- Email: [email protected]
Optical performance of the eye with progressive addition lens correction
Optik 121 (2010) 1937–1940
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Optical performance of the eye with progressive addition lens correction Agnieszka Barcik a,, Damian Siedlecki a,b,1 a b
Institute of Physics, Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50370 Wroclaw, Poland Instituto de Optica, Consejo Superior de Investigaci´ on Cient´ıficas, Serrano 121, 28006 Madrid, Spain
a r t i c l e in fo
abstract
Article history: Received 9 February 2009 Accepted 14 May 2009
The aim of this study was to simulate the retinal image of the eye with progressive addition lens (PAL), correction (without cylindrical component) and investigate its quality. The optical design and analysis software was used in simulation. The topographical data of the front surface of PAL were taken from the scanning profilometry. Also the coordinates of the points of the distant and near power measurements were taken from topographical measurements. The back surface was assumed to be spherical with a radius of curvature matched so that the optical power of PAL meets the data given by the manufacturer. In order to describe the retinal image quality, various metrics were calculated in comparison to the eye without correction. & 2009 Elsevier GmbH. All rights reserved.
Keywords: Progressive addition lenses Presbyopia Image quality metrics Eye schematic model
1. Introduction Presbyopia is a vision disorder in which the crystalline lens loses its flexibility, which is a natural part of the aging process of the eye. A presbyopic eye is not able to image close objects properly, because of the loss of accommodation ability [1]. Although it is not a disease, becomes a problem for people in their mid-40’s when reading and working with computers becomes more and more difficult. There are several ways to correct presbyopia, such as bifocal or trifocal spectacles lenses, progressive contact lenses or progressive addition lenses (PALs) [2,3]. Progressive addition lenses (PAL) have been commercially available for more than 50 years now [4,5]. During this time a huge variety of different types of PALs have been designed and manufactured [4,6–8]. They are proven to be a very successful treatment for presbyopia and are considered so far to be the best solution for people suffering from presbyopia, especially for those, who work with computers. PALs, in contrast to bifocal or trifocal spectacle lenses, are seamless multifocal lenses characterized by power changing along the umbilical line, which account for central line of progressive corridor. Optical power may change linearly or in a more complex way [6]. However, the technology of manufacturing is not well known and in most cases it usually states an object of commercial secret of the producer. The topography of the progressive surface
Corresponding author. Tel.: +487 13203774; fax: +48 71 328 36 96.
E-mail addresses: [email protected] (A. Barcik), [email protected] (D. Siedlecki). 1 Tel.: +34 91 561 68 00; fax: +34 91 564 55 57. 0030-4026/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.05.030
is complex and its exact form is also confidential. Therefore there is a need on carrying out researches that would lead to finding the optimal shape of the progressive surface in order to improve the retinal image quality and increase patient’s comfort. Some patients complain about inconvenience relating to the need of head movements [9], blurs and unnatural motion of image [10]. An eye viewing a point object in the best focus through a progression corridor or through other areas of the lens, is not be able to image a point on retina. It is well known that in PALs, the continuous change of defocus over the lens induces peripheral astigmatism which increases progressively outside the corridor [11,12]. It is not possible to produce a progressive spherical power surface without astigmatism and distortion being present at some point [11]. These makes vision through some parts of the lens degraded and may play a role in reducing the success of the adaptation process. One cannot remove these negative factors definitely, but they can be reduced due to the improvements in the progressive surface. Real PAL measure is possible by means of various methods including moire deflectometry [13], Hartmann–Shack wave front sensor [14] and automated focimeter [15]. These methods are able to measure the optical properties of the PALs, but there is no possibility for them to test only the progression surface. In order to predict the visual performance of a system including PAL and eye, some optical parameters have been proposed for assessing the image quality of visual instruments with small sizes of pupil [16], like radius or 80% encircled energy of the point spread function or frequency of modulation transfer function, where it is decreased by half. Present study constitutes a trial of the optical performance estimation for the system including eye plus commercially available PAL based on the PAL’s topographical data coming from
1938
A. Barcik, D. Siedlecki / Optik 121 (2010) 1937–1940
the scanning measurements. The quality of the retinal image is described by means of several quantities (geometrical and diffractive) in order to fully characterize the optical performance of the eye plus PAL system.
