588 J. Opt. Soc. Am. A/Vol. 25, No. 3/March 2008 Sutherland et al.
Cephalopod coloration model. I. Squid chromatophores and iridophores
Richard L. Sutherland,1,2,* Lydia M. Mäthger,3 Roger T. Hanlon,3 Augustine M. Urbas,1 and Morley O. Stone1 1Air Force Research Laboratory, 3005 Hobson Way, Wright-Patterson Air Force Base, Ohio 45433-7707, USA 2Science Applications International Corporation, 4031 Colonel Glenn Highway, Dayton, Ohio 45431, USA 3Marine Resources Center, Marine Biological Laboratory, 7 MBL Street, Woods Hole, Massachusetts 02543, USA *Corresponding author: [email protected]
Received September 11, 2007; revised November 28, 2007; accepted December 3, 2007; posted December 21, 2007 (Doc. ID 87405); published February 7, 2008 We have developed a mathematical model of skin coloration in cephalopods, a class of aquatic animals. Cepha- lopods utilize neurological and physiological control of various skin components to achieve active camouflage and communication. Specific physical processes of this coloration are identified and modeled, utilizing avail- able biological materials data, to simulate active spectral changes in pigment-bearing organs and structural iridescent cells. Excellent agreement with in vitro measurements of squid skin is obtained. A detailed under- standing of the physical principles underlying cephalopod coloration is expected to yield insights into the be- havioral ecology of these animals. © 2008 Optical Society of America OCIS codes: 000.1430, 050.7330, 160.1435, 240.0310, 260.3160, 330.1690.
1. INTRODUCTION experimental spectra or combined to illustrate coloration effects in different environments. Color is an important and ubiquitous property in biology. Our objective is to provide answers to questions such as Although animal pigments have long been studied, struc- the following: Given a particular light source, what does a tural coloration in biology has become a subject of recent intense interest, with many examples of exotic photonic patch of cephalopod skin look like to an observer? Particu- structures discovered in both aquatic and terrestrial sys- larly, what is the color rendition of the skin in a spatial tems [1–3]. Since most animals are visual, color plays a resolution element (i.e., a pixel)? What are the physical significant role in, for example, predator–prey relation- properties of the skin that lead to this coloration? In an- ships, mating, and the generation of visual signals be- swering such questions we must consider the multiple op- tween conspecific or interspecific animals. A crucial de- tical interactions of various skin components (pigments fense mechanism for many animals is camouflage. and structural elements), which can undergo physiologi- Cephalopods—a class of aquatic invertebrates (Phylum cal or neurological modification, and the fact that within Mollusca) that includes squid, octopus, and cuttlefish— an aquatic environment the nature of ambient light can have developed a highly sophisticated system of dynamic range from highly collimated to diffuse. Results of this camouflage [4,5]. They adapt their color and body pattern study should enhance our understanding of how these to various visual features of the immediate background. animals employ dynamic camouflage to adapt to their Color is controlled through a combination of both pig- background. ments and photonic structures in an elaborate skin con- We have developed a model of the reflective properties figuration. This produces a rich repertoire of spectra and for a generic cephalopod skin consisting of any number of patterns that is highly adaptive to changing backgrounds layers with various optical elements, both structural and and situations. Current biological research is focused on pigmented. The model incorporates multiple internal re- understanding how cephalopods neurologically perform flections among the various skin components and consists these adaptations [6–9]. also of various submodels describing the optical proper- Although cephalopod skin has been widely studied and ties of each skin element individually. In this paper we characterized biologically, there has been relatively little present these submodels for two components, namely, the work done in modeling its optical properties. Masthay pigmented chromatophores and the structural iri- presented a quantitative model of pigment absorption in dophores, for which the spectral reflectance and transmit- animals, including cephalopods, based solely on Beer’s tance can be changed by the animal, either neurologically law, but did not include scattering or diffuse reflection or physiologically, to achieve dynamic camouflage. We [10]. Denton and Land discussed the mechanism of irides- compare these models with in vitro experimental mea- cent reflections from fish and cephalopods in terms of surements of skin components in squid. Materials proper- quarter-wave thin-film stacks [11], and Mäthger and Den- ties, such as index of refraction, absorption and scattering ton described theoretical calculations of spectra that they coefficients, and dispersion are obtained either from the related qualitatively to the iridescence of loliginid squid literature or through comparison of the models with ex- [12]. In none of these previous works, however, were de- perimental data. A future paper will combine these re- tailed calculations of spectra compared quantitatively to sults with those of submodels for static skin components
1084-7529/08/030588-12/$15.00 © 2008 Optical Society of America Sutherland et al. Vol. 25, No. 3/March 2008/J. Opt. Soc. Am. A 589 to investigate multilayer effects on the total animal re- scattering, we introduce the forward scattering ratio , flectance with a generic cephalopod skin. which is defined as the ratio of light scattered into the for- ward hemisphere to the total scattered light [16]. We treat the chromatophore ideally as a disk of uni- 2. THEORY form thickness d and elliptical cross section. Electron mi- The topmost layers of a squid skin are pigment-bearing croscopy data for an expanded chromatophore indicate ϳ organs called chromatophores. Each chromatophore com- that a typical pigment granule size is 300 nm and the Ͼ prises a sacculus containing pigment granules of a spe- mean center-to-center spacing of granules is 300 nm cific color (brown, red, or yellow) and surrounded by a se- [17]. Applying the standard Rayleigh criterion, the chro- ␦ Ͻ ries of radial muscles. When the muscles contract, the matophore would be optically smooth if h /8n, where ␦ sacculus is expanded, thus spreading out the pigment h is the root-mean-square roughness height measured ␦ granules. The sacculus is elastic so that when the muscles from some reference plane, which would imply h Ͻ relax, the chromatophore retracts. The pigment colors are 55 nm for =600 nm and n=1.33. Since the surface is ordered, going from yellow to red to brown in skin depth not optically smooth, we make the simplifying assump- Ϸ [9]. Subjacent to the chromatophores are structural color tion that Rcc 