Absorption-assisted mode transformation in butterfly compound eyes

Abstract

The ommatidium of the butterfly's afocal apposition eye exhibits angular performance that can only be achieved by transforming the diffraction pattern of its corneal lens into the fundamental mode of its rhabdom waveguide. A graded index model of the ommatidium has been proposed and verified but the efforts to extract the transformation's underlying physics from it have been hindered by its extreme complexity. Here we numerically investigate the ommatidium model and reveal that the current model, involving only the graded index distribution, is insufficient for the transformation. We further find that adding spatially varying absorption to the existing model dramatically improves its transformation performance, producing near-perfect mode matching with overlap integral exceeding 0.96. Such a combined action of spatially varying index and absorption for microscale mode transformation is new to researchers in optics and biology and will benefit both disciplines.

Introduction

Insects' compound eyes achieve their wide field-of-view using hemispherically arrayed optics¹. In the apposition type, one corneal lens (CL) and one rhabdom (Rh) form a lens-waveguide pair which, along with other parts between the two, is collectively called an ommatidium (Fig. 1a). For optical isolation between ommatidia, the level of light coupled into Rh depends critically on the incidence angle θ_i. The angular response typically peaks at normal incidence (θ_i = 0) and tapers off as θ_i increases. The acceptance angle θ_A marks the full-width half-maximum point of the response curve^1,2,3. In the superposition type, in contrast, each CL collects rays over a wide angular range. Then the crystalline cone (CC) collimates them for redistribution over multiple photoreceptors (Fig. 1b)^1,4.

Figure 1

Features of afocal apposition eye and its model.

(a,b) Schematic diagrams of (a) apposition and (b) superposition compound eyes. The former consists of CL-Rh pairs. Each pair maximally accepts only the normally impinging ray with the level of acceptance tapering off as the incidence angle increases. The full-width half-maximum point of the curve marks the acceptance angle θ_A. In the latter, in contrast, Ph collects rays from multiple CLs. Light collimation by CC is essential for propagation across CZ. (c) The ommatidium of the afocal apposition eye. CL focuses light inside CC, not at the Rh entrance as in (a). Similar to the superposition eye, its CS collimates the defocusing rays. (CL: corneal lens, CC: crystalline cone, CS: cone stalk, Rh: rhabdom, Ph: photoreceptors, CZ: clear zone).

Nilsson et al. found a third type in butterflies⁵. Structurally, it is an apposition eye comprising a large number of ommatidia. But it also resembles the superposition eye in collimating its output using the cone stalk (CS) as the collimation lens (Fig. 1c). These mixed characteristics positioned this third type, called afocal apposition eye for its perfect collimation, as the link between the apposition and superposition eyes^5,6.

At question was how an apposition eye came to employ collimation which does not give clear advantages while requiring very complex CS architectures. For instance, with its unique index distribution, Heteronympha merope's CS produces 0.2 mega-diopters of magnification from just ~10 μm in length, making itself “the most powerful known to man.⁵”

Nilsson et al. provided a clue by revealing another role of CS⁷. As mentioned above, ommatidial isolation requires reductions in θ_A. Optically, θ_A minimizes when the electric field pattern hitting the Rh waveguide entrance exactly matches that of the waveguide's fundamental mode (LP₀₁). However, CL generates Airy-like patterns that are different from LP₀₁. In typical apposition eyes, the mismatch broadens θ_A by ~15% from its theoretical minimum^6,7. In contrast, the afocal apposition eye's θ_A turns out to match the theoretical minimum, indicating that CS transforms the Airy-like pattern into LP₀₁. Collimation is just a part of the mode transformation process.

The search for the transformation mechanism has produced a graded index model of the ommatidium⁸ and multiple hypotheses. One interpreted CS as a waveguide mode coupler⁷. Since Airy pattern closely resembles the sum of the two lowest axis-symmetric modes, LP₀₁ and LP₀₂, CL practically excites both modes into CS. Totally transferring LP₀₂'s mode power towards LP₀₁ will realize the LP₀₁ transformation. Another suggested that CS functions as a mode mixer which synthesizes LP₀₁ from the local modes of the tapered, index-graded CS⁸. So far, theoretical efforts to verify the hypotheses have been thwarted by the model's extreme complexity.

Here we analyze the LP₀₁ transformation mechanism numerically. We find that the existing graded index model is insufficient for the task. Complementing it with spatially varying absorption, however, has directly led to successful LP₀₁ transformations. Such a combined action of non-uniformly distributed index and absorption gives the afocal ommatidium a new interpretation as an absorption-assisted mode shaper, rather than a coupler or mixer. The new findings will enrich the knowledgebase of insect optics¹ and bio-inspired optics^9,10 alike.

