1. Introduction

Negative refraction was the subject of a large amount of papers in the last years [1–7], also in view of very interesting new phenomena like the super-lens effect [8–16]. A negative refractive index arises in meta-materials where inhomogeneities are much smaller than the wavelength of the incoming radiation and it has been demonstrated at microwave wavelengths. On the other hand, it has been shown that using photonic crystals (PhC’s) it is possible to obtain a negative refraction behaviour also at optical wavelengths [17–27], i.e., a refracted beam going to the “wrong direction” compared to the direction expected on the base of the classical refraction laws.

Most of the theoretical analysis of negative refraction in PhC’s makes use of the equi-frequency surface (EFS), (ex. Refs. [18, 19, 23, 25, 26, 27]) which are contour diagrams in the wavevector space for a constant frequency. The normal to such curves gives the Poynting vector direction, indicating the energy flow in PhC’s. In such an analysis, using the conservation of the parallel wavevector component across the boundary, conditions are determined under which negative refraction behaviour appears in a PhC even without a negative refraction index.

We follow here the classical approach of Dynamical Diffraction Theory (DDT) [28, 29] making also use of EFS (dispersion surfaces, adopting the DDT terminology). We will show however that, in general, it is not correct to select among the different possible wavevectors in the crystal. In general the possible different wavevectors in the crystal coexist, and it is really their difference which explains the modulation: this is the base of the phenomenon that we analyze in this paper. In the following we will show that, at the exit surface of the crystal, the intensity will be at a maximum in the positive (forward diffracted), or in the negative (diffracted) direction depending on the difference of the two wavevectors inside the crystal: this is a manifestation of the Pendellösung effect.

Pendellösung is a fundamental result of DDT, as worked out by Ewald at beginning of the 20th century [30–32]. This phenomenon is a periodic exchange of energy between two interfering beams inside the crystal, as between two coupled pendulums, hence the name given to that effect by Ewald [31] who also gave a mechanical analogy [32]. It is not surprising that we find the same result for PhC’s. The optical properties of PhC’s [33–36] are commonly derived in analogy with the electron band theory in solid state physics; on the other hand, the analogy with the x-ray diffraction in the crystals has been evident since the publications of the first papers on PhC’s [37–38].

In the next sections we illustrate the essential characteristics of the dispersion surfaces (or EFS), analytically derived within DDT, and also applied to the negative refraction analysis [39]. Starting from dispersion surfaces, we will show that it is possible to understand the Pendellösung phenomenon and to quantify the thickness (Pendellösung distance) where the periodic exchange of energy happens. Finally a numerical study, based on a Finite Element Method (FEM) will prove the existence of this phenomenon for the PhCs and the validity of the parameters calculated by DDT.

2. Dispersion surface and Pendellösung phenomenon for photonic crystals

In vacuum, as in any homogeneous material, the dispersion relation in reciprocal space is simply a sphere of radius k=2π/λ, where λ is the wavelength in the medium (in vacuumk=ω/c). In photonic crystals, like for the real crystals in x-ray diffraction, a periodic spatial variation of dielectric properties has to be considered. To a first approximation, when the medium can be considered as homogeneous (i.e., in the long wavelength limit) and λ_av is the average wavelength in the medium, dispersion surfaces are spheres of radius k=2π/λ_av centred in each reciprocal lattice point (Fig.1(a)). Increasing the energy, λ_av decreases and the medium cannot be more considered as homogeneous because spheres approach each others. As a consequence, around the point L ₀ where the Bragg law is satisfied, the spheres partly split resulting in a more complex dispersion surface where a Bragg gap appears (Fig. 1(b)).

In the following we will limit ourselves to consider only the “two-beam case”, i.e., when only two Fourier components of the fields in the crystal are significant [28], which is usually the case in x-ray diffraction. This assumption is well satisfied for PhCs if the contrast, in the dielectric spatial distribution, is not very high. In such a case when the Bragg condition is satisfied for a considered reciprocal lattice vector, $\overrightarrow{h}=\overrightarrow{\mathit{HO}}$ (see the Fig. 1), the dispersion surfaces are only affected by $\overrightarrow{h}$ and the associated component in the Fourier expansion of the field. In the two-beam approximation, the influence on the Bragg diffraction from other reciprocal lattice vectors can be neglected.

Fixing the origin in L ₀ and indicating by $\overrightarrow{k}$ _B the wavevector which satisfies the Bragg law for the considered reciprocal lattice vector $\overrightarrow{h}$ (i. e. $\overrightarrow{k}$ _B·$\overrightarrow{h}$ =-h ²/2) the modulus of which is equal to the average wavenumber in the crystal, the dispersion surface can be conveniently described by the displacement $\overrightarrow{\delta }$ from the point L ₀.