0. Although this does not entirely rule out elements called iridophores. These consist of several iri- the presence of a specular component, it is likely to be dosomes that have a multilayer structure consisting of small and not contribute significantly to the total skin re- thin, transparent protein platelets [13] sandwiched be- flectance. Hence we consider a model for the isolated chro- tween spacers of cytoplasm. The iridophores produce iri- matophore as illustrated in Fig. 1. Let a prime symbol in- descent colors upon illumination by white light. The in- dicate the derivative with respect to z, where z is the tensity and color of iridophores can change in the distance measured from the front of the chromatophore presence of the neuromodulator acetylcholine [14]. (see Fig. 1). Designating I as the forward flux and J as the Ultimately, we wish to relate the various components of backward flux, the radiation transport equations may radiant flux (both collimated and diffuse) at the bound- then be written as [16] aries of the various skin layers. To this end we define the Ј ͑␣ ͒ ͑ ͒ I =− + Ic, 1a partial reflectances and transmittances for each layer. By c partial we mean the reflectance or transmittance that Ј ␣ ͑ ͒ ͑ ͒ ͑ ͒ would be measured in an experiment for a completely iso- Id =− Id − 1− Id + 1− Jd + Ic, 1b lated layer using either pure collimated or pure diffuse Ј ␣ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ light, and separately detecting specularly and diffusely Jd = Jd + 1− Jd − 1− Id − 1− Ic, 1c reflected or transmitted light. For example, if we desig- which is a variant of the Kubelka–Munk model [18]. In nate Jd←c as the diffuse component of the total reflected light for a purely collimated input I , then the ratio R Eqs. (1b) and (1c) we have introduced the effective path c cd length coefficient for diffuse light, defined by =J ← /I is the partial reflectance. A similar definition d c c =͗1/cos ͘, where is an angle of propagation with re- holds for the partial transmittance Tcd. We also define partial reflectance and transmittance for collimated light spect to the z axis and the brackets indicate an average produced from collimated light (R and T ) and diffuse over the full solid angle in the forward direction weighted cc cc by the angular distribution of the diffuse radiation light produced from diffuse light (Rdd and Tdd). For the models presented here we have made some sim- [16,18]. Note that Ic represents the flux of ballistic (un- plifying assumptions. All elements, as well as the entire scattered) photons and thus has an effective path length skin itself, are embedded in a watery environment. Hence coefficient of one. we can ignore interfacial Fresnel reflections except those This system of coupled differential equations consti- that arise from components of interest. These include tutes what is known as a three-flux model and may be specular reflections from iridophores and scattering from solved exactly. The solutions are similar in many respects pigment granules. The scattering elements are taken to to those for the two-flux model (forward and backward dif- behave as ideal Lambertian surfaces, yielding only diffuse fuse light flux only) given by Kubelka [18], and the four- reflectance. They do, however, have some specular trans- flux model (forward and backward collimated and diffuse mittance (i.e., regular transmittance, light that is neither light flux) given by Maheu et al. [16]. In both of these ear- absorbed nor scattered) as well as diffuse transmittance. lier works reflection from a background was also included. Therefore, specular reflection arises only from smooth This type of reflection is ignored here since we are consid- photonic structural elements. We assume that there are ering an isolated chromatophore. Appendix A gives the no index heterogeneities in these components. Therefore, detailed solution of this particular three-flux model, they have diffuse reflectance or transmittance only when irradiated by diffuse light.
A. Chromatophore Layers The pigment granules of chromatophores absorb and scat- ter light. We thus define absorption and scattering coeffi- cients ␣ and , respectively, both having units of inverse length. (We note that these parameters are commonly designated by a and b, respectively, in the ocean optics Fig. 1. Three-flux model for computing transmittance and re- community [15].) To distinguish forward and backward flectance of a chromatophore. 590 J. Opt. Soc. Am. A/Vol. 25, No. 3/March 2008 Sutherland et al.
which differs to some extent from these previous develop- determined parameters are in the form of products Sd ments. The results for the collimated–collimated and and Kd, where S and K are proportional to pigment con- diffuse–diffuse components are simple: centration and hence inversely proportional to the volume of the sacculus. Although the volume, and hence the pig- T = exp͓− ͑␣ + ͒d͔, ͑2͒ cc ment concentration, is conserved as the sacculus expands (i.e., S and K are constants), the path length d is reduced, producing changes in the reflectance and transmittance T = , ͑3͒ dd cosh d + ͓␣ + ͑1−͔͒sinh d as indicated by the explicit dependence of Eqs. (2)–(4), (6), and (7) on d. Consequently, the extracted experimental ͑1−͒ sinh d parameters Sd and Kd are inversely proportional to the ͑ ͒ Rdd = , 4 cross sectional area of the chromatophores. For constant cosh d + ͓␣ + ͑1−͔͒sinh d volume, the cross sectional area and thickness are recip- rocally related, and an increase in one by some factor re- where sults in a decrease in the other by the same factor. Assum- = ͱ␣2 +2͑1−͒␣. ͑5͒ ing that the shape of the chromatophore remains an elliptical disk as its volume changes, we introduce a scal- The results for the collimated–diffuse components can be ing factor for the radius (or geometric mean radius) of expressed analytically but are rather complex. They may the elliptical cross section. Therefore, to simulate expan- be simplified, however, by using an approximation. For sion and retraction of the chromatophore, which dynami- pure collimated light incident on a particle with size com- cally changes the reflectance and transmittance, the pa- parable to or greater than the wavelength of light, the rameters Sd and Kd are scaled as −2. The default scattering is peaked sharply in the forward direction, and parameters ͑=1͒ are determined by the size of the chro- the effective path length coefficient is not much larger matophores for which Rcd and T are measured. than one [16]. Typical pigment granule size is ϳ300 nm Finally, note that the total scattering coefficient is [17]. Hence for visible collimated light we make the ap- needed for calculating Tcc and is given by =S/͑1−͒. proximation Ϸ1. Then the expressions for the Hence, values for the forward scattering ratio must be as- collimated–diffuse components reduce to (see Appendix A) sumed for modeling purposes. Some guidance for select- ing these values may be found in the literature [16,20]. Ϸ ͓ ͑␣ ͒ ͔ Tcd − exp − + d , Vargus and Kinlasson