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Physical Review A

Extraresonances in degenerate four-wave mixing induced by sequential decay

Harry Friedmann, A. D. Wilson‐Gordon, M. Rosenbluh

Published Mar 1, 1986• 14 citations
Publisher

Abstract

Collisionless extraresonances (ER) in degenerate four-wave mixing (DFWM) are predicted when the upper state \ensuremath{\Vert}b〉 of the \ensuremath{\Vert}a〉\ensuremath{\rightarrow}\ensuremath{\Vert}b〉 transition, excited by near-resonant incident laser beams, decays to the lower state \ensuremath{\Vert}a〉 both directly (\ensuremath{\Vert}b〉\ensuremath{\rightarrow}\ensuremath{\Vert}a〉) and via one or two intermediate states: \ensuremath{\Vert}b〉\ensuremath{\rightleftarrows}\ensuremath{\Vert}d〉\ensuremath{\rightarrow}\ensuremath{\Vert}a〉 or \ensuremath{\Vert}b〉\ensuremath{\rightarrow}\ensuremath{\Vert}c〉\ensuremath{\rightarrow}\ensuremath{\Vert}d〉\ensuremath{\rightarrow}\ensuremath{\Vert}a〉. The width of the new ER, which depends on the relative decay rates in the cascade process, may be very narrow leading to very strong DFWM signals. Dephasing collisions affect the intensity of the new ER and also produce the usual, pressure-induced ER.

DOI
10.1103/physreva.33.1783

Related papers

Enhancement of near-resonant degenerate four-wave-mixing signals by excitation of a coupled transition

A. D. Wilson‐Gordon, Harry Friedmann

It is shown that intense collisionless Rayleigh-type extraresonant signals in degenerate four-wave mixing (DFWM) can be obtained when the $| a 〉\ensuremath{\rightarrow}| b 〉$ transition which produces the DFWM signal is coupled to another transition ($ | a 〉\ensuremath{\rightarrow}| d 〉$) which is excited near-resonantly by an additional laser beam. Two effects contribute to the Rayleigh-type extraresonant signal at ${\ensuremath{\omega}}_{2}={\ensuremath{\omega}}_{1}$. The first effect is resonance-enhanced three-photon scattering (TPS), which leads to the population of state $| b 〉$ and consequently contributes a term in ${\ensuremath{\rho}}_{\mathrm{ba}}$, the off-diagonal density-matrix element, which is characterized by the linewidth ${(\frac{1}{{T}_{1}})}_{\mathrm{ba}}$, the inverse lifetime of the state $| b 〉$. The second effect derives from the effective lifetime acquired by the ground state $| a 〉$ due to its strong coupling to state $| d 〉$. This leads to a term in ${\ensuremath{\rho}}_{\mathrm{ba}}$ characterized by a linewidth $\frac{1}{2}{(\frac{1}{{T}_{1}})}_{\mathrm{da}}$, which is half the inverse lifetime of the state $| d 〉$. When ${(\frac{1}{{T}_{1}})}_{\mathrm{da}}\ensuremath{\gg}{(\frac{1}{{T}_{1}})}_{\mathrm{ba}}$, the TPS effect dominates and the extraresonant DFWM signal is narrow and intense. When ${(\frac{1}{{T}_{1}})}_{\mathrm{da}}\ensuremath{\simeq}{(\frac{1}{{T}_{1}})}_{\mathrm{ba}}$, the signal is less intense and interference between the two effects is evident.

Physical Review A•Sep 1, 1984•5 citations

Degenerate four-wave mixing spectroscopy with short-pulse lasers: theoretical analysis

