) ∗103 ((450 −7.9 − 6.00)−1 − 0.4 1450) = −450 ((450 −8 − 6.00)−1 − 0.000275862) = −450 ∗ (−0.016158452) = 7.271mm

ALTERNATIVE METHOD OF CALCULATION OF ANTERIOR OPTIC RADIUS ra0 WHEN CENTRE THICKNESS (tC) IS KNOWN (OR CHOSEN), GIVEN r0, REFRACTIVE INDEX (n) AND BVP (Fv′) Formula XXXVI An alternative approach to the calculation of ra0is to add a correcting value to the radius of the ‘thin lens’ front surface. The correcting value is part of a formula derived by Bennett (1985) for afocal lenses, adapted here for powered lenses. tC must still be chosen empirically for plus lenses when using Formula XXXVI, but see Formula XXXVII below. (n2 − n1 ) ∗ tC n2 = (n − 1) ∗ tC n for a lens of refractive index n in air

Bennett’s value =

(where n2 is the refractive index of the lens and n1 is air). This represents the amount by which the radius of the ‘thick lens’ front surface exceeds that of its ‘thin lens’ front surface equivalent, so it is added to the ‘thin lens’ front surface radius. The ‘thin lens’ power of the front surface: F1 ‘thin’ = Fv′ − F2 The power of the back (concave) surface: (n 2 − n1 ) (from III ) r0 (see below) 1− n = r0

F2 =

Care must be taken when calculating F2; n2 = air and n1 = the lens. Elsewhere in this calculation n1 − n2 relates to the front surface when n2 = the lens and n1 = air (see below). (n − n ) ∗103 ra 0 ‘thin’ (in mm) = 2 1 (from modified IV ) F1 ‘thin’ (n − 1) ∗103 = (r in metres) 1− n 0 Fv′ − r0

ra 0 ‘thick’ (in mm) =

157.e23

(n − 1) ∗ 103 (n − 1) ∗ tC + 1− n n Fv′ − r0 − − − − − − XXXVI

= (n − 1) ∗ 103 /(Fv′ − (1 − n) r0 ) + (n − 1) ∗ tC n − − − − XXXVI The lens in Example 17 has r0 = 7.90 mm; n = 1.45; tC = 0.4 mm; F ′v = +6.00 D ra0 = (1.45 − 1.00) ∗103 (6.00 − (1.00 − 1.45)* 103 7.9 + (1.45 − 1.00) ∗ 0.4 4 1.45 = 450 (6.00 − ( −56.96)) + 0.18 1.45 = 450 62.96 + 0.1241 = 7.14717 + 0.1241 = 7.27113 mm ( ≈ 7.27 mm ) (from XXXVI ) The lens in Example 18 has r0 = 8.00 mm; n = 1.448; tC = 0.2 mm; Fv′ = −10.00 D ra0 = 448 ( −10.00 − ( −448) 8 + (1.448 − 1.00) ∗ 0.2 1.448 = 448 ( −10.00 − ( −56.00)) + 0.0896 1.448 = 448 46.00 + 0.061878 = 9.739130 + 0.061878 = 9..801009 mm (≈ 9.80) (from XXXVI )

CALCULATION OF AXIAL EDGE THICKNESS (tEA) tEA + sa 0 = s p + tC for full aperture lenses. where sa0 = sagitta of ra0 at ØT for full aperture lenses; sp = primary sag (see Figs 7.25 and 7.26; sags from XI). tEA + sa 0 = s∅a 0p (s p at ∅a 0 ) + tC , for positive lenticulars with a parallel carrier (see Fig. 7.27, where sa0 = sØa0ra0). If the lens in Example 17 has the back surface: C3 7.90:7.70/8.70:8.20/9.80:9.20 sp = s∅0r0 + s∅1 r 1 − s∅0r 1 + s∅Tr2 − s∅1r2 = 1.3779 mm (see Fig. 7.26) tC = 0.4 mm sa0 = s∅Tra 0 = 1.6400 mm (from ra 0 = 7.2713 and Ø T = 9.20, see above and XI ) tEA = s p + tC − s a0 = 1.3779 + 0.4 − 1.6400 = 0.1379 mm ( ≈ 0.14 mm ) (see Fig. 7.25) If the back surface of the lens in Example 18 is: C3 8.00:7.70/8.85:8.20/10.30:9.20 t EA = s p + tC − sa 0 = 1.3461 + 0.2 − 1.61 = 0.3995 mm (≈ 0.40 mm) (calculated as for Example 17). The front surface of this lens would probably be lenticulated to give tEA = 0.15 mm.

157.e24

SECTION 3  •  Instrumentation and Lens Design

For the lens in Example 24:

For calculations:

C3 8.00:7.70/8.85:8.20/10.30:9.20 tEA = s p + tC − sa 0 = 1.3461 + 0.15 − 1.2933 = 0.20 mm (calculated as above).

CALCULATION OF ANTERIOR OPTIC RADIUS (ra0) AND (tC) FOR PLUS LENSES, GIVEN Fv′, REFRACTIVE INDEX (N) AND BACK SURFACE SPECIFICATION Formula XXXVII The calculation of (ra0) is more complicated if (tC) is not known, which is often the case for plus lenses. An appropriate tEA is normally chosen for full aperture plus lenses or tJA for lenticular lenses. The centre thickness (tC) so produced is then calculated from tC = Sa0 + tEA −sp (see Fig. 7.27). First the sagitta of the required front surface (sao) must be found. F2 is calculated from III, then, assuming the lens to be infinitely thin, the F2 is calculated from III, then, assuming the lens to be infinitely thin, the power F1‘thin’ is: F1‘thin’ = Fv′ − F2. The corresponding focal length, f1‘thin’ = 1/F1‘thin’. This establishes the position of the second principal focus of the front surface relative to the back vertex. f1‘thin’ is then modified to maintain this relationship, taking account of the change in vergence due to the thickness of the finished lens, i.e. tEA (or tJA) + the (tc) imposed by the refractive index. This is achieved using a modified version of a formula published by Jalie (1988): Sao =    103  (n − 1)  n − (s p − t EA ) 1− n  Fv′ −    r0 − n − 2(n − 1)   Ø 2   103 − (s p − t EA ) −  T  n(n − 2(n − 1)) (n − 1)  n  2 1− n  Fv′ −    r0 n − 2(n − 1) The formula in this form is for full-aperture lenses, symbols as above. For lenticular lenses tEA would be replaced by tJA and ØT by Øa0; also, sp would be calculated at Øa0 instead of ØT. Jalie (2016) simplified the formula to: y ∗n − − − XXXVII 2−n 2