2. Materials and methods A progressive addition lens of a Rodenstock’s line LIFE having the following correction: Sph: 0.00D, Cyl: 0.00D, Add: +1.75D with an anti-reflective coating Solitaire SC was undergone the progressive surface topography measurement using Renishaw Cyclone 2 scanning system (with the axial resolution equal to 1 mm). This repeatability seems inadequate to determine higher order aberration, but purpose of this work is research of low-order aberration, because their influence on the quality of vision is significantly higher than the higher order aberrations [17]. Then the data points were approximated with an extended polynomial defined by the following formula: z¼
cr2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ q1 x þ q2 x2 þ q3 x3 þ q4 x4 þ q5 x5 1 þ 1 ðc2 þ r 2 Þ þ q6 x6 þ q7 x7 þ q8 x8 þ q9 y þ q10 y2 þ q11 y3 þ q12 y4 þ q13 y5 þ q14 y6 þ q15 y7 þ q16 y8
ð1Þ
where c is the curvature (the reciprocal of the radius), r is the radial coordinate in lens units and qi is the coefficient on the ith extended polynomial term (i ¼ 1–16). As a result of approximation we were given continuous change of curvature in each points of surface. Next the formula was implemented into the Zemax optical design software [18], where posterior surface was calculated assuming its spherical and centered shape. PAL was combined with the Gullstrand–Le Grand schematic eye model [19] with 4 mm pupil diameter. For the analysis the area with 40 mm in diameter and wavelength equal to 589 nm were chosen. This eye model is one of the most commonly used in ophthalmology and due to its compromise between accuracy and relative simplicity. Construction of this model was based on the biometric data. Many researches modified this model by incorporating the aspherical surfaces, gradient refractive index distribution in the crystalline lens medium, etc. [20–22]. There were 5 chosen points (‘‘far’’ and ‘‘near’’ points in which the power for far and near vision were measured, and 3 intermediate points from the progression corridor; see Fig. 1), for which the retinal image quality was estimated. In order to do so, the progressive addition lens was rotated around the z-axis in order for the considered point to have positive y-coordinate, the x-coordinate to be equal to 0. Then the PAL was rotated again, this time around the rotation point of the eye (which was assumed to
Fig. 1. The result of the approximation of the topographical data points of the anterior surface of the lens. 5 points for which the performance was estimated are marked.
be placed 12 mm behind the anterior surface of the cornea) in order for the considered point to be located on the optical axis of the eye model. The same scheme was applied for all of the 5 points mentioned above. In this process also the pantoscopic angle of the spectacles and the convergence angle of the eye were taken into account. Such scheme means that the angle of inclination of the PAL is larger for the near points and smaller for area of far vision (it can be observed in the Fig. 2), which of course takes place in real ‘‘eye plus spectacle lens systems.’’ All the simulations were performed for monochromatic light (l ¼ 589 nm).
3. Results Estimation of image quality was made by means of various techniques. Wavefront aberrations were described using Zernike polynomials expansions. Spot diagram presents a point image changed by aberrations (especially spherical aberration and astigmatism). For the image quality estimation the root-meansquare error (RMS) of the spot diagram was chosen as one the essential metric. The RMS spot radius means the root-meansquare radial size. The distance between each ray and the reference point is squared, and averaged over all the rays, and then the square root is taken [18]. Point spread function and the modulation transfer function were used for analysis, as well. Power and astigmatism contour diagrams were not used for analysis, because they override previous surface influence on vision [23]. However, relationship between them is an important issue in designing optical process, exemplary to describe prismatic effect in binocular vision. We have compared the retinal image quality for five points lying in different regions of vision area on progressive addition lens (Fig. 1) using the metrics mentioned in the previous paragraph. For the intermediate points from the progression corridor (points nos. 2–4) an object distances was estimated, for which the aberration spots on the retina was the smallest (the smallest RMS radius). The object distances are shown in the first column in Table 1. Fig. 3 presents RMS radius (in mm) for spot diagrams for all the considered points in comparison to the RMS radius calculated for the naked eye. Spot size depends of course on the wavefront aberrations. RMS for far point is most similar to the naked eye. On the Fig. 4 several spot diagrams are shown. The symmetry with respect to vertical axis is a consequence of the lens rotation scheme assumed and applied. On the other hand, in horizontal direction there is asymmetry which comes out because of the lens inclination. The smaller rotation of lens in relation to the eye rotation point the larger symmetry occurs. On these pictures (Fig. 4) a relatively large defocus and spherical aberration can be observed, as well. Strehl ratio (Fig. 5) in all cases achieves better values than those for a naked eye. They fit into range (0.02, 0.16) – in the literature these results are acknowledged [24].