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The use of picosecond lasers for degenerate four-wave mixing (DFWM) diagnostics of atmospheric pressure flames has been suggested as a means of overcoming the complicated collision-rate dependence of the DFWM signal in the nanosecond-laser pulse-length regime. In this paper DFWM spectroscopy in the regime in which the laser pulse length (τL) is less than the characteristic collisional time (τC) is investigated theoretically. DFWM signal levels for the phase-conjugate geometry (counterpropagating pump beams) are calculated by integration of the time-dependent density-matrix equations at numerous grid points along the phase-matching axis and by summing of the polarization contribution from each of these grid points. Both purely homogeneously broadened resonances and resonances that are both collision broadened and Doppler broadened are considered. DFWM signal generation for these two types of resonances in the fully transient regime (τL≪τC) is strikingly different. For a homogeneously broadened resonance, the DFWM signal persists long after the laser beam has passed through the medium. The decay rate of the DFWM signal is determined by the collision rate in the medium, and the integrated DFWM signal is still dependent on the collision rate even in this fully transient regime. However, when the Doppler broadening of the resonance is much greater than the collisional broadening, the DFWM signal decays with a fall time close to that of the laser pulse. For τL≪τC, we find that it is the laser spectral width, rather than the collisional width, that dictates the range of velocity groups with which the pulses effectively interact. The signal decays with a fall time close to that of the laser pulse because the excited velocity groups radiate out of phase with one another. In a medium in which the Doppler width is much greater than the collisional width, the DFWM reflectivity is independent of collision rate for τL≪τC, but the signal level is dependent on the Doppler width. Finally, we investigate the effect of different laser pulse lengths on reflectivity saturation curves for resonances with different levels of Doppler broadening.

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Theory for polarization-dependent four-wave-mixing signals in quantum wells

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In degenerate four-wave-mixing (DFWM) experiments the polarization of the incident laser pulses can have a strong influence on the measured signal. For GaAs quantum wells a number of DFWM experiments were performed by using either parallel or perpendicular polarization of the incident laser pulses. Experimentally, it turned out that some characteristic features of the signal differ appreciably for the two configurations. In particular, different dephasing times, phase shifts of the quantum beats, different time-resolved behavior (echoes versus free-induction decay), and intensity dependence of the measured signal have been reported.1–3

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Absorption and saturation effects on degenerate four-wave mixing in excited states formed during collisions

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Degenerate four-wave mixing (DFWM) in excited atoms is investigated as a probe of the excited-state density produced by collision-assisted transitions. Sodium atoms are excited to the 3P state by absorption of light in the extreme blue wing of the D lines in the presence of He, Ne and Ar perturbers. The rare-gas pressure dependence of the DFWM signals, resonantly enhanced by the 3P-4D transition, is studied to determine the regime where the signals give an unambiguous measure of the excited-state (3P) density. The observed behaviour is compared with a simple model of DFWM in absorbing media. Good qualitative agreement is obtained and the results illustrate the role of absorption, saturation and finite laser bandwidth. The technique allows small excited-state densities ( approximately 10 9 cm -3 ) to be detected with good temporal and spatial resolution.

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Saturation effects in gas-phase degenerate four-wave mixing spectroscopy: nonperturbative calculations

Robert P. Lucht, Roger L. Farrow, David J. Rakestraw

Saturation effects in gas-phase degenerate four-wave mixing (DFWM) are investigated for conditions of interest for diagnostic applications in flames and plasmas. In particular, the case in which Doppler and collisional broadening are comparable, as is often the case for flame species such as NO and OH, is investigated. DFWM line shapes and signal intensities are calculated nonperturbatively and compared with high-resolution laser measurements. In the nonperturbative calculations the time-dependent density-matrix equations for a two-level system interacting with three laser fields are integrated directly on a grid of spatial locations along the phase-matching axis. The electric-field amplitude for the DFWM signal is determined by multiplying the time-varying, laser-induced polarization at each grid point by the appropriate phase factor and then by summing over all grid points. The calculations are in excellent agreement with measurements of DFWM line shapes and signal intensities for NO in a buffer gas of He over a wide range of He pressure. Under saturation conditions the pressure dependence of the DFWM signal is reduced greatly compared with the unsaturated case. The signal level is still dependent on the ratio of pure dephasing to quenching collisions, even for saturation conditions. From the standpoint of minimal dependence on collisional processes it appears that operation with pump-laser intensities approximately equal to the saturation intensity is optimal. The dependence of the DFWM line shape and signal intensity on the ratio of probe to pump-laser intensity is also investigated.

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Degenerate four-wave mixing from laser-populated excited states

Peter Ljungberg, Ove Axner

Degenerate four-wave mixing (DFWM) from laser-populated excited states, i.e., two-step DFWM (2S-DFWM), has been performed to investigate the possibility of increasing signal quality (i.e., strength or signal-to-noise ratio) when species with low transition probabilities or far-UV transitions are to be detected or when large beam areas are used. Gold atoms, aspirated into an air-acetylene flame, were chosen as a suitable species for this investigation. The 2S-DFWM signal strength was found to be comparable to the ordinary (one-step) DFWM signals for moderately high UV-light intensities but substantially better for low UV-light intensities. This finding implies that DFWM detection of species with lower transition probabilities in the first step as compared with gold (<10(-8) s(-1)) can benefit from the 2S-DFWM technique when moderate or low UV-light intensities are available. Additional possible advantages of using 2S-DFWM are also discussed.

