Page 64 - Phase Space Optics Fundamentals and Applications

CHAPTER2

Ambiguity Function

in Optical Imaging

Jean-Pierre Guigay

European Synchrotron Radiation Facility (ESRF) Grenoble, France

2.1 Introduction
The concept of the ambiguity function (AF) was introduced by
1
Woodward in the theory of signal processing of radar or sonar mea-
surements; it bears in its name the idea that it is impossible to perform
arbitrarily accurate measurements of both the distance and the veloc-
ity of a moving target. The well-known reason is that the width of
a signal in the time domain is inversely proportional to its width in
the frequency domain (a narrower signal has a wider spectrum and
inversely). From the mathematical point of view, this is just a property
of the Fourier transformation; from the physical point of view, this is
a common property in wave mechanics, for any kind of waves, and is
the basis of the uncertainty relations in quantum physics.
This concept can be introduced in optics as an extension of the
spectral analysis of images; the images are two-dimensional intensity
patterns I (x, y) which can often be analyzed in a convenient way by
considering their Fourier transform (intensity spectrum) written as

˜ I(u, v) = dx dy I (x, y) exp [−i2 (ux + vy)]
or

˜ I(f) = dx I (x) exp [−i2 x · f] (2.1)

where two-dimensional vectors are used; the variables (u, v) are the
spatial frequencies conjugate to the coordinates (x, y).

45