Understanding Cephalopod Camouflage By Use of a Novel 3D Vector
Radiative Transfer Code
Meng Gao, Yu You, Sergio Dagach, and George W. Kattawar
Department of Physics and Astronomy, Texas A&M University
College Station, TX 77843
ABSTRACT
One of the great enigmas of cephalopods is their remarkable ability to camouflage with
astonishing speed and precision. They are capable of matching the colors, patterns and
even the textures of their surroundings and are also responsive to the polarization of the
light field. The fundamental processes involved are the light interaction with the
coloration cells (chromatophores, iridophores, and leucophores) in their skin. To
understand this complicated optical process, we obtained the Muller matrix, which
contains all the informations one can obtain from a scattering particle, for each cell by
numerical calculations dependent on the size of the particle and includes techniques such
as the Improved Geometric Optics Method (IGOM), the Finite Difference Time Domain
Method (FDTD) and the Discrete Dipole Approximation (DDA) method. The influences
of size, shape, orientation, and interaction between multiple cells on the scattering
properties are discussed and justified. These parameters are then used to calculate and
predict the polarized reflectance distribution of light from cephalopod skin using a Monte
Carlo simulation, which is based on 3D vector Radiative Transfer equations. This method
can be used to then calculate the dynamic behavior of camouflage by the dynamic
behavior of both chromatophores and iridophores. We will present results of spectral skin
reflectance distributions for various compositions of skin structure and underwater light
conditions. The coloration of skin will be obtained by use of the tristimulus values of the
human eye combined with the spectral reflectance of the skin to calculate the
chromaticity coordinates. To our knowledge this is the first fully polarized 3-dimensional
radiative transfer code to be reported specifically for this purpose.
INTRODUCTION
The skin of cephalopods is highly sophisticated, which includes various coloration cells:
chromatophores, iridophores and leucophores, which absorb and scatter the light field.
Both the resultant reflectance spectrum and polarized light distribution are important for
the camouflage and communication for cephalopods. It is also important to understand
how the cephalopod skin responds in various environmental light fields. The multiple
scattering is characterized by the use of a three-dimensional (3-D) vector radiative
transfer equation (VRTE) for the tissue. Although this method has been applied widely in
the atmosphere and atmosphere-ocean systems [1, 2, 3], it has yet to be applied to light
transport in the skin system of cephalopods. There are other methods available for the
skin coloration mechanisms [4, 5], but they can only deal with an idealized skin structure,
which is different from the real structure with a distribution of cell size, orientation, and
inter cellular distances. The interplay between cells and the inhomogeneities within the
skin make the Monte Carlo model the most plausible approach to solve these realistic
systems.
SINGLE CELL MODELS
The scattering properties for each cell are calculated by using a combination of IGOM,
FDTD and DDA methods. The selection of the method depends on the size and optical
properties of the scatterers.
Chromatophore:
Chromatophores are primarily the absorbers, which reflect very little of the incident light,
but transfer most of the light forwardly and absorbs it as it passes through the cell. The
size of each chromatophore cell when expanded is on the order of a millimeter, which is
very large compared with the wavelength of visible light. Therefore, the IGOM is used
for the chromatophores. Absorption spectra of the three kinds of colored chromatophore
cells are needed to calculate the attenuation of the light field through each cell.
The chromatophores are assumed to have an elliptical cylinder shape, with absorption
dependent on the wavelength of the incident light. The aspect ratio and thickness is
determined by the expansion state of the chromatophores. To account for the rough
surface of the sacculus which encloses the absorber, random orientations and a tilted facet
method are used to calculate their phase function. For the tilted facet method, the surface
of the cylinder is composed by numerous rough surface facets, and their spatial
orientations are sampled by the 2-D Gaussian distribution [10].
Iridophore:
Inside an iridophore there are several units called iridosomes with multilayered structures
called lamellae which provide finer control on the scattering spectrum by coherent
scattering. We use the DDA (ADDA) and FDTD (Meep) to calculate the Mueller matrix
of individual iridosomes, which can show strong interference effects. The color of
reflected light is mainly determined by the thickness and orientation of the iridosomes.