Applied Optics•Jan 20, 1995•9 citations
Collisionless extraresonances (ER) in degenerate four-wave mixing (DFWM) are predicted when the upper state \ensuremath{\Vert}b〉 of the \ensuremath{\Vert}a〉\ensuremath{\rightarrow}\ensuremath{\Vert}b〉 transition, excited by near-resonant incident laser beams, decays to the lower state \ensuremath{\Vert}a〉 both directly (\ensuremath{\Vert}b〉\ensuremath{\rightarrow}\ensuremath{\Vert}a〉) and via one or two intermediate states: \ensuremath{\Vert}b〉\ensuremath{\rightleftarrows}\ensuremath{\Vert}d〉\ensuremath{\rightarrow}\ensuremath{\Vert}a〉 or \ensuremath{\Vert}b〉\ensuremath{\rightarrow}\ensuremath{\Vert}c〉\ensuremath{\rightarrow}\ensuremath{\Vert}d〉\ensuremath{\rightarrow}\ensuremath{\Vert}a〉. The width of the new ER, which depends on the relative decay rates in the cascade process, may be very narrow leading to very strong DFWM signals. Dephasing collisions affect the intensity of the new ER and also produce the usual, pressure-induced ER.
It is shown that intense collisionless Rayleigh-type extraresonant signals in degenerate four-wave mixing (DFWM) can be obtained when the $| a 〉\ensuremath{\rightarrow}| b 〉$ transition which produces the DFWM signal is coupled to another transition ($ | a 〉\ensuremath{\rightarrow}| d 〉$) which is excited near-resonantly by an additional laser beam. Two effects contribute to the Rayleigh-type extraresonant signal at ${\ensuremath{\omega}}_{2}={\ensuremath{\omega}}_{1}$. The first effect is resonance-enhanced three-photon scattering (TPS), which leads to the population of state ∣b〉| b 〉∣b〉 and consequently contributes a term in ${\ensuremath{\rho}}_{\mathrm{ba}}$, the off-diagonal density-matrix element, which is characterized by the linewidth (1T1)ba{(\frac{1}{{T}_{1}})}_{\mathrm{ba}}(T1​1​)ba​, the inverse lifetime of the state ∣b〉| b 〉∣b〉. The second effect derives from the effective lifetime acquired by the ground state ∣a〉| a 〉∣a〉 due to its strong coupling to state ∣d〉| d 〉∣d〉. This leads to a term in ${\ensuremath{\rho}}_{\mathrm{ba}}$ characterized by a linewidth 12(1T1)da\frac{1}{2}{(\frac{1}{{T}_{1}})}_{\mathrm{da}}21​(T1​1​)da​, which is half the inverse lifetime of the state ∣d〉| d 〉∣d〉. When ${(\frac{1}{{T}_{1}})}_{\mathrm{da}}\ensuremath{\gg}{(\frac{1}{{T}_{1}})}_{\mathrm{ba}}$, the TPS effect dominates and the extraresonant DFWM signal is narrow and intense. When ${(\frac{1}{{T}_{1}})}_{\mathrm{da}}\ensuremath{\simeq}{(\frac{1}{{T}_{1}})}_{\mathrm{ba}}$, the signal is less intense and interference between the two effects is evident.
The use of picosecond lasers for degenerate four-wave mixing (DFWM) diagnostics of atmospheric pressure flames has been suggested as a means of overcoming the complicated collision-rate dependence of the DFWM signal in the nanosecond-laser pulse-length regime. In this paper DFWM spectroscopy in the regime in which the laser pulse length (τL) is less than the characteristic collisional time (τC) is investigated theoretically. DFWM signal levels for the phase-conjugate geometry (counterpropagating pump beams) are calculated by integration of the time-dependent density-matrix equations at numerous grid points along the phase-matching axis and by summing of the polarization contribution from each of these grid points. Both purely homogeneously broadened resonances and resonances that are both collision broadened and Doppler broadened are considered. DFWM signal generation for these two types of resonances in the fully transient regime (τL≪τC) is strikingly different. For a homogeneously broadened resonance, the DFWM signal persists long after the laser beam has passed through the medium. The decay rate of the DFWM signal is determined by the collision rate in the medium, and the integrated DFWM signal is still dependent on the collision rate even in this fully transient regime. However, when the Doppler broadening of the resonance is much greater than the collisional broadening, the DFWM signal decays with a fall time close to that of the laser pulse. For τL≪τC, we find that it is the laser spectral width, rather than the collisional width, that dictates the range of velocity groups with which the pulses effectively interact. The signal decays with a fall time close to that of the laser pulse because the excited velocity groups radiate out of phase with one another. In a medium in which the Doppler width is much greater than the collisional width, the DFWM reflectivity is independent of collision rate for τL≪τC, but the signal level is dependent on the Doppler width. Finally, we investigate the effect of different laser pulse lengths on reflectivity saturation curves for resonances with different levels of Doppler broadening.
