Macroscopic optical response and photonic bands
We develop a formalism for the calculation of the macroscopic dielectric response of composite systems made of particles of one material embedded periodically within a matrix of another material, each of which is characterized by a well defined dielectric function. The nature of these dielectric functions is arbitrary, and could correspond to dielectric or conducting, transparent or opaque, absorptive and dispersive materials. The geometry of the particles and the Bravais lattice of the composite are also arbitrary. Our formalism goes beyond the longwavelenght approximation as it fully incorporates retardation effects. We test our formalism through the study the propagation of electromagnetic waves in 2D photonic crystals made of periodic arrays of cylindrical holes in a dispersionless dielectric host. Our macroscopic theory yields a spatially dispersive macroscopic response which allows the calculation of the full photonic band structure of the system, as well as the characterization of its normal modes, upon substitution into the macroscopic field equations. We account approximately for the spatial dispersion through a local magnetic permeability and we analyze the resulting dispersion relation, obtaining a region of left-handedness.
The propagation of light within homogeneous materials is completely characterized by their electromagnetic response. In the optical regime, magnetic effects are typically negligible so that knowledge of the dielectric response is sufficient for the study of wave propagation . However, when the system has spatial inhomogeneities, the behaviour of light is not trivial and has captured the attention of many researchers [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The possibility of controlling the propagation of photons by producing artificially structured materials has been widely demonstrated in recent years. Novel effects have been obtained, such as negative refraction, inverse Doppler effect, optical invisibility cloaking, optical magnetism etc. [12, 13, 14, 15].
Effective medium theories [16, 17, 18, 19, 20, 21, 22, 23, 24] have been proposed to describe inhomogeneous systems, such as diluted colloidal suspensions in the quasistatic or long wavelength limit, in terms of a homogeneous macroscopic response. Further developments in homogenization of composites have been proposed [25, 26, 27, 28, 29, 30, 31, 32, 33] and their limits of validity have have been discussed [2, 34, 35].
In this paper we obtain the macroscopic optical response using a computationally efficient reformulation of a procedure [5, 6] original developed to account for the local field effect of systems with spatial fluctuations. Our main result is that for any system, the macroscopic response to an external excitation is simply the average projection of the corresponding microscopic response. We apply this general result to Maxwell equations in order to obtain an explicit expression for the macroscopic dielectric response of binary composites. We illustrate its use with numerical results for a 2D periodic photonic crystal. As the wavelength of the electromagnetic field becomes comparable to the characteristic microscopic spatial lengthscale of the system, the macroscopic response becomes highly non-local, with a strong dependence on the wavevector besides its usual dependence on frequency. When the full spatial dispersion is taken into account, the photonic band structure of the system can be obtained and analyzed from its macroscopic response.
The paper is organized as follows: In Sec. 2 we formulate a general theory for the macroscopic dielectric response of composite systems. In Sec. 3 we adapt our formulation to periodic systems made of two alternating materials, each characterized by well defined dielectric functions. In Sec. 4 we develop an efficient computational scheme which allows the numerical calculation of the macroscopic response without recourse to explicit operations between large matrices. In Sec. 5 we obtain the non-local macroscopic dielectric tensor and use it to calculate macroscopically the full band structure of a 2D photonic crystal made up of non-dispersive dielectric components; we compare it to exact results as well as to results within a local approximation that partially accounts for spatial dispersion through an effective magnetic permeability. Finally, in Sec. 6 we present our conclusions.
2 General theory
Consider a non-magnetic inhomogeneous system characterized by its dielectric response , defined through the constitutive equation , where and are the microscopic displacement and electric fields respectively. We designate these fields as microscopic as they have spatial fluctuations due to the inhomogeneities of the material. They are not microscopic in the atomic scale, but rather, on the scale of the spatial inhomogeneities of the system. We use a caret () over a symbol to denote its operator nature and we leave implicit the dependence of the dielectric response on position as well as on frequency. Our purpose is to obtain the macroscopic dielectric response of the system, relating the macroscopic displacement and electric fields, from which the fluctuations have been removed.