The iridosome is modeled as a cylinder, with layered inner structure of high and low
refractive index material inside. The refractive indices of the two possible materials
composing the lamellae, namely, chitin and reflectin, are 1.56 and 1.59, respectively. The
spacing material is cytoplasm, whose refractive index is close to water, which is 1.33.
Thus, we can safely choose the relative refractive index as 1.2.
The backward scattering efficiency is defined as the integral of the phase function over
scattering angles from 90 to 180 degrees, divided by the particle projected area. It will be
used extensively as an indicator to show the discrepancy between scattering properties of
the small scatterers and those of semi-infinite multilayer plates. When the cylinder radius
approaches infinity, the backward scattering efficiency is the reflectance as computed
from geometric optics.
To generalize our results, the size parameter x (= kd = 2d/) is used to characterize the
dimension of iridophores. It is the phase change of the light along the length d in the
medium. For example, the typical thickness of one plate is 100 nm, therefore, the visual
band from 380 nm to 780 nm (in vacuum) will correspond to size parameters (in water)
from 1.1 to 2.2.
Leucophore:
The leucophore is a broad-spectrum reflector, consisting of numerous small transparent
spherical granules each with a high refractive index. Since the granules are close to each
other, there can be significant interaction between them, such that their combined
scattering properties deviate significantly from a single granule. In our simulation, we
modeled the leucophore cell as a cylinder filled by an assembly of small granules with
sizes on the order of a wavelength. ADDA is used for this simulation.
MODEL RESULTS FOR CELLS
Chromatophore:
Fig. 1. The averaged phase function of an elliptical cylinder with fixed orientation,
random orientations and one with tilted facets similar to a wind ruffled sea surface.
Refractive index is chosen as nr=1.3+i 0.5. The size parameters along the minor and
major axes are 150 and 300, respectively.
The large size of chromatophores makes them amenable to IGOM. The color of
chromatophores comes from the selective absorption of the light at certain wavelengths,
and the three kinds of chromatophores; brown, red and yellow, are determined by their
absorption spectra. The expanded chromatophores are rather transparent for the other part
of visible spectrum which makes it easy to mix the filtered light spectrum by different
kinds of chromatophores. To understand the refraction and diffraction properties, we
choose a refractive index with a large imaginary value to account for the high absorption.
Fig. 1 shows the phase functions calculated by IGOM. We see a strong forward
diffraction peak which is in general only size dependent, and a relatively strong backward
peak if the chromatophore had a smooth surface. The scattering becomes approximately
isotropic in the backward direction when random orientations or the tilted facet averaging
is done, as shown in Fig.1.
Iridophore:
Here, we discuss several factors that are important for the characterization of iridophores.
We have created a database of Mueller Matrices, which also include the phase function
and the backward scattering efficiency, for various sizes, number of layers, and
orientations of the iridosomes,
Fig. 2. Phase functions of the tilted 5-layered structure in the y-z plane, with the optical
thickness of the plate and spacing equal, and tilted angle  from 00 to 900 as shown in
the legend. Radius r = 5d, where d is the thickness of one layer. The relative refractive
index is nr = 1.2.
The cylinder is tilted along the x axis parallel to its upper surface by an angle . Shown
here in Fig. 2 are results for a 5-layer system, with the optical thickness of each plate and
spacing a quarter of a wavelength. This of course implies that the physical thickness of
the plate and spacing are different since they have different refractive indices. We use
the phase function to describe the distribution of scattered light for an unpolarized light
source.
We can see a strong backward scattering peak due to constructive interference just as for
semi-infinite plates. When = 0, the light is at normal incidence, the backward scattering
peak is at 1800. For the finite-sized structure, with the increase of the tilt angle, the
projected geometric cross section of the scatterer is decreased by a factor of cos. At the
same time, the optical path through the cylinder is increased, so in order to preserve
maximum interference, a shorter wavelength is required. This effect can be seen in Fig. 2
by the rapid decrease of back scattering peaks at the angle 180 - 2.
