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The influence of reorientational collisions on coherent optical processes
Volume 8, number 1
OPTICS COMMUNICATIONS
THE INFLUENCE
OF REORIENTATIONAL
ON COHERENT
May 1973
COLLISIONS
OPTICAL PROCESSES
*
Frederic A. HOPF t Optical Sciences Center, University of Arizona, Tucson, Arizona 85721, USA
and Charles K. RHODES Lawrence Livermore Laboratory, Livermore, California 94550, USA
Received 22 March 1973
Numerical calculations of pulse transmission in SF 6 are performed to appraise the role of reorientational collisions. No support is found for the recent postulate calling for thermally equilibrated magnetic subtevels in sell-induced transparency.
In a recent article [ 1] describing coherent pulse propagation studies in attenuating media (SF 6 gas), the authors concluded that a simple nondegenerate two-level model of the radiating system properly accounted for their experimental findings. |n order to arrive at this determination, it was necessary to postulate [ I, 2] the existence of a collisional mechanism which effectively thermalized the magnetic sublevels of the degenerate states, but did n o t otherwise perturb the coherent interaction (phase relationships preserved) of the radiators with the electromagnetic field **. Since collisions normally perturb both the amplitude and phase of a radiating system (both components contribute to the homogeneous width), such a result is not generally anticipated. In this paper, we present the results of a numerical calculation in which we compare degenerate against nondegenerate media using input data for the attenua* Work performed under the auspices of the U.S. Atomic Energy Commission. ~-Work supported in part by the U.S. Air Force, Kirtland Air Force Base, AFWL. ** Such a collisional mechanism calls for a change in the diagonal matrix elements of the density matrix without a corresponding modification of the off-diagonal matrix elements. 88
tor that correspond to the experimental configuration of ref. [ 1]. It is found that the results for the two cases are insensitive to medium degeneracy, and we consequently conclude that their collisional hypothesis cannot be demonstrated on the basis of their findings. An example of the influence of degeneracy on pulse propagation without sublevel equilibration [3] is illustrated in fig. 1. In each case for fig. la these pulses illustrate the waveform occurring at maximum pulse delay. These curves are to be compared to the corresponding experimental waveform for maximum delay shown in fig. lb. This comparison indicates one important reason for the use of the Q-branch transition to model the experiment described in ref. [3]. Further details of this model are given in ref. [4], and the waveforms shown here are generated by using that model in conjunction with an input pulse shape that was similar to the experimental input pulse in ref. [3]. It is observed that the delay time as indicated by the location of the peak pulse intensity is only slightly affected while the pulse waveform undergoes a more substantial modification. The decrease in the reradiated intensity is caused by the partial dephasing of the effective dipole moments of the degenerate levels [3, 4]. In a P-branch transitions, the distribution of dipole
Volume 8, number 1
OPTICS COMMUNICATIONS
(a) Experimentalpulse shapes
f-,,
t
\ ~ \
/
Nondegeneratetwo-level system
c
Time
(b)
Time
Fig. 1. Figure (a) iUustrates theoretical pulse waveforms (intensity versus time) corresponding to maximum pulse delay for the experiment of ref. [3]. The precise conditions for these calculations are given in ref. [3] except that in this example greater effort was made to tailor the theoretical input pulses to conform more closely to the actual experimental pulse shapes. For identical input data, the results for three different cases are shown; nondegenerate, P(6), and Q(6). Note that the delay as indicated by the peak intensity is the same in all three examples, but that the peak power is substantially different. Figure (b) shows an experimental waveform taken from ref. 113]which represents very nearly the condition of maximum delay. This resulting waveform most closely resembles the Q-branch case as opposed to the P-branch or nondegenerate transitions. moments tends to cluster near the maximum value. This effectively reduces the amount of dephasing and the waveform more nearly resembles the nondegenerate case. It is observed, that under the experimental conditions of ref. [3], :it was possible to discern the influence of medium degeneracy. However, incoherent dephasing processes rapidly mask this effect, and as we demonstrate below for the case of ref. [1], the value of T 2 was too short for these subtle features to be apparent. The theoretical treatment used for this calculation
May 1973