where n − 1 1000n  ∗ − q 2 − n  F1’thin ’  F1 ‘thin’ = Fv′ − F2, q = s p − tEA (s∅a 0p − t JA for lenticular) where s p = s Ø0r0 + s Ø1r 1 − s Ø0r 1 + ….. s ØN rN − s ØN−1rN and y = Ø T 2 (see XI and Fig. 7.26).

F2 = (1.00 − 1.45) 0.008 = −56.25 D (from III ) F1 ‘thin’ = Fv′ − F2 = +8.00 –56.25 = +64.25 D q = s p – tEA = 1.4681 – 0.15 = 1.3181 mm (for s p see Figs 6.28 and 6.29) y = Ø T 2 = 9.6 2 = 4.8 mm b = [(n − 1) (2 − n)] ∗ [1000n F1‘thin’ − q] = [0.45 0.55] ∗ [1450 64.25 − 1.3181] = 0.8182 ∗ 21.2500 = 17.3864 Enter the subvalues ‘b’ and ‘y’ into Formula XXXVII

= 17.3864 − 302.2868 − 23.04 ∗1.45 0.55

2

b=

Example 25.  What is the radius of the front surface (ra0) and the centre thickness (tC) of the C3 lens 8.00:7.50/ 8.90:8.60/10.00:9.60 having a refractive index 1.45, BVP +8.00D and tEA 0.15 mm ? A corneal lens with BVP +8.00D would normally be made in lenticular form but it will be designed first as a full-aperture lens to show the method and also to illustrate the reduction in (tC) achieved by using the lenticular form. First the precalculations:

sa0 = 17.3864 − (17.38642 − 4.82 ∗1.45 (2 − 1.45)

2

sa 0 = b − b 2 −

sa 0 = b − (b2 − (∅T 2)2 ∗ n (2 − n))0.5 − − − − XXXVII and b = [(n − 1) (2 − n)] ∗ [(1000n F1’thin ’ ) − q ]

= 17.3864 − 241.545 = 17.3864 − 15.5417 = 1.84468 mm (frrom XXXVII) (when calculated with ’floating’ decimal point) rao = ( y2 + s 2 ) (2 ∗ s) = (4.82 + 1.83162 ) (2 ∗1.8316) = (23.04 + 3.4028) 3.6894 = 26.4428 3.6894 = 7.1673 mm (from XVII) tC = sa 0 + t EA − s p = 1.8447 + 0.15 − 1.4681 = 0.5266 mm (see above) Using these values, which are correct to 4 decimal places, Fv′ checks as +8.0002 D. Using ‘floating decimal mode’ for the calculations, Fv′ checks as exactly +8.00D.

CALCULATION OF ANTERIOR SURFACE SPECIFICATION AND tC FOR PLUS LENSES IN LENTICULAR FORM, GIVEN F′V, REFRACTIVE INDEX (n) AND BACK SURFACE SPECIFICATION Formulae XVII, XXVIII, XXIX, XXXVII and XXXVIII Example 26.  The front surface of the lens in Example 25 will now be redesigned in lenticular form with a negative surfaced carrier portion. Ideally the anterior optic diameter

7  •  Optics and Lens Design

(Øa0) of any lenticular lens should be greater than Ø0 to avoid ‘designed in’ junction flare caused by light, having passed through the anterior optic surface, converging within the thickness of the lens to a diameter less than Ø0. Worst case calculations using r0 = 6.25 mm and F′v = +18.00D indicate that, if Øa0 exceeds Ø0 by 0.08 mm or more, no flare will emanate from this cause. The tolerance allowed in BS EN ISO 18369-2: (2017) for optic and peripheral diameters is ± 0.2 mm so the front surface will be designed with Øa0 0.3 mm greater than Ø0 on the grounds that a lens produced within tolerance will not create junction flare, and no detrimental effect will be caused (other than a very slight increase in tC) if Øa0 is 0.5 mm greater than Ø0. All the values for Formula XXXVII remain the same except y and q, and of course, the value b because it contains q. The anterior optic diameter, Øa0 = Ø0 + 0.3 = 7.8 mm y = Øa 0 2 = 7.8 2 = 3.90 mm q = s Øa 0p – t JA

157.e25

(a)

tJA

Xa1 X

ø0

tEA

tJA øa0

øI

(b) tJA Xa1

X

øa0

tEA

øT

Xa1

tEA

X

tEA

X

tJA ø0

Xa1

øl

øT

Fig. 7.33  (a) Positive lenticular lens with tricurve back surface and Øa0 > Ø0. Left-hand side of figure shows a parallel carrier zone with tJA = tEA and xa1 = x. Right-hand side of figure shows a negative carrier zone with tJA tEA and xa1 > x

