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Stabilization of stokes pulse energies in the nonlinear regime of stimulated Raman scattering
Volume 53, number 2
OPTICS COMMUNICATIONS
15 February 1985
STABILIZATION OF STOKES PULSE ENERGIES IN THE NONLINEAR REGIME OF STIMULATED RAMAN SCATTERING I.A. WALMSLEY and M.G. RAYMER The Institute of Optics, University of Rochester, Rochester, NY 14627, USA
T. SIZER II, I.N. DULING III, J.D. KAFKA * Laboratory for Laser Energetics and The Institute of Optics, University of Rochester, Rochester, N Y 14627, USA Received 21 November 1984
An experimental study of quantum mechanical pulse energy fluctuations in stimulated Raman scattering from hydrogen is presented. The probability density function P(W) for Stokes pulse energy Wis measured for highly transient scattering in both the linear and nonlinear gain regimes. While large pulse energy fluctuations (100%) occur in the linear gain regime, the fluctuations are reduced to about 20% in the nonlinear regime where the laser pulse is depleted significantly. The results are in excellent agreement with the theoretical predictions of Lewenstein.
1. Introduction
It is now understood that the energy of light pulses generated by stimulated Raman scattering (SRS) fluctuates from pulse to pulse because of the quantum noise associated with the spontaneous initiation of the scattering [ 1 - 3 ] . Described simply, several spontaneously scattered photons, of fluctuating number, are simplified to a macroscopic level to produce the observed Stokes pulse. If the amplification takes place in the linear regime, where neither laser field nor molecular population are depleted, then the statistical properties of the initiating photons are preserved. But if the amplification takes place in the nonlinear regime, then it is expected that the statistical properties of the initiating photons will be modified during amplification. Experiments measuring the probability density function P(W) for Stokes pulse energy W in the linear regime o f SRS have been reported [2,3]. In these experiments P(W) was found to be close to a negative exponential function, in agreement with the theoretical * Present address: Spectra-Physics, Inc., 1250 West Middlefield Rd., Mountain View, CA 94042, USA 0 030-4018/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
prediction for this regime [1]. In ref. [3] the scattering took place under transient conditions, that is, I~TL "~ gZ, where F is one-half the spontaneous Raman linewidth, TL is the laser pulse duration, g is the Raman gain coefficient, and L is the length of the medium. In ref. [2] the conditions were intermediate between transient and steady-state, i.e. Ur L ~ gL. In both experiments the Fresnel number of the pumped volume was near unity, meaning that a single spatial mode is dominantly excited. In the case of transient scattering, the exponential form of P(W) can be understood in a simple way [4]. The quantum noise responsible for the initiation of the spontaneous scattering obeys gaussian statistics [1 ;5]. Thus the spontaneously emitted field can be thought of as a gaussian random process having zero mean. The energy emitted during a short time interwil "rL is proportional to the square of the field, and is thus distributed according to P(IV) = ( I g ) - I exp(-W/(W)),
(1)
where (W) is the mean value of the energy. When the sampling time interval r L becomes comparable to the coherence time (~gL/P) of the emitted field, the above argument no longer holds and a more detailed 137
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OPTICS COMMUNICATIONS
theory must be considered [6,7]. Furthermore, if the Fresnel number is larger than unity, then many spatial modes can contribute, also leading to a non-exponential distribution function [8]. Recently two theoretical studies have been carried out on the form o f the probability distribution in the nonlinear regime o f SRS. The nonlinearity arises because o f depletion o f the pumping laser field. The study o f Lewenstein [9] is based on the numerical solution o f the Maxwell-Bloch equations for SRS, using random initial conditions to simulate the microscopic quantum noise. Many trajectories were computed, and from the resulting Stokes pulse energies, the distribution P(W) was estimated. The other recent theoretical study is that by Trippenbach and Rzazewski [10], in which a simplified set o f equations is used to model the SRS process in the nonlinear regime. This set o f equations was solved and an analytic expression was found for the distribution POe). In b o t h o f these studies it was found that in the linear regime the distribution P(W) is a negative exponential function, while in the nonlinear regime P ( g 0 is strongly peaked at the mean energy. This corresponds to a stabilization of the Stokes pulse energy in the nonlinear regime. The result can be understood b y analogy with a saturated amplifier, in which the output is insensitive to variations o f the input. The purpose of the present paper is to describe the results o f an experiment [11] in which the Stokes pulse energy distribution P(W) was measured in both the linear and nonlinear regimes. The scattering process t o o k place in hydrogen gas under highly transient conditions, I'rL/gL ~ 10 -3, so that temporal incoherence effects were negligible. In addition the Fresnel number was equal to about 0.3, so that spatial incoherence effects were minimal. These conditions are ideal for studying the effects of saturation on the fluctuation statistics o f the Stokes pulse energies. The results o f the experiment are in agreement with the above-mentioned theories; that is, P(W) goes from a negative exponential function to a well peaked function as the SRS process becomes saturated.
