1. Introduction

Photoreceptor inner and outer segments (IS, OS) are highly evolved for capturing and guiding light. First discovered by Stiles and Crawford [1], this biological specialization is highly sensitive to the direction of the rays entering and exiting the photoreceptor. The process has been extensively studied experimentally both psychophysically and in reflection, the former coined the Stiles‐Crawford effect (SCE) and the latter the optical SCE [2–11]. As normal photoreceptor directionality requires normal morphology, the SCE and optical SCE are of significant clinical interest. The directional properties have been used in preliminary studies to indicate the stage and degree of various retinal abnormalities as listed in Gao, et al. [7].

Parallel to these experimental measures of photoreceptor directionality, extensive theory has been developed for modeling light propagation in photoreceptors based on the principles of optical waveguides, i.e., tiny optical fibers or light pipes [12–18]. These elegant models predict not only the directional sensitivity, but also the existence of so‐called optical modes, which represent the spatial distribution of the light energy as it propagates through the photoreceptor. This modal representation provides a more fundamental descriptor of the photoreceptor’s ability to capture light than directionality with the number of modes depending crucially on the physical properties of the photoreceptor IS and OS. The modes exquisite sensitivity to the photoreceptor properties has led to considerable interest in their use to define visual performance at the photoreceptor level and to detect perturbations in the photoreceptor properties associated with pathology. Unfortunately, theoretical models have remained largely abstractions, owing in part to the lack of experimental tests of their predictions [8] and their exquisite sensitivity to refractive index [12]. Even though multimode behavior was reported many decades ago in animal models in postmortem [19], only recent evidence has suggested similar behavior might also occur in human photoreceptors [20–23]. Human evidence was obtained using adaptive optics (AO) ophthalmic methods that provide sub-cellular imaging of the retina. Even with these latest findings, however, uncertainty surrounds even the most basic question as to whether cone photoreceptor segments support just one mode (single mode behavior) or more than one mode (multimode behavior).

In this paper, we take advantage of the micron-level 3D resolution afforded by AO and optical coherence tomography (OCT) to probe the waveguide properties of cone photoreceptors in the living human eye. To do this, we illuminate cones through a large pupil at the eye (6.7 mm) and collect the reflected cone signal through the same large pupil. In this way the cone numerical aperture (NA), which is narrower than the Stiles-Crawford effect [3], is overfilled thereby increasing the likelihood to excite waveguided modes, and to detect modes owing to the increased lateral resolution. Note that this approach is limited to measuring the mode profile only after exiting the cone segments in reflection, a process that is influenced by both forward and backward propagation of the modes and optical properties of the reflective site in the segments. This constraint is not unique to this study, however, as photoreceptors are most commonly studied in reflection, and commercial optical fibers are sometimes assessed in a similar manner. Our AO-OCT experiment was designed to test two hypotheses. One, cone IS and OS have optical properties consistent of waveguides, which depend on segment diameter and refractive index. Two, cone IS and OS support different optical modes. We tested both by taking advantage of the monotonic increase in cone diameter with retinal eccentricity and by imaging at the focal plane that gave best cone image quality. We compared these experimental findings to theoretical predictions of cone modes using a step-index optical fiber model of the cone. Discrepancies between prediction and measurement motivated us to improve the model, and this involved the test of two additional hypotheses in the Discussion section. Note that an early derivation of this work has been published in a conference proceeding [24].

2. Methods

Methods is divided in five sections. Section 2.1 describes the Indiana AO-OCT system used in this study to image cone photoreceptors. Section 2.2 presents the experimental protocol for acquiring cone images over a range of retinal eccentricities in two normal subjects. Data processing of the cone images and second moment analysis of modal content are described in section 2.3 and 2.4, respectively. Section 2.5 presents a theoretical model to predict waveguide modes of cones and to which the second moment analysis was applied.

2.1. AO-OCT imaging system

A detailed description of the AO-OCT system can be found in Liu, et al. [22]. Important to this study was the superluminescent diode (SLD) (BLM-S-785, Superlum, Ireland) with central wavelength at λ_c = 785 nm and bandwidth of Δλ = 47 nm that was used for both OCT imaging and wavefront sensing. The source provided a nominal axial resolution in retinal tissue (n = 1.38) of 4.2 μm. The detection channel of the SD-OCT system was designed around a Basler Sprint camera (SPL4096-140km, Exton, PA, USA) whose line rate was maximized for the source spectral bandwidth, in this case achieving an A-line rate of 250 KHz using the central 832 pixels. Empirically we found this speed a reasonable compromise for cone imaging. The axial resolution and signal-to-noise were sufficient for individuating and detecting the three primary reflections of cone photoreceptors (external limiting membrane (ELM), inner segment and outer segment junction (IS/OS), and cone outer segment tip (COST)), which was a key requirement of the study. The high acquisition speed (250 KHz) reduced eye motion artifacts, minimizing motion-induced degradation of cone spatial content.

New to this study, accuracy and speed of centroiding the Shack-Hartmann wavefront sensor (SHWS) spots were improved. To do this, we developed an adaptive algorithm with windowing based on that of Yin, et al. [25]. Our method accounted for spot brightness and operated on a reduced search box centered on the brightest pixel. For faster execution, the C + + code was converted to MATLAB Executable file (MEX) and called directly by Matlab.

2.2 Experiment

Two normal subjects free of ocular disease were recruited for the study. All procedures on the subjects adhered to the tenets of Helsinki declaration and approved by the Institutional Review Board of Indiana University. Written informed consent was obtained after the nature and possible risks of the study were explained. Both subjects had best corrected visual acuity of 20/20 or better. Age, spherical-equivalent refractive error, and axial length were 36 and 47 years; −3 and −2.5 D; and 26.07 and 25.4 mm. Eye length was measured with the IOLMaster® (Zeiss, Oberkochen, Germany). Maximum power delivered to the eye was 350 µW, measured at the cornea and within safe limits established by ANSI [26]. The right eye was cyclopleged and dilated with one drop of Tropicamide 0.5% for imaging and maintained with an additional drop every hour thereafter. The eye and head were aligned and stabilized using a bite bar mounted to a motorized XYZ translation stage.

Because cone image quality is sensitive to system focus, it was critical to find the focal plane that provided best cone quality at each retinal location imaged. To achieve this, we systematically focused the AO-OCT system over a narrow range, from approximately −0.15 diopters (D) to + 0.2 D relative to the zero focus reported by SHWS after the AO converged to a stable correction. Step size was 0.025 D, equivalent to just 9 µm (using conversion of 1 D = 370 µm) in retinal depth and realized by adding desired amount of defocus to SHWS reference. The 0.35 D (~130 µm in retinal depth) range was large enough to traverse the plane of best cone quality, defined as the plane that maximized power at the cone fundamental frequency, i.e., produced maximum cone contrast. Optimizing cone image quality in this way avoided potential subjective biases and facilitated consistent comparison across retinal locations. At each focus step, ten 0.5° × 0.5° volumes were acquired in 2.3 seconds. Dense A-scan sampling (0.6 μm/pix in both lateral dimensions across the retina) assured spatial mode detail in the cone image was not sampling limited by the AO-OCT. A fast B-scan rate of 1,042 Hz reduced eye motion artifacts. Volumes were acquired at 4.3 Hz and at eight retinal locations (0.6°, 2°, 3°, 4°, 5.5°, 7°, 8.5°, and 10°) temporal to the fovea, a range over which the cone IS and OS diameters increase monotonically with retinal eccentricity (see Fig. 1(b)). As a further step to minimize image quality variations that might mask mode appearance, volumes were excluded whose AO correction as reported by the SHWS was not diffraction limited (>0.07 waves RMS).

 

Fig. 1 Cone model. (a) Simplistic cone model composed of two cylinders, IS and OS, of homogeneous refractive index and surrounded by interphotoreceptor matrix. (b) Cone segment diameters (solid curves) [30, 31] and theoretical V numbers (dashed curves) increase monotonically as function of retinal eccentricity for IS (red) and OS (blue). V number was calculated from $V=\frac{\pi d_{i,o}}{\lambda }\sqrt{n_{i,o}^2-n_s^2}$, where d_i,o is the cone IS or OS diameter, n_i,o is the refractive index of IS or OS, n_s is the refractive index of interphotoreceptor matrix, and λ is the wavelength, 785 nm for this study. V number predicts the mode cutoff frequencies of the cone model, as for example when V<2.405, all modes with exception of fundamental LP₀₁ mode are cut off. Linear polarized (LP) modes are a common descriptor for circularly symmetric waveguides and often used to characterize cone photoreceptor models [13], as was done here.

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Important for the analysis of cone images, we confirmed the image aspect ratio was unity for the lateral (XY) dimensions of the AO-OCT volumes. This was realized using a model eye and a calibrated grid target as retina.

