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

The general solutions for can be derived as follows [Saleh and Teich, 1991]:

12.5.2Comparison of Coupled Mode Theory and BPM Computations

To show that the BPM is sufficiently accurate for simulation, the wave intensities in two different directional coupler structures were computed with the BPM and compared with the analytical results obtained from the coupled mode theory [Pojanasomboon, Ersoy, 2001]. According to Eqs. (12.5-4) and (12.5-5), the analytical results for the case of zero initial intensity in wave guide 2 ðA2 ¼ 0Þ can be expressed as

Figure 12.3. Analytical results for power exchange between synchronous wave guides.

12.5.2.1 Case 1: Synchronous Wave guides. The directional coupler in this case has the same refractive index values in both guides, namely, n1 ¼ n2. Because b1 and b2 depend on the refractive index in the guides, this case yields b1 ¼ b2 or b ¼ 0. The intensities in the wave guides are given by

The parameters used in the simulations were A1 ¼ 1, n1 ¼ n2 ¼ 1:1, d1 ¼ d2 ¼ s ¼ 1. The power exchange determined from the analytical expressions is illustrated in Figure 12.3. According to Eqs. (12.5-6) and (12.5-7), complete power exchange can be achieved at z ¼ p=2g.

In the BPM simulation, power at any z in each wave guide is calculated as

ð

jUðx; zÞj2dx:

guide

The results are shown in Figure 12.4. It is observed that the BPM simulation results with the same parameters yield the same general response as in Figure 12.3.

Figure 12.4. The BPM simulation of power exchange in directional coupler with synchronous wave guides.

Figure 12.5. Analytical results for power exchange between nonsynchronous wave guides.

13

Holography

13.1INTRODUCTION

Holography involves recording a modulated form of a desired (object) wave. It is also known as wave front reconstruction. The resulting device is called a hologram. Two major types of holography can be called analog and digital holography. Analog holography deals with continuous-space waves [Farhat, 1975], [Stroke, 1975]. Digital holography discussed in Chapters 15 and 16 results when the wave fields are sampled, and the information carried in amplitude and/or phase of the wave is coded with special algorithms. Digital holography is more commonly known as diffractive optics. Some other terminologies used for diffractive optics are computer-generated holography, diffractive optical elements (DOEs), and binary optics.

Holography was first discovered by Dennis Gabor in 1948, which is before the invention of the laser [Gabor]. Being a communications engineer, he recognized that the intensity resulting from the sum of a desired wave and a reference wave carries the information on both the amplitude and the phase of the object wave. After the invention of the laser as a coherent source, Gabor’s ideas became a practical reality.

This chapter consists of six sections. The basic mechanism of holography also called coherent wave front recording and the Leith–Upatnieks hologram, the first type of hologram successfully implemented with a laser setup, are discussed in Section 13.2. A number of different types of holograms are described in Section 13.3.

As holography is a well-defined mathematical process, it can be simulated in the computer, and the results of holographic reconstruction can be displayed graphically. How this can be done is described in Section 13.4. Holographic imaging depends on a number of parameters such as wavelength and size. If these change, so do the properties of the reconstructed images. Analysis of holographic imaging and magnification as a function of these parameters are discussed in Section 13.5. As in optical imaging systems, aberrations limit the quality of holographic images. Different types of aberrations in the case of holographic imaging are discussed in Section 13.6.

Diffraction, Fourier Optics and Imaging, by Okan K. Ersoy

Copyright # 2007 John Wiley & Sons, Inc.

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