ELECTRONICS LETTERS 21st June 2001 Vol. 37 No. 13 M odelling w aveguiding photonic bandgap structures by leaky mode propagation method A. Giorgio, A.G. Perri and M.N. Armenise A model of one-dimensional fully etched waveguiding photonic bandgap (FWPBG) structures, based on the leaky mode propagation (LMP) method, is proposed for the first time, to perform a complete analysis of the propagating characteristics including the radiation loss for a structure of finite extension. Introduction: In this Letter we propose a model of the fully etched waveguiding photonic bandgap (FWPBG) structures based on the leaky mode propagation (LMP) method [1], modified to perform a complete investigation of the propagating characteristics of a one-dimensional FWPBG structure with deep grooves and of finite length by taking in to account all losses due either to a stopband (Bragg reflection) or to power leakage. Model description: Referring to the structure shown in Fig. 1, we assume: (i) an arbitrary profile with period Λ and length L for the peri- odic perturbation; (ii) a finite length along the z propagation direction and infinite length along y . The model we have developed allows any shape, i.e. sinusoidal, triangular, rectangular and more generally trape- zoidal, to be chosen. Therefore, in the grating region the permittivity profile generally depends on x and z. According to the Floquet space harmonic expansion, in the cover and substrate the solution of the well-known scalar wave equation can be expressed by where F (i) (x, z) is the appropriate electromagnetic field component (i.e. E y for TE and H y for TM polarisation), k (i) is the wave vector in the generic ith layer (i.e. i = o, s and r, respectively, for cover, substrate and PBG region); n = 0, ±1, ±2, ±3, ... is the space harmonic order and j is the imaginary unit. The amplitude coefficients C n (+) and D n (–) and the propagation vector components k x n (i) and k zn are to be calculated as explained later. In the PBG region the permittivity ε(x, z) can be expressed by a Fou- rier series expansion multiplied by the step function to take into account the finite length L. We have proved in [2] that for structures of finite extension (L/Λ > 15) the Fourier series development of ε(x, z) is still valid without any significant error if the coefficients ε n (x) of the Fourier series expansion are weighted. The wave equation in the PBG region is with the following solution, according to the Floquet theorem: where F PBG is the transversal field component (i.e. E y /H y for TE/TM modes, respectively) and f n (x) is an appropriate function of the depth x. The nth k zn component of the wave vector is related to k z0 , corre- sponding to the n = 0 fundamental harmonic, by the Floquet phase rela- tionship: The optical loss due to the scattering effect induced by the PBG region is taken into account by defining a complex value for k zn , i.e. where β o is the real part (Re) of k z0 = β o + jα, (> 0) is the mode ampli- tude attenuation constant and β n = β o + jα. Therefore we can write The square-root sign in eqn. 5 must be selected to satisfy the following condition [1]: Im(k xn ) > 0 for β n > 0 or Im(k xn ) < 0 for β n < 0. The com- plex nature of k xn (i) means that guided waves in periodic structures behave as leaky waves since some power is spread out. By distinguishing the cases of TE or TM polarisation and imposing the continuity conditions of the tangential components of the electro- magnetic field at each boundary between different layers along the x direction, we have determined the boundary conditions in a matrix form: where f(0) denotes a vector with elements f n (0), T(0) and T(t g ). These are matrices with elements where ξ n-m is the (n-m)th coefficient of the Fourier series expansion of the inverse of permittivity, δ mn the Dirac delta function and ε (i) = n (i) 2 where n (i) is the refractive index of the ith layer. Moreover, L o is a matrix with elements and L g is a matrix with elements The index m comes from the boundary conditions applications; it also denotes the harmonic order and for each n, m also ranges from –∞ to +∞. Using Maxwell’s equations in the PBG region we obtain a differential equation system to be solved using the boundary conditions of eqns. 6 and 7 and assuming k z0 is an unknown parameter. The numerical integra- tion of the system has been carried out by a fifth-order Runge-Kutta- Fehlberg algorithm and the search for the complex eigenvalue k z0 has been performed using a polynomial form of Muller’s routine. Having found k z0 , the other propagation constants, field amplitudes and phases and then, the power flow and reflection and transmission coefficients can be calculated. For this purpose we have developed [2] a general model of transmit- tivity and reflectivity, accounting for any arbitrary number of field har- monics, where the field continuity conditions at interfaces (i.e. at z = 0 and z = L) enable ρ and τ to be determined, which are the field transmis- sion and reflection coefficients, respectively. Thus the modal reflection coefficient R P = |ρ| 2 and the modal transmission coefficient T P = |τ| 2 can be obtained. The cover (substrate) radiation efficiency is simply given by the power density radiated in the cover (substrate) divided by the total radi- ated power density. The computer program in FORTRAN 77 language has been imple- mented on a 500 MHz PC; it performs all calculations to completely characterise the structure in a few seconds (~5). An analysis in corre- spondence of 100 operating wavelengths is completed in a few minutes (4–5). Fig. 1 Schematic diagram of investigated structure