Abstract

Many important diagnostic techniques yield the magnitude, but not the phase, of the Fourier transform of a response, and hence, may not determine the response uniquely. Even when a specific model function is known to approximate the response well, determining the parameters of the model uniquely may not be possible. Consider, for example, the commonly used temporal response, the sum of two exponentials: h_α(t) = θ(t){A exp(−t/τ_f) + B exp(−t/τ_s)}, where θ(t) is the unit step function, τ_f and τ_s are fast and slow time scales, A ≡ α/τ_f, B = (1 − α)/τ_s, and α is the relative weight. Knowledge of only the magnitude of the Fourier transform of h_α(t) allows two distinct values of α to result from a curve fit (even in the absence of noise). We have seen this ambiguity in frequency domain experiments measuring the ultrafast optical Kerr transient of carbon disulfide using a nonlinear optical technique. The ambiguity can be removed by reconfiguring the experiment to allow some coherent, quadrature phase background to add to the Fourier transform of h_α(t) before the magnitude is taken. Only one scan is necessary. For more general impulse responses consisting of sums of two arbitrary functions, we have derived expressions for (1) the condition in which relative weight ambiguity occurs and (2) the spurious solution that results. A corollary to these results reveals a general condition for determining whether coherent background removes a relative weight ambiguity. We also consider the sum of N exponentials, where we show that as many as 2^N−1 solutions for the relative weights can exist, depending on whether real or complex weights are assumed. Finally, we derive expressions for the amount of quadrature phase background necessary to definitively remove relative weight ambiguities (that is, remove even local minima in the fitting function) in sum-of-two-exponential impulse responses.

© 1989 Optical Society of America