tailieunhanh - Frontiers in Guided Wave Optics and Optoelectronics Part 14

Tham khảo tài liệu 'frontiers in guided wave optics and optoelectronics part 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Designer Laser Resonators based on Amplifying Photonic Crystals 505 3 where k is the Bloch wave vector and ii i the so-called Bloch function. We know that almost any periodic function can be written as a sum of harmonic functions. This defines the basic form for our unknown Bloch function 4 where c is the sum over all reciprocal lattice vectors and c - are the Fourier coefficients. After lengthy calculations we have simplified a set of partial differential calculations into a set of linear equations. Now the Fourier coefficients become accessible g c c x w - 2 5 G The Fourier transform of the inverse dielectric function c 1 can be calculated very efficiently with the Fast-Fourier Transformation giving us an easy-to-use tool for the design of PhCs. Limitations of the mathematical description As already mentioned due to all these simplifications the results are valid only for perfect systems. The PhC is infinitely large and consists of perfectly linear and loss-less materials. However these limitations are not very critical for our devices. Due to the excellent coupling between the mode and PhC even a small resonator behaves similar to the ideal PhC. The frequency dependent gain can be implemented by perturbation theory using a complex permittivity in the form e e is. For realistic THz-QCL it is not even necessary as the error due to the gain is negligible Nojima 1998b . Design of photonic crystals The PhC used for the resonator experiments has always the same basic structure a schematic is shown in Fig. 3 a . It consists of isolated free-standing pillars which are surrounded by air. Such structures typically show full bandgaps for TM-polarised light Johnson et al. 1999 which is the polarisation of light emitted by a QCL. The PhC is embedded in a DM waveguide for the real devices. For the simulations we use 2D simulations of infinitely high rods which significantly reduces the computational effort. As the waveguide allows only one vertical mode with an almost