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The 2D Ambiguity Function (AF) and its relation to 1D Optical Transfer Function (OTF)

The Ambiguity Function (AF) is an useful tool for optical system analysis. This post is a basic introduction to AF, and how it can be useful for analyzing incoherent optical systems. We will see that the AF simultaneously contains all the OTFs associated with an rectangularly separable incoherent optical system with varying degree of defocus [2-4]. Thus by inspecting the AF of an optical system, one can easily predict the performance of the system in the presence of defocus. It has been used in the design of extended-depth-of-field cubic phase mask system.

NOTE:

This post was created using an IPython notebook. The most recent version of the IPython notebook can be found here.

To understand the basic theory, we shall consider a one-dimensional pupil function, which is defined as:

$(1) \hspace{40pt} P(x) = \begin{cases} 1 & \text{if } |x| \leq 1, \\ 0 & \text{if } |x| > 1, \end{cases}$

The *generalized pupil function* associated with $P(x)$ is the complex function $\mathcal{P}(x)$ given by the expression [1]:

$(2) \hspace{40pt} \mathcal{P}(x) = P(x)e^{jkW(x)}$

where $W(x)$ is the aberration function. Then, the amplitude PSF of an aberrated optical system is the Fraunhofer diffraction pattern (Fourier transform with the frequency variable $f_x$ equal to $x/\lambda z_i$ ) of the generalized pupil function, and the intensity PSF is the squared magnitude of the amplitude PSF [1]. Note that $z_i$ is the distance between the diffraction pattern/screen and the aperture/pupil.