Jan 16, 2003 - another technique based on the calculation of the full Green function for a ..... [9] Fan S, Villeneuve P R and Haus H A 1998 Phys. Rev. Lett. 80.
INSTITUTE OF PHYSICS PUBLISHING
NANOTECHNOLOGY
Nanotechnology 14 (2003) 177–183
PII: S0957-4484(03)52714-3
Defect computations in photonic crystals: a solid state theoretical approach A Garc´ıa-Mart´ın, D Hermann, F Hagmann, K Busch and P W¨olfle Institut f¨ur Theorie der Kondensierten Materie, Universit¨at Karlsruhe, PO Box 6980, D-76128 Karlsuhe, Germany
Received 23 August 2002, in final form 14 October 2002 Published 16 January 2003 Online at stacks.iop.org/Nano/14/177 Abstract We describe two different approaches to efficiently analyse the optical properties of defect structures embedded in photonic crystals (PCs). The first one is based on an expansion of the electromagnetic field into optimally localized photonic Wannier functions and thus efficiently utilizes the information of the underlying PCs. The second is based on the recently developed multipole method and is specially suited to deal with finite-size structures. We demonstrate the efficiency of these approaches by considering several defect structures for TM-polarized radiation in two-dimensional PCs.
1. Introduction Since the invention of the laser, progress in photonics has been closely related to the development of optical materials that allow us to control the flow of light. Photonic crystals (PCs) elevate this principle to a new level in the sense that the photonic dispersion relation and associated mode structure may be tailored to almost any need through a judicious design of these two-dimensional or three-dimensional periodic dielectric arrays. In particular, the choice of material composition, lattice periodicity and symmetry as well as the deliberate creation of defect structures embedded in PCs allows a degree of control over the optical properties of PCs that may eventually rival the flexibility in tailoring the properties of their electronic counterparts, the semiconducting materials. The usefulness of PCs derives to a large extent from the fact that suitably engineered PCs may exhibit one or more photonic bandgaps (PBGs) [1–3]. For instance, recent experiments have verified earlier theoretical predictions that 3D-PCs such as the inverse opals [4, 5] exhibit frequency ranges over which ordinary linear propagation is forbidden irrespective of direction. The existence of these absolute PBGs allows complete control over the radiative dynamics of active material embedded in PCs such as the total suppression of spontaneous emission for atomic transition frequencies deep in the PBG [1] and leads to strongly non-Markovian effects such as fractional localization of the atomic population for atomic transition frequencies in close proximity to a 3D-PBG [6, 7]. In the linear regime, PBGs offer novel passive optical guiding characteristics through the engineering of defects such as microcavities and waveguides and their combination into
functional elements. For instance, the complete PBG of an appropriate 3D-PC effectively shields light propagating inside a PC-waveguide structure from the leaky modes of the surrounding homogeneous material or air. As a consequence, PC-waveguiding structures allow us to steer light around sharp bends without loss even when the corresponding curvature radii are comparable in size to the wavelength of the radiation itself [8]. This is in sharp contrast to standard high-index waveguiding structures that rely on total internal reflection for steering the flow of light. Therefore, PC-waveguiding structures offer great potential for the realization of highdensity integrated micro-optical circuits for a number of applications in telecommunication technology. However, the substantial effort associated with the fabrication of large-scale 3D-PCs with complete PBGs and the controlled engineering of waveguiding structures in them has recently triggered increased interest into 2D-PCs and their 2D-PBGs. Any experimental exploration as well as technological exploitation of the huge parameter space provided by PCs has to be accompanied by a quantitative theoretical analysis in order to identify the most interesting cases and help interpret the data as well as to find stable designs for successfully operating devices. In this manuscript, we provide an outline for a theoretical framework that allows us to qualitatively as well as quantitatively determine the optical properties of defect structures in PCs that is based on solid state theoretical concepts. In section 2.1 we show how defect structures can be efficiently treated with the help of photonic Wannier functions. As an illustration we discuss the determination of eigenfrequencies and the field distributions of defect structures.
