1

Efficient Analysis of Confined Guided Modes inPhoxonic Crystal Slabs

Mohammad Hasan Aram and Sina Khorasani

Abstract—Today’s standard fabrication processes are just ca-pable of manufacturing slab of photonic and phononic crystals, soan efficient method for analysis of these crystals is indispensable.Plane wave expansion (PWE) as a widely used method instudying photonic and phononic (phoxonic) crystals in full threedimensions is not suitable for slab analysis in its standard form,because of convergence and stability issues. Here, we propose amodification to this method which overcomes these limitations.This improved method can be utilized for calculation of bothphotonic and phononic modes in phoxonic slabs. While in thestandard three-dimensional PWE, Fourier series are used toestimate the field dependence across the normal component ofthe slab, we expand the fields across the third dimension usingeigenmodes of a plain unstructured slab. Despite its approximatenature, this approach is observed to be both much faster andmore accurate than the conventional PWE method and can givea very accurate estimation of confined propagating modes. Asan application example of the proposed method we investigatea non-reciprocal photonic device. This device is a phoxonic slabwaveguide which changes modes of optical waves by elastic wavesand can be used as an optical insulator or mode converter.

Index Terms—Phononic crystal, Photonic crystal, Phoxoniccrystal, Plane Wave Expansion, Photo-elastic interaction.

I. INTRODUCTION

THERE exist many different types of waves in the nature,among them elastic and electromagnetic waves being the

most famous and known ones. Because of the availabilityof sound waves as the scalar polarization of elastic pressurewaves, their behavior was well known to physicists manyyears before the formulation of Maxwell’s equations. With theadvent of electromagnetism and special relativity, the focusof study in wave phenomena shifted primarily to the realmof optics and electromagnetics and consequently photoniccrystals (PtC) were introduced [1], [2] before their phononiccounterparts [3]. They are called crystals since their con-stituents are periodic. Periodicity of permittivity in photoniccrystals, and density and stiffness in phononic crystals (PnC)may be held responsible for their characteristics. The mainproperty of these crystals is their ability to prevent somecontinuous frequency ranges of waves from propagating inside

M. H. Aram is with the School of Electrical Engineering, Sharif Universityof Technology, Tehran, Iran (e-mail: mharam@ee.sharif.ir).S. Khorasani is with the Ecole Polytechnique Federale de Lausanne(EPFL), CH-1015 Lausanne, Switzerland, on leave from School of Elec-trical Engineering, Sharif University of Technology, Tehran, Iran (e-mail:sina.khorasani@epfl.ch).This work has been supported in part by Iranian National Science Foundation(INSF) under the grant 93026841. M.H.A. is being supported through a post-doctoral fellowship at Sharif University of Technology. S.K. is being supportedby the Research Deputy of Sharif University of Technology over a sabbaticalvisit, as well as Laboratory of Photonics and Quantum Measurements (LPQM)at EPFL.

them. These frequency ranges are called forbidden photon-ic/phononic frequency gaps, which are caused by destructiveinterference of waves inside the crystal.

Because of this interesting property, photonic and phononiccrystals have found many applications in different fields ofscience and technology [4], among them are optomechanics[5]–[9], sensors [10], acoustic and optical insulators, waveg-uides, filters and high Q cavities [11]–[16]. With the helpof simultaneous phononic and photonic band gaps, one cancreate complete optical insulation [17]. Acoustic insulatorscan be used to isolate gyroscopes resonators and oscillatorsfrom ambient noise and vibrations [18]. By utilizing acous-tic waveguides, one can miniaturize acoustic delay elementswhich are used in signal processing and oscillators based ondelay line. These waveguides can also be used in impedancematching and focusing in acoustic photography like medicalultrasound [4].

By utilizing photonic crystals, power efficiency of solar cells[19]–[22] and extraction efficiency of light emitting diodes[23]–[25] can be significantly improved. It is possible toembed a layer of PtC beneath miniaturized antennas manufac-tured on chips to increase the power of radiated waves throughreflecting the waves, which would otherwise penetrate into thesubstrate [26], [27]. PtC cavities can be used to construct ultra-fast low-threshold tiny lasers [28], [29], which find use in thefield of cavity quantum electrodynamics [30] as well. Likeacoustic waveguides, optical waveguides created in photoniccrystals are also widely used [31]–[34].

Numerous methods have been developed for PnCs analysis,most of them being in common with those of PtCs. The mostwell-known methods are plane wave expansion (PWE), finiteelement (FE) and finite difference time domain (FDTD) [35].Among these methods, PWE is more popular than the othertwo since it is much simpler to set up and suitable for periodicstructures. It is based on Bloch theorem and approximatingperiodic quantities by their Fourier series. Its shortcomingarises when the structure is not periodic nor invariant in oneor more dimensions or the wave does not propagate uniformlythroughout the whole crystal. For instance, PnC slab in notperiodic nor invariant along its thickness or surface acousticwaves (SAWs) propagate near the surface of a semi-infinitecrystal. In order to utilize this method in these circumstanceswe have to make some changes either in its standard formu-lation or in our structure.

A reliable and conventional approach for dealing with PnCslabs is to sandwich the slab between two vacuum layers andcreate a 3D structure by stacking such sandwiches on top ofeach other [36]. The obtained structure is then analyzed withfull 3D PWE method. Despite its accuracy, this method is

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time consuming and its convergence is normally non-uniform.Nevertheless, its accuracy is not desirable in solid-solid crys-tals of large elastic mismatch. To resolve this problem anew formulation has been developed based on inverse rulein contrast to conventional Laurent’s rule to obtain Fouriercoefficients [37], [38].

