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Oscillatings superfluorescence: The shape and delay of the leading pulse
Volume 31, number 2
OPTICS COMMUNICATIONS
November 1979
OSCILLATING SUPERFLUORESCENCE: THE SHAPE AND DELAY OF THE LEADING PULSE J.A. HERMANN * and R.K. BULLOUGH Mathematics Department, Universityof Manchester, Institute of Science and Technology, P.O. Box 88, ManchesterM60 1QD, U.K. Received 13 July 1979
Expressions are given for the shape and delay of the leading pulse in superfluorescent emission from a spatially extended medium, within the context of a simple semiclassicaldescription. The results are in agreement with numerical "two-way" Bloch-Maxwell simulations previously reported by several authors.
Quantum-mechanical treatments of superfluores. cence (SF) from an array of two-level atoms, in which the coherent interaction is analysed in terms of planewave end-fire modes only, have been given recently by Polder, Schuurmans and Vrehen (PSV) [1,2], as well as by Glauber and Haake (GH) [3], Gronchi et al. [4], and Ikeda and Sawada [5]. In these works the common physical assumptions are (a) The atoms are positioned randomly within the pencil-shaped active volume (of cross-section S and length L); (b) The Fresnel number ~r = S/XOL is about unity (X0(<
OXT = ot sin o ,
(I)
where o(X, T) is the collective Bloch angle and X =x[L, T = ct/L are scaled laboratory-frame space and time coordinates (0 ~
sentially the superradiant time r R . Initial and boundary conditions are o(0, 7*) = o(X, 0) = a 0, where o 0 is an effective initial tipping angle. The principal limitations of this form of the semiclassical description are that oppositely-directed fields have been assumed to evolve independently (i.e., it describes the propagation in one direction; this has a degree of validity for the leading SF pulse, but does not describe the ringing process adequately), that as in all semiclassical descriptions the problem of how the SF is initiated has been avoided, and that damping effects have been ignored. The electric field envelope for the emission in a prescribed direction is ~o/ST; in order to satisfy the physical requirements that propagation is unidirectional and that there is no field initially, one imposes the conditions aT(O , 73 = O, aT(X, 0) = 0. Propagation of energy is ensured by matching aT(1 , 73 with the amplitude of the external (vacuum) field at X = 1. In the work of PSV [1 ] an expression has been derived for the mean-squared collective Bloch angle at the early stage of the motion; this expression reduces to (02) 1/2 -~ 2(aT/N) 1/2 at the end face (X = 1) when 2(a73 1/2 ~ 1, N being the total number of active atoms in the sample. The appropriate linear solution of the semiclassical equation (1)however is o = o010(0 ) where 0 = 2(aX73 1/2, and I0(0 ) = J0(i0) is a zeroth-order modified Bessel function. For 0 >> 1 this solution becomes a ~ 4~(0) = aO(27rO)-l/2eO. A rough match of the semiclassical and quantummechanical expressions at 0 "" 1 indicates that o 0 is of order N -1/2. 219
Volume 31, n u m b e r 2
OPTICS COMMUNICATIONS
