enhanced acousto-optic properties in layered media...ijkl is the full photoelastic tensor, ∂ lu k...

PHYSICAL REVIEW B 96, 064114 (2017)

Enhanced acousto-optic properties in layered media

M. J. A. Smith* and C. Martijn de SterkeCentre for Ultrahigh bandwidth Devices for Optical Systems, Institute of Photonics and Optical Science,

School of Physics, The University of Sydney, NSW 2006, Australia

C. Wolff, M. Lapine, and C. G. PoultonSchool of Mathematical and Physical Sciences, University of Technology Sydney, NSW 2007, Australia

(Received 6 February 2017; revised manuscript received 28 June 2017; published 23 August 2017)

We present a rigorous procedure for evaluating the photoelastic coefficients of a layered medium in which theperiodicity is smaller than the wavelengths of all optical and acoustic fields. Analytical expressions are given forthe coefficients of a composite material comprising thin layers of optically isotropic materials. These photoelasticcoefficients include artificial contributions that are unique to structured media and arise from the optical andmechanical contrast between the constituents. Using numerical examples, we demonstrate that the acousto-opticproperties of layered structures can be enhanced beyond those of the constituent materials. Furthermore, we showthat the acousto-optic response can be tuned as desired.

DOI: 10.1103/PhysRevB.96.064114

I. INTRODUCTION

Since the first phenomenological descriptions of the pho-toelastic effect by Pockels [1–3], acousto-optics has played asignificant role in optics and materials science. Acousto-opticeffects are critical for radio-frequency modulators [3–5], andthe photoelastic effect is frequently used to determine stressdistributions surrounding cracks and material defects [6]. Morerecently, acousto-optics has found applications in modernnanophotonics: photoelasticity is the fundamental effect thatunderpins cavity optomechanics [7,8] and stimulated Brillouinscattering (SBS), which is critical for a diverse range of devicessuch as ultranarrow linewidth filters and high-resolutionsensors [5,9,10]. These devices, however, rely on the existing,fixed, photoelastic response of the material platform, which intechnologically important cases can be small [11,12]. At thesame time, SBS is problematic for optical fiber systems [13],so there is considerable interest in both the suppression and theenhancement of photoelasticity, depending on the application.

It is well known that composite materials, such as layeredmedia, can possess aggregate quantities that are markedlydifferent from their constituents [14,15]. Recent work [16–18]has shown that this principle applies to the acousto-opticproperties of composites. In contrast to the intricate andexotic designs seen in the optical metamaterials community,layered materials are among the simplest structures to fabri-cate, yet a complete picture of the acousto-optic properties oflayered media has not yet been reported. To the best of ourknowledge, the only other study concerning the photoelastictensor of layered media is by Rouhani and Sapriel [19], whereanalytical expressions for an orthorhombic composite com-prising orthorhombic layers were derived. However, nearly allof the expressions for the effective photoelastic coefficientsare incomplete, as they do not include artificial photoelasticcontributions (discussed below).

It has been widely accepted that acousto-optic interactionsin uniform, nonpiezoelectric dielectric media are captured by

*[email protected]

the photoelastic tensor pijkl defined by

�(ε−1)ij = pijklskl, (1)where �(ε−1)ij denotes a change in the inverse permittivitytensor and skl is the linear strain tensor for small displacementsfrom equilibrium. In this definition, the photoelastic tensoris treated as symmetric with respect to the first and secondindex pairs, i.e., pijkl = p(ij )(kl). However, the definition in(1) is sufficient to describe only the interaction betweenelectromagnetic and acoustic waves in dielectrics possessingisotropic or cubic symmetry. This definition was sufficient inearly research on light-sound interactions since the first solidmaterials examined either were of sufficiently high symmetryor possessed low optical anisotropy [5]. However, Nelson andLax [20] established that this form of the photoelastic responseomitted the contributions of local rotations that arise wheneveracoustic shear waves propagate within the material; the effectsof these local rotations on the permittivity tensor vanish forisotropic and cubic materials but are nonzero for media thatpossess lower levels of structural symmetry such as tetragonallattices [21]. This roto-optic effect can be strong comparedto the symmetric photoelastic effect and is directly related tothe optical anisotropy of the material. The total photoelasticresponse of the material is given by [20]

�(ε−1)ij = Pijkl∂luk= pijklskl + rijklrkl, (2)

where Pijkl is the full photoelastic tensor, ∂luk = skl + rkldenotes the gradient of the displacement vector, p(ij )(kl) andr(ij )[kl] are the symmetric and antisymmetric componentsof Pijkl , respectively, and rkl is the infinitesimal rotationtensor (where round- and square-bracket notations representsymmetric and antisymmetric index pairs, following Nelsonand Lax [20], and we now omit bracket notation on index pairsfor convenience). The definition (2) captures the potentiallylarge influence that the antisymmetric component of the fullphotoelastic tensor (otherwise known as the roto-optic tensor)can have on the scattering of light by an acoustic shearwave [20].

2469-9950/2017/96(6)/064114(10) 064114-1 ©2017 American Physical Society

https://doi.org/10.1103/PhysRevB.96.064114

SMITH, DE STERKE, WOLFF, LAPINE, AND POULTON PHYSICAL REVIEW B 96, 064114 (2017)

The analytic form of the roto-optic tensor in uniformmaterials was given in [3,20,22], where the tensor coefficientswere found to be directly linked to the optical anisotropy ofthe medium (for materials that do not possess monoclinic ortriclinic symmetry). Subsequently, it is important to considerthe effects of both strains and rotations when studying acousto-optic interactions in optically anisotropic materials. Althougha wide selection of natural uniform materials exhibits strongoptical anisotropy (such as calcite [23]), it is also possible toachieve selective control over the optical birefringence of amedium by constructing composite materials [15].

