Reflection, Transmission and Diffraction

To better understand the optical properties of photonic crystals, consider for a moment the simplest case: a suspended dielectric membrane with a one-dimensional pattern, like the system discussed in Sec. 1.4.2: core dielectric constant ² = 12, waveguide thickness d = 0.5a and air filling ratio f = 30%. The incidence angles θ and φ are chosen to be 50^◦ and 0^◦, respectively, where φ is with respect to the axis aligned with the direction of periodicity. A TM-polarized

0.2 0.3 0.4 0.5 0.6 0.7 a/λ

0.0 0.2 0.4 0.6 0.8 1.0

Refl., Transm., Diffr.

Refl.

Transm.

Diffr. (R) Diffr. (T)

Figure 2.4 TM-polarized transmission, reflection and diffraction for the air bridge one-dimensional photonic crystal of Fig. 1.10. Parame-ters: ²= 12, d= 0.5a, and f = 30%. Incident wave: θ= 50^◦, φ = 0^◦ and TM-polarization. Diffr. (R) and Diffr. (T) mean diffraction above and below the membrane, respectively.

incident beam samples the photonic states that lie above the light line: reflection, transmission and diffraction are calculated for a/λ = ωa/2πc from 0.2 to 0.7. Because the plane of incidence is a mirror plane for the system, the whole process preserves the initial polarization, so that the TM

→ TE conversion is zero. For the same reason, the external wave couples only to photon states that have the same symmetry with respect to the plane of incidence. Then, the incident beam is reflected and transmitted. Moreover, if the frequency is high enough to be above the folded light line ω/c = |k+G|, diffraction occurs too. Notice that the polarization is conserved also in the diffraction process, because the diffracted beams lie in the plane of incidence. There are infinite diffraction cut-offsω_c, which are determined by imposing thatω_c/c=|k+G|. The cut-off frequencies can be expressed in terms of the incidence angles by the formulae (2.20): the first one is for φ= 0, whereas the second one is for the general case. n∈N represents the diffraction order and corresponds to G=n2π/a.

ω_ca

2πc = n

1 + sinθ, ω_ca

2πc = n

p1−sin²φsin²θ+ cosφsinθ. (2.20) Fig. 2.4 shows reflection, transmission and diffraction for the above mentioned initial conditions.

Since the pattern is one-dimensional, the curves have been calculated employing only 31 plane waves. Diffr.(R) represents diffraction in the top cladding, while Diffr.(T) is for diffraction in the bottom cladding. They exhibit the same cut-off (a/λ_c = 0.566), because both external media are air. Notice that T +R+D is equal to one below and above the diffraction cut-off, as it has to be. The anomalies in either reflection, transmission or diffraction correspond to the excitation of quasi-guided modes of the photonic-crystal slab; see for example the sharp resonance ata/λ'0.66, which matches a mode visible in Fig. 1.10. Notice also that diffraction is the dominant process above its cut-off.

0.2 0.3 0.4 0.5 0.6 0.7

a/λ 0.0

0.2 0.4 0.6 0.8 1.0

Refl. and Diffr.

Refl. TM Refl. TE Diffr. TM Diffr. TE

Figure 2.5 Reflection and diffraction for the air bridge one-dimensional photonic crystal of Fig. 1.10. Parameters: ²= 12,d= 0.5a, and f = 30%. Incident wave: θ= 50^◦,φ= 30^◦ and TM-polarized.

Fig. 2.5 shows the TM → TE conversion in reflection and diffraction for the same air bridge system, when the TM-polarized incident wave impinges with angles θ = 50^◦ and φ = 30^◦. Now, the plane of incidence does not represent a mirror plane any more. Furthermore, as the incidence

angles have changed, the diffraction cut-off is different from the previous case. Using Eq. 2.20, the cut-off is found to be a/λ_c = 0.63. The polarization conversion is rather relevant for certain frequency values. This is due to the strong anisotropy induced by the one-dimensional pattern; for a two-dimensional photonic crystal, the polarization conversion is overall weaker.

For two-dimensional photonic crystals, the study of diffraction is more complicated, of course, but it is conceptually identical to the one-dimensional case: the cut-offs are always given by ω_c/c=|k+G|, where now G spans two dimensions. Again, the polarization conversion is null if the plane of incidence corresponds to a mirror plane of the photonic crystal.

The SMM is now applied to two-dimensional photonic crystals and photonic-crystal slabs to study their optical properties and extracting information on the photonic band dispersion. More-over, for photonic-crystal slabs only, the method is also used to evaluate the propagation losses of the guided resonances.

2.3 Two-Dimensional Photonic Crystals

For in-plane propagation, two-dimensional photonic crystals have been shown to have Bloch modes with well defined polarization states, according to parity with respect to the plane of pe-riodicity, called H-modes (even) and E-modes (odd). Furthermore, each state can be specified by association to a certain irreducible representation of the corresponding small point group. In Sec. 1.3.2, the group theory analysis has been helpful in understanding the formation of photonic bands, with emphasis on the removal of degeneracy and on the photonic band gap. The present objective is to study how symmetry properties affect the determination of the band structure car-ried out with the VAR technique; in other words, how the external field couples to photonic-crystal modes. Also, the aim is to show that the method is suitable for obtaining the dispersion relation of two-dimensional photonic crystals, without need of embedding the system in a waveguide config-uration. This is a remarkable feature, because it goes beyond to the original argument, borrowed from grating theory, that the sharp resonances in the reflectance curve correspond to the excitation of guided resonances in a photonic-crystal slab [Astratov, V. N., et al. A (1999)]. More generally,

the anomalies in reflectance of two-dimensional photonic crystals are due to matching of the exter-nal field with a Bloch state, which behaves like a one-dimensioexter-nal critical point, whose density of state is given by photon dispersion in the vertical direction only, whereas the in-plane momentum is conserved.

For what concerns the theoretical implementation of a VAR study on two-dimensional photonic crystals, the calculation is easily accomplished by applying the SMM to a system composed by two semi-infinite layers, air and two-dimensional photonic crystal, with a common interface. The presence of the interface breaks the symmetry with respect to the plane of periodicity, so that the classification into H-modes E-modes is not rigorously valid anymore. Nevertheless, the bulk dis-persion relation is not modified at all by the surface and the SMM samples a truly two-dimensional photonic band structure.

As regards an experimental realization of the above analysis, if one wants to measure bands in the near-infrared frequency regime, macro-porous silicon is the unique system that provides high aspect ratios for obtaining a two-dimensional photonic crystal. Therefore, the possibility of a direct comparison with experimental curves and theory makes macro-porous silicon the preferable system for studying the optical properties of two-dimensional photonic crystals.