Frequency Spectra of Isolated Laser Pulse Envelopes

This paper will deal with isolated laser pulses, those which last for a finite time interval and whose envelope function is strictly zero outside of this interval. We numerically calculate the Fourier transform of this function and study its asymptotic behavior at high frequencies. This work is motivated by recent results on the probability distributions of quadratic operators in second quantized systems. An example is the density of a material which is subject to zero point fluctuations in the phonon vacuum state. These distributions can decrease very slowly, leading to a relatively high probability for large fluctuations. If the operator is measured by a laser pulse, the rate of decrease of the distribution mirrors the rate of decrease of the pulse envelope Fourier transform. We describe a model for the creation of isolated pulses in which this Fourier transform falls as an exponential of a fractional power of frequency and find examples where this fraction is in the range 0.1 to 0.2. The probability distribution for large fluctuation has the same functional form and implies a significant probability for fluctuations very large compared to the variance.