Results

Testing mode coupling/mixing hypotheses

With the numerical simulation code and ommatidium model set up as described in Methods, we first simulated the propagation of a plane wave through the afocal ommatidium to test the existing hypotheses. Figures 2a and b show that the simulation correctly reproduced the experimentally observed mid-CC focal spot formation (z ~ 58 μm). Despite the non-uniform index distribution in CC, the focal field pattern closely matched Airy pattern, both in amplitude and phase (Fig. 2c). Existing studies hypothesized that this Airy-like pattern would evolve into LP₀₁ through mode coupling or mixing as it approaches the Rh entrance at z = 68 μm^7,8.

Figure 2

Testing mode coupling/mixing hypotheses.

(a) and (b) show the intensity distribution and the change in its center axis intensity resulting from the simulated propagation of a plane wave through the ommatidium model in Methods. Both confirms the mid-CC focusing by CL near z = 58 μm. The inset in (b) shows the change in the overlap integral η between z = 55 and 68 μm. (c) The amplitude and phase profiles at the focal spot (z = 58 μm). They agree well with those of Airy pattern except for the peripheral, low-intensity part (ρ > 5 μm). (d) The amplitude and phase profiles obtained at the Rh entrance (z = 68 μm), however, exhibit significant mismatches from the Rh waveguide's LP₀₁ mode, indicating that the hypothesized mode coupling or mixing mechanism is insufficient for transforming an Airy-like pattern into LP₀₁.

However, the amplitude and phase profiles obtained at that point, shown in Fig. 2d, exhibited high-level mismatches from those of LP₀₁. The side-lobes of the Airy-like pattern persisted throughout CS, leaving severe wiggles to the final amplitude profile. The step-like phase distribution in Fig. 2c has been quasi-linearized as shown in Fig. 2d but it is still insufficient to match the flat wavefront of LP₀₁. Only the width of the field pattern matched that of LP₀₁ reasonably at the Rh entrance.

To quantify the level of mode matching, we introduced three metrics in Methods: the shape error δ_s, phase error δ_p and the overlap integral η which considers both shape and phase. The inset in Fig. 2b shows the computed η as a function of z. Even within CS, η increases very slowly, only to reach 0.77 at its end. The values of δ_s and δ_p are also large, reaching 0.0603 and 0.6068 at the Rh entrance, respectively. Simulations performed at different wavelengths with slightly varied structural parameters, including the CP thickness, CC length and CS index profile, all produced similar mismatch. It indicates that mechanisms relying solely on the graded index distribution of CS, such as mode coupling or mixing, are insufficient to transform the Airy-like pattern into LP₀₁.

Impact of distributed apodization

The intensity distribution in Fig. 2a shows that the wiggles in the final amplitude profile evolved mainly from the outermost fringe of the CL-focused light. So we attempted to suppress the wiggles by blocking the fringe with an annular absorptive region. It is analogous to the apodization process in Fourier optics which removes the peripheral, high spatial frequency portion of a diffraction pattern¹¹. Removal of the wiggles alone, however, does not guarantee LP₀₁ transformation. It hinges on whether the subsequent geometry and index distribution, already given by Nilsson et al.^5,6 and Pask et al.⁸, can correctly modify the amplitude and phase of the apodized wave.

We explored its feasibility with iterative simulations. We assumed that the absorption is mild yet broadly distributed and implemented it by making the portion of CC with ρ ≥ a and z ≥ z_o weakly absorbing in our numerical model, as shown in Fig. 3a. We empirically set the electric field attenuation at 1% per 50 nm or 0.873 dB·μm⁻¹. Given the existence of light-absorbing pigments around the ommatidium, this type of distributed absorption in CC seems plausible^12,13. Figure 3b shows the intensity distribution from an exemplary simulation. The absorptive region apparently blocked most of the outermost fringe.

Figure 3

Absorptive apodization and its impact.

(a) The setting of the hypothesized absorptive apodization. Low-level absorption (0.873 dB μm⁻¹) was implemented in the region with z > z_o and ρ > a. (b) Intensity distribution obtained from an apodized propagation simulation with (z_o, a) = (36, 2.3) μm. Blocking of the outermost fringe is evident. (c) The amplitude and phase profiles obtained at z = 67.4 μm at which the phase error δ_p becomes minimized. The dotted red curves represent LP₀₁ which also appeared in Fig. 2 d. The bulging in the amplitude profile for ρ ~ 2 μm persisted for other values of (z_o, a), necessitating additional mode shaping.