give numerical methods for esti- cosh d + ͓␣ + ͑1−͔͒sinh d mating the forward scattering ratio based on particle size ͑6͒ and complex index of refraction using Lorenz–Mie theory [21]. ͑1−͒ sinh d R Ϸ . ͑7͒ cd cosh d + ͓␣ + ͑1−͔͒sinh d B. Iridophore Layer Iridophores consist of stacked layers of iridosomes, which Note that we do not make the same approximation in Eqs. themselves comprise 2–10 layers of thin protein platelets (3) and (4) since, by definition, incident diffuse light is al- separated by thin layers of cytoplasm. The platelets are ready propagating in multiple directions. For isotropic composed of a recently identified protein family named re- diffuse light, =2 (see, for example, [18]). flectins [13], which are found in the iridophores of certain In laboratory experiments the incident light is purely squid. Platelets have been studied by electron microscopy, ϳ collimated, and Rcd and T=Tcc+Tcd are measured for and a typical thickness is 100 nm. Cytoplasmic spacer near-normal incidence. Forming the quantities a and b in thickness is of the same order [22]. Reflectin has been re- terms of measured parameters combinantly expressed and cast as thin films. A refractive index of 1.591 has been measured using a prism coupling 2 2 1+Rcd − T technique [23]. The refractive index of cytoplasm is 1.33, a = , similar to that of water [11]. Iridosomes are tightly 2Rcd packed in the iridophore cell and thus approximate a thin film, high–low-index multilayer stack with Bragg reflec- b = ͱa2 −1. ͑8͒ tion in the visible or near-infrared spectral regions. Large Equations (2) and (5)–(7) may be used to extract the back- numbers of Bragg reflectors are situated in the iridophore ward scattering and absorption coefficients of the pig- layer, forming iridescent splotches in the skin [6,14]. ment. These are often referred to in the literature as S Iridophores have been shown to have properties similar and K [19]. The results are to that of quarter-wave stacks [11,12], although the de- tailed spectra have not been previously modeled. Close 1 bRcd examination of iridophore micrographs reveals that, al- S ϵ͑1−͒ = sinh−1ͩ ͪ, ͑9͒ bd T though the platelets have reasonably uniform thickness, the cytoplasm spacers do not [23]. Moreover, the platelets and are not ideally flat but exhibit some small, apparently K ϵ ␣ = ͑1−a͒S. ͑10͒ random curvature. Hence, to model these reflectors we first calculate reflectance and transmittance by approxi- Since the chromatophores are characterized in vitro, the mating them as ideal multilayer thin-film stacks using thickness is not known. Consequently, the experimentally standard methods and then modify the results to include Sutherland et al. Vol. 25, No. 3/March 2008/J. Opt. Soc. Am. A 591
3 2 the effects of their random structure. We thus treat an en- −13.386 nm, and Cr =16.290ϫ10 nm [23]. However, this tire iridophore (consisting of one or more iridosomes) as a result was obtained from measurements done on a cast single multilayer stack, employing a 2ϫ2 matrix ap- film, whereas in vivo the protein is normally in an aque- proach to calculate the forward and backward propagat- ous solution bound within a cell membrane. In addition, ing electric field amplitudes [24], the results of which we physiologically active squid iridophores have been found summarize next. to change reversibly from a noniridescent to an iridescent The incident and reflected field amplitudes, E0 and Er, state. These active cells also demonstrate the same re- respectively, expressed in a column vector, may be related sponse in vitro to the neuromodulator acetylcholine. This to the transmitted field amplitude, Et,bya2ϫ2 transfer has been correlated by transmission electron microscopy matrix P: studies to the physical structure of the platelets. In the iridescent iridophores the platelets are thin and uni- E0 Et formly electron-dense. Noniridescent iridophore platelets ͩ ͪ = Pͩ ͪ. ͑11͒ Er 0 exhibit a flocculent or ribbonlike structure containing electron-lucent and electron-dense regions. In addition, The multilayer-film ͑mf͒ reflectance and transmittance the platelet regions are thicker in the noniridescent state are then given by [14,22]. We can simulate these properties of the iri-
2 2 dophores by assuming reflectin to be in an aqueous solu- Er P21 ͯ ͯ ͯ ͯ ͑ ͒ tion having some equilibrium volume fraction f0 of water Rmf = = , 12 E0 P11 when the iridophore is maximally iridescent. As the iri- dophore changes to a noniridescent form, we assume that 2 2 the platelet region is invaded with a waterlike fluid, ns cos s Et ns cos s 1 T = ͯ ͯ = ͯ ͯ , ͑13͒ swelling the region and creating a new equilibrium vol- mf n cos E n cos P 0 0 0 0 0 11 ume fraction fϾf0. We thus let the high-index region be characterized by a refractive index given by a simple mix- where the subscripts 0 and s refer to the input and sub- ture formula, strate (exit) media, respectively. We will assume these ͑͒ ͑͒ ͑ ͒ ͑͒ ͑ ͒ media to be identical. Also, since Rmf=1−Tmf (the iri- nH = fnw + 1−f nr , 17 dophores are nonabsorbing), we need only focus on the P11 element of the transfer matrix. and a thickness that changes linearly with the volume The multilayer consists of N alternating high-index ͑H͒ fraction of water, and low-index ͑L͒ pairs having refractive index n and n H L dh and thickness hH and hL, respectively. Assuming that ͑ ͒ ͑ ͒ hH = h0 + f − f0 . 18 each pair in the stack is identical, the transfer matrix can df be written in terms of the power of a single matrix M, i.e., N −1 In this model, h0 is the equilibrium platelet thickness for P=FLHM FLH, where M is the product of four matrices, volume fraction f0, and dh/df is the rate that the thick- ⌽ ⌽ ͑ ͒ M = HFHL LFLH. 14 ness changes with respect to an increase in the volume fraction of water. Both f0 and dh/df are treated as phe- We have taken the input and exit media to have the same nomenological parameters that can be adjusted to fit ex- refractive index as the low-index film, i.e., similar to wa- perimental data. ter. In the product M there are two phase matrices, given Next we incorporate small irregularities into the iri- by dophore structure. Although the platelets are highly or- dered, there appears to be some random structure super- exp͑− i2n h cos /͒ 0 ⌽ ͫ k k k ͬ imposed upon this order. Figure 2 gives a schematic k = ͑ ͒ , 0 exp + i2 nkhk cos k/ illustration of the platelet shape and spacing irregularity. ͑15͒ If we let hL designate the mean separation of platelets, then the spacer thickness at any point can be given by where the subscript k=H or L, is the wavelength, and k hL + , where is a position-dependent (positive or nega- is the angle of propagation with respect to the film- tive) random variable. Furthermore, it appears from elec- normal in the kth layer; and there are two Fresnel matri- tron micrographs that there is no statistical correlation in ces, given by spacer thickness from layer to layer [26]. Thus, we as- sume that each spacer is characterized by a statistically 1 1 rjk independent random variable . We also