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In particular we show how a cavity-like mode appears in the bandgap of the perfect lattice when the dielectric constant of a single rod is varied. In section 2.2 we briefly introduce another technique based on the calculation of the full Green function for a finite-sized PC and discuss its usefulness in device designing and analysing the robustness of the device in presence of disorder.
2. Defect structures in photonic crystals In electronic micro-circuits, electrical currents are guided by thin metal wires where electrons are bound within the cross section of the wire by the so-called work function (confining potential) of the metal. As a result, electrical currents follow the path prescribed by the wire without escaping to the background. The situation is very different for optical waves. Although optical fibres guide light over long distances, microcircuits for light based on fibres do not exist, because empty space is already an ideal conductor of light waves. The light in an optical fibre can easily escape into the background electromagnetic modes of empty space if the fibre is bent or distorted on a microscopic scale. PBGs in the bandstructure of PCs remove this problem by removing all the background electromagnetic modes over the relevant band of frequencies. Light paths can be created inside a PBG material in the form of engineered waveguide channels. The PBG localizes the light and prevents it from escaping the optical micro-circuit. To date, theoretical investigations of defect structures in PCs have almost exclusively been carried out using finite-difference time domain (FDTD) methods [9, 10]. However, applying general purpose methodologies such as FDTD or finite-element methods (FEMs) to defect structures in PCs largely disregards information about the underlying PC structure which is readily available from photonic bandstructure computation. In the following subsections we will consider two different and efficient approaches to analyse the field pattern in structures created inside PCs. For simplicity we will restrict ourselves to the two-dimensional case where the two polarizations, TM and TE, decouple and consider the case of TM-polarized waves. In this case the wave equation for the transversal component of the electric field in the perfect, infinitely extended lattice can be written as (∂x2 + ∂ y2 )E( r) +
ω2 p ( r )E( r ) = 0. c2
(1)
Here c denotes the vacuum speed of light and r = (x, y) denotes a two-dimensional position vector. The dielectric is periodic with respect to the constant p ( r ) ≡ p ( r + R) set R = {n 1 a1 + n 2 a2 ; (n 1 , n 2 ) ∈ Z 2 } of lattice vectors R generated by the primitive translations ai , i = 1, 2 that describe the structure of the PC. 2.1. Wannier functions as the optimal basis set A natural description of localized defect modes consists in an expansion of the electromagnetic field into a set of localized basis functions. The extended nature of Bloch functions of the defect-free PC suggests obtaining a generic set of localized 178
basis functions through a lattice Fourier transform. resulting Wannier functions are defined via VW SC Wn R ( r) = d2 k e−ik R E n k ( r ), (2π)2 B Z
The
(2)
where VW SC denotes the volume of the Wigner–Seitz cell. Furthermore, the Wannier function Wn R associated with band n and centred around lattice site R obeys the orthonormality relation d2 r Wn∗R ( r ) p ( r )Wn R ( r ) = δnn δ R R , (3) V
where the integration is over all space. To show the localization properties of the Wannier functions let us consider a lattice of silicon cylinders in air. The dielectric constant of silicon is = 11.9 and the cylinder radius is r/a = 0.2. As can be seen from an inspection of figure 1(a) it is a sobering exercise to compute the Wannier functions directly from the output of photonic bandstructure programs via equation (2). The poor localization properties and the erratic behaviour of these Wannier functions originates in a phase indeterminacy of the Bloch functions for an isolated band r ) → eiφn (k) E n k ( r ), (4) E n k ( represents a global k-dependent phase function. where φn (k) More generally, for an isolated group of N bands, the allowed transformations are given by a unitary transformation Umn (k) which reads r) → E n k (
N
E m k ( Umn (k) r ).