In recent years, it has also been established that thephotoelastic properties of structured materials exhibit a uniqueeffect known as artificial photoelasticity. This effect was firstrecognized in composite materials comprising cubic arrays ofspheres suspended in an otherwise uniform material by Smithet al. [16–18]. Artificial photoelasticity can be physicallyunderstood as follows: under an imposed strain, the differentmechanical responses of the constituent materials alter thefilling fraction and, in turn, contribute to changes in thepermittivity of the composite. Such artificial contributionshave been shown to play a significant role in the photoelasticproperties of composites [17] and cannot be omitted, even forhigh-symmetry structured materials.

The two main contributions to an acousto-optic interactionare photoelasticity, describing changes in permittivity inducedfrom bulk strains, and moving-boundary effects, describingpermittivity changes due to boundary strains (e.g., the bound-ary between a waveguide or a cavity and the surroundingair) [3,24]. There is extensive literature examining interface-motion (moving-boundary) contributions in acousto-opticsfor layered media [25–27], periodic structures [28,29], andgeneral structures [24,30], for example. However, the preciserelationship between the moving-boundary effect and artificialphotoelasticity is presently unclear. Both effects relate tointerface motions, and both require a permittivity contrast inorder to feature in an acousto-optic interaction. However, if thestiffness tensors of all layers are identical Cijkl = C ′ijkl , thenartificial photoelasticity is zero, whereas moving-boundarycontributions are not necessarily vanishing [24,25].

In place of photoelasticity and the moving-boundary effect,it is also possible to describe acousto-optic interactions interms of electrostriction, which describes bulk stresses inducedby an electromagnetic field, and radiation pressure, describingboundary stresses across dielectric interfaces [3]. Analyticalexpressions for the electrostrictive response of high symmetrystructures (arrays of spheres), under the approximation that theshear contribution is negligible, were given in Smith et al. [16],and a rigorous numerical investigation followed soon afterin Smith et al. [17,18]. In all instances, the electrostrictiveproperties of the composite were observed to be enhancedabove and beyond the intrinsic electrostrictive properties ofthe constituents, indicating that strong effects may also beobserved in structured materials with reduced symmetry, suchas layered media.

In this paper, we derive the photoelastic coefficients ofa layered medium, as shown in Fig. 1, giving the artificialcontribution to the symmetric photoelastic tensor explicitly,in addition to an explicit representation for the roto-optictensor. These expressions are obtained from the closed-

FIG. 1. Schematic of layered material investigated (infinitelyextending in the x-y plane) with periodicity along the z axis andconstituent parameters labeled.

form expressions for the effective permittivity and stiffnesstensors, where we do not consider frequency dependencein the materials properties [31]. The procedure we outlinefor the effective permittivity tensor is a generalization ofthat presented in Bergman [32], which was extended tothe effective stiffness tensor by Smith et al. [18], and isanalogous to the approach by Grimsditch [33]. We demonstratephotoelastic coefficients with values above and beyond that ofeither constituent material, strong roto-optic coefficients, andnon-negligible contributions from artificial photoelasticity fora silica-silicon and a silica-chalcogenide glass medium.

The outline of this paper is as follows. In Sec. II A wepresent the procedure for calculating the effective permittivitytensor εeffij . In Sec. II B we consider the analogous procedurefor the effective stiffness tensor Ceffijkl . In Sec. II C we determinethe symmetric photoelastic coefficients peffijkl , and in Sec. II Dwe give the antisymmetric photoelastic coefficients reffijkl . Thissection is followed by a numerical study of layered materialsin Sec. III, before concluding remarks are given in Sec. IV.

II. EFFECTIVE MATERIAL PARAMETERS

In this section, we outline a compact procedure for calcu-lating effective materials tensors, starting with the effectivepermittivity tensor [18,32] and the effective stiffness tensor[18]. In this work, the layered medium is constructed as aone-dimensional stack of optically isotropic dielectric slabs,with periodicity in z, that forms a medium with tetragonal(4/mmm) symmetry [21], as shown in Fig. 1. Results for pijklare presented explicitly for this case, although the procedure isreadily generalizable to consider layered materials made withoptically anisotropic constituents.

The effective-medium procedure outlined here essentiallyreplaces the layered material with a hypothetical effectivematerial exhibiting the same boundary information on theedges of the unit cell and possessing the same energy as thelayered material per unit cell. It is assumed that acousto-opticinteractions are the only nonlinear effect that the effectivemedium exhibits. In the derivation that follows, we use theconvention of unprimed notation for the first layer in the unitcell, primed notation for the second layer, and “eff” for theeffective medium. It is assumed that the thicknesses of thetwo layers, a and a′, are small relative to the wavelength of allelectromagnetic and acoustic fields (see Fig. 1). In other words,we examine the intrinsic bulk properties of the material in boththe optical and acoustic long-wavelength regimes.

