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Transient analysis of optical phase conjugation in absorbing media
Volume 59, number 2
OPTICS COMMUNICATIONS
15 August 1986
T R A N S I E N T A N A L Y S I S O F O P T I C A L P H A S E C O N J U G A T I O N IN A B S O R B I N G M E D I A J.P. L A V O I N E Ecole National Supbrieure de Physique, Groupe de Recherche en Photonique Appliqube, 7, rue de l'Universitb, 67000 Strasbourg cedex, France
and A.A. V I L L A E Y S Centre de Recherches Nuclbaires et Universitb Louis Pasteur, Physique des Rayonnements et Electronique Nuclbaire, 23, rue du Loess, 67037 Strasbourg cedex, France
Received 5 February 1986; revised manuscript received 8 April 1986
In the present work the transient response of a two-level phase conjugator is calculated for absorbing media. We consider the case of two cw pump beams and a weak pulsed signal wave. The analysis, valid for low intensity, considers purely radiative decay and neglect the depletion of the pump waves. All fields have same frequency. From a perturbative solution of the density matrix equation, Maxwell equations are solved with the slowly varying amplitude approximation. The time dependent solutions are discussed with respect to the line center small signal field attenuation coefficient a o and to the lifetime of the transition 1"- 1. It is shown that the quantity 4C~oc/r, where c is the light velocity in the vacuum, defines characteristic behaviors of the delta function response. Finally when 4C~oC/F > 1, oscillations appear and an attempt to explain their physical insight is given.
1. Introduction The properties o f optical phase conjugation via degenerate four wave mixing have suggested many practical applications in fields as diverse as image processing, laser fusion, adaptive optica, real-time holography, etc. Although this process has been the subject o f many works in recent years [ 1 ], a number o f problems are still unresolved concerning the study o f transient phenomena. Abrams and Lind [2,3] have developed a stationary theory for absorbing media, valid in the case o f monochromatic pump and signal waves. They gave a physical understanding of the mechanisms underlying the non-linear response o f a two-level system. Fu and Sargent [4] have generalized the expressions including signal-pump detuning and showed that the reflection band pass is limited by the linewidth o f the transition involved in the process. In addition, these calculations complemented the works of Pepper and Abrams [5] who have demonstrated the high frequency selectivity of phase conjugation. The first theoretical study o f time varying pulses in phase conjugation has been done by Marburger [6]. More recently Fisher, Suydam and Feldmann [7] have developed a classical theory to describe transient response o f phase conjugators. Starting with a pair o f coupled wave equations they established an analytical expression for the response function of a Kerr-medium for the case o f a cw pump and they consider a non-absorbing media. This classical approach requires the introduction of a phenomenological coupling coefficient. For degenerate fourwave mixing the frequency dependence of the coupling coefficient has been neglected due to high frequency selectivity of the reflectivity [5]. From this point of view, they showed the ability of generating any transient behavior from the only knowledge of the steady-state responses at all frequencies. Consequently, the knowledge of the filter function o f Pepper and Abrams [5] is sufficient. It seems desirable to analyze the case o f absorbing media and especially resonant conjugators, as well as to circumvent the non-frequency dependent 160
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Volume 59, number 2
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coupling coefficient assumption. This situation, quite different from the previous case discussed by Fisher et al., can be handled more easily in the framework of a quantum material system modelized by a two-level system. It is the aim of this note to develop an analytical expression of the conjugate response to input pulses of arbitrary form by analyzing the transient behavior for cw-pumped microscopic two-level conjugators. In a first step the polarisation is calculated by using the density matrix equations in the rotating-wave approximation. These equations are solved to third order by perturbation theory [8]. Introducing the medium polarisation as a source term in Maxwell equations, we obtain a pair of coupled wave equations with time dependent coefficients related to the parameters of the transition under consideration. The conjugate response function is obtained by solving this set of equations and we analyze the influence of the signal field attenuation coefficient a 0 and transition lifetime. Two different regimes are observed depending on the ratio between a 0 and the transition lifetime. The physical origin is discussed.
