Allan W. Snyder

Coupled-Mode Theory for Optical Fibers

A set of coupled-mode equations is derived to describe mode propagation in uniform and slightly nonuniform cylindrical optical-fiber systems. The coupling between fibers of an array made up of n identical fibers each at the vertex of a polygon and one at the center, which is not necessarily the same as its n neighbors, is determined. Examples of this array are two fibers, three fibers in a row, and a hexagonal array with a fiber in the center. Very simple expressions for the coupling coefficients are presented. Mode coupling on a lossy fiber is investigated and a simple expression for the loss of a HE11 mode is given.

Modes of optical waveguides

A simple method is presented for finding the modes on those optical waveguides with a cladding refractive index that differs only slightly from the refractive index of the core. The method applies to waveguides of arbitrary refractive index profile, arbitrary number of propagating modes, and arbitrary cross section. The resulting modal fields and their progagation constants display the polarization properties of the waveguide contained within the ∇ ∊ term of the vector wave equation. Examples include modes on waveguides with circular symmetry and waveguides with two preferred axes of symmetry, e.g., an elliptical core. Only a minute amount of eccentricity is necessary for the well-known LP modes to be stable on an elliptical core, while the circle modes couple power among themselves.

Acuity of compound eyes: Physical limitations and design

SummaryThe two fundamental limitations to resolving power of compound eyes are the wave (diffraction) and particle (photon noise) nature of light. By appreciating their interrelationship we gain insight into the design and limitation of eyes. In particular, we determine the dependence of eye design on the environmental light intensity.1.The limitations to resolving power include: the intensity of light, angular motion, receptor grain, lens-pupil blur, finite diameter of rhabdom, and neural convergence.2.Only those animals that are active in bright sunlight and normally have low angular velocity, profit by having some region of their eyes near the diffraction limit, i.e.DΔφ ≅ 0.58λ, whereD is the facet diameter,Δφ the interommatidial angle and λ the wavelength in vacuum. If these conditions are not fulfilled, it is better to have a largerDΔφ.3.The effect of an animal undergoing angular velocityu is equivalent to a reduction in light intensity by the amount exp−1.78(φtΔφ)2, where φt is the amount the animal turns in one integration time. Taking this into account, we present a possible explanation forMusca havingDΔφ about 4.5 times greater than the diffraction limit.4.Various strategies for dark-adaptation are considered with the conclusion that neural pooling combined with a widening of the acceptance angle is most effective for coping with reduced intensities.

Photoreceptor diameter and spacing for highest resolving power

The maximum center-to-center angular spacing delta phi of photoreceptors tolerated for reconstructing the highest spatial frequency passed by a diffraction-limited pupil of diameter D is delta phi - lambda/D square root 3 when the photoreceptors are packed in a hexagonal array, where lambda is the wavelength in vacuum. This spacing gives the maximum signal-to-photon noise ratio when the inner segments touch. The mean luminance required for an eye to achieve its highest resolving power is independent of eye size, provided the retina is designed to sample the highest spatial frequency passed by the diffraction-limited optics.

Structure and function of the fused rhabdom

SummaryThis paper considers the functional significance of fused rhabdoms. Since all rhabdomeres are joined tightly together, the possibility of optical and electrical coupling between retinula cells is greatly enhanced. We study the extent and consequences of this coupling in order to understand the functional significance of fused rhabdoms. Our methods include both theory and intracellular recordings. The results are as follows: Optical Coupling. Because rhabdomeres of different spectral types are fused into a common light guide, the absorption properties of each influence the manner in which light is transmitted along the composite rhabdom structure.1.Each rhabdomere acts as if it were an absorption filter in front of all others, i.e. rhabdomeres function as lateral absorption filters (Fig. 4).2.As a consequence of this filtering, the shape of the spectral sensitivity curve for each retinula cell is approximately independent of the amount of light it absorbs, i.e. independent of the rhabdomeres length and concentration of photopigment (Fig. 7). This is in direct contrast to the retinula cells of fly that have spectral sensitivity curves which become progressively flatter as more light is absorbed (Snyder and Pask, 1973). In other words, the flattening of curves by self absorption is prevented by optical coupling.3.Thus, one functional advantage of the fused rhabdom (due to optical coupling) is that each retinula cell can have a high absolute sensitivity while preserving its spectral identity (narrow spectral sensitivity curves). (Compare Fig. 5 to Fig. 6.) Thus the same receptors can operate in a high sensitivity and in a colour vision system (cf. vertebrate rods and cones).Since all spectral cell types are together in one rhabdom, the animal can have hue discrimination in a small field of view (fine grain colour vision). Thus an individual ommatidium has the potential for providing excellent spectral discrimination.4.If two cells have photopigments with absorption maxima close together, the maxima of their spectral sensitivity curves are moved further apart (Fig. 8).5.In the absence of electrical coupling polarization sensitivity (PS) can depend dramatically on wavelength. The spectral composition of the rhabdom, in addition to the direction of the microvilli, profoundly influences the polarization sensitivity vs. wavelength PS (λ) curves of individual retinula cells. This is shown theoretically for the worker bee rhabdom (Fig. 10) where (a) there is a pronounced difference in PS (λ) between cells with orthogonal microvilli and (b) green retinula cells show a large PS in the green while the UV cells show a much smaller PS in the UV (Fig. 13).