Sigalas and Economoua tried to combine PWE with platetheory to calculate PnC slabs band structure [39]. Althoughit is a good approach for thin slabs, its error increases as theslab becomes thicker. Tanaka and Tamura proposed a modifiedPWE method for SAWs analysis [40]. They supposed surfacewaves to decay exponentially by getting away from the crystalsurface. This idea was then utilized to study PnC slabs [41]since they are similar to semi-infinite crystals as they alsohave free surfaces. It is not an efficient method for crystal slabanalysis because it needs a numerical root searching procedurewhich is time consuming and not straightforward at all.

There exists another variation of the PWE method for PnCslab analysis, proposed by Hsu and Wu, which is based onMindlin’s plate theory [42]. It has been claimed that thismethod is much faster than full 3D PWE method but it isrestricted to thin slabs and low frequencies. Much research andeffort has been put into this domain to overcome the limitationsof PWE while avoiding coding complexity of fintie elementand boundary element formalisms [43]–[46].

In this article we introduce a new modified PWE methodwhich can be employed to find modes of both photonic andphononic modes of phoxonic crystal slabs which are composedof isotropic materials. It is shown to be much faster than thestandard full 3D PWE method. Then we investigate optome-chanical interactions in an optomechanical slab waveguide thatis used to design a recently proposed non-reciprocal device[47] and can be as well utilized as an optical insulator.

II. MODIFIED PWE METHOD

PWE method is based on approximating wave profile byFourier series. Although this approximation is inevitable fora real periodic crystal, it seems not to be so efficient for acrystal slab along its normal axis. In this section, we introducefunctions which can replace Fourier series along the off-plane,or normal component to the slab. We furthermore suppose thatthe wave profile along the slab normal axis does not changesignificantly when we carve out some holes or insert someinclusions inside the slab, thus enabling the estimation of waveprofile along the normal axis by that of a plain slab. Then,Fourier series is used to expand the waves along the two otherorthogonal in-plane axes.

A. Phononic Crystal SlabTo begin, we first have to find modes of a plain elastic plate.

As stated above, the plate is assumed to be made of isotropicmaterial. In this case wave equation is written as [48]

(λ+ µ)∇∇ · u + µ∇2u = ρu,

which after expanding is equivalent to3∑j=1

[(λ+ µ)

∂2uj∂xj∂xi

+ µ∂2ui∂x2

j

]= ρui, (i = 1, 2, 3). (1)

In this equation xi and ui are the i’th component of positionand displacement respectively, λ and µ are Lame constants andρ is the mass density of the plate. It is simple to show that twokinds of waves can propagate inside this plate, longitudinalcoupled with vertical transverse, and horizontal transverse. Weassume the plate normal direction to be along the z−axis andthe plate to be surrounded between z = −t/2 and z = t/2surfaces. We also assume the wave to propagate along thex−axis. So we can write u(r) = u(x, z). In the horizontaltransverse wave, displacement is just along the y−axis and inthe other one it is along x− and z−axes. By solving waveequation, horizontal transverse wave is obtained as [48]

uy(x, z) =

A1 cos(kz)e−ı(qx−Ωt) SymmetricA2 sin(kz)e−ı(qx−Ωt) Anti-symmetric

,

(2)in which Ω =

√(k2 + q2)c22 is the angular frequency, c2 =√

µ/ρ is the phase velocity of transverse wave and q is thewave number. Applying boundary conditions, we conclude,

k = nπ/t, n = 0, 1, 2, . . . .

The wave equation and boundary conditions give [48]

ux = [A2ıq cos(k1z)−B1k2 cos(k2z)] e−ı(qx−Ωt), (3a)

uz = [−A2k1 sin(k1z) +B1ıq sin(k2z)] e−ı(qx−Ωt), (3b)

tan(k2t/2)

tan(k1t/2)= − 4k1k2q

2

(q2 − k22)

2 , (3c)

A2

B1= −ı 2qk2 cos(k2t/2)

(q2 − k22) cos(k1t/2)

, (3d)

for symmetric longitudinal and vertical transverse wave. Theanti-symmetric wave becomes [48]

ux = [A1ıq sin(k1z) +B2k2 sin(k2z)] e−ı(qx−Ωt), (4a)

uz = [A1k1 cos(k1z) +B2ıq cos(k2z)] e−ı(qx−Ωt), (4b)

tan(k2t/2)

tan(k1t/2)= −

(q2 − k2

2

)24k1k2q2

, (4c)

A1

B2= ı

2qk2 sin(k2t/2)

(q2 − k22) sin(k1t/2)

. (4d)

The dispersion relation for longitudinal and vertical transversewave is then obtained implicitly as

tan(k2t/2)

tan(k1t/2)= −

[4k1k2q

2

(q2 − k22)

2

]±1

,

k21 = Ω2/c21 − q2,

k22 = Ω2/c22 − q2, (5)

where c1 =√

(λ+ 2µ)/ρ is the phase velocity of longitudinalwave. Fig. 1 shows dispersion curves of both horizontaltransverse and longitudinal coupled with vertical transversewaves for an isotropic Silicon plate.

Now, if we carve out a periodic pattern of holes inside thisplate, according to Bloch theorem we may expect displacementwave to be of the form

u(r) = e−ıq·rxy u(rxy, z) = e−ıq·rxy∑G

uG(z)e−ıG·rxy

3

(a) (b)AntisymmetricSymmetric

20 10 0 10 200

2

4

6

8

0

2

4

6

8

Normalized wave number

Normalizedfrequency

Fig. 1. Dispersion curves of an isotropic Silicon plate for (a) longitudinalcoupled with vertical transverse and (b) horizontal transverse waves. Si Lameconstants and mass density equal λ = 44.27GPa, µ = 80GPa and ρ =2329kg/m3. Plate thickness is assumed to be t = 0.5a where a is an arbitrarylength. Insets of the figure show both types of the waves propagating in theplate.