2.3 Image preparation and Fourier analysis

Software tools for image processing and analysis of the AO-OCT volumes were developed in MATLAB and ImageJ (Java plugins) [27]. To start, axial eye motion was removed from each AO-OCT volume by axially registering fast B-scans using a custom ImageJ plugin based on dynamic programming [28] in which fast OCT B-scans were shifted axially in order to align IS/OS in the slow B-scan projection view. Next the bright reflectance bands corresponding to IS/OS and COST layers were segmented and projected in en face to create 2D areal images of the cone mosaic at both depths [28]. Power spectra of the IS/OS and COST en face images were computed by Fourier transformation from which circumferential averages were determined and DC normalized. Averaging of power spectra from different AO-OCT volumes (typically ~5) acquired at the same focus and retinal location improved signal to noise. Best focus for cone imaging was determined by finding the circumferential-averaged power spectrum with maximum energy at the cone fundamental frequency. Noise was reduced by averaging the five pixels that covered the cone spectral peak. All further analysis of the AO-OCT images were limited to those acquired at this plane of best focus.

For each cone mosaic image, 50 to 100 cones were selected manually. Selection criteria was clear visibility of the cone at both IS/OS and COST layers, absence of vasculature shadows and motion artifacts, and a spatial distribution of cones that roughly covered the full AO-OCT image. Selected cones were extracted from the IS/OS and COST layers by cropping a square region centered on the cone, large enough to encompass the cone reflection, but small enough to avoid adjacent cones. For retinal eccentricities where cones are densely packed (0.6° and 2°), we found an additional circular binary mask, of same diameter as the selected square region, was necessary to remove adjacent cones that sometimes appeared in the corners of the square. Note that the XY coordinate of the selected region for a given cone was the same for both the cone’s IS/OS and COST reflections as these two reflections share the same set of AO-OCT A-lines. Selected region size was fixed for all cones at the same retinal eccentricity, but adjusted over eccentricities to account for varying cone diameter. Finally, selected regions were visually examined to assure only one cone per region.

The number of cones selected (50 to 100) depended on retinal eccentricity. Close to the fovea, it was easy to find more than 100 cones that met our criteria, owing to the high cone density and limited retinal vasculature there. Here we were less concerned about cone selection as the cones were uniform in appearance. In contrast in the peripheral retina, for example 10° from fovea, number of cones selected approached 50. Reduced cone density accompanied with increased retinal vasculature were primary contributing factors. Note that to avoid possible cone distortion created by the flyback of the fast-axis scanner, cones were selected from a narrower 0.3° × 0.5° image section.

2.4 Second moment analysis for quantifying modal content

Moment theory has been extensively used to describe quantitatively the features of images [29]. Here we apply the second moment to characterize the lateral spatial distribution of the cone intensity pattern, i.e., mode profile. The distinguishing features of this method compared to others already in the literature are discussed in Section 4.2. In general form, the second moment is expressed as

Prior to computing the second moment of cones in the AO-OCT en face images, we found it necessary to remove the noise floor that surrounded each cone. This was realized with an adaptive threshold in which pixels with an intensity value below 30% of the cone peak were removed. 30% was found empirically a good compromise between removing the noise floor and preserving the cone signal, but other comparable proportions were equally effective and yielded similar second moments.

As additional analysis to quantify cone modal behavior, we computed average and variance cone maps at each retinal eccentricity. This was accomplished by normalizing the peak reflectance of the selected 50 to 100 cones, rotating the major axes of the cones around their centroids ($\overline{x}$ and $\overline{y}$) so that they co-align (i.e., made θ zero), registering the cones to each other, and finally taking the average and variance of the cones. Second moment analysis was then re-applied to the averaged cone, but limited to the second moment, s, and circularity, c.

2.5 Evaluation of second moment analysis using cone optical model

To evaluate our second moment analysis for distinguishing cone mode behavior, we modeled the waveguide properties of cones and then applied the second moment analysis to the predicted mode pattern. Here we chose a simplistic cone model consisting of cylindrical IS and OS of diameters d_i and d_o, and homogeneous refractive indices n_i and n_o as shown in Fig. 1(a). The surrounding medium has a refractive index, n_s. While more complex models are possible, simple models such as the one chosen here are widely used and believed to capture the waveguide properties of cones. Furthermore, uncertainties in the cone physical parameters that all models require irrespective of their complexity remain a major point of debate, one that is addressed in the Discussion section by taking account of our AO-OCT findings.

For the three refractive indices in our cone model, we used n_i = 1.353 [12, 14], n_o = 1.419, and n_s = 1.347 [13]. We would have liked to use n_o (1.430) and n_s (1.340) from [12, 14], but these yielded model predictions further from our AO-OCT measurements. Despite this, it is important to note these indices from the literature were obtained for visible wavelengths and in postmortem animal tissue, sometimes in conjunction with optical theory to extrapolate to human. Thus the effect of dispersion at the longer wavelength of our AO-OCT system (785 nm) and imaging in living human tissue are important factors that could outweigh differences between the reported numbers themselves.

From 0.6° to 10° retinal eccentricity, IS diameter increased monotonically from 3.2 to 7.5 µm as measured by Curcio, et al. at the ELM [30]. OS diameter also increased monotonically, but noticeably less so, increasing from 1.5 to 2.2 µm as reported in Banks, et al. [31], which are consistent with Hoang, et al. [32]. We chose the former owing to their denser sampling of measurements across the macula. Figure 1(b) shows the IS and OS diameters as a function of retinal eccentricity along with the corresponding mode cutoff frequency (V number).

The intensity profile of LP modes for the cone IS and OS was computed by simulating wave propagation in the cone model. This was realized using commercial software COMSOL Multiphysics with the Wave Optics Module add-on (COMSOL, Burlington, MA) that is widely used for electromagnetic simulation in linear and nonlinear optical media, such as optical fiber. To simplify the optical waveguide model, we considered only the waveguide property of a single cone and thus no waveguide interaction with adjacent cones was considered. The predicted complex field distribution across the cone diameter was then exported from COMSOL, convolved with the diffraction-limited coherent point spread function of the human eye for a 6.7 mm pupil, converted to intensity by taking the modulus squared, and finally resampled to that of the AO-OCT data (0.6 μm/px). The convolution and resampling steps were added to more accurately emulate cone imaging with the AO-OCT system and to facilitate comparison with the experiment results. Finally, second moment analysis was applied to the cone intensity distribution predicted for cones between 0.6° to 10° retinal eccentricity.

Note that for both experiment and theory, we did not consider the polarization state of the imaging light nor the effect cones may have on it. While such knowledge is necessary to differentiate polarization degenerate states of the LP mode indices (e.g., degenerate states within each LP₁₁, LP₂₁, etc.), it is unnecessary to differentiate between the LP mode indices themselves and for that matter between single mode and multiple modes, a main objective of our study.

3. Results

3.1 Optimal focus for imaging IS/OS and COST layers

Best cone focus was achieved by systematically focusing the AO-OCT system, and then analyzing the power spectra of the images. Figure 2 shows the first step, representative en face projections of the same proximal patch of cones for three different levels of focus in subject 2. As apparent in the figure images, best quality cones occurred at 0.15 D with just ± 0.1 D change in focus yielding small, but appreciable reduction in cone quality.

 

Fig. 2 Representative AO-OCT en face projections through the IS/OS and COST layers at 3° retinal eccentricity for three different focus levels: (left) 0.05 D, (middle) 0.15 D, and (right) 0.25 D. Positive focus corresponds to a vitreal focal shift.

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To determine best focus for IS/OS and COST layers, power spectra analysis was applied to the layers with circumferential averaging about the zero spatial frequency to increase signal to noise. Figure 3(a) shows representative circumferential averaged power spectra for 3° temporal retina. Regardless of focus position (−0.125 D to 0.25 D), all power spectra show a monotonic decrease with spatial frequency, except near the expected fundamental frequency of the cones (highlighted by shaded violet stripe) where a cusp of energy occurs and is sensitive to focus. Peak energy at the cusp defines the plane of best focus, which in the figure occurs at the same system focus (0.15 D denoted as star in the legend) for the IS/OS and COST layers. The frequency at the cusp indicates a row-to-row cone spacing of 6 μm, which is consistent with histology [30] and in vivo measurements at this retinal eccentricity [33].

 

Fig. 3 Power spectra analysis to determine optimal focus for cone imaging. (a) Circumferentially-averaged power of the (left) IS/OS and (right) COST layers for 16 different levels of focus (−0.125 D to 0.25 D), which are color coded. Positive focus corresponds to a vitreal shift relative to zero diopters reported by the SHWS after AO correction. (b) Power at the cone fundamental frequency is plotted as a function of system focus averaged across eight retinal eccentricities on two subjects. Power for each retinal eccentricity was determined from the circumferentially-averaged power spectra of the IS/OS (red) and COST (blue) layers, as for example those in (a). Error bar denotes standard error.