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ENHANCED ACOUSTO-OPTIC PROPERTIES IN LAYERED . . . PHYSICAL REVIEW B 96, 064114 (2017)

It is also assumed that the optical and acoustic contrastbetween layers does not induce a perturbation to the mag-netic field, i.e., μij = μ′ij = μeffij = δij , where δij denotes theKronecker delta and μij is the relative permeability.

A. Effective permittivity tensor

We begin by computing the effective permittivity tensor fora layered medium and impose conventional electromagneticboundary conditions across the layers; continuity of thetangential E field and normal D field for our layered mediumrequires that

Ex = E′x, Ey = E′y, Dz = D′z, (3)where we further impose that the effective medium must takethe same static field values at all boundaries,

Eeffx = Ex = E′x, Eeffy = Ey = E′y, Deffz = Dz = D′z.(4)

We then require that the effective energy density [34]

U eff = 12Eeffi Deffi (5)is equivalent to the total energy density over the unit cell

U = 12 [f EiDi + (1 − f )D′iE′i], (6)where f = a/(a + a′) is the volume filling fraction, whichgives rise to

Deffx = f Dx + (1 − f )D′x, (7a)Deffy = f Dy + (1 − f )D′y, (7b)Eeffz = f Ez + (1 − f )E′z. (7c)

Using (4) and (7) with the constitutive relations

Deffi = ε0εeffij Eeffj , Di = ε0εijEj , D′i = ε0ε′ijE′j , (8)where εij denotes the permittivity tensor and ε0 is the vacuumpermittivity, it follows almost immediately that

εeffxx = f εxx + (1 − f )ε′xx −f (1 − f )(εxz − ε′xz)2f ε′zz + (1 − f )εzz

, (9a)

εeffyy = f εyy + (1 − f )ε′yy −f (1 − f )(εyz − ε′yz)2f ε′zz + (1 − f )εzz

, (9b)

1

εeffzz= f

εzz+ (1 − f )

ε′zz, (9c)

εeffyz =f εyzε

′zz + (1 − f )ε′yzεzz

f ε′zz + (1 − f )εzz, (9d)

εeffxz =f εxzε

′zz + (1 − f )ε′xzεzz

f ε′zz + (1 − f )εzz, (9e)

εeffxy = f εxy + (1 − f )ε′xy

− f (1 − f )(εxz − ε′xz)(εyz − ε′yz)

f ε′zz + (1 − f )εzz. (9f)

The expressions for εeffij above are equivalent to those presentedin Rouhani and Sapriel [19] and are valid for layered materialscomprising fully anisotropic layers. Despite the equivalence of

certain εeffij coefficients in a tetragonal material (i.e., εeffxx = εeffyy

and εeffyz = εeffxz = εeffxy = 0), all permittivity coefficients arerequired to determine the photoelastic coefficients for thecomposite in Sec. II C. For reference, we represent elementsof an inverse tensor by (ε−1)ij and reciprocal values by 1/εij .

B. Effective stiffness tensor

We now obtain closed-form expressions for the stiff-ness tensor of a layered material and begin by imposingconventional acoustic boundary conditions [35]: continuityof transverse velocity (or transverse displacement for time-harmonic fields in the long-wavelength limit) and continuityof the normal component of the stress field, which requiresthat

ux = u′x, uy = u′y, (10)in addition to

σxz = σ ′xz, σyz = σ ′yz, σzz = σ ′zz, (11)respectively. We then impose that the effective displacementand effective stress fields possess the same static values atthe boundary as per the conditions above, for example, ueffx =ux = u′x and σ effxz = σxz = σ ′xz. In analogy with Sec. II A, werequire that the strain energy density for the effective medium

U effs = 12σ effij seffij (12)is equivalent to the total strain energy density

Us = 12 [f σij sij + (1 − f )σ ′ij s ′ij ], (13)where sij = 12 (∂iuj + ∂jui). This is satisfied provided

∂xueffz = f ∂xuz + (1 − f )∂xu′z, (14a)

σ effxx = f σxx + (1 − f )σ ′xx, (14b)∂yu

effz = f ∂yuz + (1 − f )∂yu′z, (14c)

σ effyy = f σyy + (1 − f )σ ′yy, (14d)∂zu

effz = f ∂zuz + (1 − f )∂zu′z, (14e)

σ effxy = f σxy + (1 − f )σ ′xy, (14f)where, for convenience, we now differentiate (10) and compilethese with the derivatives of the displacement fields in (14)along with the stress fields to obtain

σ effxx = f σxx + (1 − f )σ ′xx, seffxx = sxx = s ′xx,σ effyy = f σyy + (1 − f )σ ′yy, seffyy = syy = s ′yy,σ effzz = σzz = σ ′zz, seffzz = f szz + (1 − f )s ′zz,σ effyz = σyz = σ ′yz, seffyz = f syz + (1 − f )s ′yz,σ effxz = σxz = σ ′xz, seffxz = f sxz + (1 − f )s ′xz,σ effxy = f σxy + (1 − f )σ ′xy, seffxy = sxy = s ′xy. (15)

Using (15) along with the constitutive relations

σ effij = Ceffijklseffkl , σij = Cijklskl, σ ′ij = C ′ijkls ′kl, (16)where Cijkl denotes the linear stiffness tensor, we recoverthe effective stiffness coefficients. Here the layers comprise

064114-3


optically isotropic media, from which we obtain all six uniquenonvanishing parameters of the mechanical stiffness tensorCeffijkl for an effective tetragonal (4/mmm) material [21] as

