Full Wave Modeling of Light Propagation and Reﬂection · Full Wave Modeling of Light Propagation and Reﬂection A. Musbach1, G. W. Meyer1, F. Reitich2, ... computer graphic ray

Volume xx (200y), Number z, pp. 1–14

Full Wave Modeling of Light Propagation and Reflection

A. Musbach1, G. W. Meyer1, F. Reitich2, and S. H. Oh3

1Computer Science & Engineering, 2Mathematics, and 3Electrical & Computer Engineering DepartmentsUniversity of Minnesota, Minneapolis, MN USA

[email protected], [email protected], [email protected], [email protected]

AbstractThe propagation and reflection of electromagnetic waves in a three-dimensional environment is simulated, and re-alistic images are produced using the resulting light distributions and reflectance functions. A finite difference timedomain method is employed to advance the electric and magnetic fields in a scene. Surfaces containing wavelengthscaled structures are created, the interaction of the electromagnetic waves with these nano-structured materials iscalculated, and the sub-surface interference and diffraction effects are modeled. The result is a reflectance functionwith wavelength composition and spatial distribution properties that could not have been predicted using classiccomputer graphic ray tracing approaches. The techniques are employed to reproduce demonstrations of simpleinterference and diffraction effects, and to create computer-generated pictures of a Morpho butterfly.

Categories and Subject Descriptors (according to ACM CCS): I.3.7 [Computer Graphics]: Three-DimensionalGraphics and Realism—Color, shading, shadowing, and textureKeywords: rendering, FDTD, shader, diffraction, interference, structural color, iridescence, thin film, Newton’scolors, optics, electromagnetic wave

1. Introduction

Interference and diffraction are physical optics effects thatcan occur simultaneously when light interacts with a mate-rial. Even though this is a well known fact, the few timesthat these two phenomena have been studied in computergraphics they have usually been treated as separate subjectsand have rarely been incorporated in the same simulation. Inaddition, most light reflection simulation in computer graph-ics has been limited to the geometric optics that can be ren-dered using traditional polygon pipeline and ray tracing al-gorithms. The general wave based nature of light is almostnever accommodated in contemporary realistic image syn-thesis programs.

In this paper, we introduce the finite difference time do-main (FDTD) method for computing the propagation ofelectromagnetic energy in a three-dimensional environment.FDTD is a numerical technique that is used to solve the setof coupled partial differential equations, called Maxwell’sequations, that describe electromagnetism. While the FDTDmethod is widely used in the scientific and engineering com-munities to study the behavior of light and to design newphotonic devices, FDTD provides a new and important ap-

proach to illumination modeling for the field of computergraphics.

The paper also presents a set of examples to illustratehow the results of a FDTD simulation can be used to makea computer graphic image. First the FDTD method is usedto generate pictures that duplicate demonstrations of simplediffraction and interference effects. Next the reflection frommore complicated structures with features on the order ofthe wavelength of light are investigated, and the interactionof diffraction and interference, as it relates to both the colorand the spatial distribution of the reflected light, is examined.Finally, we duplicate an existing Morpho butterfly reflectionsimulation, we improve the algorithm that was originally uti-lized, and we go one step further by generating a realisticimage with our results. The paper ends with a comparisonbetween our Morpho butterfly simulation and the best exist-ing computer graphic results to date.

2. Previous Work

Attempts to simulate physical optics effects in computergraphics have primarily been limited to the modeling, asseparate phenomena, of interference and diffraction effects

submitted to COMPUTER GRAPHICS Forum (12/2012).

2 A. Musbach & G. W. Meyer / Full Wave Modeling of Light Propagation and Reflection

using analytic expressions. Smits [SM90], Dias [Dia91],and Sun [SW08] used formulas for thin film reflectance tomake images of soap bubbles and oil slicks that includeNewton’s colors. Pearlescent paint effects that result frommica chips coated with titanium dioxide have been mod-eled by Gondek [JGN94] and Ershov [SEM01], and mul-tilayer interference in pearls has been addressed by Na-gata [NDM∗97]. Multilayer thin film models have been em-ployed by Smits [SM90], Sun [Sun06], and Sadeghi [Sad08]to simulate the variation of color with reflectance angle forthe Morpho butterfly, but the spatial spread of the reflectedlight was modeled in each case using the specular exponentin the Phong reflectance model. Stam [Sta99] was the firstperson to simulate diffraction effects, and he was followedby Agu [AH∗02] and Sun [SFDC00] who also used analyticexpressions to make renderings of light reflection from com-pact disks.

A few computer graphics researchers have built renderingsystems that keep track of wavefront information through-out a scene and use this information to perform physicaloptics calculations. Moravec [Mor81] was the first personto advance light waves in a scene using a numerical tech-nique that has some similarities to FDTD. His results werecrude and his pictures contained significant artifacts. Spher-ical and planar wavefronts were propagated in an environ-ment by Ziegler [ZCG08]. The wavefronts in this work wereprimarily used to determine occlusion, but they were alsoemployed to simulate depth-of-field, diffraction, and glareeffects.

There have also been efforts to extend classic raytracing methods to include wave-based phenomena.Gondek [JGN94] kept track of phase differences betweenrays cast in a virtual goniospectrophotometer and used thisinformation to calculate interference effects. Recent work todetermine physical optics effects as part of ray tracing hascentered on incorporating the Wigner Distribution Function(WDF) into the calculation. Zhang [ZL09] introduced theWDF and it was incorporated into the rendering process byOh [OBR09], [OKG∗10] and Cuypers [COH∗12]. Cuypersused the WDF to produce compelling pictures of diffractionand interference effects, but the utility of the WDF is limitedby its inability to easily model reflection from nonzerothickness surfaces and at incidence angles other than normalincidence.