assume that the F = ͫ ͬ, ͑16͒ k jk curvature of the platelet is small everywhere; i.e., al- tjk rjk 1 though the spacing between platelets changes continu- where jk=HL or LH, and tjk and rjk are the usual Fresnel ously with position, the rate of change or slope of the transmission and reflection coefficients for the jkth inter- platelet surface is very small. Consequently, we take the face, which depend on the polarization of light [25]. deviation of any ray trajectory through each layer from To a good approximation, we take the cytoplasm to that of an ideal, parallel layer stack to be negligibly small. have an index of refraction like water ͑nw Ϸ1.33͒. The re- Then the result of the random spacing in the iridophore is fractive index dispersion of reflectin has been fit to a to add a random phase ␦k =2nLk cos L / to the diago- 2 Cauchy equation, nr͑͒=Ar +Br /+Cr / , where is ex- nal elements of each phase matrix ⌽L in the transfer ma- pressed in nanometers, with Ar =1.56713, Br = trix P. Since the phase matrix is diagonal, this change 592 J. Opt. Soc. Am. A/Vol. 25, No. 3/March 2008 Sutherland et al.
tions p͑0͒,p͑1͒,p͑2͒,...,p͑N͒, all of which have zero means and respective standard deviations 0 ,1 ,2 ,...,N, and we assume that p͑k͒ is the same for any iridophore in the ensemble. Then the average collimated–collimated transmittance is given by
T = ͵ T ͑ ,͕ ͖͒p͑ ͒p͑ ͒p͑ ͒ ... cc mf 0 k 0 1 2
ϫ ͑ ͒ ͑ ͒ p N d 0d 1d 2 ...d N, 20
where ͕k͖ represents the set of random variables, k =1...N, for any single iridophore; Tcc is a function of wavelength, angle of incidence, and polarization; and Rcc=1−Tcc. For simplicity, we will assume that k = for all k=1...N, where is a constant ͑ 0͒. Therefore, the statistics of the iridophores are characterized by just two parameters. In Appendix B we give an approximate 2 form of Eq. (20) to order , which is valid for Ӷ. To calculate the diffuse–diffuse components, we assume a Lambertian light source with constant spectral radiance L independent of the angle of propagation. The spectral Fig. 2. Schematic illustration of the random spacing of platelets optical power incident on a point on the surface of the iri- in iridophores. The platelets (dark) have a uniform thickness hH, dophore, having an element of surface area dA, is given while the spacer thickness varies as hL + k for layer k, where 2⌽ ⍀ ͑ ͒ by d ,0=L cos dAd , where is the angle of incidence k x is a random variable. A sensor will integrate over the spatial variable x within a pixel and yield a response related to the re- between the iridophore-normal and a line connecting a flectance (or transmittance) averaged over the random variables. point centered on an element dAЈ of the light source with The mean spacer thickness is hL. the point on the surface of the iridophore, and d⍀ =sin dd is the solid angle subtended by dAЈ at the can be accomplished by making the substitution ⌽L point on the iridophore ( is the azimuth angle). The 2⌽ →Dk⌽L for all N low-index phase matrices in the transfer power transmitted by the iridophore is given by d ,t matrix product P, where =Tcc͑,͒L cos sin dddA. Dividing by dA and inte- grating over the hemispherical solid angle to obtain the exp͑− i␦ ͒ 0 ͫ k ͬ ͑ ͒ spectral (vector) irradiance I (this assumes that the Dk = ͑ ␦ ͒ , 19 0 exp + i k source is large compared to the iridophore), the diffuse– diffuse transmittance is given by the ratio
and k is now a parameter that runs from 1 to N. Note that /2 under these conditions the transfer matrix can no longer I,t T ͑͒ = = ͵ T ͑,͒sin 2d. ͑21͒ be written in terms of the simple power of a single matrix dd cc,u I,0 0 since each matrix Mk is different in general and continu- ously variable across the iridophore. A similar expression holds for Rdd, or one can be com- Everything we have discussed up to this point concerns puted and the other determined by Rdd+Tdd=1 since a single iridophore. However, within a typical spatial res- there is no absorption. We assume that diffuse light is olution element of the observer there will generally be a completely unpolarized. Since Tcc and Rcc depend on po- large ensemble of iridophores making up an iridescent larization, the collimated–collimated component in the in- splotch in the iridophore layer. It is unlikely that these tegrand of an expression like Eq. (21) is understood to iridophores are identical; i.e., they will display some small represent equal contributions of s polarization and p po- deviations in Bragg wavelength due to angular or larization, i.e., Tcc,u =͑Tcc,s +Tcc,p͒/2. platelet-separation variations. We will account for these Finally, we assume that the iridophore platelets and differences by assuming a Gaussian distribution of Bragg spacers contain no index heterogeneities. Since the curva- wavelengths over the iridophore ensemble, which can be ture of the platelets is small, so that deviations of ray tra- accomplished by the introduction of another single ran- jectories are negligible, the iridophores produce very little Ϸ Ϸ dom variable 0. Accordingly, we now modify the model by incoherently scattered light. We thus set Rcd Tcd 0. letting any single iridophore have a mean spacer thick- ness given by hL +0, where 0 is a random variable over the entire ensemble of iridophores. The mean spacer 3. EXPERIMENT thickness for the entire ensemble is then hL. Skin specimens were prepared from squid (Loligo pealeii) The transmittance of the iridophore layer now becomes for chromatophore and iridophore measurements. Small a function of N+1 random variables. A sensor will yield a dissected skin elements were pinned onto the Sylgard- response that is an average of the transmittance (or re- covered dish of a goniometer. The specimens were covered flectance) over these variables by integrating spatially with sea water for preservation of their properties during across a pixel. Assuming Gaussian statistics for each ran- the measurements. Spectral reflectance and transmit- dom variable, we define normalized probability distribu- tance measurements were obtained using an Ocean Op- Sutherland et al. Vol. 25, No. 3/March 2008/J. Opt. Soc. Am. A 593 tics fiber optic spectrometer attached to a dissecting mi- croscope. Details of the experimental apparatus and methods can be found in [6]. Spectral transmittance was measured by directing light from the bottom of the sample at normal incidence through the sample and into the microscope. The micro- scope objective restricted the angle of incidence for reflec- tance measurements to no less than ϳ10°. The angle of incidence in water was thus ϳ7.5°, satisfying near- normal incidence conditions. Reflectance measurements were performed using a black felt background beneath the skin samples. This background produced a very flat spectral reflectance of approximately 0.025 across the vis- ible spectrum. Corrections were made for Fresnel reflec- tions at the air–water and air–Petri-dish interfaces. The measurements recorded were total reflectance and trans- mittance, i.e., the sum of specular and diffuse compo- nents.