is fully documented in ref. [4]. The analysis involves the self-consistent solution of Maxwell's and Schr6dinger's equations for a plane-wave, planepolarized electromagnetic field that is slowly varying on the time scale of a single optical cycle coupled to a generalized form of the usual two-level atom model. There are some extensions of the previous model which entail the additions of collisional rates representing both rotational and reorientational processes. The details of this model will be addressed in a future publication. We note that the time constants can be regarded as free parameters, so that we are able to examine independently the effect of reorientational collisions without simultaneously altering the other parameters. In ref. [5], it was shown that for J ~,~ 6 the calculation is essentially independent of J. Hence, we have chosen to appraise the case of a P- or Rbranch transition, since they are basically equivalent with respect to the degeneracy for J = 6. These results then apply for all rotational states for which J > 6. Our numerical calculations, which assume a plane wave analysis, were performed for conditions * corresponding to the experimental data [1 ] and showed no evidence for the pulse break-up which Zembrod and Gruhl observe [1]. We point out that an apparent pulse break-up has been observed in transmission of pulses through absorbers containing a large number of absorption lengths **. Measurements of the transverse spatial intensity profile indicated that the effect was caused by significant deviations from plane wave propagation. The numerical results clearly indicate that for the experimental t value T2/'rpulse no pulse waveform modulation is expected in me plane wave theory. The experimental curve (see fig. 2 of ref. [1]) of transmission versus input energy for the CO 2 P(14) transition possesses a strong modulation. In contrast, the numerical results give a smooth variation for both degenerate and nondegenerate attenuators as illustrated in fig. 2. In these calculations we have made provisions * The low signal absorption coefficients for the P (16) and P(14) transitions were taken from data in ref. [4] and other measurements performed by Hinkley [5 ]. ** Appendix A of ref. [3] illustrates an experimental example of spurious pulse break-up for transmission of CO2 P(16) radiation in an SF6 absorber for which aL ~ 20. t" The appropriate value of T2 for SF6 at 30 mtorr was derived from photon echo data in ref. [6]. 89
Volume 8, n u m b e r 1
OPTICS COMMUNICATIONS
1.0
,~_
0.5
'~E
/.,,~,~~...... /s~/~:/
0.2 0.1
/
~ Experimental "PP (6)(6) ~ Nondegenerate
I
1
10 Input energy (Arbitrary units)
I
100
Fig. 2. The ratio of transmitted energy to incident energy versus incident energy for transmission of the CO2 P (14) transition in SF6 illustrating the data of ref. [ 1 ] as well as the
results for nondegenerate and degenerate P(6) transitions. Note that neither of the calculated curves exhibits a modulated transmissions such as shown by the data of ref. [ 1 ].
which allow the magnetic sublevels to be thermalized at a specified rate. A very rapid rate corresponds to the equilibrating collisional mechanism postulated in ref. [1]; a sufficiently slow rate pertains to the case of independent coherently excited degenerate ensembles. The smooth variation of the curve in fig. 2 is obtained in both limits. Furthermore, the lack of a pulse break-up is not influenced by the rate of magnetic sublevel thermalization. In contrast to the transmission factors, the calculated delay times are found to be in reasonable agreement with the experimental results. As in the previous case, we find that the differences between the results of the degenerate and nondegenerate calculations are so minor ( ~ 1%) that they could not be distinguished on the basis of the experiment performed in ref. [1]. This agreement has two important implications. Firstly, it demonstrates that the value of T 2 that we used in the numerical calculations is correct. In the regime of large attenuation, the maximum observed delay varies in proportion to and is approximately equal to T 2. Thus, any major discrepancy concerning the appropriate value of T 2 is unlikely. Furthermore, if we associate the experimentally determined maxi-
90
May 1973
mum delay [1] with the value of the input area 0 indicated by the calculation (0in ~ 27r for maximum delay appropriate to the experimental ratio of the pulse width to T 2 rather than 0in ~ 71"which is valid for the case where T 2 is much larger than the pulse width [7], we find that this reduces the value of the dipole moment that was estimated in ref. [1] by about a factor of two. This brings the value into accord with other determinations [3] (it ~ 3 X 10 -19 esu cm). On the basis of the comparison of our numerical findings with the experimental results of ref. [ 1], we conclude that Zembrod and Gruhl were not justified in proposing the model calling for a collisionally thermalized * ensemble of degenerate transitions. Our calculations show that under their experimental conditions the value of T 2 was sufficiently short to substantially suppress any effects arising from degeneracy. For these calculations, we gratefully acknowledge the expert computational assistance of Warren G. Cunninghana. * Intuitively such a special collisional mechanism seems quite improbable. Indeed, there exists a special class of collisions where this effect is ruled out on general grounds by consideration of rotational invariance, time reversal invariance, and parity conservation [8].
References [1] A. Zembrod and Th. Gruhl, Phys. Rev. Letters 27 (1970) 287. I2] C.L. Tang and H. Statz, Appl. Phys. Letters 10 (1967) 145. [3] C.K. Rhodes and A. Szoke, Phys. Rev. 184 (1969) 25. [4] F.A. Hopf, C.K. Rhodes and A. Sz6ke, Phys. Rev. B1 (1970) 2833. [5] E.D. ftinkley, private communication. [6] C.K.N. Patel and R.E. Slusher, Phys. Rev. Letters 20 (1968) 1087. [7] F.A. Hopf and M.O. Scully, Phys. Rev. B1 (1970) 50. [8] C.K. Rhodes and A. Sz6ke, unpublished.