When Øa 0 > Ø0 then : sØa0p = sØ0r0 + sØa 0r 1 – s Ø0r 1 = 0.9334 + 0.9000 – 0.8286 = 1.0048 mm (sags from XI ) (for sp see Fig. 7.26) tJA is chosen to be 0.15 mm, so q now = 1.0048 – 0.15 = 0.8548 mm b = [0.45 0.55] ∗ [1450 64.255 − 0.8548] = 0.8182 ∗ [22.5681 − 0.8548] = 0.8182∗ 21.7133 = 17.7655 sa0 = b − b2 − y2 ∗ n (2 − n) − − − − XXXVII sa0 = 17.7655 − 17.76552 − 3.92 * 1.45 0.57 = 1.1669 mm (from XXXVII ) This new sa0 is used to calculate the lenticular ra0:

Had a ‘positive’ carrier been required, tEA must be less than tJA, ra1 being required to provide a greater sag, xa1, than is provided by the back surface from Øa0 out to ØT. The amount by which tEA differs from tJA is a matter of choice. The maximum tEA advised is 0.24 mm, which will be used in the present example. For a ‘parallel’ carrier xa1 in Fig. 7.27 would = x (see above). For the negative carrier in this example, since tJA = 0.15 mm, ra1 must provide 0.24 – 0.15 = 0.09 mm less sag than is provided by the back surface from Øa0 to ØT, therefore in Fig. 7.33a (right-hand side of diagram) xa1 = x – 0.09 mm Because Øa0 > Ø0, x includes only part of the sag contributed by the surface of r1 and all of the sag contributed by the surface of r2. x = s ØTr2 – s Ø1r2 + (s Ø1r 1 – s Øa 0 r1)

ra0 = ( y2 + sa 02 ) (2 ∗ sa 0 ) − − − − XVII ra0 = (15.21 + 1.36164) 2.3338 = 7.1001 mm (from XVII )

Since xa1 is (DtEA − tJA) less than x,

(Note: The value given for ra0 has been determined by using ‘floating decimal mode’ for the calculations.)

(see XXXVIII below)

tC = sa 0 + t JA – sØa 0p (see above) = 1.1669 + 0.15 – 1.0048 (from above) = 0.3121mm Thus lenticulation has reduced tC by about 40% from 0.53 to 0.31 mm. CALCULATION OF FRONT PERIPHERAL RADIUS (Ra1).  (Also applies to negative lenses) The lens in Example 26 is to have a ‘negative’ carrier so tEA must be greater than tJA. The anterior peripheral radius (ra1) must therefore provide a smaller sag, xa1, between Øa0 and ØT than is provided by the peripheral geometry of the back surface from Øa0 out to ØT (Fig. 7.33a, right-hand side of diagram).

xa1 = s ØTr2 – s Ø1r2 + (s Ø1r 1 – s Øa 0 r 1) – ( DtEA – t JA ) xa1 = (10 – 102 – 4.82 ) – (10 – (102 – 4.32 )) + (8.9 – (8.92 – 4.32 ) ) – (8.9 – (8.92 – 3.92 )) – (0.24 – 0.15) Thus, xa1 = 1.2273 – 0.9717 + 1.1077 – 0.9000 – 0.09 (sags from XI ) = 0.3733 mm ra1 =

(((yØ 2 – ya 02 – xa12 ) (2∗ xa1 ))2 + yØT2 ) − − − − XXVIII T

ra1 = (((4.82 – 3.92 – 0.37332 ) (2∗ 0.3733))2 + 4.82 ) = 11.3643 mm The following calculation shows that this value for ra1 provides the desired value for tEA of 0.24 mm.

157.e26

SECTION 3  •  Instrumentation and Lens Design

The primary sag of the anterior surface,

J

y

sap = s Øa 0ra 0 + (s ØTra1 – s Øa 0ra1 )

D

= 1.1669 + (1.0634 – 0.6901) = 1.1669 + 0.3733 = 1.5402 mm

K

tEA = s p + tC – sap

øa1

= 1.4680 + 0.3121 – 1.5402 = 0.24 mm (using ’floating ’ decimal point). Fig. 7.33a (right-hand side) depicts a lenticular lens with a –ve carrier and Øa0 > Ø0. In Fig. 7.33b (left-hand side) the lens has a parallel carrier and Øa0 < Ø0. A lens with Øa0 = Ø0 is calculated with a parallel carrier (see p. 157.e3) and the use of a lens with a +ve carrier is described above. The following general formula applies to all three types of carrier and any combination of Øa0 and Ø0.

2

x A B0 B1 tc

N M

Ca1

C0

C1

Ca0

Fig. 7.34  (a) Exaggerated detail of a part of a positive lenticular lens showing the junction area. The distance JD along the radius of the first peripheral curve, centre of curvature C1, is the minimum thickness because it meets the back surface of the lens at a right angle.

sac = (s ØTrN − s ØT−1rN ) + (s ØT−1rN−1 − s ØT−2rN−1 ) + …(s Ø0r0 − s Øa 0p ) − (Dt EA – tJA)[ a ] − − − − XXXVIII where sac is the sag of the anterior surface of the carrier portion and is the value xa1 for Formula XXVIII: rN is the radius of the outermost back surface zone. rN–1 is the radius of the next back surface zone ‘inside’ (central to) rN. ØT = Total diameter. ØT–1 and ØT–2 represent respectively the diameter of the zone central to ØT and the diameter of the zone two zones in from ØT. sØa 0p is the primary sag at Øa0[b]. If sp is known, xa1 = sac = x – (DtEA – tJA)[a] where

In Fig. 7.34 (not to scale), the lens is turned on its side so that sags, etc, are measured along the horizontal x axis and vertical measurements along the y axis. The minimum thickness at the anterior junction is measured perpendicularly to the neighbouring back surface, in this case the first peripheral radius. tJM is represented by the distance JD on the radius from C1, the centre of curvature of r1. The x- coordinate of point J, xJ,is given by: s Øa1 = ra 0 − ra 02 − (Øa 0 2)2 + {(ra1 − ra12 − (Øa1 2)2 ) − (ra1 − ra12 − (Øa 0 2)2 )} while the y-coordinate, MJ = yJ is Øa1/2. The x-coordinate of C1 is given by:

if Øa 0[ b ] = Ø0, x = s p − s Ø0r0 (see Figs 7.27, 7.33) if Øa 0[ b ] < Ø0, x = s p − s Øa 0r0 [ b ] if Øa 0[ b ] > Ø0, x = s p − (s Ø0r0 + s Øa 0r 1[ b ] − s Ø0r 1 )