(5)
m=1
Although fixing the random part of the transformations in equations (4) and (5) by requiring the Bloch functions at the origin to be real valued and non-negative removes the erratic behaviour of the Wannier functions to a large extent, this fails to substantially improve their localization properties. These difficulties have led a number of authors to adapt variations of the familiar empirical tight-binding parametrization to photonic bandstructures [11, 12]. However, the success of empirical tight-binding parametrizations depends crucially on the existence of localized ‘orbitals’ of the individual ‘atoms’ that make up the crystal. As a consequence, its adaption to PCs presents major problems because bound states for a single dielectric scatterer simply do not exist and until now no tight-binding parametrization for TE-polarized radiation in 2D-PCs or electromagnetic waves in 3D-PCs has been obtained. In addition, we will show below that retaining only nearest-neighbour interaction in an empirical tightbinding parametrization may explain an apparent failure of this approach to correctly reproduce the higher photonic bands for TM-polarized radiation in 2D-PCs [11, 12]. A solution to this unfortunate situation is provided by recent advances in electronic bandstructure theory. Marzari and Vanderbilt [13] have outlined an efficient scheme for the computation of optimally localized Wannier functions by determining numerically a transformation, equation (5), that minimizes an appropriate spread functional. In view of the
Defect computations in photonic crystals: a solid state theoretical approach
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Figure 1. Absolute values of the amplitude of photonic Wannier functions for the first band obtained (a) by direct numerical integration of equation (2) and (b) by minimizing the corresponding spread functional, equation (6). The parameters of this model system are given in the text.
translational properties of the Wannier functions Wn R ( r) = this functional reads r − R), Wn 0 ( =
2 |n 0 − (n 0| r |n 0) 2 ] = min, [n 0|r
(6)
n
where the summation extends to the number of bands N forming the group under consideration. Here we have introduced a shorthand for matrix elements according notation ∗ 2 2 |n R = d r W ( r )r 2 Wn R ( r ). The result for to n R|r V nR the optimized Wannier function belonging to the first band of our model system is depicted in figure 1(b). The localization properties as well as the symmetries of the underlying PC structure are clearly visible. In addition, figures 2(a) and (b) display the optimized Wannier functions belonging to the second and third bands of our model system. The localization properties of this set of optimized Wannier functions clearly suggest that restricting an empirical tight-binding parametrization of photonic bandstructures to nearest-neighbour interaction only will most likely be insufficient. In particular, the spatial extent of the Wannier function belonging to the first band of our model system, figure 1(b), indicates that nearest-neighbour interaction may be sufficient to describe the frequency range corresponding to this band. In contrast, the Wannier functions for the second and third bands, figures 2(a) and (b), extend well into the unit cells adjacent to the central cell. Most notably, the Wannier function belonging to the third band of our model system is predominantly localized on the nearest-neighbour
Figure 2. Absolute values of the amplitude of photonic Wannier functions for (a) the second band and (b) the third band obtained by minimizing the corresponding spread functional, equation (6). The parameters of this model system are given in the text.
lattice sites and, therefore, suggests a potential failure of a nearest-neighbour interaction approximation in describing the corresponding bands [11, 12]. The set of optimally localized Wannier functions may be used as an expansion basis for the localized modes of defect structures embedded in PCs. E n R Wn R ( r ). (7) E( r) = n
R
Inserting this expansion into the wave equation for the PC with added defect dielectric function ( r) (∂x2 + ∂ y2 )E( r) +
ω2 ( p ( r ) + ( r ))E( r ) = 0, c2
(8)
leads to a generalized eigenvalue equation for the corresponding defect modes (δnn δ R R − n R||n R )E n R n
=
R 2
c δnn βn ( R − R )E n R , ω 2 n
(9)
R
is defined as where βn ( R) = βn ( R)
VW SC (2π)2
2 ωn k . c2
d2 k eik R BZ
(10)
As an illustration of the efficiency of this approach we will consider the effect of varying the dielectric constant of a single rod of an otherwise perfect lattice made of alumina ( = 8.9) 179
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Figure 3. Photonic bandstructure of a square lattice of cylindrical alumina posts (r/a = 0.38, = 8.9) in air. For TM-polarized light, this PC exhibits four 2D-PBGs.
