1. Introduction

Periodic crystals do not exist in the real world, they are all finite. It turns out that in a finite crystal one has to consider the reflection of the Bloch waves at the endfaces. This leads to the formation of standing waves, as mentioned, e.g., by [3]. Reflection amplitudes are easy to find in the long wavelength limit using effective medium theory.We construct here an effective medium theory that describes at all frequencies the light propagation through a finite crystal. Its key ingredients are the reflection and transmission coefficients for a half-space crystal, calculated using an expansion in Bloch waves. With our approach, standard formulas for a homogeneous layer can be used for a finite crystal.We consider a few important applications: the local density of states for finite or infinite systems, even with absorption, and frequencies of defect modes.

 

Fig. 1. (Left panel) complex eigenvalue λ₊=e^ika vs. frequency for the Kronig-Penney model, Eq. (9), with point scatterers of polarizability α=(0.2+0i)a. (Right panel) band structure ω(k) of the infinite crystal. The thick dispersion curves correspond to the physical Bloch momentum fixed by the requirements of causality and energy conservation. Even bands exhibit negative refraction (k<0).

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We discuss negative refraction. Our results are illustrated by exact formulas for a 1D Kronig-Penney (planar scatterer) model.

2. Transfer matrix approach

We start by recalling the transfer matrix method for a one-dimensional system made of a finite number N of unit cells.We use units with c=1 and denote ω the wavevector in the background medium with permittivity ε=1. The transfer matrix T connects the field E_n at the left edge of the nth unit cell to the field E _n+1 at the next cell. Expanding the fields in plane waves propagating to the left and right

where a is the crystal period, we have (see chap. 6 of [1])

Since detT=1, its eigenvalues λ_± can be written in the form λ_±=e ^±ika where k is the Bloch quasimomentum. An eigenvalue on the unit circle (real k) corresponds to a propagating (extended) Bloch mode whereas real eigenvalues (k imaginary) are found in the band gaps. An example is shown in Fig. 1 for the Kronig-Penney model detailed below.

The reflection and transmission amplitudes from a finite crystal are given by the Nth power of the primitive transfer matrix, e.g. r_N =-$T_{21}^N$/$T_{22}^N$. T^N is particularly simple to compute in the Bloch basis:

The ratios of the plane wave amplitudes for the Bloch states ±k are

Independent of the normalization factors N _± chosen for the Bloch eigenstates, the reflection coefficient is given by (in agreement with Refs. [3, 7])

 

Fig. 2. (Left) reflectance |r_N | for the Kronig-Penney model with α=(0.2+0i)a, N=4 (blue line). The envelope, Eq. (6), is shown as well. Solid green line: half-space approximation |r|, Eq. (11). Dashed dark green line: |r| for absorbing scatterers (α=(0.2+0.02i)a). (Right) reflection coefficient |r _i| for Bloch waves reflected from the end face of a semi-infinite crystal. Solid line: our result, Eq. (13); dashed line: proposed by Sakoda [4]. Kronig-Penney model with α=(0.2+0i)a.

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For a finite number of periods N, the reflectance |r_N | shows oscillations as a function of frequency due to the formation of standing waves between the end faces of the crystal, with a fringe spacing scalingwith 1/N. If these are not resolved due to some finite frequency resolution (or fluctuations in the crystal thickness), the envelope of the reflectance is a useful generalization. Maximizing |r_N | with respect to N for each fringe period, we find from Eq. (5) in allowed bands (see Fig. 2)

provided energy conservation holds. This envelope function is consistent with [3] and overestimates the reflectance in the band gap.

In the preceding plots, a Kronig-Penney type model has been used for definiteness. We consider a wave equation with point scatterers (see Ref. [9] for a 3D generalization)

where the polarizability α_n characterizes the strength of the nth scatterer. We focus on α_n =α to get a periodic crystal. The continuity of the electric field E and its first derivative on the scatterers leads to the transfer matrix

whose eigenvalues are given by

We have chosen the sign of the square root such that λ₊ is located on or inside the unit circle (provided α has an infinitesimally positive imaginary part).

3. Half-space-approximation

We show here that the Bloch modes for the infinite crystal can be used to define a reflection coefficient r for a semi-infinite crystal. The key idea is to use the transformation matrices M and M ^-1 occurring in Eq. (3) as transfer matrices, linking plane waves in the vacuum outside the crystal to Bloch modes inside the crystal. For a unit incident amplitude from the left, transmission and reflection amplitudes are thus given by

Independent of the eigenmode normalizations, this yields

where the second equation is valid for the Kronig-Penney model. At low frequency, we recover the standard effective medium result r≈(1-n _eff)/(1+n _eff) with the effective index n _eff=(1+α/a)^1/2=lim_ω→0 k(ω)/ω. Figure 2 shows that Eq. (11) gives, for all frequencies, a good approximation to the reflectance from a finite crystal, when |r_N | is averaged over the standing wave fringes. In the band gaps, we recover the intuitively expected perfect reflector, |r|=1.

In a similar manner, we define a coefficient r _i for the ‘interior’ reflection of a Bloch wave from a crystal end face, say the left one:

Reflection from both end faces can be described by the same coefficient. This fixes the normalization of the Bloch eigenmodes:

For the last equality, we have used the relation c ₊ c _-=1 which follows from the symmetry relation T ₁₂=-T ₂₁ of the transfer matrix for an even scatterer. Note the π phase shift between ‘exterior’ and ‘interior’ reflection amplitudes.