A different theoretical approach to SF, the wellknown mean-field theory, has been developed by Bonifacia and Lugiato and their collaborators (BL) [8,9,10]. Although this is a quantum theory, it has been reduced to the form of an averaged semiclassical theory. Solutions for the electric field have been obtained under the assumption that spatial variations of the field envelopes are negligible in the active region. These authors find that the emission intensity possesses a sech 2 shape with respect to T, and that for sufficiently large tipping angle the delay-time r D is roughly proportional to a - 1 Iln o 0 I. They have also demonstrated [I0] that a numerical solution of the twoway Bloch-Maxwell equations for r R >>Lc -1 (with appropriate boundary and initial conditions) may be fitted by a sech 2 curve, with the same width and height as the first peak in the output intensity, with considerable success. The relevance and applicability of this theory to recent SF experiments has been questioned recently by Bullough et al. [11 ], and by Vrehen and Schuurmans [16]. One purpose of this comment is to show that the apparently successful fitting of a sech 2 curve to the leading peak does not in itself constitute support for either the spatially-dependent or spatially-averaged formulations of SF. It will also be shown that recent experimental work in SF is compatibel with equation (1), if allowance is made for the dissipative processes. In a recent non-linear analysis of equation (1), it has been shown [I 2] that the emitted intensity I(T) = loT(l, T)] 2 does not possess a simple sech 2 shape, but assumes the following form in the domain of the leading pulse (0 >> 1): I(T) = 16a20-2F(O) 2 sech2x(0),
In the region where 0 ~< 1 it is necessary to use the linear solution. Apart from the predictions of the mean field theory, this appears to be the only analytical nonlinear solution in SF theory to date. According to the expression (2) the first stationary point in/(7") occurs at r D = ~tr-l{X + ½1n X
_ ~(1_~_ In ~)X-1 + O(X-2)}2,
x 10 -3
2O
x(O) = 0 - ½In 0 - X,
(3)
15
X = ln(4v~-~/o 0),
(4)
-~I0
(5)
5
- 0-2B2(~)
+ ....
1 3 + ~2 + 3@2/(1 + ~b2)} BI(O) = ~{g
(6a)
B2(~) = 1~-s{15--4~2 + 112~2/(1 + @2) _ 72~4/(1 + ff2)2}.
220
(7)
the first term of which agrees with the form of the expression for the delay reported by MF, as well as with the corresponding quantum-mechanical calculations of PSV and GH. The Bloch angle at T = r D is found to possess the value OD = rr -- 1 ( ~ _ + In X)X-1 + O ( ~ - 2 ) while if(0) has the corresponding value ~ D = 1 -~(~- + In X)X-1 + O(X-2) (cf: linearised theory would predict o D = 4ffD ). The width at half maximum intensity is r w = a - 1 ln(1 + X/~)[X + ½1n X + ½ + O(X-1)] Eq. (2) is also found to agree with the two-way numerical Bloch-Maxwell simulations by BuUough et al. [11 ] of the SF experiments with Cs vapour reported by Vrehen [13], apart from a positive shift of order 7 10% in the delay of the computed intensity which is attributable to the interaction between the oppositelydirected fiels. Note that the Bloch-Maxwell simulations by SB [6] of the HF experiment reported by Skribanowitz et al. [14] have also shown a difference between the positions of the delays in the one-way and two-way formulations of the same order of magnitude. In fig. 1 the leading peak i n / ( 7 ) has been plotted
(2)
where
F(O) = 1 - 0 - 1 B I ( ~ )
November 1979
(6b)
25
4a T
50
Fig. 1. The leading peak in I(T) as a function o f 4 a T with h = 5.410 (full line) and a sech 2 fit in which the height, width at h a l f - m a x i m u m intensity and delay have been treated as free parameters (broken line).