The apodization impacted LP₀₁ transformation immediately. Figure 3c shows the amplitude and phase profiles obtained at z = 67.4 μm, right in front of the Rh entrance, with (z_o, a) = (33, 2.4) μm. Comparison of the amplitude profile with that of LP₀₁ reveals that the apodization was very effective not only in removing the wiggles but also in reshaping the overall mode profile into that of the single-peaked LP₀₁. The computed shape error δ_s was 0.0216, a factor of 3 reduction from that of the unapodized one.

It is important to point out that the distributed absorption greatly improved the phase profile matching as well, even though it cannot directly affect phase. At the same z point (67.4 μm), the phase error δ_p was 0.0178, a factor of 34 improvement from the unapodized one. The inset of Fig. 3c visualizes the flatness of the wavefront across the radial extent of CS (ρ ≤ 1.3 μm). We attribute this improvement again to the apodization which suppressed the peripheral portion with its phase quadratically retarded by CL.

The results of iterative simulations shown in Fig. 4 further reveal that such large δ_p improvements near the Rh entrance is possible only within a narrow window with 33 < z_o < 36 μm and 2.1 < a < 2.5 μm, reaffirming the need for a precise and complete elimination of the outermost fringe. A simultaneous reduction of δ_s within the window, however, was hindered by the persistent bulging near ρ = 2 μm (Fig. 3c). Since the level of shape matching within the radial extent of CS remains excellent, the bulging must be due to the absence of absorption outside the terminal portion of CS, which is evident in Fig. 3a.

Figure 4

Parametric search for simultaneous minimization of δ_s and δ_p.

(a) Solid curves: z-positions at which the phase and shape errors minimize for the given (z_o, a). Lack of their intersection indicates that the two do not minimize simultaneously. Dashed curves: z_min,error values obtained by limiting the range for the shape error evaluation within ρ ≤ 1.3 μm, which enabled simultaneous minimization of δ_s and δ_p. (b) δ_p,min values as a function of (z_o, a). In general, δ_p grows rapidly as z_o moves away from CP. (c) Solid curves: δ_s,min values obtained when z_o = 33 ~ 36 μm, i.e., within 3 μm from the ending point of CP. Overall, δ_s values grow monotonically with a. Dashed curves: δ_s assessed only within ρ ≤ 1.3 μm. Now they exhibit definite dips, suggesting that the shape error accumulates mostly in the region with ρ > 1.3 μm, i.e., the area not covered by the distributed absorption shown in Fig. 3a.

Refinement by conformally extended absorption

So we extended the absorption region as shown in Fig. 5a. Starting from z = z_t, the exterior of CS was made absorptive all the way to the original apodization region. A thin buffer layer with thickness t was installed for fine control. For simplicity, we set the electric field attenuation within this conformally extended absorption region identical to that of the apodization region, i.e., 0.873 dB·μm⁻¹. Then we performed iterative simulations involving all four parameters: z_o (33 ~ 36 μm), a (2.1 ~ 2.5 μm), z_t (65 ~ 67 μm) and t (0 ~ 0.5 μm) and again observed additional improvements in LP₀₁ transformation. We present the results obtained with (z_o, a) = (36, 2.4) μm as examples.

Figure 5

Impact of conformally extended absorption.

(a) The setting of the conformally extended absorption which starts from z = z_t with a t μm-thick buffer. (b,c) The amplitude and phase profiles obtained at the Rh entrance when t = 0.35 μm. LP₀₁ profiles are also superimposed for comparison. (b) For z_t = 65 μm, i.e., 3 μm-long conformal absorption, the output becomes indistinguishable from LP₀₁. The inset shows how the increase in the conformal absorption reduces the mismatch in the output amplitude profile, especially near ρ = 2 μm. (c) The conformal absorption also improves the wavefront flatness within the radial extent of CS. (d) The minimum phase and shape errors for various (z_t, t) combinations. Note that δ_p and δ_s minimize simultaneously near (z_t, t) = (65 ~ 66, 0.2 ~ 0.35) μm. (e) The change in the mode power along the propagation through CC and CS. CP/CC: The interface between CP and CC.