APPROXIMATION OF MINIMUM JUNCTION THICKNESS OF LENTICULAR LENSES CALCULATED FROM JUNCTION OF ra0 AND ra1 (OR ra1 AND ra2 IF THE LENS HAS A FRONT JUNCTION RADIUS) Formula XXXIX Method Example 27.  The –10.00D lenticular lens at the beginning of this Appendix has an Axial Junction thickness (tJA) of 0.15 mm and will be used to illustrate the difference between tJA and the Minimum Junction thickness (tJM). Because it has an anterior junction radius, and the diameter of the junction Øa1 is larger than the BOZD, the first back peripheral radius has to be taken into account. The calculations can be simplified for a positive-powered lenticular lens where there is not an anterior junction radius, and even more when Øa0 matches the BOZD. [a]

In all cases, if the carrier is to be parallel surfaced, the –(DtEA – tJA) term can be omitted as DtEA – tJA = 0. In all cases, if the lens has a Junction radius, the term Øa0 is replaced by ØaJ (the diameter of the anterior Junction, see Fig. 7.29).

[b]

AC1 = AB0 + B0 B1 + B1C1 = AB0 + B0N + NB1 + B1C1 = tc + sφ0r0 − sφ0r 1 + r1 The equation to the line C1J is given by: y=

yJ yJ ∗ (x − AC1 ) = ∗ (x − AC1 ) C1M s ØaJ − AC1

which may be written as y = ax + b (eq 1), where yJ − y J ∗ AC1 a= and b = s ØaJ − AC1 s ØaJ − AC1 The equation to the circle B1KD representing the first peripheral curve of the lens is given by:

(x − AC1 )2 + y2 = r12

(eq 2)

Substituting Eq. 1 into Eq. 2, and simplifying gives the coordinate of D, the intersection of the perpendicular from J to the back surface of the lens: xD =

−(a ∗ b − AC1 ) ± (a ∗ b − AC1 )2 − (1 + a 2 )(AC12 + b2 − r12 ) 1 + a2

The value of yD may be determined by substituting xD into either Eq. 1 or 2 above.

7  •  Optics and Lens Design

The minimum thickness is then obtained comparing the coordinates of J and D:

(

Minimum thickness = (xD − s Øa1 )2 + yD − Øa1 2

)

2

XXXIX

The -10.00 DS lens was of construction 8.00:7.50/ 8.90:8.60/10.00:9.60, tC = 0.1 mm, ϕa0 = 7.50, ϕa1 = 8.10, ra0 9.86 and ra1 5.63 mm. Hence, xJ, is: s Øa1 = ra 0 − ra 0 − (Øa 0 2) + {(ra1 − r 2

2

2 a1

− (Øa1 2) )

− (ra1 − ra12 − (Øa 0 2)2 )} = 9.86 − 9.86 − (7.5 2) + {(5.63 − 5.63 − (8.1 2) ) 2

2

2

the arithmetic mean of the reciprocals of the various thicknesses is required =

2

− (5.63 − 5.632 − (7.5 2)2 )} = 0.7409 + 1.7192 − 1.4307 = 1.0295 mm while yJ = 8.10/2 = 4.05 mm. AC1 = 0.1 + 0.9333 − 0.8286 + 8.90 = 9.1048 mm. Hence a = −0.50153 and b = 4.56631. This gives xD = 1.149228 and yD = 3.98994 mm. Subtracting these from the coordinates of J gives Δx = 0.11975 and Δy = −0.06006, giving the minimum junction thickness of 0.134 mm, compared with the axial junction thickness of 0.15 mm. If this calculation is programmed into a spreadsheet and no anterior junction radius is used, then set ϕa1 = ϕa0. The thickness to the curve formed by r0 can simultaneously be calculated, and the lower value for minimum thickness used – this should be to the back optic zone if the anterior junction diameter equals the BOZD, and to the first peripheral curve if the anterior diameter is greater than the BOZD. To calculate the distance to r0, the x-coordinate of C0 is simply AC0 = tC + r0, so the equations above can be amended by using the value for AC0 instead of AC1.

CALCULATION OF HARMONIC MEAN THICKNESS (tHM) Formulae XXXX and XXXXI The oxygen permeability of a plastic material is independent of thickness. A statement of this property is normally given as a Dk value where D is the diffusion coefficient and k is the solubility of oxygen. The transmissibility of a material is dependent mainly on thickness and is given as Dk/t where t is the thickness. Since most lenses vary in t, the Arithmetic Mean (AM) of a number of thicknesses (tAM), calculated at varying diameters across the lens, has been used for the value t. (t + t + … . t N ) tAM = 1 2 N Sammons (1981) stated that the flow of gas through a substance is related to the reciprocal of the thickness (1/t) so

(1 tC + 1 t1 + 1 t2 + … . 1 t N ) N +1

and, because this calculation results in a ‘mean reciprocal’, the reciprocal of this value =

