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Self-induced transparency and zero-area pulses
4 December 1995 PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 208 (1995) 323-327
Self-induced transparency and zero-area pulses G.T. Adamashvili Tbilisi State University, Tbilisi, Georgia
Received 4 January 1995; revised manuscript received 28 July 1995; accepted for publication 6 September 1995 Communicated by V.M. Agranovich
Abstract A theory for the self-induced transparency for a zero-area pulse of a new type is constructed. It is shown that a nonlinear wave has a complicated structure and is characterized by two different oscillation frequencies. The well-known breather solution in the limited case is obtained.
1. When intensive optical radiation propagates in a condensed medium, various nonlinear phenomena occur. Among the nonlinear phenomena, of particular interest are those that lead to the formation of nonlinear waves with a stable shape. They can transfer energy without essential loss over considerable distances. Solitons or their various modifications, like breathers, kinks, etc., are basic among such nonlinear waves. In nonlinear physical theory, they play as fundamental a role as harmonic oscillations do in linear wave theory. Every above mentioned soliton has its conditions of existence and specific regularity of propagation. Revealing new types of solitons and the investigation of their properties is one of the basic problems of nonlinear wave physics. For the investigation of such waves one should look at the mechanisms for the formation of solitons. There are several different mechanisms for the formation of a soliton. The reverse statement seems to be true as well - by means of one and the same mechanism two or more different kinds of solitons can be formed. Depending on the physical situation, resonant solitons can be formed by means of the Elsevier Science B.V. SSDI 0375-9601(95)00734-2
mechanism of McCall and Hahn, i.e. under the conditions of a nonlinear coherent interaction of optical waves with impurity resonance atoms contained in the medium when the conditions of self-induced transparency (SIT) are satisfied [ 1,2], oT a 1, T. .K T,,z, where T, and o are the width and the frequency of the pulse, the T,,, are the longitudinal and transverse relaxation times of impurity atoms. For this soliton (27r-pulse) the condition can be realised when the area of the optical pulse envelope is O( z, t) = ~~1’ ,??(z, t’) dt’ > n. --co (where E = ge-‘@- 0)) is the x-component of the strength of the electrical field, ~a = 2 d/h, d is the dipole matrix element, h is the Planck constant, k is the wave number), and the breather (Orr-pulse) solution when o=
1.
(1)
The question whether the breather is the only possible Or-pulse or not has been an open one till now. It
324
G.T. A~~s~v~l~/
Physics Letters A 208 (199Si 323-327
is precisely this question we shall consider in this paper. 2. For the consideration of the SIT phenomenon it is sufficient to use the one-dimensional wave equation, a2E;
---~ az2
q2 a2E
c2 at2
=--
4~
a2p
2
89 )
12)
where c and 77are the speed of light in vacuum and the refractive index, P = nods, is the macroscopic polarization of the impurity non-interacting two-level atoms under the conditions of homogeneous broadening of the spectral line and in the absence of phase modulation, n, is the number of atoms per unit. The dimensionless amplitudes of the electric dipoles of impurity atoms s, and s2, together with the population difference between the ground and excited states sf are governed by the optical Bloch equations, a%( t> =
-q$,(t), at as2tt> = WoS,(t) + KoE( t, Z)Sg( t), at as, ( t > = -K&t, Z)s,(t), at
a20
a20
-CX~ sin 0,
law w = Ck, where CX~= is the speed of light in the medium. We can solve this equation by means of the inverse scattering transform and obtain the solution in the form of a 27r-pulse or a soliton [3] (see also Refs. [ 1,2]), 2rrn,d2u/fiq2,
i=
2 -
t-z/u
T,
’
where u is the speed of the pulse. Quite another kind of SIT was investigated by Lamb [2], who considered the propagation of pulses in a medium without any loss but with a change of shape. The pulse areas which are equal to 2nn belong to such pulses (where n = 0, 1, 2,. . . >. Zeroarea pulses play an essential role among such pulses and we can consider them as boundary conditions for pulses whose total area is equal to zero. We can obtain the analytical expression for such a pulse from Eq. (4) under condition (1). Then (4) leads to the following equation,
a20
(4)
C = c/q
sech ~
%Tu
T+C
where s,(t) = (6,(t)), (i = 1, 2, 3) B,(t) are the Pauli operators, o0 is the resonant frequency of the impurity atoms. All the relaxation times have been assumed to be infinite. The most signi~c~t effects at the interaction of an optical wave with a reson~tly absorbing medium are usually observed at exact resonance. Therefore, for simplicity, we consider Eq. (3) at exact resonance w = we. Then s2 = -@ sin 0 e-i(kz-or), where ci = rt 1, The plus sign corresponds to the initial condition, when all the impurity atoms are initially in the ground states, i.e. t + - w, s3 = - 1. The minus sign corresponds to the case when t -+ - ~0,s3 = + 1, i.e. all the impurity atoms for the initial condition are initially in the excited states. Substituting this solution in (2) and using the slowly-vying-envelop ap~oximation for the quantities E and P, Eq. (2) reduces to the well-mown sine-Gordon equation,
-p+e -=ataz
and the dispersion
a20 -= ata
-a,ZO+p203+0(05),
(6)
where p = f o,“. For the analysis of the equation we make use of the perturbation reduction method by obtaining the solution of Eq. (6) in the following form [41,
where the quantities q, 6 and cp are determined below by formulas (22) and (26) for ug = 0. Expression (7) is an analytical expression of the zero-area pulse, which is often called a pulsing solition or breather. We can see from (7) that the breather has an interior structure - in the process of the propagation in a medium, it oscillates according to the law exp[i(Qz - 0 r)]. Here the oscillation parameters R and Q are interior parameters of the breather. Additional conditions are necessary for the existence of this solution: R Ta > 1, QLB z+ 1, where La and Ta are the length and the width of the breather. These conditions are satisfied for many optical waves, for example, in solid dielectrics containing a small concentration of impurity atoms. For this the quantity
325
G.T. Adamashvili/ Physics Letters A 208 (1995) 323-327
oTa exceeds 106. This circumstance suggests to us the idea that the realization of a zero-area pulse with a more complicated interior structure, with two different ~h~~~~stic o~illation frequencies, is possible for the optical wave. In other words, we wish to consider a solution of Eq. (4) under condition (1) of the type of a zero-area pulse with two different characteristic oscillation frequencies.
