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New matrix formulation of spectacle magnification using pupil magnification. I. High myopia corrected with ophthalmic lenses
Copyright
Ophthal. Physiol. Opt. Vol. 15, No. 3, pp. 195-205, 1995 0 1995 Elsevier Science Ltd for British College of Optometrists Printed in Great Britain. All rights reserved 0275-5408195 $10.00 + 0.00
New matrix formulation of spectacle magnification using pupil magnification. I. High myopia corrected with ophthalmic M. Garcia,
C. Gorhlez
lenses
and I. Pascual
Laboratory of Optics, Department Alicante E 03080, Spain
of Optics,
University
of Alicante,
Apdo.
99,
Summary In this paper we have applied the matrix method to calculating spectacle magnification (SM) and relative spectacle magnification (RSM) for myopic eyes corrected with ophthalmic lenses (OLs). We have been able to obtain very simple expressions of the magnifications and analogous values to the ones obtained using other geometric methods. Through the paraxial matrix for spherical systems, we can calculate the correcting lens pupil 2 x 2 object-image matrix (M,) and the corrected ametropic eye pupil matrix (MC,). These two matrices, that is M, and M,,, directly provide the lateral magnification and the lens-eye system’s image focal parameters that enable us to easily calculate SM and RSM. The magnifications have been calculated for the case of myopic eyes corrected with OLs, which shows the validity of this method since its results are similar to the ones obtained using other methods. The analysis of other correcting systems such as contact intraocular lenses will be presented in two associated papers (to be published as parts II and III of this review). Ophthal.
Physiol.
Opt.
1995,
15,
195-205
The resurgence of activity in applied optics in the second half cf the twentieth century has caused this period to be label!ed an optical renaissance. Part of the renaissance has consisted of a steadily growing appreciation of the advantages of matrix methods applied to geometric optics analysis, which encompasses both the study of spherical optical systems’.2 including those of ophthalmic interest3, and the analysis of astigmatic optical systems4-’ and of different problems associated with these systems’-“. By this method, it is possible to simplify many tedious ray tracing calculations and provide an improved conceptual framework for analysing the different situations. This matrix method has already been used to calculate the spectacle magnification (SM) of astigmatic eyes”. However. the matricial relationship obtained was complicated due to the definition of the SM (SM was defined as the product of the power factor and the shape factor). In this paper we apply the matrix method to the calculation of spectacle magnification using a different definition of this
parameter. Like Bennett and Rabbettsi2, we defined SM as the ratio of a corrected eye retinal image size (y’,,) to an uncorrected (y’,,) one. Consequently, it analyses the change in the retinal image size in any given eye as a result of wearing some type of correcting system (in this case ophthalmic lenses). For the blurred image case, such as myopic eyes for distance vision, the image size can be determined by the limiting ray through the centre of the eye’s entrance pupil (E) or the eye’s exit pupil (E’). Thus, SM can be defined by the lateral pupil magnification as:
P ca
where /I,, is the lateral magnification of the uncorrected ametropic eye between the entrance pupil (E,,) and the exit pupil (E’,,,); E,, is the image of the iris formed by the cornea and EL, the image of the iris formed by the crystalline. When we consider the eye corrected with an ophthalmic lens, this lens forms an image of the uncorrected ametropic eye’s entrance pupil. This second image of the iris becomes
Received: 3 June 1994 Revised form: 5 October 1994
195
196
Ophthal. Physiol. Opt. 1995 15: No 3
the entrance pupil of the lens-eye system, i.e. of the corrected ametropic eye (EC,). Therefore &, is the lateral magnification of the corrected ametropic eye defined by the new corrected ametropic entrance pupil (EC,) and the corrected ametropic exit pupil (E’,,) (this E’,, coincides with EL, since both are the image of the iris formed by the crystalline). Since the corrected ametropic eye can be considered as an association of two subsystems, the correcting lens and the uncorrected ametropic eye, then:
PC,= PLP”,
optical system. This is due to the effect of refraction, translation and/or reflection. In the case of the eye and correcting lens-eye systems, we are only considering the translation and refraction effects due to different surfaces of the systems. For this reason, we only show the form of the translation and refraction matrix. The 2 X 2 refraction matrix, R, for a spherical surface is:
(5)
(2)
where & is the lateral magnification of the lens between the corrected ametropic eye entrance pupil (EC,) and the uncorrected ametropic eye entrance pupil (E,,); hence the SM can be determined by the following equation’*: SM=$ L
In this article we have applied the matrix theory to this new definition of the SM. Thus we obtain a formula that allows us to easily determine the values of the SM in ametropia specific cases. Another parameter that we have also studied matricially is relative spectacle magnification (RSM), which is defined as the ratio of the retinal image size in a corrected ametropic eye to that in a standard emmetropic schematic eye13. It is calculated as the division of the image focal length of a corrected ametropic eye (f’,,) by the image focal length of the standard emmetropic eye (f’,):
RSM = f’ca .f’e Once the expressions corresponding to the SM and the RSM had been obtained matricially, we applied them to 12 specific theoretical cases of high myopia which were corrected with ophthalmic lenses (OLs). These 12 cases consisted of eyes with respective axial length ranges of 27-33mm and with respective cornea1 power ranges of 42-48D. These cover almost the entire range of possible cases of high myopia. We will study SM and RSM together. This enables us to obtain the values of the retinal image sizes of the corrected ametropic eye (CA) and of the uncorrected ametropic eye (UA). Knowing this information is very interesting since it allows the study of the aniseikonia (inequality of image sizes) that provides anomalies in binocular vision14. Before doing the matrix calculations we will give a brief review of the matrix notation used in this paper.
where f’ is the image focal of the surface expressed in metres, II is the refractive index before the refractive surface, and n ’ is the refractive index behind it. The image focal of surface f’ is given by the relationship: P = n ‘If’ where P is the optical power of the surface and can also be calculated as: P = (n’ - n) /r with r being the radius of curvature. The 2 x 2 translation matrix, T, for the distance t expressed in metres is:
(6) In this paper, we use the convention that the light travels from left to right. Furthermore, the distances measured in the same direction as that in which the light is travelling are considered as positive in sign; if in the opposite direction, as negative. The system matrix is the product of the appropriate refraction and translation matrices in the correct order (right to left)*. For example, the system matrix M for a thick spherical lens is: M=R,TR,
(7)
where R, and R, are the refraction matrices for the first and second surfaces, respectively, and T is the translation matrix for the central thickness. In general, the elements of the 2 x 2 system matrix can be labelled: M=
L1 “, ;
(8)
The 2 x 2 system matrix, M, relates the parameters of a ray incident on the front of the system to the parameters of that ray as it exits at the back of the system:
Review of matrix notation The light path is modified when the light passes through an
(9)
Matrix
formulation
of spectacle
where 01is the angle of the incident ray and h is the height above the optical axis where the light hits the front of the system. The primed parameters are for the ray exiting at the back of the system. The geometry of the ray parameters is shown in Figure 1. Two conjugate points, object and image, are related matricially by the object-image matrix’ (M,,) which can be calculated as follows: M,, = TiMTo
(10)
where T, is the translation matrix from the object plane to the front vertex of the system, M is the system matrix from front to back vertex, and T, is the translation matrix from the back vertex of the system to the image plane; so that.
Mo, = The elements a,,, bo,, co, and d,, of this object-image matrix have a specific meaning: the coefficient b,, must be zero since the matrix has been calculated between conjugate points; this condition enables us to calculate the position of two conjugate points; consequently, the element a,, is the lateral magnification /3 defined between the object and image conjugate planes; co, is, in this case, the inverse of the image focal of the system l/f ‘, and d,,, is the angular magnification 6 between the conjugate planes given. Therefore the object-image matrix has the general form:
magnification.
1. High
n’
u
Sl
n II
S’2
Figure 1. A paraxial ray travelling from S, to S, in a spherical optical system: t is the distance between the anterior and posterior surface of the system; 01 and a’ are the respective incident and emergent angles of the ray; h and h’ are the height of the ray that is incident to and emergent from the system; n, n”, n’ are the refractive indices of the anterior, posterior and system medium, respectively.
M. Garcia
et al.
