Reflectance and Selection Rules

VAR from the sample surface is calculated by means of the SMM along the Γ−K and Γ−M crystal orientations, for both TE and TM polarizations, employing 151 plane waves. The material dielectric function is assumed to have a small dispersion, from 11.7 at 0.15eV up to 11.8 at 0.5eV, and no absorption. Fig. 2.7 shows the calculated reflectance for TE-polarized light incident along the Γ−Korientation; Fig. 2.8 compares the measured and calculated reflectance along Γ−K for TE and TM polarizations. Experimental data are courtesy of Galli, M., Universit`a degli Studi di Pavia, Italy. The reflectance curves of the macro-porous silicon sample display prominent features with a well defined dispersion as a function of incidence angle. There is a good overall agreement between the experimental and calculated spectra as regards to the number of structures in reflectance and their dispersion, although the experimental line-shape is more complex than the theoretical one.

The spectral strength of the structure depends on the angle θ. Most features become vanishingly weak at θ= 5^◦, where only one strong structure at 0.29eV is observed.

The results of Figs. 2.7 and 2.8 (and the analogous ones for reflectance along Γ−M, not shown here) are interpreted in the following way. When the frequencyω and the in-plane wave-vector kmatch those of a photonic mode propagating in the plane, a diffracted beam is created in the material and

Figure 2.7 Calculated reflectance for the macro-porous silicon sample of Fig. 2.6 with TE polarized light incident along the Γ−K ori-entation. The angle of incidence is varied from 5^◦ to 60^◦ with a step of 5^◦. Vertical bars mark the positions of 2D photonic modes for 5^◦ and 60^◦.

a corresponding structure appears in reflectance. This is very clear in the calculation, where the onset of a diffracted beam corresponds to a complex wave-vector component q that goes through zero and becomes real. In the work by Astratov, V. N., et al. A (1999), a similar approach was used for patterned GaAs-based waveguides. However, in the present case, there is no waveguide and a structure in reflectance marks the onset of a photonic mode, which is excited and remains propagating also for higher frequencies.

While most features in the experimental curves show a typical dispersive shape, the calculated curves exhibit a discontinuous derivative in correspondence of the onset of a photonic mode, such as for critical-point transitions [Bassani, F., et al. (1975); Cardona, M., et al. (1996)]. This

“universal” line-shape is broadened in the experiments, probably because of sample inhomogeneity.

Each critical point in reflectance is related to a singularity in the diffracted intensityD(ω), which may be calculated by interpreting the excitation of a photonic mode as an “absorption” process

0,1

0,20 0,25 0,30 0,35 0,40 0,45 0,50 0,05

0,20 0,25 0,30 0,35 0,40 0,45 0,50 TM Calculated (d)

θ= 5°

θ= 60°

Energy [eV]

0,30 0,35 0,40

Figure 2.8 (a),(c): experimental reflectance of the sample of Fig. 2.6 for light incident along the Γ−Korientation, for TE and TM polar-izations; courtesy of Galli, M., Universit`a degli Studi di Pavia, Italy. (b),(d): calculated reflectance. The angle of incidence is varied from 5^◦ to 60^◦ with a step of 5^◦. The curves at 5^◦, 10^◦ and 15^◦ are slightly off-set for clarity. Inset to (b): diffracted intensity corresponding to the allowed mode at θ = 5^◦ (onset marked by arrows).

0,0

Figure 2.9 Top panels: photonic bands of a triangular lattice of air holes in silicon witha= 2µm, r= 0.24a; (a)E-modes, (b)H-modes.

(the intensity of the diffracted beam is removed from specular reflectance and transmittance): thus D(ω) may be expressed as where |k|= (ω/c) sinθ. The parallel wave-vectorkis conserved and the out-of-plane dispersion of all bands (except close to the special point ω = 0) is quadratic inq, around q = 0 [Joannopoulos, J. D.,et al. (1995)], with a thresholdE(k,0)≡E₀. Thus Eq. (2.21) yieldsD(ω)∝(~ω−E₀)^−1/2, like for a one-dimensional density of states. The inset of Fig. 2.8b shows the calculated diffracted intensity of the allowed mode at near-normal incidence, which indeed has the form of an inverse square root close to the threshold E₀=0.29eV. A similar behavior is found for all diffracted rays, proving that each spectral feature in reflectance corresponds to a one-dimensional critical point.