=∑G

[uGx

(z)x+ uGy(z)y + uGz

(z)z]e−ı(q+G)·rxy ,

(6)

where u(rxy, z) is a periodic function of x and y for everyvalue of −t/2 < z < t/2, q is the Bloch wave vector and Gis a reciprocal lattice vector.

Two types of waves propagate inside a PnC slab: verti-cal displacement like (VD-like) and horizontal displacementlike (HD-like). Displacement of the mid-plane of the slabis perpendicular to slab surface in VD-like and parallel toslab surface in HD-like wave. By adopting wave profile oflongitudinal coupled with vertical transverse modes in a plainslab ((3a,b) for HD-like and (4a,b) for VD-like) we nowapproximate displacement components in VD-like modes as

uGx(z) ' AGx

sin(k1Gz) +BGx

sin(k2Gz),

uGy(z) ' AGy

sin(k1Gz) +BGy

sin(k2Gz),

uGz(z) ' AGz

cos(k1Gz) +BGz

cos(k2Gz), (7)

and in HD-like modes as

uGx(z) ' AGx cos(k1Gz) +BGx

cos(k2Gz),

uGy(z) ' AGy

cos(k1Gz) +BGy

cos(k2Gz),

uGz(z) ' AGz

sin(k1Gz) +BGz

sin(k2Gz). (8)

In these equations we set k21G

= Ω(|q + G|)2/c21 − |q + G|2and k2

2G= Ω(|q+G|)2/c22−|q+G|2, where Ω(q) of VD-like

mode is obtained from the first anti-symmetric dispersion bandand that of HD-like mode from the first symmetric dispersion

band of a plain slab with the same material and thickness.Since λ and µ are position dependent, (1) should be writtenas

3∑j=1

[∂

∂xi

(λ∂uj∂xj

)+

∂

∂xj

(µ∂ui∂xj

)+

∂

∂xj

(µ∂uj∂xi

)]= ρui, (i = 1, 2, 3). (9)

In a compact form it can be written as

L|u(r)〉 = Ω2R|u(r)〉,

which is a generalized eigenvalue problem. We can find anupper bound for its eigenvalues by replacing u(r) in

Ω2 =〈u(r)|L|u(r)〉〈u(r)|R|u(r)〉

,

by its approximate value from (6)-(8) and searching foroptimum values of AG and BG coefficients. It can be shownthat the optimum value of these coefficients are obtained bysolving the following eigenvalue problem,

M

AGx

BGx

AGy

BGy

AGz

BGz

G

+ Ω2N

AGx

BGx

AGy

BGy

AGz

BGz

G

= 0.

In this equation M and N are obtained by calculating innerproducts 〈u(r)|L|u(r)〉 and 〈u(r)|R|u(r)〉, respectively, andextracting AG and BG coefficients. The VD-like and HD-like band structure of a hexagonal lattice PnC slab calculatedutilizing this method, which we call it variational from nowon, is compared with the result of 3D PWE method in Fig. 2.It can be seen that both methods give similar bands for lowerfrequencies (below 20th band). This is a special crystal which,as is investigated in [49], has both photonic and phononic bandgaps. Hence it can be used as a basis for phoxonic waveguide.The advantage of our method over 3D PWE method is itsfaster convergence. This fact is illustrated in Fig. 3. In thisfigure we have calculated the phase velocities of two modesof the crystal at two high symmetry points of its irreducibleBrillouin zone, M(5) (fifth mode at M point) and K(3) (thirdmode at K point) with our variational and 3D PWE methodsand plotted them versus their calculation times. We began withthree plane waves along x− and y−axes for both 3D PWEand our method, that is we set the bound of Σ expression ofBloch wave to N = 1. The number of plane waves alongz−axis for PWE method is equal to x−axis at each point ofthe figure. We then increased the number of plane waves by 2,(N = N + 1), at each successive points of the plot to obtainmore accurate results. The right most points of PWE methodare related to 13 plane waves (N = 6) along each axis whilethe right most points of our method are related to 19 planewaves (N = 9) along x− and y− axes.

The variational method converges faster because the fieldtrial function along z− axis in our method needs just two un-known parameters to be determined while in the standard 3DPWE method more Fourier series terms (unknown parameters)

4

are needed to obtain a trial function which its similarity to theexact eigen function is comparable with our method. In otherwords, our method uses a smarter trial function along z− axisthan 3D PWE method that with fewer unknown parameterscan give an acceptable trial function. The fewer unknownparameters our problem has, the smaller the final matrix whichits eigen value should be calculated would be and the faster itis prepared.

Fig. 2. VD-like and HD-like band structures of a slab of PnC with hexagonallattice. The crystal which can be seen in the upper inset of the figure is madeof isotropic Silicon with Lame constants and mass density λ = 44.27GPa,µ = 80GPa and ρ = 2329kg/m3. Slab thickness is t = a/

√3 where a

is the lattice constant. Radius of vacuum holes equals r = 0.425a/√

3. Thelower inset shows the crystal unit cell in reciprocal lattice with the irreducibleBrillouin zone highlighted in red. The 3D PWE bands are calculated with 11plane waves along each axis, i.e. totally 113 = 1331 plane waves.

