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Optics Letters

Theory of two-photon Doppler-broadened probe absorption

Barbara A. Capron, A. S. Marathay, M. Sargent

Published Feb 1, 1986• 2 citations
Publisher

Abstract

We generalize the two-photon probe-absorption coefficient to include any amount of Doppler broadening and an arbitrarily intense saturator wave. By analytically performing integrals over a Lorentzian velocity distribution we are able to examine the transition from homogeneous broadening to extreme Doppler broadening for both zero and nonzero Stark shifts. For Doppler widths much larger than the saturator Rabi flopping frequency, the probe-absorption spectrum approaches a Doppler-free Lorentzian with width equal to the homogeneous linewidth and center displaced by the dynamic Stark shift.

DOI
10.1364/ol.11.000070

Related papers

Theory of two-photon Doppler-free saturation spectroscopy

A. S. Marathay, Barbara A. Capron, M. Sargent

Reference 1 derives the two-photon probe-absorption coefficient for a homogeneously broadened medium subjected to an arbitrarily intense saturator wave. We have generalized this theory to allow for Doppler broadening with a counterrunning probe and saturator waves. We have carried out integrals over a Lorentzian Doppler distribution analytically and have specialized to a number of limits, such as extreme Doppler broadening. In this limit, we find the simple probe absorption coefficient α 1 ≃ 2 α 0 ( γ D 1 I 2 − i ω s T 1 ) , where α0 is the unsaturated two-photon absorption coefficient, I2 is the saturator intensity, γ is the two-photon coherence decay rate constant, ω s is the Stark shift parameter, T1 is the population difference decay time, and D 1 = 1 / [ γ+ i ( ω+ω s I 2 − ν 2 − ν 1 ) ] . For vanishing Stark shifts (ω s = 0), this is the standard two-photon probe absorption coefficient, but for large ω s , the complex Lorentzian D 1 becomes detuned. In the homogeneously broadened limit, the absorption spectrum resembles the one-photon case. In this paper we show the transition between these two limits.

Annual Meeting Optical Society of America•Jan 1, 1985•0 citations

Two-photon spectroscopy. II. Effects of residual Doppler broadening

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Two-photon transitions have provided the basis for numerous useful nonlinear interactions with radiation. In this paper we calculate the absorption and coupling coefficients for one-, two-, and three-wave mixing in two-photon homogeneously broadened media. The resulting field equations form the basis for understanding saturation spectroscopy, phase conjugation,1 and single-mode instabilities2 in lasers and optical bistability in two-photon media. We assume that the wave frequencies are not resonant with any intermediate state j and that the probe and conjugate waves cannot saturate the two-photon transition. This allows us to use the two-photon two-level model and to transfer much of our knowledge of the one-photon two-level systems to the two-photon case. Two major differences occur between the two models. First, dynamic Stark shifts of the active levels play an important role in generating asymmetric spectra and nonzero dispersion (≃ω s T1) at line center. Second, high pump intensities do not bleach the absorption coefficient at the Rabi sidebands.

Annual Meeting Optical Society of America•Jan 1, 1985•0 citations

TWO-PHOTON MULTIWAVE MIXING (DOPPLER-FREE SPECTROSCOPY).