2. Experiment The experimental apparatus used to measure the Stokes pulse energy distribution is shown in fig. 1. i38
15 February 1985
532nrn
-
-
rHYOROGEN
' PD
7
D2
]-
~532nm~%5 / 3rim
c Fig. 1. Experimental setup. A frequency-doubled Nd:YAG laser pulse passes through hydrogen gas, generating a Stokes pulse at 683 nm, which is separated from the laser pulse using a dichronic mirror, a colored glass filter (F), and a monochromator. The Stokes pulse is detected by a photomultiplier tube (PMT) whose output voltage pulse is temporally integrated and digitized, the result being sent to a microcomputer (~C). The laser pulse is detected by photodiode PD1, which triggers the integration control circuit C, and by photodiode PD2, which is used to measure the laser pulse energy.
Single pulses from a mode-locked, Q-switched Nd:YAG laser [12] were amplified and frequency-doubled to produce 100 ps, 1 mJ pulses at 532 nm with a repetition rate o f 7 Hz. The beam was spatially filtered and collimated to produce a near gaussian profile with a diameter (at half-maximum intensity) o f 0.5 m m and was passed through a 100 cm-long cell f'filed with H 2 . This gave an excited volume Fresnel number A/XsL of about 0.3, where A is the beam cross-sectional area (at half-maximum intensity), k s is the Stokes wavelength and L is the cell length. Each laser pulse generated a Stokes pulse at 683 nm by scattering from the first vibrational transition o f the H 2 molecule. Each Stokes pulse was separated from the laser pulse with spectral filters and was detected by a photomultiplier, whose output current pulse was integrated, converted to digital form, and sent to a microcomputer. By measuring many Stokes pulse energies at a given laser pulse energy a histogram was built up, from which P(W) was estimated. It is important that the laser pulse be highly reproducible, since any small change in laser intensity results in large variations o f the Raman gain, thus masking the quantum fluctuations. Since the energy stability of our laser pulses was -+5% we measured the ener-
Volume 53, number 2
OPTICS COMMUNICATIONS
gy of each laser pulse to within 0.3% using a photodiode and sent this to the computer. By selecting laser energies falling into one of ten intervals of width 1%, we were able to obtain ten different Stokes energy histograms, each corresponding to a slightly different Raman gain. Each histogram has a sample size N between 5000 and 25000 measurements. The probability density function P(W) of the Stokes pulse energy was estimated from the histogram data by using the kth nearest neighbor method [13]. The method maintains a nearly constant fractional statistical precision o f the density estimate at each p o i n t W i by varying the sampling window size s i so as to keep the number of data points k i within the window nearly equal to some chosen constant k. The density estimation at a point Wi is then P ( w i) = N - 1k d s i
i
(a) v
~o.,' Q..
"."
"2
(b)
°°
a.