Figure 5b shows the amplitude profile obtained at z = 67.9 μm with the value of t fixed at 0.35 μm. When z_t = 65 μm, the bulging near ρ = 2 μm disappeared completely, enabling the amplitude profile to overlap that of LP₀₁ almost perfectly. The corresponding δ_s = 0.0049, a factor of 5.3 reduction from the Fig. 3c result. The absolute difference between the amplitude profiles of the simulation results and LP₀₁, plotted in the inset, shows that the match improves increasingly as the conformally extended absorption region becomes longer. Figure 5c shows the phase profile at the same z-point as a function of z_t. Just as the apodization did, this conformally extended absorption further reduced the phase error values by flattening the wavefront. When z_t = 65 μm, it became near-flat within the radial extent of CS. The simultaneous minimization of δ_p and δ_s enabled the overlap integral η to reach 0.9646 at the Rh entrance.

The minimum phase and shape error values (δ_p,min and δ_s,min) plotted in Fig. 5d show that simultaneous δ_p - δ_s minimization is limited to a narrow window of (z_t, t) combinations. The values of the optimal (z_t, t) combinations were also different for different (z_o, a) combinations. So, we extracted from our iterative simulation data the combinations of (z_o, a, z_t, t) which minimize both δ_p and δ_s at z > 67.5 μm, i.e., within 500 nm of the Rh entrance, while maintaining δ_s < 0.01. Table 1 lists them along with their performance metrics. It shows that the optimal values for z_o fall within a 3 μm range from the CC starting point, with the majority at z_o = 35 μm. The optimal values for the apodization pupil radius a are unanimously at or near 2.5 μm, which is approximately 42% of the CC radius at its starting point. The optimal z_t and t values exhibit only weak variations within 1 ~ 3 μm and 0.2 ~ 0.4 μm range, respectively. As shown in Fig. S1 in Supplementary Information, these configurations are effective over a wide spectral range spanning 400 to 600 nm.

Table 1 Configurations for LP₀₁ transformation and results. (z_o, a, z_t, t) combinations producing simultaneous δ_p - δ_s minimization at z > 67.5 μm and δ_s < 0.01. The corresponding δ_s and δ_p values can be found in the second column. The ratio between the final and initial mode power and the overlap integral η are given in the third column

Since the proposed LP₀₁ transformation relies on absorption, it inevitably reduces the mode power. For optimal configurations in Table 1, they can be 73 ~ 77% of the initial value. As shown in Fig. 5e, the initial apodization claims ~5 dB and the extended absorption another ~1 dB. The overall loss of 6 dB may seem high in power budget point of view. Removing the collimating CS and the absorptive mode transformation altogether, however, results in a situation in which the exact Airy pattern tries to excite LP₀₁. The corresponding coupling efficiency is¹⁴:

where x = 7.34·(r_o/r_A)², r_A is the Airy spot radius defined as 0.61·f_CL·λ_o/(n_CL·a_CL) and r_o is the modal field spot size for Gaussian-approximated LP₀₁ which is ~1.7 μm for the present Rh. The resulting η_couple is ~6.7 dB. Table 1 shows that the present model incurs slightly lower loss while adding the advantage of LP₀₁ transformation and, hence, θ_A minimization. This justifies the use of absorption for mode transformation. The last column of Table 1 also indicates that initiating the apodization early, i.e., near the CP/CC interface, can increase η slightly but also incurs significant penalty to the mode power.

Discussion

In conclusion, we utilized numerical simulations to test the existing hypotheses on how the afocal ommatidium transforms the CL-generated Airy-like pattern into LP₀₁ and push its acceptance angle towards the theoretical minimum. Our initial results showed that the hypothesized mechanisms relying solely on graded index distribution, such as mode coupling or mixing, cannot accomplish the LP₀₁ transformation. We then found that complementing the existing graded index model with spatially varying absorption can dramatically increase the level of mode matching. Interestingly, the added absorption improved not only the shape matching, which is under direct influence of loss, but also the wavefront flatness. This wavefront-flattening operation seems to develop into the collimation step of the superposition eye. Such a combined action of distributed index and absorption for microscale mode transformation is novel with no previous report. Iteratively, we identified a window in parameter space capable of realizing LP₀₁ transformation with overlap integral exceeding 96%. The 6 dB power penalty arising from the use of absorption can easily be justifiable given that the purely refractive model also imposes a comparable level of loss due to the Airy-to-LP₀₁ mode mismatch even though it cannot bring in the added advantage of θ_A minimization predicted for the absorption-assisted model.

It needs to be emphasized that the LP₀₁ transformation in this work was achieved with minimal modifications made to the already verified model of the afocal ommatidium. In biological ommatidia, those modifications, i.e., distributed absorption within and around CC, can be realized with light absorbing pigments infiltrating and surrounding the ommatidia, respectively. In soft biological structures, achieving and maintaining the μm-scale dimensional precision for optimal LP₀₁ transformation seems challenging. Given the existence of pigments capable of migrating around the ommatidium for dark/light adaptations, we can envision a possible adaptive control scheme which adjusts the pigment distribution to maximize the mode matching.

