1. Introduction

Routine imaging of the retina of the living human eye is limited due to wavefront aberrations and diffraction. Although coarse structures such as large blood vessels and gross pathology are readily visible, resolution of single cells is unachievable. This limits the understanding that clinicians can gain about a given case to the effect rather than the cause of ocular disease. While much can be learned about disease processes from histological preparations of retinal tissue, this approach is limited by tissue shrinkage and distortion, and does not allow the time course of a disease to be plotted in an individual eye. It also offers no clinical diagnostic value, and is limited to animal models.

Additionally, treatment of some diseases, such as diabetic retinopathy or macular degeneration, can require a finely focused laser to seal leaks in small capillaries. These blood vessels are often close to central fixation, in which case there is a high risk of collateral damage causing irreparable vision loss. Resolution on the cellular scale would allow the beam to be more tightly focused to minimize this risk.

To improve both diagnosis and treatment of retinal and optic nerve pathology, image degradation due to diffraction and aberrations must be overcome. The diffraction limit of the eye can be largely overcome by pharmacological dilation of the pupil, however this leads to concurrent aberration increases and so decreased image quality at mid range spatial frequencies [1]. Once measured, it is possible to correct wavefront aberrations with a deformable mirror whose precise shape can be fine-tuned in real-time. The resulting resolution is limited only by diffraction. This technology is known as adaptive optics (AO), and was first applied to successfully correct higher order aberrations of the eye in 1997 [2].

While adaptive optics successfully eliminates aberrations, it does so only for wavefronts emanating from a single point (beacon) on the retina. As the object of interest moves laterally away from this point, full wavefront aberration compensation is not achieved, so image quality decreases. Centred on the corrected point is a region where the wavefront does not change considerably – the ‘isoplanatic patch’ (Fried 1982). Outside the isoplanatic patch resolution is no longer diffraction-limited, so that image quality is appreciably degraded.

Theoretically, if the aberrations of a system arise from several flat, discrete layers (as opposed to a continuous distribution), using one deformable mirror to correct each layer it is possible to obtain diffraction-limited imagery over the entire field of view [3]. In order to determine the aberrations that each of these layers cause, it is necessary to measure the wavefronts arising from several beacons simultaneously. This is akin to tomography of the aberrating media. Each mirror can then be made conjugate to a different layer and shaped to correct the aberrations for that layer. This is known as multi-conjugate adaptive optics (MCAO).

In practice, aberrations will often not arise from discrete layers – for example, aberrations of ground-based astronomical telescopes are the result of atmospheric turbulence that is continuous in nature. However the majority of the turbulence occurs in relatively thin bands, allowing it to be approximated as 2 or more thin layers. One deformable mirror can correct for each of these layers, leading to a significant increase in isoplanatic patch size with the use of 2 or more deformable mirrors [4].

MCAO is a technology that is just emerging: there is yet to be a physical implementation of it at any large-scale astronomical observatory [5], although the race is on with many theoretical papers testifying to its potential [6, 7], and optical table test-beds that model atmospheric turbulence correction by MCAO have been built [8].

MCAO has not yet been applied to the human eye, but has exciting promise to yield diffraction-limited imagery over the widest field of view yet. As a guide, it has been shown that with 3 deformable mirrors, astronomical telescopes can expect an increase in isoplanatic patch diameter by a factor of 7–10 [9].

MCAO correction is more complicated in human eyes however. Firstly, the ocular surfaces are curved whereas turbulence layers can effectively be modeled as flat in astronomy. This may compromise the linear approximations required to accurately predict MCAO corrections. Secondly, the lens of the eye has a gradient refractive index, meaning that aberrations are accrued throughout the lens and not just at the surfaces [10]. Third, eyes feature an off-axis pupil, and the desired point on the retina to be viewed is often off-axis; these asymmetries further complicate the system.

