1. Introduction

Visual impairment from uncorrected refractive error affects over a billion people worldwide, and accounts for an estimated $200 billion in lost productivity annually [1–4]. The World Health Organization estimates that the potential gain in productivity from addressing this global health burden could be an order of magnitude greater than the direct cost of conducting such a large-scale intervention [5]. Despite growing global health initiatives, significant obstacles remain. One critical impediment is a lack of eye care providers in low-resource settings [6]. For example, parts of Sub-Saharan Africa have as few as one ophthalmologist per million people, while more developed nations have on average 80-fold more per capita [6–8]. This disparity has spurred innovation in developing technologies that increase the efficiency, reduce training requirements, and improve quality of care of current providers [2]. In particular, there are several recent efforts to introduce low-cost, portable autorefractors to improve access to refractive eye care [9–11], and a large field study has shown that accurate prescriptions can be obtained in low-resource settings with a hand-held wavefront aberrometer [12].

Several low-cost, portable autorefractors utilize a Shack-Hartmann wavefront sensor (SHWFS), which consists of a microlens array placed one focal length in front of an optical sensor. The incident portion of an electromagnetic wavefront scattered off a patient’s retina can be sampled by this equally-spaced, periodic lenslet array, and focused onto the plane of the optical sensor to create a spotfield intensity distribution. The location and displacement of each of these spots relative to a reference spotfield allows measurement of the gradient of the wavefront, which can be used to generate an eyeglass prescription [13–18]. The use of a periodic lenslet array allows for simple, rapid spotfield tracking by using centroiding algorithms to measure spot location and calculate displacement. Beyond their ophthalmic applications, SHWFSs are also broadly utilized in metrology, astronomy, and microscopy [19]. Despite undergoing significant development in software algorithms, microfabrication, and cost reduction over the last several decades, the basic operating principles of the SHWFS are the same, and the microlens array remains a central, high-cost component. Moreover, the limited dynamic range of a microlens array typically requires the addition of a moving Badal lens system to compensate primary aberrations and bring the wavefront from the eye into a measurable range [20, 21]. These additional optics and moving parts increase the cost and reduce reliability of portable wavefront autorefractors. Thus, while high-end, tabletop autorefractors, with a typical dynamic range around [−30D, +25D], are able to cover nearly 100% of the population [22–24], low-cost, portable autorefractors are typically limited to less than half this dynamic range and performance decreases at the extremes of the range [12,25].

Optical diffusers have been explored for lensless imaging, lightfield measurements, and imaging through complex media [26–29]. Wavefront sensing with a thin diffuser has also been demonstrated, relying on the principle of the diffuser memory effect [30–32]. Recently, Berto et al. demonstrated the fundamentals for using a diffuser to measure high resolution wavefronts using integrated transverse displacement maps, which were generated from non-rigid image registration of speckle pattern distortion between planar and aberrated wavefronts [30]. Further, Gunjala et al. used a statistical approach to reconstruct aberration profiles from multiple images through a weak diffuser at distinct angles of illumination [33].

In this report, we develop a diffuser wavefront sensor (DWFS) for ocular aberrometry, characterize performance tradeoffs, and compare metrics that are relevant for autorefraction. By constructing and testing a DWFS and a SHWFS in parallel, we measure refractive error in a model eye and test the performance of the two devices under three different illumination techniques: a laser diode (LD), a laser diode with a laser speckle reducer (LD+LSR), and a light emitting diode (LED). While we recognize that significant advancements in algorithmic [34–38] and lenslet microfabrication [39, 40] have been accomplished to increase the dynamic range of a SHWFS, we use a conventional SHWFS lenslet array and spot tracking algorithm where spots are confined within their lenslet’s field of view for robust measurement. We find that a holographic diffuser intrinsically provides an increased dynamic range and number of resolvable prescriptions, while maintaining similar precision and acceptable accuracy when compared to the conventional lenslet array. The performance benefits of the DWFS originate from the uniqueness of the caustic pattern produced by the diffuser. Compared to the more uniform spots produced by a lenslet array, unique features from the caustic pattern can be tracked across larger displacement on the sensor without incurring origin ambiguity that arises when a spot in a SHWFS translates out of the aperture of a lenslet. Figure 1 shows images of caustic patterns acquired by the DWFS and spotfields captured by the SHWFS.

 

Fig. 1. (a) Example DWFS caustic patterns from an emmetropic model eye (red) and 5D myopic eye (blue), and (b) corresponding spotfields from the SHWFS with the same refractive error and pseudocoloring as in (a). Note the periodic nature of the spotfield image, compared to the relatively random caustic pattern of the DWFS.

