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Physical Review A

Loss and gain signals in broadband stimulated-Raman spectra: Theoretical analysis

Upendra Harbola, Siva Umapathy, Shaul Mukamel

Published Jul 2, 2013• 42 citations
Publisher

Abstract

Stimulated optical signals obtained by subjecting the system to a narrow band and a broadband pulse show both gain and loss Raman features at the red and blue side of the narrow beam, respectively. Recently observed temperature-dependent asymmetry in these features [Mallick et al., J. Raman Spectrosc. 42, 1883 (2011); Dang et al., Phys. Rev. Lett. 107, 043001 (2011)] has been attributed to the Stokes and anti-Stokes components of the third-order susceptibility, ${\ensuremath{\chi}}^{(3)}$. By treating the setup as a steady state of an open system coupled to four quantum radiation field modes, we show that Stokes and anti-Stokes processes contribute to both the loss and gain resonances. ${\ensuremath{\chi}}^{(3)}$ predicts loss and gain signals with equal intensity for electronically off-resonant excitation. Some asymmetry may exist for resonant excitation. However, this is unrelated to the Stokes vs anti-Stokes processes. Any observed temperature-dependent asymmetry must thus originate from effects lying outside the ${\ensuremath{\chi}}^{(3)}$ regime.

DOI
10.1103/physreva.88.011801

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Stimulated optical signals obtained by subjecting the system to a narrow band and a broadband pulse show both gain and loss Raman features at the red and blue side of the narrow beam, respectively. Recently observed temperature-dependent asymmetry in these features [Mallick et al., J. Raman Spectrosc. 42, 1883 (2011); Dang et al., Phys. Rev. Lett. 107, 043001 (2011)] has been attributed to the Stokes and anti-Stokes components of the third-order susceptibility, \ensuremathχ(3){\ensuremath{\chi}}^{(3)}\ensuremathχ(3). By treating the setup as a steady state of an open system coupled to four quantum radiation field modes, we show that Stokes and anti-Stokes processes contribute to both the loss and gain resonances. \ensuremathχ(3){\ensuremath{\chi}}^{(3)}\ensuremathχ(3) predicts loss and gain signals with equal intensity for electronically off-resonant excitation. Some asymmetry may exist for resonant excitation. However, this is unrelated to the Stokes vs anti-Stokes processes. Any observed temperature-dependent asymmetry must thus originate from effects lying outside the \ensuremathχ(3){\ensuremath{\chi}}^{(3)}\ensuremathχ(3) regime.
A theory of stimulated Raman scattering (SRS) is presented which treats in a unified manner the buildup of SRS from spontaneous Raman scattering and the spatial propagation which leads to gain. Maxwell-Bloch equations describe the coupling between the Stokes field operator and the collective atomic operators, which are driven by a classical laser field under low-signal-gain conditions, where the atomic ground states and laser field remain undepleted. An analytic expression is derived for the forward-propagating Stokes field operator in a one-dimensional spatial approximation when the Fresnel number of the excited medium is unity. Steady-state and transient Stokes intensities are evaluated under low-gain (spontaneous) and high-gain (stimulated) conditions. The Stokes intensity is found to be exactly independent of a laser bandwidth arising solely from phase fluctuations. The power spectrum of steady-state SRS is evaluated, giving a gain-narrowed spectral profile when the laser is narrow band, and a spectrum identical to the laser spectrum when the laser is broad band. A strong analogy between SRS and two-level superfluorescence is found; in both cases the macroscopic, collective dipole moment is initiated by quantum fluctuations.
Strong interference of the first Stokes Raman radiation with the stimulated Brillouin radiation has been observed in an off-axis resonator: Brillouin gain decreases appreciably by the presence of the Raman radiation in a medium such as CS2, in which the Raman and Brillouin scatterings can be stimulated simultaneously. Such a type of interference can be explained as the Raman scattering of the Brillouin photons stimulated by the first Stokes Raman photons. A classical theory has shown a quantitative agreement with the experiment. The anti-Stokes Brillouin components have not been observed even though the laser excitation was intense enough to allow the observation of the multiple Stokes components which were excited directly in the off-axis resonator without the feed-back effect of the laser as an amplifier.