(2)
The fractional error in P(IVi) is then given by l/x/c~-i ~ lx/~. This estimate becomes unreliable near the two boundaries of the data, and thus we will not present the results in these regions. The experimental results are shown in figs. 2(a) and 2(b). The main parameters of interest are H 2 pressure (p), Raman half linewidth (F) in rad/s, gain coefficient (g), H 2 cell length (L), and average photon conversion efficiency (r/) from the laser to the Stokes. The predominantly collision-broadened linewidth F was obtained from data in ref. [14]. The gain coefficient was obtained from the measured peak laser intensity 1L and g = g0/L , where go = 2.0 X 10 -9 cm/W [14]. I L was determined by measuring the relative energies of the whole beam and that portion transmitted through an 80/~m pinhole placed at the center of the beam. Fig. 2(a) shows the results for P(IV) under the conditions p = 4.0 atm, Fr L = 0.093, gL = 500, r / ~ 10 -9. Since r~-L ~ gL the scattering was highly transient. Since r/is very small the gain process is unsaturated, i.e. in the linear regime. The probability density P(IV) is seen to be close to a negative exponential function. The departure at small values of IV is known to be due to the weak excitation of Stokes spatial modes other than the dominant, lowest order one. A detailed discussion of this point can be found in the recent paper by Mostowski and Sobolewska [15]. The form of P(IV) in fig. 2(a) is in agreement with the theoretical prediction for the unsaturated, transient case. Fig. 2(b) shows the results for P(W) under the con-
15 February 1985
4
(c)
3 Q..
2 I
%
2 W/
Fig. 2. Experimental and theoretical results for the probability density function P(W) of Stokes pulse energy W, plotted in terms of ( W>, the m e a n pulse energy. (a) shows the experimental m e a s u r e m e n t (points) and a simple exponential, eq. (1) (dashed line). The m e a n energy is of the order o f 10 -12 J; thus the gain is unsaturated. (b) shows the experimental measurement in the case of saturated gain, where the m e a n energy is a b o u t 10 --4 J. (c) shows the theoretical prediction of Lewenstein [9] for the same experimental parameters as those in (b). (b) and (c) show the stabilization o f Stokes pulse energy in the nonlinear regime.
ditions p = 27 atm, FT L = 0 . 5 0 , gZ = 740, r7 = 0.14. Here the scattering is still highly transient but the gain is partially saturated due to the 14% conversion efficiency. The data shows the strong narrowing of P(iv), representing the stabilization of the Stokes pulse energies in the nonlinear regime. A slight asymmetry can be seen in p(W), which falls off more rapidly on the high-W side of the mean than on the low-W side. The more rapid fall off at high energy is a result of energy conservation, i.e., P(IV) must go to zero for Stokes energies greater than the laser pulse energy. 139
Volume 53, number 2
OPTICS COMMUNICATIONS
3. Discussion Both the magnitude o f the width and the presence o f the asymmetry o f the distribution P(I¢) in fig. 2(b). are in agreement with the predictions o f Lewenstein [9]. His numerical results for the same experimental parameters as those in fig. 2(b) are shown in fig. 2(c). Thus it can be said that the phenomenon o f Stokes pulse energy stabilization in the nonlinear regime is well understood. The results o f the present paper are o f some general interest b e y o n d the field o f stimulated Raman scattering. The system studied provides an example in which microscopic quantum noise is amplified to the macroscopic regime in such a way that large fluctuations persist in the observable macroscopic variables. Another, closely related example of this type of behavior is superfluorescence from a collection o f inverted two-level atoms [5], where fluctuations are observed in the delay times o f the superfluorescent pulses [16]. The present comparison between experiment and theory o f SRS helps to confirm the theoretical techniques [5] that have been developed to study superfluorescence, since they are analogous to those used to make the predictions on SRS [9]. Finally, the experiments and theory o f SRS discussed here were restricted to the transient domain (FT-t ,~ gL), where collisional dephasing effects are negligible. It would be interesting to study nonlinear effects in the non transient domain ( F r L/> gL), where collisional dephasing acts as a source o f fluctuations in the molecular dipole moment.
Acknowledgements We wish to thank G. Mourou for making his laboratory at the Laboratory for Laser Energetics available to us for these experiments. We also acknowledge
140
15 February 1985
K. Rzazewski and J. Mostowski for helpful discussions on the subject o f this paper. We wish to thank M. Lewenstein for giving us permission to present his theoretical results in this paper. This work was supported by the Joint Services Optics Program.
