Science in China Series E: Technological Sciences
© 2007 Science in China Press
Springer-Verlag
Received January 30, 2007; accepted March 16, 2007 doi: 10.1007/s11431-007-0041-1
†Corresponding author (email: [email protected])
Supported by the National Natural Science Foundation of China (Grant No. 60438030) and the Key Research Foundation of Scientific and Technical Committee of Tianjin City of China (Grant No. 033183711)
Optical study on the vision correction and
supernormal vision based on the wave-front
aberrations of human eye
MU GuoGuang, WANG ZhaoQi†, LIU YongJi, QUAN Wei, WANG Yang & WANG Wei
Institute of Modern Optics, Nankai University, Tianjin 300071, China
In this paper we present the recent research results in the field of vision correction and supernormal vision according to the actual measurements of the wave-front aberrations and the corneal surface topography, the clinical detection of the visual function and the laser corneal refractive surgery, and the optimization of the optical system. These include the features of the aberrations of human eye with different pupil sizes, different fields of view and temporal accommodation, the influence of the polychromatic illumination of the visible wavelength on the supernormal vision, and the effect of the existing laser corneal refractive surgery on the wave-front aberrations of the eye. It is shown that the wave-front aberration of human eye is of temporal variation and of synthesis with multi impact factors. To achieve super-normal vision, an optimum engineering data for the customized laser corneal sur-gery should be firstly acquired, which may involve the dynamic free-form optical surface. Although the myopia can be corrected by the laser in situ keratomileusis (LASIK) in a certain degree, it brings about negative effects under scotopic condi-tions.
wave-front aberrations of human eye, vision correction, laser corneal refractive surgery
1 Introduction
A significant development in the visual optics and ophthalmology at the end of 20th century is the progress in the measurements of the wave-front aberrations of human eye and the advanced vision correction with the laser corneal refractive surgery. In the middle of 1990s, Liang and Grimm0H
[1]
successfully measured the wave-front aberrations of human eye with a new kind of wave-front sensing optometer, the Hartmann-Shack sensor, and characterized the aberrations in detail with the Zernike terms up to the 10th order. They revealed the influence of the high-order aberrations on the visual quality. With the adaptive optics technology, they firstly obtained the high-resolution image
of living retina and suggested the possibility of a supernormal vision by the correction of the higher-order aberrations1H
[2]
. From then on, the advanced vision engineering based on the wave-front aberrations has extremely attracted people’s attention, and a large amount of experimental and clinical researches have been developed involving the features and the impact factors of the wave-front aberrations of human eye2H
[3]
, the sources of the wave-front aberrations3H
[4]
, the relation-ship between the visual function and the wave-front aberrations4H
[2]
, the changes of the aberrations of human eye after the laser corneal refractive surgery5H
[5,6]
, and the customized vision correction techniques based on wave front aberrations including the laser corneal refractive surgery and the phakic intraocular lenses6H
[7]
.
Although the measurement of the wave-front aberrations of human eye provides more accurate and rich information of the eye’s optical system, the environment of the measurement differs from the actual environment that the human eyes deal with. With the wave-front technology, the defects and the visual functions of the eye’s optical system are investigated based on the diffraction optics, and the best vision correction is based on the optimal compensation of the wave-front. That is to say, the eye’s optical system is regarded as a coherent optical system. However, normally the visual information received is from an outer world with natural light, which is either temporally or spa-tially incoherent. Therefore the study of the advanced vision correction and the supernormal vision should be based on the incoherent optical system. Furthermore the wave-front aberration of the eye varies with various temporal and spatial factors, such as the aperture, the field of view, the temporal accommodation and the object distance. All of these factors should be taken into account when the data of the wave-front aberrations measured in a specific condition are used for the vision correc-tion.