A simple description of the stimulated Raman emission of Stokes radiation, and of the induced absorption found by Stoicheff at the anti-Stokes frequency, is based on the semiclassical theory of radiation. This theory gives the well-known formulas for absorption and stimulated emission when carried to the first order in the time-dependent perturbation due to the interaction of the light beam with the molecules. If there are two light beams of frequencies ωL and ω, the second-order perturbation may be associated with stimulated emission at the Stokes frequency ω=ω−1=ωL—ωR and with induced absorption at the anti-Stokes frequency ω=ω1=ωL+ωR, where ωL is the laser frequency and ωR a Raman-active vibrational frequency; in both cases, there is a molecular transition to the excited vibrational state. The highly directional and very sharp anti-Stokes emission at ω1 arises from the conversion by molecules in intense beams at ωL and ω−1 of two ωL photons into an ω−1, ω1 pair. Like spontaneous emission from excited states, the production of this anti-Stokes radiation requires the full quantum theory of radiation for an adequate description. The emission of ω1 and the observed directional absorption of ω−1 due to the reverse conversion ω−1+ω1→2ωL are explained. The anti-Stokes emission is proportional to the Stokes intensity and to the square of that of the laser, and to the square of the molecular number density; its sharpness is due to the absence of a molecular transition. The generation of photon pairs ω−n+ωn, where ωn=ωL+nωR, can occur for all values of n, but it is most efficient for n=1, due to resonance in the appropriate molecular hyperpolarizability and to the intensity of the Stokes beam at ω−1.
The stimulated Raman scattering (SRS), arising from the interaction between molecules of a medium and two strong light fields - a resonant field and a pumping field - is studied theoretically. The transitions of molecules between a given pair of levels is assumed to be both Raman active and infrared active. A semiclassical approach, in which the density matrix method is used for the quantum mechanical description for the medium, is employed. We expand the density matrix in various orders within which the zeroth-order is referred to as the solution of the equation of motion of the density matrix when only the resonant field is applied. -- We obtain a set of equations in second order of the density matrix element p₁₀⑵ (ω) from which the gain function can be derived. This set of equations in the special case of steady state is accidentally similar to those obtained by Mollow in his study of the power spectrum of a strong driven two-level system. We have shown that the SRS in the steady state under the resonant field interaction, is drastically reduced in general. When the flopping frequency is larger than the relaxation constants (Ω > r), we find there appears both weak positive and weak negative gains on both sides of the ordinary SRS components. Since we retain our second-order equations in a general form rather than the form for steady state, we are able to take account of the dependence on the transient zeroth solution. By employing the Laplace transform technique with the help of convolution products, we are able to solve for ρ₁₀⑵(ω) in the stationary flopping state. When the intensity of the resonant field is high, we have found theoretically that both the Stokes and anti-Stokes component become doublets and equally displaced by the amount of the flopping frequency, Ω, on the opposite sides of ordinary Stokes and anti-Stokes components, respectively. The gain maxima are approximately the order of Ω/4r times the ordinary gain of the Stokes component, where Ω/r >> 1 if the flopping exists. The gains are therefore large in comparison with the ordinary gain for SRS in the steady state.
Recent adaptive feedback control experiments demonstrated control of transient (i.e. nonimpulsive) Stokes emission from two closely spaced Raman-active modes in liquid methanol [e.g., B. J. Pearson et al., Phys. Rev. A 63, 063412 (2001)]. Optimally shaped pulses were found that selectively excited one of the two Stokes lines alone, optimized emission from both modes together, or completely suppressed all Stokes emission. Here, two general control mechanisms capable of affecting the ratio of intensities of the Stokes lines are identified. The first is operational when the duration of the pump pulse (tp)({t}_{p})(tp​) is on the order of the collisional dephasing time (td)({t}_{d})(td​). The ratio of the peak heights of the two Stokes lines can then be controlled by simply varying the duration and/or intensity of the pump pulse. The second operates when 1∕tp1∕{t}_{p}1∕tp​ is on the order of the energy separation of the two Raman modes, and hence when the two Raman modes are coupled due to overlapping nonlinear polarizations that drive the stimulated Raman scattering. In this regime, asymmetry in the spectral amplitudes within the pump pulse can control the asymmetry in the peak heights of the Stokes emission. Both these mechanisms have the same clear physical interpretation: shaping the pump pulse controls the nonlinear optical response of the medium, which in turn controls the stimulated Stokes emission, itself a ${\ensuremath{\chi}}^{(3)}$ nonlinear effect. In neither mechanism does the ratio of peak heights in the Stokes spectrum reflect directly the ratio of excited-state populations associated with the two Raman modes, as was assumed in the experiments, nor does the control involve coherent quantum interference effects.
A theory of nonstationary stimulated Raman scattering is developed, allowing for the coupling between the Stokes and anti-Stokes waves. The main equations are solved in the asymptotic approximation, which is valid for large values of the gain. The dependences of the gain on the phase mismatch Δ k , pulse duration τ p , and cell length l are discussed and the optimal angle for the observation of the anti-Stokes rings is calculated. Moreover, the spectral distributions of the Stokes and anti-Stokes waves are considered.