We start from Maxwell equations for monochromatic microscopic fields. We follow the usual procedure  to decouple the magnetic field from the electric field to obtain the second order wave equation
is the wave operator, is the wavenumber of light in vacuum, is the frequency, is the speed of light in vacuum and is the external electric current density. Here we introduced the transverse projector such that the transverse projection of a field is . For completeness, we also introduce the longitudinal projector such that , with the identity operator. Notice that Eq. (1) contains both a longitudinal and a transverse part, so it describes not only transverse waves, but also allows for the possible excitation of longitudinal waves such as plasmons .
We solve Eq. (1) formally for to obtain
As we are interested on the macroscopic response of the system, we introduce the average and fluctuation projectors, such that an arbitrary field can be written as with its average and its fluctuating part, and we identify the average field with the macroscopic field . Later, we will provide appropriate explicit definitions for and ; here we remark that they must be idempotent , , i.e., the average of the average is the average, and the fluctuations of the fluctuations are the fluctuations . Furthermore, and .
As the macroscopic response would be useless unless the external excitations are devoid of microscopic spatial fluctuations, we assume that the external current has no fluctuations, so and , where we used the idempotency of . Thus, acting on both sides of Eq. (3) with we obtain
Here we define , with for any operator . As Eq. (4) relates the macroscopic external electric current to the macroscopic electric field we may identify the macroscopic inverse wave operator ,
We summarize this results stating that the macroscopic inverse wave operator is simply the average of the microscopic inverse wave operator. This is a particular case of a more general result: the macroscopic response to an external excitation is simply the average of the corresponding microscopic response.
in analogy to Eq. (2). Thus, we have to invert the wave operator, average it, and invert it again to finally identify the macroscopic dielectric response. Notice that we have made no approximation whatsoever. We remark that as relates two fields, and , that have spatial fluctuations, we may not simply average to obtain its macroscopic counterpart, i.e. . The difference constitutes the local field effect .
Our result may easily be generalizable to other situations and other response functions. The procedure consists on first identifying the response (in our case ) to the external perturbation (i.e. ), and then averaging it to yield the corresponding macroscopic response (i.e. ), which may further be related to the desired macroscopic response operator (i.e. ).
3 Periodic binary systems
In this section we use Eq. (6) to obtain the optical properties of an artificial binary crystal made of two materials and with dielectric functions and . We assume that both media are local and isotropic so that and are simply complex functions of the frequency. For convenience, we will further assume that is real, though this assumption is easily relaxed .
We introduce the characteristic function of the inclusions, such that whenever is on the region occupied by the inclusions, and otherwise. Thus, we may write the microscopic dielectric response as
where the characteristic operator corresponds to multiplication by in real space, and we defined
which, as shown below, plays the role of a metric.
where we have taken advantage of the fact that is unrelated to the texture of the crystal, so that it does not couple average to fluctuating fields.
where is a given wavevector, is the reciprocal lattice of our crystal and the coefficients represent the field in reciprocal space. In this representation, all operators may be written as matrices with vector index pairs , besides other possible indices, such as Cartesian ones. Thus, we represent the longitudinal projector as the matrix
with the Kronecker’s delta, so the transverse projector becomes with the Cartesian identity matrix. The Laplacian operator in reciprocal space is represented by
and we can define the average projector as a cutoff in the reciprocal space 
so that average fields simply keep the contributions with vector while all other wavevectors are filtered out.
As shown by Eq. (11), we only require
to obtain the macroscopic inverse wave operator, where the subindices denote the projection onto the subspace with .
