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

Figure 19.10. The case of large number of channels (M ¼ 200; L ¼ 128; l ¼ 0:2 nm).

phased arrayed apertures, the number of channels, and the wavelength separation are represented by M; L, and l, respectively. Figure 19.10 shows the results for M ¼ 200; L ¼ 128, and l ¼ 0:2 nm. The figure consists of two parts. The top figure shows the demultiplexing properties under simultaneous multichannel operation. In this figure, we observe that the nonuniformity among all the channels are in the range of 2 dB. It is also usual in the literature on WDM devices to characterize the cross talk performance by specifying the single channel cross talk figure under the worst case. The bottom figure is the normalized transmission spectrum with respect to the applied wavelengths in the central output port. The

Figure 19.13. Sixteen-channel design with phase errors (ERR ¼ 0:25p).

19.5.5Error Tolerances

Phase errors are expected to be produced during fabrication. The phase error tolerance was investigated by applying random phase error to each array aperture. The random phase errors were approximated by uniform distribution in the range of [-ERR, ERR] where ERR is the specified maximum error. As long as the maximum error satisfies

the phasors point in similar direction so that there is positive contribution from each aperture. Hence, satisfactory results are expected. This was confirmed by simulation experiments. An example is shown in Figure 19.13, corresponding to ERR ¼ 0:25p.

19.5.63-D Simulations

The 3-D method was investigated through simulations in a similar fashion [Hu and Ersoy, 2002]. Figure 19.14 shows one example of focusing and demultiplexing on the image plane (x-y plane at z ¼ z0). The four wavelengths used were 1549.2, 1549.6, 1550, and 1550.4 nm, spaced by 0.4 nm (50 GHz). The array was generated with 50 50 apertures on a 2 2 mm square plane. The diffraction order dx in the x- direction was set to 5, while that in the y-direction, dy, was set to zero.

In Figure 19.14, part (a) shows demultiplexing on the image plane, and part (b) shows the corresponding insertion loss on the output line (x-direction) on the same plane.

It is observed that a reasonably small value of diffraction order (dx 5) is sufficient to generate satisfactory results. This is significant since it indicates that manufacturing in 3-D can indeed be achievable with current technology. A major advantage in 3-D is that the number of apertures can be much larger as compared to 2-D.

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.

Virtual array

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

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.

20.2BPM BASED ON FINITE DIFFERENCES

Consider the Helmholtz equation for inhomogeneous media given by Eq. (12.2-4) repeated below for convenience:

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

where n is the average index of refraction. The Helmholtz equation (20.2-1) becomes

where the ð nÞ2k2 term has been neglected. Next the field is assumed to vary as

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

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