Superradiance in layers of excited quantum and nonisochronous classical oscillators

Optics Communicahons 91 (1992) 140-146 North-Holland

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Superradiance in layers of excited quantum and nonisochronous classical oscillators N.S. G i n z b u r g a n d A.S. Sergeev Institute oflApphed Physws, USSR Academy of Sctences, 46 Uljanov Str , 603600 Ntzhny Novgorod, USSR Recewed 29 November 1990, revised manuscript recewed 7 August 1991

The b~dlrectlonalsuperradmnce of extend layers of quantum and nomsochronous classical oscillators is lnvesUgatedtheoretically At the linear stage of superradmnce spatial structures and gains of the elgenmodesof the system are found In the nonlinear stage the structure of the excited fields is comphcated and differs from the structure of elgenmodes The time dependence of the radiation power ~sdetermined Macroscopiccharacterlst~cs (field, polarization, mverslon) in superradlat~veprocess in quantum and classicaloscillators are shown to behave s~mllarly The cyclotron superradlance power is estimated

1. Introduction

ers of both q u a n t u m and classical oscillators.

The effects of the superradiance of ensembles of reverted atoms m q u a n t u m electronics have long been an object of theoretical [ 1-8 ] a n d experimenta! [9,10] investigations. Recently, interest was aroused in such p h e n o m e n a in ensembles of classical oscillators [ 10-16 ]. A classical analog of the superradiance effect is ra&atlve instability in space-locahzed ensembles of electron-oscillators having mfinite hfeUme. For example, superradlatwe instabilities may develop when an intense EM p u m p wave or u n d u l a t o r field acts on the m o v i n g electron cluster. In this paper we analyze superradiance m a layer of electrons rotating in a uniform magnetic field (section 3). For comparison, m section 2 we constder within a quaslclasslcal a p p r o x i m a t i o n the superradlance of a layer of q u a n t u m oscillators. It lS essential that the approach developed here takes into account the bidirectional nature of the radlatton in the layer As a result of the feedback elgenmodes having d~screte spectra are formed at the linear stage of radmUon. In the n o n h n e a r stage, when significant changes occur m the population inversion, the spatial structure of the field is gradually comphcated and an oscillatory regime of superradlaUon is developed. This behavior is typical for lay-

2. Superradiance in a layer of quantum oscillators

140

We assume that the osctllators form a layer which is u n b o u n d e d m the x, y &rectlons and has a width b along the z axis. Plane TEM waves are radiated in the + z directions. There are no external electrodynamlc systems. First, we will consider the superradiance of excited two-level q u a n t u m oscillators. The basic set of equations m the semlclasslcal approxim a t i o n includes the wave equatton for the radtated field E and the equations for the mean polarization 90 ~ d ~ x o and the population difference AN in the unit volume of the active m e d i u m [ 8 ]:

02E

1 02E

Ot 2

c2 Oz2 -

4n 0290 c20t 2 ,

(1)

02~ 090 Ot 2 + 2 T £ ' ~ +(O92 + T 2 2 ) ~ ° _ -2dtoo AN'E, h 0AN Ot -

T?IAN+

(2)

2 E0Y' h~oo Ot

(3)

Here ¢Oo is the transition frequency, Ti,2 are the Ion-

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gitudlnal and transverse relaxation times, respectively, and d ~s the dipole m o m e n t of the atom. We w111now use the forced solution o f the wave equation ( 1 ) that can be written as

Z+z

a=-i

~ f(Z'l~(Z',z-IZ-Z'l) Z--'r

xexp(-ilZ-Z'l)dZ',

(12)

z+Ct

E= - c

2~ f O~ J ~-(z',t-lz-z'[/c)dz'.

(4)

2 -- ct

One can naturally assume that the radmtion occurs in a spectral interval near the transition frequency tOo. Then the radiated field and the polarization can be represented as

E=Re[xod(Z, t) exp(ltOot) ], ~=Re[xo~(Z,

t) exp(itOot) ] .