4 Recursive method
onto a given state , where
is the Green’s operator corresponding to some Hermitian Hamiltonian and is a complex energy. In Ref.  a similar result was obtained, where the Hamiltonian was identified with the longitudinal projection of the characteristic function , the energy with the spectral variable , and with a slowly varying longitudinal wave, and it was shown that the corresponding projected Green’s function was proportional to the inverse of the longitudinal macroscopic dielectric function. Haydock’s method may be applied to obtain the projected Green’s function (17) in a very efficient way [42, 43], and it has been adapted previously  to the calculation of the optical response of nanostructured systems in the long-wavelength local limit. In this section we generalize the approach of Ref.  to arbitrary frequencies and wavevectors.
Eqs. (17) and (18) are similar to Eq. (16), although the operator product that plays the role of Hamiltonian does not correspond to a symmetric matrix. Nevertheless, it would correspond to a Hermitean operator if we use as a metric, that is, if we use instead of as the scalar product of two arbitrary states and . Then, it is easy to verify the Hermiticity condition . Notice, however, that is not positive definite.
According to Eq. (11), we need the average , where we projected onto an average state consisting of a plane wave with a given wavevector , frequency and polarization . Here, is normalized according to the metric , i.e., , and the coefficient is a real positive number chosen to guarantee that is normalized in the usual sense . Now, we define and we obtain new states by means of the recursion relation
where all the states are orthonormalized according to the metric , that is
with and the Kronecker’s delta function. The requirement of orthonormality yields the generalized Haydock coefficients , , and given the previous coefficients , and . Thus, are obtained from
where we choose the sign so that is positive and we may choose as a real positive number. In the basis , is represented by a tridiagonal matrix with along the main diagonal, along the subdiagonal and along the supradiagonal, so that
Notice that we may represent the states through the corresponding Cartesian field components for each reciprocal vector or for each position within a unit cell. Thus, the action of is a trivial multiplication in reciprocal space, while that of is a trivial multiplication in real space, and we may repeatedly compute and Haydock’s coefficients without performing any actual matrix multiplication, by alternatingly fast-Fourier transforming our representation between real and reciprocal space.
According to Eq. (11), we do not require the full inverse of the matrix (23), but only the element in the first row and first column. Following Ref. , we obtain that element as a continued fraction, which substituted into Eq. (11) yields
The right hand side of Eq. (24) denotes the macroscopic response projected onto a state with the given wavevector , frequency and polarization . Due to the translational invariance of the homogenized system, the inverse wave operator for a given corresponds simply to a tensor with Cartesian components . Thus, so far we have calculated its inner products with , as shown by the second term of Eq. (24). We may repeat the calculation above for different orientations of the (possibly complex) unit polarization vector and solve the resulting equations for all the individual Cartesian components . Finally, we perform a matrix inversion of this macroscopic tensor and we use Eq. 7 and 6 to compute the macroscopic dielectric tensor
Eqs. (24) and (25) constitute the main result of our formalism. Notice that in general, this dielectric tensor would depend on both and , that is, it is a temporal and spatially dispersive response function.
To test our formalism, we first calculate the local limit of the longitudinal macroscopic dielectric function of a 2D system made up of a square array of empty cylindrical holes () placed with lattice parameter within a dielectric medium with permittivity . The holes are of circular cross section with radius . We assume that the axes of the cylinders are parallel to the axis. This system has been frequently been used as a testground for calculations of photonic crystal properties .
In this calculation we took the limit along the direction, and we introduced the Cartesian unit vectors , and . We remark that it is important to assign a direction to even in this limit, as the transverse and longitudinal projection of differ, although, for this system, the resulting is isotropic within the plane.
In Fig. 1 we
show calculated for two different
lattice parameters nm and nm. Our calculations were
performed using the Perl Data Language 
The dispersion relation of transverse waves with TE polarization propagating along the plane can be simply written as 
Therefore, in Fig. 2 we also plotted the curve . Those points where it intersects the curves for correspond to normal TE modes of the system with frequencies , I, II, …Here, we use roman numerals to number the modes in increasing frequency order.
