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Chapter 17:
Temporal Optical Processing Based on Talbot's Effects
For a long time, optical instruments were described in terms of spatial coordinates only while their temporal aspects were neglected. Typical examples are interferometers as well as systems for imaging and information processing. Those instruments are operating in a time-stationary fashion. In the time-stationary case, the instantaneous intensity I (P, t) at a point P, with I(P,t)=|u(P,t)| 2 and u (P, t) being the optical field, is proportional to the time-averaged intensity I(P)=const〈I(P,t)〉, as described in Ref. 1. Here, t is the time coordinate and the angular brackets denote infinite time averaging. Time stationarity implies that the spatial coordinate along which propagation takes place, commonly denoted by the z coordinate, often only occurs as part of phase terms in the mathematical analysis. Hence, it is usually sufficient to describe time stationary optical systems in terms of the two spatial coordinates x and y. However, the situation changes notably when we turn to nonstationary systems and the temporal dependence becomes an issue. In that case, Eq. (17.2) is no longer valid and z and t play an explicit role in the description of the optical system. Two recent developments have made it interesting to consider “optics in four dimensions”: laser light, with its narrow temporal frequency spectrum, has been available for a few decades and, more recently, practical laser systems, which generate short pulses with durations in the pico- and femtosecond range. This matches with the need for high-speed optical devices in communications and computing. Hence, it is worthwhile to consider in more detail the temporal aspects of optical instruments in order to investigate their capabilities for temporal processing. Time does play an important role in optics. A few scientists paid attention to the temporal aspects of light already, long before it became a central issue in the context of optical communications and nonlinear optics. Already, in 1914, von Laue wrote an article entitled “Die Freiheitsgrade von Strahlenbündeln” (The degrees of freedom of bundles of light rays). There he implicitly introduces what would now be called the “time-bandwidth product” of an optical signal. Even earlier, Talbot described a diffraction experiment in white light, where he discovered a band structure across the spectrum, called after him “Talbot's bands.” (Note: This phenomenon is not to be confused with the Talbot self-imaging effect that will also be discussed below.) In 1904, Schuster came up with an explanation of Talbot's bands based on the pulse theory of white light. As a result, the occurrence of Talbot's bands has been considered to prove the pulse theory of white light. The pulse theory says that light from a broadband source consists of a sequence of uncorrelated short pulses. The Talbot band experiment is worth reviewing for historical reasons and because it offers an interesting approach to femtosecond-pulse processing.
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