Fig. 3. Backward scattering efficiency vs. number of layers with different radii. The
optical thickness of both the plate and the spacing between the plates are equal. Radii r
are shown in the legend, where d is the thickness of one layer. The relative refractive
index is nr = 1.2. The thick blue curve is the theoretical result for the semi-infinite
plates.
To simplify the example, we still keep the optical thickness of plate and spacing between
the two plates at ¼ wavelength, but observe the optical properties for various numbers of
layers and geometric cross sections. The example for the normal incident case is shown
in Fig.3. The size effect plays an important role for the small scatterers. The backward
scattering efficiency deviates more and more from the semi-infinite plate case (the thick
blue curve) when its radius is decreasing. Since the size of iridosomes is varied with a
wide range among the numerous species of cephalopods, we clearly have to consider the
influence of their size before performing further calculations.
The theoretical results for the semi-infinite plate are calculated by three methods:
Huxley’s method [7], transfer matrix method for electric field [8], and the transfer matrix
method for electric and magnetic fields [9]. We have proven that all these methods give
identical results for the same physical system.
The inner structure will dramatically change the scattering cross section, and other
scattering properties, such as the phase function. In Fig. 3, the enhancement of the
backward scattering efficiency is obtained by letting more and more layers contribute to
the constructive interference. Also, the backward scattering can be further reduced for
destructive interference.
Fig. 4. Backward scattering efficiency vs. size parameter of the thickness of one plate
for different radii. The optical thicknesses of the plate and spacing are equal. Radii r
are shown in the legend, where d is the thickness of one layer. Relative refractive index
nr=1.2.The thick blue curve is theoretically calculated for the semi-infinite plates. The
red dashed curve is the results of averaged orientations.
The habitats of many cephalopods are in shallow water, which will contain a wide range
of visible spectrum, but in deep water only the short wave (blue) will survive. Thus to
consider the color appearance of the animal, we need to obtain their responses to the
spectrum which is commensurate with their environment. The wide spectrum response of
a five-layer iridosome is shown in Fig. 4. Again, backscattering efficiency is plotted in
comparison with the semi-infinite results. With the decrease of radius, the shape of the
profile is conserved with the maximum peak shifted to the right until the radius is so
small that the geometric shape of our modeled structure is dramatically different from a
thin cylinder. For example, when r=d/2, as shown in Fig. 4, the position with maximum
backward scattering efficiency almost shifts to the wavelength with maximum destructive
interference, compared with the semi-infinite plate. Since the radius is very small in this
case, the interference effects become unimportant, and the scattering effects dominate.
The orientation of the iridosomes is not always normal to the incident light, even though
they may be parallel to the skin surface. It would be important and interesting to consider
a species with total random (or under some constraint) orientation of iridosomes. Their
color appearance can be stabilized for every observation angle. As shown in Fig.1, the
magnitude and position of the maximum backward scattering peak are modulated with
the change of tilt angle . Thus, when the effects for random orientation are averaged, as
shown in Fig. 4 (the red dashed curve), the maximum backward scattering efficiency is
moved to the shorter wavelength region (larger size parameter), and the magnitude is
decreased. The consideration of polarization of reflected light will be more complicated,
and this will be discussed when the specific light field is considered.
Leucophore:
Fig. 5. Cross section of granules vs. volume fraction. Relative refractive index of the
granule is assumed to be 1.5. Size parameter of the diameter of one granule is 2. The
dimension of the leucophore is assumed as 10 thick and 26 wide in units of the size
parameter, respectively.
Without lose of generality, we consider the leucophore granules to have a refractive index
1.5. The total cross section of the assembly of granules is calculated by ADDA. It is
compared with the total cross section without interparticle interaction, which is obtained
by multiplying the single particle cross section by the particle number. When changing
the volume fraction of the granules, thus the number of granules inside a fixed volume
cylinder, we can see the large deviation from the independent particle model. The total
reflection and transmission spectra can be calculated and will be ready to compare with
experimental measurements when the data are available. More specific modeling is in
progress.