Fig. 2. The tilt of the spectacle lens for far (a) and near (b) vision.
A. Barcik, D. Siedlecki / Optik 121 (2010) 1937–1940
1939
Table 1 Wavefront aberrations (in l) calculated for various points under consideration. Object distance
Near point (650 mm) Point no. 2 (750 mm) Point no. 3 (830 mm) Point no. 4 (855 mm) Far point (infinity) Naked eye (infinity)
Zernike term Astigmatism Z22
Coma Z1 3
Spherical Z04
0.0007 0.0024 0.0079 0.0021 0
0.0240 0.0527 0.0775 0.0434 0
0.306 0.287 0.269 0.317 0.320 Fig. 6. Retinal images for 5 points of system progressive addition lens plus eye and naked eye expressed by normalized point spread function.
Fig. 3. Values of RMS radius of spot diagram (in mm) for all points under consideration and naked model eye.
Fig. 7. Values of D08 parameter for all points under consideration in comparison to the naked model eye.
Fig. 4. Spot diagrams calculated for points of near (a), intermediate (b–d), and far (e) vision in comparison to the spot diagram of the model eye itself. The size of all of the images is 200 mm.
Fig. 8. Values of MTF50 parameter (in tangential and saggital direction) for particular points of lens and naked eye.
Fig. 5. Values of normalized Strehl ratio for all points under consideration in comparison to the naked eye.
On the Fig. 6 an exemplary point spread function for various areas in progressive addition lens is presented. Shape of the PSF depends on the combination of eyes’ aberration and appropriate regions of progressive addition lens. Relatively large spherical aberration disturbs sight of different aberration. D08 parameter (Fig. 7) defines a value of radius circle inclusive 80% energy of the light in the spot. D08 parameter for far point is comparable with a naked eye. Remaining points characterize similar values to they fit into 10–20 mm range.
Fig. 8 presents the values of the MTF50 parameter which is the frequency (in cycles/degree) for which the MTF drops to a value of 0.5. Differences between saggital and tangential directions for particular points are results of the magnification and distortion effects in the progression and near addition. The values of this parameter are below 10 cycles/degree, which is larger than for the naked model eye. For the far point the value is almost the same as for the naked eye. All these values are reasonable, because the peak of the contrast sensitivity for human eye is between 3 and 6 cycles/degree. In this case of out model the condition seems to be fulfilled. Table 1 presents wavefront aberrations in wavelengths scale (in terms of Zernike coefficients) having the most significant influence on the image quality of presented model. The aberrations coefficients were calculated using Zernike Standard Coefficients, which were computed using earlier defined notation [25]. This Zernike decomposition method conform to the standards set
1940
A. Barcik, D. Siedlecki / Optik 121 (2010) 1937–1940
in ophthalmic field for reporting the aberrations of the eye [26]. These aberrations are astigmatism Z22, coma Z1 and spherical 3 aberration Z04. Spherical aberration Z04 is similar for points coming from progressive addition lens plus eye structure and naked eye, where it occurs about 0.3l. Coma Z1 3 appears as result of the tilt of the lens and also as a result of different power existing on lower and higher pupil’s edge, induced by progression. Very absorbing is astigmatism Z22, which appears as a result of change of power in various areas of lens. Similar results were found in the literature [14,24].
