Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo
.pdf358 |
DENSE WAVELENGTH DIVISION MULTIPLEXING |
Figure 19.15. An example of 3-D design with four wavelengths and four-level phase quantization.
19.5.7Phase Quantization
In actual fabrication, phase is often quantized. The technology used decides the number of quantization levels. Figure 19.15 shows the demultiplexing results with four quantization levels, and otherwise the same parameters as in Figure 19.14. The results are satisfactory.
360 |
DENSE WAVELENGTH DIVISION MULTIPLEXING |
Virtual array
Real array |
Imaging optics |
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Figure 19.17. The visualization of a virtual holography setup in connection with Figure 19.16 to achieve desired phase modulation and size.
(d) Plane and spherical reference waves, negative phase implementation
Li ¼ aðxi þ rciÞ yi=k |
ð19:6-4Þ |
The equations above show that the method of physical generation of the negative phase appears to be more difficult to implement than the method of automatic zero-crossings in terms of waveguide length control. However, in the method of automatic zero-crossings, the positions of the apertures have to be carefully adjusted. Since the initial positions of apertures are randomly chosen, this is not expected to generate additional difficulties since the result is another random number after adjustment.
In 3-D, the disadvantage is that it may be more difficult to achieve large d. The big advantage is that there are technologies for diffractive optical element design with many apertures, which can also be used for phased array devices for DWDM. In our simulations, it was observed that d of the order of 5 is sufficient to achieve satisfactory resolution.
This can be achieved in a number of ways. One possible method is by using a setup as in Figure 19.16, together with the method of virtual holography discussed in Section 16.2. In order to achieve large d, the array can be manufactured, say, five times larger than normal, and arranged tilted as shown in Figure 19.16(b) so that Li shown in the figure is large. Then, the array (now called the real array) has the necessary phase modulation, and is imaged to the virtual array as shown in Figure 19.17, following the method of virtual holography. If M is the demagnification used in the lateral direction, the demagnification in the z-direction is M2. As a result, the tilt at the virtual array in the z-direction can be neglected. The virtual array has the necessary size and phase modulation in order to operate as desired to focus different wavelengths at different positions as discussed above.
362 |
NUMERICAL METHODS FOR RIGOROUS DIFFRACTION THEORY |
20.2BPM BASED ON FINITE DIFFERENCES
Consider the Helmholtz equation for inhomogeneous media given by Eq. (12.2-4) repeated below for convenience:
ðr2 þ k2ðx; y; zÞÞU0ðx; y; zÞ ¼ 0 |
ð20:2-1Þ |
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U0ðx; y; zÞ ¼ uðx; y; z; tÞe jwt |
ð20:2-2Þ |
and the position-dependent wave number kðx; y; zÞ is given by |
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kðx; y; zÞ ¼ nðx; y; zÞk0 |
ð20:2-3Þ |
in which k0 is the free space wave number, and nðx; y; zÞ is the inhomogeneous index of refraction.
As in Section 12.3, the variation of nðx; y; zÞ is written as
nðx; y; zÞ ¼ n þ nðx; y; zÞ |
ð20:2-4Þ |
where n is the average index of refraction. The Helmholtz equation (20.2-1) becomes
½r2 þ n2k02 þ 2n nk02&U ¼ 0 |
ð20:2-5Þ |
where the ð nÞ2k2 term has been neglected. Next the field is assumed to vary as
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in which Uðx; y; zÞ is assumed to be a slowly varying function of z, and k equals nk0. |
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Substituting Eq. (20.2-6) in the Helmholtz equation yields |
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Equation (20.2-7) is the Helmholtz equation used in various finite difference formulations of the BPM.
When the paraxial Helmholtz equation is valid as discussed in Section 12.3, Eq. (20.2-7) becomes
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This is the basic paraxial equation used in BPM in 3-D; the 2-D paraxial equation is obtained by omitting the y-dependent terms. The paraxial approximation in this form allows two advantages. First, since the rapid phase variation with respect to z is
BPM BASED ON FINITE DIFFERENCES |
363 |
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:
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366 |
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NUMERICAL METHODS FOR RIGOROUS DIFFRACTION THEORY |
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Table 20.1. Low-order Paˆde´ approximants for |
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the term k |
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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].
FINITE DIFFERENCES |
367 |
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]
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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
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uðxi þ xÞ þ uðxi xÞ ¼ 2u |
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where x3 lies in the interval ðxi x; xi þ xÞ. Rearranging the above expression yields
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Eq. (20.4-4) can be written as |
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The second partial derivative of u with respect to time is similarly given by |
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