197
(12)
Application
of the matrix method
Given the definitions of the SM and the RSM (Equations (3) and (4)), we need to determine the lateral magnification of the correcting lens (&) and the image focal of the lens-eye system (f&) in order to calculate these magnifications. By comparing Equations (11) and (12), one can note that PL and f ‘,, are respectively related to the elements a,, and co, of the object image (O-I) matrices which we are going to calculate in this paper. Since the origins of measurement considered in the definitions of the SM and RSM are the entrance pupil (E) and the exit pupil (E’), then those O-I matrices will be calculated between both conjugate points; for that reason, the matrices are labelled pupil matrices. Therefore in order to determine /3, and f ‘,, we need to respectively calculate the correcting lens pupil matrix (ML) and the lens-eye system pupil matrix (M,,). The correcting lens pupil matrix is determined through the uncorrected ametropic eye pupil matrix (M,,), the ophthalmic lens power (OL) and the corrected ametropic eye pupil matrix, that is, the lens-eye system pupil matrix (M,,). The cases of ametropia studied consist of myopic eyes with different cornea1 powers (P,) and axial lengths (L). All the other data used for these calculations will be taken from the Gullstrand-Emsley theoretical eye”. For the correction of the ametropia, we have considered an OL with a refractive index of 1.523 and a central thickness of 1 mm at a cornea1 vertex distance of 12 mm. The working or focusing distance is always taken as infinity. Uncorrected ametropic
n
myopia:
eye pupil
matrix
The uncorrected ametropic eye pupil matrix (M,,) is the matrix defined between the entrance pupil E,, and the exit pupil EL, of the uncorrected ametropic eye. Hence, to calculate M,, we must know the position of these two pupils. The position of the exit pupil EL, is fixed, it does not change with the ametropia here given (axial ametropia and refractive ametropia due to increase of the cornea1 power) and it has a value of 3.69 mm measured from the cornea”. The position of the entrance pupil E,, must be calculated for each ametropia since it depends on cornea1 power (P,). We calculate the object-image matrix M,, between the plane of the entrance pupil and the plane of the real pupil (iris) in order to determine E,,. Following the definition of the surfaces and distances in Figure 2, we calculate M,,:
M,, = TXR, To
(13)
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From Equation (18), we can obtain the position of the uncorrected ametropic eye entrance pupil measured from the cornea (to):
,ci ‘1
n
n O
to
2
=
to
0.0027 0.0027 P, - 1
(19)
When the positions of E,,, and EL, are known, matrix M,, can be determined by the following equation (Figure 3): Mu, = T, MC, T, R, To Figure 2. The cornea refracting surface and the real pupil (iris) of a myopic eye. Refraction indices, radii of curvature and distance to from entrance pupil of the uncorrected ametropic eye (E,,) to the cornea and t2 from the cornea to the real pupil (iris Es).
where T, is the translation matrix for the distance t,, from entrance pupil E,, to the cornea, R, is the refraction matrix for the cornea, and T, is the translation matrix for the distance t, from cornea to the real pupil (iris), i.e. the translation along the depth of the anterior chamber of the eye. In matrix R,, f ‘, is going to depend on cornea1 power (P,) since P, = n2/f‘,. These matrices have the following forms:
I 1
To=
' 0
-7
I--! 1
R,=
(14)
0
1
no
f'l
%
T2=[;
-p]
=
P L 1.3333 :
= [;
0.7500
where To, R, and T2 are the matrices given by Equations (14), (15) and (16), respectively, McT is the matrix for the crystalline lens, and T, is the translation matrix for the distance ts from the crystalline lens back vertex to the exit pupil E,, that has the following form: TX 6
[ 1 ’ 0
-t6
1
1
0
1
n2
I--if'3
n4
R, =
(15) I 1
I
(21)
In the calculation of t,, we have taken into account the fact that the light travels from left to right. Therefore: t6 = 0.0037 - (t2 + t,). According to Figure 3, we need the following matrices to calculate M,,:
T=4
0
1
(20)
'
0
-b
1
(22)
R, =
-p”“‘“]
(16)
Therefore the matrix M,, is: (17)
M,, = r
r
l-~ 1.3333 P, 0.0036
I
PI
0.0036
“0
p, ~1.3333 to - 0.7500
p,
- to 1
0.7500 - ~ to 1.3333
1.3333
Comparing Equation (17) with the O-I matrix general form given by Equation (12) one can see that:
0.0036
PI
~ to - 0.7500 1.3333
1to -
= 0
(18)
Figure 3. Surfaces indices, distances schematic eye.
of the considered myopic eye. Refractive and thickness of the Gullstrand-Emsley
Matrix
formulation
of spectacle
1 5.8404
=
i lk.3352
FO62Oj [:
0.9790 0 0.9416 I = [ 16.3206
1. High
myopia:
M. Garcia
et al.
199
the O-I matrix (MrR) between the far point and the retina (R), since these points are conjugate points, so that:
The crystalline lens is obtained by the product:
M,, = R,T,R,
magnification.
-Foolh]
-0.0034 0.9650
(23)
The elements of the matrix, M,,, are:
W, = Tm MuaT,, where T,, is the translation matrix for the distance rE from the far point to the UA entrance pupil E,,, M,, is the UA pupil matrix expressed by Equation (24), and T,., is the translation matrix for the distance E’R from the exit pupil EL, to the retina. These matrices become:
(24)
(32)
where: (33) a“Cl= 1.0362 - 0.0028 P,
(25)
bu,= 9 p”;o;;;3,25
(26)
where E’R is calculated as: I
c Ua=
d,, =
+ (2.5730 x 10-9) = 0
0.6797 P, + 16.3206 2412.41 - (6.2514 x lo-‘) 3333.25 - 9 P,
E’R = L - 0.0037 (27) (28)
(34)
L is the different axial lengths that we consider and 0.0037 m is the standard distance” from the cornea1 vertex to the exit pupil Ek,. Multiplying the three matrices in Equation (3 1) we obtain:
All these matrices are obtained assuming the necessary data and model of the Gullstrand-Emsley eye. The optical power of the correcting lens To calculate the back vertex power (F \) of the ophthalmic lens (OL), we apply the principle of correction that says that an eye is corrected for distance vision by a lens with its second principal focus coinciding with the eye’s far point”. If we take the pupils as origins of measurement, this principle can be written as:
Since this matrix M,, is defined between conjugate points, element b of this matrix must be zero and this enables us to obtain the position of the far point (rE). If instead of a“a, c,, and A,, we use their corresponding values given by Equations (25), (27) and (28), we obtain rE: rE=
(36) 0.001 - (3.253 X 10m? P. + 201.040 E'R - (8.327 x 10m")P. E'R
FL=
K
287.850 - 1.555 P, + 0.002P: - 4533.500E'R - 176.558P,E'R + 0.510P,'E'R
1 + t,, K
where t,, is the distance from the OL back vertex to the entrance pupil E,, which is calculated by t,, = t,, - t, with t,, being the distance from the OL back vertex to the cornea (12 mm); t,, is a negative distance expressed by Equation (19) and K is the pupil refraction of the uncorrected ametropic eye (UA) which is calculated as:
where rE is the distance from far point (r) to the UA entrance pupil (E,,). Hence, first we must determine the position of the far point. For that reason, we consider
Once rE has been calculated, it is possible to find F \, the back vertex power using Equations (29) and (30). In subsequent calculations, we will need to know the individual powers of each of the lens surfaces. They are usually calculated by predetermining the front surface power (P,) of the lens, leaving the back surface power (P,) to be calculated. In the references used here, we have found the same tables of the values assigned to the front surface power as published by Jale16and by Bennett and Rabbetts12. Table 1 shows the range of typical values that are usually assigned to the front surface (P,) as depending on the back vertex powers. We used this table because it is a generalization of the aforementioned tables.
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Physioi.
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Table 1. Values assigned to the front surface powers of the lens (P,) depending on the back vertex powers (Fk)
F; 0)
Pa(0)
-0.25 to -6.00 -6.25 to -8.00 > -8.00
+ 6.00 + 3.00 + 1.00
(40)
When F \ and P, are known, the back surface power of the lens can be determined from15: P,=F;-
IL
P,
where t, has been calculated as tf = tb -f ‘, with t, being the central thickness of the OL (1 mm), and fb the image focal length of the front surface of the lens.
where llf: = P,ln,, with f: and P, being the image focal and the power of the back surface of the OL, and n, the refractive index of the air (no = 1).
