fax: (812) 855-7045
email: thibos
indiana.edu
Address for correspondence:
Larry N. Thibos
School of Optometry
Indiana University
Bloomington, IN 47405
voice: (812) 855-9842 or 855-4475
fax: (812) 855-7045
email: thibosindiana.edu
1. Ph.D., F.A.A.O.
2. Ph.D.
We evaluated the potential of wavefront shaping with liquid-crystals for modulating the eye's refractive state. A spatial light modulator with 127 liquid crystal cells was imaged in the entrance pupil of the eye and programmed to induce prismatic, spherical, and astigmatic refractive changes. Psychophysical evaluation of these optical effects were in agreement with expectations for prisms up to about 0.8 prism diopters and for lenses up to about 1.5 diopters. These maximum dioptric values represent wavefront retardation of about 3-4 wavelengths of 584 nm light across a 3 mm diameter pupil. Optical aliasing of high-power prisms was traced to spatial undersampling of the wavefront retardation function by the discrete array of liquid crystal cells. Undersampling may also be the factor which limits the useful dioptric range of the technique.
Adaptive optical systems change their characteristics in order to compensate for changes in the environment in which they work. For example, astronomical telescopes employ adaptive optics to compensate for the dynamic optical properties of our turbulent atmosphere. Recently Liang, Williams and Miller 1 developed an adaptive optical system for use in vision research based on the deformable mirror technology used in astronomy. They have shown that it is possible to correct the eye's optical aberrations, thereby increasing retinal image quality, to a much higher level than is possible with conventional spectacles or contact lenses. With the adaptive correction in place, Liang et al. found that the contrast sensitivity of their observers for fine spatial details increased nearly tenfold over the performance level achievable with best sphero-cylinder correction. Furthermore, when the adaptive optical components were incorporated into a fundus camera, ultra-high resolution images of the retina were obtained which included photographs of individual photoreceptors in the living human eye.
Although deformable mirrors are preferred in astronomical applications, spatial light-modulators (SLM) created from nematic liquid-crystals offer an attractive alternative technology in adaptive optical systems intended for use with the eye. 2 A spatial light modulator (SLM) is an electro-optical component used to control the phase or amplitude of a light waveform independently at different spatial locations. A liquid crystal SLM consists of a thin layer of liquid crystal molecules sandwiched between two parallel glass plates. 3 To achieve spatial control, the glass plates are subdivided into a honeycomb array of control cells which divides an incident wavefront into segments. Each independent control cell contains liquid crystal molecules oriented in parallel (as in a crystal), which causes the cell to be birefringent. Consequently, when light passes through the crystal it is subjected to a different refractive index when polarized in a direction parallel to the molecules compared to when it is polarized in the orthogonal direction. The molecules are free to rotate (as in a liquid) by increasing amounts when an electric field of increasing strength is applied to transparent conductive electrodes coated on the inside of the glass plates. Consequently, an applied voltage can be used to vary the refractive index experienced by light of fixed polarization when transmitted through the crystal.
Figure 1. Principle of refraction by a spatial light modulator (SLM) fabricated from liquid crystals. Wavefront propagation from left to right (dashed lines) passing through a medium of higher refractive index (shaded region) is retarded, causing the direction of propagation (arrows) to change.
Since the liquid crystal SLM is an electronically controlled array of transparent cells of variable refractive index, it has the potential for producing computer-controlled optical elements such as prisms or spectacle lenses. The notion of using a device of fixed thickness but variable refractive index to refract light is an unconventional approach in ophthalmic optics. Conventional prisms and lenses refract light by varying the thickness of transparent material with fixed index of refraction. Figure 1 illustrates the alternative principle by which the SLM refracts light. Since the wavelength of light in the medium of the crystal is inversely proportional to the material's refractive index, the wavefront is retarded relative to the same wavefront propagating in air. Consequently the wavefront emerges with a different slope, propagating in a different direction. When this principle is applied to a device with a hundred or more independent control cells, the potential arises for correcting not just the spherocylinder refractive errors of the eye but more complex, higher-order optical aberrations as well. Thus SLMs may have a future role to play in a variety of ophthalmic applications ranging from computer controlled spectacle lenses to ultra-high resolution fundus cameras. The purpose of the present study was to take a first step in these directions by evaluating the feasibility of using a commercially available SLM to alter the refractive state of the eye.