There have been several attempts within the scientificcommunity to use FDTD techniques to model light reflec-tion from the Morpho butterfly, but most of these effortshave been two-dimensional and none have used the resultsto generate a computer graphic picture. Plattner [Pla04] wasthe first to use FDTD to demonstrate how the nanostructuredwing scale features produce the butterfly’s characteristiclight reflection patterns. Banerjee [SB07] and Zhu [DZC09]performed more systematic studies and developed an expla-nation for how each part of the wing scale architecture con-

tributes to the structural color. Finally, Lee [Lee09] extendedthese investigations into three dimensions and modeled ad-ditional butterfly species.

3. The FDTD Algorithm

The FDTD method is a time-stepping algorithm basedon Maxwell’s equations, which describe electromagnetism.Maxwell’s equations can be written in many forms, but thedifferential time-domain Maxwell’s equations needed for theFDTD method are

∇× ~H =∂~D∂t

+ ~J (1)

∇×~E =−∂~B∂t− ~M (2)

∇·~D = ρe (3)

∇·~B = ρm (4)

where ~E is the electric field strength vector, ~D is the electricdisplacement vector, ~H is the magnetic field strength vector,~B is the magnetic flux density vector, ~J is the electric currentdensity vector, ~M is the magnetic current density vector, ρeis the electric charge density vector, and ρm is the magneticcharge density vector Elsherbini [ED09].

Components of ~E and ~H are stored in the spatial arrange-ment shown in Figure 1. The components of ~E are computedat the times when one time step ends and another begins,but the components of ~H are computed at the times halfwaythrough each time step. The notation ~En(i, j,k) refers to ~E atthe time n∆t, and the notation ~Hn+1/2(i, j,k) refers to ~H atthe time (n+1/2)∆t.

A structure is modeled by associating material propertiesto cells, namely relative permittivity ε, relative permeabil-ity µ, electrical conductivity σ

e, and magnetic conductivityσ

m. For example, a dielectric (such as water) with index ofrefraction n can be modeled by finding the cells that it oc-cupies and setting the material properties in those cells toε = n2,µ = 1,σe = 0,σm = 0. Free space has the materialproperties ε = 1,µ = 1,σe = 0,σm = 0. The material prop-erties ε(i, j,k),µ(i, j,k),σe(i, j,k),σm(i, j,k) at the positionof an electric field component ~E(i, j,k) are computed by av-eraging the material properties over the four cells meetingthere. The same material properties at the position of a mag-netic field component are computed by averaging the ma-terial properties over the two cells meeting there. For thederivation of the field updating equations, and other cover-age of the FDTD method, consult Taflove [TH05].

To avoid waves reflecting from the problem space bound-aries, absorbing boundary conditions (ABCs) are neededto absorb waves as they go away from the problem space.One type of ABC, a Perfectly Matched Layer (PML),is an absorbent medium to line the boundary so that bythe time waves have passed through this medium to the


A. Musbach & G. W. Meyer / Full Wave Modeling of Light Propagation and Reflection 3

Figure 1: A cell of an FDTD problem space.

boundary and back, hopefully they have become negligiblysmall. The absorbing boundary condition employed in thiswork is a PML known as the Complex, frequency shifted,Perfectly Matched Layer (CPML), originally proposed byChew and Weedon [CW94] and independently by Rappa-port [Rap95]. This work uses the extended version proposedby Kuzuoglu [KM96].

The Morpho butterfly at the end of this paper is basedon Lee [Lee09], but in this work, the source is imple-mented in a different way for slightly better accuracy.Whereas Lee [Lee09] uses the total field/scattered fieldformulation (TF/SF formulation), originally proposed byMerewether [MFS80], we use the scattered field formu-lation, originally proposed in Holland [Hol77] and Hol-land [HSK80]. The advantage of the scattered field formu-lation is that the incident field is exact and thus travels at ex-actly the correct speed, unlike an excitation that numericallypropagates through the problem space and thus suffers fromnumerical dispersion (resulting speed is not quite correct).The primary disadvantage of the scattered field formulationis that it requires significant computing time to calculate theincident field throughout the problem space at every timestep, although there are faster ways to do it than by just us-ing the explicit formula directly.

4. Near To Far Field Transform

The NTFF (Near To Far Field) transform uses surface inte-grals over a closed surface S in the problem space to com-pute information about scattering far away from the problemspace. S must enclose all of the objects, so that there is noth-ing but free space outside of S. Usually S is a box, but in thiswork it is a flat surface above the structure, because the goalis to analyze surface reflection.

The far field values |rEθ|, |rEφ| computed with the NTFFtransform (see [TH05] to learn how) are used to compute

Figure 2: The problem space for the periodic boundary con-dition.

the reflectance factor R, which is defined as the “ratio of theflux reflected from the specimen to the flux reflected fromthe perfect reflecting diffuser under the same geometric andspectral conditions of measurements.” Specifically,

R =2(|rEθ|2 + |rEφ|2)|~Einc|2(kinc · n)A

,

where ~Einc is the incident electric field, kinc is the directionof incidence, n is normal to the surface, and A is the surfacearea. The BRDF is then computed as ρ = R/π so that thepower reflection coefficient of a perfect reflecting diffuser(which has reflectance factor 1 at all viewing angles) is∫ 2π

0

∫ π/2

0ρcos(θ)sin(θ)dθdφ =∫ 2π

0

∫ π/2

0

1π

cos(θ)sin(θ)dθdφ = 1.

5. Periodic Boundary Conditions

The Periodic Boundary Condition (PBC) originally pro-posed by Lee [Lee09] is for modeling a doubly periodicstructure. The problem space consists of a 2×2 grid of tiles,as shown in Figure 2. There is CPML in the front and in theback, not shown in the figure. All three components of thedirection of incidence kinc are nonnegative; the analyst whodesires a direction of incidence with a negative componentis expected to reformulate the geometry (rotate or reflect it)so that the desired direction of incidence has all three com-ponents nonnegative.

