Large-Scale Structures in Acoustics and Electromagnetics Proceedings of a Symposium (1996) / Chapter Skim
Currently Skimming:
8 SYNTHESIS AND ANALYSIS OF LARGE-SCALE INTEGRATED PHOTONIC DEVICES AND CICUITS
8 SYNTHESIS AND ANALYSIS OF LARGE-SCALE INTEGRATED PHOTONIC DEVICES AND CICUITS
Pages 162-182
The Chapter Skim interface presents what we've algorithmically identified as the most significant single chunk of text within every page in the chapter.
Select key terms on the right to highlight them within pages of the chapter.
From page 162... ...
The procedure discussed in this paper, as opposed to the direct method, starts with the required propagation characteristics of the guided-wave device and obtains the refractive index profile as the end result. We achieved this by transforming the wave equation for both the TE and TM modes in the planar waveguide to a Schrodinger-type equation and then applying the inverse scattering theory as formulated by Gelfand, Levitan, and Marchenko (Gelfand and Levitan, 1955; Marchenko, 1950~.
|
From page 163... ...
The planar waveguide we are considering here has a refractive index that varies continuously in the x direction. For the planar optical waveguide shown in Figure 8.1, our problem is to find the refractive index profile function in the core for a set of specified propagation constants.
|
From page 164... ...
CORE ,\ d ~ t(k) e'k~ Figure 8.1 The physical structure of an inhomogeneous symmetrical planar optical waveguide showing the reflection and transmission of electromagnetic wave.
|
From page 165... ...
INVERSE SCATTERING THEORY (8.9, The inverse scattering theory of Kay and Moses (Kay, 1955) provides us with a way to obtain the potential from the reflection coefficient that characterizes the propagation properties of the planar waveguide.
|
From page 166... ...
DESIGN EXAMPLE I: ZERO REFLECTION COEFFICIENT The reflection coefficient characterizes the propagation properties of the guided-wave optical devices. The zero reflection coefficient characterizes a system with propagating modes only, whereas a non-zero reflection coefficient characterizes a system with both guided and nonguided modes.
|
From page 167... ...
and [B] are column vectors with A,,, and Bn = An exp(Knx)
|
From page 168... ...
Whereas for TM modes, obtaining the refractive index profile is more complicated because it is a solution to a nonlinear differential equation [(8.8)
|
From page 169... ...
A-~(x) b, in which a and b are column vectors, and are given by at(x)
|
From page 170... ...
. Reconstructing refractive index profiles for nonrational reflection coefficients is not possible in analytical closed forms and so numerical techniques must be used.
|
From page 171... ...
represents the grid position along the x direction and the argument n represents the grid position along the t direction.
|
From page 172... ...
Good initial values for K(x~t) are important for the numerical iterative scheme, in particular when a bound state corresponding to the propagating mode exists.
|
From page 173... ...
. With this coordinate transformation, the partial differential equation (8.44)
|
From page 174... ...
At the origin, the Born expression provides an approximate reconstruction, even though there exists a discontinuity at the boundary. The numerical inverse scattering theory can now be summarized in the following steps: (a)
|
From page 175... ...
For a set of prescribed propagation constants, every arbitrary choice of normalization constants will produce a different potential and a corresponding refractive index profile. In order to construct a symmetric refractive index profile with single peak, we found that the normalization constants {A',|n=1,2...AT} must satisfy the following equation (Deift and Trubowitz, 1979)
|
From page 176... ...
i' -0.S 0 0.5 x (~m) Figure 8.4 The reconstructed refractive index profiles for a single prescribed TM mode with ]
|
From page 177... ...
/ 4 -3 -2 -1 0 1 2 3 4 x (Pm) Figure 8.5 Reconstructed refractive index profiles for five prescribed TM modes with correspondence to An (dashed curve)
|
From page 178... ...
The solid curve corresponds to a = 1.0, cat = 0.8, c2 = 0.499; the dashed curve corresponds to a = 1.0, cat = 0.05, c2 = 0.1. Figure 8.7 shows the refractive index profiles for TM mode in both the above discussed examples obtained by substituting the potentials into the nonlinear differential equation (~.31)
|
From page 179... ...
2.174 0 0.5 1 1.5 2 2.5 3 x (~m) 4 Figure 8.7 Reconstructed refractive index profiles corresponding to the potentials shown in Figure 8.6.
|
From page 180... ...
(8.67) This is a case of nonrational reflection coefficient.
|
From page 181... ...
This shows that the inverse technique outlined here can be used to synthesize waveguides with prescribed modes. Table 8.1 Prescribed TM Mode Spectra Used in Reconstructing Refractive Index of Planar Waveguide and Spectra Obtained by Analysis Using Finite Difference Scheme Number of Modes Mode Prescribed Number ~Mode Spectra Fy /ko 0 2.18997 0 2.20556 1 2.18417 .
|
From page 182... ...
Lin, 1993, "Synthesis and analysis of optical planar waveguides with prescribed TM modes," J
|
Key Terms
This material may be derived from roughly machine-read images, and so is provided only to facilitate research.
More
information on Chapter Skim is available.