A schematic diagram of the apparatus is shown in Fig. 2. A grating monochromator (GM) tuned to 584 nm (full bandwidth at half-maximum = 12 nm) produced a small spot of light which trans-illuminated the visual target T made from a microscope graticule backed with diffusing tape. Lettering on the target subtended approximately 7 arcmin, equivalent to Snellen 20/28 characters. Light from the target was collimated by lens L1 and vertically polarized by filter VP. The spatial light modulator (SLM, Hex-127 by Meadowlark, Inc.) contained 127 control cells in a honeycomb pattern confined to a circular region 12 mm in diameter which we masked with a circular aperture of the same size. The control cells of this device are arranged as an hexagonal lattice of cells 1 mm wide which we were able to visualize during the experiment by a second channel of the optical system consisting of beam splitter BS and a horizontal polarizing filter HP. In this configuration the SLM lies between crossed polarizers so that it acts as a variable-transmittance attenuating filter (the same mode used for LCD displays on digital clocks and calculators). A video camera (CCD) focused on the plane of the SLM was thereby able to capture a grey-scale image for display on a TV monitor. Light transmitted by the beam splitter was focused to a point by lens L2 and then re-collimated by lens L3 (ray path shown by solid lines). The purpose of the relay lens pair L2, L3 was to place the SLM optically conjugate with the entrance pupil (EP) of the eye, which was achieved by locating SLM in one focal plane of L2 and EP in a focal plane of L3 (ray path shown by dotted lines). Conceptually we think of the SLM as being optically relayed by this system of lenses so that it lies inside the eye where it is in a position to alter the eye's refractive state. The powers of the relay lenses stood in the ratio 4:1 so that the minified image of the SLM in the entrance pupil of the eye was 3mm in diameter. Refractive error of the observer was corrected with spectacle lens S.
Figure 2. Experimental apparatus. See text for description of elements and design principles.
Since the optical effect of the SLM is achieved by independently varying the index of refraction of each cell of the array, computer control of the device required knowledge of how index of refraction varies with the applied voltage. The SLM was calibrated by the manufacturer in terms of the wavefront retardance achieved by a given control voltage applied to the liquid crystal cell. The available operating range is limited to 1-10 volts because the device has a threshold (1V) below which the device is inactive and has a ceiling (10 V) beyond which further changes are insignificant. Retardance is defined as the change in optical path length through the crystal (relative to transmission through a vacuum) and is equal to the product of the thickness of the crystal and the change of index of refraction (relative to a vacuum) produced by the applied control voltage. Since we had the reverse problem of determining the control voltage required to achieve a desired retardance, we transposed the manufacturer's data as shown by the symbols in Fig. 3 . These data indicate that to achieve a dynamic range of one full wavelength of retardance requires a source wavelength of 584 nm. We therefore adjusted the grating monochromator in Fig. 2 to this value and assumed that the manufacturer's calibration obtained at 650 nm applied also at the test wavelength of 584nm. For interpolation purposes it was convenient to fit the empirical calibration curve with a power function, shown by the solid curve in Fig 3.
Figure 3. Calibration of the SLM. Symbols represent manufacturer's calibration of the device; solid curve is the best fitting power function fit to the symbols.
The optical problem of programming the SLM to produce a prismatic
deviation of the light beam is illustrated in Fig.
4. Consider a wavefront in the x-y plane which is traveling
in the z-direction. The diagram shows that to change the direction of
propagation by the angle 
requires
that the wavefront be retarded by an amount W(y) which grows linearly
with y. Thus the desired wavefront retardation function of the
system is W(y) = Py+C. To relate the constant of proportionality
P to more familiar ophthalmic descriptions of prisms, we note
that P can be recovered from W(y) by differentiation and
therefore P is equal to the slope of the wavefront. Since the
direction of propagation is orthogonal to the wavefront, P is
also the slope of the propagation ray and therefore P =
tan()
is a direct measure of strength of the prism. For example, if W(y)
changes by 0.6 µm across a pupil of diameter y = 3mm, then P =
tan ()
= 0.6/3000 = 0.02 
(prism diopters).