Volume 31, number 2
OPTICS COMMUNICATIONS
as a function of T from eq. (2), with ~ = 5.410 (i.e., = 1 - cos go = 10-3) • For the purpose of comparison, a curve of the f o r m y = a s e c h 2 b ( T - TO)with the same height and width at half maximum has been drawn in broken line. Asymmetry clearly appears only in the wings, and eq. (2) agrees with the numerical simulation of BL in this respect. (Note, however, that the inclusion of a linear loss term in eq. (1) leads to an even better sech 2 fit.) In the mean-field theory the sech 2 solution has only one free parameter for a given delay, i.e., r R. It must be concluded that a reasonable sech 2 fit should not be interpreted as evidence for the validity of either the spatiallydependent or spatially-averged theories. It should be added that a two-way formulation clearly reduces, and higher-order standing-wave effectsffurther reduce, the strength of ringing (see BuUough et al. [11]). However, ringing is always present in the undamped Bloch-MaxweU simulations for L - l c r R >~ 1 (a ~ 1), and the emission profile is unchanged for different ~'R except in the scaling, which is determined entirely by ~'R. In so far as the present theories which predict a simple sech 2 form for the output intensity seem to be inapplicable, the reasons for the suppression of ringing in the experiments of Gibbs et al. with Cs vapour [15] have yet to be unambiguously elucidated. Bonifacio et al. [10] stress another important feature of the results of the cesium experiment, namely that "CD/Zw ~ 2.2. This feature is reproduced when o 0 ~ 10 -2 - 5 X 10 -3 using both the numerical Bloch-Maxwell simulation (Bullough et al., [ 11 ]) and also the sine-Gordon equation (1), as can be seen from fig. 2, in which TD and Tw are shown as functions of g0. Note also that these results are in conflict with mean-field theory, which predicts in particular that r w has essentially no g0-dependence. A direct measurement of o 0 for the cesium experiment has very recently been reported by Vrehen and Schuurmans [16]. They find the range 10 -4 ~< o 0 ~< 2.5 X 10 -3. The value predicted for o 0 when our simple semiclassical theory is reconciled with the observed ratio TDfi-w is outside this range, however it must be remembered that dissipative processes not: accounted for in this analysis may broaden the pulses slightly. A correction to ZD/Tw (a decrease by 2 0 25% from the observed value) accordingly brings the predicted value of g0 within the above experimental range.
November 1979
48
12 /
8
32
f~)
a:
I
J
2
L
I
3
4
n
Fig. 2. The widths at half-maximum intensity (a) and delays (b) of the leading SF pulse, according to the two-way MaxwellBloch equations (full lines) and the one-way sine-Gordon equation (1) (broken line), expressed as functions of go.
The order of magnitude of such a correction is justified by considering the diversity of fluctuation and dissipation processes which can affect the intensities, delay times, and widths of the emitted pulses. In the numerical work of SB it was found that the width of the leading pulse could be increased by as much as 10% when a moderate degree of field-damping is introduced. This was accomplished by adding a linear loss term to the Maxwell field equations. A more satisfactory procedure, still to be carried out, would be to abandon the plane-wave assumption for the fields and to treat diffraction loss as a three-dimensional problem. Other factors which, when allowed for, tend to broaden the pulses and to reduce their intensities, are (a) the presence in the axial directions of higher harmonic components of the polarisation and inversion (standing-wave effect); (b) the effects of off-axial competing field modes; (c) the inclusion of decay terms in the Blochequations (homogeneous broadening); (d) the inclusion of Doppler dephasing (inhomogeneous broadening); (e) the persistence of large intensity fluctuations into the nonlinear regime (i.e., at sufficiently large times following the initiation of the SF). In the last case, it has been pointed out by Haake [17], that procedures for fitting an early-state (linear) description to a later deterministic description at a suitable intermediate time may be unsatisfactory, since it may be impossible to reliably separate the 221
Volume 31, number 2
OPTICS COMMUNICATIONS
domains o f validity for each description. Recent work by Hopf [I 8] has shown that possible "phase-wave" fluctuations, when allowed for in the equations of motion for SF emission, can also affect the characteristics of the output intensity. Although these fluctuations may play an important role in the statistics of SF, it seems that the experimental prospects for establishing the existence o f such waves are poor. The importance of such phase-wave effects in the timedomain o f the ringing has yet to be established. In conclusion, although we have never doubted the essential character o f the conclusions now emerging from two analyses recently reported [19,20], and although we recognise that the blind application o f simple semi-classical theory together with a uniform initial tipping angle gives no insight into the quantum processes initiating the emission or into the various fluctuations which from shot to shot can affect the character o f the emission in the nonlinear regime, we find that this relatively simple and numerically expedient theory still has some relevance as a description of the observations of SF. However, the absence of ringing observed in the cesium experiments for large enough r R still remains an essentially unsolved problem.