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Figure 3(b) shows average power at the cone fundamental frequency (violet stripe in Fig. 3(a)) as a function of system focus. Space limitation prevented showing separate curves for the different retinal eccentricities, but they contained the same two general trends as the average curves. First, the IS/OS and COST layers peak at the same focus location of 0.1-0.15 D, suggesting the reflections from the inner and outer segments share a common optical aperture. This would occur if both reflections are influenced by waveguiding of the IS, not unexpected given that the COST reflection originates directly posterior of IS/OS. Regardless of the source, however, this finding indicates the same system focus can be used to assess modal content of both cone segments. Second, the figure shows that optimal system focus for cones is shifted vitreally by 0.1-0.15 D relative to zero focus reported by the SHWS. We ruled out any chromatic cause as the same light source was used for imaging and wavefront sensing. Such a shift could arise from the influence on the SHWS centroids by the underlying tissue layers, namely retinal pigment epithelium (RPE) and perhaps choroid, and yielding a hyperopic bias relative to the photoreceptor contribution alone [34].

3.2 Modal content of cone IS and OS

As expected, en face images of the IS/OS and COST layers reveal a regular pattern of bright punctate reflections at best system focus whose spacing is consistent with that of cone photoreceptors [30]. As established in the literature, we interpret each bright spot as reflected light originating from a single cone (Fig. 4). For visual comparison of modal content across retinal eccentricity, four representative cones (bright spots) from each location (colored boxes) were selected and shown enlarged below the corresponding mosaic image in the figure. For COST, the reflectance profile of the cones appears Gaussian like (analogous to LP₀₁ mode, see Fig. 1) at all retinal eccentricities imaged. This profile is a hallmark of single mode behavior, supporting the view of single mode behavior in the OS. While the Gaussian shape is preserved, its size notably increases with retinal eccentricity, an amount that is quantified in Section 3.4. Considerably more variation is observed in IS/OS reflection. Cones exhibit a Gaussian profile at 0.6° (similar to COST reflection) and become increasingly more irregular with increased eccentricity (unlike COST reflection). At 3° some cones retain their Gaussian profile and others do not. At 7°, almost every cone exhibits a bimodal (analogous to LP₁₁ mode, see Fig. 1) or multimodal (highly irregular; higher order) profile. At 10° most cones show multimodal profiles, which is in sharp contrast to the Gaussian-like COST reflection of the same cones. Similar IS/OS and COST images were obtained on the second subject (not shown).

 

Fig. 4 (Visualization 1) Representative AO-OCT images of cone photoreceptors with best focus at 0.6°, 3°, 7° and 10° temporal to fovea. Top and bottom rows show the reflectance from the IS/OS and COST layers of the same volume. Colored labels denote IS/OS and COST reflections of the same cone. Note that the scale of enlarged cones below each cone mosaic image varies with retinal eccentricity. A fly through movie of the volume at 7° temporal retina is shown in Media 1, restricted to the photoreceptor and RPE layers. A projected log B-scan is included next to the linear en face, which is globally normalized.

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Next we characterized the average cone profile at each retinal eccentricity (n = 1802 over all retinal eccentricities). This involved rotating, normalizing to maximum intensity, registering, and then averaging. Results are shown in Fig. 5 (Modes) for the IS/OS and COST reflections as a function of retinal eccentricity. The figure also shows the variance across cones (Var). Consistent with the IS/OS reflection of individual cones in Fig. 4, the average IS/OS profile in Fig. 5 shows a rotationally-symmetric, Gaussian-like distribution up to about 3° retinal eccentricity and then thereafter increasing in size and asymmetry. Because each cone is normalized to its maximum value, single mode behavior produces a hot pixel (pixel value of 1) at center of the averaged mode, and as expected multimode behavior will decrease the intensity value at center. Interestingly, at 7° and 8.5° on subject 2 (and to a lesser extent on subject 1), the average mode profile is dominated by a bimode (LP₁₁). At 10°, the bimodal shape is no longer evident, likely because additional higher-order modes are more prominent. In contrast, the average COST profile shows much less change, largely maintaining a Gaussian and rotationally symmetric shape. These changes are quantified in Section 3.4.

 

Fig. 5 Average (Mode) and variance (Var) maps of the IS/OS and COST reflections. Maps were computed from the 50 to 100 cones selected at each retinal eccentricity and share the same color bar (bottom right).

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Variance maps in Fig. 5 capture spatial differences in the reflectance profiles between cones of the same retinal patch. Single mode behavior results in an annular distribution of variance. Variance is zero at the center (dark blue) as all single-mode cones have a normalized maximum at their center. Surrounding this zero variance is a ring of elevated variance (green/yellow/light blue) caused by slight differences in the size and shape of cones. On the other extreme, cone segments that exhibit multimodal behavior are expected to present an irregular distribution of variance rather than an annular one. This is because different modes are excited in different cones and have maxima at different spatial locations. Both extreme variance profiles (annular and irregular) are evident in the Fig. 5 variance maps. The COST reflection (OS) shows the characteristic annulus of single mode performance regardless of subject and retinal eccentricity. A similar annulus in variance is observed for the IS/OS reflection (IS) at retinal eccentricities ≤ 3°. Beyond, the variance becomes increasingly elevated and irregular in appearance across the cone profile, more so for subject 2.

3.3 Example of second moment analysis using cone optical model

To illustrate our second moment method to quantify differences in cone modal content, we applied it to waveguided modes predicted for cone ISs. We used the step index fiber model described in Section 2.5 (Fig. 1(a)) with cone parameters for 7° retinal eccentricity, chosen in part because the model predicts several different types of modes at this location. The V number for the IS of the 7° cone model was 3.749 (Fig. 1(b)), indicating that only the two lowest-order modes, linear polarized modes LP₀₁ and LP₁₁, are supported by the IS. For a detailed description, COMSOL wave propagation software predicted six guided modes that are supported by the 7° IS, two LP₀₁ and four LP₁₁ (Fig. 6(a)). For simplicity we assumed the OCT reference was linearly polarized, thereby confining the modes detected by our AO-OCT to E_x, the electric field oriented in one of the two principle (x, y) axes perpendicular to the cone axis. As shown, the first column contains two Gaussian LP₀₁ modes, the lowest-order guided mode. The remaining four are variant forms of LP₁₁, which show characteristic bimodal profiles. Application of the second moment analysis to these six modes is shown in the same figure (Fig. 6(b)-6(c)).

 

Fig. 6 Second moment analysis applied to the waveguided LP modes predicted for the circular ISs of cones at 7° temporal retina. (a) Shown are the theoretical en face intensity distributions of E_x for the six modes supported by cone IS. Second moment analysis gives (b) second moment value, (c) circularity, and (d) orientation. Note (b-d) results are color coded with the six modes in (a).

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Figure 6(b) shows the second moment values (s), and reveals that the LP₀₁ modes have the most compact concentration of energy of the six modes. In this case, the LP₀₁ second moment is 32% smaller than that of the four LP₁₁ modes. This difference in s is consistent with the general property that lower order modes have more concentrated energy than higher order modes, a property that is reflected by their corresponding eigenvalues (see Eqs. (1)-(5)). In theory, the four LP₁₁ modes should be identical in s, which our modeling is consistent with. Figure 6(c) reveals perfect circularity (c = 1) for LP₀₁ modes and a value about two times smaller for the four bimodal LP₁₁ modes (c = 0.4504 on average). Figure 6(b)-6(c), however, also reveals that second moment value and circularity are insufficient to distinguish modes, as for example the four LP₁₁ modes have similar second moment value (s) and circularity (c). Distinguishing these four modes requires consideration of mode orientation, which is captured by the third second moment parameter, θ (Fig. 6(d)). Note that orientation is not shown in the figure for the two LP₀₁ modes as they are rotationally symmetric.

Taken together the three second moment parameters (s, c, and θ) provide complementary information. They extract key differences in the pattern of modes, in this case modes that are predicted for cone photoreceptors imaged in this study at 7°. Note that a slight perturbation from perfect cone circularity (<1% change as modeled with COMSOL) collapses all four LP₁₁ modes to two bimodal ones, one oriented to the minor axis of the cone and the other to its major axis, a result consistent with that reported by Snyder et al [35]. Thus orientation (θ) provides a means to test for anatomical asymmetries that may exist across cones of the same retinal patch, which we consider for our AO-OCT data in Section 3.4.

