Nov 1, 1984 - Superfluorescent pulses have been observed so far mainly in gases [1]. At low pressures the relaxation times T 2 and T~ which correspond to ...
Volume 52, number 1
OPTICS COMMUNICATIONS
1 November 1984
INDUCED SUPERRADIANCE IN ACTIVATED CRYSTALS R.F. MALIKOV and E.D. TRIFONOV
Department of Theoretical Physics, Herzen PedagogicalInstitute, Leningrad, USSR Received 23 January 1984 Revised manuscript received 3 August 1984
The observation of superradiance in activated crystals induced by propagation of a short coherent pulse through an inverted sample is proposed. The theoretical estimation of the induced superradiance in Nd :YAG is given.
Superfluorescent pulses have been observed so far mainly in gases [1]. At low pressures the relaxation times T2 and T~ which correspond to homogeneous and inhomogeneous broadening are of the order of 10-100 ns. The duration of the superfluorescent pulses in these systems is usually of the same order of magnitude. Superfluorescence is suppressed by the relaxation unless the characteristic time intervals of radiation are shorter than T2 and T~. As for the activated crystals the relaxation times commonly lie in the picosecond range, so one can expect a superfluorescent pulse to be able of picosecond duration. Recently two-color superfluorescence of O~-centers has been observed in KCI crystals [2]. According to the authors of ref. [2] the delay time t d of superfluorescent pulses varies from 0.5 ns to 10 ns. Although in this system T2 and T~ are of the order of 100 ps, superfluorescence seems to survive due to a very high gain. Evidently such relation between t d and the relaxation times leads to strong suppression of the superfluorescent pulses as compared to the ideal situation when T2, T~ >>t d. The theory for this experiment is given in ref. [3]. The nanosecond superfluorescent pulse, also suppressed by the relaxation, has been observed in the diphenyl crystal with impurity ofpyrene [4]. Superfluorescence is initiated by the quantum fluctuations of atomic polarisation. As shown in ref. [5] the initial atomic polarisation is a gaussian random field. The corresponding distribution of initial tipping angles 0 of the Bloch vectors yields the mean value 00 74
= 2/V~, where N is a full number of initially inverted atoms in a cooperatively radiating system. Random initiation of superfluorescence results in variation of the shape of the pulses. For instance the delay time of the first intensity maximum has the variance as large as 12% of its mean value [6]. The latter can be estimated as t d = (rR/4)lln O0/2rrl2 ,
(1)
where r R is the superradiant time. This expression for t d has been obtained in ref. [7] (see also ref. [8]) for the case, when the Fresnel number is close to unity, the sample is shorter than cooperative length, T2 = T~ = oo and 0 is the same for all atoms. Comparison of (1) with the results of the delay time statistics obtained in ref. [7] shows that (1) gives a correct order of magnitude of t d . The delay time can be reduced by injecting a short coherent pulse of suitable area into an inverted sample. The injected pulse induces atomic polarisation which then stimulates superradiance. Such an experiment has been performed in cesium vapor by the authors of ref. [9]. In this case the tipping angle of the Bloch vector is not a random one since it is determined by the area of the driving pulse
O(r) = (g/h) /
E(t, r)dt,
(2)
--oo
where # is the matrix element of the transition dipole moment and E(t, r) is the amplitude of the electric field in the point r. If the input pulse is sufficiently
Volume 52, number 1
OPTICS COMMUNICATIONS
short (tp < rR) it only slightly modifies when passing through the medium and hence 0 is nearly the same for all the atoms. It is well known that under this condition the superradiant pulse can be described by the Burnham-Chiao automodel solution of the MaxwellBloch equations [10]. Let us consider the possibility to observe induced superradiance in Nd :YAG. The system of the Maxwell-Bloch equations for the slowly varying complex amplitudes E ± and P± of the electric field and of the polarisation can be written in the one-dimensional approximation as
Oa±/~x = (S2/o) f R ± ( t , x , u) G(u)d~,
~,
\
~
~
length L = 3.64 v~2-1 , T2 = 5 f r I , T~ = ~. (a) eo = 0.02 lr, tp = 0.05 ~ - 1 ; Co) e o = 0.02 ~r, tp = 0.2 f~-I ; (e) # o = 0.04 ~r, tp = 0.1 ~2-1 ; (d) Oo = 0.01 n, tp = 0.1 ~2-1 .
2f2A +-(t, x ) Z ( t , x , u ) ,
bZ/i}r = -~2 [A+(t, x ) R - ( t , +
,L
Fig. 1. Induced superradiant pulses for different values of tp (duration at half field amplitude) and of 0 o. The sample
oR±/or = (+_i. - r £ I ) R ±(t, x, ~) +
1 November 1984
.4-(t, x)R+(t, x,
v)]
x , ~)
,
(3)
where 6o0 is the resonance frequency, ~2 = (21rNo~6oo)l/21a/nh, v = 6o - 6oo,A ± = ~- il~Et / h f ~., R ± = P±/laN O, 2 N o Z is the inversion density,N 0 is the initial inversion density, r = t - x / v is a retarded
time, o is the velocity of light in the medium, n is the refractive index, G(v) is the inhomogeneous lineshape. In the experiment on coherent amplification in Nd:YAG at I00 K the small signal gain ~L is equal to 5, other parameters beingN 0 ~ 1016 cm -3 ,L = 6 cm, n = 1.82, T~ ~ 20 ps, T 2 ~ 100 ps [11]. From these data one can obtain r R = T~/c~L = 1/(T~-1 + T ~ - I ) a L 3 ps, ~-1
=
(rR2L/o)I/2 ~
Now let us show that under the same conditions superfluorescence is impossible. From eq. (1) we obtain that, neglecting relaxation, the mean delay time of the superfluorescent pulses are about 300 ps. This value is greater than T 2 and much greater than T~, what is unfavourable for superfluorescence. To estimate the intensity of the superfluorescent pulse we shall add a noise term in eqs. (3) and consider a station. ary regime, provided the inversion is fLxed at its maxiZ
Re A-
50 ps.