(5)

Here A (z, t) and ~ (z, t) are the slowly varying amphtudes for which, on averaging eqs. ( 2 ) - ( 4 ) , we will have

where z=toot, Z = ( t o / c ) z , ~ = ~ / d A N o , a = A / 2ndANo, n = AN/ANo, I = t o c2/ 4 t o o2, and f ( Z ) is a function that describes the distribution of the atomic density• The initial conditions for eqs. ( 1 0 ) - (12) can be written as ~ I~=o = ~o,

n[~=o=l .

(13)

We will first analyze the superradlance of a thin (on the wavelength scale) layer: f ( Z ) = B g ( Z ) ( B = (too/c)b, fi(Z) is the delta function). For this layer, eq. (12) yields a= -iB~

(14)

Taking into account the integral of eqs. (10), ( 11 ),

O~ /Ot+ T~ ~~ = i d2ANA/h ,

(6)

I~le+n2=

0AN/0t = - T7 ~A N + I m ( A ~ * ) / h ,

(7)

and employing eq. (14), we can easily solve these equations. For 10301<< 1 we will then have (cf. refs. [2,8])

z+Cl

A=-2nxtO----~° [ ~ ( z ' , t - l z - z ' l / c ) C

d z -- ct

× e x p ( -XtOo [ z - z ' l ) dz' .

n= -tanh[F(z(8)

The proposed approach to the description of bidirectional superradiance is more rigorous than the one considered m ref. [4 ]. In particular, it enables us to take into account small-scale variations in the populatxon inversmn. Let us now consider the case when the relaxation time is long as compared to the development time o f the superradiative instability, i.e.: T1, 2 >> 2rr/tOc,

1312+1 ,

(9)

where tO~= (8nd2tOoANo/h) ,/2 is the cooperative frequency and ANo is the initial inversion. The relaxatlon processes can be neglected here, and the set of equations ( 6 ) - (8) reduces to the form

O~ /Or=ilan ,

(10)

On/Oz=IIm(a~*) ,

(11)

(15)

zd) ] ,

(16)

I~1 = [ a l / B = c o s h -2 [F(Z--Zd) ] ,

(17)

where rd= 1 / 2 F l n ( 4 / [ ~ o l 2) IS the delay time and F= IB is the instability gain. According to eqs. ( 16 ), (17), the radiation intensity has the shape of the pulse which has its m a x i m u m at Z=ra. The population difference changes its sign after the pulse has passed: n( z~oo )--, - 1. We will investigate the radiation of the extended layer f ( Z ) = const, beginning with the analysis of the linear stage. Assuming the fixed population inversion n = 1, we can linearize the set of equations ( 1 0 ) (12). By representing the solution of the lineanzed set as a = ~ i ( Z ) exp(lg2Z), ~ = ~ (Z) exp(i£2z) we obtain the characteristic equation that sets the frequencies and the spatial structures of the eigenmodes (see Appendix)

•

[

exp ( 2 I / q B ) = ~ k - f ~ , )

2 '

(18)

where ke= 1 + O and k,= (k~ +2Ike/£2) 1/2 are the normalized wave numbers outside and inside the 141

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layer, respectlvcl~ Note that under the condition 1<< l this equation coincides with the characteristic equation which can be derived for the modes of a layer w~th a &electric constant

icqZ)

%

~= 1 +~o~/(oJ2-o)~). Fig. 1 shows the dependence of the gain F = IIm £21 on the layer width that is found from the numerical solutmn ofeq. ( 18 ) for symmetric, Sq, and antlsymmetric, An, modes. Apparently, the greater the layer width, the higher the mode n u m b e r at which the gain is maximal For a given mode, the layer width corresponding to the maximal gain approximately meets the condition B=Trq (or b=q2/2), i.e., the gain reaches ~ts m a x i m u m when the layer width is multiple to 2/2 (where 2 is the radiation wavelength). The spatial structures of the first four modes are presented in fig 2 Figs. 3 and 4 illustrate results of numerical simulation by means of eqs. ( 1 0 ) - ( 1 2 ) for the n o n l i n ear stage of the superradlance of a thick layer of q u a n t u m oscillators. Fig. 3 displays the time dependence of the radiation power P = l al 2lz=B/2. As distract from the single-pulse superradiance in a thin layer (see eq. ( 1 7 ) ) , the extended layer is characterized by several additional power peaks on passage of the f u n d a m e n t a l pulse. The length of the pulse train increases with the layer thickness. Fig. 4 diplcts the evolution of the field amplitude of the polarization and of the population difference along the layer The formation of the space structure of the