B. Photonic Crystal Slab

In a similar manner, we can go through the same processto find a PtC slab modes. Here, the eigenvalue problem is

K|H(r)〉 =ω2

c2|H(r)〉, (10)

in which H(r) is the magnetic field, c is the light velocity invacuum, ω is the angular frequency of electromagnetic waveand K = ∇ × (η(r)∇× (·)), where η(r) = 1/εr(r) is therelative impermeability of environment. The magnetic fieldinside and around a PtC slab can be written as

H(r) = e−ıκ·rxyH(rxy, z) = e−ıκ·rxy∑G

HG(z)e−ıG·rxy

=∑G

[HG‖(z)eG‖ +HG⊥(z)eG⊥ +HGz (z)z

]· e−ı(κ+G)·rxy , (11)

3D PWE VD-like

3D PWE HD-like

Variational VD-like

Variational HD-like

10-2 10-1 100 101 102 103

4.6

4.8

5.0

5.2

5.4

5.6

2.3

2.5

2.7

2.9

3.1

3.3

calculation time s

Velocity

ms

Fig. 3. 3D PWE versus variational convergence speed. The comparison isdone in two high symmetry points, M(5) (fifth mode of M point) and K(3)

(third mode of K point). The leftmost points are related to three plane waves(N = 1) for each direction in plane wave expansion formalism. N hasincreased by 1 for each point after that. Left and right vertical axes which showthe phase velocity of modes are related to red and blue curves respectively.All calculations have been performed on a personal computer equipped withan intel i7 processor.

where we have expanded its elements on the coordinatesystem of wave vectors. Here, this local coordinate systemis represented by the unit vectors lying in the mid-plane ofthe slab and parallel with (perpendicular to) κ + G as eG‖

(eG⊥ ). Adopting magnetic field profile around a dielectricslab waveguide, we choose our ansatz magnetic field for thetransverse electric like (TE-like) mode as

HG‖(z) = AG‖fG‖(z),

HG⊥(z) = AG⊥fG⊥(z),

HGz (z) = −ı|κ+ G|AG‖fGz (z), (12)

where AG‖ and AG⊥ are coefficients to be determined and

fG‖(z) =

sin(kG‖z) |z| ≤ t/2sin(kG‖ t/2)e−αG(z−t/2) z > t/2− sin(kG‖ t/2)eαG(z+t/2) z < −t/2

,

fG⊥(z) =

sin(kG⊥z) |z| ≤ t/2sin(kG⊥ t/2)e−αG(z−t/2) z > t/2− sin(kG⊥ t/2)eαG(z+t/2) z < −t/2

,

fGz (z) =

cos(kG‖z)/kG‖ |z| ≤ t/2sin(kG‖ t/2)e−αG(z−t/2)/αG z > t/2sin(kG‖ t/2)eαG(z+t/2)/αG z < −t/2

.

In these equations αG =√|κ+ G|2 − ω(|κ+ G|)2 and

ω(|κ + G|) is estimated from the first TE dispersion bandof a dielectric slab waveguide with the same material andthickness. This estimation is the best one for guided modes,i.e. the modes below the light cone. kG‖ and kG⊥ are alsocalculated from the following implicit relations.

αG = kG‖ tan(kG‖ t/2),

αG = −kG⊥ cot(kG⊥ t/2)/εreff ,

where εreff is the effective permittivity of the PtC slab. Thesefunctions are chosen in a manner to satisfy ∇ · H(r) = 0.

5

For transverse magnetic like (TM-like) modes, trial functionsare similar to the above functions except that fG‖(z) andfG⊥(z) should be even functions and fGz (z) should be anodd function. Also, kG‖ and kG⊥ are calculated from

αG = −kG‖ cot(kG‖ t/2)

αG = kG⊥ tan(kG⊥ t/2)/εreff .

In this mode ω(|κ + G|) is estimated from the first TMdispersion band of the dielectric slab waveguide mentionedabove. To test this method, we calculated the band structureof the crystal slab analyzed in the previous sub-section withboth 3D PWE and the proposed methods. Fig. 4a comparesthe results.

Since both of the methods are based on variational principle,each of them gives a lower frequency for a mode is moreaccurate in that mode. Inspecting Fig. 4a, reveals that ourproposed method is more accurate in plenty of modes. In TE-like mode, our method results in bands which are very similarin shape to that of 3D PWE, while in TM-like mode, shapesof bands differ slightly. However, our method is more accuratein TM-like than TE-like modes.

In Fig. 4b we have compared the mode profiles calculatedwith variational and 3D PWE methods at two high symmetrypoints of Brillouin zone. It can be seen the results are verysimilar.

III. PHOTO-ELASTIC INTERACTION IN SLABOPTOMECHANICAL WAVEGUIDES

Interaction of electromagnetic and elastic waves origi-nates from different mechanisms among them are electrostric-tion, magnetostriction, radiation pressure, piezoelectricity,photoelasticity, and interface displacement [50]. Electrostric-tion, magnetostriction and radiation pressure are mechanismsthrough which electromagnetic wave affects elastic wave. Inpiezoelectric materials electromagnetic and elastic waves canhave mutual effects on each other through piezoelectricityphenomenon. The last two mechanisms, i.e. photoelasticityand interface displacement are responsible for effects of elasticwave on electromagnetic wave [51], [52]. In general, for piezo-electric nano-structured materials, radiation pressure termswhich arise from moving boundaries [12], [47], [53] arenegligible compared to the bulk electrostrictive forces arisingfrom photo-elastic interactions. For non-piezoelectric media,radiation pressure and electrostriction have been shown tocontribute comparably, or at least within the same order ofmagnitude [12]. A coupling coefficient between electromag-netic and elastic waves can be defined for each of thesemechanisms. Different expressions have been presented for thecoupling coefficient originated from photoelastic effect [47],[51], [52]. In this section we present a new expression for thiscoefficient which can give some insight about electromagneticwave evolution in time in the presence of elastic wave. Weuse this expression to illustrate an application of our proposedmethod.