References [ 1] D. Polder, M.F.H. Schuurmans and Q.H.F. Vrehen, Phys. Rev. A19 (1979) 1192. [ 2] M.F.H. Schuurmans, D. Polder and Q.H.F. Vrehen, Superfluorescence: QM derivation of MaxweU-Bloch description with fluctuating field source (Tenth Int. Quant. Electronics Conf., Atlanta, Georgia); J. Opt. Soc. Am. 68 (1978) 699. [3] IL Glauber and F. Haake, Phys. Lett. 68A (1978) 29.
222
November 1979
[4] M. Gronchi, L.A. Lugatio and P. Butera, Phys. Rev. A18 (1978) 689. [5] K. Ikeda, Phys. Lett. 63A (1977) 246; K. Ikeda and S. Sawada, Phys. Lett. 59A (1976) 205. [6] R. Saunders and R.K. BuUough, Theory of FIR superfluorescence, in: Cooperative effects in matter and radiation, eds. Bowden, Howgate and Robl (Plenum Press, New York, 1977) p. 209. [7] J. MacGillivray and M. Feld, Phys. Rev. A14 (1976) 1169. [8] R. Bonifacio and L. Lugiato, Phys. Rev. A l l (1975) 1507; A12 (1975) 587. [9] R. Bonifacio, M. Gronchi, L. Lugiato and A. Ricca, MaxwellBloch equations and mean-field theory for superfluorescence, in Cooperative effects in matter and radiation, eds. Bowden, Howgate and Robl (Plenum Press, New York, 1977) p. 193. [10] R. Bonifacio, M. Gronehi, L. Lugiato and A. Ricca, Superfluorescence: Maxwell-Bioch equations, mean field approach and Cs experiment, in Coherence and quantum optics, IV (Proc. 4th Rochester Conference on Coherence and Quantum Optics, Univ. of Rochester, June 1977) eds. Mandel and Wolf (Plenum Press, New York, 1978) p. 939. [ 11 ] R.K. BuUough, R. Saunders and C. Feuillade, Theory of FIR superfluorescence, in Coherence and quantum optics, IV (Proc. 4th Rochester Conference on Coherence and Quantum Optics, Univ. of Rochester, June 1977), eds. Mandel and Wolf (Plenum Press, New York, 1978) p. 263. [12] J.A. Hermann, Phys. Lett. 69A (1979) 316. [13] Q.H.F. Vrehen, Single-pulse superfluorescence in cesium, in Cooperative effects in matter and radiation, eds. Bowden, Howgate and Robl (Plenum Press, New York, 1977) p. 79. [14] N. Skribanowitz, I.P. Herman, J.C. MaeGillivray and M.S. Feld, Phys. Rev. Lett. 30 (1973) 309. [15] H.M° Gibbs, Q.H.F. Vrehen and H.M.J. Hikspoors, Phys. Rev. Lett. 39 (1977) 547. [16] Q.H.F. Vrehen and M.F.H. Schuurmans, Phys. Rev. Lett. 42 (1979) 224. [17] F. Haake, Phys. Rev. Lett. 41 (1978) 1685. [ 18] F.A. Hopf, private communication. [19] F. Haake, H. King, G. Schr/Jder, J. Hans, R. Glauber and F. Hopf, Phy~ Rev. Lett. 42 (1979) 1740. [20] F. Haake, H. King, G. Schr~der, J. Hans and R. Glauber, to be published.
OPTICS COMMUNICATIONS
November 1979
OSCILLATING SUPERFLUORESCENCE: THE SHAPE AND DELAY OF THE LEADING PULSE J.A. HERMANN * and R.K. BULLOUGH Mathematics Department, Universityof Manchester, Institute of Science and Technology, P.O. Box 88, ManchesterM60 1QD, U.K. Received 13 July 1979
Expressions are given for the shape and delay of the leading pulse in superfluorescent emission from a spatially extended medium, within the context of a simple semiclassicaldescription. The results are in agreement with numerical "two-way" Bloch-Maxwell simulations previously reported by several authors.
