4 A Solid-State Theoretical Approach to the Optical Properties of Photonic Crystals K.Busch',F.Hagmann',D.Hermann',S.F.Mingaleev1,2, and M.Schillinger2 1 Institut fir Theorie der Kondensierten Materie,Universitat Karlsruhe,76128 Karlsruhe,Germany 2 Bogolyubov Institute for Theoretical Physics,03143 Kiev,Ukraine Abstract.In this chapter,we outline an efficient approach to the calculation of the optical properties of Photonic Crystals.It is based on solid state theoreti- cal concepts and exploits the conceptual similarity between electron waves prop- agation in electronic crystals and electromagnetic waves propagation in Photonic Crystals.Based on photonic bandstructure calculations for infinitely extended and perfectly periodic systems,we show how defect structures can be described through an expansion of the electromagnetic field into optimally localized photonic Wannier functions which have encoded in themselves all the information of the underlying Photonic Crystals.This Wannier function approach is supplemented by a multipole expansion method which is well-suited for finite-sized and disordered structures.To illustrate the workings and efficiency of both approaches,we consider several defect structures for TM-polarized radiation in two-dimensional Photonic Crystals. 4.1 Introduction The invention of the laser turned Optics into Photonics:This novel light source allows one to generate electromagnetic fields with previously unattain- able energy densities and temporal as well as spatial coherences.As a result, researchers have embarked on a quest to exploit these properties through perfecting existing and creating novel optical materials with tailor made properties.A particular prominent example is the development of low-loss optical fibers which form the backbone of today's long-haul telecommunica- tion systems [4.1].With the recent advances in micro-fabrication technolo- gies,another degree of freedom has been added to the flexibility in designing photonic systems:Microstructuring dielectric materials allows one to obtain control over the flow of light on lengths scales of the wavelength of light it- self.For instance,the design of high-quality ridge waveguiding structures has facilitated the realization of functional elements for integrated optics such as beamsplitters and Mach-Zehnder interferometers [4.2]. The past two decades have witnessed a strongly increased interest in a novel class of micro-structured optical materials.Photonic Crystals (PCs) consist of a micro-fabricated array of dielectric materials in two or three spatial dimensions.A carefully engineered combination of microscopic scat- K.Busch,F.Hagmann,D.Hermann,S.F.Mingaleev,and M.Schillinger,A Solid-State Theo- retical Approach,Lect.Notes Phys.642,55-74(2004) http://www.apringerlink.com/ C Springer-Verlag Berlin Heidelberg 2004
4 A Solid-State Theoretical Approach to the Optical Properties of Photonic Crystals K. Busch1, F. Hagmann1, D. Hermann1, S.F. Mingaleev1,2, and M. Schillinger2 1 Institut f¨ur Theorie der Kondensierten Materie, Universit¨at Karlsruhe, 76128 Karlsruhe, Germany 2 Bogolyubov Institute for Theoretical Physics, 03143 Kiev, Ukraine Abstract. In this chapter, we outline an efficient approach to the calculation of the optical properties of Photonic Crystals. It is based on solid state theoretical concepts and exploits the conceptual similarity between electron waves propagation in electronic crystals and electromagnetic waves propagation in Photonic Crystals. Based on photonic bandstructure calculations for infinitely extended and perfectly periodic systems, we show how defect structures can be described through an expansion of the electromagnetic field into optimally localized photonic Wannier functions which have encoded in themselves all the information of the underlying Photonic Crystals. This Wannier function approach is supplemented by a multipole expansion method which is well-suited for finite-sized and disordered structures. To illustrate the workings and efficiency of both approaches, we consider several defect structures for TM-polarized radiation in two-dimensional Photonic Crystals. 4.1 Introduction The invention of the laser turned Optics into Photonics: This novel light source allows one to generate electromagnetic fields with previously unattainable energy densities and temporal as well as spatial coherences. As a result, researchers have embarked on a quest to exploit these properties through perfecting existing and creating novel optical materials with tailor made properties. A particular prominent example is the development of low-loss optical fibers which form the backbone of today’s long-haul telecommunication systems [4.1]. With the recent advances in micro-fabrication technologies, another degree of freedom has been added to the flexibility in designing photonic systems: Microstructuring dielectric materials allows one to obtain control over the flow of light on lengths scales of the wavelength of light itself. For instance, the design of high-quality ridge waveguiding structures has facilitated the realization of functional elements for integrated optics such as beamsplitters and Mach-Zehnder interferometers [4.2]. The past two decades have witnessed a strongly increased interest in a novel class of micro-structured optical materials. Photonic Crystals (PCs) consist of a micro-fabricated array of dielectric materials in two or three spatial dimensions. A carefully engineered combination of microscopic scatK. Busch, F. Hagmann, D. Hermann, S.F. Mingaleev, and M. Schillinger, A Solid-State Theoretical Approach, Lect. Notes Phys. 642, 55–74 (2004) http://www.springerlink.com/ c Springer-Verlag Berlin Heidelberg 2004