2. Methods

We investigated the feasibility of applying MCAO to the human eye through use of a schematic eye modeled in ZEMAX. The Liou and Brennan schematic eye was chosen for its anatomical accuracy, including aspheric surfaces, a gradient index lens, a decentered pupil, and surface curvatures and separations matching normative population data [11]. In addition, the higher order aberrations and total RMS error of this model agree well with normative population data, such as the work of Cheng et al, 2004 and Porter et al, 2001 [12, 13]. To work off-axis with this eye, a retinal radius of curvature of 12mm was assumed, following the example of other investigators to adapt the Navarro eye model for off-axis use [14]. Light of 555nm was used, which corresponds to peak human visual sensitivity under bright conditions [15]. An entrance pupil of 6mm diameter was used, representing a pupil size easily obtained in less than 30 minutes with 0.5% tropicamide, an agent commonly used to enlarge the pupil [16].

There are important considerations regarding the number of beacons used and their geometry. Beacons spaced too far apart do not allow sufficient information to be gathered to represent the three dimensional aberration profile. The end result for imaging purposes may be that the isoplanatic patch fractures into smaller zones. Adding beacons can ensure the isoplanatic patch stays intact, but this will require more processing time (decreasing the ability to obtain real-time corrections) and make the system more difficult and costly to build [17]. For astronomical MCAO systems, it has been proposed that the use of 5 beacons arranged as shown in Fig. 1 gives a good trade-off between these considerations, as long as they are appropriately spaced [17].

An identical scheme is used in our modeling. Since the eye differs so markedly from astronomical systems, determining a suitable beacon separation was the first parameter to explore. As a first-pass optimization then, we ran calculations using separations between the central and peripheral beacons of 0–10°, at 1° intervals. The beacon geometry was centered on the optical axis of the eye.

 

Fig. 1. Geometry of reference beacons on retina.

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The deformable mirrors were modeled as thin phase plates that retard the wavefront to correct the aberrations. Most astronomical investigators use 2 deformable mirrors in their simulations due to practical considerations regarding the cost and alignment of a large number of mirrors, and because atmospheric turbulence can be modeled reasonably well as stemming from 2 major layers. However the eye contains 5 major components (4 refracting surfaces and a gradient index lens), and may require more mirrors to reap the full benefit of MCAO. We form the null hypothesis that MCAO can not be applied successfully to the eye, and therefore require a large enough number of mirrors so that a lack of correctors can not be blamed if the null hypothesis is proved true. For this reason we have chosen 5 correctors to match the 5 components of the eye, and also to match the number of beacons used. Future work will need to determine the optimal number and positions of the mirror conjugates.

Since the optimal corrector positions were not known, they were chosen somewhat arbitrarily. One plate was conjugate to the anterior corneal plane since this is where the majority of aberrations arise, as this is the location of greatest refractive index difference [18]. Another plate was conjugate to the exit pupil as in conventional adaptive optics, and this also corresponds to the anterior lens surface. A third plate was located 2.5mm behind this (approximating the gradient index lens). Although not theoretically ideal since they do not correspond to surfaces of refractive index change, the other 2 mirrors were 1.5mm and 3.0mm anterior to the corneal surface, in anticipation of practical difficulties in conjugating mirrors to the posterior surfaces of the cornea and lens.

It is possible to perform a full tomographic reconstruction on the wavefront measurements to determine where in space the aberrations arise and conjugate the mirrors accordingly. This is achievable in astronomy because the light rays do not deviate significantly [19]. As a rule of thumb, the deviation becomes significant when refractive index changes are 10% or greater compared with the background index [20]. Therefore due to the high refractive power of the eye, a full tomographic reconstruction is possible only if the trajectory of rays that pass through the eye is known. This is time consuming to calculate and is impractical in real human eyes, where the ray paths may change significantly between subjects.