Our expression for the interior reflection can be compared to a conjecture by Sakoda, r _i(Sakoda)=(n _g-1)/(n _g+1) where the group index n _g=dk/dω. When the standing waves inside a finite crystal are used for lasing, |r _i| determines the quality factor of this cavity [4]. Sakoda’s result qualitatively agrees with ours (see Fig. 2), but underestimates |r _i|, except very close to the band edges.

4. Applications and discussion

4.1 Effective medium theory

In the preceding section, we have seen that physically reasonable reflection and transmission amplitudes can be introduced for the endfaces of a photonic crystal, when the field inside the crystal is expanded in Bloch modes. This immediately suggests an effective medium description of a finite length crystal. The reflectance r_N , e.g. would be given by the well-known Fabry-Pérot expression [2]

where the standing wave oscillations arise by summing a multiple scattering series.

It is easy to check that r _N,FP coincides with the transfer matrix result r_N , Eq. (5), by substituting the reflection amplitudes, Eqs. (11, 13), and the corresponding transmissions. In fact, this is not surprising because according to the definitions Eqs. (10, 12) of these amplitudes, the transformation matrices M and M ^-1 can play the role of transfer matrices at the crystal endfaces. If we express them in terms of the r, r_i, t, t′, the expression Eq. (3) for T^N becomes precisely the product of transfer matrices one would write down for a homogeneous layer, and leads to Eq. (14).

 

Fig. 3. LDOS, Eq. (15), in an infinite crystal with scatterers at x=…,-a/2,a/2,… (Left) no absorption (α=0.2a). (Right) nonzero absorption (α=(0.2+0.05i)a).

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It has been previously noted that the Bloch momentum k is a useful quantity to describe the propagation inside a finite-size crystal [7, 9, 5]. In particular, Jeong et al. [5] interpret the phase of the transmission amplitude in terms of a frequency-dependent effective index, but find differences with respect to the conventional expression n _eff=k/ω close to the band edges. Our analysis indicates that this is due to the contribution of the Fabry-Pérot denominator in the expression t_N =tt′e^ikaN /(1-$r_{\mathrm{i}}^2$ e ^2ikaN).

4.2 Local density of states

The LDOS ρ(x,ω) is the key quantity for the radiative decay of a two-level system in the crystal [6], and it is well known that it is obtained from the imaginary part of the Green function G(x,x;ω) (the field radiated by a pointlike test source). If we put two (finite or semi-infinite) crystals at distances d_L,R from a test source, the Green function is easily obtained from the corresponding reflection coefficients r_L,R . Normalizing to the free space LDOS, we get in this way

Note that the distances d_L,R are defined relative to the reference planes implicit in the reflection coefficients r_L,R (located a/2 in front of the first scatterer in our example). This simple formula reproduces more involved expressions given, e.g., in [8]. For an infinite crystal with point scatterers, the LDOS is plotted in Fig. 3 and shows the characteristic inverse square root singularities close to the band edges [7]. One also recovers the well-known dielectric and air bands at the gap edges.

Our approach immediately allows for a nonzero absorption in the sample where, at least for an infinite crystal, standard band theory breaks down because all Bloch vectors become complex (extended states do not exist any more; see [10] for a detailed discussion). For finite absorbing systems, it is well-known that transfer matrix techniques still provide the required reflection and transmission spectra (see, e.g., [11]). To define consistently a half-space reflection amplitude Eq. (11), we require that r(ω) be nonsingular as a function of complex frequency in the upper half plane (as dictated by causality). For the Kronig-Penney model, we can show that it suffices to choose the eigenvalue e^ika located inside the unit circle. (This condition is consistent with the limit N→∞ of Eq. (5) for finite absorption.) The reflection coefficient for finite absorption is shown in Fig. 2: it drops below unity in the band gaps. The LDOS (Fig. 3) exhibits a smoothing out of the dielectric band edge while the singularity at the upper gap edge persists. This is due to the approximation of point-like scatterers in our Kronig-Penney model, which makes the mode functions at the air band edge insensitive to the scattering strength α.

 

Fig. 4. (Left) reflectance from two N=6 layer crystals (α=0.2a) with a defect in between (d_R =d_L =a/2). Green line with peak: active defect with scattering strength α _d/a≈-0.62-0.04i, given by Eq. (16) for ωa/2πc≈0.462. The same structure with a passive defect (α _d≈-0.62a) gives the green line with the transmission dip. Blue line: reflectance for α _d=α. (Right) defect frequency in the first band gap vs. real part of scattering strength α _d.

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4.3 Defect modes

As a final application, we investigate the situation that two crystals surround a defect scatterer with α _d≠α. Expanding the field in plane waves around the defect, we find that the defect mode frequency is determined by

where d_R,L are again the distances between the crystals and the defect. For given real ω, α _d is in general complex and can be absorbing or even active. In the latter case, the crystal backscatters more light than is incident when the defect resonance is hit (see Fig. 4). By varying α _d, the defect mode frequency can be tuned across the band gap.

4.4 Negative refraction

To conclude, we point out that the simple one-dimensional Kronig-Penney model provides an exactly soluble example of a photonic crystal with negative refraction. For a non-absorbing, semi-infinite crystal, the requirement that |r|²≤1 leads to the condition sin(ωa) sin(ka)≥0 for the ‘physical’ Bloch momentum k. This is fulfilled when an incident plane wave in an even frequency band injects a Bloch wave with negative k into the crystal. Note that this choice of k is consistent with the well-known rule that the Bloch wave should have a positive group velocity v _g=dω/dk (see, e.g. [12]), as is manifest from the thick lines in Fig. 1.