56 K.Busch et al. tering resonances from individual elements of the periodic array and Bragg scattering from the corresponding lattice leads to the formation of a photonic bandstructure.In particular,the flexibility in material composition,lattice periodicity,symmetry,and topology of PCs allows one to tailor the photonic dispersion relations to almost any need.The most dramatic modification of the photonic dispersion in these systems occurs when suitably engineered PCs exhibit frequency ranges over which the light propagation is forbidden irrespective of direction [4.3,4.4].The existence of these so-called complete Photonic Band Gaps (PBGs)allows one to eliminate the problem of light leakage from sharply bent optical fibers and ridge waveguides.Indeed,using a PC with a complete PBG as a background material and embedding into such a PC a circuit of properly engineered waveguiding channels permits to create an optical micro-circuit inside a perfect optical insulator,i.e.an optical analogue of the customary electronic micro-circuit.In addition,the absence of photon states for frequencies in a complete PBG allows one to suppress the emission of optically active materials embedded in PCs.Furthermore, the multi-branch nature of the photonic bandstructure and low group veloc- ities associated with fat bands near a photonic band edge may be utilized to realize phase-matching for nonlinear optical processes and to enhance the interaction between electromagnetic waves and nonlinear and/or optically active material. These prospects have triggered enormous experimental activities aimed at the fabrication of two-dimensional(2D)as well as three-dimensional (3D) PC structures for telecommunication applications with PBGs in the near in- frared frequency range.Considering that the first Bragg resonance occurs when the lattice constant equals half the wavelength of light,fabrication of PCs with bandgaps in the near IR requires substantial technological re- sources.For 2D PCs,advanced planar microstructuring techniques borrowed from semiconductor technology can greatly simplify the fabrication process and high-quality PCs with embedded defects and waveguides have been fab- ricated in various material systems such as semiconductors [4.5-4.10,poly- mers [4.11,4.12],and glasses [4.13,4.14].In these structures,light experiences PBG effects in the plane of propagation,while the confinement in the third direction is achieved through index guiding.This suggests that fabricational imperfections in bulk 2D PCs as well deliberately embedding defect struc- tures such as cavities and waveguide bends into 2D PCs will inevitably lead to radiation losses into the third dimension.Therefore,it is still an open ques- tion as to whether devices with acceptable radiation losses can be designed and realized in 2D PCs.However,radiation losses can be avoided altogether if light is guided within the comlete PBG of 3D PCs and,therefore,the past years have seen substantial efforts towards the manufacturing of suitable 3D PCs.These structures include layer-by-layer structures [4.15,4.16,inverse opals [4.17-4.19]as well as the fabrication of templates via laser hologra- phy [4.20,4.21]and two-photon polymerization (sometimes also referred to as stereo-lithography)[4.22-4.24]
56 K. Busch et al. tering resonances from individual elements of the periodic array and Bragg scattering from the corresponding lattice leads to the formation of a photonic bandstructure. In particular, the flexibility in material composition, lattice periodicity, symmetry, and topology of PCs allows one to tailor the photonic dispersion relations to almost any need. The most dramatic modification of the photonic dispersion in these systems occurs when suitably engineered PCs exhibit frequency ranges over which the light propagation is forbidden irrespective of direction [4.3, 4.4]. The existence of these so-called complete Photonic Band Gaps (PBGs) allows one to eliminate the problem of light leakage from sharply bent optical fibers and ridge waveguides. Indeed, using a PC with a complete PBG as a background material and embedding into such a PC a circuit of properly engineered waveguiding channels permits to create an optical micro-circuit inside a perfect optical insulator, i.e. an optical analogue of the customary electronic micro-circuit. In addition, the absence of photon states for frequencies in a complete PBG allows one to suppress the emission of optically active materials embedded in PCs. Furthermore, the multi-branch nature of the photonic bandstructure and low group velocities associated with flat bands near a photonic band edge may be utilized to realize phase-matching for nonlinear optical processes and to enhance the interaction between electromagnetic waves and nonlinear and/or optically active material. These prospects have triggered enormous experimental activities aimed at the fabrication of two-dimensional (2D) as well as three-dimensional (3D) PC structures for telecommunication applications with PBGs in the near infrared frequency range. Considering that the first Bragg resonance occurs when the lattice constant equals half the wavelength of light, fabrication of PCs with bandgaps in the near IR requires substantial technological resources. For 2D PCs, advanced planar microstructuring techniques borrowed from semiconductor technology can greatly simplify the fabrication process and high-quality PCs with embedded defects and waveguides have been fabricated in various material systems such as semiconductors [4.5–4.10], polymers [4.11,4.12], and glasses [4.13,4.14]. In these structures, light experiences PBG effects in the plane of propagation, while the confinement in the third direction is achieved through index guiding. This suggests that fabricational imperfections in bulk 2D PCs as well deliberately embedding defect structures such as cavities and waveguide bends into 2D PCs will inevitably lead to radiation losses into the third dimension. Therefore, it is still an open question as to whether devices with acceptable radiation losses can be designed and realized in 2D PCs. However, radiation losses can be avoided altogether if light is guided within the comlete PBG of 3D PCs and, therefore, the past years have seen substantial efforts towards the manufacturing of suitable 3D PCs. These structures include layer-by-layer structures [4.15, 4.16], inverse opals [4.17–4.19] as well as the fabrication of templates via laser holography [4.20, 4.21] and two-photon polymerization (sometimes also referred to as stereo-lithography) [4.22–4.24]