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2. Theory

The spot displacement in a SHWFS due to local angular tilt of a wavefront (α) is determined by the lenslet pitch (ρ_SH) and the lenslet focal length (f_SH), shown schematically in Figure 2(a). The dynamic range and sensitivity to the angular wavefront tilt (α_max and α_min) depend on the wavefront sensor characteristics, signal to noise ratio, and the spot tracking algorithm used. While sub-pixel sensitivity is obtainable with centroiding algorithms [41], we define α_min by a centroid displacement 1-pixel in magnitude for comparison of the SHWFS and DWFS. The minimum detectable spot deviation then is set by the pixel size (Δx), while the maximum displacement is set by ρ_SH/2. The magnitude of spot deviation for a given wavefront tilt scales with the focal length of the lenslets for the tan(α) ≃ α approximation, and thus the angular sensitivity and range are fundamentally opposing constraints in a SHWFS design. This can be represented as:

From Equations 1–2, it is apparent that the number of resolvable tilts, α_max/α_min, is independent of the choice of focal length:

Similarly, for an aberrometer, the dynamic range and sensitivity of curvature measurement depends on Δx, ρ_SH, and f_SH for a given pupil diameter. This tradeoff between sensitivity and range makes the number of resolvable prescriptions a useful figure of merit.

 

Fig. 2. (a) Diagram of SHWFS fundamental dynamic range and sensitivity constraints. At high wavefront tilts (α > α_max), spot displacement into an adjacent lenslet’s field of view causes origin ambiguity. At low wavefront tilts (α < α_min), spot displacement is less than a pixel. (b) A similar model is proposed for the holographic diffuser, where the effective diffuser focal length is the distance from diffuser surface to sharp caustic image formation (f_D), and the effective diffuser pitch is the measured average distance between sharp caustic intensity peaks (ρ_D).

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There are no manufacturer specifications for the 0.5° holographic diffuser used in this study that are directly analogous to those of the lenslet array. To compare the dynamic range of the DWFS to the SHWFS directly, we model the diffuser as a non-periodic lenslet array, shown schematically in Figure 2(b). We define the effective diffuser pitch (ρ_D) as the mean distance between sharp caustic intensity bands, and the effective diffuser focal length (f_D) as the distance from the diffuser to the sensor. By taking plot profiles through the caustic pattern produced by the holographic diffuser with on-axis plane wave illumination, we measure ρ_D = 338μm with a standard deviation of 21μm (see Figure 8 in Appendix Section 6.1). Note that in contrast to the fixed ρ_SH = 300μm of the SHWFS, the pitch is not uniform for the DWFS. The effective diffuser focal length was empirically chosen to be f_D = 5.15mm based on the distance from the diffuser to the sensor where the caustic pattern appeared to be in sharpest focus.

To compare the DWFS and SHWFS, one must account for differences in their characteristics. We derive criteria based on a thin lens approximation ray trace to capture the predicted sensitivity and range of the DWFS and SHWFS in the current system used, shown in Appendix Section 6.2. Table 1 organizes the relevant characteristics of each device as well as the predicted range, sensitivity, and number of resolvable prescriptions (NRP) in terms of refractive error. Here the predicted NRP for each device is calculated as the predicted dynamic range, due to the ρ/2 cutoff criteria, divided by the predicted refractive error sensitivity, considering a one-pixel spot centroid displacement. This analysis predicts that the SHWFS has a lower dynamic range and increased sensitivity as compared to the DWFS. Importantly, the ρ/2 cutoff criteria for measurement range yields a similar predicted α_max / α_min for both devices, highlighting the tradeoff between sensitivity and dynamic range. Thus, if both devices operate with the conventional ρ/2 maximum detectable displacement criteria of a lenslet array, their characteristics predict that they should be able to resolve a similar number of prescriptions. Experimentally, if the DWFS is found able to measure more prescriptions than the SHWFS, the cause must be due to either the DWFS algorithm, or the difference in information contained in the DWFS caustic pattern as compared to the SHWFS spotfield. To discern between these two potential causes of increased performance of the DWFS, we test the SHWFS spotfield intensity distributions on the DWFS algorithm (Appendix Section 6.4, Figure 12).