Figure 4. Geometry of wavefront retardation for prismatic deviation.
Since the maximum retardance that could be produced by the SLM used in this study was one wavelength (at 584 nm), the device may at first seem restricted to very small prism magnitudes. However, for quasi-monochromatic sources it should be possible to extend its range of operation by using phase-wrapping techniques. The idea here is to subtract off integer multiples of the wavelength of light and program the SLM to produce just the fractional part of the desired retardance. This is the underlying principle of Fresnel lenses and one of our experimental aims of this study was to determine empirically how far we might usefully extend the prismatic range of the device by this technique. An example of the implementation of the phase-wrapping technique is illustrated in Fig. 5.
Figure 5. Example of use of phase-wrapping technique to extend the prismatic power of the SLM to more than 1 wavelength across the eye's entrance pupil. Dashed curve shows retardance function required to produce 3 wavelengths of prism power (0.06
A similar approach to that given above for prisms applies also
when programming the SLM to shape an incident plane wave into an
arbitrary wavefront. To implement a cylindrical lens, for example, we
used the quadratic wave retardation function W(y) = Sy2
and the same basic formula was used also for a spherical lens upon
substitution of the radius variable 
for y. To express the coefficient S in dioptric terms,
we observe in Fig. 6 that the vergence of
the desired wavefront is 
diopters. Since the small distance 
(on the order of µm) is negligible when compared to the large
distance 
(on the order of mm), we may approximate the slope of the propagating
ray as 
.
To connect this result with the wavefront retardance function, we
differentiate W(y) to find that dW/dy = 2Sy. From the geometry of
Fig. 6 we interpret this derivative as the
slope of the tangent to the curve z=W(y) at the point y and
therefore tan(
)
= 2Sy. Combining these results leads to the conclusion that S = V/2.
In other words, the retardance coefficient S is half the desired
dioptric power of the SLM lens. For example, to produce +1 diopter of
defocus (i.e. make the eye myopic) requires W(y) = -1.125 µ m at
the margin of a 3 mm diameter pupil, where negative retardance
indicates the wavefront is advanced at the margin relative to the
pupil center. Just as in the prismatic example described earlier,
retardance of this magnitude requires application of the
phase-wrapping technique as illustrated in Fig.
7.
Figure 6. Geometry of wavefront retardation for the general case of shaping an incident plane wave into an arbitrary wavefront.
To evaluate the prismatic effect of the SLM, the observer viewed a distant fixation target F through the beam splitter. Target F was adjusted by the observer so that it had the same visual direction as T when the SLM was inactive. Since light from F did not pass through the SLM, any programmed prismatic effect of the SLM could be measured as an apparent displacement between the two targets F, T which could be measured by displacing F until it once again was aligned visually with T. To evaluate the change in dioptric power of the eye induced by the SLM, a conventional refraction was performed to determine the optimum spectacle lens required to minimize the subjective appearance of blur for target T. Cycloplegia was not necessary for the neo-presbyopic observer (author LT) .
Figure 7. Example of programming the SLM to introduce 1D of spherical ametropia into the eye. Dashed curve shows retardance function required to make a 1D lens. Solid curve represents the desired retardance function after phase wrapping. Symbols represent the retardance values used to program the SLM at each of the 13 control cells across the diameter.
A qualitative verification that the individual cells of the SLM
responded appropriately to a sequence of commands for prisms of
increasing power was obtained by observing the intensity-coded image
captured by the CCD camera. An example of such an image in Fig.
8 for a 3-wavelength prism (0.06 )
clearly shows the linear geometry expected of a vertical prism. No
attempt was made to quantify the relationship between gray level in
the image and index of refraction of the SLM. Nevertheless, it is
clear from Fig. 8 that, as expected for the
phase-wrapping technique, the device responded with 3 cycles of
change in the refractive index in the vertical y-direction and was
constant in the horizontal x-direction. Several anomalous control
cells in the lower portion of the figure are evident, which we
attributed to flaws in either the SLM itself, or the control
system.