Since the wave-front aberrations are successfully measured, researchers pay great deal of at-tention on the potential of the advanced vision correction, that is, how much the benefit can be achieved by the customized vision corrections. There are many factors which limit the curative effect of the vision correction including Nyquist limit and diffraction limit. Figure 1 shows the impact of different factors on the vision correction. In Figure 1, the dashed curve represents the visual acuity under the diffraction limit, while the horizontal solid line represents that under the Nyquist limit of the retina. The solid curve represents the visual acuity achieved by traditional vision correction (the correction of the defocus and astigmatism only), the normal vision. It can be seen that the visual acuity obtained by the traditional vision correction is lower than that under diffraction limit and the Nyquist limit. It is believed that this is due to the existence of the high-order aberrations of human eye, and by the correction of the high-order aberrations a visual acuity close to the Nyquist limit about 2.0 can be achieved, the supernormal vision.
We have constructed the optical system for the measurement of the wave-front aberrations of human eye based on Hartmann-Shack technology, with an international collaboration. The speci-fications of the wave-front aberrometer are as follows:
Pupil size: > 2 mm; pupil sampling: > 500; sampling duration: = 0.25 mm; Zernike order (Terms): 12 (90); refraction range: spherical: ±10D, cylindrical ± 6D; refraction precision: 0.1D; Zernike high-order precision: 0.1 μm; time: < 0.2 s.
The constructed wave-front aberrometer is shown in Figure 2.
With the constructed wave-front aberrometer, we measured the wave-front aberrations of eyes under different situations, and optically investigated the vision correction and supernormal vision according to the acquired data of the aberrations and other clinical data in optometry. The main
Figure 1 The impact of different limits on the benefit of vision correction.
Figure 2 The constructed optical system for measurement of wave-front aberrations of human eye based on Hartmann-Shack technology.
research includes the features of the aberrations of human eye with different pupil sizes, different field of view7H
[8]
, different object distance8H
[9]
and temporal accommodation9H
[10]
, the influence of the polychromatic illumination of visible light on the supernormal vision, and the impact of the ex-isting laser corneal refractive surgery on the wave-front aberrations of the eye10H
[11]
2 Characteristics of wave-front aberrations of human eye
The characteristics of the aberrations of human eye are investigated, such as the aberrations for different pupil sizes, field of view, object distance and wavelength, and the dynamics of the aber-rations.
Figure 3 shows the RMS wave-front error contributed by each Zernike order for the pupil di-ameter of 3 mm (squares) and 7 mm (triangles), respectively, averaged across 20 human eyes. Error bars indicate the standard deviation among eyes. Dashed line represents the Marechal diffraction tolerance (λ/14). The ordinate is the RMS value of the aberration in waves, and the abscissa is the Zernike order. It can be seen that for the pupil with the diameter of 3 mm, the mean RMS value of Zernike order up to the second exceeds the Marechal diffraction tolerance. For the pupil diameter of 7 mm, however, it is up to the seventh Zernike order that the mean RMS value exceeds the Marechal diffraction tolerance. It can also be seen that the RMS value for the pupil diameter of 7
－
mm lies 2 5 times higher than that for the pupil diameter of 3 mm. This illustrates the fact that the aberrations grow up with the increasing pupil size.
Aberrations of 12 human eyes for pupil diameter of 5.2 mm both temporally and nasally across 50° horizontal visual field are measured. Measurements are taken in 10° intervals between −50° temporal visual field and +50° nasal visual field. The curves from the top to bottom in Figure 4 show the mean RMS of Zernike aberrations from the third to tenth order, respectively. Dashed line shows the Marechal diffraction tolerance. The ordinate represents the RMS value of the aberration in waves, and the abscissa represents the visual angle. It can be seen that the Zernike aberrations increase as the visual angle increases. At 0° visual field, the Zernike aberration up to the fourth-order exceeds the Marechal tolerance, but it is up to the fifth-order that the Zernike tion exceeds the Marechal tolerance at ±10° visual field. At ±50° visual field, the Zernike aberra-tion up to the sixth-order exceeds the Marechal tolerance. It can also be seen that the Zernike aberration of the third-order increases up to 2.5 times for the visual angle from 0° to ±50°, that of the forth-order increases up to 2 times, that of the fifth to tenth order increases up to 1.7 to 1.3 times, respectively. Although we present here the results of the wave-front aberrations of the eye at the visual field up to ±50°, it is up to ±10° visual field that is enough for consideration in the practical applications of the wave-front aberrations due to the gazing feature of human eye.
