By Ginés Lifante Pedrola
The easy of the BPM process within the frequency area depends on treating the slowly various envelope of the monochromatic electromagnetic box below paraxial propagation, therefore permitting effective numerical computation by way of pace and allotted reminiscence. additionally, the BPM in keeping with finite alterations is a simple solution to enforce powerful and effective machine codes. This publication provides numerous ways for treating the sunshine: wide-angle, scalar procedure, semivectorial therapy, and entire vectorial remedy of the electromagnetic fields. additionally, distinct themes in BPM disguise the simulation of sunshine propagation in anisotropic media, non-linear fabrics, electro-optic fabrics, and media with gain/losses, and describe how BPM can take care of powerful index discontinuities or waveguide gratings, via introducing the bidirectional-BPM. BPM within the time area can also be defined, and the booklet contains the strong means of finite distinction time area process, which fills the distance while the traditional BPM isn't any longer applicable. as soon as the outline of those numerical suggestions were certain, the final bankruptcy contains examples of passive, energetic and practical built-in photonic units, resembling waveguide reflectors, demultiplexers, polarization converters, electro-optic modulators, lasers or frequency converters. The booklet can assist readers to appreciate numerous BPM ways, to construct their very own codes, or to correctly use the prevailing advertisement software program according to those numerical techniques.
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Extra resources for Beam Propagation Method for Design of Optical Waveguide Devices
Starting from Eq. 33b), and considering a structure in which the refractive index varies slowly along the direction of the wave propagation (∂n2/∂z ≈ 0), we obtain: ∇2 H t + 1 ∇t n2 × ∇t × Ht + n2 k02 Ht = 0 n2 1 84 And by separating this vectorial equation in two scalar equations, we finally get: ∇2 Hx − 1 ∂n2 ∂Hx ∂Hy − + n2 k02 Hx = 0; ∂x n2 ∂y ∂y 1 85a ∇2 Hy − 1 ∂n2 ∂Hy ∂Hx − + n2 k02 Hy = 0 ∂y n2 ∂x ∂x 1 85b Electromagnetic Theory of Light 19 It is worth mentioning that the Eqs. 84) are exact for z-invariant structures, for example, in straight optical waveguides.
32a): Electromagnetic Theory of Light 17 ∇2 Ε −∇ ∇ Ε + n2 k02 Ε = 0 1 71 where we have used the relationship E ∇ln ε = − ∇ E, derived from the first Maxwell equation. Next, we split the electric field in its transversal and longitudinal components in the following way: Ε r = Εt r + Ε z r uz 1 72 Εt r = Ε x r ux + Ε y r uy 1 73 being: Also, the differential operator ∇ is split in its transversal and longitudinal parts as: ∇ = ∇t + ∂ uz ∂z 1 74 the transversal operator being defined by: ∇t = ∂ ∂ ux + uy ∂x ∂y 1 75 In this way, Eq.
The first and second order derivatives that appear in the wave equation are represented by their finite difference approaches, where the field at each grid point is related to the fields at adjacent points, both in the transversal and longitudinal directions. On the other hand, in the FE-BPM the transverse plane is divided into hybrid edge/nodal elements and the transverse and longitudinal field components are expressed as a function of the edge and nodal variables for each element, shape function vectors for edge elements and shape function vector for nodal elements.