- Email: [email protected]
Spectral properties of superfluorescent pulses
SPECTRAL PROPERTIES Guntram
1 October 1981
OF’TICS COMMUNICATIONS
Volume 39, number 3
OF SUPERFLUORESCENT
PULSES
SCHRGDER and Fritz HAAKE
Gesamthochschule
Essen, Fachbereich
Physik, 4300 Essen, Germany
Maciej LEWENSTEIN and Kazimierz RZAZEWSKI Institute for Theoretical Physics, Polish Academy
of Sciences,
02-668 Warsaw, Poland
Received 8 May 1981
We present a spectral analysis of the early stage of superfluorescent pulses. For systems with a Fresnel number near unity and a length long compared to the wavelength of the emitted radiation we find no perceptible frequency shifts or chirps.
Most of the recent work on superfluorescence [ 1,2] has been based on a one dimensional model and, moreover, the slowly-varying-envelope approximation (SVEA). The latter approximation reduces the wave equation for the electric field operator to a set of two partial differential equations of first order for the envelope operators of the left- and rightgoing pulse. Naive scaling arguments suggest that the SVEA should give reliable results if the length 1 of the system is much larger than the wavelength h pertaining to the atomic transition in question. Such arguments have been corroborated by Lewenstein and Rzazewski [3] who solved the linearized Maxwell Bloch equations without employing the SVEA. The mean radiated intensity as well as the probability density of the delay time so calculated reduce to the SVEA results of Haake et al. [l] as h/l + 0. It has been suggested that superfluorescent pulses display cooperative frequency shifts in their spectra [4]. Such shifts would escape the SVEA entirely since in that approximation the envelope operators for the electric field and the polarization density are linearly related by a real integral kernel. Sizeable frequency shifts would thus signal a qualitative shortcoming of the SVEA. The investigation to be presented here shows that cooperative frequency shifts and chirps can, if present at all, only be of tiny magnitude. Our argument is based on the linearized Maxwell Bloch equations for the usual one dimensional model [5,1]. We can thus 194
not rule out the appearance of large frequency shifts at times near the maximum intensity of the pulse. Their complete absence in the early stages of the pulse makes that possibility a rather unlikely one, however. The notorious intensity ringing found in nonlinear treatments of the one dimensional model for times after the pulse maximum would presumably show up in the spectrum as an oscillatory chirp. Such an effect would have to be considered, like the intensity ringing itself, an artefact of the one dimensional model. As a physically appealing definition of the spectrum of a transient signal we adopt the one proposed by Eberly and Wodkiewicz [6]. We thus assume that the (positive frequency part of the) field E(+)(t) emerging from the right end face of the radiating system passes through a Fabry-Perot interferometer. The response function of the interferometer is characterized by a width F and a resonance frequency w and reads H(t,w)=O(t)Fexp[-(P+iw)t],
(1)
the step function O(f) expresses causality. The electric field impinging on a detector behind the interferometer can then be taken as
.@)(r) = i dt’H(t’, w)E(+l(t -co
- t’).
(2)
The detector will then emit photons at a rate proportional to
0 030-4018/81/0000-0000/$02.75
0 1981 North-Holland
S(t, w) 2
(&)(r)@(r)) t
s
=j-m dt’ _-oodr” P(t
- t’)H(t - t”)
x ue(t’) II?+)(t
(3)
Clearly, the function S(r, o) is a measure of the time dependent spectrum of the radiated pulse. The electric field E(+)(t) radiated during the early stages of the pulse has been calculated in [3]. Its temporal Laplace transform reads, if antiresonant terms are neglected and terms not contributing to normally ordered correlation functions are suppressed, +YZ E(+)(z) = $ dx G(x, z)pr;)(x), -Y2
(4)
here, e)(x) is the (positive frequency part of the) atomic polarization density; the integral kernel is given by
sinh .$x + n sinh t/2 t t cash c;/2
(5)
with 291/c v=z&,
F2=q2+izq,
1 October 1981
OPTICS COMMUNICATIONS
Volume 39, number 3
(6)
c/woZ = 2nX/Z.In the experiments of Gibbs and Vrehen [7], /3- 0.3 and VZ w 10T4. It is easy to construct an asymptotic approximation to the kernel (5) for small values of h/Zand then even to perform an inverse Laplace transform to find the time dependent kernel G(x, r) = exp[iwo@
- r)]
X @(t - px)Zo(2_)
(7)
which is, up to the plane-wave phase factor, just the SVEA result [ 5 ,1,2]. For an evaluation of the spectral density S(t, w) we need the initial state expectation value (P&)(x) X @(x’)). When the radiating atoms are taken to be initially fully inverted two-level atoms they are initially all independent and we have [5,1,2] (p6-‘(x)p6’)(x’), N 6(x - x’).