 

Fig. 1. Dispersion surfaces in crystal in the long wavelength limit: the medium can be considered as homogeneous (a). Decreasing the wavelength, the spheres approach and a Bragg gap appears (b).

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By indicating with X ₀ and X_h the projection of $\overrightarrow{\delta }$ on the spheres centred in O and H, following the classical DDT [28,29] approach, one gets a simplified analytical expression of the dispersion surface:

This is the equation of a hyperbola. In fact, the product of the distances X ₀ and X_h from the asymptotes, which are the tangent planes of the spheres of radius k=2π/λ_av centered in O and H, respectively, is a constant (see Fig. 2 caption for details). In (1) χ ₀, χ_h and χ_-h are the Fourier components of the function χ defined by ε($\overrightarrow{r}$ )=ε ₀(1+χ($\overrightarrow{r}$ )), which is equivalent to the susceptibility definition.

 

Fig. 2. The crystal dispersion surface, which determines the permitted wavevectors in the structure for a given frequency is a hyperbola (thick line) close to the Bragg gap. In the figure are also shown dispersion surfaces in the air, which are spheres. The intersections of the hyperbola asymptotes in vacuum, the Lorentz point (Lo) and the Laue point (La) respectively, are also indicated.

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The dispersion surface is the locus of permitted wavevectors, for a given frequency ω=ck, inside the crystal. In the case of the incident wave on the crystal (in vacuum for instance), it is necessary to consider the dispersion surfaces in the two half-spaces and apply the continuity conditions across the boundary of the vacuum-crystal interface. In order to study the Pendellösung phenomenon, in the following we consider the transmission geometry (also known as Laue geometry in x-ray diffraction), where the surface normal vector $\overrightarrow{n}$ is orthogonal to the vector $\overrightarrow{h}$ . In Fig. 2 the wavevector $\overrightarrow{k}$ _i of the incident plane wave is shown. Its modulus is equal to the inverse of the wavelength in vacuum: |k_i|=2π/λ=ω/c and it lies in vacuum on the dispersion surfaces which are simply spheres of radius k, centered in O and H. The conservation of the tangential wavevector component implies that the vertices of the wavevectors inside the crystal, which have to be on the dispersion surface by definition, are aligned along the normal to the surface. This assures the continuity of the tangential component of the wavevectors. As illustrated in the Fig. 2 there are always two intersections with the dispersion surfaces (P ₁ and P ₂ in the Fig. 2). Therefore, inside the crystal there are two wavevectors oriented in the O-direction, $\overrightarrow{k_0^1}=\overrightarrow{P_{1}O}$ and $\overrightarrow{k_0^2}=\overrightarrow{P_{2}O}$ and two in the H-direction $\overrightarrow{k_h^1}=\overrightarrow{P_{1}H}$ and $\overrightarrow{k_h^2}=\overrightarrow{P_{2}H}$. All these components form the wavefield in the crystal [28,29]. There is no reason to choose only one intersection with the dispersion surface [23] because both are present in the PhC’s. The difference of such wavevectors, $\overrightarrow{P_1{P}_2}$, is oriented along the normal and a modulation is expected in this direction with a period Λ, where $\mid \overrightarrow{P_1{P}_2}\mid =1⁄\Lambda$. At the exit surface we have to apply again the conservation of the tangential wavevector component. If we limit ourselves to considering the case of a crystal slab, the exit normal coincides with that at the entrance surface. In such a case, the tip of the wavevector at the exit surface will point again on I_O for an in-vacuum dispersion surface centered in O, and on I_h for an in-vacuum dispersion surface centered on H.

Applying some straightforward DDT considerations, it is possible to show that the intensity I_O transmitted to the exit surface along the O direction and I_h along the H direction, is periodically modulated throughout the thickness, and I₀ has a maximum where I_h has a minimum and vice versa [28,29]. When the incident wavevector exactly fulfills the Bragg law (i.e., I_O ≡L_a) the modulation is given by the Pendellösung distance Λ₀=Λ(θ_i=θ_B), which is the inverse of the minimum distance in reciprocal space between the hyperbola branches (Fig. 2):

