The influence of finite bandwidth excitation sources in degenerate four-wave mixing spectroscopy

Abstract

The effects of finite-bandwidth excitation are incorporated into the theoretical treatment of degenerate four-wave mixing (DFWM) spectroscopy by employing a general description of applied electromagnetic fields, thereby permitting the transient nature of typical laser sources to be taken explicitly into account. A computationally efficient perturbative (weak-field) formalism is developed by exploiting exponentially-decaying wave forms as a flexible basis for expanding the temporal envelope functions of incident quasimonochromatic pulses. This approach has the distinct advantage of yielding time-domain response integrals that can be evaluated analytically since they are almost as simple in form as their monochromatic counterparts. The resulting frequency-domain expressions for the induced DFWM signal polarization reflect the pronounced influence of finite-bandwidth excitation without needing to cobble such behavior into phenomenological rates for population decay and coherence dissipation. Spectral line shapes are examined for both stationary and nonstationary ensembles of isolated (gas-phase) target molecules, with the latter analyses demonstrating the systematic degradation of spatial discrimination introduced by nonmonochromatic implementations of the sub-Doppler (phase-conjugate) four-wave mixing scheme. In keeping with recent experimental findings, the precipitous drop in signal intensity accompanying the action of collisional dephasing processes is predicted to be tempered greatly by the use of short-duration excitation pulses.

Laser diagnostics and minor species detection in combustion using resonant four-wave mixing

“…A perturbative (weak field limit) method was developed by Vaccaro and co-workers which involved time domain solutions that could be evaluated analytically and therefore efficiently [75]. The non-perturbative analytical method of Bratfalean et al [46] can be extended to treat finite bandwidth laser interactions by modelling the interaction as an independent spectral response ISR to a set of uncorrelated monochromatic components spanning the laser bandwidth.…”

Isotopic Dependence of Excited-State Proton-Tunneling Dynamics in Tropolone Probed by Polarization-Resolved Degenerate Four-Wave Mixing Spectroscopy

“…The ensuing discussion of rotationally resolved DFWM spectroscopy builds upon our previous tensor-algebraic analyses of weak-field response, as generalized to encompass multiple excitation paths . Each electromagnetic wave is assumed to be monochromatic, with a numerical index used to distinguish among the incident ( j = 1, 2, or 3) and generated ( j = 4) beams. The electric field vector, E j ( r , t ), for a monochromatic plane wave having angular frequency ω j and propagation wavevector k j is defined by boldE j ( r , t ) = 1 2 [ E ω j e i false( boldk j · boldr − ω j t false) + E ω j * e − i false( boldk j · boldr − ω j t false) ] where the vector amplitude E ω j embodies a constant scalar amplitude, E ω j = | E ω j |, and a polarization unit vector, ε j = E ω j /| E ω j |, such that E ω j = E ω j ε j .…”

“…The angular frequency, ω αβ , and the dephasing rate, Γ αβ , for the |α⟩ ↔ |β⟩ resonance are defined by ω α β = 1 normalℏ ( E α − E β ) normalΓ α β = 1 2 ( Γ α α + Γ β β ) + normalΓ α β φ where E α and Γ αα denote the unperturbed energy and the depopulation rate for quantum state |α⟩. The additional factor of Γ αβ φ in the latter expression accounts for the action of phase-disrupting processes that destroy molecular coherence without affecting the attendant molecular populations. , For convenience, the molecular properties embodied in eqs and often are combined into a single complex quantity to yield Ω αβ = ω αβ – iΓ αβ . ,, …”

“…Building upon the canonical (weak-field) framework of time-dependent perturbation theory, ,,, the instantaneous response evoked during a resonant four-wave mixing interaction can be shown to scale in proportion to the square modulus of the third-order electric polarization vector, ε 4 * · P (3) ( t ), as induced by successive coupling of the three impinging electromagnetic waves. For frequency-domain DFWM spectroscopy performed on nonstationary (gas-phase) target molecules without spectral discrimination of the emerging signal field, the intensity observed at incident frequency ω, I 4 (ω), can be related to the Fourier transform of this pivotal quantity, ε 4 * · P (3) (ω s ), such that I 4 ( ω ) ∝ prefix∫ − ∞ + ∞ false| ⟨ false⟨ bold-italicε 4 * · boldP false( 3 false) ( ω s ) false⟩ ⟩ boldv false| 2 ⁡ d ω s where the angular brackets signify an average over translational degrees of freedom as specified by the normalized velocity distribution f ( v ), as defined below. ⟨ false⟨ bold-italicε 4 * · boldP false( 3 false) ( ω s ) false⟩ ⟩ boldv = prefix∫ boldv false[ bold-italicε …”

“…The definition of these quantities depends on the precise nature of the impinging electromagnetic waves, with the monochromatic form of eq giving in the case of D 12g D 12 normalg = δ ( ω s − ω σ ) ( Ω eg ″ − ω 1 ) ( Ω g ′ g ″ − ω 1 + ω 2 ) ( Ω e ′ g ″ − ω 1 + ω 2 − ω 3 ) where ω σ = ω 1 – ω 2 + ω 3 and the Dirac delta function, δ(ω s – ω σ ), reflects energy conservation. Wasserman et al have reported a substitution table that allows all other D ijα factors to be obtained from this expression for D 12g . The velocity averaging in eqs and demands that the common rest frequency for the incident (degenerate) fields be replaced by their Doppler-shifted counterparts, ω j → ω j – k j · v , with a collinear approximation permitting requisite integrals to be evaluated analytically by means of the plasma dispersion function …”