However it is not necessary to perform a full tomographic reconstruction. If it is assumed that the aberrations arise at particular depths (or in regions centered on particular depths), the correctors may be conjugated to those depths, with the solution acting to minimize the wavefront error at each beacon used [4]. This correction is calculated assuming linearity of the system just as with conventional adaptive optics [19]:

Where v is the vector of command signals to the correctors and w is the resulting vector of wavefront measurements at all eccentricities considered. A is the interaction matrix, which relates w and v for a particular system. A is measured by applying a known command signal to each mode of the corrector in turn and measuring the resulting wavefront at the exit pupil. The above equation can then be rearranged to determine the command signals to supply to each mode for a given set of wavefront measurements:

A singular value decomposition method is employed to calculate the pseudoinverse of the interaction matrix to solve this equation, yielding a least squares solution.

As a figure of merit, the wavefront was quantified by the point-by-point RMS difference between the wavefront and a theoretically flat (perfect) wavefront from each retinal location. Diffraction limited performance within the isoplanatic patch is defined as a wavefront RMS error of < λ/14, which is known as the Marechal criterion [21].

A factor to consider in practical MCAO corrections is the physical limitations of the correctors themselves. The ‘stroke’ of a corrector describes its maximum possible peak-to-valley deformation. It may be possible to position the correctors so that they provide a similar improvement in isoplanatic patch size, yet minimize the stroke so that larger levels of aberration may be corrected. As a preliminary attempt at optimization of MCAO for use with the eye, we examine the simplified case of a 2 corrector system. Although the ideal positions for a 2 corrector system are not directly applicable to a system with more than 2 correctors, some general characteristics of the relationship between the stroke, patch diameter and corrector positions will be obtained and prove useful as a guideline for future scenarios. We further simplify by considering only 2 beacons, in the horizontal meridian. The precise patch diameters and corrector strokes obtained in this one dimensional case are therefore not realistic for 2 dimensional corrections, but we make the assumption that the ideal mirror positions will remain relatively unchanged.

Various combinations of mirror conjugation positions across the refractive bulk of the eye are attempted, ranging from the anterior corneal surface to a point 8.00mm inside the eye (slightly behind the posterior lens), in 0.50mm steps. Both the total stroke of the correctors and the isoplanatic patch diameter are recorded for each combination. The beacons were located on the optical axis and at the fovea.

3. Results

Figure 2 shows a contour plot of the RMS of the wavefront error as a function of eccentricity for the uncorrected eye. Wavefronts were measured in 4 meridians spaced 45° apart, in 0.25° radial increments, and the results were interpolated to produce the contour map. The central point is on the optical axis.

An interesting point to note is that best image quality actually lies approximately 5 degrees off-axis on the temporal retina, which corresponds to the position of the fovea [11]. This implies that the aberration (predominately coma) that is introduced at the fovea due to it being off-axis is almost entirely ameliorated by a nasally displaced pupil. It makes sense that the fovea would be the position of highest image quality, since this is the area employed for fixation. This finding can not be accounted for by models that do not employ both a decentered pupil and an off axis fovea, since with no pupil decentration the best image point would simply be on axis.

Figure 3 shows the same function for a conventional AO correction applied on-axis, as well as a contour line demarcating the isoplanatic patch. The mean patch diameter shown is 3.12°. Figure 4 shows a slice through the horizontal meridian of this function, which makes clear the reduction in RMS error even outside the isoplanatic patch.

 

Fig. 2. Contour plot of RMS error vs eccentricity in the uncorrected Liou Brennan schematic eye. Initials denote retinal position with respect to the optical axis, where S = superior, I = inferior, N = nasal, T = temporal.

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Fig. 3. Contour plot of RMS error vs eccentricity in the Liou Brennan schematic eye corrected with a phase plate conjugate to the pupil.

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To obtain the maximum benefit from MCAO, we must determine the optimum spacing of the 5 beacons, as described in the methods. For our selected mirror configuration, Fig. 5 shows how the radius of the patch depends upon beacon separation. The patch is seen to be larger in the horizontal/vertical meridia than in the oblique meridia. This is because the beacons are aligned in the horizontal/vertical meridia. The larger the beacon spacing the more pronounced this difference becomes, so that although the mean radius is significantly greater, the shape of the patch is distorted.