(37) (41)
where t, is the central thickness of the OL and nb is the refractive index of the lens. Since we are applying the matrix method, we can also apply it in this case. Therefore we have to consider the O-I matrix between the image focal point of the front surface cf’,) of the OL and its conjugate point through the back surface of the lens (F \) that is shown in Figure 4: M,> = T,. R, T,
the three matrices in Equation (3X), we
M,+$
(38)
where T, is the translation matrix for the distance tr from image focal point of the front surface of the lens (F ‘,) to the back vertex of the OL. R, is the refraction matrix for the back surface of the lens and Tfr is the translation matrix for the distance (trJ from the “back vertex of the OL to the image focal point of the correcting lens (t,,, = l/F \). These matrices are expressed by:
TX F I 0’ -lf1 1
By multiplying obtain:
(39)
tf[?-lj -?I
(42)
1.523 - t,PC Since this matrix is defined between conjugate points, element b must be zero and this condition enables us to solve the P,. Once tf has been substituted, the expression of P, is given by: PC= F; +
1.523 0.001 +f’,
(43)
which depends only on the back vertex power of the OL and on the image focal of the front surface of the OL. The value off ‘, is obtained from Table I taking into account that P, = qlf',. Ametropic eye pupil matrix corrected with ophthalmic lens
[
Figure 4. Parameters needed power of the lens. Surfaces, ,indices of a lens.
to calculate the back surface focal lengths and refractive
The pupil matrix (M,,) of an ametropic eye corrected with an OL, is defined between the entrance pupil (EC,) of the corrected ametropic eye and the exit pupil (E’,,) of the corrected ametropic eye (CA). As we said before, the entrance pupil EC, of a corrected ametropic eye is the image of the entrance pupil of an uncorrected ametropic eye formed by the OL”. This entrance pupil will be calculated from M,,. To calculate M,,, we assume that the entrance pupil Eta and the exit pupil EL, are conjugate points through the lens-eye system. We find that (Figure 5):
Ma = MuaTuaRcT, RaTea
(44)
Matrix
formulation
of spectacle
Figure 5. Surfaces of the myopic eye corrected indices and thickness of the Gullstrand-Emsley
T,, is the translation matrix for the distance t,, from the entrance pupil EC,to the front vertex of the lens. R, is the refraction matrix for the front surface of the OL, T, is the translation matrix for the central thickness of the OL, R, is the refraction matrix for the back surface of the OL expressed by Equation (40), T,, is the translation matrix for the distance t,, from the posterior vertex of the OL to the entrance pupil E,,. which has been calculated as: t,, = (T,, - to) with t,, = 12mm, and M,, being the pupil matrix of the UA given by Equation (24). These matrices are as follows:
(45)
magnification.
1. High myopia: M. Garcia et al.
with an ophthalmic schematic eye. a ca=
201
lens. Refractive
4 x 10-6{250t0a,,(1523.f~ + looof; x lO’f’,a,, - 4819a,, + 7,)
- 1)
+f',(2.5
(50)
- 1000f~d%+d%lifQf~ b,, = 2.6264 x 10-9{250toa,,
[ 1523f;(lOOOf;
- 1000f’,(p+ 1) + PI +f'c [~(4819a,,-y,) - lOOOf; (uua(380750 t,, + 4819) - 7,) ]
- p) (51)
+ 4, [looof;(P + 1) -IlLI l/f',f'c c,,=4 x 10-6{250toc,,(1523f',+ lOOOf; - 1) + f ', [ (2.5 x 105f ;cua) - (4819~“~ + y2) ] - looof; + 44lf:f)a
(52)
d,, = 2.6264 x 10m9{250t,c,, [ 1523f:(lOOOf; - p) - lOOOf',b+ 1) f PI +f: [P(‘@~~c,,-Y,) (53) - 1000f~(c,,(380750t,, + 4819) - yz)] ++*tl000f’,(P + 1) -PI} lf',f',
(46)
being:
Tbz[:, -i;l =[; -,““]
4, = 3a,, - 250b,, & = 3c,, - 250
(48)
There.fore the pupil matrix of the corrected ametropic eye (M,,) is:
(49)
where:
(54)
(47)
d,,,
(55)
y, = 3.8075
x lO’b,,
(56)
y2 = 3.8075
x 105d,,
(57)
p = 1523t,,
(58)
Since this matrix M,, is calculated between conjugate points, the element b,, must be zero and this enables us to derive the distance t,, from the entrance pupil E,, to the front vertex of the lens, which is:
202
Ophthal.
Physiol.
Opt. 1995 15: No 3
of the matrix M, (Equation (62)). Therefore:
L = lOOOf~{250t,,a,,(1523f~
- 1) - [f:(4819a,,
- y, - &)I
(59)
1523{250t,,a,,(1523f; + lOOOf; - 1) +f; [25 x lO”f;a,,) + y, -4819a,, - @I(looofg - 1) I 1
Pupil matrix of the ophthalmic lens The pupil matrix (ML) of the correcting lens has been defined between the entrance pupil of the corrected ametropic eye (E,,) and the entrance pupil of the uncorrected ametropic eye (E,,), i.e. between conjugate points. Hence, M, will be given by the following expression: M, = T,, W T,,
(60)
with T,, and T,, being the matrices given in Equations (4.5) and (48), and M, the final matrix of the correcting lens, calculated by the following multiplication: M, = R,T,R,
(61)
where R,, T, and R, are the matrices given in Equations (46), (47) and (40), respectively. The elements of the pupil matrix (M,) of the lens are labelled as follows: (62) L
J
where: UL = 4 x 10-6{250to(1523f; + looof; +f’c [ (2.5 x 105f’,) - 48191 - 3000f’, + 3}lf’,f’,
- 1) (63)
If we consider the expression of the SM given in Equation (l), we can also relate the magnifications &, and PC, with the coefficient uua of the matrix M,, (Equation (24)) and the coefficient ucaof matrix M,, (Equation (49)). Therefore: (67) Consequently, if we know matrices M,,, M,, or M,, we can easily calculate the spectacle magnification, given that a specific element of these matrices is directly related to the SM (Equations (66) and (67)). Relative spectacle magni$cation Another concept studied is the relative spectacle magnification which is defined as the ratio of the retinal image size in the corrected eye to that in the standard schematic eye. It is calculated by Equation (4), that can also be expressed as: RSM=
+ lOOOf’, - 1) if ‘,f ‘,
dL = 65.6599 x lo6 [ 1523f;(lOOOf; - 1523 - looof;(1523t,, + 1) + 1523t,,] /f&f’,
Spectacle magnification magnification
14-k Uf',,
-
(68)
Since we are considering the matrix theory, a comparison of Equation (49) with Equation (12) shows that l/f& is the coefficient c,, of the M,, matrix (Equation (49)); therefore: RSM=
cL = O.OOl(1523f:
(66)
S”=+=L L aL
%
(69)
(64) t,,)
(65)
and relative spectacle
Spectacle magni$cation Spectacle magnification (SM) is defined as the ratio of the retinal image size in the corrected ametropic eye to that of the uncorrected ametropic eye; it is calculated as the reciprocal of the lateral magnification (&) produced by the correcting lensI (Equation (3)). Considering matrix theory and comparing Equation (62) with Equation (12), we can identify the lateral pupil magnification of the correcting lens (&) with the coefficient aL
where f k is the image focal length of the given emmetropic eye (in this case, the Gullstrand-Emsley schematic eye) hence f ', = 22.04 mm. Numerical
application
of the concepts studied
Once the expressions of spectacle magnification (SM) in Equation (66) and the relative spectacle magnification (RSM) in Equation (69) had been obtained by using the matrix method, we applied them to 12 theoretical cases of high myopia corrected with ophthalmic lenses. These 12 myopic eyes correspond to the combinations of cornea1 powers of 42, 44, 46 and 48D and axial lengths of 27, 30 and 33 mm. We have developed a BASIC language computer program to facilitate all calculations. All operations were performed considering all the decimals obtained; however, the values given in the equations and tables have
Matrix
formulation
of spectacle
been taken considering 4 or 3 decimals, respectively. The results of SM and RSM for the 12 studied eyes are shown in Table 2. This table enables us to easily compare the values of the different degrees of ametropia. For a myopic eye of fixed axial length we can see from Table 2 that both the SM and the RSM decrease when the cornea1 power increases. Figures 6 and 7 are graphic representations of the values of Tab/e 2 that clearly show the change of the SM and RSM. In Fiqure 6 we have represented the SM as a function of cornea: power for the three axial lengths previously considerec Since the cases studied correspond to myopic eyes, the correcting lenses have a negative power and hence the SM always is less than unity. In Figure 6 we can see that the SM increases when the degree of myopia decreases and SM approaches unity. That is important, since when the value of SM is closer to unity, the image sizes of an uncorrected and corrected eye are more similar. In Figure 7 we have represented the RSM as a function of correal power for three fixed axial lengths. It can be seen that relative spectacle magnification decreases when cornea1 power increases. The data on the SM and the RSM obtained matricially (Table 2) are identical to those obtained using the concept Table 2. Data obtained on spectacle magnification (SM) and the relative spectacle magnification (RSM) for 12 myopic theoretical eyes with different combinations of cornea1 powers (P,) in dioptres and axial lengths (L) in millimetres P, IL
42 44 46 48
SM
RSM
27
30
33
27
30
33
0.901 0.871 0.843 0.818
0.817 0.789 0.762 0.734
0.751 0.724 0.697 0.670
1.037 1.009 0.982 0.955
1.062 1.032 1.002 0.972
1.087 1.054 1.021 0.988
0.95
0.85 0.80
0.65 41
42
43
44 CORNEAL
45
46
POWER
47
48
49
(D)
Figure 6. Graph of spectacle magnification as a function of cornea1 power for eyes with three different axial lengths: n , L = 27mm; A, L = 30mm; x, L = 33mm.
magnification.