~}25 0,20 835

0,t0 005

Fig 1 Gain of the symmemc, Sq, and antlsymmemc, Aq, modes as functionsof the wtdth B of the quantum oscillatorlayer, I=0 1 142

i [

i

I

Fig 2 Spatial structures of the modes m the layer of quantum oscillator, 1=0 1, B=6

0,06I 0,04 0,02 40

80

~20

¢60 200 ¥

Fig 3 Radiation power of the quantum oscillator layer as function of time, 1 = 0 1, B = 6

f u n d a m e n t a l symmetric mode, which coincides with that shown in fig. 2, can be seen at the initial linear stage of the interaction z < 15. Then, in the n o n l i n e a r stage of the interaction, when the population inversion changes, the field becomes more complicated and differs from the structure of the elgenmode U n like the case of a thin layer, the evolution of the polarization amplitude and of the population difference is oscillatory. But, like the thin layer when r--,oo, we have n - . - 1, I 91 - , 0 The mechanism for the onset of an oscillatory regime is similar to that described in refs. [2,4] and is related to lnhomogeneous changes in the population inversion along the sample's length. First, some regions of the layer are de-excited rapidly and are then re-excited by radiation from other regions. This process is accompanied by oscillations of the polarization and of the EM field amplitudes.

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0.2

3. Superradiance in a layer of classical cyclotron oscillators W h e n d e a h n g w i t h the s u p e r r a d l a n c e o f a layer o f excited classtcal oscillators, we a s s u m e that such oscillators are the e l e c t r o n s r o t a t i n g in a h o m o g e n e o u s m a g n e t i c field: H=zoHo. Let the electrons h a v e equal t r a n s v e r s e m o m e n t a p • o = myo v • o and, w~thm small f l u c t u a t i o n s , to be d ~ s t n b u t e d u n i f o r m l y m the cyc l o t r o n r o t a t t o n phases at the m m a l t~me. T h e elect r o n s h a v e no t r a n s l a t i o n a l velocity. T h e layer u n d e r conslderat~on radtates c~rcular p o l a r i z e d w a v e s m the + z & r e c t l o n s . T h e p a r h c l e s ' m o h o n is d e s c r i b e d by the f o l l o w i n g e q u a h o n

3p+/Ot--wgHp+ = --eE+ (z, t) ,

(19)

w h e r e p+=px+tpy, E+=E~+IEy, ~on=eHo/mc? ts the relativistic g y r o f r e q u e n c y , a n d ~,= 1 + [p+lZ/rnc 2 ts the relauvist~c m a s s factor. T h e e q u a t t o n for the radmtxon field as s i m i l a r to eq. ( 4 ) , E+ --

2n c

y+(z',t-[z-z'l/c)

dz',

(20)

where j+=-epo(v+) is the e l e c t r o n current d e n s~ty, Po is the u n p e r t u r b e d e l e c t r o n density, v+ = vx+Wy a n d ( ) d e n o t e s an a v e r a g e o v e r the m l t m l phases o f the cyclotron rotation. We n o w assume that the e l e c t r o n s are subrelatlvtsUc, 1 e. ~,= 1 + IP+ 12/ 2mc 2, a n d r a d m t e n e a r the nonrelatxwst~c gyrofrequency. R e p r e s e n t i n g E + = A (z, t) exp (l~oH0t) and p+ = pexp(l~OHot) we r e d u c e the set o f eqs. ( 1 9 ) ( 2 0 ) to the f o r m (cf. eqs. ( 1 0 ) - ( 1 2 ) ) ~ O/~/Ot + ~U IP I :/~ = - a ,