4. Discussion The investigated PAL’s (0D correction for distant vision) optical performance for far vision is very similar to the performance of the (emmetropic) eye model itself. For near vision it is slightly improved and for intermediate object points the image quality is significantly improved, especially in terms of diffractive (Strehl ratio, MTF, encircled energy distribution) and geometrical (spot diagram) quantities rather than wavefront aberrations. Relying on these metrics one can state that the model has been set up correctly. However, it should be emphasized that the optical performance for the progression corridor depends strongly on the proper selection of the object distances. In current study it was simply assumed that these distances should be the closer to the near distance, the closer was the position of the considered point from the corridor to the ‘‘near point’’ on the lens surface. Further investigations will be focused on more complex spectacle lenses like the ones with non-zero correction for distant vision and/or cylindrical correction with use of more complex and more relevant eye schematic models [27–33]. It has been shown that the topography of the front surface of PAL can be successfully reconstructed with use of scanning profilometers and may be implemented into any of the optical software in order to estimate their quality of correction and the optical performance of the system ‘‘eye plus PAL’’. However, some assumptions need to be made, about the shape of the back surface, the thickness of the lens, etc. A special attention must be paid while choosing the object distances, particularly for intermediate distances between the far and near points. In order to do so, a decision must be taken on which optical quantity describing the performance on the retina (rather than Strehl ratio) should be optimized.
Acknowledgements Authors would like to thank Przemys"aw Kolinka from the Institute of Production Engineering and Automation, Wroclaw University of Technology, for carrying out the topographical measurements. References [1] W.N. Charman, The eye in focus: accommodation and presbyopia, Clin. Exp. Optom. 91 (2008) 207–225. [2] E.S. Bennett, Contact lens correction of presbyopia, Clin. Exp. Optom. 91 (2008) 265–278.
[3] M.J. Collins, B. Brown, K.J. Bowman, Contrast sensitivity with contact lens corrections for presbyopia, Ophthal. Physiol. Opt. 9 (1989) 133–138. [4] C. Fowler, Recent trends in progressive power lenses, Ophthal. Physiol Opt. 18 (1998) 234–237. [5] C.W. Fowler, A. Thompson, E. Bartlam, A wearer trial of two rear surface progressive addition spectacle lenses, Optom. Practice 3 (2002) 51–53. [6] G.H. Guilino, Design philosophy for progressive addition lenses, Appl. Opt. 32 (1993) 111–117. [7] D.J. Meister, S.W. Fisher, Progress in the spectacle correction of presbyopia. Part 1: Design and development of progressive lenses, Clin. Exp. Optom. 91 (2008) 240–250. [8] D.J. Meister, S.W. Fisher, Progress in the spectacle correction of presbyopia. Part 2: Modern progressive lens technologies, Clin. Exp. Optom. 91 (2008) 251–264. [9] N. Hutchings, E.L. Irving, N. Jung, L.M. Dowling, K.A. Wells, Eye and head movement alterations in na¨ıve progressive addition lens wearers, Ophthal. Physiol. Opt. 27 (2007) 142–153. [10] C.M. Sullivan, C.W. Fowler, Visual detection and adaptation to optically induced curvature distortion. Does curvature distortion govern progressive addition lens tolerance, Appl. Opt. 32 (1993) 4138–4143. ¨ ¨ [11] G. Minkwitz, Uber den Flachenastigmatisms bei gewissen symmetrischen Aspharen, Opt. Acta 10 (1963) 223–227. [12] J.E. Sheedy, C. Campbell, E. King-Smith, J.R. Hayes, Progressive powered lenses: the Minkwitz theorem, Optom. Vis. Sci. 82 (2005) 916–922. [13] E. Keren, R. Berman, M. Hochner, An optical method for mapping the power and cylinder of multifocal progressive lenses, in: Ophthalmic and Visual Optics/Noninvasive Assessment of the Visual System Technical Digest, Optical Society of America, Washington, DC, 