Table 1:. Comparison of Attributes of SHWFS and DWFS

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3. Methods

3.1. Optical System

Figure 3 shows a schematic of the optical system constructed to measure wavefronts with the DWFS and SHWFS concurrently. The phase-encoding element of each device (diffuser for DWFS and lenslet array for SHWFS) was placed conjugate to the trial lens with a 1×, 4f telescopic relay, comprised of two f =50mm lenses. The signal to each instrument was separated by a 50:50 beamsplitter. The SHWFS consisted of a Thorlabs WFS300-14AR with lenslet pitch of 300μm, focal length of 14.6mm, and pixel size 4.65μm. The DWFS consisted of a 0.5° holographic diffuser (Edmund Optics) separated from a Thorlabs DCC1545M (pixel size 5.2μm) by 5.15mm. A holographic diffuser was utilized in the DWFS for its sharp caustic pattern formation and high transparency at visible wavelengths. The characteristics of the SHWFS and DWFS are organized in Table 1, including the predicted sensitivity and dynamic range in terms of refractive error. The model eye consisted of a biconcave lens (f = 25.4mm) separated from a model retina (diffusive piece of paper) by its focal length. Three different illumination systems were tested: (1) a laser diode (LD), (2) a LD with a laser speckle reducer (LD+LSR), each at 650nm, and (3) a light emitting diode (LED - Cree XLamp XP-E Red LED, 0.5 mW, 1mm² surface emitting area [42]). Each source was shaped to a collimated pencil beam 1mm in diameter and incorporated into the system 2mm off-axis to reduce back reflections. Note that LD+LSR illumination was focused onto the Optotune LSR-3005-17S-VIS before re-collimating, which was accomplished by two f =15mm lenses separated by 2f [43]. The LED was collimated with an f =25mm aspheric condenser and an additional f =200mm focal length lens. Crossed linear polarizers were incorporated into the system to further reduce back reflection. Trial lenses from a standard optometrist’s kit were introduced adjacent to the model eye lens to induce known refractive errors. Trial lens powers with spherical equivalent power (M) from [−24D, +24D] were measured in both the DWFS and the SHWFS, recording raw images of the distorted caustics with the DWFS and measuring centroid positions from the recorded spotfield images of the SHWFS. Refractive errors exceeding ±12D were achieved with two trial lenses placed close together. In this scenario, the thin lens approximation was decreasingly accurate for higher refractive error, so the changing effective focal length and principal plane location with each trial lens introduced was modeled using a Gaussian reduction, and used to calculate the effective trial lens power measured by the DWFS and SHWFS. More information about this modeling is shown in Appendix Section 6.3. Cylindrical trial lenses from [−4D, +4D] were introduced with their axis at 0° and 45° to measure Jackson cross-cylinder values (J₀ and J₄₅) with each wavefront sensor [9,44,45]. For each trial lens tested, five measurements were obtained to compare test retest reliability between the DWFS and SHWFS for each illumination type.

 

Fig. 3. Schematic of DWFS and SHWFS experimental setup. A collimated, slightly off-axis pencil beam was linearly polarized (P1), redirected with a beamsplitter (BS), and focused by the model eye lens (MEL) onto the model retina (MR). The resulting point-source illumination (occurring in the absence of a trial lens (TL)) is re-collimated by the MEL and relayed by a 1×, 4f telescopic system (RL1 and RL2) to the lenslet array of the SHWFS and the holographic diffuser of the DWFS simultaneously. A second crossed polarizer (P2) is used to mitigate back reflection. Various TLs are introduced, at a conjugate plane to the lenslet and diffuser, to characterize sensitivity and dynamic range.

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3.2. Shack-Hartmann Wavefront Sensing Algorithm

A commercial Shack-Hartmann wavefront sensor (Thorlabs WFS300-14AR) was calibrated to a planar wavefront in the absence of a trial lens (emmetropic condition), and subsequent measurements were taken as each trial lens was introduced to induce refractive error. We analyzed the spotfield centroids calculated by the commercial SHWFS. The distances between centroids in the emmetropic and ametropic conditions were calculated, and the resulting x–y displacement map was scaled by the known pixel size and divided by the lenslet focal length (f_SH = 14.6mm). The Zernike coefficients were fit to the displacement field data in the first derivative domain using a pseudo-inverse approach [46]. The fitted Zernike coefficients were used to calculate M, J₀, and J₄₅, as described below for the DWFS.

3.3. Diffuser Wavefront Sensing Algorithm

Wavefronts were calculated from caustic pattern distortions captured with a DWFS, using a similar approach to Berto et al. [30]. An overview of these steps is shown in Figure 4.

 

Fig. 4. Outline of the DWFS algorithm. (a) A reference caustic from an emmetropic eye (red) was registered to a distorted caustic from an ametropic eye (blue). The resulting vector field displacement map (D⃗(x, y), green arrows) was scaled to produce the transverse gradient of the wavefront (∇δ⃗(x, y)). (b) 2D gradient integration provided a measurement of the wavefront (W(ρ, θ)). (c) The measured wavefront was decomposed into a Zernike basis and a refractive error was calculated.