3. Numerical results

Hereafter we study the Pendellösung phenomenon in a 2D PhC square lattice with parameters indicated in the caption of Fig. 3. FEM Simulations are performed using the commercial package FEMLAB, considering a plane monochromatic incident wave (E along the cylinder axis) forming an angle θ_i=60° with the normal and having a Gaussian transverse profile. This wave satisfies the Bragg law for the reciprocal lattice vector indicated in Fig. 3, thus implying that λ=√3a. We have considered the case of a quite low-contrast refractive index because it allows visualizing the effect in a more pictorial way. In fact, in such a case Λ₀~12 a (a is the lattice parameter). From Eq. (2) it is clear that the period of oscillation, Λ₀, depends on the function χ which is directly linked to the index contrast. If contrast is high, Λ₀ is small and can also become a fraction of the period a, so that it is not possible to give a picture as in Fig. 3. In the considered example χ₀=0.0633; χ_h=χ_-h=0.0632. The low-contrast refractive index well justifies the two-beam approximation used in the derivation of Eq. (2). It should be underlined that in such a case the PhC has not a bandgap in frequencies.

On the base of the preceding analysis we anticipate a maximum of the electric field at the exit surface in the diffracted direction (negative refraction) when the thickness is a multiple of half of the Pendellösung distance Λ ₀.

 

Fig. 3. Electric field obtained via a FEM simulation in a 2D square lattice PhC with filling factor r/a=0.1. The cylinders have a real dielectric constant ε=3 embedded in vacuum. The polarization is chosen with the electric field (shown in the figure) parallel to the cylinder axis. The wavelength λ satisfies the Bragg law in vacuum with an incidence angle θ_i=60°, i.e. I₀≡La in Fig. 2. (a) Slab thickness t=6a=Λ₀/2: the maximum intensity is in the diffracted direction and exhibits negative refraction behavior. (b) t=12a=Λ₀ ; (c) t=18a=3/2 Λ₀ ; (d) t=24a=2Λ₀. The Pendellösung exchange of energy between positive and negative refracted beam corresponds to the thickness period Λ₀ as calculated from (2).

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On the other hand intensity will be at a maximum in the forward direction (positive refraction) for a multiple of the distance Λ₀. Figure 3 shows very clearly the excellent agreement of the numerical simulation with our theoretical analysis. Actually, the beam in the positive direction has a maximum intensity for thicknesses equal to 12a and 24a, i.e., Λ₀ and 2Λ₀, respectively, while the beam has a maximum intensity in the negative refraction direction for thicknesses equal to 6a and 18a, i.e., Λ₀/2 and 3/2Λ₀, respectively.

 

Fig. 4. FEM simulation of a PhC with the same characteristics as in Fig. 3 and a thickness t=6a. Square modulus of the electric field parallel to the cylinder axis for an incident angle θ_i=55° (a), and θ_i=65° (b). The detail (c) shows the modulus of the electric field inside the PhC.

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The Pendellösung period Λ can also be changed varying the incidence angle. In this way the point I ₀ in Fig. 2 moves, and consequently the difference $\overrightarrow{P_1{P}_2}$ of the wavevectors. Having fixed the thickness t the difference of optical path (|P ₁ P ₂|t) of the wavefield component inside the crystal becomes a function of the incidence angle and Λ≠Λ₀ (see Eq.(2)). Figure 4 shows the square modulus of the electric field in the case of a thickness t=6a, equivalent to (a) in Fig. 3. Now the incident angle is θ_i=55° (Fig. 4(a)) and θ_i=65° (Fig. 4(b)). This shows clearly how the incident angle modifies the modulation inside the crystal deviating the beam at the exit surface into positive and negative direction with respect to the case where θ_i=60° (s. Fig. 3). Some similarity with the skin effect associated to fast field oscillations [40] can be noticed.

The preceding result shows that Bragg diffraction can explain the negative refraction in the context of PhCs as also put in evidence by previous analysis [39, 41].

4. Conclusions

We have shown that inside a PhC the wavefield is, in general, given by a superposition of wavevectors which can coexist. The interference between such wavevectors gives rise to the Pendellösung effect, well known in dynamical theory applied to x-ray or neutron diffraction. Using such an effect it is possible to design a PhC with suitable refraction behaviour (negative or positive). Modifying structural parameters (for instance the index contrast) it should be possible to modulate the beam direction at the exit surface. As general consideration the previous analysis prove that, in general, negative refraction in a PhC slab cannot consider as independent from the thickness.

Finally we underline that DDT has been originally developed to understand the microscopic origin of the refractive index in homogeneous materials at optical wavelengths, explaining why the phase velocity acts as a regulating device between wavelets emitted by single resonators and the total field surrounding the resonators [36]. DDT could be helpful also for understanding negative index effects in meta-materials where inhomogeneities are much smaller than the relevant wavelengths [1–7].