1993. [14] E.A. Villegas, P. Artal, Comparison of aberrations in different types of progressive power lenses, Ophthal. Physiol. Opt. 24 (2004) 419–426. [15] J.E. Sheedy, Optics of progressive addition lenses, Am. J. Optom. Physiol. Opt. 64 (2) (1987) 90–99. [16] P. Mouroulis, H.P. Zhang, Visual instrument image quality metrics and the effects of coma and astigmatism, J. Opt. Soc. Am. A 9 (1992) 34–42. [17] J.F. Castejon-Mochon, N. Lopez-Gil, A. Benito, P. Artal, Ocular wavefront aberration statistics in a normal young population, Vis. Res. 42 (2002) 1611–1617. [18] Zemax Optical Design Program, User’s Manual, Zemax Development Corporation, August 2007. [19] Y. Le Grand, S.G. El Hage, Psychological Optics, Springer – Verlag, Berlin, 1980, pp. 65–66. [20] W. Lotmar, Theoretical eye model with aspherics, J. Opt. Soc. Am. 61 (1971) 1522–1529. [21] A.C. Kooijman, Light distribution on the retina of a wide-angle theoretical eye, J. Opt. Soc. Am. 73 (1983) 1544–1550. [22] H.L. Liou, N.A. Brennan, Anatomically accurate, finite model eye for optical modeling, J. Opt. Soc. Am. A 14 (1997) 1684–1695. [23] B. Bourdoncle, J.P. Chauveau, J.L. Mercier, Traps in displaying optical performances of a progressive addition lenses, Appl. Opt. 31 (1992) 3586–3593. [24] E.A. Villegas, P. Artal, Visual acuity and optical parameters in progressivepower lenses, Optom. Vis. Sci. 83 (2006) 672–681. [25] R. Noll, Zernike polynomials and atmospheric turbulence, J. Opt. Soc. Am. 66 (3) (1976) 207. [26] L.N. Thibos, R.A. Applegate, J.T. Schwiegerling, R. Webb, Standards for reporting the optical aberrations of eyes, J. Refract. Surgery 18 (5) (2002) S652–S660. [27] D.A. Atchison, C.E. Jones, K.L. Schmid, N. Pritchard, J.M. Pope, W.E. Strugnell, R.A. Riley, Eye shape in emmetropia and myopia, Invest. Ophthalmol. Vis. Sci. 45 (2004) 3380–3386. [28] D.A. Atchison, Optical models for human myopic eyes, Vis. Res. 46 (2006) 2236–2250. [29] M. Dubbelman, G.L. van der Heijde, The shape of the aging human lens: curvature, equivalent refractive index and the lens paradox, Vis. Res. 41 (2001) 1867–1877. [30] M. Dubbelman, G.L. van der Heijde, H.A. Weeber, The thickness of the aging human lens obtained from corrected Scheimpflug images, Optom. Vis. Sci. 78 (2001) 411–416. ¨ [31] M. Dubbelman, H.A. Weeber, G.L. van der Heijde, H.J. Volker-Dieben, Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography, Acta Ophthalmol. Scand. 80 (2002) 379–383. [32] M. Dubbelman, G.L. van der Heijde, H.A. Weeber, G.F.J.M. Vrensen, Changes in the internal structure of the human crystalline lens with age and accommodation, Vis. Res. 43 (2003) 2363–2375. [33] M. Dubbelman, G.L. van der Heijde, H.A. Weeber, Change in shape of the aging human crystalline lens with accommodation, Vis. Res. 45 (2004) 117–132.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Optical performance of the eye with progressive addition lens correction Agnieszka Barcik a,, Damian Siedlecki a,b,1 a b
Institute of Physics, Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50370 Wroclaw, Poland Instituto de Optica, Consejo Superior de Investigaci´ on Cient´ıficas, Serrano 121, 28006 Madrid, Spain
a r t i c l e in fo
abstract
Article history: Received 9 February 2009 Accepted 14 May 2009
The aim of this study was to simulate the retinal image of the eye with progressive addition lens (PAL), correction (without cylindrical component) and investigate its quality. The optical design and analysis software was used in simulation. The topographical data of the front surface of PAL were taken from the scanning profilometry. Also the coordinates of the points of the distant and near power measurements were taken from topographical measurements. The back surface was assumed to be spherical with a radius of curvature matched so that the optical power of PAL meets the data given by the manufacturer. In order to describe the retinal image quality, various metrics were calculated in comparison to the eye without correction. & 2009 Elsevier GmbH. All rights reserved.