Figure 3 The feature of the wave-front aberrations for Figure 4 The feature of the wave-front aberrations for different pupil sizes. different fields of view.
The dynamic wave-front aberrations across pupil diameter of 6 mm for 12 eyes are measured at temporal frequency of 25 Hz. Figure 5 shows the temporal variation of the wave-front aberrations for one of eyes. The curves from the top to bottom represent the temporal traces of the total wave-front aberration, Zernike defocus, high-order Zernike aberration (defocus and astigmatism removed), the 3rd-order Zernike aberration, artificial eye total wave-front aberration, the 4th-order Zernike aberration, the 5th-order Zernike aberration, the 6th-order Zernike aberration, the 7th- order Zernike aberration, respectively. It can be seen that the temporal trace of the artificial eye is nearly flat with very low in magnitude of the fluctuation, indicating that artifacts are not introduced by the instrument or the data analysis. The difference of the temporal traces between the artificial and the actual eye illustrates that photon noise and CCD read noise are not responsible for the temporal behavior of the actual eye. The amplitude of the fluctuation of the total wave-front ab-erration for normal eye is 0.1 μm，and that of the high-order Zernike abab-erration (defocus and astigmatism removed) is 0.06 μm, which is larger than the Marechal diffraction tolerance (0.056 μm). This implies that by the correction of the wave-front aberration the effect of the temporal fluctuation of the ocular aberration should not be ignored. It can also be seen that the temporal traces of the total wave-front aberration and the Zernike aberrations from the 3rd to 7th order are similar to that of the defocus term. This indicates that the fluctuation of the accommodation is the primary cause of the fluctuation in the eye’s aberrations.
Figure 5 Variation of the wave-front aberrations as temporal accommodation of human eye.
For the investigation of the variation of the wave-front aberration of the eye as the object dis-tance we firstly construct the optical system of the individual eye. The establishment procedure is as follows:
a) The wave-front aberration of the individual eye is acquired with the Hartmann-Shack aber-rometer, which is then fed into the operand ZERN in the merit function to define the constructed lens with the lens design software Zemax.
b) The topography of the anterior and the posterior corneal surfaces is acquired by the corneal topographic system Orbscan, which is then transformed into the data along the optical axis and fitted with high-order aspheric surface. The data of the corneal surfaces are then fed into the lens structure.
c) The eye’s axial lengths including the depth of cornea, anterior chamber, crystalline lens and vitreous body are acquired with the medical BMF-200 A/B Ultrasonic Diagnostic Instrument, which are then fed into the lens structure.
d) The refractive indexes of the elements of the eye optics adopted for the lens structure, in-cluding the cornea, the anterior chamber, the crystalline lens and the vitreous body, are the same as these of Gullstrand eye model.
e) The radiuses of the crystalline lens of Gullstrand eye model are also adopted, and the aspheric surfaces are obtained by optimization procedure. This is done by setting the surfaces of the crys-talline lens as Zernike Fringe Sag surfaces and defining the coefficients as variables. After opti-mization the individual eye structure is obtained which possesses the same aberrations as the actual eye.
With the above procedure we construct 10 in-dividual normal eyes with the pupil diameter of 5.4 mm. By changing the object distance and making accommodation with the front curvature of the crystalline lens as variable, the influence of the object distance on the high-order aberrations is acquired as shown in Figure 6. In this research the defocus and astigmatism are corrected be- forehand. The ordinate represents the RMS value of the aberration in waves (λ=833 nm), and the abscissa represents the eyes arranged according to the value of the wave-front aberration with the object at infinite distance. The data with solid line represent the wave-front aberration for the object at infinite distance (0D), while the data with the symbols of solid square and solid triangle represent respectively the wave-front aberration for the object at 500 mm (−2D) and 250 mm (−4D). It can be seen from Figure 6 that with the defocus and astigmatism corrected the total aberration for the object at finite distance is normally higher than that at infinite distance, and for most of the eyes the aberration varies dramatically as the object distance varies. For instance, for the second eye, the total aberration is 0.42λ for an infinite object, and it is 0.8λ and 0.5λ for the object at −2D and −4D, respectively. This brings about difficulty for the higher order aberration correction: when the aberrations of the eye are totally corrected ac-cording to the data for an infinite object, a certain amount of aberrations still exist when the object is at finite distance.