0” F
1.10
5 E:
1.08
g 2
1.06
8
1.04
2 v 2
1.01 0.99
z
0.97
9 . EL
0.95
1. High
myopia:
M. Garcia
~,,,: 41
203
et al.
,,,, 42
43
44 CORNEAL
45
46
47
48
49
POWER(D)
Figure 7. Graph of relative spectacle magnification as a function of cornea1 power for eyes with three different axial lengths: H, L = 27mm; A, L = 30mm; x, L = 33mm.
of vergence. Those last data are presented in the tables in reference 14. In Appendix 1, both the definition of vergence and its application to calculate the SM and RSM are shown. Conclusions With this study, we have shown that the matrix method can be used to calculate SM and RSM in the case of myopic eyes corrected with OLs. To obtain the expressions of these parameters we have to consider some very specific definitions of the SM and the RSM. This enables us to relate them directly to the coefficients of the specific matrices: M, (pupil matrix of the correcting lens) and M,, (ametropic eye pupil matrix corrected with an OL). Therefore if we know a specific element of these matrices (a, and c,,, respectively), we can determine the SM and RSM corresponding to each specific case of ametropia corrected with ophthalmic lenses. In a BASIC language computer program we have programmed all the formulae. This enables us to obtain these two coefficients of those matrices (ML and M,,). Therefore the values of the SM and the RSM can be easily calculated for the specific cases of the studied ametropia, by fixing the different powers and axial lengths that we wish to study. In this paper we have obtained the values of the SM and the RSM for 12 specific cases of myopic theoretical eyes whose cornea1 powers and axial lengths are given in Table 2. These results are identical to those obtained using the concept of vergence of light to calculate the expressions of the magnifications. This proves the validity of the matrix theory applied to the study of magnifications. If we subsequently apply this study to both eyes of a subject we would be able to study his degree of aniseikonia for distance vision, bearing in mind that this aniseikonia will not exist when the image sizes in both corrected eyes are equal. Although the study performed in this paper has centred
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on the case of ametropic eyes corrected with ophthalmic lenses, we are also studying the application of the matrix method in the cases of correction with contact lenses and with intraocular lenses. This way we will be able to compare the results obtained to three types of correction and choose the best correction for each case.
References 1 Casas, J. Optica, Catedra de Optica-Universidad de Zaragoza, pp. 30-39 (1980) 2 Nussbaum, A. and Phillips, R. A. Contemporary Optics for Scientists and Engineers, Prentice-Hall, Englewood Cliffs, New Jersey, USA, ch. 1 (1976) 3 Rosenblum, W. M. and Christensen, J. L. Optical matrix method: Optometric applications. Am. J. Optom. Pkysiol. Opt. 51, 961-968 (1974) 4 Long, W. F. Matrix formalism for decentration problems. Am. J. Optom. Pkysiol. Opt. 53, 27-33 (1976) 5 Keating, M. P. An easier method to obtain the sphere, cylinder, and axis from an off-axis-dioptric power matrix. Am. J. Optom. Pkysiol. Opt. 57, 734-737 (1980) 6 Keating, M. P. A system matrix for astigmatic optical systems: I. Introduction and dioptric power relationships. Am. J. Optom. Pkysiol. Opt. 58, 810-819 (1981) 7 Keating, M. P. A system matrix for astigmatic optical systems. II: Corrected system and an astigmatic eye. Am. J. Optom. Pkysiol. Opt. 58, 919-929 (1981) 8 Harris, W. F. Solving the matrix form of Prentice’s equation for dioptric power. Optom. Vision Sci. 68, 178-182
(1991)
9 Harris, W. F. and Keating, M. P. Proof that the prismatic effect is perpendicular to the lens thickness contour. Optom. Vision Sci. 68, 459-460 (1991) 10 Harris, W. F. The generalized Prentice equation and the matrix equation for lens thickness solved simultaneously for dioptric power. Optom. Vision Sci. 68, 873-876 (1991)
11 Keating, M. P. A matrix formulation of spectacle magnification. Am. Opktkalmol. Pkysiol. Opt. 2, 145-158 (1982)
12 Bennett, A. G. and Rabbetts, R. B. Clinical Visual Optics, 2nd edn, Butterworths, London, UK, pp. 11, 253, 275, 278 (1989) 13 Jalie, M. The design of intra-ocular lenses. J. Pkysiol. Opt. 32, l-32 (1978) 14 Garcia, M., Gonzalez, C. and Pascual, I. Calculo mediante vergencias de las imageries retinianas y aumentos. Aplicacidn a miopes magnos corregidos con distintos sistemas opticos. Ver y Oir. 82, 15-21 (1994) 15 Le Grand, Y. and el Hage, S. G. Physiological Optics (Springer Series in Optical Sciences), vol. 13, Springer, Berlin, Germany, pp. 111, 119 (1980) 16 Jalie, M. 7lze Principles of Ophthalmic Lenses, 4th edn, ABDO, London, UK, p. 13 (1988)
Appendix It is known
1 that the term
‘reduced
distance’
denotes a
distance (or thickness of material) traversed by a pencil of rays, divided by the refractive index of the given medium. On this basis, the reciprocal of a reduced object or image
distance, such as n’ll’, is traditionally called the ‘reduced vergence’. For brevity, we shall omit the word ‘reduced’ from this term. In this paper vergence will be used to denote the reciprocal of an object or image distance (in metres) multiplied by the refractive index of the corresponding medium’*. The vergence is also called proximity, reduced convergence or convergence. This concept began to be used to facilitate calculations because to work in m-’ IS easier than working in distances. Recently this concept was used to calculate SM and RSM. In order to do this, we had to determine the vergences required at each surface of the system and so the calculations were tedious. The calculation of spectacle magnification in the case of the myope corrected with a thick lens, is given by the following equation: SM=
L; L;
~ L, L2
where L, and L2 are the vergences arriving at each surface of the lens, and L’, and L\ are the vergences leaving each surface of the lens, and L; = Ill; with 1; being the distance from the back vertex of the lens to the entrance pupil of the uncorrected ametropic eye (E,,). All these parameters are represented in Figure 8. This expression depends only on the vergences. But to calculate
them it is necessary to first know the power of
the correcting lens. That is calculated bearing in mind the data of the ametropic eyes already studied and applying the same principle of correction as in the case of the matrix method15. The RSM is calculated by the equation: RSM = F,
L2
L, L4L5
L;L;L;L’,L;
(71)
Figure 8. Step-along method of calculation to determine the SM in the case of correction with a thick ophthalmic lens using the concept of vergence. L,, vergences arriving at each surface; Li, vergences leaving each surface; E,,, centre of the entrance pupil of the uncorrected ametropic eye.
Matrix
formulation
of spectacle
magnification.
1. High
myopia:
M. Garcia
et al.
205
Figure 9. Step-along method of calculation to determine the RSM in the case of correction with a thick ophthalmic lens using the concept of vergence. Li, vergences arriving at each surface, LI, vergences leaving each surface.
where P’,,is the equivalent power of the schematic eye (in our case the Gullstrand-Emsley), L,, L,, L4 and L, are the vergences arriving at each surface, and L’, , L\, Lb, Lk and L\ are the vergences leaving each surface of the corrected eye14. Following the definition of the surfaces in Figure 9, we can identify these vergences.
The 2 x 2 translation matrix T for the distance t expressed in metres is:
Appendix
with n being the refractive index of the given medium. The object-image matrix has the form:
2
The matrix formulation in terms of reduced distances is common in the literature and this formulation has the advantage of making the translation or refraction matrix unimodular (i.e. the determinant is 1). For these reasons we are going to give the matrix notation in terms of reduced distances2. The 2 x 2 refraction matrix R for a spherical surface is: R=
:,
--;
1
(72)
where P is the optical power of the surface and can be calculated as: P = (n ’ - n) lr with r being the radius of curvature, IZ the refractive index before the refractive surface, and ~1’ the refractive index behind it.
T=
M,,=
(73)
Ilo PI1 1 p
-p
(74)
If we calculate the pupil matrix of the ophthalmic lens, M,, using Equations (72) and (73), and we label its elements the same way they are labelled in Equation (62), then SM will be equal to uL since SM = l/p,. If we calculate the ametropic eye pupil matrix corrected with ophthalmic lens M,, using Equations (72) and (73), and we label its elements the same way they are labelled in Equation (49), then RSM will be calculated as RSM = l/f’,/(-b,,ln’).