Keywords: Progressive addition lenses Presbyopia Image quality metrics Eye schematic model
1. Introduction Presbyopia is a vision disorder in which the crystalline lens loses its flexibility, which is a natural part of the aging process of the eye. A presbyopic eye is not able to image close objects properly, because of the loss of accommodation ability [1]. Although it is not a disease, becomes a problem for people in their mid-40’s when reading and working with computers becomes more and more difficult. There are several ways to correct presbyopia, such as bifocal or trifocal spectacles lenses, progressive contact lenses or progressive addition lenses (PALs) [2,3]. Progressive addition lenses (PAL) have been commercially available for more than 50 years now [4,5]. During this time a huge variety of different types of PALs have been designed and manufactured [4,6–8]. They are proven to be a very successful treatment for presbyopia and are considered so far to be the best solution for people suffering from presbyopia, especially for those, who work with computers. PALs, in contrast to bifocal or trifocal spectacle lenses, are seamless multifocal lenses characterized by power changing along the umbilical line, which account for central line of progressive corridor. Optical power may change linearly or in a more complex way [6]. However, the technology of manufacturing is not well known and in most cases it usually states an object of commercial secret of the producer. The topography of the progressive surface
Corresponding author. Tel.: +487 13203774; fax: +48 71 328 36 96.
E-mail addresses: [email protected] (A. Barcik), [email protected] (D. Siedlecki). 1 Tel.: +34 91 561 68 00; fax: +34 91 564 55 57. 0030-4026/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.05.030
is complex and its exact form is also confidential. Therefore there is a need on carrying out researches that would lead to finding the optimal shape of the progressive surface in order to improve the retinal image quality and increase patient’s comfort. Some patients complain about inconvenience relating to the need of head movements [9], blurs and unnatural motion of image [10]. An eye viewing a point object in the best focus through a progression corridor or through other areas of the lens, is not be able to image a point on retina. It is well known that in PALs, the continuous change of defocus over the lens induces peripheral astigmatism which increases progressively outside the corridor [11,12]. It is not possible to produce a progressive spherical power surface without astigmatism and distortion being present at some point [11]. These makes vision through some parts of the lens degraded and may play a role in reducing the success of the adaptation process. One cannot remove these negative factors definitely, but they can be reduced due to the improvements in the progressive surface. Real PAL measure is possible by means of various methods including moire deflectometry [13], Hartmann–Shack wave front sensor [14] and automated focimeter [15]. These methods are able to measure the optical properties of the PALs, but there is no possibility for them to test only the progression surface. In order to predict the visual performance of a system including PAL and eye, some optical parameters have been proposed for assessing the image quality of visual instruments with small sizes of pupil [16], like radius or 80% encircled energy of the point spread function or frequency of modulation transfer function, where it is decreased by half. Present study constitutes a trial of the optical performance estimation for the system including eye plus commercially available PAL based on the PAL’s topographical data coming from
1938
A. Barcik, D. Siedlecki / Optik 121 (2010) 1937–1940
the scanning measurements. The quality of the retinal image is described by means of several quantities (geometrical and diffractive) in order to fully characterize the optical performance of the eye plus PAL system.
