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**Course Description:**

Although there are many advances in the spectroscopy of atoms, molecules, impurity-doped insulators, semiconductors, plasmas, etc., very few people appreciate that laser pulses themselves are describable through the discipline of spectroscopy. In the spectroscopy of laser pulses, special techniques have been developed over the years to articulate features of pulses as they propagate through various materials. This day-long course assumes a familiarity with complex notation for light wave propagation problems. Throughout the course, much attention is given to physical insight and to simple interpretive pictures for the phenomena which are being addressed. This course should be useful for those who are dealing with pulse evolution in optical fibers, solitons, extralinear chirping, pulse squaring, cross phase modulation, and additionally to those who are studying or predicting the evolution of pulses in complicated oscillator structures. The course will be a benefit to anybody who has ever wondered why one can consider the frequency of a signal as a function of time, even though the true Fourier component in frequency can only be evaluated by integrating over all time. These "mixed" domain interpretations of signal behavior are shown to be extremely valuable, and will be found useful to a large number of researchers, engineers, etc.

The course starts by describing Fourier transforms, plane waves, and the slowly varying envelope approximation. Although Maxwell's equations are not presented, the philosophical approach to their implementation in this problem is clearly discussed, and the concept of transfer functions is pointed out. The next section deals with implementation of the numerical technique known as the fast-Fourier transform, showing the approaches to its utilization in plane-wave pulse propagation through linear optical media. Some of the pitfalls of the Fourier transform technique are indicated, and as well as recipes for minimizing difficulties.

The next topic indicates the applicability of the fast-Fourier transform technique to linear pulse propagation. Next, a section on the spectroscopy of temporal Gaussian pulses is provided, showing the "completion of the square" trick, and applying such to both chirped and unchirped pulses. The next section deals with the optical process of self-phase modulation and chirping, effects which occur in materials whose index of refraction depends upon intensity. The concept of instantaneous frequency is introduced, and is compared to spectral decomposition of signals. It is at this point that a lot of insight can be gained by thinking in both the "time domain" and "frequency domain". Next, as an example of transforming one's thinking between the time-domain and the frequency-domain, calculational techniques for mutually incompatible properties are discussed in terms of the split-step algorithm.

Next, a chapter is provided on propagation phenomena in absorbers. Topics are especially chosen to show the benefit of transforming one's thinking back and forth from the time-domain to the frequency-domain. The complex index of refraction of an absorbing medium is described, and as well the appropriate propagation constant. Examples of time-domain interpretations of free-induction decay formation for ultrashort pulses is provided, and as well, a similar analog is examined in the case of free-induction decay from a Fabry-Perot interferometer. The pulse area theorem is described, and special properties of "zero pi" pulses are described, and are interrelated to free induction decay signals and to oscillatory signals which evolve as a pulse penetrates an absorbing media. Finally, a brief section on "anti-linear" phenomena is provided, in which it is shown that with a slight amount of care for additional bookkeeping, linear optical techniques, such as the fast-Fourier transform, can be applied to such patently nonlinear problems as the transient behavior of cw-pumped Kerr-like phase conjugators.

Each idea is explained in clear and simple terms with an emphasis on underlying physical principles. Fourier analysis will be taught during the class and many examples will be provided. Pertinent references and review material will be identified. The course notes contain copies of each vuegraph used during the day so that participants can spend a maximum amount of time listening and understanding.

**Benefits**
**This course will enable you to:**

- Understand how Fourier analysis is applied to pulse waveforms.
- Use the Fast Fourier Transform (FFT) while being aware of its pitfalls and limitations.
- Evaluate the spectrum for transform-limited pulses, for chirped pulses, and for general waveforms.
- Appreciate how "time-domain" pictures can provide insight when frequency-domain pictures fail.
- Apply time-domain pictures to the interpretation of self-phase-modulation spectra.
- Apply time-domain thinking to predicting and understanding transients in absorbers, amplifiers, and interferometers.
- Apply time-domain pictures to problems in chirp reversal and in phase-conjugate reflection

Technical persons who need to study pulses as their color content evolves, who need to evaluate chirped waveforms, who need to dispersively manipulate the shape of frequency-modulated pulses. These concepts will also be of value to those in the chirped radar discipline.

Continuing Education Units (CEU's) available upon request.

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