II. NOTAS DE CARÁCTER ESPECÍFICO :
18. ACTIVOS FINANCIEROS
The analytical theory and associated results presented in the previous subsection have been verified by numerical modelling using the finite difference-time-domain-technique (FDTD). Figures 2.5(a) and 2.5(b) show the distribution of the square of electric field for waves scattered at the interface of the high contrast photonic crystal (n1 = 2.1, n2 = 1.1).
Figure 2.5(a) shows negative refraction with the frequency of the incident wave lying in the second photonic band of the photonic crystal. Figure 2.5(b) corresponds to the case when sin(K0D) = 0, resulting in the tangential component of the Poynting vector
being equal to zero and the propagation of the wave being normal to the interface.
b
(a)
(b)
(c)
(d)
Fig. 2.5: FDTD simulation results for waves scattered at the interface of high (n1=
2.1, n2 = 1.1) (a,b) and low (n1 = 1.4, n2 = 1.8) (c,d) contrast photonic crystal. (a)
- Spatial distribution of the square of electric field for the case of negative refraction (~ω = 1.6eV, angle of incidence ϕ = 55 degrees);(b) Spatial distribution of the square of electric field for the case of normal propagation (~ω = 2.0 eV, ϕ = 55 degrees); (c) Spatial distribution of the absolute value of the square of electric field for the case of spatial oscillations of the Poynting vector (~ω = 2.0 eV, ϕ = 55 degrees); (d) Spatial
Figure 2.5(c) shows the spatial distribution of the square of the electric field in a low contrast photonic crystal(n1 = 1.8, n2 = 1.4), when the frequency is again chosen
to make sin(K0D) = 0. In this case, in the analytical expression for the tangential
component of the Poynting vector, only the oscillating term in Eq. (2.27) is non-zero. As predicted by the analytical theory, the numerical results show spatial oscillations of the electromagnetic field in the direction normal to the interface. Figure 2.5(d) shows the distribution of the real part of electric field for the same case as in Fig. 2.5(c). For the low contrast photonic crystal, in the case when sin(K0D) ̸= 0, which is not shown
here, we observe a splitting of the incident beam into positively and negatively refracted beams as also predicted by the analytical theory.
2.2.3 Results and conclusions
It should be noted that the analytical model presented assumes an incident plane-wave with an infinite wave front. In an experiment, however, we would be using a light beam of finite size. The numerical calculations show that there exists a critical source size above which the refraction of wave at the interface can be described by the analytical theory. We have derived an estimate of the critical source diameter which is given by
dthr≈ D/δn, where δn is the relative contrast of the photonic crystal. That formula is
similar to the expression obtained in [37], which defines the minimum size of the Bragg reflector for the formation of a photonic stop-band suggesting that the origin of the critical source size is the existence of a minimum photonic crystal size for the formation of the photonic band structure. When the source size is significantly less than the critical value, the refraction of the wave at the side edge of the photonic crystal is determined by the effective refractive index nef f = (n1d1+ n2d2)/(n1+ n2).
The propagation of the refracted beam normal to the interface can be considered to be the result of an excitation of an array of coupled waveguide modes, where the high refractive index layers play the role of the waveguide core and coupling is achieved via evanescent waves inside the lower refractive index layers that play the role of a waveguide cladding. However, the case of normal propagation is not entirely equivalent to the excitation of waveguide modes since in general it can occur even in the case when the light propagates in both the high and low refractive index layers.
The spatial oscillations of the Poynting vector resemble the Bloch oscillations which have been predicted and observed both in superlattices, for electrons in an external electric field, and for the electromagnetic field in wedge-shaped photonic crystals[38]. In the latter case the gradient of the photonic crystal slab leads to a gradient of the quantized lateral photon momentum which can be considered as an effective potential
or force acting on the photon propagating in the structure. Bloch oscillations occurs when an electron (or photon) propagates in a periodic potential with an external force applied. The wavevector of the particle is changed by an external force, and when it reaches the edge of the Brillouin zone, it changes its wavevector by ±2π/D, where D is the period of the structure. The main difference of the oscillations in our case is the that the oscillations take place in the absence of an external field. However, the wavevector lies exactly at the edge of the Brilloin zone since the condition sin(K0D) = 0
applies. Also, the oscillations are only observed in the case of a quarter wavelength Bragg reflector, for which the second band gap of the photonic crystal becomes degenerate. The observed effect of the Poynting vector oscillations can thus be regarded as a result of the interference of two degenerate Bloch modes existing at a photonic crystal band edge. To conclude, in this section we have developed a theoretical model, which facilitates the description of refraction of electromagnetic wave at the side edge of a one-dimensional photonic crystal. Using the model we have predicted and verified two new effects: normal propagation of the electromagnetic wave and spatial oscillations of the Poynting vector inside the photonic crystal.