Superfunctions for amplifiers

Abstract

An amplifier is characterized by its transfer function T, which expresses the dependence of the output signal on the input signal. This signal may be related to power, intensity, energy of a pulse, or its fluence, or any similar physical quantity. The internal structure of the amplified signal (e.g., its spectral content, polarization, temporal behavior, and spatial distribution) is not taken into account. The amplifier is considered to be spatially homogeneous and uniformly pumped. The transfer function is supposed to be known (measured in an experiment). The problem of reconstruction of the behavior of the signal inside the amplifier is formulated. For a given transfer function T, the evolution of the signal inside is interpreted as the superfunction F, satisfying the transfer equation F(z + 1) T(F(z)), where z is of coordinate along the propagation direction, while the length of the amplifier is used as a unit of measurement. (For simplicity, distances are measured in units of the length of the amplifier.) Two examples of realistic transfer function T are considered; they correspond to amplification of continuous wave and to amplification of pulses. In these examples, the transfer function and the distribution of the signal along the amplifier can be expressed in terms of special functions. The iterative procedure is suggested as a general method of reconstructing the signal along the amplifier, if neither the transfer function T, nor the superfunction F can be expressed with a simple combination of special functions. The examples show that the iterations converge to a physically meaningful solution. This method is expected to be useful for the characterization of laser materials from the measurement of the transfer function of a bulk sample.