3.4 Second moment analysis of cone modal content

Second moment analysis was applied to cones at each retinal eccentricity to extract s, c and Ө. Results of these three parameters are shown in Fig. 7 to Fig. 9, respectively. Figure 7 shows the second moment values, s, for the subjects and the step index fiber model presented in Fig. 1. Starting with the IS, the AO-OCT measurements (red circles) show s of 2.1 µm at 0.6°, increasing monotonically with eccentricity, and plateauing at ~4 µm around 7° eccentricity. A slight decrease appears at 10°, more notable for subject 2. The measured second moment value depends little on the order computed: average of the second moment values (solid circles) or second moment value of the average (open circles). However, the latter is slightly higher, on average by 0.09 µm across both subjects and all retinal eccentricities, and likely due to the summing of noise when the major axes are coaligned first and then averaged. The IS experimental data (red circles) appear to match the theoretical prediction of single mode performance (LP₀₁, dashed red curve) across retinal eccentricities, suggesting cone IS supports single mode propagation only. This however is inconsistent with direct observation of multimode behavior at retinal eccentricities at and beyond 4° (Figs. 4 and 5). Likewise theoretical prediction of LP₁₁ mode (solid red curve) indicates a ~40% higher second moment (s) than experimental measurement at 7° and 8.5° where the LP₁₁ (bimodal) profiles are visually evident in the en face images. The discrepancy between theory and experiment suggests an inadequacy in the cone model. In the Discussion section, two hypotheses are examined: improved parameter (n_i and d_i) fitting of the model and the influence of a focusing/tapering effect (minification) in the IS ellipsoid.

 

Fig. 7 Second moment value s as function of retinal eccentricity for subjects (symbols) and predicted modes of the step index fiber model described in Fig. 1 (curves). Second moments are for the IS/OS (red) and COST (blue) reflections. Solid symbols are the average s of cones at each retinal eccentricity. Open symbols are s of the average cone. Error bars denote standard error for the selected 50 to 100 cones per retinal eccentricity. The red and blue curves are theoretical predictions of LP₀₁ (dash) and LP₁₁ (solid) modes for IS and OS, respectively. As shown, LP₁₁ for IS is not supported below 2° retinal eccentricity.

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For OS, s values of the AO-OCT measurements (solid and open blue circles) change less than those of the IS with retinal eccentricity. The measured second moment starts near 2.1 µm at the smallest retinal eccentricity (overlaps with measured IS values) and increases to 3 µm at the highest eccentricities. As with IS, the measured second moment is consistent regardless of order computed (compare solid and open blue circles). Unlike the experimental values, predictions for single mode (LP₀₁, dashed blue curve) and bimodes (LP₁₁, solid blue curve) show little dependence on retinal eccentricity. Furthermore, the predicted s is notably smaller (~2 to 3 times) than the experimental values, a trend that is opposite to that found for the IS in which the predicted LP₁₁ is greater than the measured. Difference between prediction and experiment can result from numerous sources, including inaccurate model parameters and the influence of the IS waveguide properties on the COST reflection.

To further quantify cone profile, the other two parameters of the second moment, circularity (c) and orientation (θ), were determined. Results are shown in Figs. 8 and 9. Regardless of method to compute c (solid versus open symbols), both sets of data points in Fig. 8 show similar trends. Neither reaches perfect circularity (c = 1) regardless of retinal eccentricity and segment (IS/OS, COST), even near the fovea where single mode performance is expected. Both show an overall decrease of c with retinal eccentricity with cone profiles near the fovea more circular than those in the periphery. Circularity of IS/OS and COST reflections are similar near the fovea (0.6° to 3°), slightly different but significant (p = 0.002) immediately outside the fovea (4° to 7°), and finally similar again at the largest eccentricity measured (10°). Greatest circularity and similarity of COST and IS/OS near the fovea support the view of single mode behavior of both cone segments. Immediately outside the fovea, reduced circularity of IS/OS compared to COST suggests the presence of higher-order, asymmetric modes in the IS. This asymmetry is consistent with the observation in Fig. 4 of the asymmetric LP₁₁ mode that is notably prevalent in the IS/OS en face images.

 

Fig. 8 Cone circularity (c) as function of retinal eccentricity, subject, and photoreceptor reflection: IS/OS (red) and COST (blue). Solid symbols are average of c for the selected 50 to 100 cones per retinal eccentricity. Open symbols are c of the average cone. Error bars denote standard error.

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Fig. 9 Cone orientation (θ) as function of retinal eccentricity, subject, and photoreceptor reflection: IS/OS (red) and COST (blue). Black circles denote mean for the selected 50 to 100 cones per retinal eccentricity.

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One major difference between the two averaging methods (solid versus open symbols) is a notable reduction in circularity using the second moment of the average (open symbols). To compare, circularity for the average of the second moments (solid symbols) is 0.77 ± 0.05, averaged across the eight retinal locations, both cone reflections, and both subjects. In contrast circularity for the second moment of the average (open symbols) is 0.60 ± 0.07, a decrease of 0.17 that is significant (p = 0.004). Because the latter computation assures cone asymmetries are preserved (orientation dependence is removed), this decrease suggests that at least a portion of the cones at any location, including the fovea, have dissimilar orientation. This interpretation is further supported by our analysis of second moment parameter, θ, which is presented next.

Figure 9 shows the orientation (θ) for the same cones processed for s (Fig. 7) and c (Fig. 8). Orientations are displayed as scatter points and cover the full 180° range with an average between 59° and 101°. Using chi-square (${\chi }^2$) test for uniformity, the orientation distribution for 28 of the 32 data points (8 retinal eccentricities; 2 reflections; 2 subjects) in Fig. 9 show no significant difference from uniformity (p>0.05). Of the four data points with non-uniformity, none share a common orientation nor share contiguous locations on the retina.

4. Discussion

4.1 AO-OCT method

In this study we investigated the modal structure of cone photoreceptors in the living human retina using AO-OCT. We imaged the relatively bright reflections (IS/OS and COST) that originate near the posterior ends of the cone IS and OS. For these reflections to be useful, they need to (1) capture waveguide properties of the cone segments and (2) be spatially resolved in lateral and axial dimensions by our AO-OCT.

While the optical SCE of the retina has been known for many decades only recently has it been attributed to the IS/OS and COST reflections [7]. This attribute plus numerous AO-OCT studies of the 3D reflectance distribution of cones [7, 22, 28, 36, 37] points to the IS/OS and COST reflections as being both waveguided and confined to the cone segments. A second attribute is the nature of IS/OS and COST reflections. Intensity [37], and polarization [38] measurements of cones point to the dominate optical effect being single scattering, as oppose to multiple scattering. Furthermore phase-sensitive measurements [36] indicate the single scattering is also specular. Based on this, the IS/OS and COST reflections will retain the waveguiding established in the first pass through the cone segment(s) and add to that the contribution of the second pass. Had these reflections been (non-specular) single scattering or multiple scattering, phase of the incident wavefront would be lost and the first-pass waveguide information destroyed.

The second key point is spatial resolution of our AO-OCT. Critical to investigating modal content is not only the ability to individuate IS/OS and COST reflections from each other, but also to resolve spatial details within the reflections themselves. To do so requires sufficient system axial and lateral resolution. The 4.2 μm axial resolution of our AO-OCT system in retinal tissue is sufficient to separate the IS/OS and COST reflections. Specifically the two reflections lie at opposite ends of the OS, whose length was measured to be ~45 μm at 0.6° and decreased monotonically to about 22 μm at 10°. Axial displacement of COST from the underlying RPE is smaller, but still larger than the system axial resolution. Thus the main rod reflection, which occurs at the apical RPE [23, 39], was effectively removed.

To demonstrate the lateral resolution our AO-OCT system achieves in the eye, we imaged the foveal center of both subjects, the location where cones are most densely packed and have a row-to-row spacing of 2.1-2.4 μm based on histology [30]. As confirmation, we resolved foveal cones in both subjects, a representative example is shown in Fig. 10(a). Note that the subject’s fixation (assumed foveal center) was intentionally placed at the left-inferior edge of the imaging field in order to reduce the distracting effect of the scanning beam on the subject. Direct measurement of cone spacing at the foveal center in the image yields a row-to-row spacing of 2.5 ± 0.14 μm. This value is larger than the theoretical 1.7 μm (confocal resolution = 0.88 λf/d) for a diffraction-limited 6.7 mm pupil at a wavelength of 785 nm, indicating that our AO-OCT can resolve spatial details in the cone mode profile at least as small as 2.5 ± 0.14 μm. Evidence of still smaller structures come from the circumferential averages of the spectra in Fig. 3 that reveal energy up to spatial frequencies that correspond to structures as small as about 1.8 µm. In short, central foveal imaging and power spectra analysis confirm lateral resolution of our AO-OCT is sufficient for not only distinguishing individual cones at the eight retinal locations in our study, but also modal details as small as 1.8 µm.

 

Fig. 10 En face spacing and size of the cone reflections. (a) Fovea cone projection through IS/OS and COST is shown with foveal center at bottom left corner of image. (b) Cone OS spot size is shown as measured by visual inspection (solid circles) and by converting second moment size to an equivalent diameter, $\sqrt{2}\times s$ (open circles) (see main text). For comparison the histology OS (blue trace) and IS (red trace) diameters from Fig. 1(b) are replotted. Error bars denote standard error.

