The backscattering properties of small layered plates are studied for various size parameter values with respect to the plate thickness, plate aspect ratio, number of layers, incident direction, and polarization states of the incident light. The results are compared with the analytical results for semi-infinite plates. The phase functions and the corresponding backscattering efficiencies of the small plates are computed with the Discrete Dipole Approximation method. The angular width of the reflection peak is found to depend on both the size parameter and the aspect ratio. The criteria for using the reflectance of semi-infinite plates to approximate the backscattering efficiencies of finite size plates are quantified with respect to the number of layers, incident angle, and polarization state.
© 2012 OSA
Multiple-layered plates with finite sizes are abundant in animal skin and play an important role in producing color and polarized reflection . A type of small layered structure, called an iridosome, in the skin of cephalopods has been found to play a key role in both camouflage and communication . Meanwhile, the analytical reflectance results of semi-infinite multiple-layered plates provide an elegant, accurate, and efficient way to study reflection properties [3–6]. This semi-infinite model has been widely used to approximate the reflectance of finite multiple-layered plates [1, 7–9], but neither the backscattering properties of these plates nor the criteria for the validity of the semi-infinite approximation have been studied. We will numerically study the backscattering properties of small layered plates, based on the structure of a single iridosome, and compare the results with their semi-infinite plate counterparts. From these comparisons, we will provide the criteria for using the semi-infinite approximations.
Analysis based on semi-infinite plates can offer physical insights into the understanding of finite plates. This analysis approximately predicts the angle of specular reflection and the size parameter that leads to maximum constructive and destructive interference, and establishes the Brewster angle (i.e., the incident angle at which the reflected light is completely polarized) for a finite size structure. However, iridosomes of different species are substantially different in size [10–16], including both the thickness and the aspect ratio. Therefore, we need to evaluate the influence of the size parameter, aspect ratio, and other parameters, such as number of layers and polarization state, on the backscattering properties of a single iridosome.
To accurately account for all the scattering effects beyond the semi-infinite approximation, we use the Discrete Dipole Approximation (DDA) method [17, 18] in this study. The results can be generalized to include other systems with similar structures, such as hexagonal ice crystal plates . Furthermore, the edge effect  by the small scatterers could be one reason the reflection properties of small plates deviate from those of semi-infinite plates.
2. Numerical model & theoretical background
An iridosome is modeled as a cylinder with a layered inner structure. Two examples are shown in Fig. 1 for both single-layer and 5-layer plates. Plate diameter, thickness, and the total cylinder height are respectively denoted by, , and . The direction of the incident light is at an angle β relative to the symmetry axis of the iridosome and is shown in Fig. 1(a).
The iridosome plate material is found to be a kind of protein, called reflectin, and its refractive index is measured to bewith a negligible dispersion in the visual spectrum [21, 22]. The material of the ambient tissue and that between plates are mainly cytoplasm, whose refractive index is close to that of water (). Thus, the relative refractive index of the plate () is 1.2 referenced to the cytoplasm. In our model, we choose the same relative refractive index for iridosomes. To further simplify the modeled structure, we consider ideal multiple-layered plates in which the optical length of the plates and the spacing between plates are the same, and therefore the spacing thickness is .
The size parameter is defined as , where is the wavelength in the medium. The plate thickness d of iridosomes can range from 50nm to 200nm [10–16], and, in the visual spectrum, the incident wavelength in air is from 0.4 to 0.75 μm. Therefore, the size parameter of the plate thickness in the medium is , and can vary approximately from 0.5 to 5. This size parameter range is used in our simulations in steps of 0.1.
The aspect ratio is defined as for one layer of the plate. For example, both single-layer and 5-layer plates in Figs. 1(a) and (b) have the same aspect ratio of 10. A range of aspect ratios from 1 to 20 is chosen for our study. Rather than discussing the aspect ratio variations among certain cephalopod species, we focus on the aspect ratio’s influence on the optical properties, from which a particular type of iridisome can be evaluated later on.
