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II.B Nnon-linear Spectroscopy Background

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  • II.B  Non-linear Spectroscopy Background                                  

    Applying external electric to systems will generate induced polarizations within the sample.  These induced polarizations can be expanded and classified in terms of the powers of the electric field, E




    Each nth order polarization, P(n), is expressed as a product of electric field interactions and a material susceptibility, c(n).  The first term in the expansion is the linear polarization, P(1), and the rest of the polarizations fall in the category of non-linear polarizations.  In general, the polarization is a function of both position and time, , but we assume the electric dipole approximation holds, and the polarization is considered independent of position.4 

    The directionality of the induced polarization, in an isotropic medium, is determined solely by the directionality of the incident electric fields, hence,  when .  The even order polarizations (P2n), therefore, are sensitive only to the even powers of the electric field (E2n), which are always positive, and the only way this property this can be realized is if the even order susceptibilities, c(2n) are zero.  For isotropic systems, the third order polarization, P(3) is the lowest order non-linear response and the signals originating from this non-linearity are responsible for the investigation of the solvent dynamics discussed in this manuscript (with the exception of fifth-order, Raman signals described in Appendix A).


    II.C.  Signals

    II.C.i.  Linear Signals

    The MBO model can simulate linear absorption and fluorescence spectra.  The steady state absorption spectrum for the system can be calculated from g(t) via3,10




    and similarly, the fluorescence spectrum is expressed by




    It should be noted that Equation II.7 is different from that used previously6 due to the different definition of weg.  Equations II.7 and II.8 predict mirror image spectra shifted around l

    An interesting perspective, for both calculations and interpretation, of the linear signals is observed when the g(t) function is expanded into real and imaginary components


                                                            ,                                            (II.9)


    then Equation II.7 can be expanded to give





    The complete temporal profile of M(t) is required to calculate g(t) and hence the linear spectra, thus the dynamical timescales associated with M(t) are, in principle, fully contained in the absorptions spectrum.  However practically the sensitivity of the absorption spectra is limited to the initial timescales of the line broadening function.  Since the  is a monotonically increasing function, the exponential factor in the integrand of Equation II.10, , is effectively non-zero for a short period of time (<200 fs for most liquids studied here), thus limiting the applicability of linear spectroscopy to characterize slower solvent dynamics.

    Equation II.10 can be further rearranged into a sum of two integrals,




    where the first term in Equation II.11 is simply the cosine transform of product between and  arguments, and the second term is the sine transform of the same exponential and  terms.  Expressing Equation II.7 in this manner introduces the ability to use the power and speed of the fast Fourier transform algorithm11 (FFT) in calculating absorption spectra, since the cosine and sine transforms are simply the real and imaginary components of the Fourier transform:


                                          .                          (II.12)


    All linear spectra presented in this manuscript are calculated in this manner.

    Inspection of Equation II.12 allows us to identify the origin of asymmetry in the simulated spectra.  The cosine transform maps arguments onto a cosines basis set, which consists of even basis functions (i.e. ).  Consequentially, transformed functions will necessarily be symmetric around w=0, and any asymmetry in the calculated spectra results from contribution of the sine transform component in Equation II.12 (i.e. odd basis functions).  The magnitude of the sine transform component is determined by the magnitude of , and so any asymmetry in simulated spectra.  Simulated absorption spectra for systems with significant  magnitude (e.g. low temperature systems,12,13 presence of high frequency vibrational modes14,15) display pronounced asymmetric spectra.  Correspondingly, calculations that include only the real part of g(t) predict only symmetric linear spectra.16



    II.C.ii.  Third-Order Signals

    Theoretically, the absorption spectra of chromophores in a matrix, whether liquid or solid, contain a great deal of information concerning the timescales and coupling of the environment to the chromophore.  Unfortunately, broad, featureless bands that display little dynamical information often characterize the absorption spectra of chromophores in solution.  In contrast, both high resolution gas phase and low-temperature solid phases, absorption spectra show well-resolved features that contain considerably more information.  The major difference between these two cases is the presence of dynamics that occur on multiple timescales.  If one were to assume that the timescales of motions were separable into two limits, either infinitely fast (homogenous) or infinitely slow (inhomogenous), then the observed spectra can be explained in terms of some combination of dynamical contributions and static contributions.  Explaining spectra and dynamics in such a may be applicable in some systems (e.g. low-temperature matrices), but unfortunately, the dynamics of room temperature liquies span multiple timescales, (i.e., non-Markovian) and a more complicated model is needed.

    Photon echo measurements afford experimenters the ability to observe dynamics on multiple timescales.  More importantly, the 3PEPS method lets researchers to “gate” or time-resolve the underlying dynamics.  I prefer to think of it as “gating the inhomogeneity,” since the dynamics of slower processes appear as “inhomogeneity” to faster processes, although the concept of inhomogeneity is rather blurred in such systems.  In non-reactive, solvating systems, all the information concerning the system dynamics are held in M(t). 

    The third-order nonlinear optical signals can be calculated within a response function formalism.3,4 The envelope function of the nonlinear polarization, P(3), induced within the systems by its interactions with the external electric fields is expressed by3,6,7








    where E1, E2 and E3 refer to the pulse envelops of the first, second and third pulses respectively.  The detuning parameter, wlaser -weg, is defined as the difference between the center frequency of the laser pulses and the transition frequency.  In the next section, we discuss the nature and factors needed to calculate weg.  The times, t1, t2, t3 refer to the interaction times, while t, T, and t represent the pulse timing (i.e. the time differences between the maxima of the pulses) of the first two pulses, the second two and the time between the final interaction time, t3 and the third pulse respectively (Figure II.1).  When the laser pulses do not overlap, the summation in Equation II.13a collapses to a single term (simply Equation II.13b).  The resulting expansion terms, Eq. II.13c-e, are the result of including pulse overlaps in the simulations and must be included to accurately calculate the observed nonlinear signals.



