Requirements for segmented correctors for ... - OSA Publishing

Requirements for segmented correctors for diffraction-limited performance in the human eye Donald T. Miller, Larry N. Thibos, and Xin Hong School of Optometry, Indiana University, Bloomington, Indiana 47405 [email protected]

Abstract: Wavefront correctors have yet to provide diffraction-limited imaging through the human eyeÕs ocular media for large pupils (³ 6 mm). To guide future improvements in corrector designs that might enable such imaging, we have modeled the performance of segmented piston correctors in conjunction with measured wave aberration data of normal human eyes (mean=34.2 yr; stdev= 10.6 yr). The model included the effects of pupil size and wavelength in addition to dispersion, phase wrapping, and number and arrangement of facets in the corrector. Results indicate that ² 100x100 facets are needed to reach diffraction-limited performance for pupils up to 8 mm (extrapolated) at 0.6 µm wavelength. Required facet density for the eye was found to be substantially higher at the pupilÕs edge than at its center, which is in stark contrast to the requirements for correcting atmospheric turbulence. Substantially more facets are required at shorter wavelengths with performance highly sensitive to facet fill. In polychromatic light, the performance of segmented correctors based on liquid crystal technology was limited by the naturally occurring longitudinal chromatic aberration of the eye rather than phase wrapping and dispersion of the liquid crystal. Required facets to correct defocus alone was found highly sensitive to pupil size and decentration. ©2005 Optical Society of America OCIS codes: (010.1080) Adaptive optics; (330.4460) Ophthalmic optics; (330.5370) Physiological optics; (330.4300) Noninvasive assessment of the visual system

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9.

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Received 10 November 2004; revised 25 December 2004; accepted 30 December 2004

10 January 2005 / Vol. 13, No. 1 / OPTICS EXPRESS 275

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1. Introduction Image quality in the human eye can be greatly improved by maximizing pupil size and correcting ocular aberrations across the large pupil, using for example an adaptive optics (AO) system [1,2]. AO has been successfully applied to the living human eye, mitigating the most significant aberrations and providing the eye with unprecedented image quality. However, AO has not achieved the ultimate goal of diffraction-limited imaging in eyes with large pupils (³ 6 mm). The primary bottleneck appears to be the wavefront corrector of the AO system. The corrected wavefront can be no better than the uncorrectable errors on this device, including those generated by the finite dynamic range of the device. Several types of wavefront correctors have been applied to correcting the aberrations in the human eye. These include liquid crystal spatial light modulators (LC-SLMs) and deformable, membrane, and micro-electro-mechanical system (MEMS) mirrors. The first attempt employed a 13-actuator membrane mirror in a confocal scanning laser ophthalmoscope [3]. Correction was limited to the astigmatism in one subjectÕs eye, based on a conventional prescription. Correction to higher-order aberrations was first realized with a 37-actuator deformable mirror (Xinetics) coupled to a Shack-Hartmann wavefront sensor [4]. This same mirror was later used in an AO retina camera to track the naturally-occurring temporal fluctuations of the eyeÕs aberrations and correct them up to about one Hertz [5]. The mirror was also employed in a scanning laser ophthalmoscope [6] and an optical coherence tomographic retina camera [7]. A larger version of this mirror (97 actuators) has been reported [8], while a custom-built 37-actuator deformable mirror yielded retinal images of comparable image quality to that of the 37-actuator Xinetics [9]. More recently a 37-electrode surface micromachined MEMS mirror was experimentally compared to the 37-actuator Xinetics [10]. A 37-electrode membrane micro-mirror has been explored [11,12] and has been successful at correcting temporal fluctuations in the eyeÕs wave aberrations [13,14]. A 13-electrode bimorph mirror has also been applied successfully to the eye [15]. LC-SLMs, employing 64 and 127 controllable facets, have been investigated for correction of prism, defocus and astigmatism [16], as well as the full wave aberrations of the eye [17]. A high-resolution optically addressable LC-SLM has been evaluated on a model eye [18,19] and human eyes with medium size pupils [19,20]. Each of these wavefront corrector devices has been successful at reducing the wave aberrations of the eye, providing the eye with optical quality exceeding that with which it is endowed. However, none have been reported to provide sufficient correction to yield diffraction-limited imaging for large pupils (³ 6 mm), where the aberrations are most severe and the potential benefit of AO largest. Even at smaller pupil sizes, many of the devices have not reached diffraction limited imaging. Moreover, the characteristics of the correcting device, such as actuator number and stroke, that are required to achieve diffraction-limited imaging have not been determined. This precludes optimally matching corrector performance and cost to that required of a particular imaging task in the eye. In addition, lack of corrector requirements prevents improvements in corrector design that could enable diffraction-limited imaging. To address this problem, our objective was to model their performance in conjunction with measured wave aberration data of normal human eyes for monochromatic and polychromatic light. The correctors cited above, however, fall within two broad categories: (1) piston segmented devices (e.g., LC-SLMs and segmented mirrors), which consist of an array of many independent segments with each controlling the local wavefront piston, and (2) continuous surface mirrors (e.g., Xinetics, membrane micro-mirrors, and bimorphs), which consist of a monolithic sheet of reflective material whose surface topography can be accurately warped. As the modeling for these two types is somewhat different, particularly with regard to the LC-SLM in polychromatic light, we opted to confine our attention in this paper to piston segmented devices with the continuous surface mirrors to be addressed in