Quantum-mechanical treatments of superfluores. cence (SF) from an array of two-level atoms, in which the coherent interaction is analysed in terms of planewave end-fire modes only, have been given recently by Polder, Schuurmans and Vrehen (PSV) [1,2], as well as by Glauber and Haake (GH) [3], Gronchi et al. [4], and Ikeda and Sawada [5]. In these works the common physical assumptions are (a) The atoms are positioned randomly within the pencil-shaped active volume (of cross-section S and length L); (b) The Fresnel number ~r = S/XOL is about unity (X0(<
OXT = ot sin o ,
(I)
where o(X, T) is the collective Bloch angle and X =x[L, T = ct/L are scaled laboratory-frame space and time coordinates (0 ~
sentially the superradiant time r R . Initial and boundary conditions are o(0, 7*) = o(X, 0) = a 0, where o 0 is an effective initial tipping angle. The principal limitations of this form of the semiclassical description are that oppositely-directed fields have been assumed to evolve independently (i.e., it describes the propagation in one direction; this has a degree of validity for the leading SF pulse, but does not describe the ringing process adequately), that as in all semiclassical descriptions the problem of how the SF is initiated has been avoided, and that damping effects have been ignored. The electric field envelope for the emission in a prescribed direction is ~o/ST; in order to satisfy the physical requirements that propagation is unidirectional and that there is no field initially, one imposes the conditions aT(O , 73 = O, aT(X, 0) = 0. Propagation of energy is ensured by matching aT(1 , 73 with the amplitude of the external (vacuum) field at X = 1. In the work of PSV [1 ] an expression has been derived for the mean-squared collective Bloch angle at the early stage of the motion; this expression reduces to (02) 1/2 -~ 2(aT/N) 1/2 at the end face (X = 1) when 2(a73 1/2 ~ 1, N being the total number of active atoms in the sample. The appropriate linear solution of the semiclassical equation (1)however is o = o010(0 ) where 0 = 2(aX73 1/2, and I0(0 ) = J0(i0) is a zeroth-order modified Bessel function. For 0 >> 1 this solution becomes a ~ 4~(0) = aO(27rO)-l/2eO. A rough match of the semiclassical and quantummechanical expressions at 0 "" 1 indicates that o 0 is of order N -1/2. 219
Volume 31, n u m b e r 2
OPTICS COMMUNICATIONS
A different theoretical approach to SF, the wellknown mean-field theory, has been developed by Bonifacia and Lugiato and their collaborators (BL) [8,9,10]. Although this is a quantum theory, it has been reduced to the form of an averaged semiclassical theory. Solutions for the electric field have been obtained under the assumption that spatial variations of the field envelopes are negligible in the active region. These authors find that the emission intensity possesses a sech 2 shape with respect to T, and that for sufficiently large tipping angle the delay-time r D is roughly proportional to a - 1 Iln o 0 I. They have also demonstrated [I0] that a numerical solution of the twoway Bloch-Maxwell equations for r R >>Lc -1 (with appropriate boundary and initial conditions) may be fitted by a sech 2 curve, with the same width and height as the first peak in the output intensity, with considerable success. The relevance and applicability of this theory to recent SF experiments has been questioned recently by Bullough et al. [11 ], and by Vrehen and Schuurmans [16]. One purpose of this comment is to show that the apparently successful fitting of a sech 2 curve to the leading peak does not in itself constitute support for either the spatially-dependent or spatially-averaged formulations of SF. It will also be shown that recent experimental work in SF is compatibel with equation (1), if allowance is made for the dissipative processes. In a recent non-linear analysis of equation (1), it has been shown [I 2] that the emitted intensity I(T) = loT(l, T)] 2 does not possess a simple sech 2 shape, but assumes the following form in the domain of the leading pulse (0 >> 1): I(T) = 16a20-2F(O) 2 sech2x(0),
In the region where 0 ~< 1 it is necessary to use the linear solution. Apart from the predictions of the mean field theory, this appears to be the only analytical nonlinear solution in SF theory to date. According to the expression (2) the first stationary point in/(7") occurs at r D = ~tr-l{X + ½1n X
_ ~(1_~_ In ~)X-1 + O(X-2)}2,
x 10 -3
2O
x(O) = 0 - ½In 0 - X,
(3)
15
X = ln(4v~-~/o 0),
(4)
-~I0
(5)
5
- 0-2B2(~)
+ ....