(8)
In order to save computational labor we have evaluated the spectral density S(t, w) for the exact kernel (5) and its SVEA simplification (7) only after neglecting retardation effects within the radiating volume. That approximation amounts to setting /I = 0 in eq. (7) and to replace z by -iwor everywhere on the r.h.s. of the second of eqs. (6). Obviously, for sufficient ly large t and small values of p the error thus made will be unimportant. We show in fig. 1 the spectral density pertaining to the SVEA kernel (7) for I w - wo(7 G 10, and, from bottom on upwards, t = 2,4,6,8, and 10. The wiggles in the spectrum at the earlier times reflect an intrinsic interferometer transient. They disappear at
the spatial coordinate x is meant in units of the length I and chosen so that the radiating system lies in the interval -$
We would like to point out that two of the parameters occurring in eq. (6) are smaller than unity in typical experiments [7]. One of them is related to the cooperation length I,., /I E Z/CT= (Z/Q2 [ 11, and must be chosen smaller than unity if single-pulse superfluorescence is to be observed [8]. The second smaIl parameter is the inverse dimensionless wave number,
Y
-8
-4
0
4
.Yo
8
Fig. 1. The spectrum for early times.
195
Volume 39, number 3
OPTICS COMMUNICATIONS
1 October 1981
Fig. 2. The spectrum for later times; circles for exact kernel, fidl curves for SVEA.
Fig. 3. The half width of the spectrum as a function of time; circles for “exact” kernel, fulI curve for SVEA.
t a
found the spectral density to become slightly asymmetric and its center shifted away a bit from w. at x/2 = 0.1 and t = 20. Such large values of h/l seem hardly realistic, however, for experiments with visible or infrared light.
10
since
asr=o.17-1*. Lorentz-like
we
have chosen the interferometer width At about this time S(t, w) is a
function
centered
at the atomic
frequen-
cy WO*
In fig. 2 we display the spectrum for, from the bottom curve on upwards, t = 10,20,30, and 40. The full curves are based on the SVEA kernel (7) and the centers of the circles on the “exact” kernel (5) with h/l = $ X 10F4. The remarkable agreement between the two results attests to the reliability of the SVEA and to the absence of any shifts and chirps. Not even for times at which the linearized theory begins to break down (t - 30) does the spectrum exhibit any noticeable asymmetry or displacement of its center from the atomic frequency wo. In fig. 3 we show the temporal development of the half width of the spectral density, defined by ,S(oo f A, t) = $!?(a,, t). Here again, there is no discrepancy to speak of between the SVEA result (full curve) and the “exact” one (circles). For larger values of h/Z we must, of course, expect the SVEA to become less meaningful. Indeed, we have * For 7 = 0.2 ns and wo = 6.4 X 1014 Hz in experiments of ref. [7] this corresponds to r/we = lo6 unrealistic.
196
which is not overly
We gratefully acknowledge the support of this work by a DAAD grant to Kazimierz Rzazewski.
[l] F. Haake, H. King, G. SchrGder, J. Haus, R. Glauber and F. Hopf, Phys. Rev. Lett. 42 (1979) 1740; F. Haake, H. King, G. SchrSder, J. Haus and R. Glauber, Phys. Rev. A20 (1979) 2047; Phys. Rev. Lett. 45 (1980) 558; Phys. Rev. A23 (1981) 1322. [2] D. Polder, M.F.H. Schuurmans and Q.H.F. Vrehen, Phys. Rev. A19 (1979) 1192. [ 31 M. Lewenstein and K. Rzazewski, to be published. [4] G. Banfi and R. Bonifacio, Phys. Rev. A12 (1975) 2068. (51 R. Glauberand F. Haake, Phys. Lett. 68A (1978) 29. [ 61 H.J. Eberly and K. Wodkiewicz, J. Opt. Sot. Am. 67 (1977) 1252. [7] H.M. Gibbs, Q.H.F. Vrehen and H.M.J. Kikspoors, Phys. Rev. Lett. 39 (1977) 547. [8] R. Bonifacio, P. Schwendimann and F. Haake, Phys. Rev. A4 (1971) 302, 854.