Now suppose we have an optomechanical slab waveguide inwhich both elastic and electromagnetic wave propagate along

x−axis. We start with (10) but with a perturbed operator as

(K + ∆K)H′(r, t) = − 1

c2∂2

∂t2H′(r, t), (13)

in which ∆K = ∇× (∆η(r, t)∇× (·)) and

∆η = P : ε⇐⇒ ∆ηij =

3∑k,l=1

Pijklεkl. (14)

In (14), ∆η is the change in impermeability tensor caused bystrain wave ε =

[(∇u)T +∇u

]/2 and P is the photo-elastic

tensor. With regards to (6) we can write ∆η as

∆η(r, t) = v(r)e−ı(qx−Ωt), (15)

where v(r) is a periodic tensor of x and q is the elastic wavenumber. Since K is a Hermitian operator, its eigenstates builda complete set on which we can expand H′(r, t) as

H′(r, t) =∑κ,n

αn,κ(t)Hn,κ(rxy, z)e−ı(κx−ωn,κt), (16)

where the sum is over all wave numbers in the irreducibleBrillouin zone and all eigenstates corresponding to each valueof wave number. Here, we have discretized the wavenumberjust for the sake of simplicity. It can be replaced by anintegral as

(∑κ ←→

aπ

∫ π/a0

dκ)

, where a is the length ofour crystal slab waveguide unit cell along x axis. αn,κ(t) aretime-dependent constants to be determined. We now insertH′(r, t) from (16) into (13) to obtain

∆K∑κ,n

αn,κ(t)Hn,κ(rxy, z)e−ı(κx−ωn,κt)

= − 1

c2

∑κ,n

[2ıωn,καn,κ(t) + αn,κ(t)] Hn,κ(rxy, z)

· e−ı(κx−ωn,κt)

' − 1

c2

∑κ,n

2ıωn,καn,κ(t)Hn,κ(rxy, z)e−ı(κx−ωn,κt). (17)

The second equation is obtained with the assumption,αn,κ(t) ωn,κ. By inner producting both sides of (17) bya specific eigenstate of K such as Hm,`(rxy, z)e

−ı(`x−ωm,`t)

we get

1

lx

∫V

∑κ,n

αn,κ(t)H∗m,`(r)eı(`x−ωm,`t)

·∆K[Hn,κ(r)e−ı(κx−ωn,κt)

]dr

' − 1

c22ıωm,`αm,`(t). (18)

In this equation, V is the volume in which we take the integraland should be enlarged to occupy the whole space and lx isthe dimension of V along the x−axis. We also have supposedthe eigenstates to be normalized, i.e.

1

lx

∫V

H∗m,`(r)eı`xHn,κ(r)e−ıκx dr = δn,mδκ,`,

where δn,m and δκ,` are Kronecker delta functions. By affect-

6

(a) (b)

Fig. 4. (a) Band structure of a hexagonal PtC slab made of isotropic Silicon with relative permittivity equals εr = 11.9. The slab thickness is t = a/√

3 andthe radius of vacuum holes equals r = 0.425a/

√3. Results of both variational and 3D PWE methods are plotted in this figure. Green shaded area shows the

light cone and the lower inset shows the crystal unit cell in reciprocal lattice with the irreducible Brillouin zone highlighted in red. The 3D PWE bands arecalculated with 11 plane waves along each axis, i.e. totally 113 = 1331 plain waves. (b) Magnetic field profiles at two high symmetry points of Brillouinzone computed with the variational and 3D PWE methods. Figures (b1), (b3), (b5) and (b7) are calculated with variational method and the others are resultsof PWE. (b1) and (b2) show TE-like Hz of M(4) (fourth mode in M point) in the mid-plane of the crystal unit cell. (b5) and (b6) illustrate Hz in the planeperpendicular the to crystal slab surface at the specified cross sections of (b1) and (b2) respectively. (b3) and (b4) show TM-like H of K(3) (third mode inK point) in the mid-plane of the crystal unit cell. (b7) and (b8) are Hz in the plane perpendicular to the crystal slab surface at the specified cross sectionsof (b3) and (b4) respectively.

ing ∆K, the equation becomes

− 2ı

c2ωm,`αm,`(t)

' 1

lx

∫V

∑κ,n

αn,κ(t)H∗m,`(r)eı`x · ∇ ×[v(r)e−ıqx∇×

(Hn,κ(r)e−ıκx

) ]eı(ωn,κ−ωm,`+Ω)t dr

=∑κ,n

ξm,n,`,καn,κ(t)eı(ωn,κ−ωm,`+Ω)t, (19)

where

ξm,n,`,κ =1

lx

∫V

H∗m,`(r)eı`x·

∇ ×[v(r)e−ıqx∇×

(Hn,κ(r)e−ıκx

) ]dr,

is the coupling strength between (`,m) and (κ, n) modesthrough photoelastic effect. Replacing αm,`(t) by βm,`(t) =αm,`(t)e

ıωm,`t and doing some simplifications, (19) becomes

βm,`(t) = ıc2

2ωm,`

∑κ,n

ξm,n,`,κβn,κ(t)eıΩt

+ ıωm,`βm,`(t), (20)

(20) is a system of ordinary differential equations. Supposethe system is initially in (κ1, 1) state, then the stronger the cou-

pling between an arbitrary state and this initial state, the fasterthe growth of its coefficient in time. The first step in solvingthis system of equations is to calculate the coupling strengthbetween different states. In the next subsection, we utilizeour proposed method in previous sections for this purpose todemonstrate its usefulness in computing the optomechanicalcoupling of photonic and phononic modes [47].

