Fourier Optics - Optical Systems: General Overview and Analogy With Electrical Signal Processing Systems

Optical Systems: General Overview and Analogy With Electrical Signal Processing Systems

An optical system consists of an input plane, and output plane, and a set of components that transforms the image f formed at the input into a different image g formed at the output. The output image is related to the input image by convolving the input image with the optical impulse response, h (known as the point-spread function, for focused optical systems). The impulse response uniquely defines the input-output behavior of the optical system. By convention, the optic axis of the system is taken as the z-axis. As a result, the two images and the impulse response are all functions of the transverse coordinates, x and y.

The impulse response of an optical imaging system is the output plane field which is produced when an ideal mathematical point source of light is placed in the input plane (usually on-axis). In practice, it is not necessary to have an ideal point source in order to determine an exact impulse response. This is because any source bandwidth which lies outside the bandwidth of the system won't matter anyway (since it cannot even be captured by the optical system), so therefore it's not necessary in determining the impulse response. The source only needs to have at least as much (angular) bandwidth as the optical system.

Optical systems typically fall into one of two different categories. The first is the ordinary focused optical imaging system, wherein the input plane is called the object plane and the output plane is called the image plane. The field in the image plane is desired to be a high-quality reproduction of the field in the object plane. In this case, the impulse response of the optical system is desired to approximate a 2D delta function, at the same location (or a linearly scaled location) in the output plane corresponding to the location of the impulse in the input plane. The actual impulse response typically resembles an Airy function, whose radius is on the order of the wavelength of the light used. In this case, the impulse response is typically referred to as a point spread function, since the mathematical point of light in the object plane has been spread out into an Airy function in the image plane.

The second type is the optical image processing system, in which a significant feature in the input plane field is to be located and isolated. In this case, the impulse response of the system is desired to be a close replica (picture) of that feature which is being searched for in the input plane field, so that a convolution of the impulse response (an image of the desired feature) against the input plane field will produce a bright spot at the feature location in the output plane. It is this latter type of optical image processing system that is the subject of this section. Section 5.2 presents one hardware implementation of the optical image processing operations described in this section.