Ceffxxxx = f Cxxxx + (1 − f )C ′xxxx

− f (1 − f )(Cxxyy − C′xxyy)

2

f C ′xxxx + (1 − f )Cxxxx, (17a)

Ceffxxyy = f Cxxyy + (1 − f )C ′xxyy

− f (1 − f )(Cxxyy − C′xxyy)

2

f C ′xxxx + (1 − f )Cxxxx, (17b)

Ceffxxzz =f CxxyyC

′xxxx + (1 − f )CxxxxC ′xxyy

f C ′xxxx + (1 − f )Cxxxx, (17c)

1

Ceffzzzz= f

Cxxxx+ (1 − f )

C ′xxxx, (17d)

1

Ceffyzyz= f

Cyzyz+ (1 − f )

C ′yzyz, (17e)

Ceffxyxy = f Cyzyz + (1 − f )C ′yzyz. (17f)The expressions in (17) are equivalent to those presentedin Rouhani and Sapriel [19] and Grimsditch [33], afterconsidering the symmetry properties of the constituent layers.

C. Effective symmetric photoelastic tensor

In this section we evaluate the symmetric photoelastictensor peffijkl defined by

�(ε−1eff

)ij

= peffijklseffkl , (18a)or, equivalently,

�(εeff)ij = −εeffii εeffjj peffijkl seffkl , (18b)provided the medium does not possess triclinic or monoclinicsymmetry [22]. Expressions for the effective photoelastictensor elements peffijkl are obtained by differentiating theeffective permittivity expressions εeffij in (9) with respect toindividual strain fields seffij while holding other effective strainfields constant. For example, (18a) for an effective tetragonal(4/mmm) material is given by

�εeffzz = −(εeffzz

)2 [peffzzxxs

effxx + peffzzxxseffyy + peffzzzzseffzz

], (19)

and subsequently,

∂εeffzz

∂seffxx

∣∣∣∣seffyy ,s

effzz

= −(εeffzz )2peffzzxx, (20a)∂εeffzz

∂seffyy

∣∣∣∣seffxx ,s

effzz

= −(εeffzz )2peffzzxx, (20b)∂εeffzz

∂seffzz


effyy

= −(εeffzz )2peffzzzz. (20c)Therefore, an analytical expression for peffzzzz is recovered bydifferentiating εeffzz (9c) with respect to s

effzz , with both s

effxx and

seffyy held constant. The resulting expressions are then reducedusing the tensor definitions for the constituent materials

�εij = −εii εjj pijkl skl, (21a)�ε′ij = −ε′ii ε′jj p′ijkl s ′kl, (21b)

the mechanical constitutive relations (16), and the mechanicalboundary conditions in (15). We remark that the permittivitiesεij and ε′ij are functions of their constituent strain fields alone[see Eq. (21)]. The derivation for all seven unique nonvanishingphotoelastic constants necessary to describe a layered structureis extensive, and we feel that there is little merit in providinga complete outline for all terms. Accordingly, we considerthe derivations for peffzzxx and p

effzzzz alone and present the final

expressions for all remaining coefficients.As identified in (20a) above, we now implicitly differentiate

the effective permittivity expression (9c) with respect to seffxx ,holding the strain fields seffyy and s

effzz constant, which admits

1(εeffzz

)2 ∂εeffzz

∂seffxx


effzz

= f(εzz)2

∂εzz

∂seffxx


effzz

+ (1 − f )(ε′zz)2

∂ε′zz∂seffxx


effzz

−(

1

εzz− 1

ε′zz

)∂f

∂seffxx


effzz

. (22)

The first derivative in (22) follows immediately from thedefinition in (20a). The next derivative is evaluated using thedefinition of the photoelastic tensor (21a) in the first opticallyisotropic layer. An application of chain rule then admits

∂εzz

∂seffxx


effzz

= ∂εzz∂sxx


effzz

∂sxx

∂seffxx


effzz

+ ∂εzz∂syy


effzz

∂syy

∂seffxx


effzz

+ ∂εzz∂szz


effzz

∂szz

∂seffxx


effzz

. (23)

Using the acoustic boundary conditions sxx = seffxx and syy =seffyy from (15), we have that

∂sxx

∂seffxx


effzz

= 1, ∂syy∂seffxx


effzz

= 0, (24)

respectively. The boundary condition σzz = σ effzz and theconstitutive relations for the constituent layers (16) give

Ceffxxzzseffxx + Ceffxxzzseffyy + Ceffzzzzseffzz

= Cxxyysxx + Cxxyysyy + Cxxxxszz, (25)which after implicit differentiation, where we also hold thestrain fields seffyy and s

effzz constant, takes the form

Ceffxxzz∂seffxx

∂seffxx


effzz

+ Ceffxxzz∂seffyy

∂seffxx


effzz

+ Ceffzzzz∂seffzz

∂seffxx


effzz

= Cxxyy ∂sxx∂seffxx


effzz

+Cxxyy ∂syy∂seffxx


effzz

+Cxxxx ∂szz∂seffxx


effzz

.