The phase function describes the angular distribution of scattered light for an unpolarized light source, where is the solid angle for the scattering direction. is the azimuthal angle, and is the scattering angle between the scattering direction and the incident direction. As shown in Fig. 1(a), the upper solid angle hemisphere is defined as the backscattering region. Sine the scattered light has an angular distribution described by , and as we will discuss later, even for iridosomes with large aspect ratios, there is still a finite angular width associated with the reflection peak in the phase function. In order to compare the reflectance of irradiance for the semi-infinite plates, we will integrate the phase function over the whole backscattering region . Therefore, we can define the backscattering efficiency for unpolarized incident light asFig. 2 for both single-layer and 5-layer plates. Qb indicates the irradiance reflectance for a finite iridosome. When the aspect ratio approaches infinity, Qb becomes equal to the reflectance of the semi-infinite plate. Therefore, we use the backscattering efficiency as an indicator to show the reflection deviation of the small plates from that of the semi-infinite plates.
Besides representing the reflectance of a single iridosome, Qb can also be used to calculate the diffuse irradiance reflectance for a plane parallel system with a collection of iridosomes. We assume that is the number density of the iridosomes and is the thickness of the system. For an optically thin system, the multiple scattering between iridosomes is ignored and only the scattering by each single iridosome dominates. Considering an area of A and thickness Z in this system, the total number of iridosomes would be . The total reflectance of the whole system would be the ratio of the total backscattering cross section and the total area A. Therefore, we have, where is the backscattering cross section of a single iridosome. Iridosomes can have arbitrary orientations and Cb needs to be averaged according to the orientation distributions of these iridosomes. For an optically thick system with random oriented iridosomes, can still be estimated using the randomly averaged Cb. For such a system, there is significant multiple scattering which tends to drive the reflected radiance toward an isotropic distribution, and we can use a one-dimensional model to calculate the irradiance reflectance as proposed by Kattawar et al. , where the diffuse reflectance can be obtained as . Therefore, the backscattering efficiency of one scatterer together with its geometric cross section can be related to the diffuse reflection of a bulk system. The diffuse reflectance is important in determining the optical appearance of the system, such as the color and brightness of the cephalopod skin. However, if the angular distribution of the diffuse reflection is considered, the radiative transfer equation must be solved using the full phase function .
To calculate the reflectance of a single-layer semi-infinite plate (R1) (Fig. 3 ), Airy’s formula is used for different size parameters, polarization states, and both normal and oblique incidences :Eqs. (3), (4), and are the Fresnel reflectance formulas for the perpendicular and parallel polarized components relative to the principal plane defined by the direction of the incident and reflected light, where is the incident angle and is the refracted angle. The reflectance is a periodic function of the size parameter . Our analytical results for an arbitrary number of layers are derived by using the transfer matrix method [4, 6], and the resulting reflectance remains a periodic function of the size parameter.
Additionally, the maximum constructive and destructive interferences determine the maximum and minimum reflection of a semi-infinite plate. As shown in Fig. 3, the difference in the optical lengths of the two reflection paths is , where is the refracted angle. The phase factor α in Eq. (2) can be related to the path difference by under normal incidence. Due to the phase shift at the interface, the condition for maximum constructive interference is , where is an integer. Thus, to have maximum constructive interference under normal incidence, we have and , which corresponds to and 3.9; to have maximum destructive interference, we have , which corresponds to . However, for a finite size system, the reflection maxima and minima deviate from the prediction of these size parameters, and to compare the difference we indicate , 2.6, and 3.9 in the reflection spectrum graphs.
3. Numerical results and discussion
We used the ADDA code developed by Yurkin and Hoekstra  for the DDA simulation to calculate the scattering properties of the iridosome. In this simulation, 10 dipoles are used along the thickness of every single plate, and the required relative residual norm is specified as 10−5 for the involved quasi minimal residual solver. An angular resolution of 0.25°2° for the zenith and azimuthal angle is used to calculate the backscattering efficiency. The phase functions are calculated with various size parameters, aspect ratios, incident directions, and polarization states. We discuss the angular distribution of the reflection peak in the phase functions and compare the backscattering efficiencies of the iridosome with the reflectance of the semi-infinite plates.
