A new method of picosecond and femtosecond transient spectroscopy yielding information about ultrafast relaxation processes without requiring ultrashort light pulses is presented. It is based on the present theoretical analysis of the resonant degenerate four-wave-mixing process excited by two temporally incoherent light beams with wave vectors ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{1}$ and ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{2}$ which are originated from a single beam at frequency $\ensuremath{\omega}$ but have mutual time delay $\ensuremath{\tau}$. Under the assumption that the incoherent light field has Gaussian random complex amplitude and that the resonant material consists of the usual two-level atoms, the statistically averaged intensity of the output light field with ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{3}=2{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{2}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{1}$ at $\ensuremath{\omega}$ is calculated as a function of $\ensuremath{\tau}$. Even with the light having a much longer duration than both
T1 (the longitudinal relaxation time) and
T2 (the transverse relaxation time), the correlation trace, i.e., output intensity versus $\ensuremath{\tau}$, represents a decay profile determined mainly by
T2 for both homogeneously and inhomogeneously broadened transitions, as long as the light correlation time ${\ensuremath{\tau}}_{c}$ is much shorter than the relaxation times. The correlation trace does not always represent a single-exponential decay but is sometimes slightly deformed by the
T1 effect. However, it does not cause a significant error in the determination of
T2. Moreover, as $\frac{{T}_{2}}{{T}_{1}}\ensuremath{\rightarrow}0$, the trace becomes a single-exponential decay curve determined only by
T2. The feature of the results obtained by the present method is similar to that obtained by the conventional coherent transient spectroscopy with short pulses, such as the photon echo. The time resolution in the present method, however, is limited only by ${\ensuremath{\tau}}_{c}$ much shorter than the light duration. By regarding the incoherent light as a series of random ultrashort pulses, the present four-wave-mixing process is also interpreted as the ensemble of numerous transient four-wave-mixing processes caused by various combinations of these pulses.
Time-resolved three-beam degenerate four-wave mixing with incoherent light for observation of population relaxation, originally proposed by Morita . [ J. Opt. Soc. Am. B4, 1269 ( 1987)] is examined both theoretically and experimentally. Theoretical calculation shows that a large part of the decaying signal in the time-resolved profile originates from the ground-state relaxation if the excited-state relaxation is faster than the ground-state relaxation. It is also shown that the use of a nonzero delay time between the k2 and k1 beams, in addition to that between the k3 and k1 beams (ks = k1 − k2 + k3, where k1–3 and ks are the wave vectors for the incident and the scattered beams, respectively), provides information on population relaxation, especially when the population-relaxation time is far longer than the correlation time of the incoherent light. These theoretical considerations are verified experimentally with the use of a triphenylmethane dye in solution.
Measurements of the relaxation times T1 and T2 of a two-level system by nonlinear transient spectroscopic methods are discussed, where delayed incident pulses are assumed to have correlation times much shorter than T1 or T2 but varied widths. In particular, a delayed degenerate four-wave mixing (photon echoe) and apump-and-probe type transmission measurement with incoherent long pulses are considered in detail, and a principle of their use for ultrafast spectroscopy is clarified. If the field is steady and T1>>T2, the time resolutionof T2measurement by the delayed four-wave mixlng is determined by the field correlation, therefore by the spectral width of the incident light irrespective of the pulse width. The time resoution of the T1 measurement by the pump-and-probe method is determined by the intensity correlation of the incident light, whereasthe one by the delayed four-wave mixing by a higher order correlation.