4 A Solid-State Theoretical Approach 57 Given this tremendous flexibility in the fabrication of PCs(and the cost associated with most of the fabrication techniques),it is clear that modeling of the linear,nonlinear and quantum optical properties of PCs is a crucial element of PC research.In this manuscript,we would like to outline how the far-reaching analogies of electron wave propagation in crystalline solids and electromagnetic wave propagation in PCs can be utilized to obtain a theoret- ical framework for the quantitative description of light propagation in PCs. In Sect.4.2,we describe an efficient method for obtaining the photonic band- structure which is based on a Multi-Grid technique.The results of photonic bandstructure computations are the basis for the description of defect struc- tures such as cavities and waveguides using a Wannier function approach (Sect.4.3).As an example,we illustrate the design of a near optimal PC waveguide bend.Finally,in Sect.4.4,we utilize this bend design and a multi- pole expansion technique to construct a PC beamsplitter within a finite-sized PC and discuss the role of fabricational tolerances on the performance of the device. 4.2 Photonic Bandstructure Computation Photonic bandstructure computations determine the dispersion relation of in- finitely extended defect-free PCs.In addition,they allow one to design PCs that exhibit PBGs and to accurately interpret measurements on PC samples. As a consequence,photonic bandstructure calculations represent an impor- tant predictive as well as interpretative basis for PC research and,therefore, lie at the heart of theoretical investigations of PCs.More specifically,the goal of photonic bandstructure computations is to find the eigenfrequencies and associated eigenmodes of the wave equation for the perfect PC,i.e.,for an infinitely extended periodic array of dielectric material.For the simplicity of presentation we restrict ourselves in the remainder of this manuscript to the case of TM-polarized radiation propagating in the plane of periodicity(,y)- plane of 2D PCs.In this case,the wave equation in the frequency domain (harmonic time dependence)for the z-component of the electric field reads ,而候+))+兰6m=0 1 (4.1) Here c denotes the vacuum speed of light and r=(x,y)denotes a two- dimensional position vector.The dielectric constant ep(r)=ep(r+R)is periodic with respect to the set R={na1+n2a2;(n1,n2)E Z2}of lattice vectors R generated by the primitive translations ai,i=1,2 that describe the structure of the PC.Equation (4.1)represents a differential equation with periodic coefficients and,therefore,its solutions obey the Bloch-Floquet theorem Ek(r+ai)=eika Ek(r), (4.2)
4 A Solid-State Theoretical Approach 57 Given this tremendous flexibility in the fabrication of PCs (and the cost associated with most of the fabrication techniques), it is clear that modeling of the linear, nonlinear and quantum optical properties of PCs is a crucial element of PC research. In this manuscript, we would like to outline how the far-reaching analogies of electron wave propagation in crystalline solids and electromagnetic wave propagation in PCs can be utilized to obtain a theoretical framework for the quantitative description of light propagation in PCs. In Sect. 4.2, we describe an efficient method for obtaining the photonic bandstructure which is based on a Multi-Grid technique. The results of photonic bandstructure computations are the basis for the description of defect structures such as cavities and waveguides using a Wannier function approach (Sect. 4.3). As an example, we illustrate the design of a near optimal PC waveguide bend. Finally, in Sect. 4.4, we utilize this bend design and a multipole expansion technique to construct a PC beamsplitter within a finite-sized PC and discuss the role of fabricational tolerances on the performance of the device. 4.2 Photonic Bandstructure Computation Photonic bandstructure computations determine the dispersion relation of in- finitely extended defect-free PCs. In addition, they allow one to design PCs that exhibit PBGs and to accurately interpret measurements on PC samples. As a consequence, photonic bandstructure calculations represent an important predictive as well as interpretative basis for PC research and, therefore, lie at the heart of theoretical investigations of PCs. More specifically, the goal of photonic bandstructure computations is to find the eigenfrequencies and associated eigenmodes of the wave equation for the perfect PC, i.e., for an infinitely extended periodic array of dielectric material. For the simplicity of presentation we restrict ourselves in the remainder of this manuscript to the case of TM-polarized radiation propagating in the plane of periodicity (x, y)- plane of 2D PCs. In this case, the wave equation in the frequency domain (harmonic time dependence) for the z-component of the electric field reads 1 p(r) ∂2 x + ∂2 y E(r) + ω2 c2 E(r)=0. (4.1) Here c denotes the vacuum speed of light and r = (x, y) denotes a twodimensional position vector. The dielectric constant p(r) ≡ p(r + R) is periodic with respect to the set R = {n1a1 + n2a2; (n1, n2) ∈ Z2} of lattice vectors R generated by the primitive translations ai, i = 1, 2 that describe the structure of the PC. Equation (4.1) represents a differential equation with periodic coefficients and, therefore, its solutions obey the Bloch-Floquet theorem Ek(r + ai) = eikai Ek(r), (4.2)