Because both the geometry and the excitation are periodic,

~E(~r, t) = ~E(~r−mypy−nzpz, t−mτdy−nτdz),

~H(~r, t) = ~H(~r−mypy−nzpz, t−mτdy−nτdz),

where τdy = (kinc · y)yp/c,τdz = (kinc · z)zp/c. To handle theright boundary, the values of Ex and Ez at y = yp are bufferedfor future use as values of Ex and Ez at y = 2yp; that iscalled the right buffer. Likewise, to handle the top bound-ary, the values of Ex and Ey at z = zp are buffered for fu-



ture use as values of Ex and Ey at z = 2zp; that is called thetop buffer. The left boundary and the bottom boundary arehandled simply by accepting that they contain meaninglessvalues that propagate away from them at the speed of light.When the meaningless values are about to enter tile A, ac-curate data for the whole problem space at a past time isobtained from past (and present) values in tile A, and thesimulation proceeds from that past time. That is called thecombining procedure. In this way, the data in tile A is keptaccurate, so it can be used to obtain simulation results. Thisperiodic boundary condition works for some angles of inci-dence but not others, because depending on the angle of in-cidence, the error from the left and bottom boundaries mayenter tile A before the combining procedure can be prepared.Refer to Lee [Lee09] for a suggestion of how to overcomethis limitation in future work.

Numerical dispersion makes it possible for the errorfrom the left and bottom boundaries to reach tile A morequickly than expected. Lee [Lee09] addresses this problemby adding padding to the left and bottom boundaries (notappearing in Figure 2). In this work, for angles of inci-dence within 40◦ of normal incidence, as an alternative tothe padding, the combining procedure is performed 20% ear-lier. The advantages of this approach are:

1. The simulation can run for an arbitrarily long time; nomatter how many times the combining procedure is per-formed, tile A will never be corrupted by the left and bot-tom boundaries.

2. It avoids using more memory to accomodate the padding,and spending more time to update data associated withthe padding.

The disadvantage is that it makes the periodic boundary con-dition less efficient, and it makes the angle of incidence morerestricted. For angles of incidence more than 40◦ away fromnormal incidence, 20 cells of padding is used instead.

6. Simple Physical Optics Effects

The basic FDTD algorithm defined thus far can be used toreproduce two simple physics demonstrations that are oftenemployed to illustrate the wave nature of light. In this sec-tion we compute the fringes that occur at the boundary of ashadow due to diffraction and we simulate the colorful New-ton’s rings that are the result of interference in a soap bubble.In each case, the problem space is a long, narrow channel,effectively reducing the problem from three to two dimen-sions and making it computationally tractable. Diffraction ata shadow boundary is demonstrated with a knife edge oc-cupying 1µm of a channel 5µm long, and Newton’s ringsare demonstrated with a thin water film of gradually vary-ing thickness in a channel 46µm long. Images of these twochannels are transformed to produce a picture of the shadowof a washer and a picture of a bubble wand, respectively.

To create the image of each channel, a reference plane

Figure 3: Rendered image of the shadow of a washer undera D65 illuminant.

is positioned either below the knife edge or above the wa-ter film to record the light energy that passes this area. Theenergy at the surface of the plane is found by first comput-ing the Poynting vector as the cross product of the electricfield and the complex conjugate of the magnetic field. Thenthe Poynting vector for unpolarized light is determined bytaking the average of two Poynting vectors for orthogonalpolarizations. To determine the final energy value, we takehalf the real part of the normal component of the Poyntingvector for unpolarized light.

For the shadow, we compute the Poynting vector fromthe total field, but for Newton’s rings, we use the scatteredfield because what we are interested to see is the reflectionfrom the soap bubble and not the light coming toward thesoap bubble. In both cases, we divide by the Poynting vec-tor computed from the incident field, to discount the spectraldistribution of the incoming light in the FDTD simulation; adifferent spectral distribution can be added in after the fact.

To model diffraction at a boundary of a shadow, roughly

Figure 4: Analytical result for relative positions of rings inshadow of washer under monochromatic light of the wave-length 543nm.



Figure 5: Rendered image of a bubble wand with a D65 il-luminant.

one fifth of the channel was blocked off by a knife edge mod-eled as a perfect electric conductor. The results obtained fora small portion of a shadow boundary are used to show whatthe shadow boundary would look like for a small washer un-der a D65 illuminant (Figure 3). This picture is producedby mapping distance from the washer to distance along thechannel.

The reader may notice that as one moves away from thewasher, the fringes quickly become almost evenly spaced.That is because to make the simulation computationally fea-sible, the film plane is only 1µm away from the knife edge.See Figure 4 for an analytical result of what the spacing be-tween the rings should look like at the wavelength 543nm ifthe film plane is 1µm away. The figure also shows an analyt-ical result of what the spacing between the rings would looklike at the wavelength 543nm if the film plane were 12 feetaway (not computationally feasible with the FDTD method).These analytical results are obtained by plotting the nth ringat x =

√n2λ2−d2, where d is the distance to the film plane.

Newton’s rings are illustrated by modeling a water filmwith thickness gradually varying linearly from 1µm at theleft end of the channel to 100nm at the right end of the chan-nel. The FDTD result is put to artistic use in Figure 5.

7. Complex Physical Optics Effects

With the addition to the FDTD algorithm of periodic bound-ary conditions and the near to far field transform, we cannow consider light reflection from surfaces that include com-plex interference and diffraction effects. Periodic boundaryconditions make it possible to consider periodic surfaces ofeffectively infinite extent. The near to far field transform al-lows light to be measured at a distance from the surfacewhere the fields are stable and practical observations aremade. In this section we use these techniques to analyze atropical butterfly.

