| Fourier transform theory is one of the most widely used tools in physical sciences to retrieve vital information about various physical phenomena. In general, because a Fourier transform is a complex quantity, both its phase and magnitude are required for a unique representation of the function of interest. However, in many applications, only the magnitude of the Fourier transform can be measured, and a direct measurement of its phase is rather difficult. This constitutes a fundamental limitation in many fields, since the function of interest cannot be recovered uniquely from its Fourier transform magnitude measurement alone.; In this dissertation, a special family of functions, so called the minimum-phase functions, for which the Fourier transform phase can be recovered from the Fourier transform magnitude alone, is discussed. We apply this concept to several important fields of physics, and show in each case that with a simple magnitude spectrum measurement and data processing, we can recover the function of interest unambiguously. In all cases, this approach has led to significant breakthroughs, as well as simpler and faster measurements and processing. Specifically, in the field of poled silica, it enabled us to uniquely characterize the nonlinearity profile of the thin poled region, a challenging task that was not possible with previous techniques. In the area of fiber Bragg gratings, this concept led to the recovery of the group delay spectra using only magnitude spectrum measurements. This simple yet powerful approach is potentially an important alternative to expensive test equipment such as network analyzers currently used in the industry. Furthermore, using the same concepts, a powerful solution to another challenging task, characterization of sub-picosecond pulses using spectral interferometry based on minimum-phase functions is proposed. As we illustrate with experimental results, our proposed linear characterization tool is especially well-suited for femtosecond spectroscopy. The basics of this approach are also applied to improve other fields, such as frequency domain optical coherence tomography. |