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4.2 Rationale for using second moment approach to assess modal content

An extensive literature exists for assessing modal content of optical waveguides, much directed at modal decomposition. Strategies to decompose differ fundamentally depending on whether the modes are incoherent (time-averaged mode intensities add) or coherent (complex fields add). Incoherent modes can be determined from direct measurements of the transverse beam intensity profile [40, 41]. Such measurements, however, provide non-unique solutions when the modes add coherently because both field amplitude and phase must be taken into account [42]. This is the situation that describes the interaction of modes in photoreceptors owing to their extremely short length (tens of microns). Numerous coherent methods have been reported for decomposing the complex field of the beam into complex modes supported by the waveguide. These include phase retrieval in conjunction with intensity measurements at multiple axial positions [42, 43]; correlation filters [44, 45]; and interference techniques [46, 47]. Most of these methods require the complex modes supported by the waveguide to be known precisely. For commercial waveguides, such a priori information can be readily determined from the physical parameters of the waveguides, which are typically known with a high degree of accuracy. Such accuracy however does not extend yet to photoreceptors and in fact experimental results in this study (see Section 3.4) do not support the modes predicted from cone modeling. Unfortunately, the few coherent methods that do not require a priori information are not applicable to AO-OCT cone images. For example the interferometric method based on low coherence interferometry separates modes based on group delay differences [46, 47]. To do so requires waveguides that are much greater in length (e.g., meters) than that of photoreceptors.

It should be noted that a commonly used alternative to mode decomposition is the M² parameter [48]. While M² requires no a priori information about the waveguide, it requires intensity measurements at multiple axial positions, a criteria that does not lend itself to AO-OCT imaging. Because of the constraints imposed by all these methods, our strategy was to confine our analysis to what we could characterize: the light exiting individual cones. To do so we took advantage of moment theory [29], as detailed in Section 2.4, to characterize the lateral spatial distribution of the cone intensity pattern. While this approach is limited relative to many of the methods cited above, it remains applicable to our overarching objective to distinguish single mode from multimode behavior and equally important fulfills two critical requirements that these other methods do not: (1) it can be directly applied to our AO-OCT cone images and (2) it requires no a priori information about cone parameters.

4.3 Modal content of cones

Mode presence at eight retinal eccentricities of two subjects was qualitatively assessed by visual inspection of the IS/OS and COST reflections (Fig. 4), and quantitatively measured in terms of average and variance cone maps (Fig. 5) and second moment mode size (s, Fig. 7), circularity (c, Fig. 8), and orientation (θ, Fig. 9). The second moment was also applied to modes predicted by a step index fiber model (Fig. 6). Use of the model demonstrated utility of the second moment analysis to differentiate modal content, in particular LP₀₁ and LP₁₁ modes.

4.3.1 Modal content of the IS as determined from the IS/OS reflection

Our measurements of the IS/OS reflection provide several lines of evidence that the IS supports just a single (Gaussian-like) mode for retinal eccentricities up to 3° and multiple modes of increasing number and complexity thereafter. To start, the IS/OS reflectance profile of individual cones in the en face AO-OCT images (Fig. 4) and the IS/OS profile averaged across 50 to 100 cones (Fig. 5) were both found to be rotationally symmetric and Gaussian like up to about 3°, a hallmark of single mode performance, and then thereafter increasing in size and asymmetry, an indicator of multimode performance. Similarly the variance maps (Fig. 5) computed from the same 50 to 100 cones showed an annular distribution up to about 3° (indicator of single mode behavior) and an irregular distribution thereafter (indicator of multimodal behavior). These are further substantiated by the second moment size (s) and circularity (c). Size was smallest near the fovea (as would be expected for single mode behavior) and increased monotonically with eccentricity, separating from the COST second moment and plateauing around 7° (indicative of multimode behavior). Perhaps of most significance is the clear separation of s between the IS/OS and COST reflections that occurred between 3° and 4° for both subjects in Fig. 7. We interpret this location of separation as the transition region between single and multimode behavior of the IS.

Since the 3°-4° transition was obtained for the 785 nm wavelength of our AO-OCT system, we extrapolated the result to the shorter wavelength band used for vision (400-700 nm) assuming the influence of segment diameter and wavelength on mode structure follows the theoretical V number relation, $V=\frac{\pi d_{}}{\lambda }\sqrt{n_1^2-n_2^2}$, where $\sqrt{n_1^2-n_2^2}$ is the NA of cone segments. We predict at the wavelength of peak visual sensitivity (λ = 555 nm) that the 3°-4° transition region decreases to about 2°. Thus at wavelengths most critical for vision, the central ± 2° visual field is predicted to be confined to single mode performance. At the shortest wavelength of the visual spectral range (λ = 400 nm), use of the same arguments predicts a central ± 0.5° visual field of single mode performance. As the foveal blue-free zone encompasses a somewhat smaller field (< 0.175°) [49], we expect a narrow annulus (0.175° to 0.5°) of blue cones whose ISs support only a single mode and multiple modes everywhere else, notwithstanding anatomical differences between cone classes.

The shape of the IS/OS profile was found to be highly circular (c ~0.9) near the fovea (indicative of single Gaussian-like mode) and increasingly elliptical (decreased c) with increased retinal eccentricity. This increase in ellipticity is likely due to the increased prevalence of asymmetric modes, e.g., LP₁₁. But it is important to note that the rate of decrease in c with retinal eccentricity is shallower than that computed for the theoretical LP₁₁ as measured in Fig. 6. This shallower decrease could be because no patch of retina imaged in the experiment contained a preponderance of cones with pure LP₁₁ modes, but instead a mixture of mode shapes, some more circular than LP₁₁. The asymmetric profile of the IS/OS reflection facilitated analysis of cone orientation. A preferred orientation might occur if the retina is stretched or under shear forces, or intrinsic asymmetries in the cone structure that influence the waveguide properties are oriented the same, e.g., cilium junction and ciliary stalk. Using ${\chi }^2$ to test IS/OS orientation uniformity (0° to 180°) of the Fig. 9 data, only two of the 16 retinal patches (2 subjects; 8 retinal locations) showed evidence of non-uniformity, thus providing little support for an orientation preference.

4.3.2 Modal content of the OS as determined from the COST reflection

Our measurements of the COST reflection provide several lines of evidence that the OS supports just a single (Gaussian like) mode regardless of retinal eccentricity (0.6° to 10°). This is in contrast to the IS that was shown to have both single and multimode performance. Single mode behavior for OS was indicated by the Gaussian-like profile of individual COST reflections in AO-OCT images (Fig. 4) and the COST reflection averaged across 50 to 100 cones (Fig. 5). Similarly the variance maps (Fig. 5) computed from the same 50 to 100 cones showed an annular distribution at all retinal eccentricities (indicative of single mode behavior). These were further substantiated by the second moment size (s) and circularity (c). Size followed that of the IS/OS up to 3° or 4°, but then plateaued (~3 μm) as the IS/OS continued to increase. Leveling of the COST at a lower s is suggestive of single mode behavior. Like IS/OS, the second moment circularity of the COST profile was highest at the fovea. But COST retained a more circular profile (larger c, which is more indicative of single mode behavior) at all retinal eccentricities and noticeably more so for those between 4° and 8.5° (Fig. 8). Interestingly even at the smallest retinal eccentricity of 0.6° where single mode behavior is most likely, c never reached one (<0.85), suggesting that even foveal cones are not perfectly circular. This oddity is not without precedence as asymmetries in cone cross section are visually apparent in histology images [30, 50]. Similar to IS/OS, only two of the 16 retinal patches (2 subjects; 8 retinal locations) showed evidence of non-uniformity in the COST reflection, again providing little support for an orientation preference.

Single mode performance of our OS results is consistent with predictions from step index fiber models [12, 13]. Note that to compare the two required converting the model results from visible wavelengths (as reported in these publications) to near infrared, specifically 785 nm as used in our study. It is possible that the single mode behavior we observed in the COST reflectance might originate from an optical property in the OS that is not captured by the step-index fiber model. Specifically it has been proposed that in peripheral cones the OS actually supports higher order modes [50, 51]. However because peripheral cones have a tapered OS shape, the tapering causes the higher-order modes to radiate into the surrounding tissue [50, 51], leaving only the single LP₀₁ mode to reflect and double pass through the cone segments. This explanation, if correct, may help explain observed differences in the SCE between psychophysical and optical measures, between fovea and peripheral cones, and between cone inner and outer segments [1, 3, 7]. Unfortunately it has been difficult to test this hypothesis using conventional imaging methods. Our success with AO-OCT to evaluate the modal content of individual cones, however, may provide a path to test this hypothesis and to do so in the living human retina. This remains future work.