A. Application Example

To demonstrate one of the applications of our modifiedPWE method, we compute the coupling strength due to pho-toelasticity between different modes of a PtC slab waveguidecaused by phononic modes. This phonomenon is at the heartof optomechanical interactions in phoxonic crystals and canbe theoretically investigated by coupled-mode theory oncethe propagating eigenmodes of the slab are known [47].Conventional PWE method is inappropriate here since it is toomuch time and memory consuming, and also suffers from non-uniform convergence. It means that the numerical solutionsstart to oscillate around the values ideally obtained at infinitenumber of expansion terms instead of exponential convergence[54]. At first, we need to design a simultaneous photonicand phononic (phoxonic) waveguide. For this purpose we usethe hexagonal crystal slab analyzed in previous sections. Ascan be seen in Figs. 2 and 4, this crystal has both photonicand phononic forbidden gaps in its band structures, so it is a

7

good candidate to be used as the base of our waveguide. Byinserting a gap in the mentioned crystal along ΓK direction,and following the design instructions given elsewhere [47], weobtain a phoxonic waveguide as is shown in the inset of Fig.5a.

It can be proved that VD-like elastic waves cannot coupleany two optical modes in a phoxonic slab waveguide throughphotoelasticity, i.e. the coupling strength between any twooptical modes through a VD-like elastic wave equals zero.This is actually quite expected from overlap integrals of op-tomechanical interactions [47]. Hence, we calculated photonicTM-like and phononic HD-like band structures of the designedwaveguide using our proposed variational method in this work.Figs. 5a and 5b show these band structures. To compute theoptomechanical coupling strength we need to obtain ∆η. Weused the photo-elastic tensor of Silicon reported in [55] for thispurpose. The independent photo-elastic tensor elements of Sil-icon are p1111 = −0.094, p1122 = 0.017 and p1212 = −0.051.The coupling strengths ξS1,S3 and ξS2,S3 through phononicmodes T1, T2 and T3 are gathered in Table I. These valuesare calculated after normalizing phononic waves profiles, i.e.

1

a3

∫ t/2

−t/2

∫ ∞−∞

∫ a

0

|u(r)|2 dxdy dz = a2.

The specified modes are marked in Figs. 5a and 5b.

TABLE ICOUPLING STRENGTH BETWEEN OPTICAL WAVES THROUGH ELASTIC

WAVES DUE TO PHOTOELASTIC EFFECT

ξS1,S3 ξS2,S3

T1 0.9287 + ı0.0875 0

T2 −0.0046 + ı0.7713 0

T3 0 0.1011− ı0.2290

IV. CONCLUSION

A new approach for analysis of photonic and phononiccrystal slab was proposed. This approach which is based onPWE method converges much faster than the conventional3D PWE method and is uniformly convergent. Furthermore,it lets us analyze crystal slabs with large unit cells by apersonal computer unlike 3D PWE method which needs a hugeamount of memory in such cases. An application example waspresented, where we constructed a non-reciprocal photonicdevice utilizing a phoxonic waveguide and calculated thecoupling strength between different modes with the proposedmethod. Further details and deep discussions on this subjectare being intended as a separate article to be published soon[56].

REFERENCES

[1] E. Yablonovitch, “Inhibited spontaneous emission in solid-state physicsand electronics,” Phys. Rev. Lett., vol. 58, pp. 2059–2062, May 1987.

[2] S. John, “Strong localization of photons in certain disordered dielectricsuperlattices,” Phys. Rev. Lett., vol. 58, pp. 2486–2489, June 1987.

[3] V. Narayanamurti, H. L. Stormer, M. A. Chin, A. C. Gossard, andW. Wiegmann, “Selective transmission of high-frequency phonons bya superlattice: The “dielectric” phonon filter,” Phys. Rev. Lett., vol. 43,no. 27, pp. 2012–2016, December 1979.

[4] R. H. O. III and I. El-Kady, “Microfabricated phononic crystal devicesand applications,” Meas. Sci. Technol., vol. 20, no. 1, p. 012002,November 2009.

[5] T. K. Paraıso, M. Kalaee, L. Zang, H. Pfeifer, F. Marquardt, andO. Painter, “Position-squared coupling in a tunable photonic crystaloptomechanical cavity,” Phys. Rev. X, vol. 5, p. 041024, November 2015.

[6] J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent opticalwavelength conversion via cavity optomechanics,” Nature Commun,vol. 3, p. 1196, November 2012.

[7] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity Optome-chanics: Nano- and Micromechanical Resonators Interacting with Light.Berlin, Germany: Springer, 2014.

[8] M. Aspelmeyer, T. J. Kippenberg, and F. Marquard, “Cavity optome-chanics,” Rev. Mod. Phys., vol. 86, pp. 1391–1452, December 2014.

[9] M. Kalaee, T. K. paraıso, H. Pfeifer, and O. Painter, “Design of aquasi-2D photonic crystal optomechanical cavity with tunable, large χ2-coupling,” Optics Express, vol. 24, no. 19, pp. 21 308–21 328, September2016.

[10] A. Oseev, M. Zubtsov, and R. Lucklum, “Gasoline properties determi-nation with phononic crystal cavity sensor,” Sensors and Actuators B:Chemical, vol. 189, no. 1, pp. 208–212, December 2013.

[11] A. Khelif, A. Choujaa, S. Benchabane, B. Djafari-Rouhani, andV. Laude, “Guiding and bending of acoustic waves in highly confinedphononic crystal waveguides,” Appl. Phys. Lett., vol. 80, no. 22, pp.4400–4402, May 2004.

