An analytic theory and experimental studies of the accumulated photon echo is presented. We define a temporal correlation function (\tilde{G}) of the excitation field E ( t ) as (\tilde{G}(\tau,t)=(1/T_{1})\int_{0}^{\infty}\mathrm{d}t'E(t-t')E(t-t'-\tau)\cdot\exp(-t'/T_{1})), where T 1 is the population relaxation time of the sample, and call the width of (\tilde{G}) with respect to τ a temporal correlation time (\tilde{\tau}{\text{c}}(t)). It is proved that the echo intensity is written by the temporal correlation functions when (1) T 1 and the duration of the excitation light τ p are much longer than (\tilde{\tau}{\text{c}}(t)) in order to make the function (\tilde{G}((\tau,t)) well-defined, (2) T 1 and τ p are much longer than the homogeneous dephasing time T 2 , and (3) one can find a time interval Δ t such that (\tilde{G}(\tau,t)) does not change much between t and t + Δ t , where (\varDelta t{\gg}\tilde{\tau}{\text{c}}). Then we can show that the time resolution of the accumulated photon echo is determined by the averaged (\tilde{\tau}{\text{c}}(t)) over an interval of the integration of the echo signal. This discussion is valid for any excitation source if the above conditions are satisfied, and applies to almost all cases of the practical accumulated photon echo experiment. A discussion is also made on the spurious spike on the echo decay curve near zero delay time that appeared in an experiment with a high time resolution in Nd 3+ -doped silicate glass. It is shown that the most probable cause for this spike is uncorrelated inhomogeneous broadenings of the resonant lines of neodymium ions.
Accumulated photon echoes were observed in
Nd3+-doped silicate glass by using coherent and incoherent light, and it was theoretically and experimentally verified that the time resolution is determined not by the pulse width but by the correlation time of the excitation field. Even by using a cw laser, instead of a mode-locked laser, a subpicosecond time resolution was obtained when it was operated with a broad and smooth spectrum. The broadest output spectrum, and hence the highest time resolution, was attained when the dye laser was pumped by a mode-locked
Ar+ laser, and their cavity lengths were mismatched by a few \ensuremath{\mu}m. In an experiment with a resolution time of 220 fsec, a new very fast decay was observed in the echo decay curve besides a slower decay already reported.
Since the first observation of accumulated photon echoes by Hesslink and Wiersna [1], nearly transform-limited mode-locked pulse trains have been used as the excitation source, and the resolution times have been limited by the widths of the pulses. In this paper we show theoretically and experimentally that the time resolution of the accumulated photon echoes is determine not by the pulse width but by the correlation time of the excitation field [2]. This indicates that it is possible to obtain a sub-picosecond resolution in accumulated photon echoes Sven by using a cw laser (non-mode-locked), if the laser output has a sufficiently broad and smooth spectrum.
It has been recently pointed out that incoherent light can be used to generate coherent optical transients and that there are decided advantages in using such broad-band non-laser source to achieve very high time resolution {1,2}. When a light pulse and its time-delayed replica interact with a sample to produce the non-linear optical response, the ultimate time resolution should identify with the correlation time τC of the light field rather than with the pulse duration τL. The situation where τC ≪ τL is easily obtained with broad-band incoherent light sources. Time resolution in the femtosecond range is then attainable using nanosecond light pulses. This has been recently demonstrated through the observation of the relaxation of photon echoes in Nd3+-doped silicate glass {1,2}. In these cases, attention is focused on the measurement of fast homogeneous decay rates and the inhomogeneous broadening ΩD is assumed to be much larger than the spectral width of the excitation light.
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.
Accumulated photon echoes (APE) were used to measure the dephasing rates of Nd3+ in various inorganic glasses as a function of excitation wavelength, sample temperature, and laser power. Although dephasing processes of optical transitions in solids are of considerable interest, they are poorly understood, largely due to the lack of experimental data on femtosecond and picosecond time scales. The APE technique using sub-picosecond laser sources has made it possible to measure in the time domain dephasing rates of inhomogeneously broadened transitions in amorphous systems.1,2