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IEEE Journal of Quantum Electronics

Wigner spectrograms for femtosecond pulse-shaped heterodyne and autocorrelation measurements

Shaul Mukamel, C. Ciordas‐Ciurdariu, Vadim Khidekel

Published Jan 1, 1996• 47 citations
Publisher

Abstract

We develop a mixed time-frequency representation for the calculation and interpretation of coherent optical measurements. Heterodyne and autocorrelation signals are expressed in terms of a mixed material response function and a Wigner distribution for the incoming pulses, the detected field, and the gating device. Applications to pump-probe spectroscopy, spontaneous light emission, and impulsive four-wave mixing (three-pulse echoes) are discussed.

DOI
10.1109/3.511539

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We develop a mixed time-frequency representation for the calculation and interpretation of coherent optical measurements. Heterodyne and autocorrelation signals are expressed in terms of a mixed material response function and a Wigner distribution for the incoming pulses, the detected field, and the gating device. Applications to pump-probe spectroscopy, spontaneous light emission, and impulsive four-wave mixing (three-pulse echoes) are discussed.
We give the first analysis of heterodyne experiments that includes the propagation along the waveguide. Under the assumption that the material polarization is adiabatically eliminated we derive simple analytic expressions for the coherent coupling term, exact for Gaussian pulses and a good approximation for other shapes. We find that the measured signal does not display any phase dependence close to zero pump–probe delay. A four-wave mixing band is identified in the heterodyne signal, and a new class of experiments is proposed by analogy with the spatial case.
The use of phase-locked pulses in various spectroscopic techniques related to the third-order polarization P(3) is analyzed. Using correlation function expressions for the nonlinear response function, we clarify the interrelationship among several photon echo, pump–probe, and spontaneous light emission techniques, without alluding to any specific model for the material system. By combining phase-locked pulses and heterodyne detection it becomes possible to probe separately the real and imaginary parts of the nonlinear response function. Combining two phase-locked pulse excitation with time-resolved detection of the spontaneous light emission allows direct separation of the Raman and fluorescence contributions.
The Wigner Distribution Function (WDF), defined for a phase space representation in quantum mechanics, offers an alternate form for representing signals simultaneously in Fourier reciprocal variables such as time, and frequency. A purely nonnegative function that is closely related to the WDF, is the so‐called ’’spectrogram’’. In this paper we describe three types of optical setups for the production of the spectrogram. They use coherent, partially coherent or temporally incoherent light as illumination. As application example speech signals are processed for a speaker verification.
Nonlinear optical signals from an assembly of N noninteracting particles consist of an incoherent and a coherent component, whose magnitudes scale ~ N and ~ N(N - 1), respectively. A unified microscopic description of both types of signals is developed using a quantum electrodynamical (QED) treatment of the optical fields. Closed nonequilibrium Green's function expressions are derived that incorporate both stimulated and spontaneous processes. General (n + 1)-wave mixing experiments are discussed as an example of spontaneously generated signals. When performed on a single particle, such signals cannot be expressed in terms of the nth order polarization, as predicted by the semiclassical theory. Stimulated processes are shown to be purely incoherent in nature. Within the QED framework, heterodyne-detected wave mixing signals are simply viewed as incoherent stimulated emission, whereas homodyne signals are generated by coherent spontaneous emission.
Two-pulse four-wave-mixing signals from molecular aggregates, including effects of two-exciton states, static disorder, and exciton-phonon interaction represented by arbitrary spectral densities are calculated. Three types of contributions to the signal are identified. The first, reflecting exciton self-correlation, is similar to the photon echo from disordered two-level systems and dominates the signal for long time-delays. The second is related to correlations of one-exciton states, whereas the third reflects correlations between one- and two-exciton states. The information gained by completely resolving the signal field (both amplitude and phase) is analyzed using Wigner spectrograms.