MONTE CARLO MODEL FOR RADIATIVE TRANSFER CALCULATIONS
Given the above models that describe the interactions between polarized light and various
skin cells, we can use our 3-D polarized radiative transfer model to couple the light fields
to predict the reflected light field off bulk skin tissue. The Monte Carlo model is one of
the most popular models used for simulating radiative transfer processes in atmosphere–
ocean systems. It is also the most versatile model for complex systems such as a 3-D
system.
In a radiative transfer model, the skin is represented by a layered structure that consists of
several chromatophore layers, an iridophore layer, and a leucophore layer, from top to
bottom. For the light field incident upon organism, it can either be calculated from
another Monte Carlo model for an ocean, or from measured light field data. For a
specific skin state, we can vary the incident light field and study how the animal skin
responds to different stimuli.
The chromatophore layer is considered to be purely absorbing since the scattering off a
chromatophore cell has been found to be very weak in the backward direction. In other
words, there is no multiple scattering happening between cells in this layer. We consider
three sub-layers in order to represent the layered structure of the three kinds of colored
chromatophore cells. Absorption spectra of pigments in the three kinds of
chromatophore cells are used as inputs to determine the exponential attenuation of
polarized light through the chromatophore layer. The polarization state is not altered in
this layer. In addition, it is critical to appropriately account for the spatial distribution of
chromatophore cells in the horizontal domain, which determines the color of the skin.
This distribution can be realized in the Monte Carlo model by describing the
chromatophore layers by arrays of voxels (i.e., volumetric pixels), each of which is
assigned a different absorption spectrum. Voxels with a high absorption represent
chromatophore cells, while voxels with a low absorption represent the surrounding tissue.
By adjusting the distribution of high and low voxels, we can simulate all possible
situations where each chromatophore cell is expanded or contracted. Simulation results
for a series of different expansion-contraction states will help us understand what
physiological changes are needed to emulate the incident light field.
On the other hand, the iridophore layer acts as a regular scattering layer, since the angular
distribution of reflected light off iridophore cells was found to depend on the wavelength,
the structure of the iridosome, and the angle of incidence upon the layer. Effective phase
matrices from single scattering calculations are used as inputs to characterize the multiple
scattering in this layer in the same way as that in a traditional RT model. This layer
reflects the light field that has gone through the chromatophore layers and alters the
polarization state of the reflected light field. The iridophore layer is assumed to be
horizontally homogeneous.
Finally, the leucophore layer is modeled as a Lambertian surface that isotropically
reflects the incident light field, which is the light field transmitted through the iridophore
layer.
In our 3D Monte Carlo simulation, photons will propagate with a Mueller matrix, which
can preserve the polarization information. We will present results of skin reflectance
distributions for various composition of skin structure and underwater light conditions.
The coloration of skin will then be obtained by computing the chromaticity coordinates
using the tristimulus function of the human eye.
CONCLUSIONS
We have developed a very powerful set of codes to model virtually any cephalopod skin
structure under any ambient light condition. These codes are able to model virtually all
the primary constituents of the cephalopod skin; namely, chromatophores, iridophores,
and leucophores which can then be embedded into our 3D Mont Carlo code to obtain the
reflectance as well as transmittance properties of realistic skin structures. The code is
capable of computing the “effective” Mueller matrix for a complex skin structure which
may contain several layers of chromatophores followed by one or more layers of
iriddophores and then followed by one or more layers of leucophores. Since this Mueller
matrix is independent of the ambient illuminating light on the organism, it becomes
simply a matter of taking any ambient light field and multiplying it by this “effective
Mueller matrix to obtain the reflected light on a pixel-by-pixel basis. This reflected light
can then be used with whatever eye stimulus values that we can obtain for the organism
to find out what the organism “sees” and then contrast that with what the human eye will
see by using the tristimulus values for the human eye.