cylinders with a radius of r/a = 0.38. The bandstructure corresponding to this PC is shown in figure 3. We have used the multigrid method [14] to calculate the bandstructure as well as the Bloch modes that are necessary for construct the Wannier functions. This PC exhibits three different bandgaps for low frequencies: the first at 0.247 < ωa/2πc < 0.269, the second at 0.410 < ωa/2πc < 0.454 and finally the third at 0.615 < ωa/2πc < 0.658. In figure 4(a) we exchange one single alumina rod for a silica ( = 2.1) one with the same radius. For that structure we obtain an isolated state (ωa/2πc = 0.425) within the second bandgap of the perfect lattice. The mode profile has monopole symmetry and it is strongly localized within the silica rod. In addition, figure 4(b) shows the field distribution also of an cavity mode but in this case the modified cylinder is made of silicon ( = 11.9). As in the previous case an isolated state appears in the second bandgap (ωa/2πc = 0.441) but the mode profile is completely different. Here the cavity mode exhibits a quadrupole symmetry, showing that very complicated symmetries can be accessed within this approach. These calculations used open boundary conditions on a 9 × 9 section of the underlying lattice and included the six lowest bands. This results in a 486 × 486 matrix problem and the convergence of the eigenfrequencies of these modes has been confirmed by supercell calculations. Most importantly, the localization properties of the optimized Wannier functions allow us to neglect the overlap integrals once the respective Wannier centres are sufficiently far apart, which, despite the inadequacy of nearest-neighbour interactions, nevertheless leads to a sparse matrix problem. Therefore, optimally localized Wannier functions provide an efficient tool to study large defect structures embedded in PCs such as functional elements that contain a number of basic elements that are intentionally coupled in order to perform more complex operations. 2.2. Multipole expansion approach The approach described above is particularly useful for large systems, and has no restriction on the shape of the elements constituting the system. If we restrict ourselves to cylindrical symmetry, a complementary and also extremely efficient tool can be found in the recently developed multipole expansion technique [15]. Apart from the field pattern, in 180
Figure 4. Field distribution of localized photonic defect modes for a cavity, i.e., the dielectric constant of a single cylinder of the periodic PC is varied. In (a) we show the efect of lowering the dielectric constant. This gives rise to a monopole-like state in the second bandgap. In (b) we show the oposite case, i.e. the dielectric constant of the central rod is increased, resulting in a state also in the second bandgap, but now with a quadrupole symmetry.
nanophotonics we should take care of another quantity, crucial to understand the radiation dynamics: the local density of states (LDOS) [16]. The LDOS of an infinite system vanishes for frequencies lying in the bandgap, revealing the suppression of light emission at those frequencies. However, actual devices are not of infinite extent, and, therefore, in these finite-sized structures the LDOS will be very small but non-vanishing. The multipole expansion technique has proved to be very well suited for obtaining the field pattern and the LDOS of finitesize PCs [15, 17]. In actual structures imperfections appear as a greatly undesirable but unavoidable ingredient. As well as with the Wannier function approach, with this technique we can describe a quite general dielectric function (although restricted to cylindrically symmetric elements), including variations of the position, of the radius or of the dielectric constant of the cylinders, allowing the analysis of the effect that structural imperfections have in the wave propagation through the system. The LDOS can be obtained by extracting the imaginary part of the Green tensor ρ( r ; ω) = −
2ωn 2b Im Tr[G( r , r; ω)] πc2
(11)
where n b represents the index of refraction of the background where the cylinders are embedded, and G( r , rs ; ω) is the Green tensor for a δ-source at rs and observation point at r. As in the
Defect computations in photonic crystals: a solid state theoretical approach
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Figure 5. (a) Absolute value of the field distribution of a beam splitter that has been built by removing cylinders along the principal crystal directions. The field distribution shows that this configuration would not make a working device. (b) Absolute value of the field pattern of an optimized beam splitter where some rows have been shifted to resemble a triangular lattice. Here we can see how the incoming wave couples to both output ports.