Structural color is found in many butterflies, beetles,

Figure 6: Anatomy of the nanostructure in the scales ofa Morpho butterfly, including ridges (R), lamellae (l), mi-croribs (mr), crossribs (cr), the bottom layer (bl), and pillars(p).

birds, and fish. We focus on a tropical butterfly of the Mor-pho genus. It has a metallic blue appearance when viewedfrom above. As the viewing angle shifts toward grazingwhile remaining perpendicular to the wing veins, the colorremains about the same at first, and then suddenly be-comes purple. As the viewing angle shifts toward grazingwhile remaining parallel to the wing veins, the color quicklybecomes black. When submerged in alcohol, the butterflyappears green from above and blue when viewed from agrazing angle. When submerged in toluene (which has amuch higher refractive index), it just looks dark brown. Ki-noshita [KYM08]

Figure 6 (from Potyrailo [PGV∗07]) shows the anatomyof the nanostructure in the scales of Morpho butterflies. TheMorpho butterflies derive their appearance mainly from mul-tilayer thin film interference. The reflection lobe peaks atan angle off specular because the lamellae are at an angle.Anisotropy results from the long and narrow shape of thelamellae. Wide-angle diffuse reflection is present becauserandom variation in the heights of the ridges prevents themfrom interfering with each other. The reflectivity is high be-cause there are a large number of lamellae having a largerefractive index, and because the ridges are close together.The color is also enhanced by an underlying pigment, notmodeled in this work. Kinoshita [KYM08]

7.1. Modeling the Butterfly in Two Dimensions

As a starting point, consider a simple multilayer thin filmresembling the Morpho butterfly. The multilayer thin filmexample is composed of eight layers of dielectric materialalternating with air. The air gaps are 140nm thick, and thedielectric material has dielectric constant 2.43 and is 60nmthick. The power reflection coefficient can be computed fromnear field data in a tiny problem space, so a 30nm by 30nmproblem space is used, and the near field measurementsmade from the FDTD simulation (see Figure 7) show that



the results match calculations performed using multilayerthin film theory Hecht [HZ74]. Then a long, narrow chan-nel 64µnm by 10nm is simulated and the NTFF transformis performed using a surface above the multilayer thin film;the result is shown in Figure 8. Along the long, thin channel,the problem space is long enough to obtain accurate far-fieldmeasurements which show that the reflection has no spatialspread. The spatial spread along the direction perpendicu-lar to the long, thin channel is an artifact arising becausethat tiny distance across the problem space is certainly notenough to get accurate results from the NTFF transform.

Next we cut the multilayer thin film into identical stripsthat are similar to the ridges found in the Morpho butterfly,but without the randomness that would be found in an actualbiological structure. Figure 9 illustrates the result of usingthe long narrow channel to analyze three systems that havethe same multilayer structure but a different period betweenthe strips. A simple diffraction grating will cause the re-flected light to spread out spatially in a well defined pattern.Depending on the period a of the diffraction grating, eachwavelength λ of light will reflect in a specific direction θ ac-cording to the relation asin(θ) = mλ where m is an integerthat denotes the order of the reflection. For diffraction grat-ings with a = 800nm, a = 1200nm, and a = 1600nm, theorder 1 reflection for a wavelength in the middle of the spec-trum, λ = 550nm, occurs at 43◦, 27◦, and 20◦ respectively.Figure 9 shows the results of FDTD simulations involvingdiffraction gratings with these three periods and grating el-ements that consist of eight layers of dielectric material al-ternating with air. For this regular structure, the color of thereflected light remains consistent and is accurately predictedby the above simple thin film theory. The spatial spread ofthe light, while tightly focussed into discrete directions, fol-lows the predictions of simple diffraction theory.

We now generalize the structure further to add the ran-domness that is found in a real insect like the Morpho but-

Figure 7: The FDTD method was used to compute the powerreflection coefficient of a multilayer thin film similar to aMorpho butterfly. The results are consistent with the analyticformula for multilayer thin film interference.

Figure 8: Two views of the result of applying the NTFFtransform to the simple multilayer thin film. The blue solidlines mark the boundary of the problem space, and the reddashed lines mark the boundary of the CPML. The scale ofthe reflectance lobe relative to that of the problem space isarbitrary, but the orientation of the reflectance lobe relativeto that of the problem space is meaningful.

terfly, and to show how variations in the configuration canbe used to control the spatial and spectral composition of thereflected light. To investigate the effect of the dimensionsof the structure, twelve experiments are performed with a60µm channel full of pieces of multilayer thin film. Resultsare obtained for three multi-layer interference film thick-nesses, two widths of the pieces of multilayer thin film, andtwo different gaps between the pieces of multilayer thin film,for a total of 3× 2× 2 = 12 experiments. In every case, thewidth of each multilayer stack and the gap between themis perturbed by a uniformly distributed random number be-tween −25nm and 25nm to model structural variations inthe real butterfly. Each small piece is also displaced verti-cally by a small random distance to minimize the effect of in-teractions between light reflected from each stack. The ver-tical displacement ranges from −275nm to 275nm, with atriangular probability distribution. The right half of the struc-ture is a mirror image of the left half to ensure that the NTFFtransform will give a symmetric result. See Figure 10. Noiseis present due to the random aspect of the calculation, but theresults are still interpretable. In all cases, measurements inthe far field now indicate that there is a reflectance lobe that

Figure 9: The spatial distribution of light reflectingfrom three different diffraction gratings of period 800nm,1200nm, and 1600nm. The background color of each plot isthe color of the peak reflection.



Figure 10: The spatial distribution of light reflecting fromthree different multi-layer interference film thicknesses (leftto right: 20nm thick layers spaced 180nm apart, 60nmthick layers spaced 140nm apart, 140nm thick layers spaced60nm apart) and four different ridge gaps (top to bottom:350nm wide ridges with 100nm gaps, 675nm wide ridgeswith 175nm gaps, 350nm wide ridges with 175nm gaps,675nm wide ridges with 100nm gaps). The backgroundcolor of each plot is the color of the peak reflection.

is centered about the mirror direction. This is the result oflight traveling in a wider range of directions between the lay-ers due to diffraction at the corners of each piece. A broaderreflectance lobe is observed for narrow multilayer thin film

Figure 11: Golf balls rendered with materials based on thefirst two rows of Figure 10. Note the difference in the colorof the balls and the spatial spread of the reflected light.