The results of numerical integration of eqs. (3) for different gaussian input pulses are illustrated in figs. 1,2, which show the amplified pulse and the superradiant pulse following it. The spectrum of the induced superradiant pulse (see fig. 2) shows a characteristic doublet structure. The splitting of the spectrum is in accordance with the field modulation and can be considered as an indication of coherent interaction between atoms and the field [12]. Hence one can conclude that induced superradiance of Nd :YAG can be observed for the values of homogeneous and inhomogeneous broadening given above.
t 0
A
J
\ ~..------
1 -2
2
.~Q"
Fig. 2. Influence o f inhomogeneous broadening on induced superradiance. L = 3.64 u~2-1 , e 0 = 0.02 n, tp = 0.05 ~2-1 . (a) I n p u t pulse; Co) o u t p u t pulse; (c) inversion at x = L ; (d) spectrum of the output pulse.
75
Volume 52, number 1
OPTICS COMMUNICATIONS
mum value. Then one can use a linear approximation for eqs. (3),
aA+-/Ox= (a/v) f R+-(x, .)a(.) d., 0 =(-+ iv - T ~ I ) R e ( x , v ) + A t ( x ) + A + - ( X , V ) ,
(.2 + rf2)L NG(v)
6 (x - x ' ) 6 ( v - v'). (5)
It follows from the second equation (4) that in the absence o f field the correlation function for polarisation is
0~ (x, +
.)R
--
t
(x,
.')> = ~
Z
~(x - x ' ) ~ ( . -
.'),
(6)
which coincides with the corresponding result o f ref. [6b]. F o r the sake o f simplicity we suppose G(v) to be lorentzian: G(v) = 1/~rT2 (v 2 - T~-~). Then it is easy to show that the cooperative radiation intensity (i.e. the average number o f photons radiated in unit time per atom) is I = (v/L) (A+A - ) = ( I / 2 N T ~ ) ( e 2aL - 1).
(7)
Substituting N = 1016, T~ = 17 ps, ¢~L = 5 in (7) we obtain I ~ 1 s -1 . Such intensity is three order less than that o f the usual spontaneous emission for transition R 1 --> Y1 in Nd +3 and hence superfluorescence is completely suppressed. At the same time it means that the inversion can be obtained in this system by the help o f comparatively slow pumping. This fact should be considered as a favourable condition for cartying out the experiment on induced superradiance.
76
We would like to thank K.K. Rebane, A.M. Leontovich, A.M. Mozharovsky, O.P. Varnavsky and A.S. Troshin for helpful discussions. We are also grateful to F. Haake and D. Schmid for sending the preprints o f the papers [2b] and [3b].
(4)
where A t is the g-correlated source o f the polarisation with the pair correlation function
( a + ( x , v ) A - ( X ' , v')> -
1 November 1984
References [ 1 ] Q.H.F. Vrehen and H.M. Gibbs, in: Dissipative systems in quantum optics, ed. R. Bonifacio (Springer-Verlag, 1982). [2] R. Florian, L. Schwan and D. Schmid, (a) Solid State Comm. 42 (1982) 55 ; (b) to be published. [3] F. Haake and R. Reibold, (a) Phys. Lett. 92A (1982) 29; (b) to be published. [4] P.V. Zinov'ev, S.V. Lopina, Yu.V. Naboikan, N.B. Silaeva, V.V. Samartsev and Yu.E. Sheibut, ZhETF 85 (1983) 1945. [5] R. Glauber and F. Haake, Phys. Lett. 68A (1978) 29. [6] F. Haake, H. King, G. Schr6der, J. Haus and R. Glauber, (a) Phys. Rev. A20 (1979) 2047; (b) Phys. Rev. A23 (1981) 1322. [7] N. Skribanowitz, I.P. Herman, J.C. MacGillivray and M.S. Feld, Phys. Rev. Lett. 30 (1973) 309. [8] J.C. MacGillivray and M.S. Feld, (a) Phys. Rev. A14 (1976) 1169;(b) Phys. Rev. A23 (1981) 1334. [9] Q.H.F. Vrehen and M.F.H, Schuurmans, Phys. Rev. Lett. 42 (1979) 224. [10] D.C. Burnham and R.Y. Chiao, Phys. Rev. 188 (1969) 667. [11 ] O.P. Varnavsky, A.N. Kirkin, A.M. Leontovich, R.F. Malikov, R.G. Mirzoyan, A.M. Mozharovsky and E.D. Trifonov, Optics Comm. 46 (1983) 131. [12] R.F. Malikov, V.A. Malyshev and E.D. Trifonov, Opt. i Spectr. 51 (1981) 406.
