OPTICS AND INFORMATION THEORY

Optics and information theory General observations A new era in optics commenced in the early 1950s following the impact of certain branches of electrical engineeringmost notably communication and information theory. This impetus was sustained by the development of the laser in the 1960s. The initial tie between optics and communication theory came because of the numerous analogies that exist between the two subjects and because of the similar mathematical techniques employed to formally describe the behaviour of electrical circuits and optical systems. A topic of considerable concern since the invention of the lens as an optical imaging device has always been the description of the optical system that forms the image; information about the object is relayed and presented as an image. Clearly, the optical system can be considered a communication channel and can be analyzed as such. There is a linear relationship (i.e., direct proportionality) between the intensity distribution in the image plane and that existing in the object, when the object is illuminated with incoherent light (e.g., sunlight or light from a large thermal source). Hence, the linear theory developed for the description of electronic systems can be applied to optical image-forming systems. For example, an electronic circuit can be characterized by its impulse responsethat is, its output for a brief impulse input of current or voltage. Analogously, an optical system can be characterized by an impulse response that for an incoherent imaging system is the intensity distribution in the image of a point source of light; the optical impulse is a spatial rather than a temporal impulseotherwise the concept is the same. Once the appropriate impulse response function is known, the output of that system for any object intensity distribution can be determined by a linear superposition of impulse responses suitably weighted by the value of the intensity at each point in the object. For a continuous object intensity distribution this sum becomes an integral. While this example has been given in terms of an optical imaging system, which is certainly the most common use of optical elements, the concept can be used independent of whether the receiving plane is an image plane or not. Hence, for example, an impulse response can be defined for an optical system that is deliberately defocussed or for systems used for the display of Fresnel or Fraunhofer diffraction patterns. (Fraunhofer diffraction occurs when the light source and diffraction patterns are effectively at infinite distances from the diffracting system, and Fresnel diffraction occurs when one or both of the distances are finite.) Temporal frequency response A fundamentally related but different method of describing the performance of an electronic circuit is by means of its temporal frequency response. A plot is made of the response for a series of input signals of a variety of frequencies. The response is measured as the ratio of the amplitude of the signal obtained out of the system to that put in. If there is no loss in the system, then the frequency response is unity (one) for that frequency; if a particular frequency fails to pass through the system, then the response is zero. Again, analogously the optical system may also be described by defining a spatial frequency response. The object, then, to be imaged by the optical system consists of a spatial distribution of intensity of a single spatial frequencyan object the intensity of which varies as (1 + a cos wx), in which x is the spatial coordinate, a is a constant called the contrast, and w is a variable that determines the physical spacing of the peaks in the intensity distribution. The image is recorded for a fixed value of a and w and the contrast in the image measured. The ratio of this contrast to a is the response for this particular spatial frequency defined by w. Now if w is varied and the measurement is repeated, a frequency response is then obtained.