## Abstract

A brief introduction of the layer-Korringa-Kohn-Rostoker method for calculations of the frequency band structure of photonic crystals and of the transmission and reflection coefficients of light incident on slabs of such crystals is followed by two applications of the method. The first relates to the frequency band structure of metallodi-electric composites and demonstrates the essential difference between cermet and network topology of such composites at low frequencies. The second application is an analysis of recent measurements of the reflection of light from a slab of a colloidal system consisting of latex spheres in air.

©2001 Optical Society of America

In traditional energy band structure calculations of an electron in an ordinary crystal, or of the frequency band structure of the electromagnetic (EM) field in the case of photonic crystals, one starts with a fixed value of the reduced wavevector **k**, and by some method or other (in photonic crystals this is usually the plane-wave method, but the Korringa-Kohn-Rostoker (KKR) method has also been used to this effect [1, 2, 3]) one solves the eigenvalue problem, for the given **k**, to obtain all the eigenfrequencies within a very wide frequency range, and the corresponding eigenmodes of the scalar/vector field under consideration. These eigenmodes are, in an infinite crystal, propagating Bloch waves which have the property

where ${\mathbf{R}}_{n}^{\left(3\right)}$
is any vector of the three-dimensional (3D) lattice which defines the periodicity of the infinite crystal; and a is a band index which defines the different frequency bands *ω*_{α}
(**k**) and the corresponding eigenmodes. For the Schrödinger field (electrons in a crystal) *ψ* is a scalar quantity; in the case of the EM field *ψ* is a vector quantity.

On-shell methods [4, 5, 6, 7, 8, 9, 10, 11, 12] proceed differently; the frequency is fixed and one obtains the eigenmodes of the crystal for this frequency. Therefore, these methods are ideally suited for photonic crystals consisting of strongly dispersive materials such as real metals [13, 14]. One views the crystal as a succession of layers (slices) parallel to a given crystallographic plane of the crystal. The layers have the same two-dimensional (2D) periodicity (that of the chosen crystallographic plane) described by a 2D lattice:

where a_{1} and a_{2} are primitive vectors of the said plane (which is taken to be the *xy* plane), and *n*
_{1},*n*
_{2}=0, ±1, ±2,…. We may number the sequence of layers which constitute the infinite crystal, extending from *z*=-∞ to *z*=+∞, as follows: …- 2, -1, 0,1, 2,…. The (*N*+1)th layer is obtained from the *N*th layer by a primitive translation to be denoted by a_{3}. Obviously, a_{1},a_{2}, and a_{3} constitute a basis for the 3D space lattice of the infinite crystal.

We define the 2D reciprocal lattice corresponding to Eq. (2):

where **b**
_{j}
·**a**
_{j}
=2πδ*ij*, *i,j*=1, 2. The reduced (*k*_{x}
, *k*_{y}
)-zone associated with the above, which has the full symmetry of the given crystallographic plane is known as the surface Brillouin zone (SBZ) (see, e.g., Ref. [15]). We define a corresponding 3D reduced **k**-zone as follows:

$$-\mid {\mathbf{b}}_{3}\mid \u20442<{k}_{z}\le \mid {\mathbf{b}}_{3}\mid \u20442,$$

where b_{3}=2πa_{1}×a_{2}/[a_{1}·(a_{2}×a_{3})] is normal to the chosen crystallographic plane. The reduced k-zone defined by Eq. (4) is of course completely equivalent to the commonly used, more symmetrical Brillouin zone (BZ), in the sense that a point in one of them lies also in the other or differs from such one by a vector of the 3D reciprocal lattice.

Let us now assume that we have a photonic crystal consisting of nonoverlapping spherical scatterers in a host medium of different dielectric function and let us look at the structure as a sequence of layers of spheres with the 2D periodicity of Eq. (2). A Bloch wave solution, of given frequency ω and given **k**‖, of Maxwell’s equations for the given system has the following form in the space between the *N*th and the (*N*+1)th layers (we write down only the electric-field component of the EM wave):

with

where *q* is the wavenumber, and **A**
*N* is an appropriate origin of coordinates in the host region between the *N*th and the (*N*+1)th layers. A similar expression (with *N* replaced by *N*+1) gives the electric field between the (*N*+1)th and the (*N*+2)th layers. Naturally the coefficients ${\mathbf{E}}_{\mathrm{g}}^{\pm}$(*N*+1) are related to the ${\mathrm{E}}_{\mathrm{g}}^{\pm}$(*N*) coefficients through the scattering matrices of the *N*th layer of spheres. We have:

$${E}_{gi}^{+}\left(N+1\right)=\sum _{g\text{'}i\text{'}}{Q}_{{g}^{i};{g\text{'}}^{i\text{'}}}^{\text{I}}{E}_{g\text{'}i\text{'}}^{+}\left(N\right)+\sum _{g\text{'}i\text{'}}{Q}_{{g}^{i};{g\text{'}}^{i\text{'}}}^{\mathrm{II}}{E}_{g\text{'}i\text{'}}^{-}\left(N+1\right),$$

where *i*=*x,y,z*, and **Q** are appropriately constructed transmission/reflection matrices for the layer. For a detailed description of these matrices, which are functions of *ω*, **k**‖, the scattering properties of the individual scatterer (sphere) and the geometry of the layer, see Refs.[6, 7, 8].