One unexpected oddity is the size of the COST reflection. It is too large relative to the anatomical diameter of the OS that generated it. According to Banks, et al. [31] and Hoang, et al. [32], cone OS diameters are about the size of fovea cones or even smaller depending upon retinal eccentricity. The former is plotted in Fig. 1 and replotted in Fig. 10(b). To compare, Fig. 10(b) also plots the equivalent COST spot diameter obtained by converting previous second moment (s) measurements to diameter. Conversion is simply the multiplication of s with $\sqrt{2}$, derived based on the assumption of circular cones (c = 1). This assumption is accurate for this comparison as for example $\sqrt{2}$ yields less than 1% error for c = 0.77, the average circularity of cones measured in this study. As a control, Fig. 10(b) also plots COST spot diameter realized by manually fitting an ellipse to the COST profile of 30 cones at each retinal eccentricity (0.6° to 10°). Even though the second moment and manual estimates of size are based on different metrics, Fig. 10(b) shows they are consistent with each other with the second moment diameter slightly smaller at all retinal eccentricities. Of importance here, is that the COST spot diameters in Fig. 10(b) are noticeably larger than OS diameters measured from histology at all retinal eccentricities. We tested how much of this difference is due to blurring caused by the AO-OCT beam size at the retina. To do this, we removed the blur contribution by assuming a theoretical confocal resolution of 1.7 μm (see Section 4.1) and a COST spot diameter equal to the square root of the summed squares of the resolution (1.7 μm) and the COST’s physical diameter. Based on this, we obtain an estimated COST physical diameter of 4.1 μm for cones at 10°, which is still larger (~2 times) than the histologic OS diameter (blue curve). This difference is inconsistent with the principles of optical fibers [52] in which waveguide modes are typically smaller (not larger) than the waveguide structure that supports them and this holds whether the fiber core index is homogeneous or graded with maximum at core center. Our own cone segment modeling (e.g., Fig. 6 and blue curves for COST reflection in Fig. 7) also predicts modes smaller than the OS and would predict even smaller modes if the model had used a graded-index profile (as opposed to a homogeneous one) such as that measured by Rowe, et al. [53] for cones in sunfish. We can only surmise that the COST reflection – while waveguided by the OS – must also be altered by some other intracellular mechanism of the cone before exiting the cone.

One possible candidate is the influence of the waveguiding properties of the IS since the COST reflection presumably recouples back into the IS on second pass through the cone. This means the recoupled OS modes should propagate in only those allowed by the IS. The fact that the COST reflection appears Gaussian suggests the primarily excited mode is LP₀₁, a plausible scenario if the OS mode profile is well matched to the LP₀₁ mode of the IS. An IS influence is also suggested by the similar trend in retinal eccentricity dependence of the COST spot diameters (black circles) and histologic IS diameters (red curve) in Fig. 10(b). Influence is further suggested by the similar average and variance maps of Fig. 5 and second moments of Fig. 8 of IS/OS and COST over the single-mode range (0.6° to 3°).

4.3.3 Temporal stability of the cone reflectance as imaged with AO-OCT

Testing our hypotheses requires stability of the cone mode pattern. Our general observation is that the mode pattern of individual cones is indeed stable over time. This stability appears unaffected by eye motion that translates the cone mosaic between images, causing the AO-OCT beam to illuminate and sample differently the same cone aperture. We observed the IS/OS cone reflectance to be stable not only within videos, but also between videos, which corresponds to time durations of minutes and hours. For the former, Fig. 11 shows representative cone IS/OS reflections at various time points in the same volume videos (~2.5 s duration) at three retinal eccentricities (0.6°, 7° and 10° temporal to fovea). Each video contains 11 images and while all showed similar cone behavior, cones from only six are shown due to space limitations. As evident in the figure, the reflectance patterns of individual cones are stable and in sharp contrast to the notable differences between cones, whether at the same retinal eccentricity or a different one (0.6°, 7° and 10°).

 

Fig. 11 Temporal stability of the IS/OS reflection of individual cones as imaged with AO-OCT. A total of 12 cones are shown, each imaged six times over a ~2.5 s duration. See text for details.

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4.3.4 Influence of segment length on modal content - guided and radiation modes

Two broad types of modes are possible in cones: (1) guided modes that are supported by the segment and remain confined to the segment and (2) radiation modes that are unsupported by the segment and carry (leak) energy out of the segment. The exceedingly short length of the cone segments (tens of microns) raises the question as to how much of the radiation modes leak out. A difficulty here is that AO-OCT – like all other retinal imaging systems for in vivo use – works in reflection. Therefore in first pass through the cone segment, radiation modes that escape the segment into the interphotoreceptor matrix can only be detected after reflection from deeper retinal layers, e.g., RPE, and exit the pupil of the eye. This multi-path propagation combined with the confocality of our imaging system means leaked radiation modes in the first pass are extremely difficult to detect, if at all.

Assessment of radiation modes must then be based on their presence or absence in the segment reflections, e.g., Fig. 4. The fact that LP₀₁ and LP₁₁ modes, which are indicative of guided modes, are clearly observed in these cone images while radiation modes are not supports the view that radiation modes exit the segment prior to their second pass. Related, guided modes should be distinguishable from a mixture of guided plus radiation modes as leakage of the radiation modes will result in a larger second moment. The fact that the measured IS second moment is narrower than the histologic IS diameter regardless of retinal eccentricity examined provides additional evidence that the radiation modes exit the segment prior to their second pass. It could be argued that the absence of radiation modes in our cone images implies that the radiation modes not only exit the segment, but exit it before even reaching the IS/OS in their first pass. However, testing this hypothesis would require assessing the reflectance distribution in between cone photoreceptors, i.e., halos around individual cones. This analysis was not conducted. Nevertheless interpretation of our measurements is consistent with the single pass observations reported by Enoch, et al. [19] in animal models in which specific guided modes dominated the cone image.

4.3.5 Influence of the AO-OCT scan pattern on excitation of cone modes

It is well established that mode excitation in optical waveguides is sensitive to the launch conditions of the illumination beam, including lateral offset from fiber core center, tilt, width, and wavefront curvature [54, 55]. The latter three conditions are fixed in our experiment due to the eye geometry and our focusing protocol. The remaining parameter, lateral offset, however is continuously changing as the AO-OCT beam scans over the cone aperture. To predict the impact this might have on mode excitation in cones, we modeled the AO-OCT beam (Fig. 12 (left)) and cone IS parameters specified in Section 2.5 for cones at 7° retinal eccentricity. Because non-circularity of the cone (c<1) may reduce – perhaps strongly – sensitivity of mode excitation with beam offset, we chose a circularity of 0.75 based on our measurements (see Fig. 8). To simplify the analysis, we restricted modeling to linearly polarized light of one orientation. The resulting cone model supported two waveguide modes LP₀₁ and LP₁₁ as shown in Fig. 12 (middle). Note that other LP₀₁ and LP₁₁ modes of different polarization orientation are supported as well, but are not excited by the linearly polarized light used and therefore not shown.

 

Fig. 12 Predicted coupling efficiency into cone IS modes as a function of AO-OCT beam offset in the cone IS aperture. False colored images are shown of the (left) modeled AO-OCT beam profile at the cone aperture, (middle) predicted modes LP₀₁ and LP₁₁ supported by the cone IS with c of 0.75, and (right) predicted coupling efficiency into the cone IS as a function of beam offset and mode. Solid white line denotes edge of cone IS. Double arrow indicates the polarization orientation, which is parallel to the major axis of cone. See text for details.

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Figure 12 (right) shows the predicted coupling efficiency into the two modes as a function of beam offset in the two lateral dimensions. The result strikingly shows that scan position in the cone aperture strongly influences efficiency of the modes, a result that was insensitive to IS length (lengths from 10 μm to 80 μm were investigated) and consistent with the literature for conventional optical fibers [54]. As shown in the figure, when the scan beam is near the cone aperture center (x and y offset near zero), coupling efficiency of LP₀₁ is greater and when displaced horizontally (x offset) LP₁₁ is greater. Thus scanning over the aperture to form an image results in a complex composite of both modes, which for this particular cone example yields an image with elevated intensity along the horizontal, x, meridian (not shown). The fact that our actual AO-OCT cone images (7° images in Fig. 4) do not show this pattern of reflectance and instead reveal dominate LP₁₁ modes with a dark central region suggests other mechanisms must be at play in how cone IS modes are excited. One possible culprit, which is not incorporated in our modeling, is the source of the IS/OS reflectance and how it might influence which modes are excited in reflection. This requires further investigation perhaps using ex vivo methods that would allow examining the cone IS in both transmission and reflection in order to tease apart their individual contributions.