With the constructed individual eye structure the influence of the natural light illumination on the curable effect of the customized vision correction can be investigated by changing the working wavelength of the system into visible light and making accommodation again. We construct 8 normal eyes with the pupil diameter of 5.4 mm. To simulate the correction of both the defocus and the astigmatism a spherical-cylindrical lens is set in front of the corneaand its surfaces’ curvature
Figure 6 Variation of the wave-front aberrations as object distance when the defocus and astigmatism are corrected.
radius are optimized to correct the defocus and astigmatism. For the high-order aberrations, a kind of higher-order aspheric surface, Zernike fringe sagittal surface is set on the sphere surface of the spherical-cylindrical lens and the Zernike coefficients are optimized to correct the higher-order aberrations. Figure 7 shows the modulation transfer function (MTF) of one of the 8 eyes after above correction where the dashed line shows the MTF under diffraction limit, the curve with cross symbol represents the MTF with the correction of the defocus, astigmatism and all high-order aberrations in monochromatic light, the curve with solid circle symbol represents the MTF with the correction of the defocus and astigmatism only
at 3 mm pupil size in natural light which is the achievable result for the conventional vision correction, and the curve with hollow circle symbol represents the MTF with the correction of the defocus, astigmatism and all high-order aberrations in natural light which is the achievable result for the customized vision correction. We also plot the aerial image modu-lation curve of human retina (AIM) which is represented by the solid curve in Figure 7.It is known that the AIM curve of human retina in-dicates the image modulation required in order for that image to be resolved by the visual sys-tem, and the point of the intersection between AIM curve and MTF curve represents the visual acuity.
It can be seen that the MTF curve with the correction of the defocus, astigmatism and all high-order aberrations in monochromatic light is very close to the curve under diffraction limit. However, in natural light, it is far from that under diffraction limit due to the chromatic aberrations. At lower spatial frequencies, it is even lower than that for the eye with the correction of only the defocus and astigmatism at 3 mm pupil size. From the intersection point between the MTF and the AIM we can obtain the achievable spatial frequency and the correspondent visual acuity for dif-ferent vision correction. A spatial frequency of 30 c/deg corresponds approximately to a clinic visual acuity of 1.0. It can be seen that with the customized vision correction a visual acuity of 1.9 can be achieved in monochromatic light illumination, but the achievable visual acuity is only 1.5 in natural light illumination. It is noted that with the conventional vision correction a visual acuity of 1.2 can also be obtained in natural light illumination. Thus we conclude that the benefit from the customized vision correction is not too much. For a specific eye with larger high-order aberrations the customized vision correction possesses advantage. However, most of the cases with the con-ventional vision correction a visual acuity greater than 1.2 can be achieved.
It is obvious from the above analysis that the human eye is an aberrational optical system with multi factors and dynamic variation. Therefore an optical research topic rises: how to acquire the correct engineering data of the aberrations for the customized vision correction to achieve the satisfactory curable effect.
3 The experimental study on post-LASIK wave-front aberrations
While we are conducting the study on wave-front aberrations of the human eye, a guidance issuedFigure 7 The feature of MTF of the optical system of eye with different illuminating light.
by the National Institute for Clinic Excellence11H
[12]
gets our attention. It says “The National Institute for Clinic Excellence (NICE) has today issued guidance on laser in situ keratomileusis (LASIK) for the treatment of refractive errors (short and long-sightedness) to determine whether it is safe enough and works well enough for routine use within the NHS”. “Professor Bruce Campbell, Chairman of the Interventional Procedures Advisory Committee says ‘LASIK offers improvement to people who are moderately short or longsighted. This is a problem that can easily be corrected by spectacles or contact lenses, so any risk of damage to the eye by LASIK is a real concern.’”. Therefore, our study on LASIK surgery is the evaluation on the laser corneal refractive surgery which is very popular in the market.