1 3 + ~2 + 3@2/(1 + ~b2)} BI(O) = ~{g
(6a)
B2(~) = 1~-s{15--4~2 + 112~2/(1 + @2) _ 72~4/(1 + ff2)2}.
220
(7)
the first term of which agrees with the form of the expression for the delay reported by MF, as well as with the corresponding quantum-mechanical calculations of PSV and GH. The Bloch angle at T = r D is found to possess the value OD = rr -- 1 ( ~ _ + In X)X-1 + O ( ~ - 2 ) while if(0) has the corresponding value ~ D = 1 -~(~- + In X)X-1 + O(X-2) (cf: linearised theory would predict o D = 4ffD ). The width at half maximum intensity is r w = a - 1 ln(1 + X/~)[X + ½1n X + ½ + O(X-1)] Eq. (2) is also found to agree with the two-way numerical Bloch-Maxwell simulations by BuUough et al. [11 ] of the SF experiments with Cs vapour reported by Vrehen [13], apart from a positive shift of order 7 10% in the delay of the computed intensity which is attributable to the interaction between the oppositelydirected fiels. Note that the Bloch-Maxwell simulations by SB [6] of the HF experiment reported by Skribanowitz et al. [14] have also shown a difference between the positions of the delays in the one-way and two-way formulations of the same order of magnitude. In fig. 1 the leading peak i n / ( 7 ) has been plotted
(2)
where
F(O) = 1 - 0 - 1 B I ( ~ )
November 1979
(6b)
25
4a T
50
Fig. 1. The leading peak in I(T) as a function o f 4 a T with h = 5.410 (full line) and a sech 2 fit in which the height, width at h a l f - m a x i m u m intensity and delay have been treated as free parameters (broken line).
Volume 31, number 2
OPTICS COMMUNICATIONS
as a function of T from eq. (2), with ~ = 5.410 (i.e., = 1 - cos go = 10-3) • For the purpose of comparison, a curve of the f o r m y = a s e c h 2 b ( T - TO)with the same height and width at half maximum has been drawn in broken line. Asymmetry clearly appears only in the wings, and eq. (2) agrees with the numerical simulation of BL in this respect. (Note, however, that the inclusion of a linear loss term in eq. (1) leads to an even better sech 2 fit.) In the mean-field theory the sech 2 solution has only one free parameter for a given delay, i.e., r R. It must be concluded that a reasonable sech 2 fit should not be interpreted as evidence for the validity of either the spatiallydependent or spatially-averged theories. It should be added that a two-way formulation clearly reduces, and higher-order standing-wave effectsffurther reduce, the strength of ringing (see BuUough et al. [11]). However, ringing is always present in the undamped Bloch-MaxweU simulations for L - l c r R >~ 1 (a ~ 1), and the emission profile is unchanged for different ~'R except in the scaling, which is determined entirely by ~'R. In so far as the present theories which predict a simple sech 2 form for the output intensity seem to be inapplicable, the reasons for the suppression of ringing in the experiments of Gibbs et al. with Cs vapour [15] have yet to be unambiguously elucidated. Bonifacio et al. [10] stress another important feature of the results of the cesium experiment, namely that "CD/Zw ~ 2.2. This feature is reproduced when o 0 ~ 10 -2 - 5 X 10 -3 using both the numerical Bloch-Maxwell simulation (Bullough et al., [ 11 ]) and also the sine-Gordon equation (1), as can be seen from fig. 2, in which TD and Tw are shown as functions of g0. Note also that these results are in conflict with mean-field theory, which predicts in particular that r w has essentially no g0-dependence. A direct measurement of o 0 for the cesium experiment has very recently been reported by Vrehen and Schuurmans [16]. They find the range 10 -4 ~< o 0 ~< 2.5 X 10 -3. The value predicted for o 0 when our simple semiclassical theory is reconciled with the observed ratio TDfi-w is outside this range, however it must be remembered that dissipative processes not: accounted for in this analysis may broaden the pulses slightly. A correction to ZD/Tw (a decrease by 2 0 25% from the observed value) accordingly brings the predicted value of g0 within the above experimental range.