1 October 1981
OF’TICS COMMUNICATIONS
Volume 39, number 3
OF SUPERFLUORESCENT
PULSES
SCHRGDER and Fritz HAAKE
Gesamthochschule
Essen, Fachbereich
Physik, 4300 Essen, Germany
Maciej LEWENSTEIN and Kazimierz RZAZEWSKI Institute for Theoretical Physics, Polish Academy
of Sciences,
02-668 Warsaw, Poland
Received 8 May 1981
We present a spectral analysis of the early stage of superfluorescent pulses. For systems with a Fresnel number near unity and a length long compared to the wavelength of the emitted radiation we find no perceptible frequency shifts or chirps.
Most of the recent work on superfluorescence [ 1,2] has been based on a one dimensional model and, moreover, the slowly-varying-envelope approximation (SVEA). The latter approximation reduces the wave equation for the electric field operator to a set of two partial differential equations of first order for the envelope operators of the left- and rightgoing pulse. Naive scaling arguments suggest that the SVEA should give reliable results if the length 1 of the system is much larger than the wavelength h pertaining to the atomic transition in question. Such arguments have been corroborated by Lewenstein and Rzazewski [3] who solved the linearized Maxwell Bloch equations without employing the SVEA. The mean radiated intensity as well as the probability density of the delay time so calculated reduce to the SVEA results of Haake et al. [l] as h/l + 0. It has been suggested that superfluorescent pulses display cooperative frequency shifts in their spectra [4]. Such shifts would escape the SVEA entirely since in that approximation the envelope operators for the electric field and the polarization density are linearly related by a real integral kernel. Sizeable frequency shifts would thus signal a qualitative shortcoming of the SVEA. The investigation to be presented here shows that cooperative frequency shifts and chirps can, if present at all, only be of tiny magnitude. Our argument is based on the linearized Maxwell Bloch equations for the usual one dimensional model [5,1]. We can thus 194
not rule out the appearance of large frequency shifts at times near the maximum intensity of the pulse. Their complete absence in the early stages of the pulse makes that possibility a rather unlikely one, however. The notorious intensity ringing found in nonlinear treatments of the one dimensional model for times after the pulse maximum would presumably show up in the spectrum as an oscillatory chirp. Such an effect would have to be considered, like the intensity ringing itself, an artefact of the one dimensional model. As a physically appealing definition of the spectrum of a transient signal we adopt the one proposed by Eberly and Wodkiewicz [6]. We thus assume that the (positive frequency part of the) field E(+)(t) emerging from the right end face of the radiating system passes through a Fabry-Perot interferometer. The response function of the interferometer is characterized by a width F and a resonance frequency w and reads H(t,w)=O(t)Fexp[-(P+iw)t],
(1)
the step function O(f) expresses causality. The electric field impinging on a detector behind the interferometer can then be taken as
.@)(r) = i dt’H(t’, w)E(+l(t -co
- t’).
(2)
The detector will then emit photons at a rate proportional to
0 030-4018/81/0000-0000/$02.75
0 1981 North-Holland
S(t, w) 2
(&)(r)@(r)) t
s
=j-m dt’ _-oodr” P(t
- t’)H(t - t”)
x ue(t’) II?+)(t
(3)
Clearly, the function S(r, o) is a measure of the time dependent spectrum of the radiated pulse. The electric field E(+)(t) radiated during the early stages of the pulse has been calculated in [3]. Its temporal Laplace transform reads, if antiresonant terms are neglected and terms not contributing to normally ordered correlation functions are suppressed, +YZ E(+)(z) = $ dx G(x, z)pr;)(x), -Y2
(4)
here, e)(x) is the (positive frequency part of the) atomic polarization density; the integral kernel is given by
sinh .$x + n sinh t/2 t t cash c;/2
(5)
with 291/c v=z&,
F2=q2+izq,
1 October 1981
OPTICS COMMUNICATIONS
Volume 39, number 3
(6)
c/woZ = 2nX/Z.In the experiments of Gibbs and Vrehen [7], /3- 0.3 and VZ w 10T4. It is easy to construct an asymptotic approximation to the kernel (5) for small values of h/Zand then even to perform an inverse Laplace transform to find the time dependent kernel G(x, r) = exp[iwo@
- r)]
X @(t - px)Zo(2_)
(7)
which is, up to the plane-wave phase factor, just the SVEA result [ 5 ,1,2]. For an evaluation of the spectral density S(t, w) we need the initial state expectation value (P&)(x) X @(x’)). When the radiating atoms are taken to be initially fully inverted two-level atoms they are initially all independent and we have [5,1,2] (p6-‘(x)p6’)(x’), N 6(x - x’).