2. Theory We consider the situation illustrated in fflg. 1. The medium is constituted by an ensemble of N two-level atoms. It can be characterized by a dipole moment It, longitudinal and transverse relaxation times Pi -1, p~-I respectively and transition energy ~iO~ab. This medium is pumped by two counter-propagating plane waves 6pl(r, t), 6p2(r, t), with same frequency Go. In the following it is assumed that the fields are not depleted and that they remain plane waves in the medium. Also the fields have the same intensity and are in any case much stronger than the ones of the signal and conjugate fields. They are written as
6pl(r,t)=~Epexp[-i(¢ot-ki'r)]
+cc,
i= 1,2,
(1)
where cc stands for the complex conjugate and k 1 + k r = 0. The signal and conjugate fields are represented by the expressions
61(r, t) = ~Ei(r, t)exp [ - i ( ~ t - k]-r)] + c c ,
] = s, c ,
(2)
with k s + k e = 0 and Ec(r, t) = 0 for x / > L. Because of their low intensities, we consider throughout that the medium is linear with respect to the signal or conjugate field. The geometry is chosen so that the angle 0 between k 1 and k s is small enough to write k i • r = kix , i = 1,2, s, c. We assume also that the amplitude of the signal and conjugate fields depend only upon x. Let us write now the well-known equations of motion for the elements Pi/of the density matrix [9] Pa b(t) + (P 2 -- iO~ba)Oa O(t) = --(i/h)Hab (t) w (t) ,
(3)
Pba(O + (P2 + ROba) Pba( t) = (i/h ) H b a ( t ) w ( t ) ,
(4)
~ ( t ) + Pl (w(t) - w0) = --(2i/P0 (Hba(t)Pab(t) -- Pba(t)Ha b(t)),
(5)
where w ( t ) = Pbb(t) -- Paa(t). Here w o is the equilibrium population difference between the two levels a and b involved in the transition with
~~c(r't)
~'p2(r,t) N°nl'Tarmedium]~......._~ Fig. 1. Geometry for degenerate four wave mixing. 161
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15 August 1986
no optical fields applied. Hba(t ) stands for the matrix element of the interaction energy and is given in the rotating-wave approximation by 4
Hba(t) = --½1~ba~ El(x, t)exp [-i(~ot - k/r)] = H*b(t) ,
(6)
where El(x, t)represents the complex amplitude of one of the four fields already described for a given]. Finally ~ba is the matrix element of the dipole moment operator ~. Formal integration o f equations (3), (4) and (5) yields
Pab(t) = --(i//O
f
t
Hab(t')w(t')exp [ - ( r 2 - i6Oba)(t - t')] d t ' ,
(7)
--oo t
Pba(t) = (iflO f Hba(t')w(t')exp [--(I" 2 + iWba)(t -- t')] d t ' ,
(8)
--oo t
w(t) = w 0 - (2i/h) f [Hba(t')Pab(t' ) -- Pba(t')Hab(t')] exp [ - P l ( t -
t')] dt',
(9)
--oo
with the initial conditions Pab(-oo) = Pba(-oo) = 0 and w ( - - ~ ) = w 0 = - 1 , perturbative solutions of these equations up to third order are of the form
wO(t) = w 0 = - 1 ,
(10) t
O(1)b(t) = --(i/h)w0 f Hab(tl)exp [--(F2 -- k°ba)(t -/1)] d t l ,
(11)
--oo
t
w(2)(t) = --(2/~2)Wof
t2
dt 2 f
--oo
d/1 [Hba(t2)Hab(tl)exp [iCOba(t2 - t l ) ]
--oo
+ Hba(tl)Hab(t2) exp [--i¢Oba(t 2 - t l ) ] ] exp [--P2(t2 - tl)] ex p t
p(3)b(t)=(2i/h3)WoeXp[--(P2--iCOba)t I f
dt f
[-rl(t -
t2) ] ,
(12)
t2
dt 2 f
dt 1 •
X [Hab(t3)Hba(t2)Hab(tl)ex p [i6Oba(t2 - t 1 - t3) ] +Hab(t3)Hab(t2)Hba(tl)exp [--i~ba(t 2 - t l + t3)]] X exp [ - r ' l ( t 3 - t2) ] exp [ - r 2 ( t 2 - t 1 - t3)],
(13)
witht 1
Hab(t3)Hba(t2)Hab(tl) =
4
--~U.btUl2~ ~ ~ E~(x, t3)Em(X, t~)E*.(x, tl) l=1 m =1 n=l
k m + kn)" r ] ,
X exp [i¢o(t 3 - t 2 + t l ) ] exp [ - i ( k / 4
Hab(t3)Hab(t2)Hba(tl) = •
4
1
g#abl/.t[ 2 ~ l=1
4
4
~ ~ E~(x, t3)E*(x, t2)En(X, t l ) m=l n=l
X exp [ i ~ ( t 3 + t 2 - t l ) ] exp [ - i ( k / + k m - kn)" r].
162
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15 August 1986
Taking the contribution such k I - k m + k n = k s (resp. ke), and k l +km - kn = ks (resp. ke), in th e expressions (14) and (15), the corresponding matrix elements p(al)(t,ks) and pa(2)(t,ks) (resp. p(al)(t, ke) and p(a3)(t,kc)) are of the form p(1)(t ks) = (ilzabwO/2h) exp [i(cot - k s - r)]E*(x, t) ®exp [ - ( I " 2 - iA)t]H(O ab" ' .
p(3)(t, ks) -
--ilaab hul2w0
+ EpEp
2fi 3
(16)
[ IEp 12 exp [i(cot - k s • r)] L rl(r2 - iA) E*(x, t) ® exp [-(I" 2 - iA)t] H(t)
{Ee(X, t) ®exp(-1"1t)H(t) ®exp [-(1"2 - iA)t]H(t)}
P2 - iA + Igpl 2 (g*(x, t) ® exp [--(1"2 - iA)t]n(t) ® e x p ( - P 1 OH(t) ® exp [-(1"2 - iA)t]H(t)} IEpl2 {E*(x, t) ® exp [-(1" 2 - iA)t]H(t)) + 1"1(1,2 + iA) jEp 12 +~ { E
• s (x, t) ® exp(-1,1t)H(t) ® exp [ - ( 1 , 2 - i z ~ ) t ] H ( t ) }
• * (Ec(x, t)®exp[-(F2+iA)t]H(t)®exp(-1"1t)H(t ) ® e x p t - ( r 2 - i A ) t l H ( t ) ) J , + EpEp
(17)
where H(t) is the Heaviside function. The symbol ® stands for the convolution o t?erator and A is the detuning between the transition and field frequencies, A = C%a - ¢o. The expressions for p(~J(t, ke) and p(a3b)(t,ke) are obtained by exchanging c and s in the relations (16) and (17). We are not able to write the polarisation and more precisely the polarisation Ps(t, ks) and Pc(t, ke) due to the signal and conjugate fields respectively. They are given by
Ps(t, ks) = N[~tabPba(t , ks) + I~baPab(t, ks)],
(18)
Pc(t, ke) = N[UabPba(t, ke) + labaPaO(t, ke)],
(19)
where
pao(t, ks) =
ks) +
ks),
pba(t, ks) = p*b(t, ks),
(20)
Pba(t, ke) = p*b(t,
(21)
and
Pab(t, ke) = p(lb)(t, kc) + p(3b)(t,k c ) ,
ke).
