# 12

# The Heterodyne Transform and Signal Excision

## 12.1. INTRODUCTION

The generalized concept of the heterodyne transform provides a wealth of insights into how an important class of optical signal-processing systems work. In Chapter 9 we introduced heterodyne-detection concepts that led to a discussion of the heterodyne spectrum analyzer in Chapter 10 and to the use of decimated arrays in Chapter 11. In this chapter we consider generalized heterodyne systems which can be applied to a wide range of signal-processing problems. After we introduce the heterodyne-transform, we discuss applications such as probing three-dimensional light fields and signal excision.

## 12.2. THE HETERODYNE TRANSFORM

The *heterodyne transform* leads to an unusual method for recovering a signal; we simply integrate the light intensity produced by the sum of the signal and reference functions at the Fourier plane. Consider the interferometer, shown in Figure 12.1, in which lens *L*_{3} in the reference branch creates a point source *r*(*x*, *t*) at some point *x*_{0} at plane *P*_{2}. Lens *L*_{2} creates the Fourier transform *R*_{+}(*α*, *t*) of the reference beam at plane *P*_{3}. We express the reference function in terms of a time variable to allow for general filtering operations later. The acousto-optic cell in the signal beam has length *L* and is driven by *f*(*t*) = *s*(*t*)cos(2*πf _{c}t*). As usual, the signal corresponding to the positive diffracted order is expressed as

where the notation of Chapter 7 is used. The *jm* factor that usually appears in ...

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