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CRC Press eBooks

Chapter Two-Dimensional Photon Echo Spectroscopy

Minhaeng Cho

Published Jun 16, 2009• 0 citations
Publisher

Abstract

The phase-matched photon echo signal field should then satisfy the following wave vector equality: ks = -k1 + k2 + k3. The corresponding diagrams have been called the rephasing ones. The “mirror image” echo appears at ks = k1 - k2 + k3, and the corresponding nonlinear optical transition pathways were called the nonrephasing diagrams. At short time (<τsol or τdecoherence), the rephasing echo and the mirror image (nonrephasing) echo signals, with respect to the first coherence period, τ, do not completely overlap,9 and the time difference between the two peaks was defined as the three-pulse PEPS denoted as τ*( ).T 7, 10, 11 It turned out that the PEPS as a function of the second delay time T between the second and third pulses in the three-pulse photon echo was shown to be directly related to the transition frequency-frequency correlation function, and its asymptotic value is related to the static inhomogeneity directly reflecting unequal environments around different chromophores.7, 11 The delay time T was called the waiting or population evolution time in the literatures. Usually, the conventional integrated photon echo signal depends on two time variables τ and T, and the echo field intensity (not amplitude) was detected. Therefore, the integrated photon echo measurement method cannot provide the 2D photon echo spectrum unless notable quantum beats exist.

DOI
10.1201/9781420084306-11

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•May 21, 1990•0 citations
The phase-matched photon echo signal field should then satisfy the following wave vector equality: ks = -k1 + k2 + k3. The corresponding diagrams have been called the rephasing ones. The “mirror image” echo appears at ks = k1 - k2 + k3, and the corresponding nonlinear optical transition pathways were called the nonrephasing diagrams. At short time (<τsol or τdecoherence), the rephasing echo and the mirror image (nonrephasing) echo signals, with respect to the first coherence period, τ, do not completely overlap,9 and the time difference between the two peaks was defined as the three-pulse PEPS denoted as τ*( ).T 7, 10, 11 It turned out that the PEPS as a function of the second delay time T between the second and third pulses in the three-pulse photon echo was shown to be directly related to the transition frequency-frequency correlation function, and its asymptotic value is related to the static inhomogeneity directly reflecting unequal environments around different chromophores.7, 11 The delay time T was called the waiting or population evolution time in the literatures. Usually, the conventional integrated photon echo signal depends on two time variables τ and T, and the echo field intensity (not amplitude) was detected. Therefore, the integrated photon echo measurement method cannot provide the 2D photon echo spectrum unless notable quantum beats exist.
The time-integrated three-pulse photon echo is discussed in terms of the line-shape functions which are shown to be directly related to the solvation dynamics. By using the short-time approximation to the line-shape function, an algebraic form of the three-pulse photon echo signal S PE (τ, T ) is obtained as a function of two delay times, τ and T, between pulses. On the basis of the approximate form of the photon echo signal, the echo peak shift with respect to the second delay time (population evolution period) is derived and found to be linearly proportional to the time-dependent fluorescence Stokes shift function for times greater than the bath correlation time, τ c . In the case of intermediate inhomogeneous broadening, the asymptotic echo peak shift magnitude is found to be useful in estimating the static inhomogeneous width. To illustrate the theoretical results, an analysis of the echo peak shift measurement of IR144 in a room temperature glass, PMMA, is presented.
Phase relaxation of optical excitations in condensed matter can be studied by coherent four—wave—mixing experiments in the time domain without requiring assumptions on the spectral properties of the material. For inhomogeneously broadened transitions, three laser pulses at times t = 0, t = t21 and t = T with wave vectors k1, k2 and k3, respectively, generate a stimulated photon echo pulse, emitted time—delayed at t = T+t21 in the momentum—matching direction according to k3+k2−k1. As a function of t21 the echo amplitude decays due to scattering events, described by the dephasing time constant T2 [1].
The three-pulse stimulated photon echo has been considered as one of the most fundamental transient four wave mixing processes1, where each of the three excitation pulses induces a single elementary interaction process between a laser field and an inhomogeneously broadened two-level system., resulting in a burst of the fourth wave, i.e., echo. Now if four or more optical pulses are applied to a medium, a number of photon echoes should be expected, in general. At low excitation intensities where the perturbation theory is applicable, however, the dominant nonlinear process is still the three-pulse stimulated echo and consequently each echo signal is generated only by three of the preceding pulses. It is rather a remarkable result that the echo intensity is unaffected by the presence of all the rest of the pulses other than its three generators. With this in mind, it is theoretically and experimentally shown that by applying a single optical pulse (#1) propagating along k1 followed by N pairs of counter-propagating pulses (#2j (k2j) and #3j− (k3j = −k2j); j=1,…N) in various directions, it is possible to obtain N photon echoes along the direction -k1 opposite to the first pulse #1. We will call this degenerate photon echoes, degenerate in. the sense that the echoes are spatially overlapping and temporally simultaneous. When N=l, this scheme is reduced to the ordinary backward photon echo2,3. The doubly degenerate case (N=2) is illustrated in Fig. 1. The j-th echo will be generated by the set of pulses (#1, #2j, #3j) and appear at t = t3j + t2j − t1 where t1, i2j and t3j are the arrival times of the pulses #1, #2j and #3j, respectively.
A theoretical description of one- and two-color photon echo peak shifts (PEPS) and two-dimensional (2D) photon echo spectrum (PES) of a coupled chromophore system are presented. The effects of population relaxation in the one-exciton states on both the PEPS and the 2D PES are investigated. For values of time T shorter than the population relaxation time, a finite two-color peak shift magnitude and nonzero cross peaks in the 2D PES provide evidence of electronic coupling between the chromophores. These two distinct observables, i.e., PEPS and off-diagonal peaks, both originate from the electronic coupling. However, it is shown that the PEPS and 2D PES methods can provide complementary information on the structure-dependent nonlinear optical responses of coupled chromophore systems.
A theoretical analysis is carried out for the recently developed three-pulse photon echo spectroscopy employing dual frequency combs (DFC) as the light sources. In this method, the molecular sample interacts with three pulse trains derived from the DFC, and the generated third-order signal is displayed as a two-dimensional (2D) spectrum that depends on the waiting time introduced by employing the asynchronous optical sampling method. Through the analysis of the heterodyne-detected signal interferogram using a local oscillator derived from one of the optical frequency combs, we show that the 2D spectrum closely matches the spectrum expected from a conventional approach with four pulses derived from a single femtosecond laser pulse, and the waiting time between the second and third field-matter interactions is given by the down-converted detection time of the interferogram. The theoretical result is applied to a two-level model system with solvation effect described by solvatochromic spectral density. The model 2D spectrum reproduces spectral features such as the loss of frequency correlation, dephasing, and spectral shift as a function of the population time. We anticipate that the present theory will be the general framework for quantitative descriptions of DFC-based nonlinear optical spectroscopy.
The phase relaxation of optically excited electronic states can be studied in fourwave mixing experiments where three excitation pulses at t = 0, t =t21, and t = T, respectively, interact via the third-order polarization of the medium to generate a nonlinear signal whose intensity depends on the dephasing time T2. For inhomogeneously broadened optical transitions the signal will be emitted as a time delayed photon echo pulse at t = T + t21 provided the pulse duration is short compared with T2. The echo amplitude decays as exp(−4t21/T2).1