2

157.e27

N +1 (1 tC + 1 t1 + 1 t2 + … . 1 t N )

is the ‘effective’ or Harmonic Mean Thickness (tHM). Thus, tHM is the reciprocal of the AM of the sum of the reciprocals of a series of thicknesses. For example, the AM of 1 + 2 + 3 + 4 + 5 = 15/5 = 3. The HM = 5 (1 1 + 1 2 + 1 3 + 1 4 + 1 5) = 5 2.283333 = 2.189781 tHM is always less than tAM. Therefore, since the t in Dk/t has been based traditionally on tAM, more oxygen has been supplied through contact lenses to corneas than had been supposed. When calculating the tHM of a lens, the smaller the intervals between successive calculations the more accurate the result, especially towards the periphery. There are many ways to measure the thickness of a contact lens: axially, radially normal to the back surface, radially normal to the front surface and the true thickness (measured by calipers or calculated), which is the shortest distance between the two surfaces at a given diameter. Axial thickness is easier to calculate involving simple sag relationships. ISO 18369:1 (2017) specifies that radial thickness should be used, where the thickness is measured along radii from the centre of the vertex sphere touching the back vertex of the lens, i.e. for lenses with spherical back optic zones, C0, the centre of curvature of r0. The harmonic mean thickness should be calculated from the lens centre outwards with N zones, each of equal area. As an example, a soft lens is taken with back surface specification 8.60 : 11.0 / 12.00 : 14.0, tC of 0.075 mm, BVP −7.50, and front surface lenticulated at FOZD of 9 mm. This gives ra0 = 10.3378 and ra1 = 9.9556. A procedure similar to that of the previous section on minimum junction thickness will be used to calculate the radial thicknesses. The first step is to calculate the zone radii or y coordinates. The total area of the lens in plan view, ignoring the effects of the curvature of the lens, is π(ϕT/2)2, so if the area is to subdivided into 15 parts, then the area of the central zone is given by π(ϕT/2)2/15 and the radius of the circle of this area is φT 2∗ 15 = 1.1807mm. The next zone is an annulus, having inner radius of 1.1807 mm, and an outer radius enclosing a total area of double the central zone, i.e. to φT 2∗ 2 15 = 2.5560 mm . The outer radius of the next zone includes 3 times the area, and so on till the last zone is of diameter 7 mm or ϕT/2. The second step is to calculate the equivalent x-coordinates. For the central optic zone, the standard sag formula XI is used, but remembering to add the central thickness of the

157.e28

SECTION 3  •  Instrumentation and Lens Design

lens since the origin of the coordinates is taken as the pole of the front surface. For the outer zone, x = (r1 − r12 − y2 ) + (sφ0r0 − sφ0r 1 ) + tC

A

søTr0

SP

F

The equation to the radius from C0 to the point xn, yn is given by:

lEA

G

yn ∗ (x − AC0 ) xn − AC0 which may be written as y = ax + b (eq 1), where yn a= and b = a ∗ AC0 xn − AC0

xan =

−(a ∗ b − ACa 0 ) −

IEA = FG = søTr0 – sp

and hence yan. The radial thickness is then given by: radial thickness = (xan − xn ) + ( yan − yn ) 2

r0 r0 – sP

(a ∗ b − ACa 0 )2 − (1 + a 2 )(ACa 02 + b2 − ra 02 ) 1 + a2

2

Fig. 7.35  Dimensions necessary for conversion of axial edge lift, lEA, to radial edge lift, lER, and vice versa

Table 7.14  Radial Thicknesses (tR) Normal to r0, and Their Reciprocals, Harmonic Mean Thickness No.

Sags (s) are found from Formula XI. The earlier text above gives a detailed analysis of various lens sagitta values. Figs 7.25, 7.26 and 7.35 should also be consulted. Radial edge lift (lER) is defined as the amount by which the distance between the centre of curvature of r0 and a point on the back surface of the lens at a stipulated diameter (CE in Fig. 7.35) exceeds r0 (DE in Fig. 7.35).

y = ϕ/2

tR

1/tR

Ø0

0 1 2 3 4 5 6 7 8 9

0 1.807392 2.556039 3.130495 3.614784 4.041452 4.427189 4.78191 5.112077 5.422177

0.075 0.1062271 0.13831918 0.17134016 0.20536222 0.24046734 0.31932106 0.30423422 0.3342285 0.3653829

13.33333333 9.413794056 7.229655604 5.836343371 4.869444906 4.158568846 3.131644412 3.286941277 2.991965096 2.736854924

Ø1

10 11 12 13 14 15

5.715476 5.994442 6.26099 6.516645 6.762642 7.00

0.34022453 0.29455992 0.24839023 0.20169122 0.15443655 0.1065975

2.93923549 3.39489504 4.025923274 4.958073938 6.475151103 9.381083183

Totals

3.6057826

88.16290785

Mean

tAM = 0.225

tHM = 0.181

From

tam =

CALCULATION OF AXIAL AND RADIAL EDGE LIFT OF A SPECIFIED LENS

where s p = s Ø0r0 + s Ø1r 1 − s Ø0r 1 + s Ø2r2 − s Ø1r2 + … s ØN rN − s ØN−1rN − − − − XXX XXII Ø N–1 = the diameter of the zone central to zone N.

r0 + lER

C

For the anterior peripheral curve, an allowance must be made for the difference in sags between the central and peripheral radii at the junction diameter. The results of the calculations for this example are given in Table 7.14. Since there is little diffusion of oxygen sideways under a soft contact lens, the harmonic mean thickness may be less important than the local thickness in evaluating the potential for hypoxia, whilst for corneal lenses, lens movement increases oxygenation.