If we consider the expressions (16) and (17), Pab(t, ks) (resp. Pab(t, ke) ) can be written as
Pab(t, ks) = [o(al)(t, E*) + a(3b)(t,E~) + a(a3)(t,Ee) ] exp [ i ( ~ t
- k s "r)],
(22)
with
e,*) =
e * ) exp
t-
ks-,)],
(23)
and
O(3b)(t,Es*) + a(3b)(t,Ec) = p(a3b)(t,ks)exp [--i(¢ot
- k s- r ) ] ,
(24)
where a(a3b)(t,E~) and o~a~(t, Ee) are deemed by the expressions (17) and (24) and depend ordy on E * and Ee, respectively;Besid~ if the amplitude Of Ep is low enough we can neglect the non linear term O(a3b)(t,E*) with respect to ata*b~(t,Es). Then relation (22) takes the form 163
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15 August 1986
Pab(t, ks) = [a(al)b(t,E*) + o(3)b(t, E¢)] exp[i(cot - k s. r)].
(25)
A similar expression is obtained for pab(t, ke). The space and time dependence of the signal and conjugate fields are obtained by solving Maxwell equations 32 ~s(X, t)/Ox 2 -- c -2 026 s(X, t)/Ot 2 = (47r/c2) 32Ps(t, ks)/at 2 ,
(26)
a 2 6c(x, t)/Ox 2 - c -2 a 2 6e(x , t)/Ot 2 = (4rt/c 2) a2Pe(t, ke)/at 2 .
(27)
Applying the standard methods of non linear optics [10] this set of coupled equations yields in the slowly varying amplitude approximation
aE*(x, t)/Ox + c -1 aE*(x, t)/Ot = -(2~rico/c)P+(x, t) ,
(28)
3Ee(X, t)/ax - c -1 OEe(x, O/at = -(2nico/c)Pc (x, t) ,
(29)
where
P+(x, t)= Nub ° 2Pab(t , ks)exp [-i(¢ot- ksx) ] , PC(x, t)= Nub° 2Oba(t, kc)exp [i(¢ot- kex)]. We assume that signal and conjugate fields start at t = 0. From the Laplace-transform of (28), (29) and using relations (16), (17) and (25) the set of coupled equations (28), (29) can be written as
aE*(x, p)/ax + a(p)E*(x, p) = +/3(P)Ec(x, p),
(30)
3E e(x, p )/ ax - ct*(p)'Ee(X, p) = -{3*(p ) E * (x, p ) ,
(31)
where Es(X, p) and Ee(x, p) are defined by f~*E](x, t ) e x p ( - p t ) dt for] = s, c, and ,
c~(p) - P + °t0P2p2 - iA +Pc '
,
2
a0El~E~rlr2 P + 2P 2 I~(P)= Is(P2 - i A ) (p + P l ) ( p + F2 + iA)(p + P 2 - iA)
The quantity ~0 = 21rc°Nl#12/hcF2 is the line center small signal field attenuation coefficient and I s =/i2F 1F2/ Ipl 2 is the line center saturation intensity. The set of coupled equations (30), (31) is just the Laplace-transform of the well-known coupled set given by the relation (3) in ref. [7], but as previously mentioned in the introduction, our coefficients a and # are p-dependent (i.e. time dependent) and are functions of the transition parameters. By introducing the initial conditions Es*(0, p) = F(p) and Ee(L, p) = 0, where F(p) is the Laplace-transform of the time dependent amplitude of the signal field at x = 0, the set of equations (30) and (31) is easily solved. We get Ec(0, p ) =, 2~*(p)F(p)/{ot*(,p) + or(p) + x / ~ [ 1
+ exp(-L~]/[1
- exp(-L~]
},
(32)
with 6(p) = [t~(p) + a*(p)] 2 _ 413(p)/3*fp). For mathematical convenience we assume A = 0 and a purely radiative decay (i.e. P 1 = 2P2). Then t~(p) and/3(p) have the form
a ( p ) = pie + a0r2/(p + r 2 ) ,
•
•
2
3(p) = (2aoEpEpP2/Is)(p
+p
2)
-2
•
Therefore 8 is now given by 5 = 4(a - t/3[)(a + I~[), a + [3l = p/c + [c~oP2/(p + F2) 2] [p + P 2 + (2[Ep[2/Is)P2]. If the saturation intensity I s is much bi~her than the pump intensity [Ep[ 2 then 6(p) = 4 [a(p)] $ and this condition is consistent with the resolution of the density matrix equations to third order by perturbation theory. If 2 [E112/ I s < 1, we can reduce expression (32) to Ee(0, p) = [3*(p)F(p)/2a(p)] [ 1 - e x p ( - 2 L o~(p))]. 164
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OPTICS COMMUNICATIONS
Introducing the value of a(p) and 3(P), we get (34)
Ec(O, p)/F(p) = [K/(p + r2) (t72 + pF 2 + ao F2c)] {1 - exp [ - 2L (p/c + ct0 F2/(p + F2))] },
cr22/is.