[12] B. Djafari-Rouhani, S. El-Jallal, and Y. Pennec, “Phoxonic crystals andcavity optomechanics,” Comptes Rendus Phys, vol. 17, no. 5, pp. 555–564, August 2016.

[13] A. H. Safavi-Naeini and O. Painter, “Design of optomechanical cavitiesand waveguides on a simultaneous bandgap phononic-photonic crystalslab,” Optics Express, vol. 18, no. 14, pp. 14 926–14 943, June 2010.

[14] A. Khelif, A. Choujaa, S. Benchabane, B. Djafari-Rouhani, andV. Laude, “Guiding and bending of acoustic waves in highly confinedphononic crystal waveguides,” App. Phys. Lett., vol. 84, no. 22, pp.4400–4402, May 2004.

[15] M. Torres, F. R. M. de Espinosa, and J. L. Aragon, “Ultrasonic wedgesfor elastic wave bending and splitting without requiring a full band gap,”Phys. Rev. Lett., vol. 86, no. 19, pp. 4282–4285, May 2001.

[16] S. Mohammadi, A. A. Eftekhar, W. D. Hunt, and A. Adibi, “High-Q micromechanical resonators in a two-dimensional phononic crystalslab,” Appl. Phys. Lett., vol. 94, no. 1, p. 051906, December 2009.

[17] Z. Yu and S. Fan, “Complete optical isolation created by indirectinterband photonic transitions,” nat. photon., vol. 3, pp. 91–94, February2009.

[18] Y. M. Soliman, M. F. Su, Z. C. Leseman, C. M. Reinke, I. El-Kady,and R. H. Olsson, “Phononic crystals operating in the gigahertz rangewith extremely wide band gaps,” Appl. Phys. Lett., vol. 97, p. 193502,November 2010.

[19] D. Zhou and R. Biswas, “Photonic crystal enhanced light-trapping inthin film solar cells,” J. Appl. Phys., vol. 103, no. 9, p. 093102, May2008.

[20] J. G. Mutitu, S. Shi, C. Chen, T. Creazzo, A. Barnett, C. Honsberg, andD. W. Prather, “Thin film silicon solar cell design based on photoniccrystal and diffractive grating structures,” Optics Express, vol. 16, no. 19,pp. 15 238–15 248, September 2008.

[21] L. Zeng, P. Bermel, Y. Yi, B. A. Alamariu, K. A. Broderick, J. Liu,C. Hong, X. Duan, J. Joannopoulos, and L. C. Kimerling, “Demon-stration of enhanced absorption in thin film Si solar cells with texturedphotonic crystal back reflector,” Appl. Phys. Lett., vol. 93, no. 22, p.221105, December 2008.

[22] P. Bermel, C. Luo, L. Zeng, L. C. Kimerling, and J. D. Joannopou-los, “Improving thin-film crystalline silicon solar cell efficiencies withphotonic crystals,” Optics Express, vol. 15, no. 25, pp. 16 986–17 000,December 2007.

[23] H. Ichikawa and T. Baba, “Efficiency enhancement in a light-emittingdiode with a two-dimensional surface grating photonic crystal,” Appl.Phys. Lett., vol. 84, no. 4, pp. 457–459, January 2004.

[24] M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, andE. Yablonovitch, “Spontaneous emission extraction and purcell enhance-ment from thin-film 2-D photonic crystals,” J. Lightwave Technol.,vol. 17, no. 11, pp. 2096–2112, November 1999.

8

(a) (b) (c)

Fig. 5. Band structure of the intended phoxonic waveguide for (a) photonic TM-like and (b) phononic HD-like modes. The waveguide shown in the inset of(a) and (b) is created by introducing a gap with width w = (1.5 + 1/

√3)a along ΓK direction in the hexagonal crystal studied in previous sections. The red

region in the waveguide of inset of (a) and (b) illustrate the unit cell considered for the waveguide analysis. Green shaded area in (a) shows the light coneand the blue regions are the extended modes of the crystal. Three photonic and three phononic modes are marked in (a) and (b) respectively for couplingcalculations. The wave profile of these modes in the mid-plane (z = 0) of the crystal slab is plotted in (c). The difference in wave number of S1 and S3

equals the wave number of T1, T2 and T3.

[25] S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and E. F. Schubert, “Highextraction efficiency of spontaneous emission from slabs of photoniccrystals,” Phys. Rev. Lett., vol. 78, no. 11, pp. 3294–3297, April 1997.

[26] E. R. Brown, C. D. Parker, and E. Yablonovitch, “Radiation propertiesof a planar antenna on a photonic-crystal substrate,” J. Opt. Soc. Am. B,vol. 10, no. 2, pp. 404–407, February 1993.

[27] B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas,G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna witha very high directivity,” J. Appl. Phys., vol. 87, no. 1, pp. 603–605,January 2000.

[28] H. Hirayama, T. Hamano, and Y. Aoyagi, “Novel surface emitting laserdiode using photonic bandgap crystal cavity,” Appl. Phys. Lett., vol. 69,no. 6, pp. 791–793, June 1996.

[29] H. Altug, D. Englund, and J. Vuckovic, “Ultrafast photonic crystalnanocavity laser,” Nature Physics, vol. 2, pp. 484–488, July 2006.

[30] M. H. Aram and S. Khorasani, “Scalable cavity quantum electrodynam-ics system for quantum computing,” Journal of Modern Physics, vol. 6,pp. 1467–1477, September 2015.

[31] A. Chutinan and S. Noda, “Highly confined waveguides and waveguidebends in three-dimensional photonic crystal,” Appl. Phys. Lett., vol. 75,no. 24, pp. 3739–3741, December 1999.