Ophthal. Physiol. Opt. Vol. 15, No. 3, pp. 195-205, 1995 0 1995 Elsevier Science Ltd for British College of Optometrists Printed in Great Britain. All rights reserved 0275-5408195 $10.00 + 0.00
New matrix formulation of spectacle magnification using pupil magnification. I. High myopia corrected with ophthalmic M. Garcia,
C. Gorhlez
lenses
and I. Pascual
Laboratory of Optics, Department Alicante E 03080, Spain
of Optics,
University
of Alicante,
Apdo.
99,
Summary In this paper we have applied the matrix method to calculating spectacle magnification (SM) and relative spectacle magnification (RSM) for myopic eyes corrected with ophthalmic lenses (OLs). We have been able to obtain very simple expressions of the magnifications and analogous values to the ones obtained using other geometric methods. Through the paraxial matrix for spherical systems, we can calculate the correcting lens pupil 2 x 2 object-image matrix (M,) and the corrected ametropic eye pupil matrix (MC,). These two matrices, that is M, and M,,, directly provide the lateral magnification and the lens-eye system’s image focal parameters that enable us to easily calculate SM and RSM. The magnifications have been calculated for the case of myopic eyes corrected with OLs, which shows the validity of this method since its results are similar to the ones obtained using other methods. The analysis of other correcting systems such as contact intraocular lenses will be presented in two associated papers (to be published as parts II and III of this review). Ophthal.
Physiol.
Opt.
1995,
15,
195-205
The resurgence of activity in applied optics in the second half cf the twentieth century has caused this period to be label!ed an optical renaissance. Part of the renaissance has consisted of a steadily growing appreciation of the advantages of matrix methods applied to geometric optics analysis, which encompasses both the study of spherical optical systems’.2 including those of ophthalmic interest3, and the analysis of astigmatic optical systems4-’ and of different problems associated with these systems’-“. By this method, it is possible to simplify many tedious ray tracing calculations and provide an improved conceptual framework for analysing the different situations. This matrix method has already been used to calculate the spectacle magnification (SM) of astigmatic eyes”. However. the matricial relationship obtained was complicated due to the definition of the SM (SM was defined as the product of the power factor and the shape factor). In this paper we apply the matrix method to the calculation of spectacle magnification using a different definition of this
parameter. Like Bennett and Rabbettsi2, we defined SM as the ratio of a corrected eye retinal image size (y’,,) to an uncorrected (y’,,) one. Consequently, it analyses the change in the retinal image size in any given eye as a result of wearing some type of correcting system (in this case ophthalmic lenses). For the blurred image case, such as myopic eyes for distance vision, the image size can be determined by the limiting ray through the centre of the eye’s entrance pupil (E) or the eye’s exit pupil (E’). Thus, SM can be defined by the lateral pupil magnification as:
P ca
where /I,, is the lateral magnification of the uncorrected ametropic eye between the entrance pupil (E,,) and the exit pupil (E’,,,); E,, is the image of the iris formed by the cornea and EL, the image of the iris formed by the crystalline. When we consider the eye corrected with an ophthalmic lens, this lens forms an image of the uncorrected ametropic eye’s entrance pupil. This second image of the iris becomes
Received: 3 June 1994 Revised form: 5 October 1994
195
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Ophthal. Physiol. Opt. 1995 15: No 3
the entrance pupil of the lens-eye system, i.e. of the corrected ametropic eye (EC,). Therefore &, is the lateral magnification of the corrected ametropic eye defined by the new corrected ametropic entrance pupil (EC,) and the corrected ametropic exit pupil (E’,,) (this E’,, coincides with EL, since both are the image of the iris formed by the crystalline). Since the corrected ametropic eye can be considered as an association of two subsystems, the correcting lens and the uncorrected ametropic eye, then:
PC,= PLP”,
optical system. This is due to the effect of refraction, translation and/or reflection. In the case of the eye and correcting lens-eye systems, we are only considering the translation and refraction effects due to different surfaces of the systems. For this reason, we only show the form of the translation and refraction matrix. The 2 X 2 refraction matrix, R, for a spherical surface is:
(5)
(2)
where & is the lateral magnification of the lens between the corrected ametropic eye entrance pupil (EC,) and the uncorrected ametropic eye entrance pupil (E,,); hence the SM can be determined by the following equation’*: SM=$ L
In this article we have applied the matrix theory to this new definition of the SM. Thus we obtain a formula that allows us to easily determine the values of the SM in ametropia specific cases. Another parameter that we have also studied matricially is relative spectacle magnification (RSM), which is defined as the ratio of the retinal image size in a corrected ametropic eye to that in a standard emmetropic schematic eye13. It is calculated as the division of the image focal length of a corrected ametropic eye (f’,,) by the image focal length of the standard emmetropic eye (f’,):
RSM = f’ca .f’e Once the expressions corresponding to the SM and the RSM had been obtained matricially, we applied them to 12 specific theoretical cases of high myopia which were corrected with ophthalmic lenses (OLs). These 12 cases consisted of eyes with respective axial length ranges of 27-33mm and with respective cornea1 power ranges of 42-48D. These cover almost the entire range of possible cases of high myopia. We will study SM and RSM together. This enables us to obtain the values of the retinal image sizes of the corrected ametropic eye (CA) and of the uncorrected ametropic eye (UA). Knowing this information is very interesting since it allows the study of the aniseikonia (inequality of image sizes) that provides anomalies in binocular vision14. Before doing the matrix calculations we will give a brief review of the matrix notation used in this paper.
where f’ is the image focal of the surface expressed in metres, II is the refractive index before the refractive surface, and n ’ is the refractive index behind it. The image focal of surface f’ is given by the relationship: P = n ‘If’ where P is the optical power of the surface and can also be calculated as: P = (n’ - n) /r with r being the radius of curvature. The 2 x 2 translation matrix, T, for the distance t expressed in metres is:
(6) In this paper, we use the convention that the light travels from left to right. Furthermore, the distances measured in the same direction as that in which the light is travelling are considered as positive in sign; if in the opposite direction, as negative. The system matrix is the product of the appropriate refraction and translation matrices in the correct order (right to left)*. For example, the system matrix M for a thick spherical lens is: M=R,TR,
(7)
where R, and R, are the refraction matrices for the first and second surfaces, respectively, and T is the translation matrix for the central thickness. In general, the elements of the 2 x 2 system matrix can be labelled: M=
L1 “, ;
(8)
The 2 x 2 system matrix, M, relates the parameters of a ray incident on the front of the system to the parameters of that ray as it exits at the back of the system:
Review of matrix notation The light path is modified when the light passes through an
(9)
Matrix
formulation
of spectacle
where 01is the angle of the incident ray and h is the height above the optical axis where the light hits the front of the system. The primed parameters are for the ray exiting at the back of the system. The geometry of the ray parameters is shown in Figure 1. Two conjugate points, object and image, are related matricially by the object-image matrix’ (M,,) which can be calculated as follows: M,, = TiMTo
(10)
where T, is the translation matrix from the object plane to the front vertex of the system, M is the system matrix from front to back vertex, and T, is the translation matrix from the back vertex of the system to the image plane; so that.
Mo, = The elements a,,, bo,, co, and d,, of this object-image matrix have a specific meaning: the coefficient b,, must be zero since the matrix has been calculated between conjugate points; this condition enables us to calculate the position of two conjugate points; consequently, the element a,, is the lateral magnification /3 defined between the object and image conjugate planes; co, is, in this case, the inverse of the image focal of the system l/f ‘, and d,,, is the angular magnification 6 between the conjugate planes given. Therefore the object-image matrix has the general form:
magnification.
1. High
n’
u
Sl
n II
S’2
Figure 1. A paraxial ray travelling from S, to S, in a spherical optical system: t is the distance between the anterior and posterior surface of the system; 01 and a’ are the respective incident and emergent angles of the ray; h and h’ are the height of the ray that is incident to and emergent from the system; n, n”, n’ are the refractive indices of the anterior, posterior and system medium, respectively.
M. Garcia
et al.