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another. Note that while LC-SLMs have already been applied to the eye, segmented mirrors have not due largely to technological limitations. There is considerable interest in the field, however, to do so and hopefully the results here will help guide future developments in this area. 2. Methods Analytic models for characterizing the impact of ocular aberrations on image quality are still in early development [21,22]. This is unlike the situation for imaging through atmospheric turbulence, which has well developed theoretical models that are rooted in the fundamental mechanisms of turbulence [23-25]. Without established models for the eye, our approach was to simulate the imaging process through the eye and wavefront corrector using measured wave aberration data of normal human eyes in conjunction with an optical model for the corrector, all within the mathematical framework of Fourier optics. The simulation included the effects of pupil size D (0 to 8 mm) at the eye; facet (actuator) number (0 to 72 facets across pupil diameter); facet arrangement (hexagonally- and square-packing); and mono- and polychromatic light (0.4 to 1.0 µm). The test object for the simulation was taken to be a point source, and the Strehl ratio was used as the figure of merit. Strehl is defined as the ratio of the light intensity at the peak of the diffraction pattern of an aberrated image to that at the peak of an aberration-free image. Generally, an optical system is considered diffraction-limited if it has a Strehl ³ 0.8. Other common figure of merits include root-mean-square (RMS) wavefront error and full-width-at-half-height (FWHH) of the point spread. For reasonably well corrected systems the RMS and FWHH generally provide little additional information about image quality beyond that revealed by Strehl. This is because for such systems, Strehl and RMS are highly correlated [26], while FWHH of the PSF is largely insensitive to small changes in the wave aberrations, which Strehl is not. As such, Strehl should be a reliable guide for establishing corrector requirements for diffraction-limited imaging in the human eye and is therefore used here. A laboratory Shack-Hartmann wavefront sensor described in detail elsewhere [27] measured the local wavefront slopes at 177 locations across a 6 mm pupil in 12 subjects (mean=34.2 yr; stdev= 10.6 yr) with spectacle correction in place. The aberrometer was powered by a 632.8 nm HeNe laser. Unlike other wavefront sensor applications for the eye where the static aberrations are most important [28], adaptive optics systems are intrinsically closed loop that repeatedly measure and correct both static and temporal aberrations. Dynamic changes in the aberrations due to pupil decentration, fluctuations in accommodation, and other forms of aberration instabilities are real signal, not noise, that should ideally be corrected by the wavefront corrector to accrue optimal image quality at each moment in time. As a slight compromise in order to enhance signal, three consecutive measurements (rather than one) on each subject were collected within a period of about 30 seconds during which the subject remained on the bite bar. Because the accuracy of the high Zernike orders in an individual aberration measurement has not been established in the literature, we errored on the conservative side (i.e., overestimate the aberrations) by reconstructing a large number (66) of Zernike terms. Although an overestimate, inclusion of the very high orders will likely have a small impact on the corrector requirements as the magnitude of the ocular aberrations is largely confined to the low Zernike orders (< 5th order). The 66 Zernike modes of the wave aberration were reconstructed by fitting wavefront slopes to derivatives of ZernikeÕs circle polynomials by the method of least squares. All subjects had normal corrected vision and ranged from 12 to 50 years of age. The subjectsÕ line of sight was centered along the optical axis of the sensor with the aid of a dental impression attached to an x-y-z translation stage, which stabilized the subjectÕs head. The subject fixated on a white-light target with trial lenses inserted to optimize target clarity. No topical drugs were administered. Measurements typically consisted of illuminating the eye for #5717 - $15.00 US