OPTICS COMMUNICATIONS
THE INFLUENCE
OF REORIENTATIONAL
ON COHERENT
May 1973
COLLISIONS
OPTICAL PROCESSES
*
Frederic A. HOPF t Optical Sciences Center, University of Arizona, Tucson, Arizona 85721, USA
and Charles K. RHODES Lawrence Livermore Laboratory, Livermore, California 94550, USA
Received 22 March 1973
Numerical calculations of pulse transmission in SF 6 are performed to appraise the role of reorientational collisions. No support is found for the recent postulate calling for thermally equilibrated magnetic subtevels in sell-induced transparency.
In a recent article [ 1] describing coherent pulse propagation studies in attenuating media (SF 6 gas), the authors concluded that a simple nondegenerate two-level model of the radiating system properly accounted for their experimental findings. |n order to arrive at this determination, it was necessary to postulate [ I, 2] the existence of a collisional mechanism which effectively thermalized the magnetic sublevels of the degenerate states, but did n o t otherwise perturb the coherent interaction (phase relationships preserved) of the radiators with the electromagnetic field **. Since collisions normally perturb both the amplitude and phase of a radiating system (both components contribute to the homogeneous width), such a result is not generally anticipated. In this paper, we present the results of a numerical calculation in which we compare degenerate against nondegenerate media using input data for the attenua* Work performed under the auspices of the U.S. Atomic Energy Commission. ~-Work supported in part by the U.S. Air Force, Kirtland Air Force Base, AFWL. ** Such a collisional mechanism calls for a change in the diagonal matrix elements of the density matrix without a corresponding modification of the off-diagonal matrix elements. 88
tor that correspond to the experimental configuration of ref. [ 1]. It is found that the results for the two cases are insensitive to medium degeneracy, and we consequently conclude that their collisional hypothesis cannot be demonstrated on the basis of their findings. An example of the influence of degeneracy on pulse propagation without sublevel equilibration [3] is illustrated in fig. 1. In each case for fig. la these pulses illustrate the waveform occurring at maximum pulse delay. These curves are to be compared to the corresponding experimental waveform for maximum delay shown in fig. lb. This comparison indicates one important reason for the use of the Q-branch transition to model the experiment described in ref. [3]. Further details of this model are given in ref. [4], and the waveforms shown here are generated by using that model in conjunction with an input pulse shape that was similar to the experimental input pulse in ref. [3]. It is observed that the delay time as indicated by the location of the peak pulse intensity is only slightly affected while the pulse waveform undergoes a more substantial modification. The decrease in the reradiated intensity is caused by the partial dephasing of the effective dipole moments of the degenerate levels [3, 4]. In a P-branch transitions, the distribution of dipole
Volume 8, number 1
OPTICS COMMUNICATIONS
(a) Experimentalpulse shapes
f-,,
t
\ ~ \
/
Nondegeneratetwo-level system
c
Time
(b)
Time
Fig. 1. Figure (a) iUustrates theoretical pulse waveforms (intensity versus time) corresponding to maximum pulse delay for the experiment of ref. [3]. The precise conditions for these calculations are given in ref. [3] except that in this example greater effort was made to tailor the theoretical input pulses to conform more closely to the actual experimental pulse shapes. For identical input data, the results for three different cases are shown; nondegenerate, P(6), and Q(6). Note that the delay as indicated by the peak intensity is the same in all three examples, but that the peak power is substantially different. Figure (b) shows an experimental waveform taken from ref. 113]which represents very nearly the condition of maximum delay. This resulting waveform most closely resembles the Q-branch case as opposed to the P-branch or nondegenerate transitions. moments tends to cluster near the maximum value. This effectively reduces the amount of dephasing and the waveform more nearly resembles the nondegenerate case. It is observed, that under the experimental conditions of ref. [3], :it was possible to discern the influence of medium degeneracy. However, incoherent dephasing processes rapidly mask this effect, and as we demonstrate below for the case of ref. [1], the value of T 2 was too short for these subtle features to be apparent. The theoretical treatment used for this calculation
May 1973