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To register calibration and distorted caustic patterns in the DWFS, we utilized a custom multi-level Demon’s algorithm to provide accurate non-rigid registration between the reference and distorted caustics [47]. A 5-level pyramid approach was utilized (scaling factor of 2 per level), with decreasing iterations per level to increase the speed of the algorithm. When complete, a precise x–y displacement map between the caustic patterns was generated (D⃗(x, y)), which, after scaling by the pixel size and dividing by the distance from the diffuser to the sensor (f_D), was approximately equivalent to the gradient of the transverse optical path difference induced by the introduction of refractive error:

The wavefront (W(ρ, θ)) was obtained by integration:

Note W(ρ, θ) was converted to unit-circle normalized polar coordinates, and flipped about its vertical axis to correct for the reflection off the second beamsplitter, allowing direct comparison to the SHWFS. The measured wavefront was decomposed into the Zernike basis (Equation 6),

Here R_max represents the maximum pupil radius sampled by the wavefront sensor. An eyeglass prescription can be calculated as S = ϕ₁, C = ϕ₂ − ϕ₁, and A = θ_A, where S is the spherical refractive error power (D), C is the cylindrical refractive error power (D), and A is the cylindrical axis [46]. Further, power vector notation was calculated from M = S + 0.5(C), J₀ = −0.5(C)cos(2θ_A), and J₄₅ = −0.5(C)sin(2θ_A) [44, 46]. The multi-level Demon’s algorithm was implemented in C++ using ITK and run on an Intel Core i7 2.8 GHz CPU [48]. The remainder of the DWFS algorithm was implemented in MATLAB. All code, including image registration and Zernike fitting, is available upon request.

3.4. Wavefront sensor performance Metrics

The dynamic range (DR), accuracy, and precision of refraction measurement of the DWFS and the SHWFS were evaluated over the [−24D, +24D] range of trial lenses tested for each illumination source utilized (LD, LD+LSR, and LED). The dynamic range was defined as the range of spherical refractive error over which each device produced a mean measurement within 0.25D of the correct value. Precision was defined as the mean of the standard deviation (σ̄) of the five repeated spherical refractive error measurements for each system configuration across the dynamic range. Similarly, the accuracy of each device was calculated by the mean root-mean-square-error (RMSE) of the spherical refractive error measurements over the dynamic range. The accuracy and precision were also calculated for the DWFS over a limited fine-value trial lens range determined by the SHWFS DR for each illumination type (RMSE_FV and σ̄_FV). The theoretical number of resolvable prescriptions (NRP_th) was calculated as the dynamic range divided by theoretical sensitivity, outlined in Table 1. The experimental NRP (NRP_exp) was calculated as the dynamic range divided by the RMSE. Measuring both of these metrics considers both a theoretical pixel-wise discretization of the NRP metric, which is comparable to the predicted NRP in Table 1, as well as a functional definition, which allows for the sub-pixel accuracy of centroid fitting and caustic registration that occurs experimentally.

4. Results and Discussion

4.1. Spherical error measurements

In order to assess the dynamic range of the DWFS and the SHWFS, trial lenses ranging in spherical power from [−24D, +24D] were introduced adjacent to the model eye lens. The resulting measurements from the DWFS and SHWFS are plotted in Figure 5 and the dynamic range, accuracy, sensitivity, and number of resolvable prescriptions were computed and displayed in Table 2 for each of the illumination types considered.

 

Fig. 5. M measurements and dynamic range results for (a) laser diode, (b) laser diode with laser speckle reducer, and (c) LED illumination for the DWFS (red) and SHWFS (blue) for [−24D, +24D] trial lens tested. Dashed vertical lines represent the predicted dynamic range with α_max for the SHWFS (blue) and DWFS (red). * power-limited.

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Table 2. Spherical Error Measurement Results (* power-limited)

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Though the dynamic range of the SHWFS with each illumination source is similar to the predicted values in Table 1, the dynamic range of the DWFS is significantly greater than the predicted values from the ρ/2 cutoff criteria for the LD and LD+LSR illumination. The expanded dynamic range of the DWFS can be attributed to the non-periodic nature of the caustic pattern being less susceptible to spot origin ambiguity as compared to the periodic spotfields in the SHWFS. Figure 6 highlights the ability of the DWFS to measure wavefronts from registered caustic displacements well beyond ρ_D/2. However, when the same Demon’s image registration algorithm is applied to the SHWFS spotfields, the registration begins to fail at a similar cutoff criteria to the conventional SHWFS algorithm, further indicating it is the caustic pattern providing resistance to origin ambiguity rather than the DWFS algorithm itself (see Appendix Section 6.4). The dynamic range of the DWFS is highest in the case of LD+LSR illumination, and the range of the instrument is relatively limited in the case of LED illumination. This is not due to a fundamental constraint imposed by using LED illumination, but rather due to limited optical power and a reduced signal-to-noise ratio provided by this source.