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0.2 seconds with a total exposure energy of 2 µjoules, which is about 200 times below the ANSI maximum permissible exposure [29]. The average total wave aberration variance across the 12 eyes was 0.536 and 0.0645 µm2 with and without the second order Zernike terms (i.e., defocus and astigmatism), respectively. To establish confidence that the 12 eyes are reasonably representative of a normal healthy population, Fig. 1 shows the wavefront variance decomposed by Zernike order for each of the 12 eyes along with confidence limits for the large population (200 eyes; mean=26.1 yr; stdev=5.6 yr) measured in the IU Aberration Study [30]. Most subjects in the large population were optometry students in the age range 22-35 yr. Note the log ordinate, i.e. log 10(wavefront variance). The log10(wavefront variance) representation has been suggested to be Gaussian distributed [31], while the wavefront variance by itself is not, and therefore the former facilitates comparison between the two populations. The blue dashed curves in Fig. 1 represent the mean ± two standard deviations (95% confidence limits) across the 200 eyes. The close overlap of the two data sets suggests the 12 eyes used in this study, although an average of 8 years older and covering a larger age range, do indeed reasonably follow the much larger data set, especially for the 3rd, 4th, and 5th Zernike orders. For the other orders measured (2nd, 6th, and 7th), the 12 eyes are on average slightly more aberrated. Differences in the second order probably reflect differences in the manner in which the subjects were refracted prior to the aberration measurement. Four of the 72 data points (red circular markers) from the 12 eyes lie outside the 95% confidence limits, and this is close to the statistical expectation of (0.05*72=) 3.6.

µm^2

0

Log10 (Wave variance)

1

-1

IU Aberration Study 12 eyes

-2

-3

-4

-5 1

2

3

4

5

6

7

8

Zernike Order Fig. 1. Log10(wavefront variance) plotted as a function of Zernike order. The 12 red curves represent each of the measured 12 eyes used in this study. The blue diamonds and two dashed curves represent the mean and mean ± two times the standard deviation of the log10(wavefront variance), respectively, for the 200 eyes measured in the IU Aberration Study.

The wavefront corrector model was general enough to represent both reflective and transmissive devices, such as piston segmented micromirrors and LC-SLMs, respectively. The model consisted of a two-dimensional array of segmented facets that completely covered the circular pupil of the imaging system. Each facet was restricted to modulating only the piston component of the local wavefront. Facet inter-spacing was rs. Actual segmented devices typically have physical gaps between adjacent facets due to limitations in the fabrication process. These gaps are well known to reduce performance, yet their impact on diffraction limited imaging in the eye has not been assessed. In light of this, facets were modeled with #5717 - $15.00 US

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73.5%, 86%, and 100% fill. Facet fill is defined as the ratio of total facet to corrector surface area. A fill of 100% contains no gaps between adjacent facets and provides maximum corrector efficiency. As an approximation, light incident on the gaps was assumed unperturbed by the corrector (i.e., any phase errors were left uncorrected) and passed with 100% transmission. This is reasonably representative of reflective and transmissive LC-SLMs in that the gaps do not absorb any more light than the facets themselves. For segmented mirrors, the situation is more complicated as the reflective properties of the gaps will depend greatly on the fabrication process and how the final reflective coating is applied to the mirror segments. Because transmissive gaps probably lead to more image degradation than the absorbent variant (i.e., provides a better lower bound to image quality), the former represents a more conservative choice. The former permits aberrated rays to pass uncorrected, thereby striking the image plane at the wrong location, while the latter precludes these aberrated rays from ever reaching the image plane. Facet number was specified by the number of facets across the pupil diameter, D/rs. We found by trial-and-error that a conservative array size of 2,048x2,048 pixels assured no undersampling of the pupil and point spread functions for any of the correctors modeled. For correcting a specific aberrated wavefront, the phase profile of the corrector was determined by setting the phase of each facet to minus the average wavefront phase incident on the facet [32]. Each facet was assumed to have essentially infinite phase resolution. Figure 2 illustrates the sequence of steps in the simulation to correct the wave aberrations of the eye and compute the resulting point spread and corresponding Strehl, in this example for one eye (Fig. 2(a)) and two hexagonally-packed, segmented correctors employing 126 (top row) and 2,105 (bottom row) facets. The gray-scale image in Fig. 2(a) depicts the measured wave aberration profile, φeye, across a 6 mm pupil for one eye. Figure 2(b) shows the desired phase profile of the corrector, φ corrector, for compensating the wave aberrations in Fig. 2(a). Figure 2(c) shows the residual aberrations, φresidual = (φ eye - φ corrector), after correction of the wave aberrations in Fig. 2(a) with the corresponding corrector phase in Fig. 2(b). To compute the corrected point spread, we first represent the corrected complex field, Ψ, at the pupil as |Ψ|exp(iφresidual) with the amplitude of the wavefront, |Ψ|, defined as a circular pupil with 100% transmission including across facet gaps. Next, applying a Fourier transform operation to Ψ and taking its modulus squared yields the corresponding corrected point spreads and Strehl ratios as shown in Fig. 2(d). The point spreads generated in this manner include the impact of residual aberrations, and scalar diffraction effects generated by the finite size of the pupil and gaps between facets. In this example, facet fill was 100%. For this particular eye, the 2,105 facet device delivers diffraction-limited imaging and noticeably outperforms the 126 device. However even the 126 device significantly improves image quality compared to the uncorrected (measured) PSF for this eye (Fig. 3). The Strehl is increased by a factor of 6.9 and 10.4 for the 126 and 2,105 devices, respectively. The RMS is reduced by a factor of 3.1 and 9.3 for the same two devices. The impact of polychromatic aberrations intrinsic to the eye and some corrector types also degrade image quality. Wavefront correctors based on mirror technology, such as MEMs devices, are immune to polychromatic blurring, but their reflective nature also makes these devices inherently awkward for certain vision applications, most notably electronic spectacles and phoropters. Transmissive LC devices on the other hand are ideally suited for such applications, albeit they suffer from material dispersion of the LC and phase wrapping necessary because of the finite thickness of the LC layer. To evaluate LC corrector performance in polychromatic light, the simulation included the effects of dispersion for the commonly used nematic LC material E-7 (a positive dielectric anistropic material consisting of a mixture of cyanobiphenyls and cyanoterphenyl) [33], and phase wrapping of 2π, which is almost exclusively used in such devices in order to maximize their temporal response while still permitting full correction at the design wavelength. To