References [1 ] M.G. Raymer, K. Rzazewski and J. Mostowski, Optics Lett. 7 (1982) 71. [2] I.A. Walmsley and M.G. Raymer, Phys. Rev. Lett. 50 (1983) 962. [3] N. Fabricius, K. Nattermann and D. yon der Linde, Phys. Rev. Lett. 52 (1984) 113. [4] For another explanation, see F. Haake, Phys. Lett. 90A (1982) 127. [5] R. Glauber and F. Haake, Phys. Lett. 68A (1978) 29; F. Haake, H. King, G. Schroder, J. Haus and R. Glauber, Phys. Rev. A20 (1979) 2047; D. Polder, M. Schuurmans and Q. Vrehen, Phys. Rev. A19 (1979) 1192. [6] K. Rzazewski, M. Lewenstein and M.G. Raymer, Optics Comm. 43 (1982) 451. [7] M.G. Raymer and I.A. Walmsley, in: Coherence and quantum optics Vol. V, eds. L. Mandel and E. Wolf (Plenum, New York, 1984) p. 63. [8] J. Mostowski and B. Sobolewska, Phys. Rev. A30 (1984) 610. [9] M. Lewenstein, Z. Phys. B56 (1984) 69. [10] M. Trippenbach and K. Rzazewski (unpublished, 1984). [11 ] First reported at the fifth Intern. Conf. on Coherence and quantum optics, Rochester, NY, June, 1983. See ref. [7]. [12] T. Sizer, J. Kafka, I. Duling, C. Gabel and G. Mourou, IEEE J. Quant. Electron. 19 (1983) 506. [13 ] J. Friedman, SLAC report, no. 176 (1974). [14] E. Hagenlocker, R. Minck and W. Rado, Phys. Rev. 154 (1967) 226. [15] J. Mostowski and B. Sobolewska, Phys. Rev. 30 (1984) 1392. [16] Q. Vrehen and J. der Weduwe, Phys. Rev. A24 (1981) 2857.
OPTICS COMMUNICATIONS
15 February 1985
STABILIZATION OF STOKES PULSE ENERGIES IN THE NONLINEAR REGIME OF STIMULATED RAMAN SCATTERING I.A. WALMSLEY and M.G. RAYMER The Institute of Optics, University of Rochester, Rochester, NY 14627, USA
T. SIZER II, I.N. DULING III, J.D. KAFKA * Laboratory for Laser Energetics and The Institute of Optics, University of Rochester, Rochester, N Y 14627, USA Received 21 November 1984
An experimental study of quantum mechanical pulse energy fluctuations in stimulated Raman scattering from hydrogen is presented. The probability density function P(W) for Stokes pulse energy Wis measured for highly transient scattering in both the linear and nonlinear gain regimes. While large pulse energy fluctuations (100%) occur in the linear gain regime, the fluctuations are reduced to about 20% in the nonlinear regime where the laser pulse is depleted significantly. The results are in excellent agreement with the theoretical predictions of Lewenstein.