(21)

~l~=o=exp[i(Oo+rcosOo)],

Oo=[O, 2 n ] ,

Z+r

a=1 f f(z') Z---c

Xexp(-llZ-Z'l

F~g 4. Evolution of the &strtbut~on of (a) the electric field amphtude, (b) the population differences, (c) the polarization amphtude over the layer of quantum oscillator, I=0.1, B=6

)dZ' .

(22)

~ The set of equations (21)-(22) is rather general and describes the superradmnce m a layer of the oscillators of&fferent physical nature, including acoustic oscdlators (cf ref [171) 143

'¢olumeO[

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numbc~ 1 2

The following & m e n s l o n l e s s variables are used here:

r =~o~mt, Z = ( ~om~/ c )z, P = P /P ±o, a = e A / mc°HoV ±o, I=o)~/2~o 2, ~op= ( 4ne2po/m ) 1/2 is the p l a s m a frequency,/z = ~,~ 0/2c2 is the p a r a m e t e r describing non~sochronous oscillations o f electrons in a magnetic field, and the p a r a m e t e r r << 1 characterizes the lnltml electron m o d u l a t i o n over the cyclotron rotation phases In the limiting case o f a thin layer, f ( Z ) = B d ( Z ) , eq. (22) yields, a = I B ( b ) This relation, together with the equations o f m o t i o n (21 ), describes the superradlance o f the thin layer that was studied earlier in refs [ 1 4 - 1 6 ] . Note, however, that unlike the q u a n t u m oscillators, the superradlance o f classtcal oscillators displays a multisplke character even if the layer is thm. Below, p r i m a r y attentton will be focused on the radlatton o f extended layers with a u n i f o r m distribution o f the particle density along the layer ( f ( Z ) = const. ). In the small signal a p p r o x i m a t i o n , a ~ 0 , the equations o f m o t i o n are hnearized and the set o f equations ( 2 1 ) - ( 2 2 ) can be reduced to the form 02@) Or 2 --

Oa 0r + l / z a ,

(23)

Z+z

a=l

j f(Z')@(Z',r-IZ-Z'

))

Z--T

×exp[ -i(1 -/~)IZ-e'

I ] dZ'

(24)

Representing the solution o f e q s . ( 2 3 ) and (24) m the form a = ~ i ( Z ) exp (i£2r), @) = (#(Z)) × exp(1Or), we obtain a characteristic equation which coincides with eq. ( 1 8 ) , where the normalized wave numbers outside a n d inside the layer are given by the relations k~ = - / ~ + £ 2 + 1,

k,=(k~+2o~k~),

( 26 )

F o r a small layer density I<< 1, if we neglect the second term on the left-hand side o f eq. (26), which is responsible for cyclotron absorption, we obtain for the instability gain 144

F = I l m n l = (IB,ul2) '/2 .

(27)

The gain o f s y m m e t r i c and a n t l s y m m e t r l c modes as a function o f the layer wtdth is shown in fig. 5 Here the m a x i m u m gain belongs to the first ( f u n d a m e n tal) s y m m e t r i c m o d e (when B<< 1, its gain is det e r m i n e d by eq. ( 2 7 ) ) . The gain o f other modes approaches the gain o f the f u n d a m e n t a l m o d e as the layer width increases. The spatial structures o f elgenmodes are similar to those o f fig. 2. The nonlinear stage o f the cyclotron superradlat i r e instability was mvestlgated by numerical stmulatlon o f e q s . (21 ) and (22). The time dependence o f the radiation power and the total electron efficiency, B/2

q=l-(I/B)

