Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf

factored out, the slowly varying field can be represented numerically along the longitudinal grid, with spacing which can be many orders of magnitude larger than the wavelength. This effect makes the BPM much more efficient than purely finite difference based techniques, which would require grid spacing of the order of one tenth the wavelength. Secondly, eliminating the second order derivative term in z enables the problem to be treated as a first order initial value problem instead of a second order boundary value problem. A second order boundary value problem usually requires iteration or eigenvalue analysis whereas the first order initial value problem can be solved by simple integration in the z-direction. This effect similarly decreases the computational complexity to a large extent as compared to full numerical solution of the Helmholtz equation.

However, there are also prices paid for the reduction of computational complexity. The slow envelope approximation assumes the field propagation is primarily along the z-axis (i.e., the paraxial direction), and it also limits the size of refractive index variations. Removing the second derivative in the approximation also eliminates the backward traveling wave solutions. So reflection-based devices are not covered by this approach. However, these issues can be resolved by reformulating the approximations. Extensions such as wide-angle BPM and bidirectional BPM for this purpose will be discussed later.

The FFT-based numerical method for the solution of Eq. (12.3-5) was discussed in Section 12.4. Another approach called FD-BPM is based on the method of finite differences, especially using the Crank–Nicholson method [Yevick, 1989]. Sometimes the finite difference method gives more accurate results [Yevick, 1989], [Yevick, 1990]. It can also use larger longitudinal step size to ease the computational complexity without compromising accuracy [Scarmozzino].

In the finite-difference method, the field is represented by discrete points on a grid in transverse planes, perpendicular to the longitudinal or z-direction in equal intervals along the z-direction. Once the input field is known, say, at z ¼ 0, the field on the next transverse plane is calculated. In this way, wave propagation is calculated one step at a time along the z-direction through the domain of interest. The method will be illustrated in 2-D. Extension to 3-D is straightforward.

Let uni denote the field at a transverse grid point i and a longitudinal point indexed by n. Also assume the grid points and planes are equally spaced by x and z apart, respectively. In the Crank–Nicholson (C-N) method, Eq. (12.3-5) is represented at a fictitious midplane between the known plane n and the unknown plane n þ 1. This is shown in Figure 20.1.

The representation of the first order and second order derivatives on the midplane in the C-N method is as follows:

Figure 19.2 shows the layout of the waveguides with wide angles of bending to accommodate the length variations. Figure 20.2 shows the output intensities in a 200-channel AWG design computed with the wide-angle BPM using the RSoft software called BeamPROP.

Figure 20.2. A close-up view of the output channel intensities in a 200-channel phasar design using wide-angle BPM [Lu, et al., 2003].

20.4FINITE DIFFERENCES

In the next section, the finite difference time domain technique is discussed. This method is based on the finite difference approximations of the first and second derivatives. They are discussed below.

The Taylor series expansion of uðx;tnÞ at xi þ x about the point xi for a fixed time tn is given by [Kuhl, Ersoy]

The last term is an error term, with x1 being a point in the interval ðxi; xi þ xÞ. Expansion at the point xi x for fixed time tn is similarly given by

where x3 lies in the interval ðxi x; xi þ xÞ. Rearranging the above expression yields