A generalized Bloch wave, by definition, has the property:

$$\mathbf{k}=({\mathbf{k}}_{\parallel},{k}_{z}(\omega ,{\mathbf{k}}_{\parallel}))$$

where *k*_{z}
may be real or complex. Substituting Eq. (8) into Eq. (7) we obtain:

$$\mathrm{exp}(i\mathbf{k}\xb7{\mathbf{a}}_{3})\left(\begin{array}{c}{\mathbf{E}}^{+}\left(N\right)\\ {\mathbf{E}}^{-}\left(N+1\right)\end{array}\right)$$

where **E**
^{±} are column matrices with elements: ${E}_{\text{g}1x}^{\pm}$
, ${E}_{\text{g}1y}^{\pm}$
, ${E}_{\text{g}1z}^{\pm}$
, ${E}_{\text{g}2x}^{\pm}$
, ${E}_{\text{g}2y}^{\pm}$
, ${E}_{\text{g}2z}^{\pm}$
, …. In practice we keep a finite number of g-vectors (those with |*g*|<*g*_{max}
, where *g*_{max}
is a cutoff parameter) which leads to a solvable system of equations. The number of independent components of the electric field is in fact 2/3 of the above, in view of the zero-divergence of the electric field, but we need not go into technical details here [7, 8].

Eq. (9) constitutes a typical eigenvalue problem; because the matrix on the left-hand side of Eq. (9) is not Hermitian, its eigenvalues are in general complex numbers. We remember that *ω* and **k**
_{‖} are given quantities and therefore the eigenvalues of the matrix on the left-hand side of Eq. (9) determine *k*_{z}
; depending on the number of g -vectors we keep in the calculation, we obtain a corresponding number of kz-eigenvalues for the given ω,*k*‖. These eigenvalues of *k*_{z}
looked upon as functions *k*_{z}
=*k*_{z}
(ω;*k*‖) of real *ω*, for given **k**
_{‖}, are known as the real-frequency lines in the complex *k*_{z}
-space. We refer to them as the complex band structure of the crystal associated with the crystallographic surface defined by Eq. (2). A line *k*_{z}
(*ω*;*k*
_{‖}) may be real (in the sense that *k*_{z}
is real) over certain frequency regions, and be complex (in the sense that *k*_{z}
is complex) for *ω* outside these regions. It turns out that for given **k**
_{‖} and *ω*, out of the many eigenvalues of *k*_{z}
none or, at most, a few are real; the Bloch waves, eigensolutions of Eq. (9), corresponding to them represent propagating modes of the EM field in the given crystal. The remaining eigenvalues of *k*_{z}
are complex and the corresponding Bloch waves are evanescent waves; they have an amplitude which increases exponentially in the positive or negative *z*-direction and, unlike the propagating waves, do not exist as physical entities in the infinite crystal. They are however very useful in the understanding of the optical properties of finite slabs of the crystal. For example, the attenuation of a wave of given **k**
_{‖}, incident on a slab of the material of thickness *d*, with a frequency within a region over which no propagating solution exists for the given **k**
_{‖}, is determined by that evanescent wave, which has the *k*_{z}
with the smallest in magnitude imaginary part: *q*_{I}
; the attenuation is roughly speaking proportional to exp(-*qId*). In all cases [**k**
_{‖}, *Re*(*k*_{z}
)] lies in the reduced zone defined by Eq. (4). To obtain a full picture of the ordinary frequency band structure one needs to know all the real *k*_{z}
-sections of the real frequency lines for every **k**
_{‖} in the irreducible part of the SBZ; in the remaining part of the SBZ they are determined by symmetry. The dispersion curves obtained in this fashion can always be related with those obtained in the traditional manner (using the BZ as the reduced zone of **k**). But one does not need to do so; the physical interpretation of the results is conveniently done in either representation.

The on-shell method we have described has a number of advantages over the traditional methods. The unit layer along the *z*-direction may consist not of one plane of spheres (as implied so far) but by a number of planes which may be different (the radii of the spheres and/or their dielectric functions may be different) as long as they have the same 2D periodicity; which allows us to study a variety of heterostructures with relative ease. Moreover, we can easily calculate the transmission, reflection and absorption coefficients of light incident at any angle on a slab of the photonic crystal. For this purpose we combine the **Q**-matrices of the different layers that make the slab into a final set of matrices which determine the scattering properties of the slab [6, 7, 8]. The method applies equally well to nonabsorbing systems which have a well defined frequency band structure, and to absorbing systems (with a complex dielectric function for some of the constituent materials of the crystal) which do not have a well defined frequency band structure (there can be no truly propagating waves in such systems). Finally we can deal with slabs of photonic crystals which contain impurity planes as long as the 2D periodicity parallel to the surface of the slab is retained.