For a size parameter x = 1.3, the phase functions are shown in Fig. 4 with different aspect ratios under normal incidence for a single-layer plate. For convenience in comparing backscattering properties, each phase function is normalized by as in Eq. (1). Because the light is at normal incidence on the plate, the phase function is azimuthally symmetric. For example, as shown in Fig. 4, the phase function is symmetric around = 180°, which corresponds to the direct backscattering direction. The full width at half maximum (FWHM) of the reflection peaks is denoted by and provides a way to quantify the angular spreading of the reflection. When the aspect ratio a increases, the reflection peak around = 180° becomes narrower and decreases. The FWHM of the reflection peak is shown by the horizontal red solid bar in Fig. 4. For the aspect ratio from 4 to 40, is found to decrease from 73.5° to 7.5° with an angular resolution of 0.5°. For an aspect ratio of 1 or 2, no angle corresponds to the phase function with a value half of that at 180°, and, therefore, no value for exists; the phase function has less variations compared with the one with large aspect ratio particles.
For size parameters in the range from 0.5 to 5, the values of are summarized in Fig. 5 with the same 0.5° angular resolution for the single-layer plate as just discussed. decreases as the aspect ratio increases throughout all the considered size parameters. Note that is sensitive to the size parameter around x = 2.6, which corresponds to maximum destructive interference for the semi-infinite plate. A jump of the value can be observed for a = 4 around x = 2.6. We further studied with a finer size parameter resolution, and observed that a continuous peak existed for the value of around x = 2.6. For aspect ratios a = 1 and 2 and size parameters approximately less than 2.6, no value exists for . For size parameters larger than 2.6 and a = 1, can be found between 44° to 96°, and for a = 2, decreases to between 36° and 46°. Furthermore, when a = 4, is between 130° and 18.5° for size parameters from 0.5 to 5. When a = 40, further decreases to between 18.5° and 2° for the same range of size parameters. For the plates with a thickness of 5 and a = 40, the reflection peak is at least 2° in width. The spreading becomes less prominent when the aspect ratio is large. However, for the semi-infinite plates, the reflection is confined to a single direction determined by Snell’s law. The angular broadening of the reflection can be explained by the ray-spreading effect . When the angular distribution is considered either for a phase function or the diffuse reflection of a random system, the influence of the spreading of the reflection peak needs to be evaluated.
The backscattering efficiency Qb (Eq. (1)) is sensitive to the aspect ratio and the number of layers of an iridosome. As the aspect ratio increases, Qb will undergo a metamorphosis from a small particle to a semi-infinite layered plate. In Fig. 6 , Qb is shown for various numbers of layers and aspect ratios at x = 1.3. When the radius increases, Qb approaches the reflection of the semi-infinite plates (red asterisks in the graph) for each number of layers. However, the dependency of Qb on the aspect ratio is not monotonic. Qb first increases for the aspect ratio from a = 1 to 6, then decreases from a = 6 to 20. This can be understood by looking at a particular size parameter, x = 1.3, in the spectrum of Qb shown in Fig. 7(b) ; with the decreasing of the aspect ratio, the size parameter for the maximum Qb will shift toward larger size parameters. When a = 1 or 2 and the number of layers is larger than 1, Qb is much smaller than the reflection of semi-infinite plates. When a = 4, Qb will decrease as the number of layers increase when the number of layers is larger than 7. In general for a>4, as shown in Fig. 6, adding more layers of plates into the scattering will increase the backscattering efficiencies as predicted by the semi-infinite plates. The size effect plays a significant role for small scatterers in determining the backscattering efficiency. Meanwhile, Qb can be larger than 1 due to the interference effects of the scattered waves similar to the extinction efficiency .
For normal incidence, the backscattering efficiency Qb versus the size parameter for a single-layer and a 5-layer plate is compared with the semi-infinite results in Fig. 7. With large aspect ratios, such as a = 20, the backscattering efficiency can be approximated with great accuracy by using the results from the semi-infinite plates. When the aspect ratio decreases, two maximum reflection peaks remain observed, but the positions shift to shorter wavelengths and the values generally differ more from the semi-infinite results. For example, when a = 1, as shown in Fig. 7(a) and (b), one of the Qb maxima around x = 2.6 becomes the minimum for the semi-infinite plate. Since the radius of the plates is very small, unlike the semi-infinite plates, the interference effects between the top and bottom interface of the plates become unimportant in determining Qb.