There are 84 eyes of the 42 patients are selected for bi-LASIK surgery. Of them 20 (48%) are male and others (52%) are female. All results are gotten 1－4 months later after the operation. The mean average age is 22 (between 20 and 38). The mean average refractive error is −5.92D (between −2.75D and −10.5D). All the patients are in good physical condition and go through complete ophthalmic checking including visual acuity, contrast sensitive function and wave-front aberra-tions. The patients are divided into four groups according to their post-LASIK visual symptoms. The symptom group includes starburst group (8 eyes), double-vision group (8 eyes) and moisture group (14 eyes). Eyes which do not suffer from visual symptoms are called reference group (54 eyes).
The absolute values of each Zernike coefficient for each group are shown in Figure 8(a)－(d). The ordinate is the absolute value of Zernike coefficient in micrometer; the abscissa represents the
－
1 35 Zernike order. Error bars indicate the standard deviation among eyes in each group. It can be seen from these figures that defocus still exists in the post-LASIK eye and it contributes more to the whole wave-front aberrations than other Zernike aberrations do. However, the visual acuity of the eye has been improved to above 1.0 by the correction of the defocus which is the main cause of the refractive error. After the LASIK surgery, the contribution of the Zernike coma (Z7, Z8), Zernike
spherical aberration (Z12) and Zernike astigmatism (Z3, Z5) to the whole wave-front aberrations of
the eye increases compared to that of the eye without LASIK surgery. This means the Zernike coma and Zernike spherical aberration increase after the refractive surgery. Zernike coma is caused by the asymmetry of the cornea which is induced by the refractive surgery. After LASIK surgery, horizontal Zernike coma (Z7), vertical Zernike coma (Z8) and Zernike spherical aberration (Z12) are
larger than Zernike astigmatism (Z3, Z5).
Figure 8(a) and (b) also show that the differences among Zernike coefficients in the reference group are rather small compared to those in other groups. Especially there are very little differences among the Zernike orders higher than Z13. The Zernike coefficient differences are quite large in the
starburst group in the contrary. The differences among the higher order Zernike coefficients in the starburst group are relatively larger than those in the other three groups. As the higher order ab-errations of the eye suffered from visual symptom are larger than those in the reference group, it is reasonable to suppose that the visual symptoms are related to the higher order aberrations.
The absolute values of the Zernike coefficient of the eye without LASIK surgery are shown in Figure 9. The ordinate is the absolute value of Zernike coefficient in micrometer and the abscissa represents the 1－35 Zernike order. It shows that the defocus of the eye without the LASIK surgery is quite large whereas the other Zernike coefficients are quite small. The differences among the higher order coefficients are small compared to those of the eyes undergoing the LASIK surgery. Whether the eye suffers from visual symptoms or not, the higher order aberrations of the eye in-crease, which is due to the roughness of the cornea after the LASIK surgery.
Figure 8 The absolute values of Zernike coefficient. (a) For the reference group; (b) for the starburst group; (c) for the dou-ble-vision group; (d) for the moisture group.
Figure10 shows the RMS wave-front error contributed by each Zernike order for the pupil di-ameter of 5.8 mm for each group respectively, averaged across all the eyes of each group. The ordinate is the RMS value of the aberration in micrometer, and the abscissa is the Zernike order. The square represents reference group, the circle represents starburst group, the upward triangle represents double-vision group and the downward triangle represents moisture group. Dashed line in the figure indicates the Marechal diffraction tolerance.
Figure 9 The absolute values of Zernike coefficient of the eye Figure10 The feature of the wave-front aberrations for each without refractive surgery. group.
It can be seen that the RMS wave-front error for each group is higher than the Marechal dif-fraction tolerance, which means the post-LASIK eye is not an ideal optical system. It can also be seen that the RMS wave-front error in the reference group is less than that in the other groups. The third and fourth order RMS wave-front errors in the starburst group are about two times larger than the corresponding ones in the reference group. Except for the RMS wave-front error of the second order, other order RMS wave-front error in the symptom group is larger than the corresponding one in the reference group. The figure shows little difference of the RMS wave-front error between the double-vision and reference group. The comparison above clearly shows the differences between the reference group and the symptom group.
MTF curve is capable of estimating the quality of an optical system. So the MTF of each group is analyzed. The MTF is calculated from the wave-front aberrations without the second order ab-errations (defocus and Zernike astigmatism), averaged across all the eyes in each group. The one-dimensional MTF is obtained by averaging the two-dimensional MTF across all directions (radial average).