(26a)

064114-4


The boundary conditions in (15) and constant field require-ments evident from (20a) reduce (26a) to the form

Ceffxxzz = Cxxyy + Cxxxx∂szz

∂seffxx


effzz

(26b)

and ultimately admits

∂szz

∂seffxx


effzz

= Ceffxxzz − Cxxzz

Cxxxx. (26c)

Substituting (24) and (26c) into the derivative (23) gives

∂εzz

∂seffxx


effzz

= −(εzz)2pxxzz − (εzz)2pxxxx[Ceffxxzz − Cxxzz

Cxxxx

],

(27a)

after using (21a), and analogously, we have that

∂ε′zz∂seffxx


effzz

= −(ε′zz)2p′xxzz − (ε′zz)2p′xxxx[Ceffxxzz − C ′xxzz

C ′xxxx

],

(27b)

for the second optically isotropic layer, following the boundarycondition σ ′zz = σ effzz (15).

The derivative of the filling fraction in (22), through anapplication of the chain rule, gives rise to

∂f

∂seffxx


effzz

= a ∂f∂a


effzz ,a

′

∂szz

∂seffxx


effzz

+ a′ ∂f∂a′


effzz ,a

∂s ′zz∂seffxx


effzz

, (28a)

following from the definitions f = a/(a + a′), �szz = �a/a,and �s ′zz = �a′/a′. Using (26c) and the analogous expressionfor the second layer, we obtain

∂f

∂seffxx


effzz

= f (1 − f )[Ceffxxzz − Cxxyy

Cxxxx− C

effxxzz − C ′xxyy

C ′xxxx

].

(28b)

Substituting (20a), (27a), (27b), and (28b) into expression (22)gives

peffzzxx = fpxxyy + (1 − f )p′xxyy

− f (1 − f )(pxxxx − p′xxxx)(Cxxyy − C ′xxyy)

f C ′xxxx + (1 − f )Cxxxx

−f (1 − f )(

1

εzz− 1

ε′zz

)(Cxxyy − C ′xxyy


).

(29)

Having determined the analytical expression for peffzzxx, we nowturn to the derivation of the peffzzzz coefficient. Similarly, implicitdifferentiation of the effective permittivity expression (9c) withrespect to seffzz , with s

effxx and s

effyy held constant, admits

− 1(εeffzz

)2 ∂εeffzz

∂seffzz


effyy

=− f(εzz)2

∂εzz

∂seffzz


effyy

− (1−f )(ε′zz)2

∂ε′zz∂seffzz


effyy

+(

1

εzz− 1

ε′zz

)∂f

∂seffzz


effyy

, (30)

where the first derivative is given by (20c). In a procedureanalogous to above, we have that

∂εzz

∂seffzz


effyy

= ∂εzz∂sxx


effyy

∂sxx

∂seffzz


effyy

+ ∂εzz∂syy


effyy

∂syy

∂seffzz


effyy

+ ∂εzz∂szz


effyy

∂szz

∂seffzz


effyy

, (31)

where the boundary conditions (15) and constant-field require-ments give rise to

∂sxx

∂seffzz


effyy

= ∂syy∂seffzz


effzz

= 0. (32)

The remaining boundary condition σzz = σ effzz and constitutiverelations (16) give the expression (25) once more. Implicitdifferentiation with respect to seffzz and the new constant-fieldrequirements admits

Ceffxxzz∂seffxx

∂seffzz


effyy

+ Ceffxxzz∂seffyy

∂seffzz


effyy

+ Ceffzzzz∂seffzz

∂seffzz


effyy

= Cxxyy ∂sxx∂seffzz


effyy

+ Cxxyy ∂syy∂seffzz


effyy

+ Cxxxx ∂szz∂seffzz


effyy

,

(33)

and ultimately, we find that

∂szz

∂seffzz


effyy

= Ceffzzzz

Cxxxx. (34)

Subsequently, substituting (32) and (34) into (31), we obtain

∂εzz

∂seffzz


effyy

= −(εzz)2pxxxx(

Ceffzzzz

Cxxxx

), (35a)

after using (21a), and analogously, we have

∂ε′zz∂seffzz


effyy

= −(ε′zz)2p′xxxx(

Ceffzzzz

C ′xxxx

). (35b)

The derivative of the filling fraction in (30) takes the form

∂f

∂seffzz


effyy

= f (1 − f )[

∂szz

∂seffzz


effyy

− ∂s′zz

∂seffzz


effyy

]

= f (1 − f )[

Ceffzzzz

Cxxxx− C

effzzzz

C ′xxxx

], (36)

following (34) and the corresponding expression for the secondlayer. Substituting (35a), (35b), and (36) into (30), we obtain

peffzzzz

Ceffzzzz= f

(pxxxx

Cxxxx

)+ (1 − f )

(p′xxxxC ′xxxx

)

+ f (1 − f )(

1

εzz− 1

ε′zz

)(1

Cxxxx− 1

C ′xxxx

). (37)

The expressions for peffzzxx in (29) and peffzzzz in (37)

are presented below along with all other remaining

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coefficients:

(εeffxx )2peffxxxx = f (εxx)2pxxxx + (1 − f )(ε′xx)2p′xxxx

− f (1 − f )(Cxxyy − C′xxyy)[(εxx)

2pxxyy − (ε′xx)2p′xxyy]f C ′xxxx + (1 − f )Cxxxx

+ f (1 − f )(εxx − ε′xx)(Cxxyy − C ′xxyy)

f C ′xxxx + (1 − f )Cxxxx, (38a)

peffzzzz

Ceffzzzz= f

(pxxxx

Cxxxx

)+ (1 − f )

(p′xxxxC ′xxxx

)+ f (1 − f )