197
(12)
Application
of the matrix method
Given the definitions of the SM and the RSM (Equations (3) and (4)), we need to determine the lateral magnification of the correcting lens (&) and the image focal of the lens-eye system (f&) in order to calculate these magnifications. By comparing Equations (11) and (12), one can note that PL and f ‘,, are respectively related to the elements a,, and co, of the object image (O-I) matrices which we are going to calculate in this paper. Since the origins of measurement considered in the definitions of the SM and RSM are the entrance pupil (E) and the exit pupil (E’), then those O-I matrices will be calculated between both conjugate points; for that reason, the matrices are labelled pupil matrices. Therefore in order to determine /3, and f ‘,, we need to respectively calculate the correcting lens pupil matrix (ML) and the lens-eye system pupil matrix (M,,). The correcting lens pupil matrix is determined through the uncorrected ametropic eye pupil matrix (M,,), the ophthalmic lens power (OL) and the corrected ametropic eye pupil matrix, that is, the lens-eye system pupil matrix (M,,). The cases of ametropia studied consist of myopic eyes with different cornea1 powers (P,) and axial lengths (L). All the other data used for these calculations will be taken from the Gullstrand-Emsley theoretical eye”. For the correction of the ametropia, we have considered an OL with a refractive index of 1.523 and a central thickness of 1 mm at a cornea1 vertex distance of 12 mm. The working or focusing distance is always taken as infinity. Uncorrected ametropic
n
myopia:
eye pupil
matrix
The uncorrected ametropic eye pupil matrix (M,,) is the matrix defined between the entrance pupil E,, and the exit pupil EL, of the uncorrected ametropic eye. Hence, to calculate M,, we must know the position of these two pupils. The position of the exit pupil EL, is fixed, it does not change with the ametropia here given (axial ametropia and refractive ametropia due to increase of the cornea1 power) and it has a value of 3.69 mm measured from the cornea”. The position of the entrance pupil E,, must be calculated for each ametropia since it depends on cornea1 power (P,). We calculate the object-image matrix M,, between the plane of the entrance pupil and the plane of the real pupil (iris) in order to determine E,,. Following the definition of the surfaces and distances in Figure 2, we calculate M,,:
M,, = TXR, To
(13)
198
Ophthal.
Physiol.
Opt. 1995 15: No 3
From Equation (18), we can obtain the position of the uncorrected ametropic eye entrance pupil measured from the cornea (to):
,ci ‘1
n
n O
to
2
=
to
0.0027 0.0027 P, - 1
(19)
When the positions of E,,, and EL, are known, matrix M,, can be determined by the following equation (Figure 3): Mu, = T, MC, T, R, To Figure 2. The cornea refracting surface and the real pupil (iris) of a myopic eye. Refraction indices, radii of curvature and distance to from entrance pupil of the uncorrected ametropic eye (E,,) to the cornea and t2 from the cornea to the real pupil (iris Es).
where T, is the translation matrix for the distance t,, from entrance pupil E,, to the cornea, R, is the refraction matrix for the cornea, and T, is the translation matrix for the distance t, from cornea to the real pupil (iris), i.e. the translation along the depth of the anterior chamber of the eye. In matrix R,, f ‘, is going to depend on cornea1 power (P,) since P, = n2/f‘,. These matrices have the following forms:
I 1
To=
' 0
-7
I--! 1
R,=
(14)
0
1
no
f'l
%
T2=[;
-p]
=
P L 1.3333 :
= [;
0.7500
where To, R, and T2 are the matrices given by Equations (14), (15) and (16), respectively, McT is the matrix for the crystalline lens, and T, is the translation matrix for the distance ts from the crystalline lens back vertex to the exit pupil E,, that has the following form: TX 6
[ 1 ’ 0
-t6
1
1
0
1
n2
I--if'3
n4
R, =
(15) I 1
I
(21)
In the calculation of t,, we have taken into account the fact that the light travels from left to right. Therefore: t6 = 0.0037 - (t2 + t,). According to Figure 3, we need the following matrices to calculate M,,:
T=4
0
1
(20)
'
0
-b
1
(22)
R, =
-p”“‘“]
(16)
Therefore the matrix M,, is: (17)
M,, = r
r
l-~ 1.3333 P, 0.0036
I
PI
0.0036
“0
p, ~1.3333 to - 0.7500
p,
- to 1
0.7500 - ~ to 1.3333
1.3333
Comparing Equation (17) with the O-I matrix general form given by Equation (12) one can see that:
0.0036
PI
~ to - 0.7500 1.3333
1to -
= 0
(18)
Figure 3. Surfaces indices, distances schematic eye.
of the considered myopic eye. Refractive and thickness of the Gullstrand-Emsley
Matrix
formulation
of spectacle
1 5.8404
=
i lk.3352
FO62Oj [:
0.9790 0 0.9416 I = [ 16.3206
1. High
myopia:
M. Garcia
et al.
199
the O-I matrix (MrR) between the far point and the retina (R), since these points are conjugate points, so that:
The crystalline lens is obtained by the product:
M,, = R,T,R,
magnification.
-Foolh]
-0.0034 0.9650
(23)
The elements of the matrix, M,,, are:
W, = Tm MuaT,, where T,, is the translation matrix for the distance rE from the far point to the UA entrance pupil E,,, M,, is the UA pupil matrix expressed by Equation (24), and T,., is the translation matrix for the distance E’R from the exit pupil EL, to the retina. These matrices become:
(24)
(32)
where: (33) a“Cl= 1.0362 - 0.0028 P,
(25)
bu,= 9 p”;o;;;3,25
(26)
where E’R is calculated as: I
c Ua=
d,, =
+ (2.5730 x 10-9) = 0
0.6797 P, + 16.3206 2412.41 - (6.2514 x lo-‘) 3333.25 - 9 P,
E’R = L - 0.0037 (27) (28)
(34)
L is the different axial lengths that we consider and 0.0037 m is the standard distance” from the cornea1 vertex to the exit pupil Ek,. Multiplying the three matrices in Equation (3 1) we obtain:
All these matrices are obtained assuming the necessary data and model of the Gullstrand-Emsley eye. The optical power of the correcting lens To calculate the back vertex power (F \) of the ophthalmic lens (OL), we apply the principle of correction that says that an eye is corrected for distance vision by a lens with its second principal focus coinciding with the eye’s far point”. If we take the pupils as origins of measurement, this principle can be written as:
Since this matrix M,, is defined between conjugate points, element b of this matrix must be zero and this enables us to obtain the position of the far point (rE). If instead of a“a, c,, and A,, we use their corresponding values given by Equations (25), (27) and (28), we obtain rE: rE=
(36) 0.001 - (3.253 X 10m? P. + 201.040 E'R - (8.327 x 10m")P. E'R
FL=
K
287.850 - 1.555 P, + 0.002P: - 4533.500E'R - 176.558P,E'R + 0.510P,'E'R
1 + t,, K
where t,, is the distance from the OL back vertex to the entrance pupil E,, which is calculated by t,, = t,, - t, with t,, being the distance from the OL back vertex to the cornea (12 mm); t,, is a negative distance expressed by Equation (19) and K is the pupil refraction of the uncorrected ametropic eye (UA) which is calculated as:
where rE is the distance from far point (r) to the UA entrance pupil (E,,). Hence, first we must determine the position of the far point. For that reason, we consider
Once rE has been calculated, it is possible to find F \, the back vertex power using Equations (29) and (30). In subsequent calculations, we will need to know the individual powers of each of the lens surfaces. They are usually calculated by predetermining the front surface power (P,) of the lens, leaving the back surface power (P,) to be calculated. In the references used here, we have found the same tables of the values assigned to the front surface power as published by Jale16and by Bennett and Rabbetts12. Table 1 shows the range of typical values that are usually assigned to the front surface (P,) as depending on the back vertex powers. We used this table because it is a generalization of the aforementioned tables.
200
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Table 1. Values assigned to the front surface powers of the lens (P,) depending on the back vertex powers (Fk)
F; 0)
Pa(0)
-0.25 to -6.00 -6.25 to -8.00 > -8.00
+ 6.00 + 3.00 + 1.00
(40)
When F \ and P, are known, the back surface power of the lens can be determined from15: P,=F;-
IL
P,
where t, has been calculated as tf = tb -f ‘, with t, being the central thickness of the OL (1 mm), and fb the image focal length of the front surface of the lens.
where llf: = P,ln,, with f: and P, being the image focal and the power of the back surface of the OL, and n, the refractive index of the air (no = 1).