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This dissertation examines aspects of the interaction of multiple coherent light fields for the two-photon two-level model. In this model the interacting energy levels are not connected by an atomic dipole and a two-photon transition between them is necessary. We employ the density matrix formalism allowing easy comparison between the one- and two-photon two-level models. Significant differences are found due to dynamic Stark shifts and conjugate scattering off the pump-induced two-photon coherence. Averages over Doppler broadening are performed and the new upper-level relaxation mechanisms of decay to an intermediate nonresonant level and ionization from the upper state are included. The new relaxation mechanisms, introduced to the theory to better model experiments, are similar except that ionization is intensity dependent. They cause the resulting probe absorption spectra to become more complex and in general asymmetric. Doppler broadening is also important in experiments using gases. We analytically average over a Lorentzian velocity distribution for both co- and counterpropagating pump and probe beams. For copropagating fields the results are similar to those for the one-photon case averaged over inhomogeneous broadening, whereas counterpropagating pump and probe fields yield the so-called Doppler-free configuration that is normally only modelled to third order in the pump amplitude. We consider the pump field amplitude to all orders and find that as long as the width of the Doppler velocity distribution is significantly larger than the two-photon Rabi frequency the results are Doppler-free. The final part of the dissertation treats the question of two-photon squeezed states. This requires quantized sidemodes. Squeezed states are minimum uncertainty states with unequal variances in the two quadratures of the electromagnetic field amplitude. One way to generate these states is via multiwave mixing and we present here the first calculation for nondegenerate two-photon multiwave mixing as it applies to squeezed states. We find that in general two-photon squeezed states require lower intensities and detuning than those predicted by the one-photon model.

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Nondegenerate two-photon phase conjugation

Barbara A. Capron, Murray Sargent

Our semiclassical theory of two-photon multiwave mixing1 is a generalization of the degenerate four-wave mixing case2 to treat nondegenerate pump/signal/conjugate frequencies. We apply that theory to four-wave mixing in two-photon media. We present explicit formulas for the nondegenerate signal/conjugate absorption and coupling coefficients for cases for which the two-photon Stark shifts are sufficiently small compared to the homogeneous linewidths. More general cases require that the spatial hole burning averages be performed numerically. We find intricate dependencies of the transmission and reflection spectra on the pump intensity and on dynamic Stark shifts. In particular, the reflection coefficients do not necessarily bleach to zero as the pump intensity is increased indefinitely, and interferences occur between pure index and mixed absorption/index contributions