Formulae XXXXII and XXXXIII The calculation of both axial and radial edge lifts depend upon Pythagoras’ theorem (CE2 = BC2 + BE2 in Fig. 7.35). Axial edge lift (lEA) is defined as the amount by which the sag of the back central optic zone radius (r0) of a lens, calculated for a stipulated diameter (ØN), exceeds the primary sag (sp) of the lens at the same diameter l EA = s ØN r0 − s ØN p

D

lER = DE = √(BC2 + BE2)– r0

y=

and hence the intercept, xan, with the central front curve evaluated from

E

y

B

3.6057826 16

tHM =

16 88.16290785

Values for a soft lens with back surface specification 8.60: 11.0 / 12.00: 14.0, tC of 0.075 mm, BVP −7.50, and front surface lenticulated at FOZD (Øa0) of 9 mm.

lER = (BC 2 + BE2 ) − r0; where BC = r0 − s P; BE = y = the chosen diameter 2 lER = ((r0 − s p )2 + y2 ) − r0 − − − − − XXXXIII Example 28 (for the C4 Lens From Example 23) 7.98:7.5/8.17:8.5/9.18:9.3/11.54:9.8

7  •  Optics and Lens Design

An approximate conversion of axial to radial edge lift may be obtained from:

s ØT r0 = 7.98 − (7.892 − 4.92 ) = 7.98 − (63.6804 − 24.01)

lER ≈ l EA ∗ (1 − (Ø2 (8 ∗ r02 ))) − − − − XXXXV

= 7.98 − (39.6704) = 7.98 − 6.2984

where Ø = Total diameter

s ØT r0 = 1.6816 mm (from XI ) s p = 7.98 −

(7.982 − 3.752 )

) ( )} {( + {(9.18 − (9.18 − 4.65 )) − (9.18 − (9.18 − 4.25 ) )} + {(11.54 − (11.54 − 4.9 ) ) − (11.54 − (11.54 − 4.25 ) )} + 8.17 − (8.172 − 4.252 ) − 8.17 − (8.172 − 3.752 ) 2

2

2

2

2

2

2

= 0.936 + (1.1924 − 0.9115) + (1.2648 − 1.0431) + (1.0920 − 0.9783) (The brackets in the line above are not essential and are included for clarity only) = 0.936 + 0.2810 + 0.2218 + 0.1136 = 1.5524 s p = 1.5524 mm (from XXXXII ) lEA = SØTr0 − s p = 1.6816 − 1.5524 = 0.1292 lEA = 0.1292 mm lER = ((r0 − s p )2 + y2 ) − r0 = ((7.98 − 1.5524)2 + 4.92 ) − 7.98 = (6.4276 + 4.9 ) − 7.98 2

157.e29

2

= (41.3141 + 24.01) − 7.98 = (65.3241) − 7.98 = 8.0823 − 7.98 = 0.1023 mm (fro om XXXXIII)

CONVERSION OF AXIAL TO RADIAL AND RADIAL TO AXIAL EDGE LIFTS Formulae XXXXIV, XXXXV, XXXXVI and XXXXVII Conversion of axial to radial and radial to axial edge lifts is enabled by transposing the above formulae lER = r02 + l EA 2 + ((2l EA ) r02 − y2 )) − r0 − − − − XXXXIV or lER = (r02 + l EA 2 + 2 ∗ l EA ∗ (r02 − y2 )0.5 )0.5 − r0 − − − − XXXXIV Example 29 (Using the C4 Lens From Example 23) lER = (7.982 + 0.12922 + 2 ∗ 0.1292 ∗ (7.982 − 4.92 )) − 7.98 = (63.6804 + 0.0167 + 0.2583 ∗ (63.6804 − 24.01)) − 7.98 = (63.6971 + 0.2583 ∗ (39.6704)) − 7.98 = (63.6971 + 0.2583 ∗ 6.2984) − 7.98 = (63.6971 + 1.6270) − 7.98 = (65.3241) − 7.98 = 8.0823 − 7.98 = 0.1023 lER = 0.1023 mm (from XXXXIV)

2

lER ≈ 0.1292∗ (1 − 9.82 (8 ∗7.982 )) = 0.1292 ∗ (1 − 96.04 (8 ∗ 63.6804)) = 0.1292∗ (1 − 96.04 509.4432) = 0.1292∗ (1 − 0.1885) = 0.1292∗ 0.8115 ≈ 0..1048 mm (from XXXV ) Accurate value = 0.1023 mm

Conversion from radial to axial lift: accurate conversion. lEA = ((r0 + l ER )2 − y2 ) − (r02 − y2 ) − − − − XXXXVI lEA = ((7.98 + 0.1023)2 − 4..92 ) − (7.982 − 4.92 ) = (65.3236) − 24.01) − (63.6804 − 24.01) = (41.3136) − (39.6704) lEA = 6.4276 − 6.2984 = 0.1291 mm (from XXXXVI ) Approximate conversion: lEA ≈ l ER ∗ (1 + (Ø2 (8 ∗ r02 ))) − − − − XXXXVII For calculators: lEA ≈ l ER ∗ (1 + (Ø2 (8 ∗ r02 ))) − − − − XXXXVII where Ø = Total diameter lEA = 0.1023 ∗ (1 + 9.82 (8 ∗7.982 )) = 0.1023 ∗ (1 + 96.04 / (8 ∗ 63.6804)) = 0.1023 ∗ (1 + 96.04 / 509.4432) = 0.1023 ∗ (1 + 0.1885) = 0.1023 ∗1.1885 ≈ 0.1216 mm (from XXXXVII ) Accurate value = 0.1292 mm

CALCULATION OF AXIAL AND RADIAL EDGE CLEARANCE OF A SPECIFIED LENS Formula XXXXVIII The axial clearance CA between a point on the back surface of a rigid lens centred on an aspheric cornea may be calculated using equation XXXIV, as shown in examples 21 to 23. The radial distance to the aspheric cornea is more difficult to calculate because the point N at which the normal to the cornea passes through the edge of the lens cannot easily be determined. Burek and Douthwaite (1993) derived both approximate and accurate methods to convert axial to radial clearance or vice-versa. The approximate method is as follows. Fig. 7.36a shows a cross section through an aspheric cornea, assumed to be an ellipse, with its apex at the origin of the x-y axes, P is a point on the back surface of the lens (not necessarily the edge), Q a point on the cornea at the same semi-diameter as

157.e30

SECTION 3  •  Instrumentation and Lens Design

Hence

(a) Q

P

 7.96992 − 0.8 ∗ 4.92  C R = 0 .1 ∗  2 2 7.9699 − (0.8 − 1.0) ∗ 4.9 

N

 63.519 − 19.208  = 0. 1 ∗   63.519 − ( −0.2) ∗ 24.01 

yp

44.311  = 0.1 ∗   68.321  = 0.08 mm.