f

(I/5]2)dZ,

B/2 are given in fig. 6 for various thicknesses o f the layer. A c o m p a r i s o n o f figs. 6a and 6b demonstrates that the peak power and the pulse d u r a t i o n o f superradtance grow with the layer thickness. Fig. 7 displays the f o r m a t i o n o f the spatial structure o f the f u n d a m e n t a l s y m m e t r i c m o d e at the lnttlal lmear stage of the radiative process ( r < 60), and the appearance of trregular behavior o f this structure In the nonlinear stage ( r > 60). The variation o f the mean square o f the transverse electron m o m e n t u m , ( I/~2(Z, r) l ) , along the layer, 1.e. the extent to which the lnverston is eliminated, is shown in fig. 7b Fig. 7c displays the evolution o f the a m p l i t u d e of the H F electron current, I @ ( Z , r ) ) l These dependences p t8~ ~,

(25)

where a = I ( - £2 + jz ) / £22. F o r the case o f the thin layer B<< 1, eqs (18) and (25) yield

,(22 _ tlBE2+ l l d B = 0 .

1 J u l y 1992

i O

2

q

6

8

t0

t2

t~

Fig 5 Gain of the first symmetric S~ and the first antlsymmetrlc A~ modes as functions of the width B of the classical oscillator layer, I=0 1,#=0 1

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P

(a)

2 (8 (6

l~

1 July 1992

lal 0A

t0°:

[2 03

10'

/. fO0

200

300

400

500

p ~o~

"C

z

q,

Z8

(b)

2,4 2 46

0,6

[2 0,8



0,2

0'4 j 0

(a)

05 ~00

200

500

400

500

-c

Fig 6 Radmtlon power and electron efficiency as a functmn of time, (a)B=6, (b)B=12,1=0 1,/~=0 1.

are, on the whole, quite similar to the corresponding dependences for the layers of q u a n t u m oscillators (cf. figs. 3 and 4). Finally we estimate numerically the peak power of the cyclotron superradiance a n d the pulse duration. Let the magnetic field strength be equal to 100 kOe, the ra&aUon frequency o J = 2 X 1012 s - I (the wavelength 2 = 1 r a m ) , the electron density Po = 2 X 10~4 c m - 3 (~op= 8 X 1011 s - ~), and the rotational velocity of electrons v±0 = 0.45c. These values of the parameters of the system correspond to I = 0.1 a n d / l = 0.1. From fig. 3 for the specified ra&atlon power, the layer width being b = 2 2 ( B = 12), we obtain P ~ 2 . 8 X 10 -3. The power radmted by 1 cm 2 of the layer surface a m o u n t s to 1.4 G W in d i m e n s i o n a l magnitudes. The pulse duration of the superradiance is about 1.5 X 10- ~1 s at the level e - l of the peak power. Note that it is possible to increase the superradlance power essentially a n d to shift the frequency to the shortwave part of the spectrum by imparting translational velocity (approximating the light velocity) to the electrons [ 15 ].

t1<~>1

Fig 7 Evolution of the distribution of (a) the electric field amplitude, (b) the mean square value of transverse electron pulse, (c) the HF electron current amplitude over the layer of classical oscillators, I=0 1,/~=0 1, B=6

Appendix We will derive the characteristic equation (18) from the system of equations (10), ( 11 ) assuming a fixed population inversion, n = 1. Representing the field amplitude and polarization as 145

Volume 91, number 1,2 d = ~ i ( Z ) exp(lg?Z),

OPTICS COMMUNICATIONS /)=,)(Z)

exp(lg2Z)

(A1)

we t r a n s f o r m eqs ( 1 0 ) , ( 11 ) to the f o r m ( k e = 1 +£2) B

~= -1 f f(Z') ,~(Z')

exp( -lk e 12-2'

I ) dZ' ,

0

~=la/O.