Annual Meeting Optical Society of America•Jan 1, 1987•1 citation
We generalize the two-photon probe-absorption coefficient to include any amount of Doppler broadening and an arbitrarily intense saturator wave. By analytically performing integrals over a Lorentzian velocity distribution we are able to examine the transition from homogeneous broadening to extreme Doppler broadening for both zero and nonzero Stark shifts. For Doppler widths much larger than the saturator Rabi flopping frequency, the probe-absorption spectrum approaches a Doppler-free Lorentzian with width equal to the homogeneous linewidth and center displaced by the dynamic Stark shift.
Reference 1 derives the two-photon probe-absorption coefficient for a homogeneously broadened medium subjected to an arbitrarily intense saturator wave. We have generalized this theory to allow for Doppler broadening with a counterrunning probe and saturator waves. We have carried out integrals over a Lorentzian Doppler distribution analytically and have specialized to a number of limits, such as extreme Doppler broadening. In this limit, we find the simple probe absorption coefficient α 1 ≃ 2 α 0 ( γ D 1 I 2 − i ω s T 1 ) , where α0 is the unsaturated two-photon absorption coefficient, I2 is the saturator intensity, γ is the two-photon coherence decay rate constant, ω s is the Stark shift parameter, T1 is the population difference decay time, and D 1 = 1 / [ γ+ i ( ω+ω s I 2 − ν 2 − ν 1 ) ] . For vanishing Stark shifts (ω s = 0), this is the standard two-photon probe absorption coefficient, but for large ω s , the complex Lorentzian D 1 becomes detuned. In the homogeneously broadened limit, the absorption spectrum resembles the one-photon case. In this paper we show the transition between these two limits.
For pt.I see ibid., vol.8, p.1795 (1975) where the theory of Doppler-free two-photon transitions was discussed for the case of one single laser frequency. Here, the effects of two different frequencies are considered. When the residual Doppler width of the two-photon transition is less than the natural linewidth, the previous results are essentially regained. For large residual Doppler shifts, the absorption profile is a power-broadened Lorentzian if the frequency of the probe field is larger than the frequency of the strong field. In the opposite case, the absorption peak splits because of the AC Stark effect. The high-intensity distortions are minimized when equal frequencies can be used. Power-dependent shifts do not ordinarily occur with a resonantly tuned intermediate level. Some modifications of the spectra are introduced by phase-disturbing collisions.
Two-photon transitions have provided the basis for numerous useful nonlinear interactions with radiation. In this paper we calculate the absorption and coupling coefficients for one-, two-, and three-wave mixing in two-photon homogeneously broadened media. The resulting field equations form the basis for understanding saturation spectroscopy, phase conjugation,1 and single-mode instabilities2 in lasers and optical bistability in two-photon media. We assume that the wave frequencies are not resonant with any intermediate state j and that the probe and conjugate waves cannot saturate the two-photon transition. This allows us to use the two-photon two-level model and to transfer much of our knowledge of the one-photon two-level systems to the two-photon case. Two major differences occur between the two models. First, dynamic Stark shifts of the active levels play an important role in generating asymmetric spectra and nonzero dispersion (≃ω s T1) at line center. Second, high pump intensities do not bleach the absorption coefficient at the Rabi sidebands.
This dissertation examines aspects of the interaction of multiple coherent light fields for the two-photon two-level model. In this model the interacting energy levels are not connected by an atomic dipole and a two-photon transition between them is necessary. We employ the density matrix formalism allowing easy comparison between the one- and two-photon two-level models. Significant differences are found due to dynamic Stark shifts and conjugate scattering off the pump-induced two-photon coherence. Averages over Doppler broadening are performed and the new upper-level relaxation mechanisms of decay to an intermediate nonresonant level and ionization from the upper state are included. The new relaxation mechanisms, introduced to the theory to better model experiments, are similar except that ionization is intensity dependent. They cause the resulting probe absorption spectra to become more complex and in general asymmetric. Doppler broadening is also important in experiments using gases. We analytically average over a Lorentzian velocity distribution for both co- and counterpropagating pump and probe beams. For copropagating fields the results are similar to those for the one-photon case averaged over inhomogeneous broadening, whereas counterpropagating pump and probe fields yield the so-called Doppler-free configuration that is normally only modelled to third order in the pump amplitude. We consider the pump field amplitude to all orders and find that as long as the width of the Doppler velocity distribution is significantly larger than the two-photon Rabi frequency the results are Doppler-free. The final part of the dissertation treats the question of two-photon squeezed states. This requires quantized sidemodes. Squeezed states are minimum uncertainty states with unequal variances in the two quadratures of the electromagnetic field amplitude. One way to generate these states is via multiwave mixing and we present here the first calculation for nondegenerate two-photon multiwave mixing as it applies to squeezed states. We find that in general two-photon squeezed states require lower intensities and detuning than those predicted by the one-photon model.
We describe the influence of the intensity variation of the light in a standing wave on the lineshape of a Doppler-free multi-photon transition. We show that as long as the light shifts and the saturation are small compared to the Doppler width, a motional narrowing effect occurs and the line remains Lorentzian, the centre being shifted by an amount equal to the mean value of the dynamic Stark effect. We consider also the case of transients. We show that at resonance, it is possible to observe different responses according to the light intensity. In particular, it is possible to observe Rabi oscillations only if the Rabi frequency is larger than the natural width and smaller than the Doppler width of the transition.
Our semiclassical theory of two-photon multiwave mixing1 is a generalization of the degenerate four-wave mixing case2 to treat nondegenerate pump/signal/conjugate frequencies. We apply that theory to four-wave mixing in two-photon media. We present explicit formulas for the nondegenerate signal/conjugate absorption and coupling coefficients for cases for which the two-photon Stark shifts are sufficiently small compared to the homogeneous linewidths. More general cases require that the spatial hole burning averages be performed numerically. We find intricate dependencies of the transmission and reflection spectra on the pump intensity and on dynamic Stark shifts. In particular, the reflection coefficients do not necessarily bleach to zero as the pump intensity is increased indefinitely, and interferences occur between pure index and mixed absorption/index contributions