In order to test our theory, we also plot the TE modes obtained by solving the eigenvalue equation 
through an implicitly restarted Arnoldi iteration  , applying in real space and in reciprocal space . We further verified our results by recalculating them with the freely available package MPB  and comparing it to previous results  for the same system. The agreement between these calculations shows that it is feasible to use the macroscopic dielectric response of the system to obtain its photonic band structure. However, to succeed, we have to account fully for the spatial dispersion of .
We remark that although we obtained the correct photonic bands even for large wavevectors, going all the way around the first BZ, there are some regions where we failed to obtain all of the normal modes. For example, our procedure did not produce the third band (labeled III in the figure) along the line, and neither the fourth band (IV) along the line, between and . Furthermore, it did not yield the second band (II) which is almost degenerate with the third band (III) along . It is easily shown that the microscopic field corresponding to the missing band along is antisymmetric with respect to the reflection about the line. Similarly, the missing bands along the region are antisymmetric with respect to the reflection about the line. Thus, the reciprocal vector does not contribute to the electric field for those bands , i.e., the missing modes have no macroscopic field components and thus, apparently they may not be obtained from the roots of the macroscopic dispersion relation (26).
Nevertheless, within our formalism, the wavevector is actually not restricted to lie within the first BZ. Thus, we may calculate the macroscopic dielectric function for wavevectors beyond the first BZ, and obtain the modes in the extended scheme. In Fig. 2 we show part of the macroscopic dielectric function with a wavevector that has been shifted out of the first BZ from along the direction by the reciprocal vector , i.e., . We remark that is not a periodic function of , as it corresponds to a specific plane wave, not to a microscopic Bloch’s wave. The intersection of the curve with yields the corresponding macroscopic modes. For example, the diamonds in Fig. 2 illustrate two of the normal modes that could be obtained using this shifting procedure. One of these modes is identical to the mode labelled IV that was obtained previously using the unshifted response . Nevertheless, another mode, labelled III appears just below and it corresponds to one of the previously missing modes. We may shift it back into the first BZ in order to display it in the reduced zone scheme. Proceeding in this fashion, we have obtained all of the previously missing bands, shown by diamonds in Fig. 3. Thus, we have shown that we can calculate the full photonic band structure for the TE modes from the macroscopic non-local dielectric function of the composite system.
A similar procedure to that discussed above can be employed also for the TM modes of the system, with the electric field within the plane. In Fig. 4
we show the in-plane macroscopic dielectric functions and of the system for the same wavevector as in Fig. 2. This wavevector and the system are symmetric under reflections. Therefore, and we may identify the transverse and longitudinal responses as and respectively. The normal modes are then obtained from the transverse dispersion relation (26) and the longitudinal dispersion relation 
These are identified with circles and diamonds in the figure.
In Fig. 5 we do a similar calculation but along the line, for the wavevector . Along this line, the wavevector and the system are symmetric under the reflection , and thus . We may now identify the transverse and longitudinal responses as and respectively. The upper panel of Fig. 5 shows , and their intersections (circles) which yield the transverse modes, while the lower panel displays and its zero (diamond) which correspond to a longitudinal mode.
Along the line, from to , there is no crystal symmetry operation that leaves invariant the wavevector, and thus transverse and longitudinal fields mix among themselves, i.e., there are no longitudinal nor transverse modes. Nevertheless, we may obtain the frequencies of the mixed modes through the singularities of the wave operator matrix
as illustrated in Fig. 6
for a wavevector along the line.
From calculations such as those illustrated by Figs. 4, 5 and 6, or more generally, from the zeroes of for arbitrary wavevectors , we can obtain the full TM photonic band structure, as shown by Fig. 7.
In order to test our theory, we also plot the TM modes obtained by solving the eigenvalue equation 
using similar methods [48, 38] as for the TE case and comparing them successfully to previous results  for the same system. The agreement between these calculations shows that it is also feasible to use the macroscopic dielectric response of the system to obtain its photonic band structure in the TM case.