where K = OtoE~E~ From the inverse Laplace transform, the amplitude of the conjugate field at x = 0 is given by Ec(O, t) = R(t)
® F(t)
(35)
,
where F(t) is the input pulse and R(t) the conjugate response deffmed by ff e x p ( - F 2 t )
R(t) = K [~
a-b
exp(-at) ~ b(b-a)
e a(b-a) x p ( - b t ) ]] H(t) - A t - 2L/c)H(t - 2L/c) 1 ,
(36)
with a = (F2/2) [1 + (1 - 4aoC/F2)l/2],
b
(F2/2) [1 - (1 - 4aoC/r2)l/2].
=
The quantity f(t) is defined by 2LbOr'2 )
(-2LaOr2/b)n ~
exp(-F2t)~ n =0
2L
1 a) e x p ( - b t ) e x p ( a(b --
(bt) k
k =0
_~0 ) e x p ( - r 2 t ) * * ( - 2 L a o F 2 / a ) n ~ ( a t ) k r2 + a ( b - a ) ~-~ n! k! n =O k=O
(37)
3. Discussion
From the analytical expression (34), we see that the response function is proportional to t~0 and (E~) 2. This is in agreement with the Abrams and Lind theoretical results [2,3] where the limit IEpl2/ls ~ 1 is taken. Our results IRI
20-"f Arb.
(b)
Unit
IRI
F
Arb. Unit.
1.5
~o=10m_1 I I ,x%=sm-1
(a) '~
2L/c=lOns. %=0.1rn-I
1.0
15 --
iiiiiill" I.t~I.I
lo_
!
2L/c=lOns. r~=o2.s:'
\
i
I
0.5
I 5
I 10
I 15
I 20
I 25 t(ns)
5
10
15
20
25 t(ns)
Fig. 2. We h'ave sketched the variations of IR(t)l as a function oft for different values ofa 0. The cases 4aoc/F < 1 (a) and 4aoc[F > 1 (b) are considered. 165
Volume 59, number 2
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15 August 1986
IRI
Arb, Unit,
2L/c=lOns. ~o =1 m71
20 I
/
\
w a o
15
. i
i
12=0'2ns-" 1
l
J 1
10
5
/
12= 0 . 1 n s - . 1
l
Fig. 3. The time dependence of the response function is reI
I
I
I
I
5
10
15
20
25
t(ns)
presented for various relaxation times r2.
show that for 4Soc/P 2 > 1, R(t) exhibits oscillations. Fig. 2 contains plots of IR(t)l versus t for several values of s 0. The cases 4SoC < r 2 and 4 s 0 > P2 are represented by figs. 2a and 2b respectively. To understand why this quantity play such a role in the response function and why R(t) exhibits oscillations let first consider the simple model of a grating v~hich corresponds to the interference figure formed by the two pump fields on the non-linear medium. Assume the conjugate wave of the form exp(-1` 0 when the signal wave impinges on a maximum of the interference figure. Then at x = 0, the conjugate signal can be written as e x p ( - F t ) + exp ( - s o d ) exp [-1`(t - 2d/c)] + e x p ( - 2 s 0 d ) e x p [-1`(t - 4d/c)] + ..., where d is the distance between two maxima. If sod ,~ rd/c (i.e. SoC ,~ F) we have only a non-interacting superposition of pulses. In the other case corresponding to SoC >> P we have an interfering contribution resulting from the superposition of all the pulses. When the delta function interacts with a maximum, the conjugate spectrum is reduced due to the lifetime of the transition. When all pulses overlap, oscillations occur. For a given 1` and with the condition SoC > F the efficiency of this collective interaction is more important than for small s 0 and the bandwidth of the conjugate wave decreases with s 0. This can explain the time behavior of R(t) with respect to s 0 as seen in fig. 2. The influence of 1"2 is plotted in fig. 3 where we observe that the time evolution of the response function is smoother when 1" decreases. This results from the diminution of the bandwidth with 1"2 [10].
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
166
See for example Optical Engineering 21 (1982) 2. R.L, Abrams and R.C. Lind, Optics Lett. 2 (1978) 94. R.L. Abrams and R.C. Lind, Optics Lett. 3 (1978) 203. T.Y. Fu and M. Sargent III, Optics Lett. 4 (1979) 52. D.M. Pepper and R.L. Abrams, Optics Lett. 4 (1979) 366. J.H. Marburger, Appl. Phys. Lett. 32 (1978) 372. R.A. Fisher, B.R. Suydam and BJ. Feldman, Phys. Rev. A23 (1981) 3071. N. Bloembergen, Nonlinear optics (Benjamin, New York, 1965)• DJ. Hatter and R.W.Boyd, IEEE JQE-16 (1980) 1126. A. Yariv, Quantum electronics (Wiley, NewYork, 1975). J. Nilsen and A. Yariv, AppL Optics 18 (1979) 143.