58 K.Busch et al. where i=1,2.The wave vector k E 1.BZ that labels the solution is a vector of the first Brillouin zone (BZ)known as the crystal momentum.As a result of this reduced zone scheme,the photonic bandstructure acquires a multi-branch nature that is associated with the backfolding of the dispersion relation into the 1.BZ.This introduces a discrete index n,the so-called band index,that enumarates the distinct eigenfrequencies and eigenfunctions at the same wave vector k. The photonic dispersion relation wn(k)gives rise to a photonic Density of States (DOS),which plays a fundamental role for the understanding of the quantum optical properties of active material embedded in PCs [4.25].The photonic DOS,N(w),is defined by "counting"all allowed states with a given frequency w d2k 6(w-wn(k)). (4.3) Other physical quantities such as group velocities vn(k)=Vkwn(k)can be calculated through adaption of various techniques known from electron bandstructure theory.For details,we refer to [4.26]and [4.27]. A straightforward way of solving(4.1)is to expand all the periodic func- tions into a Fourier series over the reciprocal lattice 9,thereby transforming the differential equation into an infinite matrix eigenvalue problem,which may be suitably truncated and solved numerically.Details of this plane wave method (PWM)for isotropic systems can be found,for instance,in [4.26,4.28 and for anisotropic systems in [4.29].While the PWM provides a straight- forward approach to computing the bandstructure of PCs,it also exhibits a number of shortcomings such as slow convergence associated with the trunca- tion of Fourier series in the presence of discontinuous changes in the dielectric constants.In particular,this slow convergence makes the accurate calculation of Bloch functions a formidable and resource-consuming task.Therefore,we have recently developed an efficient real space approach to computing pho- tonic bandstructures [4.27].Within this approach,the wave equation,(4.1), is discretized in a single unit cell in real space (defined through the set of space points r=ria1+r2a2 with r1,r2 E [-1/2,1/2]),leading to a sparse matrix problem.The Bloch-Floquet theorem,(4.2),provides the boundary condition for the elliptic partial differential equation(4.1).In addition,the eigenvalue is treated as an additional unknown for which the normalization of the Bloch functions provides the additional equation needed for obtaining a well-defined problem.The solution of this algebraic problem is obtained by employing Multi-Grid (MG)methods which guarantee an efficient solu- tion by taking full advantage of the smoothness of the photonic Bloch func- tions [4.27,4.30](see also the chapter of G.Wittum in this volume).Even for the case of a naive finite difference discretization,the MG-approach easily outperforms the PWM and leads to a substantial reduction in CPU time. For instance,in the present case of 2D systems for which the Bloch func- tions are required we save one order of magnitude in CPU time as compared
58 K. Busch et al. where i = 1, 2. The wave vector k ∈ 1.BZ that labels the solution is a vector of the first Brillouin zone (BZ) known as the crystal momentum. As a result of this reduced zone scheme, the photonic bandstructure acquires a multi-branch nature that is associated with the backfolding of the dispersion relation into the 1. BZ. This introduces a discrete index n, the so-called band index, that enumarates the distinct eigenfrequencies and eigenfunctions at the same wave vector k. The photonic dispersion relation ωn(k) gives rise to a photonic Density of States (DOS), which plays a fundamental role for the understanding of the quantum optical properties of active material embedded in PCs [4.25]. The photonic DOS, N(ω), is defined by “counting” all allowed states with a given frequency ω N(ω) = n 1.BZ d2k δ(ω − ωn(k)). (4.3) Other physical quantities such as group velocities vn(k) = ∇kωn(k) can be calculated through adaption of various techniques known from electron bandstructure theory. For details, we refer to [4.26] and [4.27]. A straightforward way of solving (4.1) is to expand all the periodic functions into a Fourier series over the reciprocal lattice G, thereby transforming the differential equation into an infinite matrix eigenvalue problem, which may be suitably truncated and solved numerically. Details of this plane wave method (PWM) for isotropic systems can be found, for instance, in [4.26,4.28] and for anisotropic systems in [4.29]. While the PWM provides a straightforward approach to computing the bandstructure of PCs, it also exhibits a number of shortcomings such as slow convergence associated with the truncation of Fourier series in the presence of discontinuous changes in the dielectric constants. In particular, this slow convergence makes the accurate calculation of Bloch functions a formidable and resource-consuming task. Therefore, we have recently developed an efficient real space approach to computing photonic bandstructures [4.27]. Within this approach, the wave equation, (4.1), is discretized in a single unit cell in real space (defined through the set of space points r = r1a1 + r2a2 with r1, r2 ∈ [−1/2, 1/2]), leading to a sparse matrix problem. The Bloch-Floquet theorem, (4.2), provides the boundary condition for the elliptic partial differential equation (4.1). In addition, the eigenvalue is treated as an additional unknown for which the normalization of the Bloch functions provides the additional equation needed for obtaining a well-defined problem. The solution of this algebraic problem is obtained by employing Multi-Grid (MG) methods which guarantee an efficient solution by taking full advantage of the smoothness of the photonic Bloch functions [4.27,4.30] (see also the chapter of G. Wittum in this volume). Even for the case of a naive finite difference discretization, the MG-approach easily outperforms the PWM and leads to a substantial reduction in CPU time. For instance, in the present case of 2D systems for which the Bloch functions are required we save one order of magnitude in CPU time as compared