is fully documented in ref. [4]. The analysis involves the self-consistent solution of Maxwell's and Schr6dinger's equations for a plane-wave, planepolarized electromagnetic field that is slowly varying on the time scale of a single optical cycle coupled to a generalized form of the usual two-level atom model. There are some extensions of the previous model which entail the additions of collisional rates representing both rotational and reorientational processes. The details of this model will be addressed in a future publication. We note that the time constants can be regarded as free parameters, so that we are able to examine independently the effect of reorientational collisions without simultaneously altering the other parameters. In ref. [5], it was shown that for J ~,~ 6 the calculation is essentially independent of J. Hence, we have chosen to appraise the case of a P- or Rbranch transition, since they are basically equivalent with respect to the degeneracy for J = 6. These results then apply for all rotational states for which J > 6. Our numerical calculations, which assume a plane wave analysis, were performed for conditions * corresponding to the experimental data [1 ] and showed no evidence for the pulse break-up which Zembrod and Gruhl observe [1]. We point out that an apparent pulse break-up has been observed in transmission of pulses through absorbers containing a large number of absorption lengths **. Measurements of the transverse spatial intensity profile indicated that the effect was caused by significant deviations from plane wave propagation. The numerical results clearly indicate that for the experimental t value T2/'rpulse no pulse waveform modulation is expected in me plane wave theory. The experimental curve (see fig. 2 of ref. [1]) of transmission versus input energy for the CO 2 P(14) transition possesses a strong modulation. In contrast, the numerical results give a smooth variation for both degenerate and nondegenerate attenuators as illustrated in fig. 2. In these calculations we have made provisions * The low signal absorption coefficients for the P (16) and P(14) transitions were taken from data in ref. [4] and other measurements performed by Hinkley [5 ]. ** Appendix A of ref. [3] illustrates an experimental example of spurious pulse break-up for transmission of CO2 P(16) radiation in an SF6 absorber for which aL ~ 20. t" The appropriate value of T2 for SF6 at 30 mtorr was derived from photon echo data in ref. [6]. 89
Volume 8, n u m b e r 1
OPTICS COMMUNICATIONS
1.0
,~_
0.5
'~E
/.,,~,~~...... /s~/~:/
0.2 0.1
/
~ Experimental "PP (6)(6) ~ Nondegenerate
I
1
10 Input energy (Arbitrary units)
I
100
Fig. 2. The ratio of transmitted energy to incident energy versus incident energy for transmission of the CO2 P (14) transition in SF6 illustrating the data of ref. [ 1 ] as well as the
results for nondegenerate and degenerate P(6) transitions. Note that neither of the calculated curves exhibits a modulated transmissions such as shown by the data of ref. [ 1 ].
which allow the magnetic sublevels to be thermalized at a specified rate. A very rapid rate corresponds to the equilibrating collisional mechanism postulated in ref. [1]; a sufficiently slow rate pertains to the case of independent coherently excited degenerate ensembles. The smooth variation of the curve in fig. 2 is obtained in both limits. Furthermore, the lack of a pulse break-up is not influenced by the rate of magnetic sublevel thermalization. In contrast to the transmission factors, the calculated delay times are found to be in reasonable agreement with the experimental results. As in the previous case, we find that the differences between the results of the degenerate and nondegenerate calculations are so minor ( ~ 1%) that they could not be distinguished on the basis of the experiment performed in ref. [1]. This agreement has two important implications. Firstly, it demonstrates that the value of T 2 that we used in the numerical calculations is correct. In the regime of large attenuation, the maximum observed delay varies in proportion to and is approximately equal to T 2. Thus, any major discrepancy concerning the appropriate value of T 2 is unlikely. Furthermore, if we associate the experimentally determined maxi-
90
May 1973
mum delay [1] with the value of the input area 0 indicated by the calculation (0in ~ 27r for maximum delay appropriate to the experimental ratio of the pulse width to T 2 rather than 0in ~ 71"which is valid for the case where T 2 is much larger than the pulse width [7], we find that this reduces the value of the dipole moment that was estimated in ref. [1] by about a factor of two. This brings the value into accord with other determinations [3] (it ~ 3 X 10 -19 esu cm). On the basis of the comparison of our numerical findings with the experimental results of ref. [ 1], we conclude that Zembrod and Gruhl were not justified in proposing the model calling for a collisionally thermalized * ensemble of degenerate transitions. Our calculations show that under their experimental conditions the value of T 2 was sufficiently short to substantially suppress any effects arising from degeneracy. For these calculations, we gratefully acknowledge the expert computational assistance of Warren G. Cunninghana. * Intuitively such a special collisional mechanism seems quite improbable. Indeed, there exists a special class of collisions where this effect is ruled out on general grounds by consideration of rotational invariance, time reversal invariance, and parity conservation [8].
References [1] A. Zembrod and Th. Gruhl, Phys. Rev. Letters 27 (1970) 287. I2] C.L. Tang and H. Statz, Appl. Phys. Letters 10 (1967) 145. [3] C.K. Rhodes and A. Szoke, Phys. Rev. 184 (1969) 25. [4] F.A. Hopf, C.K. Rhodes and A. Sz6ke, Phys. Rev. B1 (1970) 2833. [5] E.D. ftinkley, private communication. [6] C.K.N. Patel and R.E. Slusher, Phys. Rev. Letters 20 (1968) 1087. [7] F.A. Hopf and M.O. Scully, Phys. Rev. B1 (1970) 50. [8] C.K. Rhodes and A. Sz6ke, unpublished.