Symmetry Properties

The eigenvalues of the master equation (1.5a) are interpreted in terms of Bloch vector and band index within the photonic band picture. Moreover, for in-plane propagation in two-dimensional photonic crystals, the polarization is used as a further classification of the eigenfrequencies. All of that is founded on certain symmetry properties of the system. The Bloch vector is a consequence of the periodicity of the dielectric function, the separation intoH-modes and E-modes stems from a mirror symmetry. In the last paragraph of Sec. 1.2.2, it has already been mentioned how symmetry can be exploited to work out the irreducible Brillouin zone. The aim of this section is to introduce the use of group theory in studying the photonic band structure, in analogy to what has been done for the symmetry analysis of electronic states in solids [Bassani, F., et al. (1975)]. The system under investigation is always the two-dimensional photonic crystal of Fig. 1.3; however, in this case, the hole radius is r = 0.2a, the lattice constant is given, a= 3µm, and the energies are expressed in electron-Volt (eV).

Since the Bloch theorem already accounts for the translation invariance, the attention will be focussed only on thepoint group of the system, which contains all the symmetry operations with a fixed point. The photonic bands with Bloch vector kcan be classified according to thesmall point group atk. The small point group is the subgroup that leaveskinvariant (apart from a reciprocal lattice vector). The symmetry of a state H_k(r) is determined by the transformation rule given in Eqs. (1.8), where Tˆ_R is replaced by the operator associated to a transformation belonging to the small point group. The electric field and the magnetic field components transform like a vector and a pseudo-vector respectively. The symmetry properties related to the spatial coordinaterare

0 ,0 0

Figure 1.7 Empty lattice bands for the photonic crystal shown in Fig. 1.3.

The average dielectric constant is²_k= 10.4, the lattice constant is a = 3µm and the energies are in electron-Volt (eV). The symmetry labels refer toE-modes.

found looking at the profile of H_k(r) and E_k(r). The notation is taken from Koster, G. F., et al. (1963).

The point group for a triangular lattice with cylindrical holes isD_6h, which is useful to view as the direct product of C_6v and C_s. C_6v has a six-fold rotation axis (z) and six mirror planes that form 60^◦ angles among each other. C_s contains the identity and the reflection σ_xy. The small point groups at the main symmetry points are D_2h at M and D_3h at K. The small point group along the symmetry lines is C_2v for the Γ−M, Γ−K and M −K directions. Notice that the twofold axis ofC_2v differs for the three cases. For off-plane propagation, the specular reflectionσ_xy is not a symmetry operation anymore and the point group remains C_6v only. The discussion will be restricted to in-plane propagation.

For studying the electronic states in solids, it is often useful to work with the empty-lattice scheme, which consists of the free-electron levels folded in the Brillouin zone for a given Bravais lattice. A similar methodology can be applied to photonic bands, choosing the free-photon dispersion given for an homogenous medium with the effective dielectric tensor of Eq. (1.28). Since the system is

0 ,0 0

Figure 1.8 Photonic bands of the 2D photonic crystal of Fig. 1.3: (a) E-modes, (b) H-modes. The lattice constant is a= 3µm, the hole radius is r = 0.2a and the material dielectric constant is

²= 12.

uniaxial, H-modes andE-modes have to be studied separately. Nevertheless, since the air fraction is small (f = 14.5%), H-modes can be treated with the same dispersion of E-modes with good approximation: ²_k ∼²_⊥.

Fig. 1.7 shows the empty-lattice photonic bands for the considered structure, whose average dielec-tric function is given by Eq. (1.26), that is ²_k = 10.4. The folding of the free-photon dispersion gives rise to degeneracies at the symmetry points and lines. The symmetry labels in Fig. 1.7 refer to E-modes. They are derived by finding the characters of the symmetry operations of the small point group acting on degenerate plane waves, and decomposing the resulting representation into irreducible representations. For clarity, the procedure for the states at the Γ point is reported in detail.

The first state (ω = 0) corresponds to the plane wave with G= (0,0), which transforms like the identical representation. Taking into account the symmetry of the transformation properties of the electric field componentE_z, the symmetry of this state is Γ⁻₂. The next state at Γ consists of the plane waves with G=b(0,±1), b(±^√₂³,±¹₂), where b= ^√4π_3a =|g_i|, and is six-fold degenerate. The characters of the symmetry operations, taking into account the spatial dependence of the plane waves as well as the transformation properties of the E_z field component, are

E C₂ 2C₃ 2C₆ 3C₂⁰ 3C₂⁰⁰ I σ_h 2S₆ 2S₃ 3σ_v 3σ_d

6 0 0 0 −2 0 0 −6 0 0 2 0

and can be decomposed into the irreducible representations Γ⁺₄ +Γ⁺₅ +Γ⁻₂ +Γ⁻₆. In a similar manner one classifies the other states.

Fig. 1.8 displays the calculated photonic band structure for²= 12 and r = 0.2a. Most of the degeneracies of the empty lattice have been removed because of the potential due to the dielectric function. Only the degeneracies imposed by the irreducible representations of the small point group remain. The photonic bands at Γ and atK are non-degenerate or twofold degenerate; the bands at the M point and along the lines are non-degenerate. In particular, notice that the twofold degen-erate states at Γ can have Γ⁺₅ or Γ⁻₆ symmetry forE-modes and Γ⁻₅ or Γ⁺₆ symmetry forH-modes.

This originates from the different transformation properties of the electric and magnetic field under the inversion operation. Along the lines, bands with the same symmetry anti-cross, while bands

with different symmetry cross each other.

In conclusion, the photonic band structure of a two-dimensional photonic crystal can be classified according the the irreducible representations of the small point group. For in-plane propagation, the main consequence of symmetry is the decoupling of the electromagnetic field intoH-modes and E-modes. The use of group theory is helpful in understanding the formation of the photonic band structure, with emphasis on the removal of degeneracy and on the crossing/anti-crossing of photonic bands. These results are also important for the interpretation of the optical properties of photonic crystals; for instance, the coupling of the Bloch waves with the external field.

As regards the photonic band gap, a two-dimensional photonic crystal made of a triangular lattice of air holes in a dielectric medium exhibits a band gap for H-modes in a wide range of the hole radius r. A complete band gap is found only for large filling ratios. Controlling light propagation by means of a full band gap in a two-dimensional photonic crystal is quite a problem, because of the absence of vertical confinement. This issue might preclude the use of two-dimensional photonic crystals for certain functionalities. Moreover, a practical realization of a two-dimensional photonic crystal in the near infrared is available only with macro-porous silicon [Lehmann, V., et al. (1990); Birner, A., et al. (1998)]. Also, the fabrication process of macro-porous silicon photonic crystals does not easily allow the integration with electronic circuitry or the inclusion of active layers, for instance, making its technological potential somewhat limited.

A plausible solution to all of these problems is the concept of finite-height two-dimensional photonic crystal [Meade, R. D., et al. (1994)]. The idea is to etch a two-dimensional photonic crystal in a planar dielectric waveguide so that the in-plane confinement is provided by the band gap, while the vertical confinement is given by the conventional total internal reflection. Such structures can be fabricated with top-down processes using lithographic techniques, which allow the integration of metal contacts, quantum wells and other nano-structures. However, these promising statements involve new issues that has to be understood, like the modification of the photonic band structure with respect to the ideal two-dimensional case or the occurrence of out-of-plane diffraction losses.

The next section is aimed to formalize the new concepts and study the band structure of two-dimensional photonic-crystal slabs.