OPTICS COMMUNICATIONS
15 August 1986
T R A N S I E N T A N A L Y S I S O F O P T I C A L P H A S E C O N J U G A T I O N IN A B S O R B I N G M E D I A J.P. L A V O I N E Ecole National Supbrieure de Physique, Groupe de Recherche en Photonique Appliqube, 7, rue de l'Universitb, 67000 Strasbourg cedex, France
and A.A. V I L L A E Y S Centre de Recherches Nuclbaires et Universitb Louis Pasteur, Physique des Rayonnements et Electronique Nuclbaire, 23, rue du Loess, 67037 Strasbourg cedex, France
Received 5 February 1986; revised manuscript received 8 April 1986
In the present work the transient response of a two-level phase conjugator is calculated for absorbing media. We consider the case of two cw pump beams and a weak pulsed signal wave. The analysis, valid for low intensity, considers purely radiative decay and neglect the depletion of the pump waves. All fields have same frequency. From a perturbative solution of the density matrix equation, Maxwell equations are solved with the slowly varying amplitude approximation. The time dependent solutions are discussed with respect to the line center small signal field attenuation coefficient a o and to the lifetime of the transition 1"- 1. It is shown that the quantity 4C~oc/r, where c is the light velocity in the vacuum, defines characteristic behaviors of the delta function response. Finally when 4C~oC/F > 1, oscillations appear and an attempt to explain their physical insight is given.
1. Introduction The properties o f optical phase conjugation via degenerate four wave mixing have suggested many practical applications in fields as diverse as image processing, laser fusion, adaptive optica, real-time holography, etc. Although this process has been the subject o f many works in recent years [ 1 ], a number o f problems are still unresolved concerning the study o f transient phenomena. Abrams and Lind [2,3] have developed a stationary theory for absorbing media, valid in the case o f monochromatic pump and signal waves. They gave a physical understanding of the mechanisms underlying the non-linear response o f a two-level system. Fu and Sargent [4] have generalized the expressions including signal-pump detuning and showed that the reflection band pass is limited by the linewidth o f the transition involved in the process. In addition, these calculations complemented the works of Pepper and Abrams [5] who have demonstrated the high frequency selectivity of phase conjugation. The first theoretical study o f time varying pulses in phase conjugation has been done by Marburger [6]. More recently Fisher, Suydam and Feldmann [7] have developed a classical theory to describe transient response o f phase conjugators. Starting with a pair o f coupled wave equations they established an analytical expression for the response function of a Kerr-medium for the case o f a cw pump and they consider a non-absorbing media. This classical approach requires the introduction of a phenomenological coupling coefficient. For degenerate fourwave mixing the frequency dependence of the coupling coefficient has been neglected due to high frequency selectivity of the reflectivity [5]. From this point of view, they showed the ability of generating any transient behavior from the only knowledge of the steady-state responses at all frequencies. Consequently, the knowledge of the filter function o f Pepper and Abrams [5] is sufficient. It seems desirable to analyze the case o f absorbing media and especially resonant conjugators, as well as to circumvent the non-frequency dependent 160
0 030-4018/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 59, number 2
OPTICS COMMUNICATIONS
15 August 1986
coupling coefficient assumption. This situation, quite different from the previous case discussed by Fisher et al., can be handled more easily in the framework of a quantum material system modelized by a two-level system. It is the aim of this note to develop an analytical expression of the conjugate response to input pulses of arbitrary form by analyzing the transient behavior for cw-pumped microscopic two-level conjugators. In a first step the polarisation is calculated by using the density matrix equations in the rotating-wave approximation. These equations are solved to third order by perturbation theory [8]. Introducing the medium polarisation as a source term in Maxwell equations, we obtain a pair of coupled wave equations with time dependent coefficients related to the parameters of the transition under consideration. The conjugate response function is obtained by solving this set of equations and we analyze the influence of the signal field attenuation coefficient a 0 and transition lifetime. Two different regimes are observed depending on the ratio between a 0 and the transition lifetime. The physical origin is discussed.