4. RESULTS AND DISCUSSION A. Chromatophores Reflectance and transmittance spectra were measured for 6–8 chromatophores of each color. Average spectra were then computed for brown, red, and yellow chromato- phores, and K͑͒ and S͑͒ were extracted from these av- erages using Eqs. (8)–(10). These spectral properties are plotted in Fig. 3. With these values of K͑͒ and S͑͒ we can scale the reflectance and transmittance of individual chromatophores of variable size (area and thickness) us- ing Eqs. (2), (6), and (7) ͑Ϸ1͒ and by adjusting the scale factor . Examples of this scaling for individually selected chromatophores, compared to data, are given in Figs. 4 and 5. We should point out that at present we are limited to this semiempirical method of predicting R and T spec- tra for the chromatophores based solely on their size, us- ing experimentally extracted K and S values. Detailed modeling from first principles (e.g., Mie theory) would re- Fig. 3. (Color online) Spectral absorption and backscattering co- quire knowledge of n and k spectra (real and imaginary efficients (normalized to thickness d) derived from reflectance and transmittance measurements of the brown, red, and yellow parts of the complex index of refraction, respectively) for chromatophores of the squid Loligo pealeii. (a) Absorption (K͒ the pigments. This information is not currently unavail- spectra. (b) Backscattering ͑S͒ spectra. able. Although it may appear that the results discussed here are circular, i.e., measured R and T spectra are used to determine K and S spectra, which in turn are used to are shifted toward the blue, giving them their character- predict R and T, it is important to note that one cannot istic colors. Backscattering coefficient spectra, S͑͒, for simply scale R and T spectra for the varying size of chro- the brown and red chromatophores are fairly flat, rising matophores. This scaling must be done within the argu- slightly toward the red end of the spectrum; S͑͒ for the ments of the hyperbolic functions in Eqs. (3), (4), (6), and yellow chromatophore exhibits an inverse dependence on (7). Also, it is not possible to compute Tcc,Eq.(2), without wavelength, more like a typical Mie scatterer. Again, knowledge of S and K. This becomes particularly impor- since the absolute thickness of these chromatophores is tant when dealing with multilayer skin effects (e.g., chro- unknown, we cannot comment on the relative magnitudes matophores over iridophores; see Subsection 4.C), where of the backscattering coefficients. the specular reflection from underlying elements depends It is interesting to compare these S and K spectra to on the specular transmittance of the chromatophores. those of inorganic paint pigments. A survey of common Moreover, knowledge of S and K spectra also gives us colorants is given in [20]. Three of these are visually simi- valuable information about the optical properties of the lar in color to the chromatophores studied here: Red Iron pigments themselves, as we discuss below. Oxide (brownish), Acra Red, and Cadmium Yellow Light. The brown chromatophore exhibits the largest absorp- (Photos of these pigments in [20] can be compared with tion, but we cannot ascertain whether this is due to a the color photos of L. pealeii chromatophores in [6].) Red larger absorption coefficient, a larger thickness, or both. Iron Oxide has strong absorption in the blue with a rolloff Its spectrum is similar to that of a melanin pigment. The toward the red beginning at about 550 nm, similar to the absorption edges of the red and yellow chromatophores K͑͒ spectrum for the brown chromatophore in Fig. 3(a). 594 J. Opt. Soc. Am. A/Vol. 25, No. 3/March 2008 Sutherland et al.
