By Igor A. Sukhoivanov
The nice curiosity in photonic crystals and their purposes some time past decade calls for a radical education of scholars and execs who can essentially practice the data of physics of photonic crystals including abilities of self sufficient calculation of easy features of photonic crystals and modelling of assorted photonic crystal components for software in all-optical verbal exchange structures. This booklet combines uncomplicated backgrounds in fiber and built-in optics with particular research of mathematical types for 1D, 2nd and 3D photonic crystals and microstructured fibers, in addition to with descriptions of actual algorithms and codes for useful cognizance of the models.
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Extra info for Photonic Crystals: Physics and Practical Modeling
5) Thus, the dispersion relation inside the uniform medium is represented by linear function. Let us consider a situation when a medium which is periodic in one dimension is taken instead of a uniform medium. The medium is assumed to be uniform within all other dimensions. 2). 6) In this expression, however, functions h(x) and ε(x) are not constants anymore, because of periodicity of the permittivity distribution. Therefore, solution of such kind of problems is much more complex than it was in the case of the uniform medium and, consequently, the dispersion relation will take a more complex form.
When an electric ﬁeld is present, it separates the positive and negative charges. This charge separation results in an additional electric ﬁeld. This additional electric ﬁeld is called the induced polarization. Various media respond diﬀerently to a given external electric ﬁeld. 21) P = ε0 · χE, where χ is called the electric susceptibility. It is the proportion coeﬃcient between polarization and electric ﬁeld. 4 Waves in Dielectric Media 21 Linear susceptibility is, in general, a second-rank tensor but reduces to a scalar for an isotropic medium such as silica glass.
3. Possible choice of primitive lattice vectors where l, m, n are integers and a1 , a2 , a3 are primitive lattice vectors. 7) is written in the most general form for 3D lattice. 2D and 1D PhC require two and one primitive lattice vectors, correspondingly. The choice of the lattice vectors as well as the choice of the unit cell is not standardized. The examples of diﬀerent sets of primitive lattice vectors for the PhC with square lattice is shown in Fig. 3. The only requirement is the possibility of the unit cell translation.