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arXiv:1905.11687 · Journal of the Optical Society of America B

Theory of three-pulse photon echo spectroscopy with dual frequency combs

Jonggu Jeon, JunWoo Kim, Tai Hyun Yoon, Minhaeng Cho

Published Oct 25, 2019• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1364/josab.36.003196

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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.
Dual frequency comb (DFC) nonlinear spectroscopy is an emerging technique that can be used to study a variety of molecular nonlinear responses by exploiting the automatic pulse-to-pulse time delay scanning and fast data acquisition characteristics of the DFC techniques. Here, we propose a DFC-based photon echo spectroscopy (DFC-PES), where two optical frequency comb lasers have slightly different repetition rates and are also allowed to have different carrier frequencies. We first demonstrate our theoretical approach for DFC linear spectroscopy. Then, the signals expected from the proposed DFC-PES are theoretically calculated. The slight offset in the pulse repetition rates enables asynchronous optical sampling and automatic scanning of the time intervals between different field–matter interaction events, making frequency tuning unnecessary. Analytic expression of the third-order photon echo signal is demonstrated in two frequency dimensions for a simple two-level model system. The signal has a well-defined and simple connection to the underlying third-order response function and exhibits the expected down-conversion features from the optical frequency into the radio frequency region with a down-conversion factor that is experimentally controllable.
The connection between dephasing of optical coherence and the measured spectral density of the pure solvent is made through measurements and calculations of photon echo signals. 2-pulse photon echo measurements of a cyanine dye in polar solvents are presented. Signals are recorded for both phase matched directions enabling accurate determination of the echo signal time shift. Echo signals are calculated by two approaches that employ the response function description of nonlinear spectroscopy; (i) a single Brownian oscillator line shape model, and (ii) the line shape obtained using the solvent spectral density. The strongly overdamped Brownian oscillator model incorporates only a single adjustable parameter while the experimental data present two fitting constraints. The second model incorporates the measured solvent spectral density. Both give very good agreement with the experimental results. The significance of the second method lies in this being a new approach to calculate nonlinear spectroscopic signals, for comparison with experimental data, that uses directly the measured spectrum of equilibrium fluctuations of the solvent. This approach also provides a better conceptual perspective for deriving insight into the nature of the solute–solvent coupling mechanism. Comparing the parameters for the strength of interaction in a variety of polar solvents it is found that the coupling involves the solvent polarizability and not the solvent polarity. The interaction mechanism cannot be deduced from the Brownian oscillator calculations.
Two-dimensional (2D) line shapes derived from four types of coherent third-order nonlinear spectroscopies as a function of two independent time or frequency variables are compared. The signals scattered into two wave-vector-matching directions in the limits of high-time or high-frequency resolution are calculated for an inhomogeneously broadened ensemble of two-level systems with phenomenological dephasing and population relaxation times. While one-dimensional (1D) line shape analysis for this model is ambiguous, the contours of these 2D line shapes are shown to be characteristic of the relative time scales of the three relevant line-broadening parameters. These broadening mechanisms are also apparent by comparing 2D spectral profiles along the diagonal and antidiagonal frequency axes. For time domain experiments, in analogy with NMR correlation spectroscopy, the radiated signal is observed as a function of the two coherence periods and is 2D Fourier transformed to obtain a line shape. For experiments based on the photon echo, the ellipticity of the absolute value 2D line shape is related to the ratio of the inhomogeneous width to the dephasing rate, whereas experiments based on the transient grating method are shown to be functionally 1D. For frequency domain experiments, the contours of the 2D line shape can yield information on static and both dynamic broadening mechanisms. While the addition of spectral diffusion to such calculations will modify these results for frequency domain experiments, characterizing these dynamics in the time domain naturally leads to a 3D experiment analogous to the spin diffusion experiment in NMR.
In recent years, frequency-comb-based multidimensional coherent spectroscopy has emerged as a very powerful optical method for studying samples with long dephasing times. It enables the measurement of comb resolution multidimensional coherent spectra in seconds as opposed to hours or even days required by traditional mechanical stage-based spectrometers. However, for some samples (e.g., cold atoms and molecules) the dephasing times could be longer than the repetition periods of the excitation lasers which means that the generated signal will not fully decay between the pulses. Depending on the relative phase or time between the pulses, the signals generated by the subsequent laser pulses can constructively or destructively interfere with each other. In this paper we study this behavior by solving the optical Bloch equations and provide a theoretical description of frequency-comb-based multidimensional coherent spectroscopy.
The role of weakly chirped pulses (time bandwidth product, DeltanuDeltatau<0.61) on three-pulse photon echo signals has been systematically studied. Pulses with varying chirp were characterized with frequency resolved optical gating (FROG) and used to measure spectrally resolved three-pulse photon echoes of a dye in solution. The weakly chirped pulses give rise to markedly different echo signals for population times below approximately 100 fs. The chirped pulses can decrease or enhance spectral signatures of an excited state absorption transition in the echo signal. Furthermore, the observed dephasing dynamics depend on the phase of the electric fields. Simulations based on a three-level model and the electric fields retrieved from the FROG traces give a good agreement for photon echo experiments with both transform limited and chirped pulses. The simulations also allow for a numerical investigation of effects of chirp in two-dimensional spectroscopy. For a two-level system, the chirped pulses result in nonelliptical two-dimensional spectra that can erroneously be interpreted as spectral heterogeneity with frequency dependent dephasing dynamics. Furthermore, chirped pulses can give rise to "false" cross peaks when strong vibrational modes are involved in the system-bath interaction.
Summary form only given. A very high temporal resolution and a broad bandwidth are but two advantages provided by the use of extremely short sub-5-fs pulses in a nonlinear spectroscopic experiment. However, the applicability of the standard theoretical description becomes questionable for the pulses that consist merely of a couple of optical oscillations. Far instance, the conventionally employed slowly varying envelope approximation, implying that the change of the pulse amplitude on the duration of an optical cycle is negligible compared to the magnitude of the amplitude itself, can no longer be maintained. Furthermore, the phase-matching bandwidth that is limited due to dispersion in the nonlinear medium rapidly gains importance with the increase of the spectral width of the pulse. Another point of serious concern is the frequency-dependent variation in the sensitivity of signal photodetectors. In combination, these features result in what is known as a spectral-filter effect. Finally, artificial lengthening of the experimental transients is a direct consequence of the noncollinear geometry employed in spectroscopic experiments. We present a theoretical analysis which thoroughly reexamines the formalism of ultrafast photon echo spectroscopy. We obtain a general expression for the echo signal, which is valid even for single-cycle-pulse applications. The derived formalism is applied to photon-echo spectroscopy on the hydrated electron with 5-fs pulses.