seen in Fig. 3(b) for the squid chromatophore. The absorp- tion spectra of both Cadmium Yellow Light and the squid yellow chromatophore have a rolloff toward the red begin- ning at ϳ500 nm, although it is much steeper for the in- organic pigment. The S͑͒ spectra are much different for these two, however, rising nearly monotonically toward the blue in Fig. 3(b), but low in the blue end of the spec- trum and rising sharply at ϳ500 nm for Cadmium Yellow Light. Again, this pigment was evidently designed as a high yellow reflector (ϳ40% for an 11 m film [20], whereas typically RϽ5% for the yellow chromatophore of L. pealeii [6]). Finally, assuming dϳ1 m as a typical thickness for expanded chromatophores [17], the maxi- mum absorption coefficient would be approximately 3 m−1, which is as great as an order of magnitude larger than K for the inorganic pigments. However, the back- scattering coefficients would be in the range of ϳ0.05–0.08 m−1 in the region of minimum absorption, which is generally smaller than the maximum S of the in- organic pigments (with the exception of Acra Red, a low- Fig. 4. (Color online) Examples of model fits to specific chro- reflectance pigment) [6,20]. We conclude that the chro- matophore reflectance measurements of the squid Loligo pealeii matophores function more effectively as color absorptive obtained by using K and S values in Fig. 3 and adjusting the filters than colored reflectors. The biological significance scaling parameter . of this is possibly that, in some patterns, chromatophores filter light reflected from structures beneath them to fine- Likewise, both pigments exhibit a rise in backscattering tune a particular body pattern. coefficient at about the same wavelength. However, for Since the absorption of the brown pigment is large, it ͑͒ Red Iron Oxide S increases by about an order of mag- strongly filters light transmitted to and reflected from nitude between 550 and 600 nm. Clearly, this commercial other elements beneath it. Hence, regardless of the num- ϳ pigment is designed to yield high reflectance ( 30% for a ber of elements in the skin structure, the brown chro- 26 m thick film [20]), whereas the cephalopod brown matophore largely determines the reflected spectrum Ͻ chromatophore reflectance is very low, typically 2% [6]. when it is present (i.e., the skin will appear brown). Light Acra Red and the squid red chromatophore both exhibit a is only partially filtered by the red and yellow chromato- ϳ maximum absorption in the visible at 500 nm. They phores. Consequently, components beneath them more ͑͒ ϳ both also display a relative minimum in S at 550 nm prominently affect the overall reflection spectrum. followed by a relative maximum at ϳ600 nm. However, S͑͒ changes by about a factor of 5 in this spectral range for Acra Red compared to the approximate 10% change B. Iridophores Reflection spectra were collected for the red dorsal iri- dophore of the squid L. pealeii. The iridophores in this area of the mantle are parallel to the skin [12]. The angle of incidence in air was approximately 10°. A comparison of iridophore model calculations to these experimental data is given in Fig. 6. The angle of incidence in water is taken to be 7.5°. For these calculations we have selected N=20, a typical number of platelets in the iridophores of squid. We also let hL =75 nm and hH =140 nm, which are within the ex- perimental error of red iridophore spacer and platelet measurements performed by transmission electron mi- croscopy on the squid Lolliguncula brevis [23]. Other pa- rameter selections were f0 =0.4, =2.4 nm, and 0 =7.6 nm. We see that Ӷ and that the standard devia- tion 0 of spacer thickness over the ensemble of iri- dophores is about 10% of the mean. The theoretical plots in Fig. 6 illustrate the effects of the random variables. The ideal stack yields a peak that is too large, having several sidelobes and a general shape that is inconsistent with the data. Introducing a random Fig. 5. (Color online) Examples of model fits to specific chro- spacing over the ensemble of iridophores washes out the matophore transmittance measurements of the squid Loligo pea- leii obtained by using K and S values in Fig. 3 and adjusting the sidelobes and lowers the peak reflectance. Further ran- scaling parameter . These data are not from the same samples domizing the layer spacings of each iridophore has a mini- given in Fig. 4. mal effect on the peak and width of the reflection band Sutherland et al. Vol. 25, No. 3/March 2008/J. Opt. Soc. Am. A 595
Fig. 6. (Color online) Fit of iridophore model to reflectance data Fig. 7. (Color online) Theoretical chromaticity coordinates for from the red dorsal iridophore of the squid Loligo pealeii. The the iridophore of Fig. 6 for various values of the statistical pa- dotted curve gives the calculated spectrum of an ideal thin-film rameters 0 and . Also shown is the CIE 1931 chromaticity dia- stack (Bragg reflector), while the dashed curve illustrates the ef- gram. CIE standard daylight illuminant D65 was used for these fect of averaging over an ensemble of Bragg reflectors with ran- calculations. Circles represent various 0 values, ranging from domly variable mean spacer thickness. The solid curve shows the 0 to 12 nm, for a constant =2.4 nm. Squares represent various effects of additionally averaging over individual random spacing values, ranging from 0 to 4.8 nm, for a constant =7.6 nm. of each spacer layer and provides a good fit to the data (circles). 0 ⌬ The inset shows the dependence of bandwidth on 0 and the ratio of out-of-band to in-band integrated reflectance on . for squid near the surface of clear water. For deeper wa- ter, the illuminant spectrum would need to be modified based on the spectral attenuation coefficients of the water but raises the baseline. The net result is in excellent [27]. agreement with the data (circles). A partially active dorsal iridophore was treated with The statistical parameters and have a direct im- 0 acetylcholine in vitro, and the change in the reflection pact on coloration effects of the iridophores. As the inset spectrum was observed over time. These data are shown to Fig. 6 shows, the bandwidth ⌬ of the calculated reflec- in Fig. 8. Over a period of 60 s the peak reflection doubles tance, defined as the full width of the reflection notch at and shifts ϳ50 nm to the blue. We model this using Eqs. half-maximum reflectance (FWHM), increases as 0 in- creases for constant . For reference, ⌬=50.8 nm for the ideal Bragg reflector. The random platelet spacing has a somewhat different effect. Since increasing causes the baseline reflectance to rise while having a very minor ef- fect on the peak, we consider the integrated spectral re- flectance in band and out of band. The former is obtained by integrating Rcc͑͒ over ⌬ about the peak reflectance, while the latter is given by the integration over the re- mainder of the spectrum, excluding ⌬. The inset to Fig. 6 shows how the ratio of the out-of-band to in-band inte- grated reflectance varies with for constant 0. For ref- erence, this ratio is 21.2% for the ideal Bragg reflector. We also note that these statistical parameters have relatively minor polarization effects, and these can only be observed at high angles of incidence. To assess the coloration effects, we plot the chromatic- ity coordinates for these cases of variable 0 and in Fig. 7. For these calculations we have used the CIE 1931 color matching functions and D65 standard daylight illumi- nant. Variation of 0 results in a color perception shift from orange-pink to orange, while that for manifests a color change from reddish orange to pink. The pink color Fig. 8. (Color online) Theoretical reflectance plots and experi- mental data for an active iridophore treated with acetylcholine will get fainter as continues to increase, and we antici- (ACh). The points show the measured spectra at various times pate that the iridophore will take on a silvery appearance after application of ACh. The curves are fits to the data obtained for large . We emphasize that these coloration effects from a dynamic iridophore model by adjusting the volume frac- correspond to what a standard human observer would see tion f of water in the platelet regions of the iridophore. 596 J. Opt. Soc. Am. A/Vol. 25, No. 3/March 2008 Sutherland et al.