[32] S.-Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos,“Experimental demonstration of guiding and bending of electromagneticwaves in a photonic crystal,” Science, vol. 282, pp. 274–276, October1998.

[33] S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linearwaveguides in photonic-crystal slabs,” Phys. Rev. B, vol. 62, no. 12, pp.8212–8222, September 2000.

[34] M. Loncar, T. Doll, J. Vuckovic, and A. Scherer, “Design and fabricationof silicon photonic crystal optical waveguides,” J. Lightwave Technol.,vol. 18, no. 10, pp. 1402–1411, October 2000.

[35] C. M. Reinke, A. A. Jafarpour, B. Momeni, M. Soltani, S. Khorasani,A. Adibi, Y. Xu, and R. K. Lee, “Nonlinear finite-difference time-domain simulation of χ(2) and χ(3) effects in two-dimensional photoniccrystals,” Journal of Lightwave Technology, vol. 24, no. 1, pp. 624–634,January 2006.

[36] Z. Hou and B. M. Assouar, “Modeling of lamb wave propagation inplate with two-dimensional phononic crystal layer coated on uniformsubstrate using plane-wave-expansion method,” Phys. Lett. A, vol. 372,pp. 2091–2097, March 2008.

[37] Y. Cao, Z. Hou, and Y. Liu, “Convergence problem of plane-waveexpansion method for phononic crystals,” Phys. Lett. A, vol. 327, pp.247–253, June 2004.

[38] M. G. Baboly, Y. Soliman, M. F. Su, C. M. Reinke, Z. C. Leseman,and I. El-Kady, “Enhanced plane wave expansion analysis for the bandstructure of bulk modes in two-dimensional high-contrast solidsolidphononic crystals,” Photonics and Nanostructures Fundamentals andApplications, vol. 12, pp. 487–492, November 2014.

[39] M. M. Sigalas and E. N. Economou, “Elastic waves in plates withperiodically placed inclusions,” J. Appl. Phys., vol. 75, pp. 2845–2850,March 1994.

[40] Y. Tanaka and S. ichiro Tamura, “Surface acoustic waves in two-dimensional periodic elastic structures,” Phys. Rev. B, vol. 58, pp. 7958–7965, September 1998.

[41] C. Charles, B. Bonello, and F. Ganot, “Propagation of guided elasticwaves in 2D phononic crystals,” Ultrasonics, vol. 44, pp. e1209–e1213,December 2006.

[42] J.-C. Hsu and T.-T. Wu, “Efficient formulation for band-structure cal-culations of two-dimensional phononic-crystal plates,” Phys. Rev. B,vol. 74, p. 144303, October 2006.

[43] S. A. H. Nekuee, M. Akbari, and K. Mehrany, “A novel method for bandstructure analysis of photonic crystal slabs,” IEEE Photonics Journal,vol. 3, no. 6, pp. 1111–1122, December 2011.

[44] S. Shi, C. Chen, and D. W. Prather, “Plane-wave expansion methodfor calculating band structure of photonic crystal slabs with perfectlymatched layers,” J. Opt. Soc. Am. A, vol. 21, no. 9, pp. 1769–1775,Sepember 2004.

[45] M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett., vol. 81,no. 7, pp. 1163–1165, August 2002.

[46] S. Shi, C. Chen, and D. W. Prather, “Revised plane wave method fordispersive material and its application to band structure calculations of

9

photonic crystal slabs,” Appl. Phys. Lett., vol. 56, no. 4, p. 043104,January 2005.

[47] S. Khorasani, “Coupled mode theory of optomechanical crystals,” IEEEJournal of Quantum Electronics, vol. 52, no. 10, p. 6100406, October2016.

[48] K. Graff, Wave Motion in Elastic Solids, ser. Dover Books on PhysicsSeries. 31 East second street, Mineola, N. Y. 11501: Dover Publications,1975.

[49] S. Mohammadi, A. A. Eftekhar, A. Khelif, and A. Adibi, “Simultaneoustwo-dimensional phononic and photonic band gaps in opto-mechanicalcrystal slabs,” Optics Express, vol. 18, no. 9, pp. 9164–9172, April 2010.

[50] Y. Pennec, V. Laude, N. Papanikolaou, B. Djafari-Rouhani, M. Oudich,S. E. Jallal, J. C. Beugnot, J. M. Escalante, and A. Martnez, “Modelinglight-sound interaction in nanoscale cavities and waveguides,” Nanopho-tonics, vol. 3, pp. 413–440, September 2014.

[51] M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter,“Optomechanical crystals,” Nature, vol. 462, no. 7269, pp. 78–82,October 2009.

[52] B. Djafari-Rouhani, S. El-Jallal, M. Oudich, and Y. Pennec, “Optome-chanic interactions in phoxonic cavities,” AIP Advances, vol. 4, no. 12,p. 124602, December 2014.

[53] K. C. Balram, M. I. Davano, J.-D. Song, and K. Srinivasan, “Coherentcoupling between radiofrequency, optical and acoustic waves in piezo-optomechanical circuits,” Nature Photon., vol. 10, pp. 346–352, March2016.

[54] M. H. Aram and S. Khorasani, “Novel variational approach for analysisof photonic crystal slabs,” Mater. Res. Exp., vol. 2, no. 5, p. 056201,April 2015.

[55] D. K. Biegelsen, “Photoelastic tensor of silicon and the volume de-pendence of the average gap,” Phys. Rev. Lett., vol. 32, no. 21, pp.1196–1199, May 1974.

[56] M. H. Aram and S. Khorasani, “Optical wave evolution due to inter-action with elastic wave in a phoxonic crystal slab waveguide,” to bepublished.