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electrically control birefringence, E-7 (∆n = 0.23 at 20ûC and 589 nm wavelength) is typically aligned between two conducting glass plates. Also included in the model was the longitudinal chromatic aberration (LCA) of the eye as specified by the Indiana schematic eye [34]. Although the Indiana eye was derived for wavelengths over the visible spectrum, a more recent study reported that it is also a good predictor of chromatic defocus in the near-infrared [35]. Transverse chromatic aberration of the eye was set to zero, the average amount along the visual axis in the normal population [36]. 126 Facets (12 across) Strehl = 0.62

6 mm pupil rms = 0.12 λ

rms = 0.37 λ

F{ }

2,105 facets (48 across)

Strehl = 0.94

rms = 0.04 λ

A

B

C

D

Fig. 2. Compensation of ocular aberrations across a 6 mm circular pupil using two hexagonally-packed correctors with 12 (top row) and 48 (bottom row) facets across their pupil diameter. Wavefront phase is represented by a gray-scale image (black and white tones depict minimum and maximum phase, respectively). (a) shows the measured uncorrected wave aberrations for one subjectÕs eye with defocus and astigmatism removed. The phase RMS is 0.37λ. (b) shows the desired phase profile across two correctors for compensating the subjectÕs wave aberrations in (a). (c) shows the residual aberrations after correction of the wave aberrations in (a) with the corrector phase profile in (b). The residual phase RMS is 0.12 λ (top) and 0.04λ (bottom). The corresponding corrected point spreads and Strehl values are given in (d) that were computed using scalar diffraction theory that incorporated the residual wave aberrations and a circular pupil. λ is 0.6 µm.

Strehl = 0.09

Fig. 3. SubjectÕs point spread calculated from the measured wave aberration shown in Fig. 2(a)

The residual wave aberration error after correction, φresidual(λ), is defined by #5717 - $15.00 US

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φ residual ( λ) = φ eye ( λ) − φ SLM ( λ) ,

(1)

where φeye(λ) is the uncorrected wave aberrations of the eye at wavelength λ, and φSLM(λ) is the wavefront correction provided by the LC-SLM at the same wavelength. φeye(λ) and φSLM(λ) are mathematically defined by

φ eye ( λ) = φ eye ( λdesign ) and

φ SLM ( λ) = φ SLM ( λdesign )

λdesign − φ LCA λ

λdesign ∆n ( λ) , ∆n ( λdesign ) λ

(2)

(3)

respectively. λdesign is designated as the wavelength of optimal correction in that chromatic aberration in the eye and SLM are zero and 2π phase wrapping is exact at this wavelength. φLCA refers to the induced defocus caused by the longitudinal chromatic aberration of the eye as specified by the Cornu hyperbolic formula given by n(λ) = a + b/(λ-c), where a=1.320535, b=0.004685, and c=0.214102 [34]. Higher-order chromatic effects were assumed negligible [37]. λdesign/λ and ∆n(λ)/∆n(λdesign) represent wavelength scaling of the aberrations and the impact of material dispersion of the SLM, respectively. For the E-7 liquid crystal, we used the fitting function