(17) and (18) in the following way. First, we determine pa- chromaticity diagram of Fig. 10 illustrates the type of dy- rameters for the case of optimum reflection (i.e., at 60 s). namic color changes that squid can use for camouflage For this we set h0 =139.6 nm, f0 =0.28, =2 nm, and 0 and communication. As the chromatophore expands over =11 nm to develop a good fit to the data. As before, hL the iridophore (given by the circles in Fig. 10 for increas- =75 nm, inc=7.5°, and N=20. Notice that the volume ing , assuming that the chromatophore always fills the fraction of water is smaller and the standard deviation 0 field of view), the skin color gradually changes from pink is larger than those for the fit in Fig. 6. The reason for ͑=0.5͒ to purplish pink ͑=1͒. This is also in agreement this is that spectra in Fig. 8 are somewhat broader. Next with color photos of this chromatophore-iridophore combi- we selected a value of dh/df=69.1 nm. This is consistent nation [6]. As the chromatophore continues to expand, the with a cutoff of iridophore reflectance at ϳ700 nm, i.e., color would shift to purplish blue and eventually to blue the peak wavelength of the iridophore reflection spectrum for =2 or 3. These changes may occur within a fraction of just before it is extinguished [6]. Finally, we adjusted the a second. Note that the color for =3 is virtually indistin- volume fraction f, as given in Fig. 8, to fit the calculated guishable from that of a bare iridophore. However, the spectra to the data. The results are in good agreement overlaid chromatophore will produce a diffuse component with the measured spectra. There does appear to be some of the blue reflection, due to scattering in the chromato- asymmetric broadening of the spectral reflection data to- phore, that is relatively insensitive to viewing angle, un- ward the red as the peak decreases, not accounted for in like the bare iridophore, which produces a purely specular the model, which could possibly indicate the development reflection. A physiological change of the iridophore to a of a quadratic chirp in the grating [28]. Since Eqs. (17) noniridescent state, which occurs over several seconds, and (18) are in simultaneous agreement with all of the would produce the color change given by the squares in data sets using reasonable fitting parameters, we con- Fig. 10, with the parameter f increasing in equal incre- clude that, to a good approximation, the refractive index ments from 0.3 to 0.9 for =1.8. In this case, the skin color and the thickness of the platelets depend linearly on the passes through various areas (blue, green, yellow, orange, volume fraction of water (or waterlike fluid) present in pink) of the chromaticity diagram. The case of f=0.9 is the platelet regions. virtually indistinguishable from that of a bare chromato- phore. Combinations of other iridophores, some situated C. Chromatophore Over Iridophore at large angles with respect to the skin, and chromato- We give now an example of a combination spectrum by phores lead to a rich repertoire of intensity, color, and overlaying a red chromatophore on a blue iridophore. The body patterns available to the animal for adaptation to results are shown in Fig. 9. For these calculations we se- different backgrounds and situations. lected =1 and =0.7 for the red chromatophore. This value of the forward scattering ratio is typical for diffuse 5. CONCLUSIONS reflectors [19,21]. For the iridophore hH =90 nm, hL In summary, we have presented coloration models appli- =71 nm, f0 =0.3, N=20, =2.4 nm, and 0 =7.6 nm. The cable to the pigment-bearing chromatophores and irides- total reflection spectrum (specular plus diffuse) is in good cent iridophores of squid skin. Dynamic effects have been qualitative agreement with optical measurements of this type of chromatophore–iridophore combination [6]. The
Fig. 10. (Color online) Theoretical chromaticity coordinates for a blue iridophore overlaid with a red chromatophore illustrating color changes for various values of and f. Also shown is the CIE Fig. 9. (Color online) Calculated reflection spectra for a red 1931 chromaticity diagram. CIE standard daylight illuminant chromatophore, a blue iridophore, and the combination of a red D65 was used for these calculations. Circles represent various chromatophore over a blue iridophore. The chromatophore and values, ranging from 0.5 to 3, for a constant f=0.3. The leftmost total reflectance spectra have been multiplied by a factor of 5 for circle represents a bare iridophore. Squares represent various f ease of viewing. values, ranging from 0.3 to 0.9, for a constant =1.8. Sutherland et al. Vol. 25, No. 3/March 2008/J. Opt. Soc. Am. A 597