 

Fig. 6. Example caustic images (0D red, +16D blue) from the DWFS before registration (a), and after registration (b). Full field of view (left) and outlined regions of interest (right). The green arrow shows an example registered feature exceeding the ρ_D/2 cutoff criteria that limits a conventional SHWFS algorithm.

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The data in Table 2 demonstrate that NRP_th for the SHWFS is similar to the predicted NRP in Table 1, however NRP_th is significantly higher for the DWFS with LD and LD+LSR illumination. Although the SHWFS out-performs the DWFS in overall accuracy (RMSE), the increased dynamic range of the DWFS predominates to produce a significantly increased NRP_exp. The LD+LSR illumination offered the highest dynamic range and number of resolvable prescriptions. This is likely due to the relatively large power throughput and high signal-to-noise ratio allowed with the LD in combination with speckle noise reduction with the LSR.

The differences in RMSE between the SHWFS and the DWFS are likely due to the inherent predicted differences in sensitivity outlined in the Theory section in Table 1, the greater error that comes with registering highly distorted caustics near the limit of the DWFS dynamic range, and the effects of speckle noise on each algorithm. Despite having lower accuracy than the SHWFS across its full dynamic range, the DWFS is still able to produce measurements over its dynamic range with an RMSE ≤ 0.125D for LD+LSR illumination, and is capable of providing accurate eyeglass prescriptions (typically prescribed in 0.25D increments). Additionally, the DWFS demonstrates comparable σ̄ with LD illumination, and improved repeatability with LD+LSR illumination as compared to the SHWFS over its full dynamic range, indicating the DWFS algorithm has high test-retest repeatability. We note that for both the SHWFS and the DWFS, the measured sensitivity was better than the sensitivity predicted from a minimum 1-pixel displacement, indicating that both systems measure displacement with sub-pixel accuracy.

The DWFS repeatability and accuracy was also calculated over the restricted dynamic range defined by the SHWFS DR at each illumination type (σ̄_FV and RMSE_FV). In calculating precision with this restricted range, the DWFS shows an improved precision in fine-value measurements for LD and LD+LSR illumination. It also demonstrates an increase in accuracy for fine-value measurements with LD+LSR illumination, while performing similarly with LD illumination. Taken together, these results suggest that measurement of high Diopter refractive errors tends to cause more variability and error than fine-value measurements in speckle-reduced sources. This is expected considering the more distorted caustics should be more difficult to register due to larger displacement and intensity spreading. Interestingly, in measuring the fine-value range of refractive error dictated by the SHWFS DR, we observe that the DWFS with LD+LSR illumination outperforms the SHWFS in both accuracy and repeatability.

The effects of speckle noise on measurement accuracy and precision are evident in both instruments. For the SHWFS, the RMSE_FV was highest with laser diode illumination due to difficulty of centroid finding in the presence of speckle noise, a phenomenon that has been previously reported [49]. As expected, as the amount of speckle noise decreases with the introduction of the LSR, the SHWFS RMSE_FV decreases significantly. The RMSE_FV for the DWFS also followed this trend, though to a lesser extent. This is due to the structure of the multi-level Demon’s algorithm. When the registration algorithm initiates at the lowest level, the effects of speckle noise are suppressed by sampling, and as the algorithm progresses towards higher resolution images, it has already found a relatively accurate minima among the speckle noise.

4.2. Cylindrical error measurements

To generate a complete eyeglass prescription, the sphero-cylindrical refractive error must be determined accurately. In this section, we compare the ability of the SHWFS and DWFS to refract a model eye with astigmatism by introducing cylindrical trial lenses over the range of [−4D, +4D] at angles of 0° and 45°. This experiment tests a range of Jackson cross-cylinder values of J₀ and J₄₅ ranging from [−2D, +2D] (Figure 7). Table 3 summarizes the results of these measurements.

 

Fig. 7. J₀ (top row) and J₄₅ (bottom row) measurements for (a) laser diode, (b) laser diode with laser speckle reducer, and (c) LED illumination for the DWFS and SHWFS for [−4D, +4D] cylindrical trial lens tested at 0° and 45° axis.

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Fig. 8. (a) Diffuser caustic from emmetropic model eye with LD+LSR illumination. Red dashed line represents the location of single plot profile taken across the diffuser caustic. (b) plot profile produced of red-dashed line in (a), with threshold intensity of 40 counts marked by the blue dashed line. 20 distinct peaks extend above this threshold over a distance of 6656μm.

