Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Sideband interactions in homogeneously broadened saturable absorbers

H. L. Fragnito, E. Palange, C. H. Brito Cruz, F. DeMartini

Published May 1, 1984• 4 citations
Publisher

Abstract

A theoretical study of amplitude (AM) and frequency (FM) modulation components of an optical wave interacting resonantly with a homogeneously broadened two-level system is presented. It is shown that the AM component is usually less attenuated than the FM component; however, under conditions which depend on the modulation frequency, relaxation times, and the saturation parameter, the reverse may occur. In strongly saturated media the AM component can be amplified, a process limited by the depletion of the carrier intensity as the wave propagates in the absorber. The two- and three-wave cases of unidirectional saturation spectroscopy are considered and the effects of depletion are discussed. In the two-wave (single-probe) case a third wave is generated which substantially modifies the absorption profile for the probe. An analysis is included of the sideband evolution in a laser with an intracavity saturable absorber. By analyzing the problem in the frequency domain, some new insights concerning the role of the saturable absorber in passively mode-locked lasers are presented.

DOI
10.1103/physreva.29.2716

Related papers

Propagation of modulated optical fields through saturable-absorbing media: a general theory of modulation spectroscopy

Mark A. Kramer, Robert W. Boyd, Lloyd W. Hillman +1 more

The propagation of a weakly modulated light beam through a nonlinear material is treated. The optical field representing such a beam consists of a strong carrier-frequency component and two weak, symmetrically displaced sidebands that combine to form a field, which may be purely amplitude modulated (AM), frequency modulated (FM), or some combination of these two modulation forms. It is shown that, for any optical nonlinearity, two modulation forms exist that have the property that the form of modulation is invariant under propagation of the beam. If a modulation form other than one of these natural modes is injected into the nonlinear medium, the modulation form will change as the beam propagates, asymptotically approaching the natural mode that experiences the lower attenuation. Explicit expressions for these natural modes are presented for the case in which the nonlinear medium can be modeled as a collection of two-level atoms. For the special case of an on-resonance pump beam, the natural modes correspond to pure amplitude modulation and pure frequency modulation. This formalism provides a general description of saturation spectroscopy for both AM and FM fields. Formulas are derived for the rate at which a pure FM beam is converted to an AM beam owing to its interaction with a two-level atomic medium. We also consider the variation that is due to propagation of the depth of modulation of an AM wave interacting resonantly with two-level atoms. The formalism predicts the existence of spectral features whose shape depends sensitively on the internal relaxation processes of the material. Experimental spectra are presented for ruby, alexandrite, and fluorescein in glass and are interpreted.

Journal of the Optical Society of America B•Sep 1, 1985•22 citations

Absorption and depletion effects on degenerate four-wave mixing in homogeneously broadened absorbers

Wilbur P. Brown

The coupled equations satisfied by the pump, probe, and signal waves in a four-wave interaction in a two-level saturable absorber are derived, taking into account the effects of absorption and depletion. Numerical solutions of these equations are presented in two limits; one in which the probe-wave amplitude is small compared with that of the pump waves and another in which the probe- and pump-wave amplitudes are comparable. The results indicate that absorption effects are important when the wave amplitudes are less than the saturation amplitude of the absorption process. The importance of the depletion effects depends on the type of four-wave interaction. The most significant effects occur when the undepleted theory predicts probe-wave reflectivities exceeding unity.

Journal of the Optical Society of America•May 1, 1983•50 citations

Theory of two-photon multiwave mixing

Shlomo Ovadia, Murray Sargent, Mao Hong Lu

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

Theory of optical heterodyne three-level saturation spectroscopy via collinear non-degenerate four-wave mixing in coupled Doppler-broadened transitions