∆n ( λ) = G

(λλ* ) 2

λ2 − λ* 2

,

(4)

where G=3.06e-6 nm-2, λ*=250 nm, and λ is in units of nanometers [33]. φSLM(λdesign) was 2π phase wrapped about λdesign = 0.6 µm. As the final step of our analysis of piston segmented devices, we investigated their monochromatic performance for the restricted case of correcting (or generating) defocus only. This scenario reflects the primary role of essentially all ophthalmic devices (spectacles, contact lenses, phoropters) where the correction of defocus (sphere) is of highest concern. Performance was evaluated for a range of facet number (0 to 72 facets across pupil diameter) and defocus (0 to 4 diopters) for two pupil sizes (3 and 6 mm). Performance was assessed using corrected Strehl at a wavelength of 0.6 µm and with a 100% facet fill. 3. Aberration correction in monochromatic light Figure 4 shows the predicted corrected Strehl for hexagonally- and square-packed segmented correctors as a function of D/rs for a 6 mm pupil and 0.6 µm wavelength. The error bars represent ±1 standard deviation across the 12 subjects. The top two curves do not include defocus and astigmatism (Z20, Z2-2, and Z22), the residual 2nd order aberrations left uncorrected by the trial lenses. The bottom two curves include the residual defocus and astigmatism associated with quantization of spectacle lens power (0.25D) and the subjective criterion for optimum focus in the presence of higher-order aberrations [38]. All four curves exhibit a similar shape, monotonic and positive sloped. With zero facets, the corrected Strehl reflects the image quality of the well-focused natural eye. Over a D/rs range of 0 to 24 facets, significant improvement in image quality is predicted as the Strehl rises sharply with increased facet number. Small changes in the number of facets lead to noticeable changes in corrected image quality. This increase is likely due to the effective correction of low order aberrations, which is where the vast majority of the eyeÕs wave aberrations reside. The error bars are relatively large over this range, reflecting the large variability in corrector performance between the worst and best eyes. For the range D/rs > 24, the previously

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observed sharp increase in Strehl is replaced with a gradual rise converging to one at large facet numbers. The diminishing improvement in corrected image quality makes larger and more expensive correctors increasingly less attractive. The error bars over this range are noticeably smaller than before, indicating that consistently high image quality can be achieved across many eyes.

1.0

without astig & defocus

Corrected Strehl

0.8 0.6

with astig & defocus

0.4 hexagonal packing square packing

0.2 0.0 0

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# Facets across Pupil Diameter (D/rs ) Fig. 4. Corrected Strehl ratio for (red) hexagonally- and (green) square-packed segmented correctors as a function of facet number. Pupil diameter and λ were set to 6 mm and 0.6 µm, respectively. Top two curves do not include the impact of the residual defocus and astigmatism, which were left uncorrected by trial lenses. Bottom two curves include defocus and astigmatism. Error bars represent ±1 standard deviation across the 12 subjects.

The figure also shows the pronounced debilitating effect of the residual astigmatism and defocus. For example, a corrected Strehl of 0.8 required four times more facets with astigmatism and defocus present than without. These results suggest three plausible scenarios to approach diffraction-limited imaging. First to avoid the significant additional facets needed to correct the second order terms, the least expensive approach is to employ a medium faceted corrector (e.g., D/rs = 24) that primarily corrects third order and higher aberrations. Second order aberrations can then be substantially reduced, although not eliminated, by cyclopleging the subject (which freezes the ocular microfluctuations) and meticulously employing trial lenses in conjunction with continuously adjustable lenses to avoid quantization errors. This approach was chosen for the first successful attempt to correct the higher-order aberrations in the eye [4]. The second approach is to substitute cyclopleging and translating lenses with a low-order corrector, specifically designed to correct second-order aberrations. The cascade of the low- and high-order correctors represents a woofer-tweeter combination. The most elegant and clinically conducive approach is to simply rely on a large faceted corrector for compensating all the aberrations, including those of second order. At least two reflective liquid crystal devices are commercially available that contain facet numbers much larger than that predicted (Fig. 4) for diffraction-limited imaging in the eye. This suggests that a single device may be a plausible strategy for correcting all of the aberrations. For a Strehl of 0.8, D/rs is predicted to be slightly higher for the square configuration (21.7 and 49.2 square facets versus 20.1 and 45.0 hexagonal facets for the two astigmatic and defocus cases). This is not unexpected as facet density is lower for the square array for the same D/rs value. The total number of facets across the whole pupil is found to be essentially #5717 - $15.00 US