1. Introduction
It is now understood that the energy of light pulses generated by stimulated Raman scattering (SRS) fluctuates from pulse to pulse because of the quantum noise associated with the spontaneous initiation of the scattering [ 1 - 3 ] . Described simply, several spontaneously scattered photons, of fluctuating number, are simplified to a macroscopic level to produce the observed Stokes pulse. If the amplification takes place in the linear regime, where neither laser field nor molecular population are depleted, then the statistical properties of the initiating photons are preserved. But if the amplification takes place in the nonlinear regime, then it is expected that the statistical properties of the initiating photons will be modified during amplification. Experiments measuring the probability density function P(W) for Stokes pulse energy W in the linear regime o f SRS have been reported [2,3]. In these experiments P(W) was found to be close to a negative exponential function, in agreement with the theoretical * Present address: Spectra-Physics, Inc., 1250 West Middlefield Rd., Mountain View, CA 94042, USA 0 030-4018/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
prediction for this regime [1]. In ref. [3] the scattering took place under transient conditions, that is, I~TL "~ gZ, where F is one-half the spontaneous Raman linewidth, TL is the laser pulse duration, g is the Raman gain coefficient, and L is the length of the medium. In ref. [2] the conditions were intermediate between transient and steady-state, i.e. Ur L ~ gL. In both experiments the Fresnel number of the pumped volume was near unity, meaning that a single spatial mode is dominantly excited. In the case of transient scattering, the exponential form of P(W) can be understood in a simple way [4]. The quantum noise responsible for the initiation of the spontaneous scattering obeys gaussian statistics [1 ;5]. Thus the spontaneously emitted field can be thought of as a gaussian random process having zero mean. The energy emitted during a short time interwil "rL is proportional to the square of the field, and is thus distributed according to P(IV) = ( I g ) - I exp(-W/(W)),
(1)
where (W) is the mean value of the energy. When the sampling time interval r L becomes comparable to the coherence time (~gL/P) of the emitted field, the above argument no longer holds and a more detailed 137
Volume 53, number 2
OPTICS COMMUNICATIONS
theory must be considered [6,7]. Furthermore, if the Fresnel number is larger than unity, then many spatial modes can contribute, also leading to a non-exponential distribution function [8]. Recently two theoretical studies have been carried out on the form o f the probability distribution in the nonlinear regime o f SRS. The nonlinearity arises because o f depletion o f the pumping laser field. The study o f Lewenstein [9] is based on the numerical solution o f the Maxwell-Bloch equations for SRS, using random initial conditions to simulate the microscopic quantum noise. Many trajectories were computed, and from the resulting Stokes pulse energies, the distribution P(W) was estimated. The other recent theoretical study is that by Trippenbach and Rzazewski [10], in which a simplified set o f equations is used to model the SRS process in the nonlinear regime. This set o f equations was solved and an analytic expression was found for the distribution POe). In b o t h o f these studies it was found that in the linear regime the distribution P(W) is a negative exponential function, while in the nonlinear regime P ( g 0 is strongly peaked at the mean energy. This corresponds to a stabilization of the Stokes pulse energy in the nonlinear regime. The result can be understood b y analogy with a saturated amplifier, in which the output is insensitive to variations o f the input. The purpose of the present paper is to describe the results o f an experiment [11] in which the Stokes pulse energy distribution P(W) was measured in both the linear and nonlinear regimes. The scattering process t o o k place in hydrogen gas under highly transient conditions, I'rL/gL ~ 10 -3, so that temporal incoherence effects were negligible. In addition the Fresnel number was equal to about 0.3, so that spatial incoherence effects were minimal. These conditions are ideal for studying the effects of saturation on the fluctuation statistics o f the Stokes pulse energies. The results o f the experiment are in agreement with the above-mentioned theories; that is, P(W) goes from a negative exponential function to a well peaked function as the SRS process becomes saturated.
2. Experiment The experimental apparatus used to measure the Stokes pulse energy distribution is shown in fig. 1. i38
15 February 1985
532nrn
-
-
rHYOROGEN
' PD
7
D2
]-
~532nm~%5 / 3rim
c Fig. 1. Experimental setup. A frequency-doubled Nd:YAG laser pulse passes through hydrogen gas, generating a Stokes pulse at 683 nm, which is separated from the laser pulse using a dichronic mirror, a colored glass filter (F), and a monochromator. The Stokes pulse is detected by a photomultiplier tube (PMT) whose output voltage pulse is temporally integrated and digitized, the result being sent to a microcomputer (~C). The laser pulse is detected by photodiode PD1, which triggers the integration control circuit C, and by photodiode PD2, which is used to measure the laser pulse energy.