An approximate band structure could be obtained for transverse waves in the region of small wavevectors by expanding the LHS of Eq. (26) up to second order in and solving for . The result is a local dispersion relation
where the non-locality of is partially accounted for through an effective magnetic permeability 
In Fig. 8 we show the resulting approximate bands for the TE case.
We can see that the acoustic band within the local approximation is indistinguishable from the exact result for small wavevectors. This local acoustic band extends up to the frequency for which has a pole. The local approximation also reproduces the upper band of the exact calculation (labelled V in Fig. 3), but only in the immediate vicinity of , close to the zero of . For larger wavevectors it acquires a larger positive dispersion. As discussed above, the band labelled III in Fig. 3 doesn’t couple to macroscopic fields for along the line, while the band labelled IV doesn’t couple along the line. Thus, neither band couples to macroscopic fields at and as a consequence, these bands are not reproduced at all by the local approximation (31).
In the region the local approximation predicts a curious band that displays backbending. Its lower part begins at the pole of , which corresponds to a double zero of . This part of the local dispersion relation is spurious, as the second order Taylor expansion of the dielectric function that yields Eqs. (32) and (31) starting from Eq. (26) would be meaningless at a pole. Consequently, this part of the local band doesn’t correspond to any band within the exact calculation. On the other hand, the upper part of the backbending band starts at a simple zero of which arises from a different kind of singular behavior, consisting of a pole in at which is suppressed at as its weight is approximately proportional to . This part of the local dispersion relation agrees with band II of Fig. 3 at and displays a negative dispersion as does band II. Furthermore, for this band, both and are negative. Thus, our photonic crystal, made up of holes within a dispersionless dielectric, behaves for some frequencies as a left-handed metamaterial [52, 53]. Nevertheless, caution should be exercised when using Eqs. (32) and (31) close to singularities of .
We have shown that the macroscopic inverse wave operator of composite systems with arbitrary spatial fluctuations is simply given by the average projection of the corresponding microscopic operator. These operators may be related to the macroscopic and macroscopic dielectric functions, thus yielding a general procedure to incorporate the effects of spatial fluctuations in the calculation of all the components of the macroscopic dielectric tensor of the system. We have extended Haydock’s recursive scheme in order to calculate very efficiently the response of periodic two-component systems in terms of a continued fraction whose coefficients may be obtained without recourse to operations with the large matrices that characterize the microscopic response and the field equations. We illustrated and tested our results through the calculation of the wavevector and frequency dependent macroscopic dielectric tensor of a 2D dielectric system with non-dispersive non-dissipative components. We identified the transverse and longitudinal components of the response for wavevectors along symmetry lines and we obtained all the components of the tensorial response for the case of reduced symmetry. The macroscopic response has a series of poles, related to resonant multiple coherent reflections and by accounting for its spatial dispersion we obtained the full photonic band structure of the system from the dispersion relations corresponding to the homogenized material. Besides yielding the correct bands from a macroscopic approach, our scheme allowed us to classify the polarization of each mode as either transverse, longitudinal or mixed. Even though the macroscopic approach might fail to yield some modes, namely, those which have no coupling to the macroscopic field due to symmetry, they may be recovered by extending the notion of macroscopic state, allowing its wavevector to lie beyond the first Brillouin zone. The non-locality of the dielectric response may be partially accounted for through a magnetic permeability which can then be employed in the calculation of the optical properties of the system. We compared the band structure obtained through this local approximation to the exact results and we obtained partial agreement close to for those bands that do couple to long-wavelentgh fields. In particular, we showed that this approach can yield a negative dispersion in frequency regions where both the permitivity and permeability are negative. Nevertheless, we discussed some shortcommings of the local approach. Although here we presented results for dispersionless transparent dielectrics, our calculation does not require that the materials that make up the system be non-dispersive nor non-dissipative, so that calculations for real dielectrics and metals may be performed with the same low computational costs.
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