4 A Solid-State Theoretical Approach 59 0.8 206 0.2 (a) 0.0 N(o)(a.u.) Fig.4.1.Density of States (a)and photonic band structure (b)for TM-polarized radiation in a square lattice (lattice constant a)of cylindrical air pores of radius Rpore =0.475a in dielectric with e=12 (silicon).This PC exhibits a large fun- damental gap extending from w=0.238 x 2mc/a to w=0.291 x 2c/a.A higher order band gap extends from w=0.425 x 2rc/a to w=0.464 x 2rc/a. to PWM.Additional refinements such as a finite element discretization will further increase the efficiency of the MG-approach. In Fig.4.1(b),we show the bandstructure for TM-polarized radiation in a 2D PC consisting of a square lattice (lattice constant a)of cylindrical air pores (radius rpore =0.475a)in a silicon matrix (ep =12).This structure exhibits two 2D PBGs.The larger,fundamental bandgap(20%of the midgap frequency)extends between w=0.238 x 2mc/a to w=0.291 x 2mc/a and the smaller,higher order bandgap (8%of the midgap frequency)extends from w=0.425×2πc/atow=0.464×2πc/a.Furthermore,.in Fig..4.1(a)we depict the DOS for our model system,where the photonic band gaps are manifest as regions of vanishing DOS.Characteristic for 2D systems is the linear behavior for small frequencies as well as the logarithmic singularities, the so-called van Hove singularities,associated with vanishing group velocities for certain frequencies inside the bands (compare with Fig.4.1(a)) 4.3 Defect Structures in Photonic Crystals To date,the overwhelming majority of theoretical investigations of cavities and waveguiding in PCs has been carried out using Finite-Difference Time- Domain(FDTD)and/or Finite-Element (FE)techniques.However,applying general purpose methodologies such as FDTD or FE methods to defect struc- tures in PCs largely disregards information about the underlying PC struc- ture which is readily available from photonic bandstructure computation.As a result,only relatively small systems can be investigated and the physical insight remains limited
4 A Solid-State Theoretical Approach 59 N(ω) (a.u.) 0.0 0.2 0.4 0.6 0.8 Frequency ( ωa/2πc) Γ X M Γ -π/a 0 π/a Γ X M ky kx x y a (a) (b) Fig. 4.1. Density of States (a) and photonic band structure (b) for TM-polarized radiation in a square lattice (lattice constant a) of cylindrical air pores of radius Rpore = 0.475a in dielectric with ε = 12 (silicon). This PC exhibits a large fundamental gap extending from ω = 0.238 × 2πc/a to ω = 0.291 × 2πc/a. A higher order band gap extends from ω = 0.425 × 2πc/a to ω = 0.464 × 2πc/a. to PWM. Additional refinements such as a finite element discretization will further increase the efficiency of the MG-approach. In Fig. 4.1(b), we show the bandstructure for TM-polarized radiation in a 2D PC consisting of a square lattice (lattice constant a) of cylindrical air pores (radius rpore = 0.475a) in a silicon matrix (εp = 12). This structure exhibits two 2D PBGs. The larger, fundamental bandgap (20% of the midgap frequency) extends between ω = 0.238 × 2πc/a to ω = 0.291 × 2πc/a and the smaller, higher order bandgap (8% of the midgap frequency) extends from ω = 0.425 × 2πc/a to ω = 0.464 × 2πc/a. Furthermore, in Fig. 4.1(a) we depict the DOS for our model system, where the photonic band gaps are manifest as regions of vanishing DOS. Characteristic for 2D systems is the linear behavior for small frequencies as well as the logarithmic singularities, the so-called van Hove singularities, associated with vanishing group velocities for certain frequencies inside the bands (compare with Fig. 4.1(a)). 4.3 Defect Structures in Photonic Crystals To date, the overwhelming majority of theoretical investigations of cavities and waveguiding in PCs has been carried out using Finite-Difference TimeDomain (FDTD) and/or Finite-Element (FE) techniques. However, applying general purpose methodologies such as FDTD or FE methods to defect structures in PCs largely disregards information about the underlying PC structure which is readily available from photonic bandstructure computation. As a result, only relatively small systems can be investigated and the physical insight remains limited
60 K.Busch et al. 4.3.1 Maximally Localized Photonic Wannier Functions A more natural description of localized defect modes in PCs consists in an expansion of the electromagnetic field into a set of localized basis functions which have encoded into them all the information of the underlying PC. Therefore,the most natural basis functions for the description of defect struc- tures in PCs are the so-called photonic Wannier functions,WnR(r),which are formally defined through a lattice Fourier transform WaR(r)=(2)Joz Vwsc Pke-ikR Enk(r) (4.4) of the extended Bloch functions,Enk(r).The above definition associates the photonic Wannier function WaR(r)with the frequency range covered by band n,and centers it around the corresponding lattice site R.In addition, the completeness and orthogonality of the Bloch functions translate directly into corresponding properties of the photonic Wannier functions.Computing the Wannier functions directly from the output of photonic bandstructure programs via(4.4)leads to functions with poor localization properties and erratic behavior (see,for instance,Fig.2 in [4.31]).These problems origi- nate from an indeterminacy of the global phases of the Bloch functions.It is straightforward to show that for a group of Nw bands there exists,for every wave vector k,a free unitary transformation between the bands which leaves the orthogonality relation of Wannier functions unchanged.A solution to this unfortunate situation is provided by recent advances in electronic bandstruc- ture theory.Marzari and Vanderbilt [4.32]have outlined an efficient scheme for the computation of maximally localized Wannier functions by determin- ing numerically a unitary transformation between the bands that minimizes an appropriate spread functional F Nw F=∑[aolr2no)-((n0))月=Mim. (4.5) Here we have introduced a shorthand notation for matrix elements according to (nRlf(r)ln'R)=/dPrWiR(r)f(r)ep(r)WwR(r), (4.6) for any function f(r).For instance,the orthonormality of the Wannier func- tions in this notation read as (nRIIn'R')=d'rWaR(r)Ep(r)Ww'R(r)=6nmRR, (4.7） /R2 The field distributions of the optimized Wannier functions belonging to the six most relevant bands of our model system are depicted in Fig.4.2 (see