(1

εzz− 1

ε′zz

)(1

Cxxxx− 1

C ′xxxx

), (38b)

(εeffxx

)2peffxxzz

Ceffzzzz= f (εxx)

2pxxyy

Cxxxx+ (1 − f ) (ε

′xx)

2p′xxyyC ′xxxx

− f (1 − f )(εxx − ε′xx)(

1

Cxxxx− 1

C ′xxxx

), (38c)

(εeffxx

)2peffxxyy = f (εxx)2pxxyy + (1 − f )(ε′xx)2p′xxyy

− f (1 − f )(Cxxyy − C′xxyy)[(εxx)

2pxxyy − (ε′xx)2p′xxyy]f C ′xxxx + (1 − f )Cxxxx

+ f (1 − f )(εxx − ε′xx)(Cxxyy − C ′xxyy)

f C ′xxxx + (1 − f )Cxxxx, (38d)

peffzzxx = fpxxyy + (1 − f )p′xxyy

− f (1 − f )(pxxxx − p′xxxx)(Cxxyy − C ′xxyy)

f C ′xxxx + (1 − f )Cxxxx− f (1 − f )

(1

εzz− 1

ε′zz

)(Cxxyy − C ′xxyy


), (38e)

εeffyy peffyzyz

Ceffyzyz= f εyypyzyz

Cyzyz+ (1 − f )ε

′yyp

′yzyz

C ′yzyz, (38f)

εeffxx εeffyy p

effxyxy = f εxxεyypxyxy + (1 − f )ε′xxε′yyp′xyxy. (38g)

We remark that in the expressions above, the photoelastic co-efficients possess the form peffijkl = αqrst pqrst + α′qrst p′qrst +partijkl , where αqrst and α

′qrst are functions of material parameters

but may be regarded as weightings for the photoelasticcoefficients of the constituent layers. Following the conventionestablished in earlier work [16], the final contribution partijkl istermed artificial photoelasticity, as this represents a nontrivialcontribution to the photoelastic properties of the compositewhen pqrst = p′qrst = 0. These artificial contributions aredirectly proportional to the contrast in relevant componentsof the permittivity and stiffness tensors and have been shownto play a significant role in the photoelastic properties of othersubwavelength structured designs [17,18]. Note that for peffyzyzand peffxyxy above, there is no artificial photoelastic component,as shear waves do not change the volume of the unit cell whenthe constituent and effective material are oriented with theCartesian coordinate frame, i.e.,

∂f

∂seffyz= ∂f

∂seffxz= ∂f

∂seffxy= 0; (39)

however, we emphasize that this result holds only for high-symmetry composites.

The derivation outlined in this section gives results forthe symmetric photoelastic strain tensor; expressions for thesymmetric photoelastic stress tensor may be found through astraightforward application of Hooke’s law [21]. However,the photoelastic strain coefficients peffijkl may be expressedin terms of both effective stress and strain fields; if chosenappropriately, the photoelastic coefficients are then derivedin terms of acoustic fields that are everywhere continuous inthe layered medium. Such an approach is analogous to thatoutlined in Rouhani and Sapriel [19]. For example, substituting

one line of the constitutive relation (16) into the photoelastictensor definition (19), we obtain

�εeffzz = −(εeffzz

)2 [{peffzzxx −

Ceffxxzz

Ceffzzzzpeffzzzz

}seffxx

+{peffzzxx −

Ceffxxzz

Ceffzzzzpeffzzzz

}seffyy +

peffzzzz

Ceffzzzzσ effzz

], (40)

where photoelastic strain coefficients are now obtainedthrough differentiation (as before), but with effective stress andeffective strain fields held constant. However, such a proceduregives final expressions for peffijkl identical to those presented in(38). As a final remark, the pijkl coefficients presented in (38),with partijkl = 0, are identical to those tabulated in Rouhani andSapriel [19] after considering symmetry reductions of tensorcoefficients [21].

D. Effective antisymmetric photoelastic tensor

In this section, we evaluate the antisymmetric componentof the photoelastic tensor reffijkl defined by

�(ε−1eff

)ij

= reffijklreffkl , (41)

where rkl = 12 (∂luk − ∂kul) denotes the infinitesimal rotationtensor. The derivation for the roto-optic tensor of a uniformmaterial is given in [3,20] and extends to the case of asubwavelength structured material as

reffijkl = 12[(

ε−1eff)ilδkj +

(ε−1eff

)ljδik

− (ε−1eff )ikδlj − (ε−1eff )kj δil]. (42a)064114-6


FIG. 2. (a) Symmetric photoelastic coefficients peffijkl corresponding to simple strains, (b) roto-optic coefficient reffyzyz and symmetric

photoelastic coefficients peffyzyz and peffxyxy, (c) artificial photoelastic terms p

artijkl , (d) stiffness tensor coefficients C

effijkl , (e) permittivity coefficients

εeffij , and (f) symmetric electrostriction coefficients γeffijkl as a function of filling fraction f for silica and Si [100] layers with a + a′ = 50 nm and

labels in Voigt notation.