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Table 3. Measured J₀ and J₄₅ mean standard deviation (σ̄) and root-mean-square-error (RMSE) for the SHWFS and DWFS.

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For all three illumination types and for both J₀ and J₄₅, the SHWFS measured cylindrical powers with < 0.06D RMSE. The DWFS, while having slightly worse accuracy for the LD and LED illumination, is as accurate as the SHWFS with the introduction of the laser speckle reducer. Similar to the M measurements before, the DWFS demonstrates a lower σ̄ and improved test-retest repeatability than the SHWFS for LD+LSR and LED illumination, and comparable precision for LD illumination. From these results we can conclude that the DWFS is capable of providing astigmatism measurement accurate enough for an eyeglass prescription.

4.3. Optimized DWFS for ocular aberrometry

Based on the comparison of the SHWFS and DWFS under the three illumination types presented here, we can theorize what an ideal, optimized DWFS setup would be for the field of ocular aberrometry. We measured that the 0.5° holographic diffuser produces a sharp caustic pattern at f_D = 5.15mm beyond the diffuser and has an effective diffuser pitch of ρ_D = 338μm. The dynamic range of the diffuser utilized here was large and the sensitivity acceptable to provide eyeglass prescriptions when utilized with LD+LSR or LED illumination. However, similar to choosing lenslet arrays in order to trade off dynamic range for sensitivity in a SHWFS, a different holographic diffuser could be utilized to increase the sensitivity of the DWFS. Changing from the 0.5° holographic diffuser used here to a 0.2° holographic diffuser with similar ρ_D should provide a longer f_D, an increased sensitivity comparable to the SHWFS used here over its full dynamic range, and a similar improvement in number of resolvable prescriptions. Additionally, by assessing performance from sources with a range of speckle noise, we find that sources with limited speckle noise significantly improve DWFS accuracy. The utilization of a laser diode with a laser speckle reducer provided the best balance of a large dynamic range and high accuracy. Further optimizing the system for higher power throughput and increased signal-to-noise ratio with LED illumination could also produce an inexpensive, accurate system with an improved dynamic range. If cost is not a primary concern in designing a DWFS-based ocular aberrometer, a super-luminescent diode would provide an excellent compromise of high power with low speckle noise. However, one of the advantages of using a holographic diffuser instead of a lenslet array is decreased cost - the diffuser used in this experiment is nearly 40× less expensive when comparing cost per clear aperture area. Lastly, despite the advantages of the DWFS, the SHWFS outperforms the DWFS significantly with regards to speed. Though the data collection occurs in the same time frame and the memory requirements are nearly equivalent, the post-processing of the DWFS data takes approximately 2 minutes per measurement, while the SHWFS data processing occurs in a few seconds. In this report we did not optimize our registration algorithm for speed, and processing time could be significantly reduced by using an optical flow-based or other feature-based tracking algorithm, and by fitting the Zernike coefficients directly in the displacement domain. Due to the computational complexity of this specific algorithm, optimization for improved performance on a CPU is not practical. However, it is possible to sacrifice some accuracy for improved speed by using discrete Fourier transform-based registration methods [50]. For a hand-held device, the algorithm presented here could be implemented on a low power field-programmable gate array (FPGA) that can handle massively parallel calculations. Since FPGAs can be expensive, the cost can be reduced by mass producing the hard-coded architecture as an application-specific integrated circuit (ASIC). Alternatively, a cloud-based system could be used, where the computational burden can be off loaded to a non-local server.

5. Conclusions

In this report we apply diffuser wavefront sensing to the field of ocular aberrometry, demonstrating that a DWFS is capable of providing accurate, high dynamic range refractive error measurements in a model eye by registering highly distorted caustics to reference images. We find that the non-periodic nature of the intensity distribution encoded by a holographic diffuser renders it resistant to spot origin ambiguity, which plagues conventional SHWFS algorithms. Compared to a SHWFS approach, we show that a DWFS achieves significantly increased dynamic range and number of resolvable prescriptions. Additionally, the diffuser is nearly 40× less expensive per area than a lenslet array, making it ideal for affordable, large-range autorefraction devices. The implementation of this approach in modern portable autorefractors could lead to greater refractive care accessibility by enabling large-range wavefront sensing without any moving parts. Additionally, with recent advances in diffuser-based imaging and digital refocus technology, a diffuser-based approach may enable combined autorefraction and ocular imaging in the same device.