S. Le Boiteux, Daniel Bloch, M. Ducloy

We present a theoretical analysis of high-frequency optical heterodyne saturation spectroscopy in Doppler-broadened coupled three-level systems. The saturating beam, which is assumed to be amplitude modulated at frequency δ (double-sideband suppressed carrier amplitude modulation) is resonant for one of the transitions. Through non-degenerate four-wave mixing processes, this modulation is transferred to a probe beam, resonant for the coupled transition. Saturating and probe fields may be either co-propagating or counter-propagating. The lineshape of the probe modulation is analysed via a third-order perturbation expansion of the atomic density matrix with respect to the incident field amplitudes. The resulting signal is integrated over velocities, in the Doppler limit approximation. Both population saturation effects and coherent (Raman-type, or two-photon) processes contribute to the signal. The various contributions appear as Lorentzian-type resonance doublets. We show that, in the absence of relaxation processes (collisional dephasing, or radiative cascades), destructive interferences between population saturation and coherent two-photon processes are responsible for the disappearance of one resonance doublet. Phase-interrupting collisions are thus predicted to lead to the existence of a « pressure-induced extra-resonance » (PIER) doublet, which could yield information on collisional processes in the impact regime. The properties of the predicted PIER doublets are analysed in relation with other types of PIER signals studied in four-wave mixing, and in time-resolved saturation spectroscopy.

Journal de physique•Jan 1, 1986•10 citations

Theory of forward degenerate four-wave mixing in two-level saturable absorbers

Dai‐Hyuk Yu, Jai-Hyung Lee, Joon-Sung Chang +1 more

A theoretical description for a forward degenerate four-wave mixing process in homogeneously broadened two-level absorbers with arbitrary intensities of all four beams is presented. Assuming no pump absorption and depletion, we obtain an analytical solution of forward degenerate four-wave mixing signal intensity in the limit of the strong pumps and weak probe and signal. In the case of low absorption or large detuning, this solution is shown to become equivalent to that of the phase-conjugate degenerate four-wave mixing derived by Abrams and Lind [Opt. Lett.2, 94 (1978)]. For arbitrary beam intensities and absorption parameters, the signal intensity is numerically calculated by solution of the coupled equations of complex wave amplitudes. Comparing the results of the numerical calculation with those of the analytical solution, we discuss the validity of using the analytical solution in practical experiments and show the saturation behavior that is due to the strong probe beam. The line shapes under the various ratios among the input beams and absorption parameters are also obtained and discussed.

Journal of the Optical Society of America B•Aug 1, 1999•11 citations

WAVE INTERACTIONS IN SATURABLE ABSORBERS

S. E. Schwarz, Teng Yong Tan

The interaction between waves in an idealized saturable absorber is discussed. It is shown that the presence of a strong wave will cause the line to appear to be ``hole-burned'' when probed by a second wave, even though the broadening mechanisms are conventionally homogeneous. This effect would be expected to work against the tendency of dye-switched ruby lasers to operate in a single mode. Related effects should play a role in optical pulse generators and in other devices using saturable absorbers.