The code is only as good as the data we obtain from modeling the individual
chromatophores, iridophores, and leucophores. Due to the large size of chromatophores,
when in the expanded state, we used the IGOM to calculate their Mueller matrix for
single scattering using realistic morphology and optical properties which are woefully
lacking for the constituents of these cells. They are shown to scatter light strongly in the
forward direction, the small amount of light that is backscattered is fairly isotropic when
the surface of the cell is rough, which justifies treating its surface as a Lambertian
reflector. The primary roll of the chromatophores is to absorb radiation by different
amounts depending on the color of the chromatophores. The ADDA method is used to
obtain the Mueller matrices for the iridophores for various sizes, number of layers, and
orientations of the iridosomes contained in it. One of the very important things we have
discovered is that the iridodsomes comprising the iridophore can only be modeled by the
semi-infinite plate theory under very limited circumstances. We do not need to make any
fidelity reducing assumptions about the actual shape of these constituents to model them
correctly. We have found that the backward scattering efficiency deviates further away
from the semi-infinite plate when the aspect ratio of the iridosome decreases. The number
of layers inside of iridosomes also plays an important role in maximizing interference
effects, and the orientations provide another control on the scattering properties of the
spectrum and distribution. The granules inside of leucophores are shown to have strong
multiple scattering between each other but the entire leucophore can be approximated as
a highly reflecting Lambertian surface in the simulation.
Based on those complexities, we are convinced that the realistic light response of the
cephalopod skin has to include all these variants: size, shape, orientation, and
intracellular structure. As we have shown, it takes a vast amount of different approaches
to gain the necessary information to model cephalopod skin in a realistic way. It should
also be noted that these important tools we have developed can also be effectively used to
model camouflage in teleosts as well.
ACKNOWLEDGMENTS
This research was partially supported by the ONR MURI program N00014-09-1-1054.
Put a thanks to Lei Bi for giving us his IGOM code
REFERENCES
1.
S. Chandrasekhar, Radiative Transfer (Dover, 1960).
2.
Mishchenko, M. I., L. D. Travis, and A. A. Lacis, 2006: Multiple Scattering of
Light by Particles: Radiative Transfer and Coherent Backscattering, Cambridge
University Press, Cambridge.
3.
P.-W. Zhai, G. W. Kattawar, and P. Yang, "Impulse response solution to the
three-dimensional vector radiative transfer equation in atmosphere-ocean systems.
I. Monte Carlo method," Applied Optics 47, 1037-1047 (2008).
4.
P.-W. Zhai, G. W. Kattawar, and P. Yang, "Impulse response solution to the
three-dimensional vector radiative transfer equation in atmosphere-ocean systems.
II. The hybrid matrix operator-Monte Carlo method," Appl. Opt. 47, 1063-1071
(2008).
5.
Richard L. Sutherland, Lydia M. Mäthger, Roger T. Hanlon, Augustine M. Urbas,
and Morley O. Stone , "Cephalopod coloration model. I. Squid chromatophores
and iridophores", JOSA A, Vol. 25, Issue 3, 588-599 (2008)
6.
Richard L. Sutherland, Lydia M. Mäthger, Roger T. Hanlon, Augustine M. Urbas,
and Morley O. Stone, " Cephalopod coloration model. II. Multiple layer skin
effects" JOSA A, Vol. 25, Issue 8, 2044-2054 (2008)
7.
8.
9.
10.
A. F. Huxley, "A theoretical treatement of the reflexion of light by multilayer
structures", J. Exp. Biol. 48, 227-245 (1968)
B E. A. Saleh and M Carl Teich, "Fundamentals of Photonics", p243, 2nd edition,
Wiley (2007)
Eugene Hecht, "Optics", p426, 4th edition, Addison Wesley (2002)
C. Cox and W. Munk, "Measurement of the roughness of the sea surface from
photographs of the sun's glitter," J. Opt. Amer. Soc., vol. 44, no. 11, pp. 838-850,
Nov. 1954.