60 K. Busch et al. 4.3.1 Maximally Localized Photonic Wannier Functions A more natural description of localized defect modes in PCs consists in an expansion of the electromagnetic field into a set of localized basis functions which have encoded into them all the information of the underlying PC. Therefore, the most natural basis functions for the description of defect structures in PCs are the so-called photonic Wannier functions, WnR(r), which are formally defined through a lattice Fourier transform WnR(r) = VWSC (2π)2 BZ d2k e−ikR Enk(r) (4.4) of the extended Bloch functions, Enk(r). The above definition associates the photonic Wannier function WnR(r) with the frequency range covered by band n, and centers it around the corresponding lattice site R. In addition, the completeness and orthogonality of the Bloch functions translate directly into corresponding properties of the photonic Wannier functions. Computing the Wannier functions directly from the output of photonic bandstructure programs via (4.4) leads to functions with poor localization properties and erratic behavior (see, for instance, Fig. 2 in [4.31]). These problems originate from an indeterminacy of the global phases of the Bloch functions. It is straightforward to show that for a group of NW bands there exists, for every wave vector k, a free unitary transformation between the bands which leaves the orthogonality relation of Wannier functions unchanged. A solution to this unfortunate situation is provided by recent advances in electronic bandstructure theory. Marzari and Vanderbilt [4.32] have outlined an efficient scheme for the computation of maximally localized Wannier functions by determining numerically a unitary transformation between the bands that minimizes an appropriate spread functional F F = NW n=1 n0| r2 |n0 − (n0| r |n0) 2 = Min . (4.5) Here we have introduced a shorthand notation for matrix elements according to nR| f(r)|n R = R2 d2r W∗ nR(r) f(r) εp(r) WnR (r) , (4.6) for any function f(r). For instance, the orthonormality of the Wannier functions in this notation read as nR| |n R = R2 d2r W∗ nR(r) εp(r) WnR (r) = δnmδRR , (4.7) The field distributions of the optimized Wannier functions belonging to the six most relevant bands of our model system are depicted in Fig. 4.2 (see
4 A Solid-State Theoretical Approach 61 19 Fig.4.2.Photonic Wannier functions,Wno(r),for the six bands that are most relevant for the description of the localized defect mode shown in Fig.4.3(a).These optimally localized Wannier functions have been obtained by minimizing the cor- responding spread functional,(4.5).Note,that in contrast to the other bands,the Wannier center of the eleventh band is located at the center of the air pore.The parameters of the underlying PC are the same as those in Fig.4.1. also the discussion in Sect.4.3.3).Their localization properties as well as the symmetries of the underlying PC structure are clearly visible.It should be noted that the Wannier centers of all calculated bands (except of the eleventh band)are located halfway between the air pores,i.e.inside the dielectric (see [4.32]for more details on the Wannier centers).In addition,we would like to point out that instead of working with the electric field [4.33,4.31, (4.1),one may equally well construct photonic Wannier functions for the magnetic field,as recently demonstrated by Whittaker and Croucher [4.34. 4.3.2 Defect Structures via Wannier Functions The description of defect structures embedded in PCs starts with the corre- sponding wave equation in the frequency domain 2 V2E(r) ( (Ep(r)+6s(r))E(r)=0. (4.8) Here,we have decomposed the dielectric function into the periodic part, Ep(r),and the contribution,s(r),that describes the defect structures. Within the Wannier function approach,we expand the electromagnetic field according to E(r)=∑EnRWnR(r), (4.9) n.R
4 A Solid-State Theoretical Approach 61 n=1 n=2 n=3 n=5 n=11 n=19 Fig. 4.2. Photonic Wannier functions, Wn0(r), for the six bands that are most relevant for the description of the localized defect mode shown in Fig. 4.3(a). These optimally localized Wannier functions have been obtained by minimizing the corresponding spread functional, (4.5). Note, that in contrast to the other bands, the Wannier center of the eleventh band is located at the center of the air pore. The parameters of the underlying PC are the same as those in Fig. 4.1. also the discussion in Sect. 4.3.3). Their localization properties as well as the symmetries of the underlying PC structure are clearly visible. It should be noted that the Wannier centers of all calculated bands (except of the eleventh band) are located halfway between the air pores, i.e. inside the dielectric (see [4.32] for more details on the Wannier centers). In addition, we would like to point out that instead of working with the electric field [4.33, 4.31], (4.1), one may equally well construct photonic Wannier functions for the magnetic field, as recently demonstrated by Whittaker and Croucher [4.34]. 4.3.2 Defect Structures via Wannier Functions The description of defect structures embedded in PCs starts with the corresponding wave equation in the frequency domain ∇2E(r) + ω c 2 (εp(r) + δε(r)) E(r)=0 . (4.8) Here, we have decomposed the dielectric function into the periodic part, εp(r), and the contribution, δε(r), that describes the defect structures. Within the Wannier function approach, we expand the electromagnetic field according to E(r) = n,R EnR WnR(r) , (4.9)