Figure 8. Visualization of the array of liquid crystal cells for producing a vertical prism which retards wavefronts by 3 wavelengths (0.06 prism diopters) over the extent of the 3mm entrance pupil of the eye.
Rough verification of the magnitude of prismatic deflection
produced by the 0.06
prism was done by visual comparison with a 1 prism diopter glass
prism. Deflection produced by the SLM was clearly much smaller than
that of the glass prism. The deflection produced by a 1 wavelength
prism (0.02 )
was near visual threshold for detection and was clearly above
threshold for stronger prisms. Since a 1 wavelength prism should
refract light by about 40 arcsec, these results are not unreasonable.
Psychophysical measurement of the deflection produced by a 3
wavelength prism gave a mean value of 0.06
(s.d. = 0.01 ,
n=3) as expected.
Visual inspection of the target revealed a deterioration of image
quality as the magnitude of prism was increased beyond 3 wavelengths
(0.06 prism diopters). This deterioration rendered the test letters
on the target unreadable for prisms of more than 4 wavelengths (0.08
).
Figure 9. Visualization of the array of liquid crystal cells for producing a spherical lens which retards wavefronts by 1 wavelength (0.5 diopters) at the margins of the 3mm entrance pupil of the eye.
An example of the intensity-coded image of the SLM is shown in Fig. 9 for a 0.5 diopter spherical lens. Except for the few anomalous control cells mentioned earlier, the pattern shows the circular symmetry expected of a spherical lens. When viewing the test target through the SLM in this configuration the details of the target were obviously blurred. We verified that the blur was refractive in nature by introducing trial lenses of either positive or negative power and noticing that lenses of one sign made the target even more blurred while lenses of the opposite sign made the target appear clearer.
Since the SLM was optically conjugate to the entrance pupil of the eye in our test arrangement, we think of the SLM as being located inside the eye where it is in a position to alter the eye's refractive state. Accordingly, we performed a standard clinical refraction to determine the optimum spectacle lens needed to correct the ametropia induced into the eye by the SLM. The results of this experiment for various SLM lens powers are shown in Fig. 10. We adopted the usual optometric convention of expressing the refractive error of the eye in terms of the lens required to correct that refractive error. For example, if the SLM is programmed to produce a lens of positive power +S diopters, then the subject's eye will appear to be myopic with a refractive error of -S diopters. According to this convention, we expect the scatter plot of data points in Fig. 10 to lie along the dotted line of slope +1. As may be seen from Fig. 10, this expectation was confirmed for spherical as well as cylindrical lenses up to a maximum of 1.5 diopters ametropia. The measured axis of cylindrical refractive error was no more than 10 degrees from the expected value in this range. However, when the SLM was programmed to introduce more than 1.5 D of ametropia, the resulting refractive state of the eye could not be corrected with spectacle lenses of any power. In this case the subject reported that the best image quality was obtained for zero spectacle correction.
Figure 10. Experimental determination of refractive correction for an eye which has either spherical (circles) or cylindrical (squares) ametropia induced by the SLM. Dashed line is the prediction.
The results of this study demonstrate the successful use of a
liquid crystal spatial-light-modulator to alter the refractive state
of the eye. For the device we used, which was capable of varying
retardance by 584 nm independently at each of 127 cells arranged in
an hexagonal mosaic, dioptric power could be varied over a 3 D range
(-1.5 to +1.5 D) for spherocylinder lenses and 0.08
for prism. This dioptric range corresponds to a phase retardation of
about ± 3 wavelengths of 584 nm light at the edge of a 3 mm
diameter pupil, which is more than enough to compensate for the wave
aberrations of the typical eye with a medium-sized pupil. 4-6
Consequently, we anticipate that the liquid crystal SLM currently
available may be a useful component of an adaptive optics system for
correcting the optical aberrations of the eye with a medium sized
pupil. However, the main benefits of correcting the eye's aberrations
are expected to accrue for larger, dilated pupils. Doubling the image
diameter of our SLM in the eye's entrance pupil (by changing the
ratio of focal powers of lenses L2, L3 in Fig.