2. Theory We consider the situation illustrated in fflg. 1. The medium is constituted by an ensemble of N two-level atoms. It can be characterized by a dipole moment It, longitudinal and transverse relaxation times Pi -1, p~-I respectively and transition energy ~iO~ab. This medium is pumped by two counter-propagating plane waves 6pl(r, t), 6p2(r, t), with same frequency Go. In the following it is assumed that the fields are not depleted and that they remain plane waves in the medium. Also the fields have the same intensity and are in any case much stronger than the ones of the signal and conjugate fields. They are written as
6pl(r,t)=~Epexp[-i(¢ot-ki'r)]
+cc,
i= 1,2,
(1)
where cc stands for the complex conjugate and k 1 + k r = 0. The signal and conjugate fields are represented by the expressions
61(r, t) = ~Ei(r, t)exp [ - i ( ~ t - k]-r)] + c c ,
] = s, c ,
(2)
with k s + k e = 0 and Ec(r, t) = 0 for x / > L. Because of their low intensities, we consider throughout that the medium is linear with respect to the signal or conjugate field. The geometry is chosen so that the angle 0 between k 1 and k s is small enough to write k i • r = kix , i = 1,2, s, c. We assume also that the amplitude of the signal and conjugate fields depend only upon x. Let us write now the well-known equations of motion for the elements Pi/of the density matrix [9] Pa b(t) + (P 2 -- iO~ba)Oa O(t) = --(i/h)Hab (t) w (t) ,
(3)
Pba(O + (P2 + ROba) Pba( t) = (i/h ) H b a ( t ) w ( t ) ,
(4)
~ ( t ) + Pl (w(t) - w0) = --(2i/P0 (Hba(t)Pab(t) -- Pba(t)Ha b(t)),
(5)
where w ( t ) = Pbb(t) -- Paa(t). Here w o is the equilibrium population difference between the two levels a and b involved in the transition with
~~c(r't)
~'p2(r,t) N°nl'Tarmedium]~......._~ Fig. 1. Geometry for degenerate four wave mixing. 161
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15 August 1986
no optical fields applied. Hba(t ) stands for the matrix element of the interaction energy and is given in the rotating-wave approximation by 4
Hba(t) = --½1~ba~ El(x, t)exp [-i(~ot - k/r)] = H*b(t) ,
(6)
where El(x, t)represents the complex amplitude of one of the four fields already described for a given]. Finally ~ba is the matrix element of the dipole moment operator ~. Formal integration o f equations (3), (4) and (5) yields
Pab(t) = --(i//O
f
t
Hab(t')w(t')exp [ - ( r 2 - i6Oba)(t - t')] d t ' ,
(7)
--oo t
Pba(t) = (iflO f Hba(t')w(t')exp [--(I" 2 + iWba)(t -- t')] d t ' ,
(8)
--oo t
w(t) = w 0 - (2i/h) f [Hba(t')Pab(t' ) -- Pba(t')Hab(t')] exp [ - P l ( t -
t')] dt',
(9)
--oo
with the initial conditions Pab(-oo) = Pba(-oo) = 0 and w ( - - ~ ) = w 0 = - 1 , perturbative solutions of these equations up to third order are of the form
wO(t) = w 0 = - 1 ,
(10) t
O(1)b(t) = --(i/h)w0 f Hab(tl)exp [--(F2 -- k°ba)(t -/1)] d t l ,
(11)
--oo
t
w(2)(t) = --(2/~2)Wof
t2
dt 2 f
--oo
d/1 [Hba(t2)Hab(tl)exp [iCOba(t2 - t l ) ]
--oo
+ Hba(tl)Hab(t2) exp [--i¢Oba(t 2 - t l ) ] ] exp [--P2(t2 - tl)] ex p t
p(3)b(t)=(2i/h3)WoeXp[--(P2--iCOba)t I f
dt f
[-rl(t -
t2) ] ,
(12)
t2
dt 2 f
dt 1 •
X [Hab(t3)Hba(t2)Hab(tl)ex p [i6Oba(t2 - t 1 - t3) ] +Hab(t3)Hab(t2)Hba(tl)exp [--i~ba(t 2 - t l + t3)]] X exp [ - r ' l ( t 3 - t2) ] exp [ - r 2 ( t 2 - t 1 - t3)],
(13)
witht 1
Hab(t3)Hba(t2)Hab(tl) =
4
--~U.btUl2~ ~ ~ E~(x, t3)Em(X, t~)E*.(x, tl) l=1 m =1 n=l
k m + kn)" r ] ,
X exp [i¢o(t 3 - t 2 + t l ) ] exp [ - i ( k / 4
Hab(t3)Hab(t2)Hba(tl) = •
4
1
g#abl/.t[ 2 ~ l=1
4
4
~ ~ E~(x, t3)E*(x, t2)En(X, t l ) m=l n=l
X exp [ i ~ ( t 3 + t 2 - t l ) ] exp [ - i ( k / + k m - kn)" r].
162
(14)
(15)
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15 August 1986
Taking the contribution such k I - k m + k n = k s (resp. ke), and k l +km - kn = ks (resp. ke), in th e expressions (14) and (15), the corresponding matrix elements p(al)(t,ks) and pa(2)(t,ks) (resp. p(al)(t, ke) and p(a3)(t,kc)) are of the form p(1)(t ks) = (ilzabwO/2h) exp [i(cot - k s - r)]E*(x, t) ®exp [ - ( I " 2 - iA)t]H(O ab" ' .
p(3)(t, ks) -
--ilaab hul2w0
+ EpEp
2fi 3
(16)
[ IEp 12 exp [i(cot - k s • r)] L rl(r2 - iA) E*(x, t) ® exp [-(I" 2 - iA)t] H(t)
{Ee(X, t) ®exp(-1"1t)H(t) ®exp [-(1"2 - iA)t]H(t)}
P2 - iA + Igpl 2 (g*(x, t) ® exp [--(1"2 - iA)t]n(t) ® e x p ( - P 1 OH(t) ® exp [-(1"2 - iA)t]H(t)} IEpl2 {E*(x, t) ® exp [-(1" 2 - iA)t]H(t)) + 1"1(1,2 + iA) jEp 12 +~ { E
• s (x, t) ® exp(-1,1t)H(t) ® exp [ - ( 1 , 2 - i z ~ ) t ] H ( t ) }
• * (Ec(x, t)®exp[-(F2+iA)t]H(t)®exp(-1"1t)H(t ) ® e x p t - ( r 2 - i A ) t l H ( t ) ) J , + EpEp
(17)
where H(t) is the Heaviside function. The symbol ® stands for the convolution o t?erator and A is the detuning between the transition and field frequencies, A = C%a - ¢o. The expressions for p(~J(t, ke) and p(a3b)(t,ke) are obtained by exchanging c and s in the relations (16) and (17). We are not able to write the polarisation and more precisely the polarisation Ps(t, ks) and Pc(t, ke) due to the signal and conjugate fields respectively. They are given by
Ps(t, ks) = N[~tabPba(t , ks) + I~baPab(t, ks)],
(18)
Pc(t, ke) = N[UabPba(t, ke) + labaPaO(t, ke)],
(19)
where
pao(t, ks) =
ks) +
ks),
pba(t, ks) = p*b(t, ks),
(20)
Pba(t, ke) = p*b(t,
(21)
and
Pab(t, ke) = p(lb)(t, kc) + p(3b)(t,k c ) ,
ke).