62 K.Busch et al. with unknown amplitudes EnR.Inserting this expansion into the wave equa- tion (4.8)and employing the orthonomality relations,(4.7),leads to the basic equation for lattice models of defect structures embedded in PCs ∑{nrr+D}EnR=()'∑AER (4.10) n n'R The matrix A depends only on the Wannier functions of the underlying PC and is defined through A2=- d2r WiR(r)V2Ww'R(r). (4.11) R2 The localization of the Wannier functions in space leads to a very rapid decay of the magnitude of matrix elements with increasing separationR-Rbe- tween lattice sites,effectively making the matrix A sparse.Furthermore, it may be shown that the matrix A is Hermitian and positive definite. Similarly,once the Wannier functions of the underlying PC are determined, the matrix DR depends solely on the overlap of these functions,mediated by the defect structure: d2r WaR(r)6E(r)Ww'R(r). (4.12) R3 As a consequence of the localization properties of both the Wannier functions and the defect dielectric function,the Hermitian matrix D,too,is sparse. In the case of PCs with inversion symmetry,Ep(r)=Ep(-r),the Wannier functions can be chosen to be real.Accordingly,both matrices,A and DR become real symmetric ones. Depending on the nature of the defect structure,we are interested in (i)frequencies of localized cavity modes,(ii)dispersion relations for straight waveguides,or (iii)transmission and reflection through waveguide bends and other,more complex defect structures.In the following,we consider each of these cases separately. 4.3.3 Localized Cavity Modes As a first illustration of the Wannier function approach,we consider the case of a simple cavity created by infiltrating a single pore at the defect site Rdef with a material with dielectric constant edef,as shown in the inset of Fig.4.3(a).In this case,we directly solve (4.10)as a generalized eigen- value problem for the cavity frequencies that lie within the PBG,and recon- struct the cavity modes from the corresponding eigenvectors.In Fig.4.3(a) we compare the frequencies of these cavity modes calculated from (4.10) with corresponding calculations using PWM-based super-cell calculations
62 K. Busch et al. with unknown amplitudes EnR. Inserting this expansion into the wave equation (4.8) and employing the orthonomality relations, (4.7), leads to the basic equation for lattice models of defect structures embedded in PCs n,R δnnδRR + Dnn RR EnR = c ω 2 n,R Ann RREnR . (4.10) The matrix Ann RR depends only on the Wannier functions of the underlying PC and is defined through Ann RR = − R2 d2r W∗ nR(r) ∇2 WnR (r) . (4.11) The localization of the Wannier functions in space leads to a very rapid decay of the magnitude of matrix elements with increasing separation |R − R | between lattice sites, effectively making the matrix Ann RR sparse. Furthermore, it may be shown that the matrix Ann RR is Hermitian and positive definite. Similarly, once the Wannier functions of the underlying PC are determined, the matrix Dnn RR depends solely on the overlap of these functions, mediated by the defect structure: Dnn RR = R2 d2r W∗ nR(r) δε(r) WnR (r) . (4.12) As a consequence of the localization properties of both the Wannier functions and the defect dielectric function, the Hermitian matrix Dnn RR , too, is sparse. In the case of PCs with inversion symmetry, εp(r) ≡ εp(−r), the Wannier functions can be chosen to be real. Accordingly, both matrices, Ann RR and Dnn RR become real symmetric ones. Depending on the nature of the defect structure, we are interested in (i) frequencies of localized cavity modes, (ii) dispersion relations for straight waveguides, or (iii) transmission and reflection through waveguide bends and other, more complex defect structures. In the following, we consider each of these cases separately. 4.3.3 Localized Cavity Modes As a first illustration of the Wannier function approach, we consider the case of a simple cavity created by infiltrating a single pore at the defect site Rdef with a material with dielectric constant εdef, as shown in the inset of Fig. 4.3(a). In this case, we directly solve (4.10) as a generalized eigenvalue problem for the cavity frequencies that lie within the PBG, and reconstruct the cavity modes from the corresponding eigenvectors. In Fig. 4.3(a) we compare the frequencies of these cavity modes calculated from (4.10) with corresponding calculations using PWM-based super-cell calculations.
4 A Solid-State Theoretical Approach 63 0.2 (2/m 0.28 (a) 0.27 02 0.24 0.23 0 .6 1012 Defect pore permitivity,Eer Fig.4.3.(a)Frequencies of localized cavity modes created by infiltrating a single defect pore with a material with dielectric constant eder (see inset).The results of the Wannier function approach (diamonds)using Nw=10 Wannier functions per unit cell are in complete agreement with numerically exact results of the super- cell calculations (full line).The parameters of the underlying PC are the same as those in Fig 4.1.(b)Electric field distribution for the cavity mode with frequency w=0.290 x 2mc/a,created by infiltrating the pore with a polymer with eder =2.4. Upon increasing edef,a non-degenerate cavity mode with monopole symme- try emerges from the upper edge of the bandgap.The results of the Wannier function approach using the Nw 10 most relevant Wannier functions per unit cell in (4.10)are in complete agreement with numerically exact results of the super-cell calculations.In Fig.4.3(b),we depict the corresponding mode 10 15 20 n Fig.4.4.The strength Vn of the individual contributions from the Wannier func- tions of the lowest 20 bands (index n)to the formation of the cavity modes depicted in Fig.4.3.The Wannier functions with V 10-3 may be safely leaved out of ac- count.Arrows indicate the six must relevant Wannier functions depicted in Fig.4.2. The parameters of the underlying PC are the same as those in Fig.4.1
4 A Solid-State Theoretical Approach 63 024 6 8 10 12 14 Defect pore permitivity, εdef 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 Frequency, ( ωa/2πc) (a) (b) Fig. 4.3. (a) Frequencies of localized cavity modes created by infiltrating a single defect pore with a material with dielectric constant εdef (see inset). The results of the Wannier function approach (diamonds) using NW = 10 Wannier functions per unit cell are in complete agreement with numerically exact results of the supercell calculations (full line). The parameters of the underlying PC are the same as those in Fig 4.1. (b) Electric field distribution for the cavity mode with frequency ω = 0.290 × 2πc/a, created by infiltrating the pore with a polymer with εdef = 2.4. Upon increasing εdef, a non-degenerate cavity mode with monopole symmetry emerges from the upper edge of the bandgap. The results of the Wannier function approach using the NW = 10 most relevant Wannier functions per unit cell in (4.10) are in complete agreement with numerically exact results of the super-cell calculations. In Fig. 4.3(b), we depict the corresponding mode 1 5 10 15 20 n 10-6 10-4 10-2 100 V n Fig. 4.4. The strength Vn of the individual contributions from the Wannier functions of the lowest 20 bands (index n) to the formation of the cavity modes depicted in Fig. 4.3. The Wannier functions with Vn ≤ 10−3 may be safely leaved out of account. Arrows indicate the six must relevant Wannier functions depicted in Fig. 4.2. The parameters of the underlying PC are the same as those in Fig. 4.1