4.4 Improving cone optical model by incorporating AO-OCT measurements

Second moment analysis is found to be a sensitive indicator of cone modes, both in terms of theoretical predictions based on a step index fiber model (Figs. 6 and 7) and experimental measurements with AO-OCT (Figs. 7, 8, and 9). Using this metric, discrepancies were found between the predictions and measurements (Fig. 7), thus motivating consideration as to how the cone model based on step-index fiber optics could be improved to better match our experimental findings. While it would have been appropriate to apply this strategy to both cone segments, we confined our investigation to the IS as a proof of concept and motivated by the importance of the IS as it influences both IS/OS and COST reflections as per our AO-OCT results. For IS, Fig. 7 shows two discrepancies between theory and measurement. First, theory predicts multimode behavior of the IS to occur closer to the foveal center than was measured experimentally (2° compared to ≥4°). Second, theory predicts a larger second moment s for IS than was measured experimentally (compare Fig. 7 solid red curves to red targets). In an attempt to resolve these discrepancies, we systematically searched the parameter space that defined the IS in conjunction with certain constraints (described below), to obtain predictions consistent with experiment. To do so required modifying the model as depicted in Fig. 13(a) to better capture the geometric shape of cones in particular the variation in diameter of the segments [32]. Using this model as a guide, two hypotheses were tested, the second an extension of the first. Both are presented below.

 

Fig. 13 (a) Modified cone model to test hypotheses one and two. (b-c) Test of hypothesis one by determining values of IS diameter, d, and refractive index, n_i, that allow the predicted and measured second moment size s to match. Three retinal eccentricities were selected. Predicted size was computed from a step-index fiber model. (b) Colored curves are model predictions of s for (dashed) LP₀₁ and (solid) LP₁₁ modes. Each color denotes a unique IS diameter (2.5-7.4 μm). Not all combinations of d and n_i support LP₀₁ and LP₁₁ modes, thus the curves are of different lengths. The horizontal gray bands correspond to the measured second moments at 0.6°, 3°, and 8.5° eccentricities from Fig. 8 with band widths equal to the stand errors in the same figure. For comparison of d values, the histologic IS diameters at the three eccentricities are given in parentheses. (c) To summarize the results of (b), colored polygons superimposed on the d-versus-n_i plot enclose the combinations of d and n_i values in (b) that enable the predicted s to equal the measured s. The two black curves are the theoretical V number cutoff for LP₁₁ (solid) and LP_{21, 02} (dashed) modes.

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4.4.1 Hypothesis 1

We hypothesize that support of optical modes in the IS – as observed in the IS/OS reflection – is determined by IS’s physical parameters (d, n_i) and surround (n_s), which collectively define the IS V number $\left(V=\frac{\pi d}{\lambda }NA\right)$. To test this hypothesis, we systematically varied d and n_i of the step-index fiber model (Fig. 13(a)). This was then followed by matching the predicted second moment size for single and multimode profiles to that of our experimental findings with AO-OCT, i.e., those plotted in Fig. 7 as a function of retinal eccentricity. Several constraints were implemented to make testing of the hypothesis more tractable. First, we limited the range of NA to straddle values commonly used in the literature [12, 13]. Second, because the physical meaning of NA, is somewhat abstract, we chose to fix n_s at its stated value (1.347) in Section 2.5 and vary n_i from 1.353 to 1.361, a combination that provided the intended range. Third, because of the lack of evidence in the literature that IS refractive index varies with retinal eccentricity, we assumed n_i to be constant regardless of retinal location. Last, the IS diameter was allowed to vary from its histologic value given in Fig. 1(b) down to 2.5 µm. Larger diameters were not considered because the histologic diameters were measured at the external limiting membrane [30] where the IS is widest due to its conic shape, as depicted in Fig. 13(a). This range (histology measurement down to 2.5 µm) covered expected diameters of the IS, important because of uncertainty as to which axial portion of the IS provides the defining d for the optical modes.

For each V number value, we predicted energy distribution across the cone diameter using COMSOL and then applied second moment analysis following the protocol established in Section 2.5. Figure 13(b) shows the predicted second moment result for LP₀₁ (dashed colored lines) and LP₁₁ modes (solid colored lines) along with the experimental results (horizontal gray bars) at three selected retinal eccentricities: 0.6°, 3°, and 8.5°. These eccentricities were selected as two (0.6° and 3°) were experimentally found to be dominated by single mode behavior, LP₀₁, and the third (8.5°) dominated by multimode behavior, LP₁₁ and possibly higher order modes. Thus in the figure the objective was to find the overlap of the experimental 0.6° and 3° results (gray bars) with the predicted LP₀₁ modes (solid lines) and the experimental 8.5° results with the predicted LP₁₁ modes (dashed lines). Additional details are given in the figure caption.

The solid star symbol in the top, left corner of Fig. 13(b) denotes the second moment value (LP₁₁) predicted for cone ISs at 8.5° based on the original step index fiber model as given in Section 2.5 (Fig. 1(a)). As shown the star is clearly well above the measurements (gray bar), giving a predicted second moment of >5 μm compared to the experimental value of ~4 μm. It is also evident that the model predicts (and as expected) a decrease in the IS diameter or an increase in IS refractive index results in a monotonic decrease in the second moment. For the cone example at 8.5°, an IS diameter range of 4.5 to 5.5 μm and n_i range of 1.354 to 1.3585 (fixed n_s = 1.347) gives a predicted LP₁₁ second moment that overlaps with the experimental value (gray band). This region of overlap is extracted and outlined in red in the accompanied diameter-versus-refractive-index plot in Fig. 13(c). Likewise for cone ISs at 0.6°, the overlapping region between model prediction and experiment for the LP₀₁ mode is outlined in green and for 3° it is outlined in blue.

The black solid and dashed curves in Fig. 13(c) denote the theoretical cutoffs for multimodal LP₁₁ and LP_{21, 02} performance as a function of IS diameter (d) and refractive index values (n_i). We see that the 0.6° (green) and 8.5° (red) regions that share overlapping n_i values fall within the correct single and multimode space in the plot as indicated by the black solid and dashed curves. That is, the model and experiment give the same second moments and yield the same mode profile (LP₀₁ or LP₁₁): LP₀₁ for 0.6° and LP₁₁ for 8.5°. Consistency of the model, however, fails for 3° cones. While the model parameters were adjusted to give the same second moment values as that of the experiment (blue curve), the resulting IS diameter is too large, supporting multiple modes (LP₁₁) in contrast to the experiment that was dominated by single mode (LP₀₁). Similar inconsistencies were found for cones at adjacent retinal locations 2° and 4° (not shown). Furthermore, if we hold to our assumption of n_i being independent of retinal eccentricity, then the only n_i range common to all three retinal eccentricities (0.6°, 3°, and 8.5°) is 1.354 to 1.358. Across this range, Fig. 13(c) shows that the model at 3° and 8.5° strongly overlap, meaning cones at these two retinal eccentricities must have identical d and n_i, yet from our AO-OCT measurements must support different numbers of modes. This discrepancy violates the underlying principles of optical fibers, and furthermore anatomical studies show the physical diameters of cone ISs at these two locations to be different. Taken together, our test of hypothesis one suggests the modified step-index fiber model does not properly capture the modal behavior of the cone IS across the range of retinal eccentricities evaluated here. This failure led to hypothesis two in which our step index fiber model was further adjusted, this time in an unconventional manner.

4.4.2 Hypothesis 2

Failure of hypothesis one in the transition region between single and multiple mode performance resulted from the simultaneously need for a large IS model diameter to match the size of the empirical mode (i.e., second moment), and a small IS diameter to match the shape of the empirical mode (i.e., LP₀₁ and LP₁₁). Hypothesis two attempts to address this discrepancy. The hypothesis states that confinement of modes in the IS – as observed in the IS/OS reflection – is determined in the apical region of the IS myoid (near ELM), but the size of modes is reduced (minified) as it propagates to the IS basal end due to a focusing or tapering effect of the ellipsoid [13, 32, 56] without loss of modes. Therefore the complete mode profile established in the myoid is preserved with only its size minified. In this way the size and profile of the mode(s) are decoupled with the size defined by the IS focusing/tapering effect and the profile by the IS myoid refractive index and diameter. Because diameter of the myoid is known from histology [30], testing of this hypothesis required varying refractive index of the cone IS model (Fig. 13(a)), determining empirically minification of the focusing/tapering effect, and then finally comparing to the AO-OCT results following the protocol of the first hypothesis.

Like hypothesis one, we systematically varied NA by fixing n_s at its stated value (1.347) and searching over n_i. Because the original step index fiber model predicts multimode behavior for cone ISs of narrower diameter than what was measured, we compensated for this in the new model by extending the range of n_i to include smaller values in accord with the relation of d and n_i in the V number expression $\left(\frac{\pi d}{\lambda }NA\right)$. Specifically the range of n_i was 1.350 to 1.361 instead of 1.353 to 1.361 as was used to test hypothesis one. This new range corresponds to NA between 0.09 and 0.19. The lower refractive index demand may reflect a dispersion effect of the retina tissue. Using this model, Fig. 14(a) shows predicted and measured results for cone ISs at 0.6°, 3°, and 8.5° with layout of the plot identical to that of Fig. 13(b). Figure 14(a) shows no overlap of the solid red curve with the 8.5° gray band, no overlap of the dashed blue curve with the 3° gray band, and no overlap of the dashed green curve with the 0.6° gray band. Thus theory and experiment do not match, however, we have not yet applied the focusing/tapering effect of the ellipsoid as stated in hypothesis two.