0 .5

0.5

0 .5

= 0.1∗ 0.6486

To obtain the axial clearance from the radial value, the rearranged formula can be used:

(b) CA

P

Q

0 .5

XXXXIX

Burek and Douthwaite (1993) also give an iterative approach to a more accurate conversion to or from axial clearance – today, the “goal seek” function on a spreadsheet would also be appropriate. From the practical point of view, the approximate expression is probably sufficiently precise, and was used by Rabbetts (1993) in spreadsheet programs for calculating the edge lift or back surface geometry of contact lenses.

CR N’ N

Fig. 7.36  Approximate conversion of axial clearance, CA (= PQ), to the cornea from a point P on the back surface of a contact lens to radial clearance, CR (= PN), and vice versa. b) shows an enlarged view of the clearance area, with the line QN’ being the tangent to the ellipsoidal cornea at Q.

P, and N is the foot of the normal to the cornea. Fig. 7.36b shows the region around PQN in greater detail. The slope of the tangent to the ellipse at Q is given by differentiating the equation for the ellipse: slope of tangent =

(r0 K 2 − pyP2 )0.5 yP

where r0K is the apical radius of the ellipse and yP is the semidiameter at the point P. As an approximation, the line PN’ is dropped to meet the tangent perpendicularly at N’. If the distances are small, then the distance PN’ ≈ PN =CR, the radial clearance. From the triangle PQN’, the angle PQN’ is given by CR (C A 2 − C R 2 )0.5 and combining these two equations and simplifying gives ˆ ’= tan PQN

 r 2 − p ∗ yP 2  C R = C A ∗  20 K 2  r0 K − ( p − 1) ∗ yP 

 r 2 − ( p − 1) ∗ yP2  CA = CR ∗  0K 2 2   r0 K − p ∗ yP 

0.5

XXXXVIII

In example 21, p = 0.8. r0K = 7.9699 mm, the design axial edge clearance CA was 0.1 mm and the total diameter was 9.80 mm.

Archived From the Printed Chapter in the Previous Edition BENNETT’S WORK ON FLEXURE This was briefly mentioned on page 152 of the printed text. The calculations needed to demonstrate the effects of flexure on lens power for Fig. 7.19 were based on the following method and example, as described by Bennett (1976). Figs 7.37 and 7.38 illustrate this flexure on a positive and a negative lens, respectively. Example.  In order to assist calculation, a positive lens of knife-edge form is illustrated, but the same arguments can be applied to a lens of any specified edge thickness. A lens has BOZR 9.50 mm, TD 14.50 mm, BVP +6.00D, n = 1.44. This lens must have tc 0.616 mm and FOZR 8.599 mm to give a knife edge (Jalie 1988) (Fig. 7.37). The volume of this lens can be calculated, as explained by Bennett (1976), by subtracting the volume of the spherical cap bounded by the back surface from the volume of the spherical cap bounded by the front surface. The volume of a spherical cap = (π/3)(3r – s)s2 where r = radius and s = sag. For the unstressed lens defined above, the volumes are thus: Front spherical cap = 361.113 mm3 Back spherical cap = 297.379 mm3 Volume of lens, thus = 63.734 mm3 If this lens is applied to a spherical cornea of radius 8.00 mm, the semi-diameter y (7.25 mm) is reduced to some new value y′, and since the BOZR is assumed to equal the corneal radius, the sag s2′ of the new back surface can be calculated, as therefore can the volume. Because (for a knifeedge lens), s1′ = s2′ + tc and the semi-diameter y′ is common

7  •  Optics and Lens Design

157.e31

Fig. 7.37  Steepening of a positive soft lens with knife edge: solid line before steepening and broken line after steepening. Before steepening: edge E, centres of curvature of front and back surfaces C1 and C2 with their radii of curvature r1 and r2 and sagitta s1 and s2 respectively, and semi-diameter y. After steepening these become: E′, C1′, C2′, r1′, r2′, s1′, s2′ and y′, respectively. Centre thickness tc is the same before and after steepening

Fig. 7.38  Steepening of a negative soft lens: solid line before steepening and broken line after steepening. Before steepening: front and back surface edges E1 and E2 with semi-diameters y1 and y2 intersecting the primary axis at J and K and subtending an angle θ at C2. After steepening these become: E1′, E2′, y1′, y2′, J′, K′ and θ′ at C2′, respectively. Other symbols are as in Fig. 7.37.

to both front and back surfaces (Fig. 7.37), the new front surface radius r1′ is obtainable from the expression relating radius to sag and semi-diameter, namely: r=

y2 + s 2 (see Formula XVII) 2s

The volume of the new front surface spherical cap can also be calculated. The value of y′ must be found iteratively, the correct value being the one which gives the flexed lens the same volume as the unstressed lens. In this example: y′ = 6.9 mm, r1′ = 7.496 mm, BVP = +5.21D, showing a power change of –0.79D.

After steepening, the volumes become: Front spherical cap = 391.951 mm3 Back spherical cap = 328.217 mm3 Flexed lens (= that of unstressed lens) = 63.734 mm3 From the above it can be seen that whereas the BOZR has shortened by 1.5 mm (from 9.5 to 8.0 mm) the FOZR has shortened by only 1.103 mm. If both radii had shortened by 1.5 mm the volume of the lens would have increased to: Front spherical cap = 460.939 mm3 Back spherical cap = 328.217 mm3 Volume of lens, thus = 132.722 mm3(!) and the centre thickness would have increased to 1.479 mm!