Figure 12: The NTFF transformation surface that the spe-cial NTFF transform in this work is loosely based on.

stacks, but the size of the gap between the multilayer thinfilm stacks is not important to the spatial spread of the re-flected light. The color is primarily related to multilayer thinfilm interference, but it is visibly affected by the widths ofboth the multilayer thin film stacks and the gaps betweenthem. This complex physical optics problem involving bothdiffraction and interference could not have been modeled us-ing traditional computer graphics ray tracing techniques.

Finally, a shader is used to clarify the FDTD sim-ulation results. The data from an experiment is usedto populate a lookup table with the BRDF value forwavelengths 200nm,201nm, . . . ,800nm and viewing angles0◦,0.1◦, . . . ,90◦. Data is linearly interpolated between theviewing angles. To cope with the two-dimensional simula-tion, isotropic reflection is assumed. To handle data that onlyrefers to normal incidence, it is assumed that as the lightingangle shifts one way, the reflection lobe shifts the other wayat the same rate. The shader is implemented in LuxRender,and a golf ball is rendered. See Figure 11.

7.2. NTFF for the Butterfly

The long, thin channel amounts to a two-dimensional model.A full three-dimensional model is computationally feasibleif the length of one period across the ridges is taken to beonly one ridge. In this case the PBC cannot account for therandom variation in the heights of the ridges, but accordingto Lee [LS09], the effect of this random height variation isapproximated by the way that the limited size of this problemspace affects the NTFF transform. This claim is justified bycomparison with the surface in Figure 12.

The bottom of the surface in Figure 12 is neglected be-cause it mostly contributes to transmission; what matters isreflection. The vertical parts of the surface are neglected be-cause they are relatively tiny. What remains is a flat surfaceabove each ridge.

The significance of the random vertical displacement ofthe ridges is that the scattering from these flat surfaces addsincoherently, and in a statistical sense it is equivalent to thescattering from just one of these flat surfaces. So the NTFFtransform is simply performed using a rectangular surfacethat only covers one tile. In terms of the PBC, the NTFF



transform is performed using a rectangular surface that fitsperfectly into tile A.

Refer to Lee [LS09] for experimental evidence that thisway of doing the NTFF transform gives accurate results.

7.3. Setting Up the Problem Space

For the full three-dimensional butterfly model, the coordi-nate system used to capture the reflected light is chosen sothat the z-axis runs parallel to the ridges (the negative x-direction is normal to the surface of the butterfly, and they-axis runs perpendicular to the ridges), and the awkwardpart of the sampling pattern is where the results show notmuch to see anyway, as shown in Figure 13. In this coordi-nate system, the power reflection coefficient is∫ 3π/2

π/2

∫ π

0−ρsin2(θ)cos(φ)dθdφ.

For angles of incidence where kinc · y is small, the PBC canbe made more efficient by treating two ridges as one pe-riod, so that the combining procedure can be done less often.On the other hand, then the problem space is represented byabout twice as much data. This strategy was adopted for allangles of incidence in the zx-plane beyond 20◦ from normalincidence. The NTFF transform was still performed over justone ridge.

7.4. FDTD Simulation Results

In Lee’s [Lee09] work, many variations on the model aresimulated to explore how all the various aspects of the struc-ture affect the observed appearance of the butterfly. In thework reported here, the structure employed has offset lamel-lae without tapering, and no microribs. A snapshot of anFDTD simulation is shown in Figure 14. Notice that the re-flection consists of curved wavefronts. The results are shownin Figures 15 and 16a. Results for angles of incidence in

Figure 13: The sampling pattern that arises from the coor-dinate system employed in this work.

Figure 14: One frame of an animation depicting the z-component of the electric field in a cross-section of the Mor-pho nanostructure.

the zx-plane within 20◦ of grazing would take an inordinateamount of computing time. The base layer, though includedin the model for Lee’s [Lee09] work, is omitted from themodel for this work, because it increases computing timesubstantially and has little influence on the result (see Fig-ure 17).

The results show two reflection lobes when the angle ofincidence is in the zx-plane, but only one reflection lobewhen the angle of incidence is in the xy-plane (see Fig-ure 16a). As the lighting angle departs from normal inci-dence while remaining perpendicular to the ridges, intenseblue light continues to shine in a direction not far from nor-mal, and a hint of purple begins to emerge. As the lightingangle departs from normal incidence while remaining par-allel to the ridges, reflection diminishes dramatically. Theseresults are consistent with the physical observations that aMorpho butterfly is purple as viewed from a grazing angleperpendicular to the wing veins, but black as viewed froma grazing angle parallel to the wing veins. The results forangles of incidence near grazing suggest the Fresnel effect.

Figure 15: The power reflection coefficient for some anglesof incidence (θ,φ).



Figure 16: (a) The results of FDTD simulations for differentangles of incidence (θ,φ). (b) How the shader behaves fordifferent angles of incidence (θ,φ).

The ray-based approach prominent in current computergraphics would not give the broad reflection lobes predictedby the wave-based simulation. For the wing structure modelemployed in this research, ray tracing could only be expectedto predict reflection in a finite number of directions approxi-mating the peaks of those lobes. Ray tracing can only predictwide reflection lobes if the ridges are modeled with slightsagging that varies from ridge to ridge.

By contrast, wave-based optics can explain the broad re-flection lobes as a consequence of circular wavefronts aris-ing as light hits corners. These circular wavefronts meanlight reflected in a wide range of directions. This explo-sion of light complicates the multilayer thin film interfer-ence, producing some color variation over individual reflec-tion lobes.