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the same for the two packing arrangements for 0.8 Strehl performance (440 and 1,860 hex facets compared to 460 and 1,890 square facets). This is in agreement with the general rule that wavefront correction is primarily determined by the total degrees of freedom of the corrector with other factors, such as facet arrangement, playing a secondary role. Pupil size and wavelength are two key parameters that can strongly influence corrector performance. To guide the design of correctors for use with the eye at various pupil sizes and wavelengths, simulations were conducted to predict corrector performance along two orthogonal axes in D-λ space. The two axes chosen were λ = 0.6 µm and D = 6 mm. For each λ, the monochromatic aberrations of the eyes, φeye(λ), were adjusted using Eq.(2) with φLCA set to zero (φLCA is used in Section 4). Fig. 5 shows the number of facets needed to achieve a corrected Strehl of 0.8 along these two axes. At pupil diameters less than 4 mm, D/rs need not be larger than 15. Fitting the simulation results to a third-order polynomial, allows the D/rs requirement to be interpolated and extrapolated to any pupil diameter ranging from zero to the maximum physiological size of 8 mm. Note that while the third-order fit is appropriate for reflecting the impact of ² 3rd order Zernike aberrations, it will fail to fully capture ³ 4th order Zernike aberrations, which are most significant at the edge of large pupils (e.g., 8 mm). This fitting error is reduced in the eye, however, due to a monotonic drop in the magnitude of the aberrations with order (see Fig. 1) with the most significant aberrations residing at the lowest orders. In light of this, the polynomial fit for pupils larger than 6 mm should be viewed as an underestimate of facet number with the true value likely lying somewhat higher. As shown in Fig. 5, the polynomial fit is not linear and therefore does not follow the atmospheric scenario (imaging through Kolmogorov turbulence) where a linear relationship holds and for which many of the commercial wavefront correctors are designed. The nonlinear fit implies that if a FriedÕs parameter (the metric of choice for quantifying image quality for viewing through atmospheric turbulence) could be developed for the human eye, it would depend on pupil position with a larger FriedÕs parameter near the pupil center and progressively smaller values at increasingly larger pupil eccentricities. Interestingly, this variation across the pupil was not predicted by the recent statistical modeling of the eyeÕs wave aberrations [22]. This variation suggests non-uniform facet dimensions with larger facets in the middle and smaller at the edge. This will have a direct impact on retinal imaging applications in which the full pupil of the eye is utilized. Vision applications on the other hand will likely benefit less owing to the natural apodization of the pupil due to the Stiles-Crawford effect. The second plot of Fig. 5 shows the dependence of D/rs with wavelength. As expected, wavefront correction at shorter wavelengths requires more facets than at longer wavelengths to achieve the same imaging performance. For example at the edges of the wavelength band examined, nine times more facets are required to fill the pupil at 0.4 µm than at 1 µm. The data closely fit a λ-6/5 curve with a slightly better fit achieved with an exponential function and a slightly worse one with a 1/λ curve. It is not known if D/rs for the eye can be mathematically expressed as separate D and λ functions, i.e. D/rs = F1(D)*F2(λ), as is the case for Kolmogorov turbulence [25]. If so, the fitted curves along the two orthogonal axes in Fig. 5 could be combined to determine D/rs for any D and λ combination. As an initial attempt to assess the legitimacy of F1(D)*F2(λ), we obtained estimates of D/rs using both the simulation and F1(D)*F2(λ) expression for seven D and λ combinations. The combinations included D and λ values ranging from 2 to 6 mm and 0.4 to 1 µm, respectively, with none of the seven coinciding with the D = 6 mm or λ = 0.6 µm axes. The percent difference between estimates was found to be no greater than 20%, lending support for separate D and λ functions for at least the D−λ combinations considered here.

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Facets across Pupil Diameter

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Fig. 5. Number of facets required to achieve diffraction-limited imaging (Strehl = 0.8) along two orthogonal axes in D −λ space. The two axes are λ = 0.6 µm (left) and D = 6 mm (right). Simulation results were fit to λ-6/5 and third-order polynomial functions.

Results in Figs. 4 and 5 are based on ideal facets that completely cover the circular pupil of the imaging system (100% fill). Fig. 6 shows the degrading impact of gaps for facets with 73.5%, 86%, and 100% fill, the performance of the latter is duplicated from that in Fig. 4. The results support the expected monotonic trend of reduced Strehl with reduced fill. More specifically, the 73.5% and 85% curves fail to reach diffraction limited performance even with the maximum number of facets evaluated (D/rs=72). The appearance of an asymptotic limit for these two curves, suggests a tremendous number of facets would be required to offset their somewhat poor fill. Interestingly, the simulated results in Fig. 6 are only somewhat predicted by a 1st order geometric optics approach where the corrected Strehl is approximated as the corrected Strehl for 100% fill multiplied by the actual percent fill. This 1 st order approximation assumes that light falling between facets is absorbed and makes no contribution to Strehl. As mentioned in the methods section, one might anticipate that transmissive gaps should yield a more conservative estimate of Strehl than the absorbent variant as the former permits aberrated rays to pass uncorrected and to therefore strike the image plane at the wrong location. In support of this for example at D/rs = 72, the geometric optics approach predicts a higher corrected Strehl of (0.896*73.5% =) 0.66 and (0.896*86% =) 0.77 for 73.5% and 86% fill, while the simulation results in Fig. 6 show reduced Strehls of 0.50 and 0.68, respectively. In general, facet fill is a critical parameter that impacts performance; results in Fig. 6 suggest a fill well above 90% is highly desirable for diffractionlimited imaging in the eye. 4. Aberration correction in polychromatic light Figure 7 shows the Strehl performance of four types of SLMs used to correct the wave aberrations of the 12 subjects for 3 and 6 mm pupils. Wavelength range was from 0.4 to 0.8 µm. The corrector types correspond to the four possible combinations of dispersion and phase wrapping. The effect of facet number was eliminated by reducing facet size to a single pixel. As shown in Fig. 7 (right), perfect correction for a 6 mm pupil occurred for the SLM type having no phase wrapping and no dispersion. An example would be an SLM whose performance is identical to that of a piston segmented mirror with actuator stroke greater than the peak-to-valley of all 12 aberrated wavefronts (4.63 µm). Introducing the liquid crystal