If we consider the expressions (16) and (17), Pab(t, ks) (resp. Pab(t, ke) ) can be written as
Pab(t, ks) = [o(al)(t, E*) + a(3b)(t,E~) + a(a3)(t,Ee) ] exp [ i ( ~ t
- k s "r)],
(22)
with
e,*) =
e * ) exp
t-
ks-,)],
(23)
and
O(3b)(t,Es*) + a(3b)(t,Ec) = p(a3b)(t,ks)exp [--i(¢ot
- k s- r ) ] ,
(24)
where a(a3b)(t,E~) and o~a~(t, Ee) are deemed by the expressions (17) and (24) and depend ordy on E * and Ee, respectively;Besid~ if the amplitude Of Ep is low enough we can neglect the non linear term O(a3b)(t,E*) with respect to ata*b~(t,Es). Then relation (22) takes the form 163
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15 August 1986
Pab(t, ks) = [a(al)b(t,E*) + o(3)b(t, E¢)] exp[i(cot - k s. r)].
(25)
A similar expression is obtained for pab(t, ke). The space and time dependence of the signal and conjugate fields are obtained by solving Maxwell equations 32 ~s(X, t)/Ox 2 -- c -2 026 s(X, t)/Ot 2 = (47r/c2) 32Ps(t, ks)/at 2 ,
(26)
a 2 6c(x, t)/Ox 2 - c -2 a 2 6e(x , t)/Ot 2 = (4rt/c 2) a2Pe(t, ke)/at 2 .
(27)
Applying the standard methods of non linear optics [10] this set of coupled equations yields in the slowly varying amplitude approximation
aE*(x, t)/Ox + c -1 aE*(x, t)/Ot = -(2~rico/c)P+(x, t) ,
(28)
3Ee(X, t)/ax - c -1 OEe(x, O/at = -(2nico/c)Pc (x, t) ,
(29)
where
P+(x, t)= Nub ° 2Pab(t , ks)exp [-i(¢ot- ksx) ] , PC(x, t)= Nub° 2Oba(t, kc)exp [i(¢ot- kex)]. We assume that signal and conjugate fields start at t = 0. From the Laplace-transform of (28), (29) and using relations (16), (17) and (25) the set of coupled equations (28), (29) can be written as
aE*(x, p)/ax + a(p)E*(x, p) = +/3(P)Ec(x, p),
(30)
3E e(x, p )/ ax - ct*(p)'Ee(X, p) = -{3*(p ) E * (x, p ) ,
(31)
where Es(X, p) and Ee(x, p) are defined by f~*E](x, t ) e x p ( - p t ) dt for] = s, c, and ,
c~(p) - P + °t0P2p2 - iA +Pc '
,
2
a0El~E~rlr2 P + 2P 2 I~(P)= Is(P2 - i A ) (p + P l ) ( p + F2 + iA)(p + P 2 - iA)
The quantity ~0 = 21rc°Nl#12/hcF2 is the line center small signal field attenuation coefficient and I s =/i2F 1F2/ Ipl 2 is the line center saturation intensity. The set of coupled equations (30), (31) is just the Laplace-transform of the well-known coupled set given by the relation (3) in ref. [7], but as previously mentioned in the introduction, our coefficients a and # are p-dependent (i.e. time dependent) and are functions of the transition parameters. By introducing the initial conditions Es*(0, p) = F(p) and Ee(L, p) = 0, where F(p) is the Laplace-transform of the time dependent amplitude of the signal field at x = 0, the set of equations (30) and (31) is easily solved. We get Ec(0, p ) =, 2~*(p)F(p)/{ot*(,p) + or(p) + x / ~ [ 1
+ exp(-L~]/[1
- exp(-L~]
},
(32)
with 6(p) = [t~(p) + a*(p)] 2 _ 413(p)/3*fp). For mathematical convenience we assume A = 0 and a purely radiative decay (i.e. P 1 = 2P2). Then t~(p) and/3(p) have the form
a ( p ) = pie + a0r2/(p + r 2 ) ,
•
•
2
3(p) = (2aoEpEpP2/Is)(p
+p
2)
-2
•
Therefore 8 is now given by 5 = 4(a - t/3[)(a + I~[), a + [3l = p/c + [c~oP2/(p + F2) 2] [p + P 2 + (2[Ep[2/Is)P2]. If the saturation intensity I s is much bi~her than the pump intensity [Ep[ 2 then 6(p) = 4 [a(p)] $ and this condition is consistent with the resolution of the density matrix equations to third order by perturbation theory. If 2 [E112/ I s < 1, we can reduce expression (32) to Ee(0, p) = [3*(p)F(p)/2a(p)] [ 1 - e x p ( - 2 L o~(p))]. 164
(33)
Volume 59, number 2
15 August 1986
OPTICS COMMUNICATIONS
Introducing the value of a(p) and 3(P), we get (34)
Ec(O, p)/F(p) = [K/(p + r2) (t72 + pF 2 + ao F2c)] {1 - exp [ - 2L (p/c + ct0 F2/(p + F2))] },
cr22/is.