(A2)

D i f f e r e n t i a t i o n o f the d e p e n d e n c e a ( z ) o v e r the long i t u d i n a l c o o r d i n a t e yields

I July 1992

R e p r e s e n t i n g the s o l u t i o n ( A 3 ) in the f o r m

a=Cl exp( lk, Z) + C2 e x p ( - ik, Z) , w h e r e k,= (k~ + 2keI/g2) 1/2, C~,2 are a r b i t r a r y constants, a n d taking into a c c o u n t the b o u n d a r y con& t i o n s ( A 4 ) we o b t a i n the c h a r a c t e r i s t i c e q u a t i o n (18)

ke-k,]

exp(21k, B ) = 1,

or R 2 exp(21k, B ) = 1 , (A5)

Z

a= -1 f .~(Z' ) exp[

-lke(Z-Z'

) ] dZ'

w h e r e R is e f f e c t i v e r e f l e c n v l t y c o e f f i c i e n t at the die l e c t r i c - v a c u u m interface

0 B

k e +k, k~-k,

- i f ~ (Z') e x p [ l k e ( Z - Z ' ) ] dZ' ,

R= - -

Z

-

( 1 +21/£2ke)1/2+ 1 (1+ 2I/£2ke)'/2-1

(A6)

/

da _ -k~ f ~ (Z') exp[ dZ

-lke(Z-Z'

) ] dZ' References

0 B

[1 ] R H Dlcke, Phys Rev 93 (1954) 99 [2] J C MacGllllvray and M S Feld, Phys Rev A 14 (1976)

+ke f ,~'(Z' ) e x p [ i k d Z - Z ' ) 1 dZ' , Z

deer dZ 2-

1169

- 2k~ ~ ( Z ) /-

+ l k ~ f ,Y~(Z' ) e x p [

--lk~(Z--Z')l dZ'

0 B

+lk~ f ~ ( Z ' ) e x p [ t k e ( Z - Z ' ) l dZ' . Z

C o n s e q u e n t l y , ~ m e e t s the e q u a n o n d2a + k e 2 a = - 2 k ~ - ~ - dZ 2 w i t h the endfaces:

2kel

£2 a ,

following boundary

(A3) conditions

at

the

B

d~

d Z z=o

=k¢ f ~ ( Z ' )

exp(-lk~Z)

dZ'

=1k¢8(0)

0 B

da

=k~ f ,~(Z' ) expOk~Z' ) dZ' Z=B

0

= -lk~gl(B)

146

(A4)

[3 ] R Saunders, R K Bullough and S S Hassan, J Phys A 9 (1976) 1725 [4] R K Bullough, R. Saunders and C Feuillade, Coherence quantum optics IV, eds L Mandel and E. Wolf (Plenum Press, New York, 1978) p. 263 [51 R Bonlfaclo and LA Luglato, Phys Rev A 11 (1975) 1507 [61 R. Bonlfaclo and LA Luglato, Optics Comm 47 (1983) 79 [71 A B Andreev, V I Emeljanov and Yu A lljmsky, Usp Fiz Nauk 131 (1980) 653 [81 VV Zheleznyakov, V V Kocharovsky and V1 V Kocharovsky, Usp. Flz Nauk 159 (1989) 194 [91 N Scnbanowitz, I P Herman, J C MacGllhvray and M S Feld, Phys Rev Lett 30 (1973) 309 [101 Q H F Vrehen, H M J Hlkspoors and H M. Gibbs, Phys Rev Lett 38 (1977) 764 [ l l ] R H Bonlfaclo, C Maroll and N Plovella, Optics Comm. 68 (1988) 369 [12] VV Zheleznyakov, VV Kocharovsky and VI V Kocharovsky, Izv VUZ Radlofiz. 24 (1986) 1095 [131 N S. Glnzburg, Pls'ma ZhTF 14 (1988) 440 [141 Yu A IIjmsky and N S Maslova, ZhETF 94 (1988) 171 [151 N S Ginzburg and I V Zotova, Pis'ma ZhTF 15 (1989) 83 [161 N S Gmzburg and Yu V Novozhllova, Pls'ma ZhTF 15 (1989) 60 [171 Yu V. Kobelev, L A Ostrovsky and I A Soustova, Izv VUZ Radlofiz 24 (1986) 1129