2. Materials and methods A progressive addition lens of a Rodenstock’s line LIFE having the following correction: Sph: 0.00D, Cyl: 0.00D, Add: +1.75D with an anti-reflective coating Solitaire SC was undergone the progressive surface topography measurement using Renishaw Cyclone 2 scanning system (with the axial resolution equal to 1 mm). This repeatability seems inadequate to determine higher order aberration, but purpose of this work is research of low-order aberration, because their influence on the quality of vision is significantly higher than the higher order aberrations [17]. Then the data points were approximated with an extended polynomial defined by the following formula: z¼
cr2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ q1 x þ q2 x2 þ q3 x3 þ q4 x4 þ q5 x5 1 þ 1 ðc2 þ r 2 Þ þ q6 x6 þ q7 x7 þ q8 x8 þ q9 y þ q10 y2 þ q11 y3 þ q12 y4 þ q13 y5 þ q14 y6 þ q15 y7 þ q16 y8
ð1Þ
where c is the curvature (the reciprocal of the radius), r is the radial coordinate in lens units and qi is the coefficient on the ith extended polynomial term (i ¼ 1–16). As a result of approximation we were given continuous change of curvature in each points of surface. Next the formula was implemented into the Zemax optical design software [18], where posterior surface was calculated assuming its spherical and centered shape. PAL was combined with the Gullstrand–Le Grand schematic eye model [19] with 4 mm pupil diameter. For the analysis the area with 40 mm in diameter and wavelength equal to 589 nm were chosen. This eye model is one of the most commonly used in ophthalmology and due to its compromise between accuracy and relative simplicity. Construction of this model was based on the biometric data. Many researches modified this model by incorporating the aspherical surfaces, gradient refractive index distribution in the crystalline lens medium, etc. [20–22]. There were 5 chosen points (‘‘far’’ and ‘‘near’’ points in which the power for far and near vision were measured, and 3 intermediate points from the progression corridor; see Fig. 1), for which the retinal image quality was estimated. In order to do so, the progressive addition lens was rotated around the z-axis in order for the considered point to have positive y-coordinate, the x-coordinate to be equal to 0. Then the PAL was rotated again, this time around the rotation point of the eye (which was assumed to
Fig. 1. The result of the approximation of the topographical data points of the anterior surface of the lens. 5 points for which the performance was estimated are marked.
be placed 12 mm behind the anterior surface of the cornea) in order for the considered point to be located on the optical axis of the eye model. The same scheme was applied for all of the 5 points mentioned above. In this process also the pantoscopic angle of the spectacles and the convergence angle of the eye were taken into account. Such scheme means that the angle of inclination of the PAL is larger for the near points and smaller for area of far vision (it can be observed in the Fig. 2), which of course takes place in real ‘‘eye plus spectacle lens systems.’’ All the simulations were performed for monochromatic light (l ¼ 589 nm).
3. Results Estimation of image quality was made by means of various techniques. Wavefront aberrations were described using Zernike polynomials expansions. Spot diagram presents a point image changed by aberrations (especially spherical aberration and astigmatism). For the image quality estimation the root-meansquare error (RMS) of the spot diagram was chosen as one the essential metric. The RMS spot radius means the root-meansquare radial size. The distance between each ray and the reference point is squared, and averaged over all the rays, and then the square root is taken [18]. Point spread function and the modulation transfer function were used for analysis, as well. Power and astigmatism contour diagrams were not used for analysis, because they override previous surface influence on vision [23]. However, relationship between them is an important issue in designing optical process, exemplary to describe prismatic effect in binocular vision. We have compared the retinal image quality for five points lying in different regions of vision area on progressive addition lens (Fig. 1) using the metrics mentioned in the previous paragraph. For the intermediate points from the progression corridor (points nos. 2–4) an object distances was estimated, for which the aberration spots on the retina was the smallest (the smallest RMS radius). The object distances are shown in the first column in Table 1. Fig. 3 presents RMS radius (in mm) for spot diagrams for all the considered points in comparison to the RMS radius calculated for the naked eye. Spot size depends of course on the wavefront aberrations. RMS for far point is most similar to the naked eye. On the Fig. 4 several spot diagrams are shown. The symmetry with respect to vertical axis is a consequence of the lens rotation scheme assumed and applied. On the other hand, in horizontal direction there is asymmetry which comes out because of the lens inclination. The smaller rotation of lens in relation to the eye rotation point the larger symmetry occurs. On these pictures (Fig. 4) a relatively large defocus and spherical aberration can be observed, as well. Strehl ratio (Fig. 5) in all cases achieves better values than those for a naked eye. They fit into range (0.02, 0.16) – in the literature these results are acknowledged [24].
Fig. 2. The tilt of the spectacle lens for far (a) and near (b) vision.