Spatial Information Capacity of Compound Eyes

SummaryThe capacity of the compound eye to perceive its spatial environment is quantified by determining the number of different pictures that can be reconstructed by its array of retinula cells. We can then decide on the best compromise between an animals capacity for fine detail and contrast sensitivity. The theory accounts for imperfect optics, photon noise, and angular motion limitations to acuity.1.There is an optimum parameterp = D Δ φ, whereD is the facet diameter andΔ φ is the interommatidial angle, for each mean luminance, angular velocity and mean object contrast. We find that this value ofp is approximately that found by Snyder (1977) for threshold resolution of a sinusoidal grating at the ommatidial sampling frequency.2.A diffraction limited eye (D Δ φ ≅λ/√¯3) is the optimum design only for those animals that are active in the brightest sunlight, and have a region of their eye that normally experiences low angular velocity, otherwise it is better to have a largerD Δ φ. λ is the wavelength of light in vacuum.3.The design of the flyMusca is consistent with that of an animal with high angular velocity.

Self-induced optical fibers: spatial solitary waves

To be self-guided, a beam must exactly equal the mode of the linear-optical fiber that it induces. From this elementary consistency condition we can borrow solutions and their associated physics directly from the familiar literature of linear-optical waveguides. By considering a nonlinear medium characterized by ideal saturation, we present what is to our knowledge the first exact analytical solution of a two-dimensional self-guided beam. This beam is the familiar fundamental mode of a step-profile fiber. The stability of the beam is also determined.

Polarization sensitivity of individual retinula cells

SummaryThis paper elucidates the influence of the structure of a rhabdom on the polarization sensitivity of its retinula cells. The terminology polarization sensitivity (PS) and dichroic sensitivity (Δ) needs clarification. Δ expresses the directional property of the local microvillar medium and is independent of the gross morphology of the rhabdom. The PS of a retinula cell is that found by single cell electrophysiology and depends strongly on the gross morphology of the rhabdom. Both Δ and PS are ratios of the effects of theE vector of linear polarized light parallel to, to that perpendicular to the microvilli. From the theoretical analysis and its correlation with experiments the following is concluded.

Information capacity of eyes

The capacity of an eye to perceive the visual environment is quantified by determining the number of different pictures that can be reconstructed by its array of photoreceptors. There is an optimum density of photoreceptors for each mean luminance and contrast. This is determined by the wave and particle nature of light (diffraction and photon noise). Anatomical and psychophysical data are consistent with the hypothesis that the human retina maximizes the reconstruction of different pictures over the range in luminance required for day and night vision.

Power transfer between optical fibers

Previous papers have treated power transfer between HE11, TE01, and TM01 modes propagating on identical cylindrical fibers. Here we extend the theory to include power transfer between modes of any order propagating on uniform circular fibers of different radii and dielectric constant. A simple analytical expression for the coupling coefficient is derived. The error in using the decoupled two-mode form of the coupled-mode equations is determined. Examples are given to illustrate the extension of the two-fiber results to arrays of fibers with different properties. All results are presented in a dimensionless form applicable to circularly cylindrical fibers of arbitrary physical parameters.