Single pulses from a mode-locked, Q-switched Nd:YAG laser [12] were amplified and frequency-doubled to produce 100 ps, 1 mJ pulses at 532 nm with a repetition rate o f 7 Hz. The beam was spatially filtered and collimated to produce a near gaussian profile with a diameter (at half-maximum intensity) o f 0.5 m m and was passed through a 100 cm-long cell f'filed with H 2 . This gave an excited volume Fresnel number A/XsL of about 0.3, where A is the beam cross-sectional area (at half-maximum intensity), k s is the Stokes wavelength and L is the cell length. Each laser pulse generated a Stokes pulse at 683 nm by scattering from the first vibrational transition o f the H 2 molecule. Each Stokes pulse was separated from the laser pulse with spectral filters and was detected by a photomultiplier, whose output current pulse was integrated, converted to digital form, and sent to a microcomputer. By measuring many Stokes pulse energies at a given laser pulse energy a histogram was built up, from which P(W) was estimated. It is important that the laser pulse be highly reproducible, since any small change in laser intensity results in large variations o f the Raman gain, thus masking the quantum fluctuations. Since the energy stability of our laser pulses was -+5% we measured the ener-
Volume 53, number 2
OPTICS COMMUNICATIONS
gy of each laser pulse to within 0.3% using a photodiode and sent this to the computer. By selecting laser energies falling into one of ten intervals of width 1%, we were able to obtain ten different Stokes energy histograms, each corresponding to a slightly different Raman gain. Each histogram has a sample size N between 5000 and 25000 measurements. The probability density function P(W) of the Stokes pulse energy was estimated from the histogram data by using the kth nearest neighbor method [13]. The method maintains a nearly constant fractional statistical precision o f the density estimate at each p o i n t W i by varying the sampling window size s i so as to keep the number of data points k i within the window nearly equal to some chosen constant k. The density estimation at a point Wi is then P ( w i) = N - 1k d s i
i
(a) v
~o.,' Q..
"."
"2
(b)
°°
a.
(2)
The fractional error in P(IVi) is then given by l/x/c~-i ~ lx/~. This estimate becomes unreliable near the two boundaries of the data, and thus we will not present the results in these regions. The experimental results are shown in figs. 2(a) and 2(b). The main parameters of interest are H 2 pressure (p), Raman half linewidth (F) in rad/s, gain coefficient (g), H 2 cell length (L), and average photon conversion efficiency (r/) from the laser to the Stokes. The predominantly collision-broadened linewidth F was obtained from data in ref. [14]. The gain coefficient was obtained from the measured peak laser intensity 1L and g = g0/L , where go = 2.0 X 10 -9 cm/W [14]. I L was determined by measuring the relative energies of the whole beam and that portion transmitted through an 80/~m pinhole placed at the center of the beam. Fig. 2(a) shows the results for P(IV) under the conditions p = 4.0 atm, Fr L = 0.093, gL = 500, r / ~ 10 -9. Since r~-L ~ gL the scattering was highly transient. Since r/is very small the gain process is unsaturated, i.e. in the linear regime. The probability density P(IV) is seen to be close to a negative exponential function. The departure at small values of IV is known to be due to the weak excitation of Stokes spatial modes other than the dominant, lowest order one. A detailed discussion of this point can be found in the recent paper by Mostowski and Sobolewska [15]. The form of P(IV) in fig. 2(a) is in agreement with the theoretical prediction for the unsaturated, transient case. Fig. 2(b) shows the results for P(W) under the con-
15 February 1985
4
(c)
3 Q..
2 I
%
2 W/
Fig. 2. Experimental and theoretical results for the probability density function P(W) of Stokes pulse energy W, plotted in terms of ( W>, the m e a n pulse energy. (a) shows the experimental m e a s u r e m e n t (points) and a simple exponential, eq. (1) (dashed line). The m e a n energy is of the order o f 10 -12 J; thus the gain is unsaturated. (b) shows the experimental measurement in the case of saturated gain, where the m e a n energy is a b o u t 10 --4 J. (c) shows the theoretical prediction of Lewenstein [9] for the same experimental parameters as those in (b). (b) and (c) show the stabilization o f Stokes pulse energy in the nonlinear regime.