(8)
In order to save computational labor we have evaluated the spectral density S(t, w) for the exact kernel (5) and its SVEA simplification (7) only after neglecting retardation effects within the radiating volume. That approximation amounts to setting /I = 0 in eq. (7) and to replace z by -iwor everywhere on the r.h.s. of the second of eqs. (6). Obviously, for sufficient ly large t and small values of p the error thus made will be unimportant. We show in fig. 1 the spectral density pertaining to the SVEA kernel (7) for I w - wo(7 G 10, and, from bottom on upwards, t = 2,4,6,8, and 10. The wiggles in the spectrum at the earlier times reflect an intrinsic interferometer transient. They disappear at
the spatial coordinate x is meant in units of the length I and chosen so that the radiating system lies in the interval -$
We would like to point out that two of the parameters occurring in eq. (6) are smaller than unity in typical experiments [7]. One of them is related to the cooperation length I,., /I E Z/CT= (Z/Q2 [ 11, and must be chosen smaller than unity if single-pulse superfluorescence is to be observed [8]. The second smaIl parameter is the inverse dimensionless wave number,
Y
-8
-4
0
4
.Yo
8
Fig. 1. The spectrum for early times.
195
Volume 39, number 3
OPTICS COMMUNICATIONS
1 October 1981
Fig. 2. The spectrum for later times; circles for exact kernel, fidl curves for SVEA.
Fig. 3. The half width of the spectrum as a function of time; circles for “exact” kernel, fulI curve for SVEA.
t a
found the spectral density to become slightly asymmetric and its center shifted away a bit from w. at x/2 = 0.1 and t = 20. Such large values of h/l seem hardly realistic, however, for experiments with visible or infrared light.
10
since
asr=o.17-1*. Lorentz-like
we
have chosen the interferometer width At about this time S(t, w) is a
function
centered
at the atomic
frequen-
cy WO*
In fig. 2 we display the spectrum for, from the bottom curve on upwards, t = 10,20,30, and 40. The full curves are based on the SVEA kernel (7) and the centers of the circles on the “exact” kernel (5) with h/l = $ X 10F4. The remarkable agreement between the two results attests to the reliability of the SVEA and to the absence of any shifts and chirps. Not even for times at which the linearized theory begins to break down (t - 30) does the spectrum exhibit any noticeable asymmetry or displacement of its center from the atomic frequency wo. In fig. 3 we show the temporal development of the half width of the spectral density, defined by ,S(oo f A, t) = $!?(a,, t). Here again, there is no discrepancy to speak of between the SVEA result (full curve) and the “exact” one (circles). For larger values of h/Z we must, of course, expect the SVEA to become less meaningful. Indeed, we have * For 7 = 0.2 ns and wo = 6.4 X 1014 Hz in experiments of ref. [7] this corresponds to r/we = lo6 unrealistic.
196
which is not overly
We gratefully acknowledge the support of this work by a DAAD grant to Kazimierz Rzazewski.
[l] F. Haake, H. King, G. SchrGder, J. Haus, R. Glauber and F. Hopf, Phys. Rev. Lett. 42 (1979) 1740; F. Haake, H. King, G. SchrSder, J. Haus and R. Glauber, Phys. Rev. A20 (1979) 2047; Phys. Rev. Lett. 45 (1980) 558; Phys. Rev. A23 (1981) 1322. [2] D. Polder, M.F.H. Schuurmans and Q.H.F. Vrehen, Phys. Rev. A19 (1979) 1192. [ 31 M. Lewenstein and K. Rzazewski, to be published. [4] G. Banfi and R. Bonifacio, Phys. Rev. A12 (1975) 2068. (51 R. Glauberand F. Haake, Phys. Lett. 68A (1978) 29. [ 61 H.J. Eberly and K. Wodkiewicz, J. Opt. Sot. Am. 67 (1977) 1252. [7] H.M. Gibbs, Q.H.F. Vrehen and H.M.J. Kikspoors, Phys. Rev. Lett. 39 (1977) 547. [8] R. Bonifacio, P. Schwendimann and F. Haake, Phys. Rev. A4 (1971) 302, 854.