previous sections, for the sake of simplicity, we will restrict ourselves to the two-dimensional case and E-polarization, where only the G zz component is needed and satisfies the wave equation ω2 r )G zz = δ( r − rs ). (∂x2 + ∂ y2 )G zz + 2 ( c
Figure 6. (a) Field distribution of a disordered beam splitter (positional disorder). The disorder is introduced by randomly shifting the position of the cylinders by up to 20% of the lattice constant. The field distribution shows that disorder strongly perturbs the response of the device. (b) The same as (a) but when the source of disorder is the indeterminacy of the cylinder radii.
or 0, respectively) and H0(1) and Jm are Hankel’s and Bessel’s functions, respectively. Similarly, we can write an expression for G zz outside any of the Nc cylinders 1 ext rs )H0(1) (k| r − rs |) χ ( 4i Nc ∞ + Cmq Hm(1) (k| r − rs |)eimarg(r −rs ) .
G ext r , rs ) = zz ( (12)
(14)
q=1 m=−∞
The multipole method consists in expanding the Green function G zz in cylindrical harmonics, outside and inside the cylinders that build up the lattice, and subsequently obtaining the expansion coefficients through the continuity relations across the cylinder surface and the boundary conditions at infinity. Let us consider that our cluster of Nc cylinders with a refractive index n l is placed in air (n b = 1). The case of n b = 1 can be easily obtained by the appropriate rescaling of k (k → kn b ) and n l (n l → n l /n b ). Inside the lth cylinder G zz is given by G int r , rs ) = zz,l ( +
∞
1 int rs )H0(1) (kn l | r − rs |) χ ( 4i l
Bml Jm (kn l | r − rl |) eimarg(r −rl )
(13)
m=−∞
where rl is the cylinder position, the value of χlint indicates whether rs lies inside or outside the lth cylinder (χlint = 1
Again χ ext accounts for the position of the source and takes the value χ ext = 1, if rs is outside any of the cylinders, whereas χ ext = 0 otherwise. As mentioned above the coefficients Bml of equation (13) q and Cm in equation (14) are linked by the continuity conditions across the surface of the cylinders. These give rise to the full multiple-scattering problem whose details can be found in [18]. To show the feasibility of this approach as a designing tool we consider a beam splitter as a model device. This functional element is made of 240 silicon cylinders (n l 3.4) with radius r/a = 0.18. For the perfect lattice, bandstructure calculations show the existence of a bandgap for TM modes at frequencies in the interval 0.302 < ωa/2πc < 0.444. We will assume that our external source is radiating at ωa/2πc = 0.356, i.e. at a frequency very well inside the bandgap. The source is located on the left of the system, therefore the left-hand side of the structure will be the input port whereas the right-hand side will act as the output port. 181
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In figure 5(a) we show the field distribution of a trial beam splitter. It has been made by simply removing certain silicon rods along the principal directions of the crystal. Although the wave penetrates in the device through the input waveguide, this beam splitter is unable to guide any radiation to any of the output ports and all the incoming radiation is reflected back. This failure of a naive example of beamsplitter manifests the need of a more thorough investigation of the parameters to build up a working device. A more complicated example of a beam splitter is depicted in figure 5(b). This device is designed from the previous one but adjusting the position of some cylinders around the corners of the splitter. Some rows of cylinders have been shifted so that the structure where the waveguides intersect now looks like a triangular array rather than the original square lattice. By these simple modifications the field distribution shows that the wave will travel through a now working device. So far, we have been considering devices built within perfect lattices. This, unfortunately, is far from the experimental situation. Defects or imperfections are always present and they greatly influence the response of any actual device. It is thus important to characterize the effects of disorder in the behaviour of our devices. As an illustration we are considering the working beam splitter of figure 5(b) as the perfect device and suppose that during the fabrication process two different situations could occur: (a) the position of the cylinders can be specified with an accuracy of 80% of the lattice constant, i.e. there is a variation up to 20% in the position of the cylinders, and (b) the radii of the cylinders can be controlled up to 20%, i.e. the cylinder radius can be up to either 20% bigger or 20% smaller. The field distribution corresponding to a typical realization of these two situations is depicted in figure 6. The response of the device under positional disorder is shown in figure 6(a) whereas the same device with disordered radii can be seen in figure 6(b). In both cases the functionality of the device is strongly affected: firstly the relative transmission through the output ports can be as large as 200%, and secondly the total transmission is greatly reduced.