These findings will interest both optical engineers and biologists. The former may get excited at the prospect of realizing highly miniaturized mode converters that are coupled with microscale focusing lenses and/or formed in-fiber. The recent trend in optical communication to exploit various modes supported by multiple mode fibers²¹ indeed necessitates new paradigms in optical mode conversion^22,23 and the new absorption-assisted scheme will certainly contribute to the efforts. The possible involvement of absorption in the operation of insects' ommatidia would motivate the latter to examine additional roles of absorptive pigments. Numerically reproducing the angular response of the afocal apposition ommatidium using a fully 3-dimensional model will give the present work an ultimate validation. Here we emphasize that the optimal configurations found in this work are by no means ultimate or exhaustive. They can be further refined once the currently unknown refractive index and loss distributions around CC become available and the possibility of finding purely refractive models capable of performing the required transformation, as suggested in Ref. 23, should not be ruled out either.

Methods

Numerical simulation of wave propagation

To efficiently simulate the optical wave propagation in the ommatidium, we utilized the finite-difference beam propagation method (FD-BPM)¹⁵. Specifically, we chose a version of FD-BPM customized for scalar wave propagation in an axially symmetric cylindrical space^{16,17,18,19,20} and implemented it with Matlab. Azimuthally changing modes, such as LP₁₁ or LP₂₁, were inherently excluded. More universal methods exist for vectorial, azimuthally varying wave propagation but their computational burden was prohibitive for iteratively simulating a structure measuring over 15 μm in radius and 70 μm in length. Throughout our work, the radius of the simulation space was 20 μm. We used the transparent boundary condition to suppress spurious reflections from the edges of computation domain. The spatial resolution was fixed at 50 nm in both radial and axial directions. A full propagation typically took 15 ~ 30 seconds on computers with a xeon processor and 4 GB RAM. We validated the FD-BPM code by simulating the formation of Airy pattern by a spherical lens (Supplementary Figure S2). The wavelength λ_o was varied between 400 and 600 nm.

Heteronympha merope ommatidium model

Most structural and optical parameters in the ommatidium model were from Nilsson et al.^5,6 and Hateran et al.⁷: CL radius: a_CL = 12.25 μm, CL thickness: 23 μm, CL radius of curvature: 20 μm, n_CL = 1.52. f_CL ~ 54 μm. CP was modeled as a 10 μm-thick plate with n_CP = 1.34. The Rh parameters are also close to those of Nilsson et al.^5,6: Rh diameter = 2.07 ~ 3 μm, n_{Rh_amb} = 1.35 and n_Rh = 1.36 ~ 1.38. For the graded index profile and dimensions of CC and CS, we adopted the model proposed by Pask et al.⁸: n_CC² = n_o²·[1 + 2·Δ·(θ₁² − θ²)/R²] where n_o = 1.4, Δ = 131.5, θ₁ = 0.1 rad. R and θ were computed at each position within the CC. With the length of CC set to 35 μm⁸, its radius decreases linearly from 6.1 to 1.3 μm as it approaches Rh. Pask et al. verified this model⁸ by reproducing the experimental optical magnification data of Nilsson et al.⁵Supplementary Figure S3 shows the numerical model schematically.

Quantification of mode matching

To quantify how closely a simulated electric field distribution matches that of LP₀₁, we utilized three metrics. The shape error δ_s is defined as:

where E_sim and E_ref represent the electric field distributions obtained from the simulation and the exact LP₀₁ mode solution, respectively. Since our FD-BPM presumes total axis-symmetry, the cross-sectional integration was actually performed only along the radial direction. Two different integration ranges were used. For overall shape error, the intensity distributions within ρ = 0 ~ 7 μm range was compared. When comparing only the center portion of the intensity distribution, the range was reduced to ρ = 0 ~ 1.3 μm. Since LP₀₁ exhibit a flat phase distribution across its mode profile, we assessed the level of match in the phase with the phase error δ_p which is the standard deviation of the simulated phase response. Because CS manipulates only the center portion of the wave, δ_p was evaluated within the ρ = 0 ~ 1.3 μm range. To consider the impacts of both the shape and phase, we utilized the overlap integral η defined as

Since η was used primarily for very similar field profiles, the integration range was fixed to ρ = 0 ~ 7 μm.