November 1979
48
12 /
8
32
f~)
a:
I
J
2
L
I
3
4
n
Fig. 2. The widths at half-maximum intensity (a) and delays (b) of the leading SF pulse, according to the two-way MaxwellBloch equations (full lines) and the one-way sine-Gordon equation (1) (broken line), expressed as functions of go.
The order of magnitude of such a correction is justified by considering the diversity of fluctuation and dissipation processes which can affect the intensities, delay times, and widths of the emitted pulses. In the numerical work of SB it was found that the width of the leading pulse could be increased by as much as 10% when a moderate degree of field-damping is introduced. This was accomplished by adding a linear loss term to the Maxwell field equations. A more satisfactory procedure, still to be carried out, would be to abandon the plane-wave assumption for the fields and to treat diffraction loss as a three-dimensional problem. Other factors which, when allowed for, tend to broaden the pulses and to reduce their intensities, are (a) the presence in the axial directions of higher harmonic components of the polarisation and inversion (standing-wave effect); (b) the effects of off-axial competing field modes; (c) the inclusion of decay terms in the Blochequations (homogeneous broadening); (d) the inclusion of Doppler dephasing (inhomogeneous broadening); (e) the persistence of large intensity fluctuations into the nonlinear regime (i.e., at sufficiently large times following the initiation of the SF). In the last case, it has been pointed out by Haake [17], that procedures for fitting an early-state (linear) description to a later deterministic description at a suitable intermediate time may be unsatisfactory, since it may be impossible to reliably separate the 221
Volume 31, number 2
OPTICS COMMUNICATIONS
domains o f validity for each description. Recent work by Hopf [I 8] has shown that possible "phase-wave" fluctuations, when allowed for in the equations of motion for SF emission, can also affect the characteristics of the output intensity. Although these fluctuations may play an important role in the statistics of SF, it seems that the experimental prospects for establishing the existence o f such waves are poor. The importance of such phase-wave effects in the timedomain o f the ringing has yet to be established. In conclusion, although we have never doubted the essential character o f the conclusions now emerging from two analyses recently reported [19,20], and although we recognise that the blind application o f simple semi-classical theory together with a uniform initial tipping angle gives no insight into the quantum processes initiating the emission or into the various fluctuations which from shot to shot can affect the character o f the emission in the nonlinear regime, we find that this relatively simple and numerically expedient theory still has some relevance as a description of the observations of SF. However, the absence of ringing observed in the cesium experiments for large enough r R still remains an essentially unsolved problem.
References [ 1] D. Polder, M.F.H. Schuurmans and Q.H.F. Vrehen, Phys. Rev. A19 (1979) 1192. [ 2] M.F.H. Schuurmans, D. Polder and Q.H.F. Vrehen, Superfluorescence: QM derivation of MaxweU-Bloch description with fluctuating field source (Tenth Int. Quant. Electronics Conf., Atlanta, Georgia); J. Opt. Soc. Am. 68 (1978) 699. [3] IL Glauber and F. Haake, Phys. Lett. 68A (1978) 29.
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November 1979
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