In Lee’s work, this butterfly is only simulated under nor-mal incidence, and there is a small amount of reflection go-ing straight up. That is what happens if the lamellae are ex-tended beyond the stem and then truncated horizontally, re-sulting in horizontal “plateaus” (see Figure 18) that unsur-prisingly reflect some light straight up (see Figure 17). Thisshape seems uncharacteristic of a biological structure, andthe reflection straight up is not evident in the previously pub-lished results that Lee’s work uses for comparison.

See Table 1 for the parameters used in the FDTD algo-rithm.

7.5. The Shader

Shading the butterfly requires more than just a lookup ta-ble, because the periodic boundary condition only worksfor a limited range of lighting angles. On the other hand,

Figure 17: (a) The result of a simulation with the baselayer included. (b) The result of a simulation without thebase layer included. (c) The result of a simulation with“plateaus” included (and no base layer).



Figure 18: ”Plateaus” result if the lamellae are extended beyond the stem and then truncated horizontally.

Figure 19: A bumpy surface rendered with the shader: fivelighting angles (θ,φ) with constant viewing angle (70◦,0◦).

the data has no apparent pattern that would facilitate fit-ting an analytical model to it and dispensing with lookuptables completely, though that would be nice. Two distinct

Figure 20: A morpho butterfly rendered in LuxRender usinga shader derived from full wave simulations.

reflectance lobes are present. The angle of incidence has anunusual effect on the angle, shape, and magnitude of each re-flectance lobe, and it also impacts the color shift seen in eachreflectance lobe. An analytical formulation that accuratelycaptures all of this complexity without the use of any lookuptable would be inordinately complicated. Fortunately, thereis still a pattern that can be exploited to get by with lessthan a huge lookup table that simply gives a BRDF valuefor every lighting angle, viewing angle, and wavelength. Re-ferring to Figure 16a, one can notice that data for all anglesof incidence in the zx-plane can be obtained (to a fair ap-proximation) from the data for normal incidence by rotatingthe reflection lobes away from the light and scaling them asdescribed in Table 2.

The shader is implemented in LuxRender for full spec-tral ray tracing. It obtains BRDF data for the wavelengths200nm,201nm, . . . ,800nm at an arbitrary angle of inci-dence by rotating and scaling reflection lobes from anglesof incidence in the xy-plane. Specifically, FDTD simulationdata is taken from the projection of the desired angle ofincidence onto the xy-plane, and the reflection lobe fromthis FDTD simulation data is rotated and scaled accord-ing to the projection of the desired angle of incidence ontothe zx-plane. The FDTD simulation data employed is forthe lighting angles (90◦,280◦),(90◦,290◦), . . . ,(90◦,80◦)and viewing angles (θ,φ) where θ = 0◦,5◦, . . . ,180◦,φ =90◦,95◦, . . . ,270◦. Data is bilinearly interpolated acrossthese lighting and viewing angles. The scaling factors in theabove table are linearly interpolated as well.

The behavior of the shader is shown in Figure 16b, to becompared with Figure 16a. A bumpy surface rendered withthe shader is shown for five lighting angles in Figure 19. Theviewing direction for all of these bumpy surface images is(70◦,0◦) so that there is something to see as the reflectionpeaks at 20◦ off specular. An image of a morpho butterflyis shown in Figure 20. The color of the wing scale in thispicture was synthesized using the shader.



Figure 21: A golf ball rendered with a shader similar to thatof Sun [Sun06], where the color is given by a multilayer thinfilm interference calculation and the spatial spread of re-flected light is given by the Phong model with three differentspecular exponents (left to right: 5, 25, 100).

7.6. Comparison with Existing Results

The color of our Morpho butterfly simulation compares fa-vorably with existing computer graphic attempts to modelthis phenomenon, and our approach is the first effort thateven tries to compute the spatial distribution of the re-flected light. The three published papers (Smits [SM90],Sun [Sun06], and Sadeghi [Sad08]) that we are aware of alluse an approximation to compute the spectral reflectance ofthe multi-layer thin film system in the Morpho butterfly. Weshowed in Section 7.1 that our FDTD simulations accuratelyreproduce the spectral reflectance for a multi-layer thin filmsystem. In addition, our FDTD simulations include diffrac-tion effects that determine the spatial distribution of the lightreflected from the Morpho butterfly (see Section 7.1) whilethe existing computer graphic approaches all rely upon an ar-bitrary choice for the specular reflection cosine exponent inthe Phong reflection model. Although our approach is com-putationally more expensive than existing methods, it accu-rately computes interference effects and offers the first con-crete justification in the computer graphics literature for thespatial spread of the reflected light.

The shader of Sun [Sun06] simply uses the analytical re-sult for multilayer thin film interference to obtain a color,and the Phong model to obtain the spatial spread of reflectedlight. Figure 21 depicts a sphere rendered in this way (afull spectral shader implemented in LuxRender, using wave-lengths 200nm,201nm, . . . ,800nm) for the specular expo-nents 5, 25, and 100. See Figure 22 for a comparison be-

Figure 22: Comparison between a shader similar to thatof Sun [Sun06] with specular exponent 73 (left) and theanisotropic butterfly shader of this work (right).

Figure 23: Disparity between the butterfly reflection modelof Sun and that of this work, verses the Phong specular ex-ponent employed in Sun’s model.

tween that Morpho butterfly shader and the Morpho butterflyshader in this work. The shader of Sun [Sun06] is isotropic,whereas the shader of this work gives two distinct reflectionlobes and is not even close to isotropic; thus, the shader inthis work is more detailed. The shader of Sun [Sun06] re-quires a choice of specular exponent that must be manipu-lated to obtain visually acceptable results, and no particularvalue of that specular exponent is easy to justify theoreti-cally, whereas the shader of this work gives results that aretheoretically justified for a given model of a nanostructure.