(37) (41)
where t, is the central thickness of the OL and nb is the refractive index of the lens. Since we are applying the matrix method, we can also apply it in this case. Therefore we have to consider the O-I matrix between the image focal point of the front surface cf’,) of the OL and its conjugate point through the back surface of the lens (F \) that is shown in Figure 4: M,> = T,. R, T,
the three matrices in Equation (3X), we
M,+$
(38)
where T, is the translation matrix for the distance tr from image focal point of the front surface of the lens (F ‘,) to the back vertex of the OL. R, is the refraction matrix for the back surface of the lens and Tfr is the translation matrix for the distance (trJ from the “back vertex of the OL to the image focal point of the correcting lens (t,,, = l/F \). These matrices are expressed by:
TX F I 0’ -lf1 1
By multiplying obtain:
(39)
tf[?-lj -?I
(42)
1.523 - t,PC Since this matrix is defined between conjugate points, element b must be zero and this condition enables us to solve the P,. Once tf has been substituted, the expression of P, is given by: PC= F; +
1.523 0.001 +f’,
(43)
which depends only on the back vertex power of the OL and on the image focal of the front surface of the OL. The value off ‘, is obtained from Table I taking into account that P, = qlf',. Ametropic eye pupil matrix corrected with ophthalmic lens
[
Figure 4. Parameters needed power of the lens. Surfaces, ,indices of a lens.
to calculate the back surface focal lengths and refractive
The pupil matrix (M,,) of an ametropic eye corrected with an OL, is defined between the entrance pupil (EC,) of the corrected ametropic eye and the exit pupil (E’,,) of the corrected ametropic eye (CA). As we said before, the entrance pupil EC, of a corrected ametropic eye is the image of the entrance pupil of an uncorrected ametropic eye formed by the OL”. This entrance pupil will be calculated from M,,. To calculate M,,, we assume that the entrance pupil Eta and the exit pupil EL, are conjugate points through the lens-eye system. We find that (Figure 5):
Ma = MuaTuaRcT, RaTea
(44)
Matrix
formulation
of spectacle
Figure 5. Surfaces of the myopic eye corrected indices and thickness of the Gullstrand-Emsley
T,, is the translation matrix for the distance t,, from the entrance pupil EC,to the front vertex of the lens. R, is the refraction matrix for the front surface of the OL, T, is the translation matrix for the central thickness of the OL, R, is the refraction matrix for the back surface of the OL expressed by Equation (40), T,, is the translation matrix for the distance t,, from the posterior vertex of the OL to the entrance pupil E,,. which has been calculated as: t,, = (T,, - to) with t,, = 12mm, and M,, being the pupil matrix of the UA given by Equation (24). These matrices are as follows:
(45)
magnification.
1. High myopia: M. Garcia et al.
with an ophthalmic schematic eye. a ca=
201
lens. Refractive
4 x 10-6{250t0a,,(1523.f~ + looof; x lO’f’,a,, - 4819a,, + 7,)
- 1)
+f',(2.5
(50)
- 1000f~d%+d%lifQf~ b,, = 2.6264 x 10-9{250toa,,
[ 1523f;(lOOOf;
- 1000f’,(p+ 1) + PI +f'c [~(4819a,,-y,) - lOOOf; (uua(380750 t,, + 4819) - 7,) ]
- p) (51)
+ 4, [looof;(P + 1) -IlLI l/f',f'c c,,=4 x 10-6{250toc,,(1523f',+ lOOOf; - 1) + f ', [ (2.5 x 105f ;cua) - (4819~“~ + y2) ] - looof; + 44lf:f)a
(52)
d,, = 2.6264 x 10m9{250t,c,, [ 1523f:(lOOOf; - p) - lOOOf',b+ 1) f PI +f: [P(‘@~~c,,-Y,) (53) - 1000f~(c,,(380750t,, + 4819) - yz)] ++*tl000f’,(P + 1) -PI} lf',f',
(46)
being:
Tbz[:, -i;l =[; -,““]
4, = 3a,, - 250b,, & = 3c,, - 250
(48)
There.fore the pupil matrix of the corrected ametropic eye (M,,) is:
(49)
where:
(54)
(47)
d,,,
(55)
y, = 3.8075
x lO’b,,
(56)
y2 = 3.8075
x 105d,,
(57)
p = 1523t,,
(58)
Since this matrix M,, is calculated between conjugate points, the element b,, must be zero and this enables us to derive the distance t,, from the entrance pupil E,, to the front vertex of the lens, which is:
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Opt. 1995 15: No 3
of the matrix M, (Equation (62)). Therefore:
L = lOOOf~{250t,,a,,(1523f~
- 1) - [f:(4819a,,
- y, - &)I
(59)
1523{250t,,a,,(1523f; + lOOOf; - 1) +f; [25 x lO”f;a,,) + y, -4819a,, - @I(looofg - 1) I 1
Pupil matrix of the ophthalmic lens The pupil matrix (ML) of the correcting lens has been defined between the entrance pupil of the corrected ametropic eye (E,,) and the entrance pupil of the uncorrected ametropic eye (E,,), i.e. between conjugate points. Hence, M, will be given by the following expression: M, = T,, W T,,
(60)
with T,, and T,, being the matrices given in Equations (4.5) and (48), and M, the final matrix of the correcting lens, calculated by the following multiplication: M, = R,T,R,
(61)
where R,, T, and R, are the matrices given in Equations (46), (47) and (40), respectively. The elements of the pupil matrix (M,) of the lens are labelled as follows: (62) L
J
where: UL = 4 x 10-6{250to(1523f; + looof; +f’c [ (2.5 x 105f’,) - 48191 - 3000f’, + 3}lf’,f’,
- 1) (63)
If we consider the expression of the SM given in Equation (l), we can also relate the magnifications &, and PC, with the coefficient uua of the matrix M,, (Equation (24)) and the coefficient ucaof matrix M,, (Equation (49)). Therefore: (67) Consequently, if we know matrices M,,, M,, or M,, we can easily calculate the spectacle magnification, given that a specific element of these matrices is directly related to the SM (Equations (66) and (67)). Relative spectacle magni$cation Another concept studied is the relative spectacle magnification which is defined as the ratio of the retinal image size in the corrected eye to that in the standard schematic eye. It is calculated by Equation (4), that can also be expressed as: RSM=
+ lOOOf’, - 1) if ‘,f ‘,
dL = 65.6599 x lo6 [ 1523f;(lOOOf; - 1523 - looof;(1523t,, + 1) + 1523t,,] /f&f’,
Spectacle magnification magnification
14-k Uf',,
-
(68)
Since we are considering the matrix theory, a comparison of Equation (49) with Equation (12) shows that l/f& is the coefficient c,, of the M,, matrix (Equation (49)); therefore: RSM=
cL = O.OOl(1523f:
(66)
S”=+=L L aL
%
(69)
(64) t,,)
(65)
and relative spectacle
Spectacle magni$cation Spectacle magnification (SM) is defined as the ratio of the retinal image size in the corrected ametropic eye to that of the uncorrected ametropic eye; it is calculated as the reciprocal of the lateral magnification (&) produced by the correcting lensI (Equation (3)). Considering matrix theory and comparing Equation (62) with Equation (12), we can identify the lateral pupil magnification of the correcting lens (&) with the coefficient aL
where f k is the image focal length of the given emmetropic eye (in this case, the Gullstrand-Emsley schematic eye) hence f ', = 22.04 mm. Numerical
application
of the concepts studied
Once the expressions of spectacle magnification (SM) in Equation (66) and the relative spectacle magnification (RSM) in Equation (69) had been obtained by using the matrix method, we applied them to 12 theoretical cases of high myopia corrected with ophthalmic lenses. These 12 myopic eyes correspond to the combinations of cornea1 powers of 42, 44, 46 and 48D and axial lengths of 27, 30 and 33 mm. We have developed a BASIC language computer program to facilitate all calculations. All operations were performed considering all the decimals obtained; however, the values given in the equations and tables have
Matrix
formulation
of spectacle
been taken considering 4 or 3 decimals, respectively. The results of SM and RSM for the 12 studied eyes are shown in Table 2. This table enables us to easily compare the values of the different degrees of ametropia. For a myopic eye of fixed axial length we can see from Table 2 that both the SM and the RSM decrease when the cornea1 power increases. Figures 6 and 7 are graphic representations of the values of Tab/e 2 that clearly show the change of the SM and RSM. In Fiqure 6 we have represented the SM as a function of cornea: power for the three axial lengths previously considerec Since the cases studied correspond to myopic eyes, the correcting lenses have a negative power and hence the SM always is less than unity. In Figure 6 we can see that the SM increases when the degree of myopia decreases and SM approaches unity. That is important, since when the value of SM is closer to unity, the image sizes of an uncorrected and corrected eye are more similar. In Figure 7 we have represented the RSM as a function of correal power for three fixed axial lengths. It can be seen that relative spectacle magnification decreases when cornea1 power increases. The data on the SM and the RSM obtained matricially (Table 2) are identical to those obtained using the concept Table 2. Data obtained on spectacle magnification (SM) and the relative spectacle magnification (RSM) for 12 myopic theoretical eyes with different combinations of cornea1 powers (P,) in dioptres and axial lengths (L) in millimetres P, IL
42 44 46 48
SM
RSM
27
30
33
27
30
33
0.901 0.871 0.843 0.818
0.817 0.789 0.762 0.734
0.751 0.724 0.697 0.670
1.037 1.009 0.982 0.955
1.062 1.032 1.002 0.972
1.087 1.054 1.021 0.988
0.95
0.85 0.80
0.65 41
42
43
44 CORNEAL
45
46
POWER
47
48
49
(D)
Figure 6. Graph of spectacle magnification as a function of cornea1 power for eyes with three different axial lengths: n , L = 27mm; A, L = 30mm; x, L = 33mm.
magnification.