In degenerate four-wave-mixing (DFWM) experiments the polarization of the incident laser pulses can have a strong influence on the measured signal. For GaAs quantum wells a number of DFWM experiments were performed by using either parallel or perpendicular polarization of the incident laser pulses. Experimentally, it turned out that some characteristic features of the signal differ appreciably for the two configurations. In particular, different dephasing times, phase shifts of the quantum beats, different time-resolved behavior (echoes versus free-induction decay), and intensity dependence of the measured signal have been reported.1–3
Degenerate four-wave mixing (DFWM) in excited atoms is investigated as a probe of the excited-state density produced by collision-assisted transitions. Sodium atoms are excited to the 3P state by absorption of light in the extreme blue wing of the D lines in the presence of He, Ne and Ar perturbers. The rare-gas pressure dependence of the DFWM signals, resonantly enhanced by the 3P-4D transition, is studied to determine the regime where the signals give an unambiguous measure of the excited-state (3P) density. The observed behaviour is compared with a simple model of DFWM in absorbing media. Good qualitative agreement is obtained and the results illustrate the role of absorption, saturation and finite laser bandwidth. The technique allows small excited-state densities ( approximately 10 9 cm -3 ) to be detected with good temporal and spatial resolution.
Saturation effects in gas-phase degenerate four-wave mixing (DFWM) are investigated for conditions of interest for diagnostic applications in flames and plasmas. In particular, the case in which Doppler and collisional broadening are comparable, as is often the case for flame species such as NO and OH, is investigated. DFWM line shapes and signal intensities are calculated nonperturbatively and compared with high-resolution laser measurements. In the nonperturbative calculations the time-dependent density-matrix equations for a two-level system interacting with three laser fields are integrated directly on a grid of spatial locations along the phase-matching axis. The electric-field amplitude for the DFWM signal is determined by multiplying the time-varying, laser-induced polarization at each grid point by the appropriate phase factor and then by summing over all grid points. The calculations are in excellent agreement with measurements of DFWM line shapes and signal intensities for NO in a buffer gas of He over a wide range of He pressure. Under saturation conditions the pressure dependence of the DFWM signal is reduced greatly compared with the unsaturated case. The signal level is still dependent on the ratio of pure dephasing to quenching collisions, even for saturation conditions. From the standpoint of minimal dependence on collisional processes it appears that operation with pump-laser intensities approximately equal to the saturation intensity is optimal. The dependence of the DFWM line shape and signal intensity on the ratio of probe to pump-laser intensity is also investigated.
Degenerate four-wave mixing (DFWM) from laser-populated excited states, i.e., two-step DFWM (2S-DFWM), has been performed to investigate the possibility of increasing signal quality (i.e., strength or signal-to-noise ratio) when species with low transition probabilities or far-UV transitions are to be detected or when large beam areas are used. Gold atoms, aspirated into an air-acetylene flame, were chosen as a suitable species for this investigation. The 2S-DFWM signal strength was found to be comparable to the ordinary (one-step) DFWM signals for moderately high UV-light intensities but substantially better for low UV-light intensities. This finding implies that DFWM detection of species with lower transition probabilities in the first step as compared with gold (<10(-8) s(-1)) can benefit from the 2S-DFWM technique when moderate or low UV-light intensities are available. Additional possible advantages of using 2S-DFWM are also discussed.