Figure 11 shows MTF curves for each group at pupil diameter of 3 and 5.8 mm respectively. The ordinate is the logarithmic coordinate of the MTF and the abscissa represents the spatial frequency. The square represents reference group, the circle represents starburst group, the upward triangle represents double-vision group and the downward triangle represents moisture group. Dashed line corresponds to the MTF value of 0.1. It can be seen from the figures that the MTF value for the reference group is larger than that for the symptom group, which means the quality of the eye in the reference group is better than that in the symptom group. The MTF for the starburst group is the lowest among the four groups which is due to the higher third order Zernike aberrations as shown in Figure 10. It can also be seen that for small pupil size the MTF differences among the groups are relatively small, whereas for large pupil size the differences are becoming obvious, which means that the quality of the symptomatic eye is getting worse under scotopic conditions.
Figure 11 MTF for the referent group and symptom group for the pupil diameter of 3 mm (a) and 5.8 mm (b).
4 Conclusions
It is obvious from the above analysis that the human eye is an aberrational optical system with multi factors, and with dynamic variation. Just like the aberrations of the normal optical system, the
wave-front aberrations of the human eye are affected by the aperture size, the field of view, the object distance and the wave length, and vary temporally as well. To achieve a certain degree of supernormal vision, the optimal engineering data for the customized laser corneal surgery should be acquired. However, according to the analysis above, the customized vision correction is very complicated, which may involve dynamic free-form optical surface.
Although the myopia can be corrected by LASIK in a certain degree, it brings out negative ef-fects under scotopic conditions.
The experimental study on post-LASIK wave-front aberrations was conducted as a collaborative effort with the Tianjin Eye Hospital.
1 Liang J Z, Grimm B, Goelz S, et al. Objective measurement of wave aberrations of the human eye with the use of Hart-mann-Shack wave-front sensor. J Opt Soc Am A, 1994, 11(7): 1949－1957
2 Liang J Z, Williams D R. Aberrations and retinal image quality of the normal human eye. J Opt Soc Am A, 1997, 14(11): 2873－2883
3 12HYoon G Y, Williams D R. Visual performance after correcting the monochromatic and chromatic aberrations of the eye. J Opt
Soc Am A, 2002, 19(2): 266－275
4 He J C, Gwiazda J, Thorn F, et al. Wave-front aberrations in the anterior corneal surface and the whole eye. J Opt Soc Am A, －
2003, 20(7): 1155 1163
5 Mierdel P, Kaemmerer M, Krinke H E, et al. Effects of photorefractive keratectomy and cataract surgery on ocular optical errors of higher order. Graefes Arch Clin Exp Ophthalmol, 1999, 237: 725－72913H[DOI]
6 Guirao A, Gonzalez C, Redondo M, et al. Average optical performance of the human eye as a function of age in a normal population. Invest Ophthalmol Visual Sci, 1999, 40: 203－21314H[DOI]
7 15HBellucci R, 16HMorselli S, 17HPucci V. Spherical aberration and coma with an aspherical and a spherical intraocular lens in normal
age-matched eyes. J Cataract Refract Surg, 2007, 33(2): 203－209
8 Wang Y，Wang Z Q, Liu M, et al. Study on the aberrations of human eyes at wide field of view based on individual eye model. Acta Opt Sin, 2006, 26(11): 1727－1733
9 Wang Y，Wang Z Q. Study on accommodation of human eyes based on individual eye model. Chin J Sci Instr, 2006, 27(6): －
1049 1051
10 Wang Y, Wang Z Q, Guo H Q, et al. Wavefront aberrations in the accommodated human eye based on individual eye model. Optik, in press
11 Liu Y J, Mu G G, Wang Z Q, et al. Correlation between post-LASIK starburst symptom and ocular wavefront aberrations. Chin Phys Lett, 2006, 23(6): 1498－150018H[DOI]
12 National Institute for Clinical Excellence. NICE issues guidance on laser eye surgery (LASIK) for treating refractive errors, http: // guidance.nice.org.uk/IPG102/?C=91506