    Figure II.1. Temporal schematic of third order signals collected with three, distinct laser pulses.  The time variables: t, T and t, refer to pulse timing (of the peak), while the time variables: t1, t2, t3 refer to actual interaction times.  The finite duration of each laser pulse, leads to an ambiguity of interaction times that is accounted for in the convolution over a delta function response (Equation II.20). 





    Within the cumulant expansion,3 the response functions in Equation II.13 can be expanded in terms of the line broadening function, g(t):




    where the RII and RIII responses represent the rephasing pathways responsible for a non-zero peak in the polarization and the RI and RIV responses represent the non-rephasing contributions to the signal and peak at t3=0.6,7,17 The RI and RII response functions describe the system when it propagates along the ground state during the population period, and RIII and RIV describe the dynamics along the excited state.  The separation of the signals into the ground and excited state contributions is convenient in analyzing the nonlinear signals. 

    The phase matching condition in the experiment requires that each interaction occurs from each of the different pulses,3 but does not require that the interactions occur in the sequential order with the pulses.  For example, the first interaction with the light field can originate from the second pulse and the second interaction with the first pulse and finally the third interaction occurs from the third and final impinging pulse (represented by the sequence: 2,1,3).  The respective pulse-interaction combinations for P1-P4 are (1,2,3), (2,1,3), (1,3,2), and (3,1,2); the other two possible combinations, (2,3,1) and (3,2,1), will not contribute to the signals generated from two-level systems in the k3+k2-k1 phase matched direction and are not included in the simulations.

    The first polarization component, P1, represents the polarization responsible for the photon echo signal with the three pulses interactions occurring from the sequential pulse combination (1,2,3).  The next component in the summation, P2, represents a free induction decay-like response (FID).  The subsequent contributions, P3 and P4, originate from the rephasing and non-rephasing contributions with different interactions of the pulses respectively.

    The evolution of the polarizations during the three time durations can be interpreted as either a propagation in a coherence state, where the system either dephases or rephases (t1 and t3), or a population state (t2), with no dephasing nor rephasing.  The first interaction creates a coherence in the system, where the system will evolve as a superposition between the ground and excited states.  If the system evolves with no further interactions with the E-field, the system will completely dephase and lead to a FID in a similar manner as with NMR experiments.  The next interaction then creates a population state, where the system will evolve either in the ground or excited states.  During this state, the system ceases to dephase electronically and evolves via the population dynamics associated with the respective electronic states.  The third and last interaction creates another coherence state.  When this second coherence state is the complex conjugate of the first, the system has the capacity to rephase and can generate an optical photon echo, e.g. P1 and P3.  In contrast, when the system evolves in the same manner as the first coherent period, it continues to dephase, leading to a FID.  The rephasing capability of the system is stored in the population state and subsequently destroyed by spectral diffusion process (i.e. solvation).  When the population period is long, the system will lose its capacity to rephase upon the last interaction, then during the third period, the system cannot rephase and the photon echo polarizations, P2 and P4, more resemble the FID polarizations than conventional echo signals.  The memory of the system to rephase is directly related to the spectral diffusion involved in solvation dynamics and is the property that is directly probed with 3PEPS experiments.  We refer to the time delay between the first and second pulses as the coherence period (t) and the time delay between the second and third pulse as the population period (T), the third period is not experimentally controlled, though several groups have characterized the evolution of the nonlinear signals during this period.18,19

    The integrated echo signal measured in the laboratory is expressed in terms of P(3) by




    In the peak shift experiment, we measure the maximum of the photon echo intensity for a fixed delay T as a function of t  (Figure II.2).  The peak of the photon echo trace at, constant T, directly measures of the system’s ability to rephase (to cause an echo) after spending time T in a population state.  As shown previously, the peak shift is directly related to the transition frequency correlation function, and hence gives direct information on the optical dephasing, time scale and amplitude of the fluctuations of the environment and inhomogeneity of the system.6,7,17

                With Transient Grating measurements, the first and second pulses overlap completely in time (t=0) and the TG signal is collected by scanning the T delay.  The resulting 1-dimensional TG signal is calculated from the following relation.




    The response functions in Equation II.14 can also be categorized into whether the population period evolve on the ground state and excited state and similarly the P(3) in Equations II.13 can be also expanded.  Since the intensity of TG signals is given by the square modulus of P(3), in addition to the square modulus of the polarizations in the ground, Pgg, and the excited states, Pee, we have an interference term between the two states:




    The interference term is then given by




    As will be displayed, the detailed analysis of the TG signals in terms of the constituent dynamics on the respective electronic states requires the inclusion of this interference term.

    In the Transient Absorption (TA) measurements, the first and second interactions originate from the same pulse.  The signal is measured in the same direction of the third pulse.  The pulse sequence is same as in the Transient Grating measurement. 

    The TA signals are calculated from the following relations:





    where wpr and Epr are the carrier frequency and the field envelop of the probe pulse, respectively.  The signal intensity in the TA experiment is directly proportional to the induced optical polarization and is a sum of the ground state (photo bleach) and excited state (stimulated emission).  In contrast to the TG signals, the TA signals do not contain an interfering term (Eq. II.23a and Eq. II.22b).20 The principle difference between the TG and the TA signals is that the TA signal is intrinsically heterodyned against the third pulse,3 while the TG is a homodyned signal. Though, both signals are sensitive to P(3), they differ in the manner of probing the nonlinear polarization.



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