A. Barcik, D. Siedlecki / Optik 121 (2010) 1937–1940
1939
Table 1 Wavefront aberrations (in l) calculated for various points under consideration. Object distance
Near point (650 mm) Point no. 2 (750 mm) Point no. 3 (830 mm) Point no. 4 (855 mm) Far point (infinity) Naked eye (infinity)
Zernike term Astigmatism Z22
Coma Z1 3
Spherical Z04
0.0007 0.0024 0.0079 0.0021 0
0.0240 0.0527 0.0775 0.0434 0
0.306 0.287 0.269 0.317 0.320 Fig. 6. Retinal images for 5 points of system progressive addition lens plus eye and naked eye expressed by normalized point spread function.
Fig. 3. Values of RMS radius of spot diagram (in mm) for all points under consideration and naked model eye.
Fig. 7. Values of D08 parameter for all points under consideration in comparison to the naked model eye.
Fig. 4. Spot diagrams calculated for points of near (a), intermediate (b–d), and far (e) vision in comparison to the spot diagram of the model eye itself. The size of all of the images is 200 mm.
Fig. 8. Values of MTF50 parameter (in tangential and saggital direction) for particular points of lens and naked eye.
Fig. 5. Values of normalized Strehl ratio for all points under consideration in comparison to the naked eye.
On the Fig. 6 an exemplary point spread function for various areas in progressive addition lens is presented. Shape of the PSF depends on the combination of eyes’ aberration and appropriate regions of progressive addition lens. Relatively large spherical aberration disturbs sight of different aberration. D08 parameter (Fig. 7) defines a value of radius circle inclusive 80% energy of the light in the spot. D08 parameter for far point is comparable with a naked eye. Remaining points characterize similar values to they fit into 10–20 mm range.
Fig. 8 presents the values of the MTF50 parameter which is the frequency (in cycles/degree) for which the MTF drops to a value of 0.5. Differences between saggital and tangential directions for particular points are results of the magnification and distortion effects in the progression and near addition. The values of this parameter are below 10 cycles/degree, which is larger than for the naked model eye. For the far point the value is almost the same as for the naked eye. All these values are reasonable, because the peak of the contrast sensitivity for human eye is between 3 and 6 cycles/degree. In this case of out model the condition seems to be fulfilled. Table 1 presents wavefront aberrations in wavelengths scale (in terms of Zernike coefficients) having the most significant influence on the image quality of presented model. The aberrations coefficients were calculated using Zernike Standard Coefficients, which were computed using earlier defined notation [25]. This Zernike decomposition method conform to the standards set
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in ophthalmic field for reporting the aberrations of the eye [26]. These aberrations are astigmatism Z22, coma Z1 and spherical 3 aberration Z04. Spherical aberration Z04 is similar for points coming from progressive addition lens plus eye structure and naked eye, where it occurs about 0.3l. Coma Z1 3 appears as result of the tilt of the lens and also as a result of different power existing on lower and higher pupil’s edge, induced by progression. Very absorbing is astigmatism Z22, which appears as a result of change of power in various areas of lens. Similar results were found in the literature [14,24].
4. Discussion The investigated PAL’s (0D correction for distant vision) optical performance for far vision is very similar to the performance of the (emmetropic) eye model itself. For near vision it is slightly improved and for intermediate object points the image quality is significantly improved, especially in terms of diffractive (Strehl ratio, MTF, encircled energy distribution) and geometrical (spot diagram) quantities rather than wavefront aberrations. Relying on these metrics one can state that the model has been set up correctly. However, it should be emphasized that the optical performance for the progression corridor depends strongly on the proper selection of the object distances. In current study it was simply assumed that these distances should be the closer to the near distance, the closer was the position of the considered point from the corridor to the ‘‘near point’’ on the lens surface. Further investigations will be focused on more complex spectacle lenses like the ones with non-zero correction for distant vision and/or cylindrical correction with use of more complex and more relevant eye schematic models [27–33]. It has been shown that the topography of the front surface of PAL can be successfully reconstructed with use of scanning profilometers and may be implemented into any of the optical software in order to estimate their quality of correction and the optical performance of the system ‘‘eye plus PAL’’. However, some assumptions need to be made, about the shape of the back surface, the thickness of the lens, etc. A special attention must be paid while choosing the object distances, particularly for intermediate distances between the far and near points. In order to do so, a decision must be taken on which optical quantity describing the performance on the retina (rather than Strehl ratio) should be optimized.
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