The selection rules for specular reflectance are discussed starting from the symmetry proper-ties of the system. The photonic band structure of the photonic crystal under investigation, for in-plane propagation, is classified intoH-modes andE-modes, as displayed in Fig. 2.9. Recall that

the point group of the triangular lattice is D_6h, view as direct product of C_6v and C_s. The small point group is C_2v for the Γ−M, Γ−K and M−K directions, though the twofold axis of C_2v differs in the three cases. Recall also that the twofold degenerate levels at Γ can have Γ⁺₅ or Γ⁻₆ symmetry forE-modes and Γ⁻₅ or Γ⁺₆ forH-modes; Γ⁺₅ (Γ⁻₅) is the symmetry of thexy component of a pseudovector (vector), see Sec. 1.3.2. The surface of the crystal breaks mirror symmetry with respect to the x−y plane, so that the reflection σ_xy is not a symmetry operation anymore. The photonic modes should then be classified according to the subgroup C_6v of the point group at Γ and the corresponding subgroups at other k points. Along the Γ−M and Γ−K directions the small point group becomesC_s, i.e. specular reflection with respect to the plane of incidence is the only symmetry operation besides the identity.

The general selection rule can be stated as follows: a photonic band can appear in reflectance only if it has the same symmetry of the incident electromagnetic field. At normal incidence, the electric field (E_x, E_y) as well as the magnetic field (H_x, H_y) transform similar to the twofold degenerate representation Γ₅ of C_6v. The irreducible representations ofD_6h that reduce to this representation are Γ⁺₅ and Γ⁻₅, which implies that only states with symmetries Γ^±₅ can appear in reflectance. Such selection rule is obeyed in reflectivity curves of Figs. 2.7 and 2.8; in particular, the strong structure around 0.29eV at θ= 5^◦ corresponds to the allowed band with symmetry Γ⁻₅, see Fig. 2.9.

Concerning now selection rules along Γ−M and Γ−K, notice that these are the only orientations for which the plane of incidence is also a mirror plane of the structure: the photonic bands can be classified as even or odd with respect to this mirror symmetry. A TE wave is odd for specular re-flection with respect to the plane of incidence, while a TM wave is even. Therefore, a TE-polarized wave interacts with photonic bands that are odd for specular reflection in the vertical mirror plane, while a TM-polarized wave interacts only with even bands. Odd photonic bands correspond to Σ₃ and Σ₂ representations of C_2v for Γ−M (T₃ and T₂ for Γ−K), while even bands correspond to Σ₁ and Σ₄ for Γ−M (T₁ andT₄ for Γ−K).

Notice that an incident plane wave can interact with bothE−andH-modes of the photonic struc-ture. For this reason, it is appropriate to compare the photonic bands extracted from a reflectivity experiment not with those of E- and H-modes, but rather with those of the same parity with respect to a specular reflection in the plane of incidence. Such comparison is shown in Fig. 2.10.

Τ Κ

Figure 2.10 Measured dispersion of the photonic bands (points), derived from the structures in reflectance curves; the solid and dashed lines are the same photonic bands of Fig. 2.9, separated ac-cording to parity with respect to the plane of incidence: (a) TE polarization, odd modes, (b) TM polarization, even modes.

The open triangles in (b) represent diffraction in air and must be compared with the folded free-photons dispersion (dotted lines).

It can be seen that some non degenerate bands “stop” at the Γ point for a given polarization and

“restart” in the other polarization: this peculiar behavior is due to the fact that the mirror plane changes when turning from the Γ−M to the Γ−K direction. The experimental points agree very well with the calculated photonic bands of the proper parity. Anti-crossings are seen to occur between bands of the same symmetry, e.g. between two Σ₂states and between twoT₂ states around 0.3-0.36eV.