We endeavour to find the Phong specular exponent thatbest fits the model of Sun [Sun06] to that of this work. Thiscalculation is performed in a coordinate system where the x-axis runs parallel to the ridges, the y-axis runs perpendicularto the ridges, and the z-axis is in the normal direction. At-tention is focused on normal incidence, and the shape of thereflection lobe from Sun [Sun06] is compared to the shapeof the larger reflection lobe from this work, which peaks atthe viewing angle (30◦,135◦).

To prepare for direct comparison, the reflectance distri-bution from this work is rotated −135◦ about the z-axisand then −30◦ about the y-axis, so that the larger reflec-tion lobe peaks in the normal direction. Then the smallerreflection lobe is removed by setting the BRDF data to zeroat all wavelengths for viewing angles where θ > 39◦ and104◦ < φ < 166◦. Finally, for each specular exponent that isconsidered as a candidate, the reflectance distribution fromSun [Sun06] is rescaled so that it peaks at the same value asthe reflection lobe from this work, and the focus is on fittingthe shape of it.

Disparity is measured as∫ 2π

0

∫ π/2

0(YSun−YMusbach)

2 sin(θ)dθdφ,

where Y is the luminance of the reflected light. It is com-puted for the specular exponents between 50 and 100, andthe result it plotted in Figure 23. The specular exponent 73



gives the best fit. We recommend this value to those whochoose to employ the model of Sun [Sun06]. See Figure 22.

8. Conclusions

In this paper we have introduced a new way for computergraphics practitioners to model how light interacts with ob-jects in a synthetic scene. The FDTD method of solvingMaxwell’s Equations was used to simulate the reflection ofelectromagnetic energy from a surface that contains struc-tures that are on the order of a wavelength of light. While theeffect of interference and diffraction of light has been stud-ied in computer graphics using analytical expressions andextensions to ray tracing, it has never been computed usingas general an approach as the one employed here. The resultsof this work show that physical optic effects can change boththe color of the light reflected from a surface and its spa-tial distribution. Predicting the spatial spread of the reflectedlight is a result that could not have been as easily achievedusing existing analytic modeling and ray tracing techniques.As a practical result we used our simulation technique tosuggest a plausible value for the parameter that determinesthe spatial spread of reflected light in an existing computergraphic Morpho reflection model.

The research reported here also makes small contributionsoutside the field of computer graphics. In the PBC, for an-gles of incidence not more than 40◦ away from normal in-cidence, numerical dispersion is accounted for by doing thecombining procedure earlier, instead of padding the problemspace. Morpho simulations are performed for a comprehen-sive assortment of lighting angles, including lighting anglesnot previously simulated with the FDTD method. In partic-ular, FDTD simulation data is reported for lighting anglesin the plane of the structure that supports the lamella, andreflection intensity is found to vary among these lighting an-gles. Previously published FDTD simulation results are fromtwo-dimensional FDTD simulations referring to a plane per-pendicular to the supporting structure, and such FDTD sim-ulations could not predict a change in reflection intensity asthe lighting angle departs from that plane. A previously pub-lished suggestion could further expand the range of lightingangles for which data is obtainable. There is no shortage ofFDTD literature that could be drawn on for additional func-tionality, such as material dispersion.

This work only begins to demonstrate how this very gen-eral method for modeling light can be used in computergraphics. The world contains many other biological struc-tures with appearance properties that can only be simulatedby using a full wave approach. The manufactured environ-ment includes an increasing array of materials and paintsthat have color reflection characteristics that are determinedby physical optics effects. Realistic image synthesis can playa role in understanding the appearance of existing iridescentobjects and in designing new coatings with heretofore un-seen visible optical properties.

9. Acknowledgements

The authors acknowledge the help of Seth Berrier in imple-menting the shader for the Morpho butterfly and Lian Duanfor her efforts in using the shader to model and render thebutterfly in its natural habitat. The authors would especiallylike to thank Shuxia Zhang for her assistance in making themost out of the resources available at the Minnesota Super-computing Institute. This research was funded by NationalScience Foundation grant 0941537 and by the MinnesotaSupercomputing Institute.

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Appendix A: Details of the FDTD Algorithm

Field Updating Equations

Assume that all materials under consideration are isotropic,meaning that they have the same properties along all direc-tions. Assume that all materials under consideration are lin-ear, meaning ~D = ε~E and ~B = µ~H. Assume that all materi-als under consideration are nondispersive, meaning that theirproperties do not depend on wavelength—or at least assumethat their properties are constant to a somewhat reasonableapproximation over the range of wavelengths under consid-eration.

The divergence equations (3) and (4) are unused. Theycan be employed as a reality check: if the FDTD algorithmis working then the divergence equations should be satisfied.The algorithm itself does not actually make use of them.

Rewrite the curl equations (1) and (2).

∇× ~H = ε∂~E∂t

+σe~E + ~Ji

∇×~E =−µ∂~H∂t−σ

m~H− ~Mi

Work with components individually. Only the derivation forthe field updating equation for Ex is shown here; the othersare similar.

∂Ex

∂t=

1εx

(∂Hz

∂y− ∂Hy

∂z−σ

exEx− Jix

)Use the central difference approximation

f ′(x)≈ f (x+∆x/2)− f (x−∆x/2)∆x

for the partial derivatives.

En+1x (i, j,k)−En

x (i, j,k)∆t

=

1εx(i, j,k)

Hn+1/2z (i, j,k)−Hn+1/2

z (i, j−1,k)∆y

− 1εx(i, j,k)

Hn+1/2y (i, j,k)−Hn+1/2

y (i, j,k−1)∆z

− σex(i, j,k)

εx(i, j,k)En+1/2

x (i, j,k)

− 1εx(i, j,k)

Jn+1/2ix (i, j,k)

Data for ~E is only available at integer time steps, soEn+1/2

x (i, j,k) is approximated by averaging Enx (i, j,k) and

En+1x (i, j,k), like so:

En+1/2x (i, j,k) =

En+1x (i, j,k)+En

x (i, j,k)2



Plug that in and solve for En+1x (i, j,k) to obtain the updating

equation for Ex.