A region of frequency constitutes an absolute (omnidirectional) frequency gap if no propagating wave exists in the infinite crystal over this region whatever the value of **k**
_{‖}. This can be ascertained from the projection of the frequency band structure on the SBZ of the given surface. In Fig. 1 we show two examples of such band-structure projections. Both examples refer to so-called metallodielectric structures. Nowadays, it is possible to fabricate well-defined ordered metallodielectric composites consisting of tailored mesoscopic building blocks. Such nanocrystals provide opportunities for optimizing optical properties of materials and offer possibilities for observing new, and potentially useful physical phenomena [16, 17, 18]. In both cases considered here the crystal is an fcc one and is viewed as a sequence of planes parallel to the (001) surface. In the first case (an example of cermet topology in relation to the metallic component) the crystal consists of metallic spheres, described by a Drude dielectric function

in gelatine. In the second case, which is typical of network topology in relation to the metallic component, the crystal consists of nonoverlapping silicon spheres in a metal described by Eq. (10). We have deliberately disregarded absorption by the metallic component, in order to be able to calculate a frequency band structure in an unambiguous manner. The shaded areas in Fig. 1 correspond to allowed regions of frequency; for a given **k**
_{‖}, these regions extend over those frequencies for which there is at least one propagating wave in the infinite crystal; the blank regions represent frequency gaps for the given **k**
_{‖}. Obviously an absolute gap exists only when a blank region of frequency is common to all **k**
_{‖} in the SBZ. As a rule one shows the projection of the band structure for **k**
_{‖} along symmetry lines of the SBZ as in Fig.1; but one must check that a gap over these lines extends to the whole of the SBZ. In Fig. 1 we indicate the positions of the 2
^{l}
-pole plasma resonances (for *l*=1,2,3) of an individual scatterer (a metallic sphere in gelatine in the first case, and a silicon sphere in a metal in the second case). It is evident that in each case things develop about these resonances; in the second case one clearly obtains the allowed frequency bands about these levels. Evidently absorption will occur over the allowed regions of frequency and to some lesser degree beyond these regions into the frequency gaps, when *ε*(*ω>*) of Eq. (10) is replaced by the complex Drude dielectric function: *ε*(*ω*)=1-${\omega}_{p}^{2}$
/*ω*(*ω*+*iτ*
^{-1}) [13]. One thing is worth pointing out, and this concerns the very different behavior of the cermet and network topologies at low frequencies. In the cermet topology the composite behaves like a dielectric in the long wavelength limit, allowing the propagation of EM waves through it; in the network topology the composite behaves like a metal (its DC conductivity does not vanish as *ω*→0) and does not allow the propagation of EM waves through it.

In the remaining of this paper we shall demonstrate the usefulness of our method in a di erent situation which relates to the reflection of light from a slab of a colloidal system [19]. This can be described as an fcc (111)-oriented slab of closed-packed poly-methylmethacrylate (PMMA) spheres in air. The slab does not consist of a perfect monocrystal, and deviations from periodicity (stacking faults, dislocations, point defects, etc.), which are not well defined, must be born in mind in any comparison with a theoretical calculation of the reflectance from a perfectly periodic slab. The reflectance of light incident normally on a slab of the material, as measured by Allard *et al*. [19], is shown by the dotted curve in Fig. 2c. In Fig. 2a, we show the band structure of the corresponding infinite crystal along the normal to the (111) surface (**k**
_{‖}=**0**). Next to the band structure we show the reflectance evaluated for a perfect slab with no absorption. The calculation was done for a slab 32-layers thick, but we note that the reflectance, when absorption is taken into account (see below), does not depend on the thickness of the slab when this exceeds 30 layers or so. The reflectance equals unity over the Bragg gap about λ≈1085 nm. The oscillations on either side of the peak are of the Fabry-Perot type and are due to the finite size of the slab. The fine structure of the reflectance at higher frequencies (λ<600 nm) reflects the complexity of the frequency band structure in this region. Some of these bands (denoted by red lines in Fig. 2a) are nondegenerate, non optically active bands (they do not couple with the incident EM field) and the rest (denoted by black lines) are active, doubly degenerate bands, and it is through them that light flows into the crystal. Evidently, the theoretical reflectance of the perfect nonabsorbing slab relates, at a qualitative level, to the measured reflectance of the imperfect and absorbing (at least to a small degree) slab of the material. At this stage we could take into account absorption and disorder in an approximate manner, by adding an imaginary part independent of frequency to the dielectric constant of the PMMA spheres, which effectively removes light from the coherent beam. The results are shown by the solid line in Fig. 2c, and one can see that with the use of this single parameter one reproduces satisfactorily the essential features of the experimental curve.

## Acknowledgements

We thank M. Allard and coworkers for sending us a copy of their paper prior to publication. V. Yannopapas is supported by the State Foundation (I. K. Y.) of Greece.

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