64 K.Busch et al. structure for a monopole cavity mode created by infiltration of a polymer with edef =2.4 into the pore.The convergence properties of the Wannier function approach should depend on the nature and symmetry properties of the cavity modes under consideration.To discuss this issue in greater detail, it is helpful to define a measure V of the strength of the contributions to a cavity mode from the individual Wannier function associated with band n via Vn =REnR2.In Fig.4.4 we display the dependence of the parame- ter Vn on the band index n for the cavity modes shown in Fig.4.3,for two values of the defect dielectric constant,Edef =2.4(solid line)and def =8 (dashed line),respectively.In both cases,the most relevant contributions to the cavity modes originate from the Wannier functions belonging to bands n =1,2,3,5,11,and 19,and all contributions from bands n 20 are negligi- ble.These most relevant Wannier functions for our model system are shown in Fig.4.2.In fact,fully converged results are obtained when we work with the 10 most relevant Wannier functions per unit cell(for a comparison with numerically exact super-cell calculations see Fig.4.3(a)). 4.3.4 Dispersion Relations of Waveguides The efficiency of the Wannier function approach is particularly evident when considering defect clusters consisting of several defect pores.In this case the defect dielectric function,6(r),can be written as a sum over positions,Rm, of the individual defect pores,so that (4.12)reduces to a sum D=∑D(m)RR,r-Rm, (4.13) m over the matrix elements D(m)of the individual defects(see discussion in [4.31]for more details).Therefore,for a given underlying PC structure, it becomes possible to build up a database of matrix elements,D(m)R, for different geometries(radii,shapes)of defect pores,which allow us highly efficient defect computations through simple matrix assembly procedures. This is in strong contrast to any other computational technique known to us. Arguably the most important types of defect clusters in PCs are one or several adjacent straight rows of defects.Properly designed,such defect rows form a PC waveguide which allows the efficient guiding of light for frequencies within a PBG [4.35,4.36].Due to the one-dimensional periodicity of such a waveguide,its guided modes,Ep(=.)WR(),obey the 1D Bloch-Floquet theorem (w)k()(w), (4.14) and thus they can be labeled by a wave vector,kp(w),parallel to the waveg- uide director,sw=w1a1+w2a2,where a1=(a,0)and a2=(0,a)are the prim- itive lattice vectors of the PC,and integers w and w2 define the direction
64 K. Busch et al. structure for a monopole cavity mode created by infiltration of a polymer with εdef = 2.4 into the pore. The convergence properties of the Wannier function approach should depend on the nature and symmetry properties of the cavity modes under consideration. To discuss this issue in greater detail, it is helpful to define a measure Vn of the strength of the contributions to a cavity mode from the individual Wannier function associated with band n via Vn = R |EnR| 2. In Fig. 4.4 we display the dependence of the parameter Vn on the band index n for the cavity modes shown in Fig. 4.3, for two values of the defect dielectric constant, εdef = 2.4 (solid line) and εdef = 8 (dashed line), respectively. In both cases, the most relevant contributions to the cavity modes originate from the Wannier functions belonging to bands n = 1, 2, 3, 5, 11, and 19, and all contributions from bands n > 20 are negligible. These most relevant Wannier functions for our model system are shown in Fig. 4.2. In fact, fully converged results are obtained when we work with the 10 most relevant Wannier functions per unit cell (for a comparison with numerically exact super-cell calculations see Fig. 4.3(a)). 4.3.4 Dispersion Relations of Waveguides The efficiency of the Wannier function approach is particularly evident when considering defect clusters consisting of several defect pores. In this case the defect dielectric function, δε(r), can be written as a sum over positions, Rm, of the individual defect pores, so that (4.12) reduces to a sum Dnn RR = m D(m) nn R−Rm,R−Rm , (4.13) over the matrix elements D(m)nn R,R of the individual defects (see discussion in [4.31] for more details). Therefore, for a given underlying PC structure, it becomes possible to build up a database of matrix elements, D(m)nn R,R , for different geometries (radii, shapes) of defect pores, which allow us highly efficient defect computations through simple matrix assembly procedures. This is in strong contrast to any other computational technique known to us. Arguably the most important types of defect clusters in PCs are one or several adjacent straight rows of defects. Properly designed, such defect rows form a PC waveguide which allows the efficient guiding of light for frequencies within a PBG [4.35, 4.36]. Due to the one-dimensional periodicity of such a waveguide, its guided modes, E(p) (r | ω) = n,R E(p) nR(ω) WnR(r), obey the 1D Bloch-Floquet theorem E(p) nR+sw (ω) = eikp(ω)sw E(p) nR(ω) , (4.14) and thus they can be labeled by a wave vector, kp(ω), parallel to the waveguide director, sw=w1a1 +w2a2, where a1=(a, 0) and a2=(0, a) are the primitive lattice vectors of the PC, and integers w1 and w2 define the direction
