1. Introduction

Propagation properties in Photonic Crystals (PhCs) [1–2], in particular those related to dispersion such as negative refraction [3] self-collimating [4] or superprism effect [5], are usually analyzed using dispersion surfaces, also known as Equi-Frequency Surfaces (EFS).

The electromagnetic field in a PhC can be decomposed using a Bloch waves expansion. EFS are the locus of wavevectors, in the reciprocal space and for a given frequency, associated to the propagating modes of the Bloch expansion. EFS represent the wavevectors of the self-consistent optical field in the unbounded structure and are obtained considering the PhC as an infinitely periodic structure.

The wavevectors excited inside the PhC are determined applying the continuity of the tangential component of the wavevectors across the boundary between the PhC and the external medium [3]. No propagating modes are excited inside the PhC when there are no points on the EFS that satisfy the continuity condition. In such case, the incident wave experiences a photonic gap (a partial gap when it happens in a limited range of incident angles) and the wave is totally externally reflected by Bragg diffraction.

From a general point of view the application of wavevectors tangential continuity ensures the linear momentum conservation and, following the Jackson [6] classification, it describes a kinematic property i.e. it determines the light propagation direction.

On the other hand the application of continuity of the tangential component of the electric and magnetic fields determines the dynamic properties of the electromagnetic field, i.e. it determines the intensities of the refracted and reflected-diffracted beams.

Actually it is not surprising that a dynamic relationship, such as Fresnel relations that are obtained for homogeneous media, cannot generally be applied to PhCs [7], even when their kinematic properties, such as refraction-like behaviour [3], can be applied to PhCs. For instance uncoupled modes that do not match the symmetry between the incident wave and the modes in PhC, can give zero transmission even without a band-gap [8]–[9]. This is a typical kinematic property that cannot be analysed simply using EFS even if other phenomena, such as changes of refraction angles obtained by rotating directors of liquid crystals, usefully can be studied with help of EFS [10]. Essentially, even if inside the first Brillouin zone the EFS has the shape of a homogenous medium (circle in 2D, sphere in 3D), the PhC does not generally behave as an effective refractive medium. In addition to the previously mentioned symmetry mismatch, also the propagation inside the crystal depends strongly on the interface termination that plays a fundamental role for surface localised states at PhC boundaries [11]–[14], that is useful to improve focusing with negative refraction PhC [14–16] or the beaming effects [18]–[19].

In the following sections we show that an incident wave inclined respect to the surface of a PhC can completely be back-reflected in a counter-propagating direction. This can be interpreted as an enhancement of a diffraction order, where the specular reflected wave is completely suppressed.

The surface termination plays a key-role in the distribution of energy of the incident wave between the specularly reflected wave and the back-diffracted wave in a counter-propagating direction. Simply changing the surface termination the wave can be completely back reflected or completely specularly reflected.

The observed effect is somehow related to the strong variation of diffraction efficiency experimentally observed in Ref. [20] using the Littrow geometry, which substantially is the same configuration of the present study. However in Ref. [20] the external wave can couple with internal Bloch waves because in such case authors works far from a bandgap. Then in Ref. [20] this behaviour is interpreted as an interference effect between the light diffracted by the surface corrugation and the light scattered from the Bloch modes of the bulk photonic crystal. The surface termination in Ref. [20] acts like a very special antireflection coating.

In our case we are involved with a partial bandgap and the interpretation of the phenomenon cannot be performed in analogy with [20]. As explained earlier, EFS can simply give information on the propagation but cannot distinguish between two different surface terminations. However also other approaches, even taking into account the whole Bloch expansion including real-propagating and complex-evanescent wavevectors [8] cannot simply explain the surface termination dependence of the reflected power.

We will show that an analysis of the energy flow allows an interpretation of the influence of the surface termination in terms of matching between symmetry of the energy flow inside and outside the PhC.

Finally we underline that the energy inside the PhC close to the interface, is distributed over vortexes reproducing the hexagonal symmetry of the PhC, which is itself a special relevant result that will be analysed in a forthcoming paper.

2. EFS design procedure: the bulk waves

We have analysed a 2D PhC with a background dielectric ε_Si =11.95 (silicon at telecom infrared wavelength) and air-holes arranged on a hexagonal lattice. We have limited our attention to the TM polarization (electric field out of the plane, directed along the holes axis).