En+1x (i, j,k) =

2εx(i, j,k)−∆tσex(i, j,k)

2εx(i, j,k)+∆tσex(i, j,k)

Enx (i, j,k)

+2∆t

(Hn+1/2

z (i, j,k)−Hn+1/2z (i, j−1,k)

)(2εx(i, j,k)+∆tσe

x(i, j,k))∆y

−2∆t

(Hn+1/2

y (i, j,k)−Hn+1/2y (i, j,k−1)

)(2εx(i, j,k)+∆tσe

x(i, j,k))∆z

− 2∆t2εx(i, j,k)+∆tσe

x(i, j,k)Jn+1/2

ix (i, j,k)

The updating equations for the other field components arederived similarly.

Boundary Conditions

The field updating equations cannot be used to update thecomponents of ~E at the problem space boundaries, becausethat would require components of ~H outside the problemspace. The choice of how to handle the boundary is calleda boundary condition. The simplest way is to just not up-date the components of ~E at the problem space boundaries.That is called a PEC (Perfect Electric Conductor) boundarycondition because the components of ~E at the boundary, tan-gential to the boundary are initialized to zero and they stayzero thereby implementing a perfect electric conductor at theboundary.

The PEC boundary condition causes the boundary of theproblem space to reflect outgoing waves back into the prob-lem space, which is usually undesirable. Absorbing bound-ary conditions (ABCs) are needed to absorb waves as theygo away from the problem space. One type of ABC, a Per-fectly Matched Layer (PML), is an absorbent medium to linethe boundary so that by the time waves have passed throughthis medium to the boundary and back, hopefully they havebecome negligibly small. The phrase “perfectly matched”refers to matching the impedance of the PML to that of theenclosed problem space so that the PML itself does not re-flect waves (which would defeat the purpose).

In the PML proposed by Bérenger [B94], every field com-ponent was artificially split into two components, therebyintroducing extra degrees of freedom with which to designthe PML. By judiciously choosing different conductivitiesfor these two artificial parts of each field component, he isable to develop a PML that is perfectly matched at all anglesof incidence. His experimental results show that for somesimulations the results are more accurate and for other sim-ulations less computing resources are needed.

It turns out that Bérenger’s PML is equivalent to using thetransformation x = x(1 + σ/ jωε0) to change an unwanted

wave e− jkx into an exponentially decaying wave

e− jkx = e− jkxe−σkx/ωε0 = e− jkxe−σx/cε0 ,

where σ is a spatially varying constant controlling the rateof decay. This finding was established by Chew and Wee-don [CW94], and independently by Rappaport [Rap95]. Thisnew formulation of Bérenger’s PML is called the Complex,frequency shifted, Perfectly Matched Layer (CPML). It ismuch easier to understand and to edit, and it also lends itselfto SIMD parallel computation.

Kuzuoglu and Mittra [KM96] propose using

x = x(

κ+σ

α+ jωε0

),

where α is a spatially varying constant that can be adjustedto help with absorbing long wavelengths, and κ is a spatiallyvarying constant that can be adjusted to help with absorbingevanescent waves. This version of the CPML reduces to theoriginal equivalent to Bérenger’s PML when α = 0F/(m ·s),κ = 1. It is the ABC used in this work.

Far-Field Source

To implement a far-field source such as incoming light, theelectric and magnetic fields are viewed as the sum of theincident field and the scattered field. The incident field isdefined as what would happen if there were only the far-fieldsource and nothing for it to interact with. The scattered fieldis defined as the total field minus the incident field, and it isthe field to which an ABC applies.

The incident field is known in the sense that there is anexplicit formula for it. Maxwell’s equations refer to the totalfield, but by writing this total field as the sum of the inci-dent field and the scattered field, one obtains equations thatcan be viewed as describing the scattered field in terms ofthe incident field. Then one can derive special field updat-ing equations for use in the FDTD algorithm to simulate thescattered field instead of the total field.

In this work, the far-field source (incoming light) isimplemented using what is known as the scattered fieldformulation, originally proposed in Holland [Hol77] andHolland [HSK80]. In this implementation, the field valuesthroughout the problem space represent the scattered field,and they are updated using the corresponding special fieldupdating equations. An explicit formula gives the incidentfield throughout the problem space for use in those specialfield updating equations. There are more efficient ways tocompute the incident field for this purpose, but in this work,for simplicity, the explicit formula was used directly.



Cell dimensions 10 nm× 10 nm× 10.0153 nmTime step duration 1.7341× 10−17 sDistance to NTFF surface 1 cell (10 nm)Distance to CPML 25 cells (250 nm)Thickness of CPML 20 cells (200 nm)CPML order 4CPML σ 0 F/(m · s) to 1.0610× 106 F/(m · s)CPML κ constant at 1CPML α constant at 0 F/(m · s)

Source waveform e−

(t−1.5033×10−15 s

3.3407×10−16 s

)2

Duration of simulation one pass of excitation through problem space

Table 1:

rotation scaling rotation scaling(30◦ ,0◦) 0.4698 (100◦ ,0◦) 0.6052(40◦ ,0◦) 0.5077 (110◦ ,0◦) 0.2877(50◦ ,0◦) 0.6837 (120◦ ,0◦) 0.1910(60◦ ,0◦) 0.9966 (130◦ ,0◦) 0.1681(70◦ ,0◦) 1.2857 (140◦ ,0◦) 0.1594(80◦ ,0◦) 1.2832 (150◦ ,0◦) 0.2438(90◦ ,0◦) 1.0000

Table 2:


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