To quantify the error of using the semi-infinite results to approximate the backscattering efficiency (Qb), we defined a quantity Qerror by comparing the value of the maxima between the finite and semi-infinite plates: Qerror = |(Qb,max-Rmax)/ (Qb,max + Rmax)|, where Qb,max is the maximum backscattering efficiency for the finite plates and Rmax is the maximum reflectance for the semi-infinite plates. A smaller Qerror means less difference between the maxima of these two kinds of plates, and, as observed from Fig. 7, when the maxima are close, the wavelengths for the locations of those maxima are also close to each other. We choose a Qerror of less than 10% as an acceptable criterion to use the semi-infinite approximation. Therefore, from Fig. 7, for single-layer plates, the aspect ratio must be , and for 5-layer plates, the aspect ratio must be . A larger layer number requires a larger aspect ratio.
With an aspect ratio a = 10, for a single-layer plate, Rmax = 0.033 at both x = 1.3 and 3.9, Qb,max = 0.035 at x = 1.4, and Qerror = 4%; for a 5-layer plate, Rmax = 0.52 at both x = 1.3 and 3.9, Qb,max = 0.60 at x = 4.0, and Qerror = 7%. At this aspect ratio, both the single-layer plate and the 5-layer plate have Qerror <10% and, therefore, to further study the dependency of backscattering efficiency on the incident angles we use a = 10 for both the single and 5-layer plates.
The angular dependence of the backscattering efficiency (Qb) can be used to study the response of different incident polarizations on the small plates backscattering. The results are shown in Figs. 8 and 9 for angle intervals of 10° and an aspect ratio of 10 for both perpendicular and parallel polarization states. To have Qerror ≤ 10% for the single-layer plate, we need an incident angle β≤ 30° for a parallel polarized component, and β≤ 50° for a perpendicular component. However, for the 5-layer plate to have the same Qerror≤10%, we need β≤10° for both polarizations due to the large fraction of edge compared to the total projected area. Therefore, the perpendicular component provides a better approximation compared with the parallel polarized component for both the single-layer and 5-layer results. The reflectance of semi-infinite plates approaches unity for all size parameters as the incident angle approaches 90°. The flat line in the top of Figs. 8 and 9 indicates a 100% reflection for β = 90° calculated by using Eq. (2). Since the light is incident directly on the edges of the plates, the backscattering efficiency is completely different from the reflection. The large value of Qb is mainly due to the small projected area as shown in Fig. 2. We cannot use the reflection of semi-infinite plates to approximate Qb for large incident angles.
The semi-infinite plates will respond differently to different incident polarization states; the reflectance of the parallel-polarized incident light will vanish at the Brewster angle, leaving the reflected light totally polarized in the direction perpendicular to the principal plane. As shown in Figs. 8 and 9, we found similar behaviors for both the single and 5-layer finite size plates. We compared the maximum reflectance for each incident angle, and found the minimum value for single-layer plates to be 0.013 and for 5-layer plates to be 0.06, and both to occur at β = 40°. The angle is smaller than the Brewster angle of 50.2° for nr = 1.2. Moreover, these two values are 37% and 10% of their corresponding maximum Qb at normal incidence, while for the semi-infinite plates the value of the minimum reflectance of the parallel, polarized light will be zero at the Brewster angle. This constitutes another prominent difference between finite and semi-infinite plates.
We studied the backscattering properties of small layered plates as a model for iridosomes, including both the backscattering efficiency and the shape of the reflection peak in the phase function. The width of the reflection peak is shown to depend on both the aspect ratio and the size parameter. The criteria for using the reflection of the semi-infinite plate to approximate the backscattering efficiency of small plates are discussed with respect to the aspect ratio, number of layers, and incident angle. For normal incidence and with an accuracy Qerror ≤ 10%, the semi-infinite plate assumption can work for a single-layer plate when the aspect ratio satisfies and for 5-layer plates when . For plates with small aspect ratios and large incident angles, the scattering properties must be calculated beyond the semi-infinite approximation. Furthermore, the backscattering efficiency can be used to estimate the diffuse reflectance of a random system, and therefore, determine its optical appearance.
This research was partially supported by the ONR MURI program N00014-09-1-1054. Drs. George Kattawar and Ping Yang also acknowledge support from the National Science Foundation (OCE-1130906) and (ATMO-0803779) respectively. We also thank the two anonymous reviewers for their contributions on improving the readability of the paper.
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