The PhC is cut along the direction ΓK and the normal of the PhC with external homogenous silicon is along the ΓM direction (Fig. 1). In the next sections we have studied the influence of the cut position, in particular the cut in two relevant symmetric positions: in the middle and on the limit of the elementary cell, as defined in Fig. 1(a). The incident plane wave, with a Gaussian transverse profile, incomes from the silicon background with a direction of 30° respect to the normal of the PhC. The considered incident wavevector is highly symmetric respect to the hexagonal lattice symmetries, Fig. 1(b). A partial gap appears when there are no intersections between EFS and the line intersecting the end of the k_in vector normal to the PhC along the ΓM direction. In Fig. 2(a) we studied the dependence of this partial gap by the normalised hole radius r/a.

The incident wave satisfies the Bragg condition respect to the h⃗₁₁ = (1,1) lattice vector as defined in Fig. 1(b):|k_in| ≡ n_Si2π/λ = 2π/a so that normalized frequency is ω_n ≡ a/λ = 1/n_Si = 0.289. In fact the wavevector k⃗_nr satisfies the Bragg relation with the h⃗₁₁ = (1,1) vector of the reciprocal lattice : k⃗_i = k⃗_nr + h⃗₁₁, Fig. 1(b).

In correspondence of the normalised frequency ω_n=0.289, for TM polarization, a partial gap opens in the range 0.29<r/a<0.42. In particular for r/a<0.33 small regions of EFS appear in correspondence of the K points in the reciprocal space, as a consequence of the partial overlap between the first and the second band, the green circles in Fig. 2(b). As a general property the exponential decay of the evanescent wave inside a gap is at maximum in the middle of the gap and increases when the gap increases in the reciprocal space [21]. Because of the bands overlap, shown in Fig. 2(b), the partial gap is quite small in the reciprocal space in the range 0.29<r/a<0.33. In this range the exponential decay is small too and the evanescent wave penetrates significantly in the PhC for a length equivalent to many lattice constants. This enhances the interaction of the incident wave with the PhC.

 

Fig. 1. (a). Direct space elementary cell of the hexagonal lattice with position of two surface terminations used in this paper: on the limit of elementary cell, position 1, or in the middle of the air hole, position 2. (b) The reciprocal space with the first Brillouin zone (dotted), the incident and the specular reflected wavevectors across the circle (dashed blue) which represent the dispersion surface in silicon centered in the Γ point (0,0) of the reciprocal lattice. Red dotted circle represent the dispersion surface centered in points (1,1).

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3. Negative reflection

To maximize the penetration of the wave in the PhC as previously explained, we have fixed r/a=0.31. The incident wave incomes from the homogeneous silicon; the incident wavevector k_i is defined in Fig. 1(b) and Fig. 2(b). Two different PhC surface terminations will be considered. First a termination as in Fig. 1(a) was analyzed, i.e. with a truncation of the PhC at the end of an elementary cell without any modification of the holes along the surface line. We found out that - as expected - the wave penetrates in the PhC for many periods and, after an interaction-time, it is completely reflected in the silicon, according to the partial gap for the selected radius.

In Fig. 3(a) a movie versus time is shown, obtained using a Finite Difference Time Domain (FDTD) code, of a wave packet centred on the wavelength λ = a/0.289 and with a packet length of 12 λ. We have observed that more than 80 % of the incident energy is transferred to the diffracted wave that propagates in the same direction as the incident wave but they have opposite orientations k⃗_nr: in the following, we refer to this wave as a negatively reflected beam. We have also observed that a very small part of the energy is transferred to the specularly reflected beam, which actually coincides with the family of all the diffracted waves centred in reciprocal lattice nodes (0, n), where n=0,-1, 1,-2, 2.

However if the surface is terminated as in position 2 of Fig. 1(a), i.e. cutting in the middle the holes of the surface line, the wave is totally reflected in a specular direction, Fig. 3(b).

4. Analysis of the surface termination influence on the negative reflection

Using analysis of section 2 we found modes that can propagate in PhC and external homogeneous silicon satisfying the tangential wavevector continuity (i.e. that satisfy the photon linear momentum continuity). EFS are a bulk property of media and such analysis couples bulk properties inside and outside the PhC. However, EFS analysis cannot determine the distribution of energy across each mode. This requires the additional application of the tangential continuity of the electric and magnetic fields, as for instance in Ref. [8] or [21].