 

Fig. 14 Test of hypothesis two. Testing centered on determining a common n_i that allows predicted second moment size s and mode profile to match those of the measured IS/OS reflection for three retinal eccentricities: 0.6°, 3° and 8.5°. For n_i axis, n_s was fixed at 1.347. (a) Predicted size was computed from a step-index fiber model. Colored curves are model predictions of s at the three retinal eccentricities and for (dashed) LP₀₁ and (solid) LP₁₁ modes. The horizontal gray bands correspond to the measured second moments at 0.6°, 3°, and 8.5° eccentricities from Fig. 7 (s) with band widths equal to the stand errors in the same figure. (b) Plot is identical to that in (a) except minification factor and error bars were applied to the predicted curves. See text for how minification factor was computed. Because the minification factor has measurement error, predicted colored curves consist of two lines, the separation being the standard error calculated from OS diameter measurements in Fig. 10(c). (c) To summarize the results of (b), colored lines superimposed on the d-versus-n_i plot represent the combinations of d and n_i values in (b) that enable the predicted s to equal the measured s. The two black curves are the theoretical V number cutoff for LP₁₁ and LP_{21, 02} modes.

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To do so we determined the magnitude of the focusing/tapering effect (extent of minification), defined as the ratio of the measured OS to predicted IS/OS reflection size. The predicted size was determined using COMSOL, the histologic IS diameter, and n_i along its specified range. Since the focusing/tapering effect should be mode independent, we restricted the ratio to the single mode case (LP₀₁), thereby avoiding contamination from other modes, i.e., additional modes in the IS/OS reflection. While LP₀₁ size can be straightforwardly predicted, the corresponding measurement is masked by the presence of other modes, as for example at 4° and higher. To circumvent this problem, we made use of our findings that (1) the LP₀₁ mode is the same size for the IS/OS and COST reflections and (2) the measured COST is single mode for all retinal eccentricities considered. The former finding was discussed in Section 4.3.2 in terms of the IS influence on COST and substantiated by Figs. 5 and 7 measurements that show the two reflections are consistent in size over the single-mode range (0.6° to 3°). Note that because of the presence of other IS/OS modes at and above 4°, we could not confirm this relation at these locations. As reference, the solid markers plotted in Fig. 10(b) are the COST reflection diameters. Appling the focusing/tapering effect to the predicted s in Fig. 14(a) results in the adjusted predictions of Fig. 14(b), our final result for comparing prediction and experiment.

For the cone example at 8.5° in Fig. 14(b), a n_i range of 1.3505 to 1.354 gives a predicted LP₁₁ second moment that overlaps with the experimental value (gray band). This region of overlap is extracted and shown in red in the accompanied diameter-versus-refractive-index plot in Fig. 14(c). For cone ISs at 0.6°, the overlapping region between model prediction and experiment for the LP₀₁ mode is shown in green and for 3° it is shown in blue. Also in Fig. 14(c) are black solid and dashed curves that denote the theoretical cutoffs for single (LP₁₁) and multimodal (LP_21,02) performance as a function of IS diameter (d) and refractive index (n_i). Looking at this plot, we see now that the 0.6° (green), 3° (blue), and 8.5° (red) lines share overlapping n_i values and fall within the correct single and multimode space as indicated by the black curves. That is, the model and experiment give the same second moments and yield the same mode profile (LP₀₁ or LP₁₁): LP₀₁ for 0.6° and 3°, and LP₁₁ for 8.5°. Note that 3° straddles the LP₁₁ cutoff, where the mode profile is still dominated by the LP₀₁ owing to its high transmission efficiency and LP₁₁ begins to propagate. The n_i range common to all three retinal eccentricities is ~1.3505 to 1.3510. Note that these values are less than the 1.353 value that was in our original optical model for n_i (Section 2.5). This decrease in n_i is consistent with that associated with dispersion for imaging at a longer wavelength. Taken together, our test of hypothesis two supports the possibility that modes are established at the apical end of the IS (near ELM) and reduced in size by a tapering effect in the ellipsoid region, a reduction on average of about 30% across the three retinal eccentricities considered.

4.4.3 Hypotheses discussion

We tested two hypotheses to address the mismatch between cone IS modes observed in the AO-OCT experiment and those predicted by a conventional step index fiber model using physical parameters reported in the literature (Fig. 7). Modifying the model allowed testing the hypotheses. For the first hypothesis we found that by varying IS diameter and refractive index of the modified step index fiber model, combinations could be found whose predictions agreed with experiment, but failed in the transition region between single and multi-mode performance (Fig. 13(b)-13(c)). To correctly predict modal behavior across the spectrum of mode profiles observed (single mode near the fovea, multimode further away, and a transition region in between), hypothesis two confined the modes in the myoid and minified them (without mode loss) in the ellipsoid. This solution worked, matching prediction to experiment (Fig. 14). Note that other solutions might exist, which we did not test for. In short conventional optical models based on homogeneous step index fibers appear to oversimplify the behavior of light inside the IS. Thus more elaborate models are needed.

Note also that hypothesis two - while supported by our AO-OCT measurements – does not address as to how the light propagates from IS apical to basal ends. Of particular longstanding interest is the mechanism by which the IS couples light into the narrower OS, a critical step for efficient use of the captured light for vision. Efficient coupling requires matching modes of the IS and OS [57] at their interface, e.g., IS/OS. Two general optical mechanisms are possible: focusing and waveguide tapering. Focusing could be realized by the high concentration of mitochondria in the ellipsoid - more than is metabolically needed – that acts as an optical lens that focuses light onto the OS [32]. In this way the mitochondrial lens rescales the mode profile, but without altering mode content.

The second mechanism is waveguide tapering that could be realized by maintaining total internal reflection while narrowing of the IS. Similar to focusing, theory and experiment with tapered optical fibers demonstrate that a taper can rescale the mode profile without altering mode content [57]. However, tapers can also perform notably differently by filtering out modes and/or coupling energy between modes. The former radiates energy from the fiber (loss of energy), while the later maintains energy in the fiber core, but converted to different modes. Conversion in this way is useful for efficient coupling, as for example converting higher-order modes of a multimode optical fiber into the single fundamental mode of an adjoined single mode fiber [58].

It is intriguing that our AO-OCT measurements seem to rule out the possibility that the tapered ellipsoid acts as a mode filter or mode converter. The fact that the IS/OS reflection profile in our AO-OCT images appears multi-mode at retinal eccentricities greater than 4° suggests modes established in the IS are not converted into single modes, at least up to the basal end of the IS where the IS/OS reflection is believed to occur [37]. If mode filtering or conversion was the dominant effect then it would be plausible to expect the IS/OS reflection to be single mode or at least nearly so as this is the only mode we found to be supported by the OS. However this interpretation may oversimplify what optically occurs in the OS, which itself is known to be tapered, not cylindrical, with increasing distance from the foveal center. Because the IS and OS should have identical diameters where they join (e.g., near IS/OS) and the OS is known to have a larger refractive index than IS, any multimode exiting the basal tip of IS should survive entrance into the OS. The fact that our data show only OS single mode behavior (COST reflection) regardless of retinal eccentricity could be evidence – as predicted by Miller and Snyder [50] – that higher-order modes are radiated out from the OS as the modes propagate down OS with only the fundamental mode (LP₀₁) preserved at COST.

If the conic ellipsoid indeed reduces the mode size without altering mode content, it will be difficult to determine if this effect is attributed to ellipsoid tapering or ellipsoid focusing. In favor of the latter, however, cones within 10° of the fovea have an ellipsoid length of just ~21 μm and a tapered region even shorter (a few microns) [32]. Over this length the IS must constrict from 4 to 7 μm to ~2 μm (compare red and blue curves in Fig. 10). The consequence is a steep taper, not a shallow one that has been found necessary to achieve low loss in manufactured fiber tapers [58]. Thus if manufacturing know-how of fiber-based tapers is on par with the biological evolution of cone photoreceptors, ellipsoid tapering is an unlikely option. Obviously more work is needed to test this interpretation.

5. Conclusion

Modal behavior of cone photoreceptors was investigated at near infrared wavelengths using state-of-the-art imaging technology in AO-OCT in conjunction with second moment analysis and theoretical wave propagation in cone models. Hypothesis-driven investigations such as this one not only deepens our understanding of how photoreceptors capture the retinal image, but are also critical for establishing new, more sensitive biomarkers to assess photoreceptor health, in this case based on waveguide modes. Mode biomarkers present an attractive alternative owing to their exquisite sensitivity to the physical parameters of the photoreceptor and temporal stability, at least over the durations we observed here.