where K = OtoE~E~ From the inverse Laplace transform, the amplitude of the conjugate field at x = 0 is given by Ec(O, t) = R(t)
® F(t)
(35)
,
where F(t) is the input pulse and R(t) the conjugate response deffmed by ff e x p ( - F 2 t )
R(t) = K [~
a-b
exp(-at) ~ b(b-a)
e a(b-a) x p ( - b t ) ]] H(t) - A t - 2L/c)H(t - 2L/c) 1 ,
(36)
with a = (F2/2) [1 + (1 - 4aoC/F2)l/2],
b
(F2/2) [1 - (1 - 4aoC/r2)l/2].
=
The quantity f(t) is defined by 2LbOr'2 )
(-2LaOr2/b)n ~
exp(-F2t)~ n =0
2L
1 a) e x p ( - b t ) e x p ( a(b --
(bt) k
k =0
_~0 ) e x p ( - r 2 t ) * * ( - 2 L a o F 2 / a ) n ~ ( a t ) k r2 + a ( b - a ) ~-~ n! k! n =O k=O
(37)
3. Discussion
From the analytical expression (34), we see that the response function is proportional to t~0 and (E~) 2. This is in agreement with the Abrams and Lind theoretical results [2,3] where the limit IEpl2/ls ~ 1 is taken. Our results IRI
20-"f Arb.
(b)
Unit
IRI
F
Arb. Unit.
1.5
~o=10m_1 I I ,x%=sm-1
(a) '~
2L/c=lOns. %=0.1rn-I
1.0
15 --
iiiiiill" I.t~I.I
lo_
!
2L/c=lOns. r~=o2.s:'
\
i
I
0.5
I 5
I 10
I 15
I 20
I 25 t(ns)
5
10
15
20
25 t(ns)
Fig. 2. We h'ave sketched the variations of IR(t)l as a function oft for different values ofa 0. The cases 4aoc/F < 1 (a) and 4aoc[F > 1 (b) are considered. 165
Volume 59, number 2
OPTICS COMMUNICATIONS
15 August 1986
IRI
Arb, Unit,
2L/c=lOns. ~o =1 m71
20 I
/
\
w a o
15
. i
i
12=0'2ns-" 1
l
J 1
10
5
/
12= 0 . 1 n s - . 1
l
Fig. 3. The time dependence of the response function is reI
I
I
I
I
5
10
15
20
25
t(ns)
presented for various relaxation times r2.
show that for 4Soc/P 2 > 1, R(t) exhibits oscillations. Fig. 2 contains plots of IR(t)l versus t for several values of s 0. The cases 4SoC < r 2 and 4 s 0 > P2 are represented by figs. 2a and 2b respectively. To understand why this quantity play such a role in the response function and why R(t) exhibits oscillations let first consider the simple model of a grating v~hich corresponds to the interference figure formed by the two pump fields on the non-linear medium. Assume the conjugate wave of the form exp(-1` 0 when the signal wave impinges on a maximum of the interference figure. Then at x = 0, the conjugate signal can be written as e x p ( - F t ) + exp ( - s o d ) exp [-1`(t - 2d/c)] + e x p ( - 2 s 0 d ) e x p [-1`(t - 4d/c)] + ..., where d is the distance between two maxima. If sod ,~ rd/c (i.e. SoC ,~ F) we have only a non-interacting superposition of pulses. In the other case corresponding to SoC >> P we have an interfering contribution resulting from the superposition of all the pulses. When the delta function interacts with a maximum, the conjugate spectrum is reduced due to the lifetime of the transition. When all pulses overlap, oscillations occur. For a given 1` and with the condition SoC > F the efficiency of this collective interaction is more important than for small s 0 and the bandwidth of the conjugate wave decreases with s 0. This can explain the time behavior of R(t) with respect to s 0 as seen in fig. 2. The influence of 1"2 is plotted in fig. 3 where we observe that the time evolution of the response function is smoother when 1" decreases. This results from the diminution of the bandwidth with 1"2 [10].
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
166
See for example Optical Engineering 21 (1982) 2. R.L, Abrams and R.C. Lind, Optics Lett. 2 (1978) 94. R.L. Abrams and R.C. Lind, Optics Lett. 3 (1978) 203. T.Y. Fu and M. Sargent III, Optics Lett. 4 (1979) 52. D.M. Pepper and R.L. Abrams, Optics Lett. 4 (1979) 366. J.H. Marburger, Appl. Phys. Lett. 32 (1978) 372. R.A. Fisher, B.R. Suydam and BJ. Feldman, Phys. Rev. A23 (1981) 3071. N. Bloembergen, Nonlinear optics (Benjamin, New York, 1965)• DJ. Hatter and R.W.Boyd, IEEE JQE-16 (1980) 1126. A. Yariv, Quantum electronics (Wiley, NewYork, 1975). J. Nilsen and A. Yariv, AppL Optics 18 (1979) 143.