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material E-7 caused the corrected Strehl to fall sharply at wavelengths below 0.55 µm. A gradual monotonic decrease occurred at wavelengths longer than λdesign (0.6 µm). 2π phase wrapping present in the final two corrector types was found to be substantially more limiting, reducing the full width at half height (FWHH) of the corrected Strehl to about 125 nm. Unlike material dispersion in which image quality essentially decreases monotonically for wavelengths further from λ design, 2π phase wrapping produces perfect correction at discrete wavelengths that are integer fractions of λdesign, i.e. at λdesign/n where n is an integer. For the model described here this corresponds to wavelengths of 0.3 µm, 0.2 µm, 0.15 µm, etc. Although these fall outside the range shown in Fig. 7, a Strehl of one was confirmed for the 0.3 µm case. 1.0

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# Facets across Pupil Diameter Fig. 6. Corrected Strehl ratio for facet fill of 73.5%, 86%, and 100% as a function of facet number for a square-packed segmented corrector. Pupil diameter and λ were 6 mm and 0.6 µm, respectively. Residual defocus and astigmatism was present. Error bars represent ±1 standard deviation across the 12 subjects

The black curve in Fig. 7 (right) reflects the performance of the diffraction-limited eye corrupted only by the typical amount of LCA and no wavefront corrector present. The FWHH of this curve is only 20 nm, five times less than that of a phase-wrapped SLM. This suggests that in polychromatic light, SLM performance is limited by the naturally occurring LCA of the eye rather than 2π phase wrapping and material dispersion of the corrector. As shown in Fig. 7 (left), reduction of the 6 mm pupil to 3 mm effectively reduces the degrading impact of SLM dispersion and phase wrapping. In fact E-7 dispersion by itself can be neglected for wavelengths above 450 nm, a range over which Strehl > 0.8. 2π phase wrapping is again more degrading than material dispersion, but does not dominate to the extent observed for the 6 mm pupil. The most degrading combination is phase wrapping with E-7 material dispersion that yields a FWHH of the corrected Strehl of at least 350 nm, extending from 450 nm to beyond 800 nm. The black curve in Fig. 7 (left) represents the performance of the diffraction-limited eye corrupted only by the typical amount of LCA. Its FWHH is 87 nm, 4.4 times wider than that for the 6 mm pupil. Again, SLM performance is limited by the LCA of the eye rather than phase wrapping and material dispersion of the corrector.

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1.0

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Fig. 7. Average SLM performance on human eyes using polychromatic light for 3 (left) and 6 mm (right) pupils. The color-coded curves represent the performance of four SLM types that correspond to the four possible combinations of dispersion and phase wrapping. Dispersion was that of the liquid crystal material E-7. Phase wrapping was modulo 2π for λdesign = 0.6 µm. Facet size was set to a single pixel. The black curves represent the performance of the diffraction-limited eye corrupted only by the eyeÕs naturally occurring longitudinal chromatic aberration. The design wavelength, λdesign, was 0.6 µm.

5. Defocus correction in monochromatic light Fig. 8 shows the corrected Strehl for various amounts of defocus as a function of facet number for two pupil sizes typical for bright and dark viewing conditions. For the smaller 3 mm pupil, a refractive error of at least 1 diopter can be effectively corrected with a sparse segmented corrector (12 ² D/rs ² 18). The largest array size examined (D/rs = 72) provides diffraction limited imaging up to 4 diopters. For the larger pupil size, corrector performance falls substantially with sparse arrays (D/rs ² 18) effective only for dioptric powers ² 1/4. The largest array size can handle only up to 1 diopter, where it just reaches a Strehl of 0.8. This substantial reduction in performance with pupil size stems from the increasingly steep (quadratic) shape of the defocus wavefront error at the pupilÕs edge that cannot be adequately sampled by the corrector facets. It reveals a significant limitation of segmented piston correctors for diluting or creating large amounts of low-order aberrations, in this case defocus. The above defocus results are confined to the ideal case of on-axis correction where the optical axis of the system (eye) is aligned to the optical axis of the corrector (i.e. passes through the physical center of the corrector). Eye rotation relative to the stationary corrector (for example an SLM spectacle lens) causes the pupil of the eye to project through an off-axis location of the corrector. A typical displacement of the eyeÕs pupil relative to the spectacle axis can be on the order of millimeters for rudimentary visual tasks such as reading. As a first step, we assessed the impact of off-axis imaging for a specific pupil decentration of 3 mm relative to the optical axis of the corrector that remained fixed at the physical center of the corrector. Although this displacement corresponds to a relatively small eye movement, it noticeably increases the steepness of the defocus error to be corrected and requires substantially more corrector facets to maintain a given corrector performance relative to that for the equivalent on-axis case. For example for a 4 mm pupil, a 3 mm pupil decentration, and λ=0.58 µm, a segmented corrector requires roughly four times more facets to correct 2 diopters to a Strehl of 0.4 than for the same case with no decentration. It should be noted that our choice of corrector geometry (a uniform, regular array of facets) is unlike the conventional geometry of diffractive optics in which a discrete number of