This simplifies to the form [22]

reffijkl =1

2

(1

εeffjj− 1

εeffii

)(δikδjl − δilδjk), (42b)

provided the layered material does not possess triclinic ormonoclinic symmetry. For our tetragonal (4/mmm) layeredstructure, there are only eight nonvanishing rijkl terms, whichall take the same value modulo a sign change that arises fromthe antisymmetric nature of the tensor rijkl = −rij lk . From theexpressions for the effective permittivity presented in Sec. II A,we have that

reffxzxz =1

2

(f ε′zz + (1 − f )εzz

εzzε′zz− 1

f εxx + (1 − f )ε′xx

)(43)

for layers of optically isotropic media.

III. NUMERICAL EXAMPLES

In this section, we present the effective permittivity,stiffness, and photoelastic tensors for a selection of materialcombinations, where constituent parameter values are takenfrom Table 1 of Smith et al. [17]. Here values are presentedat a vacuum wavelength of λ = 1550 nm and for a total layercell width of a + a′ = 50 nm.

We begin by considering the material properties of afused silica and silicon [100] layered medium in Fig. 2. InFig. 2(a) we present the symmetric photoelastic coefficientspeffxxxx (blue curve), p

effxxyy (cyan curve), p

effxxzz (solid red curve),

peffzzzz (black curve), and peffzzxx (dashed red curve) as a function

of filling fraction. Here the coefficients exhibit a varieddependence on filling fraction, with enhancement in thepeffxxxx and p

effzzzz elements beyond either of the constituent

values to peffzzzz = 0.135 at f = 0.275 and peffxxxx = −0.121at f = 0.32. For reference, we describe such behavior asextraordinary enhancement (i.e., when a composite materialpossesses material values that are beyond the values for eitherof the constituents). Interestingly, the off-diagonal elementspeffxxyy, p

effxxzz, and p

effzzxx do not demonstrate extraordinary

enhancement for this material combination. We also havepeffxxxx = 0 at f = 0.045 along with peffzzzz = 0 at f = 0.87,which implies that longitudinal acoustic waves traveling alongx at f = 0.045 and longitudinal acoustic waves travelingalong z at f = 0.87 will not alter the optical properties ofthe medium. Reassuringly, symmetry-required degeneraciesare recovered at f = 0 and f = 1, where the layered mediumreturns to a uniform material (i.e., pxxxx = pzzzz and pxxyy =pxxzz = pzzxx in a cubic or isotropic medium). In Fig. 2(b)we show the remaining symmetric photoelastic coefficientspeffyzyz and p

effxyxy as functions of filling fraction, in addition

to the roto-optic tensor coefficient reffxyxy. Here we observe astrong roto-optic effect in the layered material, which reachesa maximum of reffxyxy = 0.081 at f = 0.295. This value isgreater than |pSixyxy| = 0.051 and |pSiO2xyxy| = 0.075 and is alsodifferent in sign, which demonstrates that the roto-optic effectcan significantly alter the predicted change in permittivityfor acoustic shear waves and should not be omitted a priori

064114-7


FIG. 3. (a) Symmetric photoelastic coefficients peffijkl corresponding to simple strains, (b) roto-optic coefficient reffyzyz and symmetric

photoelastic coefficients peffyzyz and peffxyxy, (c) artificial photoelastic terms p

artijkl , (d) stiffness tensor coefficients C

effijkl , (e) permittivity coefficients

εeffij , and (f) symmetric electrostriction coefficients γeffijkl as a function of filling fraction f for silica and As2S3 glass layers with a + a′ = 50 nm

and labels in Voigt notation.

without careful consideration. The two photoelastic shear con-stants also exhibit extraordinary enhancement, taking extremalvalues of peffyzyz = −0.04 at f = 0.545 and peffxyxy = −0.109 atf = 0.117, and possess the correct degeneracy at f = 0 andf = 1 (i.e., pyzyz = pxyxy in a cubic or isotropic medium).

In Fig. 2(c) we present the artificial contributions to thephotoelastic tensors shown in Fig. 2(a), which all exhibitsignificant, positive-valued contributions to the effective sym-metric photoelastic coefficients. This artificial contribution[obtained by substituting pijkl = p′ijkl = 0 in (38)] diminishesthe extreme range of peffxxxx and shows that the individualweightings for the constituent coefficients can take values|αqrst |,|α′qrst | > 1. This demonstrates that extraordinary en-hancement is possible without artificial photoelasticity. In thecase of a layered medium, we remark that partxxxx = partxxyy sincethese terms both arise from in-plane strains and are relatedto the same in-plane permittivity and stiffness contrast. Forreference, maximum values are listed as follows: partxxxx =partxxyy = 0.031 at f = 0.17, partzzzz = 0.072 and partzzxx = 0.04 atf = 0.592, and partxxzz = 0.056 at f = 0.17. In Figs. 2(d) and2(e), we present the effective stiffness and permittivity coeffi-cients as functions of filling fraction, following their explicitdefinitions in (9) and (17). Here the simple dependences on fare visible, extraordinary enhancements are not observed, andthe required material symmetries are recovered at f = 0 andf = 1. In effect, for the permittivity and stiffness tensors,all in-plane terms are given by volume averaging, and allout-of-plane terms are given by the inverse of volume-averagedreciprocal values.

In Fig. 2(f) we give the corresponding symmetric elec-trostriction coefficients, defined as γijkl = εiiεjjpijkl . In amanner analogous to that for the photoelastic coefficients, theelectrostrictive coefficients of a composite material also exhibita nontrivial dependence on f , in addition to extraordinaryenhancement. Maxima of γ effxxyy = 2.96 at f = 0.74, γ effxxzz =3.626 at f = 0.68, and γ effzzxx = 4.87 at f = 0.89 are observed,demonstrating that the choice of polarization and propagationdirection can have important implications for SBS experimentsin layered media.