6. Appendix

6.1. Calculation of effective diffuser pitch (ρ_D)

In order to quantify the effective diffuser pitch, the number of peaks above a threshold of 40 counts was quantified and averaged across five plot profiles of a diffuser caustic produced by an emmetropic eye. An example of one of these measurements showing 20 peaks across a sensor width of 6656μm (ρ = 333μm) is shown in Figure 8. A mean value of ρ_D = 338μm with a standard deviation of 21μm was calculated from the five repeated measurements.

6.2. Trial lens dynamic range and sensitivity calculation

While the angular dynamic range and sensitivity of the SHWFS can be calculated by Equations 1–2, the direct conversion of these angles to a myopic and hyperopic dynamic range and sensitivity is not as straightforward. To calculate these metrics, we model the system using a thin lens approximation, shown schematically for an example myopic refractive error in Figure 9.

 

Fig. 9. Schematic of an example myopic refractive error measurement in the SHWFS path. Note the thin lens approximation, similar right triangles defined by α, and Equations 1–2 allow a derivation of predicted dynamic range and sensitivity of the SHWFS and DWFS in terms of refractive error. The resulting inequality is shown below in Equation 10.

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While the distance from the model eye lens to the model retina stays fixed (s), the focal length of the model eye changes when a trial lens is introduced (f_ME(P_tl)). For a myopic eye (P_TL > 0D), the rays converge before the model retina, and thus f_ME(P_tl > 0) < |s|, while for a hyperopic eye (P_TL < 0D), the rays converge behind the retina, and thus f_ME(P_tl < 0) > |s|. These two conditions result in a converging wavefront and real image, or diverging wavefront and virtual image, respectively. The final convergence point of the wavefront (s′(P_tl)) after propagation through the model eye lens and the 4f telescopic relay assembly can be calculated by applying multiple iterations of the thin lens equation, using the spot on the retina as the initial object. With this approach, using similar right triangles defined by α seen in Figure 9 and combining this with the SHWFS angular sensitivity and dynamic range criteria of Equations 1–2, we derive the minimum and maximum measurable refractive error in terms of the wavefront sensor focal length (f), the pixel size (Δx), the sensor radius (R_S = 2.381 mm and R_S = 2.662 for the SHWFS and DWFS, respectively), and the pitch (ρ), shown below in Equation 10. A similar approach focusing on local wavefront criteria has been offered previously by Campbell [51], but here we add a fixed R_S dependency for the conventional SHWFS algorithm applied in this report.

(10)$$\mathrm{\Delta }x<\frac{f}{\left|s^{\prime }(P_{tl})-f_{RL2}-f\right|}*R_S<\frac{\rho }{2}$$

The middle term of the inequality is the maximum displacement experienced by a spot of the wavefront sensor at the periphery of the sensor, where f refers to lenslet or diffuser focal length. As introduced with Equations 1–2, this displacement must be larger than Δx and smaller than ρ/2 to be detectable within the aperture of a given lenslet. With these criteria, we calculate a predicted dynamic range of [−4.30D, +4.30D] and [−12.30D, +12.30D] and a predicted sensitivity of 0.13D and 0.37D, for the SHWFS and DWFS, respectively. Figure 10 shows example caustic patterns and spotfield images at the predicted dynamic range and sensitivity extrema for each device, providing experimental verification of the validity of Equation 10.

 

Fig. 10. Example spotfields (a)–(b) and caustics (c)–(d) demonstrating SHWFS and DWFS refractive error sensitivity and dynamic range predicted by Equation 10. For each image, red corresponds to an emmetropic eye, blue to myopic refractive error, and green to hyperopic refractive error. In each panel, a full frame 1024×1024 image is presented (left), with a 256×256 ROI outlined in yellow and magnified (right). (a) SHWFS spotfield image showing sensitivity to ±0.25D refractive error. Though spot intensity overlap occurs, centroid displacement is greater than Δx, visible by the color separation in the top right spot. (b) SHWFS spotfield image showing ±4.5D refractive error near the limit of the instrument’s dynamic range. Note spotfield displacement towards the periphery becomes (ρ_SH/2), visible where the green and blue spots overlap between two red spots. (c) DWFS caustics with ±0.50D refractive error, near the predicted sensitivity of the instrument. Color separation is just barely visible towards the periphery of the image. (d) DWFS caustics with ±12D refractive error, near the predicted dynamic range of the instrument. Note towards the edge of the pupil caustic displacement becomes nearly (ρ_D/2) visible where green and blue caustics overlap halfway between two red caustics.