Applied Physics Letters•Jan 1, 1967•116 citations
A theoretical study of amplitude (AM) and frequency (FM) modulation components of an optical wave interacting resonantly with a homogeneously broadened two-level system is presented. It is shown that the AM component is usually less attenuated than the FM component; however, under conditions which depend on the modulation frequency, relaxation times, and the saturation parameter, the reverse may occur. In strongly saturated media the AM component can be amplified, a process limited by the depletion of the carrier intensity as the wave propagates in the absorber. The two- and three-wave cases of unidirectional saturation spectroscopy are considered and the effects of depletion are discussed. In the two-wave (single-probe) case a third wave is generated which substantially modifies the absorption profile for the probe. An analysis is included of the sideband evolution in a laser with an intracavity saturable absorber. By analyzing the problem in the frequency domain, some new insights concerning the role of the saturable absorber in passively mode-locked lasers are presented.
The propagation of a weakly modulated light beam through a nonlinear material is treated. The optical field representing such a beam consists of a strong carrier-frequency component and two weak, symmetrically displaced sidebands that combine to form a field, which may be purely amplitude modulated (AM), frequency modulated (FM), or some combination of these two modulation forms. It is shown that, for any optical nonlinearity, two modulation forms exist that have the property that the form of modulation is invariant under propagation of the beam. If a modulation form other than one of these natural modes is injected into the nonlinear medium, the modulation form will change as the beam propagates, asymptotically approaching the natural mode that experiences the lower attenuation. Explicit expressions for these natural modes are presented for the case in which the nonlinear medium can be modeled as a collection of two-level atoms. For the special case of an on-resonance pump beam, the natural modes correspond to pure amplitude modulation and pure frequency modulation. This formalism provides a general description of saturation spectroscopy for both AM and FM fields. Formulas are derived for the rate at which a pure FM beam is converted to an AM beam owing to its interaction with a two-level atomic medium. We also consider the variation that is due to propagation of the depth of modulation of an AM wave interacting resonantly with two-level atoms. The formalism predicts the existence of spectral features whose shape depends sensitively on the internal relaxation processes of the material. Experimental spectra are presented for ruby, alexandrite, and fluorescein in glass and are interpreted.
The coupled equations satisfied by the pump, probe, and signal waves in a four-wave interaction in a two-level saturable absorber are derived, taking into account the effects of absorption and depletion. Numerical solutions of these equations are presented in two limits; one in which the probe-wave amplitude is small compared with that of the pump waves and another in which the probe- and pump-wave amplitudes are comparable. The results indicate that absorption effects are important when the wave amplitudes are less than the saturation amplitude of the absorption process. The importance of the depletion effects depends on the type of four-wave interaction. The most significant effects occur when the undepleted theory predicts probe-wave reflectivities exceeding unity.
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.
We present a theoretical analysis of high-frequency optical heterodyne saturation spectroscopy in Doppler-broadened coupled three-level systems. The saturating beam, which is assumed to be amplitude modulated at frequency δ (double-sideband suppressed carrier amplitude modulation) is resonant for one of the transitions. Through non-degenerate four-wave mixing processes, this modulation is transferred to a probe beam, resonant for the coupled transition. Saturating and probe fields may be either co-propagating or counter-propagating. The lineshape of the probe modulation is analysed via a third-order perturbation expansion of the atomic density matrix with respect to the incident field amplitudes. The resulting signal is integrated over velocities, in the Doppler limit approximation. Both population saturation effects and coherent (Raman-type, or two-photon) processes contribute to the signal. The various contributions appear as Lorentzian-type resonance doublets. We show that, in the absence of relaxation processes (collisional dephasing, or radiative cascades), destructive interferences between population saturation and coherent two-photon processes are responsible for the disappearance of one resonance doublet. Phase-interrupting collisions are thus predicted to lead to the existence of a « pressure-induced extra-resonance » (PIER) doublet, which could yield information on collisional processes in the impact regime. The properties of the predicted PIER doublets are analysed in relation with other types of PIER signals studied in four-wave mixing, and in time-resolved saturation spectroscopy.
A theoretical description for a forward degenerate four-wave mixing process in homogeneously broadened two-level absorbers with arbitrary intensities of all four beams is presented. Assuming no pump absorption and depletion, we obtain an analytical solution of forward degenerate four-wave mixing signal intensity in the limit of the strong pumps and weak probe and signal. In the case of low absorption or large detuning, this solution is shown to become equivalent to that of the phase-conjugate degenerate four-wave mixing derived by Abrams and Lind [Opt. Lett.2, 94 (1978)]. For arbitrary beam intensities and absorption parameters, the signal intensity is numerically calculated by solution of the coupled equations of complex wave amplitudes. Comparing the results of the numerical calculation with those of the analytical solution, we discuss the validity of using the analytical solution in practical experiments and show the saturation behavior that is due to the strong probe beam. The line shapes under the various ratios among the input beams and absorption parameters are also obtained and discussed.
The interaction between waves in an idealized saturable absorber is discussed. It is shown that the presence of a strong wave will cause the line to appear to be ``hole-burned'' when probed by a second wave, even though the broadening mechanisms are conventionally homogeneous. This effect would be expected to work against the tendency of dye-switched ruby lasers to operate in a single mode. Related effects should play a role in optical pulse generators and in other devices using saturable absorbers.