 

Fig. 4. Plot of RMS error vs eccentricity in horizontal meridian, for the uncorrected Liou Brennan eye, the same eye corrected with a phase plate at the exit pupil, and the theoretical diffraction limit given by the Marechal criterion.

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Fig. 5. Radius of the isoplanatic patch depends on the angular separation of the beacons. The zero degree separation case indicates the patch obtained with conventional adaptive optics. The mean radius was calculated from the radii of the whole patch (i.e. not just from the average of the horizontal/vertical and oblique meridia).

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This distortion may be considered detrimental if increasing the beacon separation decreases the patch radius in one meridian. As can be seen from the graph, this occurs for beacon separations greater than 3°. Additionally, the majority of the potential improvement in patch radius seems to have been reached by this stage, and the oblique and horizontal/vertical meridia are still relatively similar. For these reasons we have chosen 3° to be the optimum beacon spacing for this setup.

As discussed earlier, the central patch size decreases dramatically at large angular separations. This is because when the beacons are too far apart, the points in between them are no longer corrected, causing ‘island’ patches to break away from the central patch.

Figure 6 shows a contour plot for the MCAO case with 5 beacons spaced 3° apart and 5 phase plates, as well as a contour demarcating the patch. The smaller dotted contour line shows for reference the patch size from the conventional AO correction. The mean patch diameter is now 7.61°, an increase by a factor of 2.44. This corresponds to an area increase by a factor of 5.95. Figure 7 depicts a slice through the horizontal meridian.

 

Fig. 6. Contour plot of RMS error vs eccentricity in the Liou Brennan schematic eye for the MCAO case (solid line) compared to the conventional AO case (dashed line).

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Fig. 7. Plot of RMS error vs eccentricity in the horizontal meridian, for conventional AO at the pupil, the same eye corrected via MCAO with 5 mirrors and 5 beacons, and the theoretical diffraction limit given by the Marechal criterion.

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Figure 8 shows a color map of the total corrector stroke as a function of conjugation positions of the simplified 2 mirror system operating in one dimension. Figure 9 shows a color map of the isoplanatic patch diameter as a function of the same positions. It is necessary to compare these 2 figures, for it seems that often an improvement in one parameter can come only at the cost of a deterioration in the other. For example, Fig. 8 shows that the positions of least stroke occur when the correctors are maximally separated. But Fig. 9 shows that the largest isoplanatic patch diameters occur when the correctors are close together, near the posterior lens.

From a practical standpoint, if one had access to correctors of sufficiently wide stroke, the best positions to conjugate the 2 mirrors to seem to be at 6.00mm and 8.00mm into the eye, corresponding roughly to the middle and the posterior of the lens. If stroke were limited in a system, it is best instead to move the mid-lens corrector forward to the anterior cornea (at positions posterior to this, stroke and patch size are both less), while keeping the other corrector at the posterior lens. Note that it appears that isoplanatic patch size will keep increasing at positions even further behind the posterior lens than calculated here (i.e., towards the image plane), but that this will be commensurate with further increasing the required stroke of the correctors.

 

Fig. 8. Plot of corrector position combinations vs total corrector stroke for the simplified case of 2 correctors and 2 beacons. Combinations were measured in 0.5mm increments and the results interpolated to produce the color map. The black region represents redundant combinations of corrector positions.

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Fig. 9. Plot of corrector position combinations vs patch diameter in the horizontal meridian for the simplified case of 2 correctors and 2 beacons. Combinations were measured in 0.5mm increments, and wavefronts were measured in 1.25° increments. The results were interpolated to produce the color map. The black region represents redundant combinations of corrector positions.

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4. Discussion

Clearly MCAO has the potential to provide the highest resolution wide-field optical images in living human eyes to date. In our simulations we show a significant increase in isoplanatic patch size for the 5 mirror case, by a factor of 5.95 in area. This improvement is likely to be an underestimate of the potential performance of the model, since we have not yet optimized the multitude of parameters associated with an MCAO correction.