included. Calculated spectra agree very well with in vitro (1b) and differentiate the resulting equation. After com- optical measurements of squid skin components, and com- bining the result with Eqs. (1b) and (1c), a simple second- bination reflection spectra, for example, a chromatophore order differential equation is obtained: overlaid on an iridophore, also agree qualitatively with Љ 22  ͑ ͒ experimental data. Chromatophores yield low diffuse re- Id − Id =− IIc, A2 flectance due to small backscattering coefficients, but where their strong absorption allows fine-tuning of spectral re-  ͕͑␣ ͒ ͓␣ ͑ ͔͖͒ ͑ ͒ flections from skin components beneath them. The inten- I = + + + 1− . A3 sity of these reflections can change in fractions of a second Performing a similar set of operations on Eq. (1c),weob- by expansion or retraction of the chromatophore, which tain the second-order differential equation for J : can be modeled by the adjustment of a single scaling pa- d rameter ͑͒. The color and intensity of iridophores can be Љ 22  ͑ ͒ Jd − Jd =− JIc, A4 changed reversibly by a mechanism that seemingly allows invasion of the platelet regions with a waterlike fluid. with These changes depend simply on the volume fraction ͑f͒ of  = ͑ −1͒͑1−͒͑␣ + ͒. ͑A5͒ this fluid. Cephalopods exert direct control over these pa- J rameters. Varying just these two control parameters ( Equations (A2) and (A4) have the general solutions and f), we have shown that skin color for a  I chromatophore–iridophore combination can be changed I c I = A e z + A e− z + , ͑A6͒ dramatically in a time scale from a fraction of a second to d 1 2 ͑͒2 − ͑␣ + ͒2 several seconds. A future paper will combine these prop- erties of squid with models and data from cuttlefish for and other skin components to obtain a full-skin coloration  JIc model for a generic cephalopod. This study provides the z −z ͑ ͒ Jd = B1e + B2e + 2 2 , A7 physical basis for an improved understanding of the be- ͑͒ − ͑␣ + ͒ havioral ecology of these animals. respectively. Integration constants A1, A2, B1, and B2 can be found by applying the boundary conditions Id͑0͒=Id,0, APPENDIX A ͑ ͒ Ј͑ ͒ Ј͑ ͒ Jd d =0, and the expressions for Id 0 and Jd 0 resulting In this appendix we present the solutions to Eqs. from Eqs. (1b) and (1c) to Eqs. (A6) and (A7). After some (1a)–(1c), yielding expressions for the partial transmit- long and tedious but straightforward algebra, we find an tances and reflectances of the chromatophores. Equation expression for Jd,0=Jd͑0͒ and form the ratio (1a) can be integrated immediately, giving ͑ ͒ ͓ ͑␣ ͒ ͔ ͑ ͒ Jd,0 Ic z = Ic,0 exp − + z . A1 ϵ͑ ͒ ͑ ͒ Rd = 1−q Rdd + qRcd, A8 Ic,0 + Id,0 Setting z=d in Eq. (A1) and forming the ratio Ic͑d͒/Ic͑0͒, we obtain Eq. (2) for Tcc. Now substitute Eq. (A1) into Eq. where q=Ic,0/͑Ic,0+Id,0͒. Here Rdd is given by Eq. (4), and
b1 cosh d − b2 sinh d R = , ͑A9͒ cd ͓͑͒2 − ͑␣ + ͒2͔͕ cosh d + ͓␣ + ͑1−͔͒sinh d͖
b =  ͕ − ͓ + ␣ + ͑1−͔͒e−͑␣+−͒d͖, ͑A10͒ I ͑d͒ 1 J d ϵ͑ ͒ ͑ ͒ Td = 1−q Tdd + qTcd, A12 Ic,0 + Id,0 b = ͕ ͓␣ + − ͑ + ␣ + ͑1−͒͒e−͑␣+−͒d͔ 2 J where Tdd is given by Eq. (3), and ͑ ͓͒͑͒2 ͑␣ ͒2͔͖ ͑ ͒ 2 − 1− − + . A11 Tcd = a1 sinh d − a2 cosh d + a3 sinh d  −͑␣+͒d →  → Ie In the limit 1, J 0 and Eq. (A9) reduces to Eq. (7). + , ͑A13͒ Similarly, forming the ratio ͑͒2 − ͑␣ + ͒2
͓͑͒2 ͑␣ ͒2͔  ͑␣ ͒  ͑ ͒ − + + I + J 1− a = + 1 ͓͑͒2 − ͑␣ + ͒2͔ ͓͑͒2 − ͑␣ + ͒2͔
͕␣ + − e−͑␣+−͒d + ͓␣ + ͑1−͔͒͑1−e−͑␣+−͒d͖͒sinh d ϫ ͭ1−e−͑␣+−͒d − ͫ ͬͮ, ͑A14͒ cosh d + ͓␣ + ͑1−͔͒sinh d 598 J. Opt. Soc. Am. A/Vol. 25, No. 3/March 2008 Sutherland et al.
 I Hence, many more terms in the expansion would have to a = , ͑A15͒ be included. It is thus simpler to integrate over numeri- 2 ͑͒2 − ͑␣ + ͒2 0 cally. The final expression for Tcc is then given by
2 2 ͗ ͘ ͑*͗⌬͒͘ ͑ ͒ ͑1−͒ Tcc = Tmf = T0 +2Re 0 , B5 a = . ͑A16͒ 3 ͓␣ ͑ ͔͒ cosh d + + 1− sinh d ϱ T = ͵ T͑ ͒p͑ ͒d , ͑B6͒ In the limit →1, Eq. (A13) reduces to Eq. (6). 0 0 0 0 −ϱ
͖͒ ͑ ͕⌬2ץ N ϱ APPENDIX B 1 0, k *͗⌬͘ = ͵ *͑ ͒ͯ ͯ 2p͑ ͒d , ͚ 2 0 0ץ In Eq. (20) we gave an expression for the collimated– 0 0 0 2 k=1 −ϱ k ͕ ͖ collimated transmittance of an iridophore in terms of an k =0 integral over N+1 random variables. Here we present an ͑B7͒ approximation that reduces Eq. (20) to an integration with over a single random variable. Let =1/P11 be the trans- mission coefficient, and =͑ ,͕ ͖͒ is a function of the 2 0 k 1 0 N+1 random variables. Then if we let ⌬͑ ,͕ ͖͒ p͑ ͒ = expͩ− ͪ. ͑B8͒ 0 k 0 ͱ2 22 =͑0 ,͕k͖͒−0, where 0 =͑0 ,͕k͖=0͒, we can write the 0 0 multilayer-film transmittance as T ͑ ,͕ ͖͒ = T +2Re͑*⌬ + ⌬*⌬͒, ͑B1͒ mf 0 k 0 0 ACKNOWLEDGMENTS * where T0 = 0 0 (for ns =n0) and the asterisk indicates com- We gratefully acknowledge DARPA and the Air Force Re- plex conjugate; T0 is the multilayer-film transmittance of search Laboratory for their support of this work. We also an ideal stack, where the low-index layer has a thickness thank Phil McFadden, Rajesh Naik, and Timothy Bun- hL +0 and is computed using Eqs. (13)–(18). ning for stimulating discussions. We now expand ⌬ in a Taylor series,
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