157.e32

SECTION 3  •  Instrumentation and Lens Design

If the front surface radius steepened more than the back surface, both volume and centre thickness would increase to an even more ridiculous extent. Because the front surface radius of a positive contact lens is shorter than that of the back surface, the front surface radius must alter by substantially less than the BOZR if the volume and thickness of the lens are to remain the same. Negative lenses are treated in much the same way, except that edge thickness must be taken into account, and so it is necessary for the convenience of calculation to make one further assumption. It is assumed that the lens has a conical edge, with the apex of the cone at C2 (see Fig. 7.38). The edge E1E2 is therefore normal to the back surface and it is assumed that E1E2 and its relationship to the back surface remain unchanged after flexure. The volume of the lens is the volume of the front spherical cap plus the volume of the frustum of the cone bounded by E1E2, minus the volume of the back spherical cap. (Volume of frustum of cone = (π/3)([y12 × JC2] – [y22 × KC2]); Bennett 1976.) Although highly unlikely to become flattened, hydrophilic lenses acquire additional positive power when in this state. The change of power due to flexure is given by: ∆F ′ V =

1 −(n − 1)  1 t − 2 2 n  (r2′) r2 

(n = refractive index and other symbols are as in Fig. 7.37.).

FUSED BIFOCALS Formulae VII–X should be consulted in connection with this section. In the formulae rCS ≡ rBCS or rFCS (depending on whether the fused segment is a back surface or front surface segment) while the alternative symbols, based on BS EN ISO 18369-1:2017, are used. Thus r1 ≡ ra0, r2 ≡ r0 and subscripts for powers F are similarly denoted. The contact surface FCS must have a positive power (FBCS) with a back surface fused segment (Fig. 7.32) and a negative power (FFCS) when the segment is on the front surface. Since the back surface is a negative surface and the segment has the higher refractive index, there is actually a gain in negative power at the back surface of 1.490 − 1.560 r2 and where r2 is in millimetres this expression becomes −70 D r2 For example, if the BOZR (r2) is 8.00 mm, this gives a power of –70/8 = –8.75D. Alternatively, the back surface powers in tears are: F2DP =

(1.336 − 1.490)1000 D = −19.25 D 8.00

(see Formula III) F2NP =

(1.336 − 1.560)1000 D = −28.00 D 8.00

The difference is F2NP – F2DP = –8.75D as before. The power difference due to the segment is the same whether the power is determined in air or in tears, because the BOZR, (r2), is the same throughout (as distinct from the back surface solid bifocal where r2DP < r2NP). This is easily shown using the same value for r2 as above. The back surface powers in air are (1.490)1000 D = −61.25 D 8.00 (1 − 1.560)1000 D = −70.00 D = 8.00

F2DP = F2NP

The difference, F2NP – F2DP = –8.75D. This is exactly the same as when the back surface powers were determined in tears (see above). Because the back surface addition is the same measured in air as in tears, the addition read on a focimeter is the same as that on the eye. Since the fused segment creates a gain in negative power on the back surface, it needs a gain in positive power at the contact surface, sufficient to overcome the negative gain as well as to provide the near addition, i.e. if the power of the contact surface is FCS then for the refractive indices already given 70  70  FCS = Addition −  −  = Addition +  r2  r2 Now FCS (which is convex for the medium of higher refractive index) =

1.560 − 1.490 70 D= D (where rCS is in mm) rCS rCS And since FCS = Addition +

70 r2

(see Formula IX) rCS =

70 Add +

70 r2

(where the addition is in dioptres, and r2 and rCS are in millimetres). Where the segment is fused on the front surface, Formulae VIII and X apply (see Example 13). The above example illustrates a PMMA fused bifocal but RGP bifocal lenses have been available with similar straighttop segments (see Chapter 14).

References Bennett, A.G., 1976. Power changes in soft contact lenses due to bending. Ophthal. Optician 16, 939–945. Bennett, A.G., 1985. Optics of Contact Lenses, fifth ed. Association of Dispensing Opticians, London. Bennett, A.G., 1988. Aspherical and continuous curve contact lenses: Parts 1–4. Optom. Today 28, 11–14, 140–142, 238–242, 433–444. BS EN ISO 18369-1 (2017) Ophthalmic optics – Contact lenses – Part 1: Terminology. BS EN ISO 18369-2 (2017) Ophthalmic optics – Contact lenses – Part 2: Tolerances. Burek, H., Douthwaite, W.A., 1993. Axial-radial interconversion. J. Br. Contact Lens Assoc. 16 (1), 5–13.

7  •  Optics and Lens Design Campbell, C., 1987. A method for calculating the tear volume between the cornea and a hard contact lens with a spherical base curve. J. Br. Contact Lens Assoc. 10 (1), 29–35. Douthwaite, W.A., 2006. Contact Lens Optics and Lens Design, third ed. Elsevier Butterworth-Heinemann, Oxford. Jalie, M., 1988. The Principles of Ophthalmic Lenses, fourth ed. Association of British Dispensing Opticians, London, pp. 334–336. Jalie, M., 2016. The Principles of Ophthalmic Lenses, fifth ed. ABDO, London, pp. 278–279. Lehman, S.P., 1967. Corneal areas used in keratometry. Optician 154, 261–264.

157.e33

Mackie, I.A., 1973. Design compensation in corneal lens fitting. In: Symposium on Contact Lenses, Transactions of the New Orleans Academy of Ophthalmology. Mosby, St Louis. Rabbetts, R.B., 1993. Spreadsheet programs for contact lens back surface geometry. J. Br. Contact Lens Assoc. 16, 129–133. Sammons, W.A., 1981. Thin lens design and average thickness. J. Br. Contact Lens Assoc. 4 (3), 90–97. Young, G., 1988. Computer-assisted contact lens design. Optician 196 (5171), 32–33, 37, 39.