Of primary importance are the number of mirrors used, and the positions to which they are conjugated. The more mirrors that are employed, the greater the performance of the system, but the number of mirrors needs to be kept at a minimum to keep processing time down as well as ensuring that the system is robust, compact and affordable. The mirrors also need to be conjugated to positions such that they have enough physical stroke to produce diffraction limited imagery, while maximizing the size of the isoplanatic patch. There is often a trade-off between these considerations. Given a simple 2 corrector system, a good compromise between these factors based on our preliminary optimization is to have one mirror located at the anterior cornea and the other at the posterior surface of the lens. Full optimization of a system with larger numbers of correctors will be explored in the future.

Also important is the spacing of the beacons. Here we have shown an optimal spacing for the 5 mirror configuration, however for a set of mirrors conjugate to different positions a different beacon spacing may well be indicated. It is also possible that the beacon geometry was not ideal – the large differences between the horizontal/vertical and the oblique meridia for some beacon spacings may indicate that a 3×3 grid pattern of beacons would provide a more radially symmetric correction.

A consideration for application to conventional flood-based illumination techniques is that the resolution of the images that we capture is limited by the resolution of the camera used. Current CCD cameras of sufficient sensitivity have ~ 0.5 - 1 Megapixels. In order to make use of all the information that an MCAO system provides, the camera needs to contain a sufficient number of pixels. Using the Rayleigh criterion for resolution of 2 points coupled with Nyquist sampling theory [21], pixels need to be spaced twice as close together as the half width of the point spread function (PSF). Using ZEMAX, the PSF half width at the Marechal criterion is ~ 0.0073°. Therefore pixels on our camera need to be spaced ~ 0.0036° apart. Since the CCD array will be square, the diameter of the array needs to be equal to the longest diameter of the isoplanatic patch, which is 8.98°. Therefore the necessary number of pixels along the diameter of the array is 8.98/0.0036 = 2,472 pixels. This corresponds to a total of 6.11 Megapixels. Current 6 Megapixel cameras lack the sensitivity required for AO ocular imaging – it can be seen that flood-illuminated MCAO imaging of the eye is likely to require an improvement in the standard of cameras used to record the image.

Of course, scanning systems such as scanning laser ophthalmoscopy (SLO) could be employed immediately to take advantage of the increased corrected field sizes, up to the limitation imposed by their raster scan rates and the temporal sampling and sensitivity of their photodetectors.

Also a limiting factor for practical applications is that with traditional wavefront sensing methods, 5 wavefront sensors would be required to measure the wavefronts coming from each position for MCAO. Since each of these would include a Hartmann-Shack lenslet array and a CCD camera, the cost of such a system may be prohibitive. It has been suggested that a ‘field lenslet array’ and relay system could be utilized to instead image the pupils from each beacon adjacent to each other on a single Shack-Hartmann lenslet array and hence a single CCD camera [8].

An interesting advantage of MCAO is that it is effectively a tomographic reconstruction of the ocular media. This means that the corrections for each mirror reflect the aberrations of the surface to which that mirror is conjugate. In this way, the aberrations arising due to the different components of the eye may be elucidated – although this is somewhat limited due to the gradient index of the lens. Determining the aberrations of a particular layer is impossible using conventional means since the only individual surface whose aberrations may be measured is the anterior cornea, by analyzing the shape through topography. The other ocular surfaces produce too little reflectance for an accurate shape map to be constructed [22]. Obtaining information on the aberrations of individual ocular components is crucial in determining the best form of a custom designed intra-ocular lens to replace a patient’s existing lens, for example after cataract surgery [23].

5. Conclusions

MCAO has the potential to significantly improve the size of the isoplanatic patch in the human eye. The magnitude of this improvement will depend upon a number of factors relating to the configuration of the deformable mirrors and the beacons on the retina. These factors still need to be optimized. An improvement in CCD camera technology is likely to be necessary to reap the full benefits of MCAO.