In Fig. 3 we present the material properties for a layeredmedium comprising fused silica and As2S3 glass layers, in amanner analogous to Fig. 2. In Fig. 3(a) we show a selectionof symmetric photoelastic constants for the composite, whereit is observed that all peffijkl corresponding to simple strainsexhibit extraordinary enhancement. The enhancement of thepeffxxzz coefficient to p

effxxzz = 0.428 at f = 0.184 is remarkable

when compared to pSiO2xxyy = 0.27 and pAs2S3xxyy = 0.24 (i.e., anenhancement of 59% and 78%, respectively). In Fig. 3(b) weshow the dependence on filling fraction for the remainingsymmetric photoelastic constants, in addition to the roto-optic coefficient. For silica and chalcogenide glass layers, amaximum of reffyzyz = 0.036 is achieved at f = 0.38. Here weobserve peffyzyz = 0 at f = 0.535 and peffxyxy = 0 at f = 0.678,revealing that only roto-optic contributions participate inacousto-optic interactions at these filling fractions.

In Fig. 3(c) we present the artificial contribution to the totalsymmetric photoelastic coefficients shown in Fig. 3(a). Hereit is apparent that artificial terms contribute negatively to the

064114-8


photoelastic properties of the layered medium, reducing thepeffxxzz and p

effzzzz coefficients significantly. From this figure, we

also determine that the extraordinary enhancement in peffxxzzis due to |αqrst |,|α′qrst | > 1. For reference, a maximum valueof partxxzz = −0.141 is achieved at f = 0.17. In Fig. 3(d) wepresent the stiffness tensor coefficients for the layered medium,and in Fig. 3(e) we show the permittivity tensor coefficientsas functions of filling fraction. Both of these figures exhibita behavior qualitatively similar to that in Figs. 2(d) and 2(e)with the absence of extraordinary enhancement. In Fig. 3(f)we present the electrostriction constants as a function of thefilling fraction for this material combination for completeness.Despite the large value for peffxxzz observed in Fig. 3(a), thecorresponding γ effxxzz term is smoothed by the much strongergrowth in (εeffxx )

2. Also, we observe zero values for γ effyzyz andγ effxyxy following Fig. 3(b) along with γ

effxxxx and γ

effzzzz following

Fig. 3(a).Following earlier works by some of the authors [17,18]

on the numerical study of photoelasticity in compositescomprising arrays of spheres, we now briefly compare resultsfor a layered structure of silicon and chalcogenide glasswith a corresponding cubic lattice structure. The numericalprocedure for the sphere configuration determines the effectivebulk photoelastic response (including artificial contributions)by comparing the change in the effective permittivity tensorrelative to a small mechanical strain imposed on the unit-cellboundary.

In Fig. 4 we compare the photoelastic constants obtainedwith silicon and chalcogenide glass as a function of fillingfraction when they are structured in the form of a cubic arrayof spheres (cub) and as a layered material (tet). The valuesfor the cubic material are obtained using an extended finite-element simulation procedure [17,18] in which we restrict ourattention to 0 < f < 0.5 as this range approaches the spheretouching limit and, subsequently, the extent of the numericalprocedure. In Figs. 4(a) and 4(b) we observe that values for thelayered medium act as approximate bounds for the cubic latticeand suggest that our closed-form expressions may be used toobtain estimates of the photoelastic constants for an arbitrarymaterial pair. The limit behavior of these curves also differsconsiderably, with only ptetxxxx ≈ pcubxxxx and ptetxxyy ≈ pcubxxyy forvanishing filling fraction. However, we remark that furtherinvestigation is needed to determine bounds on the photoelasticproperties of composite materials.

IV. CONCLUDING REMARKS

We have presented an accurate procedure for determiningthe acousto-optic properties of layered media, fully accountingfor artificial photoelasticity and the roto-optic effect. Themethods outlined in this work are easily generalizable tolayered media with anisotropic constituents. This study opensthe path for exploring the acousto-optic properties of highly

FIG. 4. Comparison of symmetric photoelastic coefficients(a) peffxxxx and (b) p

effxxyy as a function of filling fraction for a cubic

array (cub) of As2S3 glass spheres embedded in Si [100] (where f =4πr3/[3(a + a′)3] and r is the radius of a sphere) with correspondingterms for a layered medium (tet) comprising the same materials [withf = a/(a + a′)], with a + a′ = 50 nm.

anisotropic media, such as hyperbolic metamaterials [36] andthin film composites [15].

We have shown that the symmetric photoelastic constantspeffijkl of a layered material are nontrival functions of fillingfraction, can exhibit extraordinary enhancement, and can betuned as desired for applications.

We have also demonstrated that roto-optic coefficientscan take values comparable to the symmetric photoelasticcoefficients. This result has important implications for acousticshear-wave propagation in optically anisotropic media. Fur-thermore, the tunable photoelastic response offered by layeredmaterials may have important implications for SBS structures.

ACKNOWLEDGMENTS

This work was supported by the Australian Re-search Council: CUDOS Centre of Excellence Project No.CE110001018 and Discovery Projects No. DP150103611, No.DP160101691.

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enhanced acousto-optic properties in layered media...ijkl is the full photoelastic tensor, ∂ lu k...

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