0” F
1.10
5 E:
1.08
g 2
1.06
8
1.04
2 v 2
1.01 0.99
z
0.97
9 . EL
0.95
1. High
myopia:
M. Garcia
~,,,: 41
203
et al.
,,,, 42
43
44 CORNEAL
45
46
47
48
49
POWER(D)
Figure 7. Graph of relative spectacle magnification as a function of cornea1 power for eyes with three different axial lengths: H, L = 27mm; A, L = 30mm; x, L = 33mm.
of vergence. Those last data are presented in the tables in reference 14. In Appendix 1, both the definition of vergence and its application to calculate the SM and RSM are shown. Conclusions With this study, we have shown that the matrix method can be used to calculate SM and RSM in the case of myopic eyes corrected with OLs. To obtain the expressions of these parameters we have to consider some very specific definitions of the SM and the RSM. This enables us to relate them directly to the coefficients of the specific matrices: M, (pupil matrix of the correcting lens) and M,, (ametropic eye pupil matrix corrected with an OL). Therefore if we know a specific element of these matrices (a, and c,,, respectively), we can determine the SM and RSM corresponding to each specific case of ametropia corrected with ophthalmic lenses. In a BASIC language computer program we have programmed all the formulae. This enables us to obtain these two coefficients of those matrices (ML and M,,). Therefore the values of the SM and the RSM can be easily calculated for the specific cases of the studied ametropia, by fixing the different powers and axial lengths that we wish to study. In this paper we have obtained the values of the SM and the RSM for 12 specific cases of myopic theoretical eyes whose cornea1 powers and axial lengths are given in Table 2. These results are identical to those obtained using the concept of vergence of light to calculate the expressions of the magnifications. This proves the validity of the matrix theory applied to the study of magnifications. If we subsequently apply this study to both eyes of a subject we would be able to study his degree of aniseikonia for distance vision, bearing in mind that this aniseikonia will not exist when the image sizes in both corrected eyes are equal. Although the study performed in this paper has centred
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Opt. 1995 15: No 3
on the case of ametropic eyes corrected with ophthalmic lenses, we are also studying the application of the matrix method in the cases of correction with contact lenses and with intraocular lenses. This way we will be able to compare the results obtained to three types of correction and choose the best correction for each case.
References 1 Casas, J. Optica, Catedra de Optica-Universidad de Zaragoza, pp. 30-39 (1980) 2 Nussbaum, A. and Phillips, R. A. Contemporary Optics for Scientists and Engineers, Prentice-Hall, Englewood Cliffs, New Jersey, USA, ch. 1 (1976) 3 Rosenblum, W. M. and Christensen, J. L. Optical matrix method: Optometric applications. Am. J. Optom. Pkysiol. Opt. 51, 961-968 (1974) 4 Long, W. F. Matrix formalism for decentration problems. Am. J. Optom. Pkysiol. Opt. 53, 27-33 (1976) 5 Keating, M. P. An easier method to obtain the sphere, cylinder, and axis from an off-axis-dioptric power matrix. Am. J. Optom. Pkysiol. Opt. 57, 734-737 (1980) 6 Keating, M. P. A system matrix for astigmatic optical systems: I. Introduction and dioptric power relationships. Am. J. Optom. Pkysiol. Opt. 58, 810-819 (1981) 7 Keating, M. P. A system matrix for astigmatic optical systems. II: Corrected system and an astigmatic eye. Am. J. Optom. Pkysiol. Opt. 58, 919-929 (1981) 8 Harris, W. F. Solving the matrix form of Prentice’s equation for dioptric power. Optom. Vision Sci. 68, 178-182
(1991)
9 Harris, W. F. and Keating, M. P. Proof that the prismatic effect is perpendicular to the lens thickness contour. Optom. Vision Sci. 68, 459-460 (1991) 10 Harris, W. F. The generalized Prentice equation and the matrix equation for lens thickness solved simultaneously for dioptric power. Optom. Vision Sci. 68, 873-876 (1991)
11 Keating, M. P. A matrix formulation of spectacle magnification. Am. Opktkalmol. Pkysiol. Opt. 2, 145-158 (1982)
12 Bennett, A. G. and Rabbetts, R. B. Clinical Visual Optics, 2nd edn, Butterworths, London, UK, pp. 11, 253, 275, 278 (1989) 13 Jalie, M. The design of intra-ocular lenses. J. Pkysiol. Opt. 32, l-32 (1978) 14 Garcia, M., Gonzalez, C. and Pascual, I. Calculo mediante vergencias de las imageries retinianas y aumentos. Aplicacidn a miopes magnos corregidos con distintos sistemas opticos. Ver y Oir. 82, 15-21 (1994) 15 Le Grand, Y. and el Hage, S. G. Physiological Optics (Springer Series in Optical Sciences), vol. 13, Springer, Berlin, Germany, pp. 111, 119 (1980) 16 Jalie, M. 7lze Principles of Ophthalmic Lenses, 4th edn, ABDO, London, UK, p. 13 (1988)
Appendix It is known
1 that the term
‘reduced
distance’
denotes a
distance (or thickness of material) traversed by a pencil of rays, divided by the refractive index of the given medium. On this basis, the reciprocal of a reduced object or image
distance, such as n’ll’, is traditionally called the ‘reduced vergence’. For brevity, we shall omit the word ‘reduced’ from this term. In this paper vergence will be used to denote the reciprocal of an object or image distance (in metres) multiplied by the refractive index of the corresponding medium’*. The vergence is also called proximity, reduced convergence or convergence. This concept began to be used to facilitate calculations because to work in m-’ IS easier than working in distances. Recently this concept was used to calculate SM and RSM. In order to do this, we had to determine the vergences required at each surface of the system and so the calculations were tedious. The calculation of spectacle magnification in the case of the myope corrected with a thick lens, is given by the following equation: SM=
L; L;
~ L, L2
where L, and L2 are the vergences arriving at each surface of the lens, and L’, and L\ are the vergences leaving each surface of the lens, and L; = Ill; with 1; being the distance from the back vertex of the lens to the entrance pupil of the uncorrected ametropic eye (E,,). All these parameters are represented in Figure 8. This expression depends only on the vergences. But to calculate
them it is necessary to first know the power of
the correcting lens. That is calculated bearing in mind the data of the ametropic eyes already studied and applying the same principle of correction as in the case of the matrix method15. The RSM is calculated by the equation: RSM = F,
L2
L, L4L5
L;L;L;L’,L;
(71)
Figure 8. Step-along method of calculation to determine the SM in the case of correction with a thick ophthalmic lens using the concept of vergence. L,, vergences arriving at each surface; Li, vergences leaving each surface; E,,, centre of the entrance pupil of the uncorrected ametropic eye.
Matrix
formulation
of spectacle
magnification.
1. High
myopia:
M. Garcia
et al.
205
Figure 9. Step-along method of calculation to determine the RSM in the case of correction with a thick ophthalmic lens using the concept of vergence. Li, vergences arriving at each surface, LI, vergences leaving each surface.
where P’,,is the equivalent power of the schematic eye (in our case the Gullstrand-Emsley), L,, L,, L4 and L, are the vergences arriving at each surface, and L’, , L\, Lb, Lk and L\ are the vergences leaving each surface of the corrected eye14. Following the definition of the surfaces in Figure 9, we can identify these vergences.
The 2 x 2 translation matrix T for the distance t expressed in metres is:
Appendix
with n being the refractive index of the given medium. The object-image matrix has the form:
2
The matrix formulation in terms of reduced distances is common in the literature and this formulation has the advantage of making the translation or refraction matrix unimodular (i.e. the determinant is 1). For these reasons we are going to give the matrix notation in terms of reduced distances2. The 2 x 2 refraction matrix R for a spherical surface is: R=
:,
--;
1
(72)
where P is the optical power of the surface and can be calculated as: P = (n ’ - n) lr with r being the radius of curvature, IZ the refractive index before the refractive surface, and ~1’ the refractive index behind it.
T=
M,,=
(73)
Ilo PI1 1 p
-p
(74)
If we calculate the pupil matrix of the ophthalmic lens, M,, using Equations (72) and (73), and we label its elements the same way they are labelled in Equation (62), then SM will be equal to uL since SM = l/p,. If we calculate the ametropic eye pupil matrix corrected with ophthalmic lens M,, using Equations (72) and (73), and we label its elements the same way they are labelled in Equation (49), then RSM will be calculated as RSM = l/f’,/(-b,,ln’).