 

Fig. 2. Partial gaps along MM’ as a function of the hole radius. The eyes line is in correspondence of the frequency ω_n=0.289 (a). The correspondent EFS for r/a=0.31. The blue circle are EFS corresponding to the first band (0 corresponds to the lowest valence band), whereas green EFS correspond to the second band are located around the K points (b).

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Fig. 3. (2.52 Mb) Movie versus time of FDTD simulation for TM polarization of an incident wave-packet modulated by Gaussian transversal profile (σ=15a) and a Gaussian longitudinal length (σ=12/λ) for a surface termination, with reference to Fig. 1, as in position 1, i.e. no cut in holes (a) and in position 2, i.e. holes cut exactly in the middle (b). [Media 1]

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On the other hand, an energy flow picture gives a lot of interesting information useful for the interpretation of the surface termination dependence of reflected beams. Figure 4 shows the time-average Poynting vector calculated for both the studied surface terminations close to the interface. In Fig. 4 the incident wave has the same characteristics of Fig. 3 except for a continuous wave excitation, instead of the finite length pulse considered in Fig. 3.

Inside the PhC, which is in the upper part of Fig. 4(a) and Fig. 4(b), the energy flow is distributed along the vortex, arranged following the hexagonal symmetry. In the middle of the holes the flow is parallel to the interface with external medium.

 

Fig. 4. Time average Poynting vector close to the interface between the PhC (air-holes are colored in magenta), in the upper part of figure, and the external homogenous silicon (red color). When the PhC is terminated without any holes cut the vortexes generated inside are preserved also outside the PhC (a). When the PhC is terminated in the middle of holes, where the energy flux is parallel to the interface this is preserved in the external silicon, enhancing the specular reflected beam (b).

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In Fig. 4 we can see that symmetry of the energy flow is preserved across the interface. When the PhC surface terminates in the middle of holes, the energy flow is parallel to the interface in the PhC, Fig. 4(b), and this flow is conserved in the external homogenous silicon. This enhances the specular reflected beam: in fact in the external medium the sum of the incident with the specular reflected beam originate a net flow parallel to the surface, whereas the components along the normal to the interface are opposite each other and their interference establishes a standing wave in the external medium along the normal direction.

Conversely when surface termination does not cut the holes, the vortex arrangement of the energy flow in the PhC close to the interface is preserved in the external medium. This arrangement of the time average Poynting vector gives a zero flow across the lines perpendicular to the incident wavevector. Such topology property clearly help an enhancement of the negative reflected beam, where incident and reflected beam travel in an opposite direction, Fig. 4(a). Considering the plane waves decomposition of the wavefield inside and outside the PhC the vortex position depends nonlinearly on the amplitude and on the direction of the plane wave components, and will be deeply analyzed in a forthcoming paper. We underlined here that a vortex close to the surface inside and outside the PhC reveals the complexity of the wavefield in the interface region that cannot simply be resumed to the interference of two plane waves travelling in opposite directions [22].

The energy flow topology suggests the way to further enhance the negatively reflected beam. Looking carefully at Fig. 4(a) we see that the vortexes inside the PhC and outside are not exactly aligned. The misalignment is slight and displacing the holes line along the interface by a quantity equal to 0.2 a we obtain a totally reflected beam, as shown in Fig. 5. The displacement of last holes lines is a blazing effect, that concentrates all the energy across a diffraction order [23]. Electromagnetic waves crossing the photonic crystal interact strongly with the boundary, but it’s still a difficult problem to establish a rigorous link between finite and infinite PhCs [24]–[25].

 

Fig. 5. (3.07 Mb) Movie versus time of FDTD simulation for TM polarization of an incident plane wave modulated by Gaussian transversal profile (σ=15a) for a surface termination as in position 1 of Fig. 1, i.e. no cut in holes. [Media 2]

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5. Conclusions

We have shown that surface termination strongly influences the reflection from a PhC. In particular a counter-propagating reflected beam can be obtained also for an oblique incident angle. A such negatively reflected beam is very sensitive to the PhC termination. We have proved that a study of the energy flow close to the interface is extremely useful to analyze this surface termination influence. In the case of a negatively reflected beam the energy flow is distributed across vortexes arranged with the same hexagonal symmetry of the PhC. When the surface termination breaks the symmetry of the energy flow, the negatively reflected beam is suppressed. Conversely, if the termination is slightly modified in accordance with the symmetry of the vortex flow, the negatively reflected beam is further enhanced.