2) would require re-scaling the x-axis of Figs. 5
and 6 so that the same retardance function
with the same number of sample points spans twice the distance. Under
this scenario, however, the same dynamic range of ± 3
wavelengths corresponds to only 0.4 D of dioptric power. Therefore,
to achieve the same useful range of focusing power as obtained here
(± 1.5 D) for a 6 mm pupil would require a device with 4 times
as many control cells.
Although spherocylinder refractive errors up to 1.5 D induced by the SLM could be corrected with conventional spectacle lenses, the observer noted that the quality of the retinal image was visibly degraded by what appeared to be diffractive effects. It is encouraging that these effects were not so severe as to reduce visual acuity below the 20/30 level of our test chart. However, no attempt was made to measure the maximum visual performance attainable when refractive errors of the eye are corrected with the SLM as compared to conventional spectacle lenses. Further research is required to determine the extent to which the finite number of SLM control cells and their limited dynamic range of retardance may nullify any improvements gained in image quality when using the device to correct refractive errors or other ocular aberrations.
Figure 11. Undersampling of the phase-wrapped retardance-function for a 9-wave prism leads to optical aliasing as a 3-wave prism when rendered by the SLM.
Our attempt to expand the useful dioptric range beyond 1.5 D was
not successful. In this case retinal image quality was significantly
reduced and could not be corrected with spectacle lenses of any
power. We suspect that finite size of control cells is the primary
factor which limits the useful dioptric range of the SLM. To see why,
consider Fig. 11 which illustrates the
practical difficulty which arises when attempting to implement a
large (9-waves = 0.18 )
prism. Since the desired retardance function W(y) is so steep, the
phase-wrapped version oscillates rapidly. As for any sampled
function, if there are fewer than 2 sample points per cycle of the
highest frequency component of an oscillating function, then the
function will be misrepresented. Consequently, when the phase-wrapped
control signal is significantly undersampled by the discrete array of
cells as in Fig. 11, the SLM values will
misrepresent the desired control function. The reader will notice
that the sample values shown in Fig. 11
form a pattern of their own which is quite different from the
intended pattern. This is the well-known phenomenon of aliasing
appearing in an unusual optical context. A comparison of Fig.
11 with Fig. 5 would suggest that
this aliased control signal represents a prism of about 3 waves (0.06
)
in magnitude, and of the opposite sign (i.e. the alias is base-down
if W(y) is base-up). We verified empirically that the SLM was indeed
producing an alias of the intended prism by visually comparing the
effect of a 3-wave prism and a 9-wave prism. Both control programs
produced a deflection of the target by about the same magnitude, but
in opposite directions.
Figure 12. Undersampling of the phase-wrapped retardance-function for a 3 diopter spherical lens produces unstructured wavefront alterations when rendered by the SLM.
We expect that optical aliasing caused by undersampling the phase-wrapped retardance function will be even more acute when attempting to correct focusing errors and other aberrations of the eye. This is because a polynomial retardance function of the second degree is required to correct focus, third degree for coma, and fourth degree for spherical aberration. As the degree of polynomial increases, the phase-wrapped version of the retardance function oscillates with increasing frequency near the margin of the pupil, thus increasing the likelihood of undersampling by the discrete SLM. An example is shown in Fig. 12 for a quadratic retardance function required to produce 3D of vergence change in the wavefront. Undersampling of the desired function beyond 1mm pupil radius leads to a discrete control function which appears to lacks the systematic trends required to mimic a spectacle lens. This would account for our observation that conventional spectacle lenses could not correct the refractive changes induced by an SLM programmed to deliver 3D of refractive error. Such problems will only increase when attempting to correct higher-order aberrations of the eye. To overcome these technical limitations will require the future development of SLMs with more control cells or perhaps an alternative geometry tailored for specific applications.
This project was supported by a research grant from Indiana University to LNT and AB and by grant R01 EY05109 from the National Eye Institute to LNT. We appreciate the technical advice and support of Mike Anderson and Alan Graham of Meadowlark, Inc.
