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Journal of Physics B Atomic and Molecular Physics

Saturation of optical susceptibilities in strong amplitude-modulated fields

G. S. Agarwal, N. Nayak

Published Oct 28, 1986• 32 citations
Publisher

Abstract

Optical non-linearities and spectroscopy in fields that are strongly modulated and also intense are examined by obtaining the solution of the optical Bloch equations in modulated fields as a matrix continued fraction. Numerical results for emission and absorption in such fields are presented. The saturation optical susceptibilities and the different types of spectra show resonances at the Rabi frequency if the modulation index is small. With increase in this index, additional resonant features appear at submultiples of the Rabi frequency. These additional features become more and more prominent as the modulation index increases further. For modulated fields on resonance, a dressed-atom formulation is developed to explain the results obtained from the numerical evaluation of the continued fraction. The results are also applicable to non-linear spectroscopic measurements such as Raman gain when the probe field is modulated.

DOI
10.1088/0022-3700/19/20/022

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International Science and Technology Journal•Oct 5, 2024•0 citations
Optical non-linearities and spectroscopy in fields that are strongly modulated and also intense are examined by obtaining the solution of the optical Bloch equations in modulated fields as a matrix continued fraction. Numerical results for emission and absorption in such fields are presented. The saturation optical susceptibilities and the different types of spectra show resonances at the Rabi frequency if the modulation index is small. With increase in this index, additional resonant features appear at submultiples of the Rabi frequency. These additional features become more and more prominent as the modulation index increases further. For modulated fields on resonance, a dressed-atom formulation is developed to explain the results obtained from the numerical evaluation of the continued fraction. The results are also applicable to non-linear spectroscopic measurements such as Raman gain when the probe field is modulated.
The dynamical behavior of a two-level system in an intense frequency-modulated field is obtained without recourse to the weak-modulation assumption. Solutions to the optical Bloch equations are expressed in terms of continued fractions involving 2\ifmmode\times\else\texttimes\fi{}2 matrices. Such matrix-continued fractions are numerically evaluated to obtain a wide variety of absorption and fluorescence spectra in frequency-modulated fields. Results in various special cases involving combinations of weak fields, weak modulation, strong modulation, and strong fields are given. Analytical results for weak fields but with an arbitrary degree of modulation are also presented. The spectra exhibit additional resonances for strong modulation. The main resonance shifts and broadens considerably due to strong modulation and saturation effects.
A nonlinear response theory of a quantum-mechanical system undergoing arbitrary relaxation and interacting with fields, some of which may be strong enough to saturate optical transitions, is developed. Explicit expressions for second-order and third-order susceptibilities are obtained. If the fields are weak, then these expressions show the existence of additional resonant contributions to ${\ensuremath{\chi}}^{(n)}$ which arise due to inelastic collisions. Various applications of these ${\ensuremath{\chi}}^{(n)}$'s to modulation spectroscopy, four-wave mixing, and pump-probe experiments are discussed. The general structure suggests how the additional resonances can be used to determine inelastic rates in a Doppler-broadened medium. For saturating fields, ${\ensuremath{\chi}}^{(n)}$'s become dependent on the intensity of such fields and can be formally obtained from weak-field ${\ensuremath{\chi}}^{(n)}$'s, provided proper identification of unperturbed eigenfunctions, eigenvalues, and relaxation times is made. Such intensity-dependent ${\ensuremath{\chi}}^{(n)}$'s have resonant denominators which lead to resonances at Rabi frequencies and submultiples of these frequencies, the widths of which are also dependent on the strength of the field. In the special cases intensity-dependent ${\ensuremath{\chi}}^{(1)}$ agrees with the work of Cohen-Tannoudji and co-workers.
The paper reports investigations of refractive index enhancement (RIE) with zero absorption in a three-level Λ-type atom subject to a drive field, a saturation field and a probe field. As RIEs lie in the alternate points between emission and absorption peaks, one can predict their positions by employing dressed state theory and optical Bloch equations. The numerical calculations with dressed state analysis indicate that high refractive index enhancement appears mainly at two conditions: (i) off-resonance detuning of drive and saturation frequencies have the opposite sign; (ii) the saturation Rabi frequency is larger than the drive Rabi frequency. The analysis can be extended to multi-field multi-level atomic systems.
We have investigated the interaction between bichromatic optical fields and atomic transitions. We have considered both the interaction of a saturating bichromatic field in the form of a 100% amplitude-modulated field resonantly interacting with a single atomic transition as well as the interaction of a pair of 100% amplitude-modulated fields with two coupled atomic transitions. In our studies of a single atomic transition resonantly excited by a 100% amplitude-modulated field we have used a fully quantum-electrodynamical analysis to model the Autler-Townes effect induced by a bichromatic optical field. We find that the effect induced by a bichromatic field is markedly different from the effect produced by a monochromatic exciting field. We have derived analytic expressions for the bichromatic Autler-Townes absorption spectrum. The absorption spectrum consists of equally spaced lines spaced by the modulation frequency. The linewidths of neighboring lines alternate as functions of the field strength. Experimentally, we observed this spectrum for both resonant and detuned 100% amplitude-modulated driving fields by monitoring the absorption of a weak monochromatic field as it was tuned about the resonance frequency of the 3²P₃/₂(F = 3, mF = 3) ↔ 4²D₅/₂(F = 4, mF = 4) transition while driving the 3²S₁/₂(F = 2, mF = 2) ↔ 3²P₃/₂(F = 3, mF = 3) transition with a saturating 100% amplitude-modulated field. We found excellent agreement between our predicted and measured spectra. We then investigated the resonant interaction between a pair of 100% amplitude-modulated fields and two atomic transitions sharing a common atomic level. We considered both a three-level atomic cascade system and a three-level atomic lambda system. We examined the effect that the relative phase between the two modulations has on the rate at which the modulated fields are absorbed. We find that the saturated time-averaged field absorption exhibits parametric resonances in both the cascade and lambda systems. The time-averaged absorption is strongly dependent upon the relative phase of modulation between the two fields and for certain phase differences the time-averaged absorption can be enhanced over the maximum saturated absorption experienced by cw fields of the same time-averaged intensities. We also find that the stationary atomic response exhibits periodic oscillations and for the cascade system periodic two-photon inversions of up to 40% can be achieved. We further find dark absorption resonances as a function of phase in the lambda system. We analyze the results from the point of view of semi-classical adiabatic dressed states.
The effect of changing the value of Rabi frequency has been studied in six level atomic system, Optical Bloch equations where used as the main tool to get the final shape of the model to treat the model. In this work we present a theoretical study of the non-linear optical properties of materials as a result of their interaction with laser radiation in resonance state. Density matrix was used in our formulation of Optical Bloch Equation. These equations were solved using Matlap program for Six-levels Inverted-Y Atomic Systems. A formula for calculation of electrical susceptibility was obtained. With increasing the real and imaginary parts of the electric susceptibility have also increased gradually which effects the index of refraction of the medium (real part), and the absorption window (described by the imaginary part). Moreover, when increasing the real and imaginary parts of the electric susceptibility decreased and we noticed that with increasing the value of the signal laser field related to transition |2⟩ ↔ |6⟩a distortion starts to appear on both real and imaginary parts. Key Words: Rabi frequency, Density matrix, electrical susceptibility, laser, Optical Bloch equations.