ditions p = 27 atm, FT L = 0 . 5 0 , gZ = 740, r7 = 0.14. Here the scattering is still highly transient but the gain is partially saturated due to the 14% conversion efficiency. The data shows the strong narrowing of P(iv), representing the stabilization of the Stokes pulse energies in the nonlinear regime. A slight asymmetry can be seen in p(W), which falls off more rapidly on the high-W side of the mean than on the low-W side. The more rapid fall off at high energy is a result of energy conservation, i.e., P(IV) must go to zero for Stokes energies greater than the laser pulse energy. 139
Volume 53, number 2
OPTICS COMMUNICATIONS
3. Discussion Both the magnitude o f the width and the presence o f the asymmetry o f the distribution P(I¢) in fig. 2(b). are in agreement with the predictions o f Lewenstein [9]. His numerical results for the same experimental parameters as those in fig. 2(b) are shown in fig. 2(c). Thus it can be said that the phenomenon o f Stokes pulse energy stabilization in the nonlinear regime is well understood. The results o f the present paper are o f some general interest b e y o n d the field o f stimulated Raman scattering. The system studied provides an example in which microscopic quantum noise is amplified to the macroscopic regime in such a way that large fluctuations persist in the observable macroscopic variables. Another, closely related example of this type of behavior is superfluorescence from a collection o f inverted two-level atoms [5], where fluctuations are observed in the delay times o f the superfluorescent pulses [16]. The present comparison between experiment and theory o f SRS helps to confirm the theoretical techniques [5] that have been developed to study superfluorescence, since they are analogous to those used to make the predictions on SRS [9]. Finally, the experiments and theory o f SRS discussed here were restricted to the transient domain (FT-t ,~ gL), where collisional dephasing effects are negligible. It would be interesting to study nonlinear effects in the non transient domain ( F r L/> gL), where collisional dephasing acts as a source o f fluctuations in the molecular dipole moment.
Acknowledgements We wish to thank G. Mourou for making his laboratory at the Laboratory for Laser Energetics available to us for these experiments. We also acknowledge
140
15 February 1985
K. Rzazewski and J. Mostowski for helpful discussions on the subject o f this paper. We wish to thank M. Lewenstein for giving us permission to present his theoretical results in this paper. This work was supported by the Joint Services Optics Program.
References [1 ] M.G. Raymer, K. Rzazewski and J. Mostowski, Optics Lett. 7 (1982) 71. [2] I.A. Walmsley and M.G. Raymer, Phys. Rev. Lett. 50 (1983) 962. [3] N. Fabricius, K. Nattermann and D. yon der Linde, Phys. Rev. Lett. 52 (1984) 113. [4] For another explanation, see F. Haake, Phys. Lett. 90A (1982) 127. [5] R. Glauber and F. Haake, Phys. Lett. 68A (1978) 29; F. Haake, H. King, G. Schroder, J. Haus and R. Glauber, Phys. Rev. A20 (1979) 2047; D. Polder, M. Schuurmans and Q. Vrehen, Phys. Rev. A19 (1979) 1192. [6] K. Rzazewski, M. Lewenstein and M.G. Raymer, Optics Comm. 43 (1982) 451. [7] M.G. Raymer and I.A. Walmsley, in: Coherence and quantum optics Vol. V, eds. L. Mandel and E. Wolf (Plenum, New York, 1984) p. 63. [8] J. Mostowski and B. Sobolewska, Phys. Rev. A30 (1984) 610. [9] M. Lewenstein, Z. Phys. B56 (1984) 69. [10] M. Trippenbach and K. Rzazewski (unpublished, 1984). [11 ] First reported at the fifth Intern. Conf. on Coherence and quantum optics, Rochester, NY, June, 1983. See ref. [7]. [12] T. Sizer, J. Kafka, I. Duling, C. Gabel and G. Mourou, IEEE J. Quant. Electron. 19 (1983) 506. [13 ] J. Friedman, SLAC report, no. 176 (1974). [14] E. Hagenlocker, R. Minck and W. Rado, Phys. Rev. 154 (1967) 226. [15] J. Mostowski and B. Sobolewska, Phys. Rev. 30 (1984) 1392. [16] Q. Vrehen and J. der Weduwe, Phys. Rev. A24 (1981) 2857.