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zones composed of discrete steps are strategically created with the boundary between zones occurring at the pupil position where 2π phase shift occurs. More so the fineness of discrete steps increases with increased eccentricity so as to maintain sufficient sampling of the wavefront, which is unlike that for the regular array used here. 0 diopters

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Fig. 8. Corrected Strehl for various amounts of defocus (0, 1/16, 1/8, 1/4, 1/2, 1, 2, and 4 diopters) as a function of facet number for 3 mm (left) and 6 mm (right) pupils. Facets were hexagonally-packed with 100% fill, and λ was 0.6 µm.

6. Conclusion AO has been successfully applied to the human eye using a variety of corrector types, none of which have provided sufficient compensation to yield diffraction-limited imaging through large pupils. In this paper we focused our attention on piston segmented devices and modeled their performance in conjunction with measured wave aberration data collected on normal human eyes. The model included the effects of pupil size and wavelength in addition to dispersion, phase wrapping, and number and arrangement of facets in the corrector. For medium size pupils (4 mm), 15x15 facets were found necessary to reach diffractionlimited performance at λ=0.6 µm with residual defocus and astigmatism present. For pupils up to 8 mm (extrapolated), ² 100x100 facets are predicted. Hexagonally- and square-packed configurations performed very similarly with essentially equal number of facets. Interestingly, the prediction of 100x100 facets suggests that the 1024x768 element Hamamastu LC-SLM, which has already been applied to the eye [19,20], is likely excessive for ocular applications. Corrector designs have been driven historically by atmospheric turbulence applications in which the atmospheric aberrations are uniformly distributed across the telescope pupil. Ocular aberrations, however, do not follow this distribution, but instead are substantially stronger at the pupil edge than at pupil center. This incidentally is not an artifact of our convention to use the pupil center as the reference. The non-uniform distribution infers a required facet density substantially higher at the pupilÕs edge than at its center. Similar to atmospheric turbulence, ocular aberrations can be effectively compensated using less facets at longer wavelengths. For example, nine times more facets are needed at 0.4 µm than 1 µm. The prior results all assume a 100% facet fill, which frequently does not occur with actual devices. Corrector performance is highly sensitive to facet fill and initial results suggest a fill well above 90% is highly desirable for diffraction-limited imaging in the eye. In polychromatic light, performance is further degraded by material dispersion and phase wrapping of certain types of correctors as well as the normal chromatic aberrations of the human eye. Our polychromatic analysis revealed that for 3 and 6 mm pupils, the combined #5717 - $15.00 US

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effect of phase wrapping and dispersion of E-7, a commonly used nematic LC material, limits the FWHH spectral range to (at least) 350 nm and 125 nm, respectively. In both cases, the effect of phase wrapping dominates dispersion. However, wrapping and dispersion are substantially less degrading than the chromatic aberrations of the eye. In this context 2π limitations of SLMs are relatively unimportant and suggest that SLM correctors will perform very similar to that of piston segmented mirrors that have larger stroke. As the final step of our analysis, we investigated the monochromatic performance of piston segmented devices to compensate (or create) a defocused wavefront covering a range of dioptric values. This is relevant for such applications where dynamic correction of defocus is important as for example for the presbyopic eye and patient refraction. In this study, correction was found highly sensitive to pupil size and decentration. For a small 3 mm pupil, a sparsely segmented device (12 ² D/rs ² 18) is sufficient for handling one diopter of defocus, while larger pupils require substantially more facets, especially for off-axis imaging. Acknowledgments The authors thank Nathan Doble and Huawei Zhao for helpful discussions as well as two anonymous reviewers who provided important recommendations. Financial support was provided by the National Eye Institute grant 5R01 EY014743 and R01-EY05109. This work has been supported in part by the National Science Foundation Science and Technology Center for Adaptive Optics, managed by the University of California at Santa Cruz under cooperative agreement No. AST-9876783. Send all correspondence to Donald T. Miller.

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