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6.3. Effective focal length and principal plane correction

To a first approximation, the power of the model eye lens with refractive error induced by adjacent placement of trial lenses can be well modeled with a thin lens, zero distance separation approximation:

(11)$$P_{\mathit{ME}}=P_{\mathit{MEL}}+P_{\mathit{TL}}$$

Where P_ME is the power of the model eye with refractive error, P_MEL is the power of the emmetropic model eye lens, and P_TL is the power of the trial lens(es) introduced. In practice, this approximation works well for small refractive error, but requires correction for larger refractive error. Here we model the back focal length (BFL), effective focal length (EFL), and principal plane (PP) location using a thin lens Gaussian reduction for each combination of trial lens used to better approximate the refractive error introduced. Figure 11(a)–(b) provides a schematic of how these characteristics change with P_TL > 0D and P_TL < 0D trial lenses, the resulting calculations of EFL, BFL, and PP location (c), and the effect these have on the expected trial lens power measurement in the system (d).

 

Fig. 11. (a) Schematic of the effective focal length (EFL), back focal length (BFL), principal plane (PP) location, and conjugate plane for the 1×, 4f telescopic relay (CP) for P_TL > 0D (myopic condition). (b) EFL, BFL, PP, and CP for P_TL < 0D (hyperopic condition). Note that the distance between CP and PP is larger than in (a). (c) Calculation of changing EFL, BFL, and PP location for each of the [−24D, +24D] trial lens tested. Curves generated with Equation 11 thin lens, 0mm separation approximation are denoted with “d = 0” subscript. Curves generated from Gaussian reduction and correction for moving principal plane are denoted with “GR” subscript. (d) Simulated trial lens power (TLP) measurements with thin lens d = 0mm approximation (red dashed line) and with Gaussian reduction correction (blue solid line). M measurements from DWFS with LD+LSR illumination are plotted over these curves vs. TLP_d=0 in blue x’s before correction, and vs. TLP_GR in red o’s after correction.

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In Figure 11(d) we can see that the approximation of Equation 11 begins to fail for high power trial lenses. The error is asymmetrically skewed towards the hyperopic range due to the alignment of the 1×, 4f telescopic relay conjugate with the trial lens housing. When a hyperopic trial lens is placed in front of the model eye lens the principal plane is pushed behind the model eye lens, further from the conjugate plane of the telescopic relay (Figure 11(b)). This introduces a larger error than the myopic condition, where the principal plane is pushed forward to be more closely aligned with the telescopic relay conjugate plane (Figure 11(a)). The result of this correction is clearly demonstrated in Figure 11(d), where the measured M DWFS LD+LSR data is plotted vs. the simple approximation initially considered in Equation 11 (blue x’s) and vs. the predicted effective trial lens power when considering the changing EFL, BFL, and PP (red circles). Though this correction is important for validation of the DWFS, we note that the change principal plane location for a human eye with refractive error will be significantly less than for the trial lenses considered here. Much of the error introduced is simply due to physical constraints of the housing of the trial lenses being placed adjacent to the model eye. In a real human eye with refractive error, the cornea and lens, are physically much closer together.

6.4. Demon’s registration applied to SHWFS spotfields

In order to further confirm that the non-periodic nature of the DWFS caustics was the cause of the instrument’s increased dynamic range, we applied the same multi-level Demon’s non-rigid registration algorithm to the SHWFS spotfield images from [−10D, +10D]. The results of this registration are shown in Figure 12.

 

Fig. 12. Registration results of multi-level Demon’s algorithm run on SHWFS spotfield images from [−10D, +10D]. Each image includes the 0D emmetropic spotfield (red), the P_tl > 0 trial lens spotfield after registration (blue), and the P_tl < 0 trial lens after registration (green). Note that after ±4D the registration algorithm begins to fail. This is approximately the same dynamic range found for the ρ_SH/2 criteria. This is in contrast to the DWFS caustic intensity pattern, which is able to register beyond ρ_D/2, as seen in Figure 6.

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We see from this data that the multi-level Demon’s algorithm applied to the SHWFS spotfields begins to fail after ±4D, indicating the periodic intensity distribution created by a lenslet array is prone to spot origin ambiguity. Application of the DWFS algorithm to the SHWFS spotfield data would yield approximately the same dynamic range as the conventional SHWFS algorithm confining the spotfield displacement to the lenslet’s field of view. Thus, we can be confident that it is truly the non-periodic nature of the caustic intensity distribution that affords the DWFS resistance to origin ambiguity and an extended dynamic range (see Figure 6).

Funding

Johns Hopkins Medical Science Training Program Fellowship; NIH (R21 EB024700 and R43 EB024299).

Disclosures

The authors are co-inventors on a provisional patent application assigned to Johns Hopkins University. They may be entitled to future royalties from intellectual property related to the technologies described in this article.

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