Not all bands that are allowed by symmetry appear in reflectance curves. This is not in contrast with the selection rule: an allowed band may have a nonzero, yet very weak, spectral strength.

Indeed, theoretical simulations with a very fine mesh indicate that weaker structures are present, which in the experiments fall below the signal-to-noise ratio. It is interesting to remark that most

measured photonic bands correspond to H-modes. This may be understood since, at normal inci-dence, the Γ⁻₅ mode at 0.29eV is much more intense than Γ⁺₅. At oblique incidence, the photonic bands that are forbidden atk= 0 gain spectral strength by mixing with allowed bands: since only one strong feature Γ⁻₅ is present below 0.5eV, most photonic bands that appear in reflectance in this energy range have H-mode character.

The experimental points, marked by open triangles in Fig. 2.10b, have a steep dispersion and do not match any photonic band of the silicon material. However, they match the dispersion of light in air, folded in the Brillouin zone: the corresponding structures in reflectance mark the onset of diffraction in air. These structures depend only on the Bravais lattice (not on the pore shape or depth) and would be present also for a shallow grating [Wood, R. W. (1902)]. In the present con-text, these “Wood anomalies” represent photonic bands in the upper half-space and are intermixed with photonic bands of the macro-porous silicon crystal.

In conclusion, the photonic bands of a two-dimensional photonic crystal can be determined by variable-angle reflectance: the spectral features, which yield the energy position of a photonic mode at q = 0, are interpreted as one-dimensional critical points. Only bands with the same symmetry of the incident electromagnetic field can appear in reflectance. The selection rules derived from symmetry show that the photonic modes behave similar to other elementary excitations in solids.

The same analysis is now applied to two-dimensional photonic-crystal slabs.

2.4 Two-Dimensional Photonic-Crystal Slabs

As already mentioned, the VAR technique has been first performed on weak index-contrast photonic crystal slabs [Astratov, V. N., et al. A (1999)]. Both theory and experiment can be conducted very much the same as shown for two-dimensional photonic crystals. The SMM has now to deal with two semi-infinite layers (air and substrate) separated by a certain number of finite-thickness layers, which form the desired planar waveguide. Then, each layer has to be patterned, completely or partially, to obtain the photonic crystal structure. It is intuitive that the core layer must be completely perforated in order to attain well defined resonances; a more detailed study on

that will be reported in Sec. 2.5. As discussed in Sec. 1.4, there is too much freedom in choosing the structure parameters, for thinking of a systematic and comprehensive study of all possible heterostructure geometries. That is why, the idea is again to group several systems into classes that exhibit similar features and restrict the study to representative cases only. Moreover, while in Sec. 1.4 the study has been focussed on photonic-crystal slabs characterized by infinite etch-depth, with the SMM it is possible to extend the study to more realistic systems, with finite etch-depth and also material dispersion and/or absorption.

In the present section, two cases will be shown: the air bridge system, corresponding to a high-index-contrast photonic-crystal slab, and a GaAs-based photonic crystal, which is for the weak-index contrast case. The SMM is used to extract the photonic band structure existing above the light line, as seen for macro-porous silicon photonic crystals. The aim is also to show that the band-structure picture is meaningful for interpreting the guided resonances, or quasi-guided modes, as stated in Sec. 1.4.2, and that the numerical method by Andreani, L. C. (2002) gives quantitative results also for modes above the light line. Furthermore, from the analysis of reflectance spectra, one finds again the selection rules described in the previous section for two-dimensional photonic crystals. Indeed, the symmetries involved in the coupling process to the external field depend only on the two-dimensional pattern, whereas the coupling strength may depend on the vertical geometry and on the hole etch-depth too.

As a final remark, while for a two-dimensional photonic crystal each mode retains a dispersion in the vertical direction, thus providing a behavior typical of a one-dimensional critical point in the reflectance structures, for a two-dimensional photonic-crystal waveguide, the modes have only in-plane dispersion, so that coupling to the external field appears as a resonant process, where ω and kof the incident wave match those of a guided-resonance. This results in a typical dispersion-like line-shape [Fan, S., et al. (2002)], in contrast to the “absorption-like” line-shape described in Sec. 2.3.2.