3. Conclusions In summary, we have outlined a framework based on solid state theoretical methods that allows us to quantitatively calculate the optical properties of defect structures in PCs. In the first case we have described a ‘divide and conquer’ approach, where photonic bandstructure computation of the infinitely extended PC provides as input photonic Bloch functions that are subsequently processed into maximally localized Wannier functions. These Wannier functions contain all the information of the underlying PC and are, therefore, ideally suited as expansion basis for the localized modes of defect structures whose eigenfrequencies lie inside 2D- or 3D-PBGs. Due to the localized nature of the photonic Wannier functions, the corresponding matrix systems will be sparse. However, in contrast to most lattice models of electronic systems, the lattice models for photonic systems cannot be restricted to nearestneighbour interaction only. In contrast to semi-empirical tightbinding description of PCs, the present approach allows us to 182
obtain lattice models from first principles and is solely based on the input of photonic Bloch functions that are obtained from efficient bandstructure programs. Therefore, it is straightforward to extend our photonic Wannier function approach to the TE polarization in 2D-PCs as well as to fully three-dimensional systems. The matrix systems obtained from the expansion into Wannier functions are sparse and much smaller than the corresponding sparse matrix systems obtained via standard all-purpose discretization techniques such as FDTD or FEM methods. For a given PC basis structure, the pre-processing necessary to obtain optimally localized photonic Wannier functions has to be carried out only once and, therefore, the associated dramatic reduction in numerical load allows us to treat much larger systems as compared to FDTD and FEM methods. Perhaps even more important is the fact that using the Wannier function approach allows us to efficiently explore huge parameter space for the design of defect structures embedded in a given PC basis structure. This is of particular importance for obtaining robust designs for functional elements by considering the tolerances inherent to any fabrication process. Our second approach relies on the efficiency of the multipole expansion when the cylindrical symmetry is preserved. We have shown that this approach is very well suited for finite-sized PCs and can be the ideal complement to the Wannier function approach. The usefulness of this multipole expansion manifests itself when we are considering efficient designs for actual devices. The naive approximation to device designing has proved to fail, raising the necessity to explore a diverse range of parameters to have a working device. Unfortunately, the experimental situation is far from ideal and the lattice where the design is realized is not defect free. We introduced disorder in a model structure and analysed its response. Under these conditions the device clearly lost its functionality, showing the importance of finding designs that are as robust as possible under the influence of fabrication imperfections.
Acknowledgments We are grateful to S Mingaleev for interesting and fruitful discussions. We acknowledge the support by the DFGForschungszentrum Centre for Functional Nanostructures (CFN) at the University of Karlsruhe. KB and AGM acknowledge support from the Deutsche Forschungsgemeinschaft (DFG) under Bu 1107/2-2 (Emmy-Noether Programme).
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Defect computations in photonic crystals: a solid state theoretical approach
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