where
3. To this end we will search the solution of (6) in the form
w-0,
O( z, r) =
Substituting (11) in (10) we obtain the equation
i X,F,(Z m= --m
T),
The use of this expansion is possible following conditions are satisfied,
the
KBQ,
1afifl,,/a7 I ~3: n I f:y;
(8)
beCaUse
I,
I af:ll,,/az
I c f2 I fLYi) I.
where im(rn-
X,=e
wr)
,
Z=z-u,t,
T=t,
ug=;.
Herewith the following inequalities are satisfied,
a’,a”mv
k>> K,
WSW,
Xf(l) m m Substi~ting (8) in 6) we obtain
cm
%I
k
(_
s,n
J m,n =Q[qz----4
(10)
n
,_
r
f(P'?f("% in ,n s,r
en’+a” -co
I
=o,
(12)
-n(fl+
- fiq,
+ nQ2q, - naQq,
+ nfl*),
3ug + 2nQq, Qu,)qs + 2nRu,],
H m.n =Q2(qcq~Ug+$),
q1 =t?Z2W(KC-W),
h m.m = q3 + nQq5 - 2nfi.
m(2Wl.4, - WC - CKtl,),
q5 = c -
_
n’,r=
$n.*= 41 + WPI,“~
qr + iq,% + i&G
q,=RZ(KC-2W),
ppos= --m
-cc
where
W “,n = -n(Qq2
C& =
c
c
z
-p2
To determine the values of fA:nf we set the terms corresponding to the same powers of E equal to zero. As a result, we obtain a chain of equations: in the first order in E
q4=Ug(Ug-c),
2u,.
For the solution of Eq. (10) we use the perturbative reduction method according to which the quantity F,,, can be represented in the following form,
(%., f LYf)f;‘J! = 0, in the second order in E
F,( 2, T) = g ? &“Ynti;:( 5, T), n-1 n---m
(131
(11)
(14
326
G.T. Aabzashuili/Physics
and in the third order in E
Letters A 208 f1995) 323-327
where the quantities p, the expressions
and y are determined from
q
2 ,
q=2p
p
=
H%Al
nOQ2$,,n, ’
nohn 0.n”
Z= \/3;y+ugT.
(22) = 0.
(16)
This is the well-known nonlinear Schriidinger equation (NSE), which under the condition PQ > 0 has the soliton solution
To begin with we consider the situation when m = n = n, = - 1. Then it follows from Eq. (14) that
Pno= no X 2in
“n,,., +(X,2=0.
where
(17)
w-4 -icp,no) ch(2r)q)
(23)
’
From this relation follows the connection between w and K, (CK-W)W+Ct;=o.
(18)
and the relation between 0 and Q is determined from
Qq2-@+ni,(Q*cr
flQqs +n*)
=O.
(19)
Such a “separation” of Eq. (17) into two relations is not arbitrary. The connection between the quantities R and Q should be such as to guarantee the carrying out of Eq. (15), i.e. for m = n = n, the quantity J no,“o should be equal to zero. Indeed, taking into account the definition of the quantity us one can easily observe that the following relation occurs,
Q
J %.“O = ---=
no
a%,.,,
o
aQ
*
Consequently, if relations (18) and (19) occur for cn) then Eq. (15) is satisfied. Accordany value of L,.,,, ing to Eq. (14), only the following terms of all the quantities fi’,! differ from zero: fn($O. We have to note that relations (17)-(20) occur also when the condition m = n = no = + 1 is satisfied. TO guamntee the reality of the quantity 0, we set 0 = 0 * and consenquently we obtain f
(21)
The quantities 5, r), ‘p. and y. are the scattering data that arise when the NSE is solved by the inverse scattering transform [3]. Relations (21)-(24) are valid under the condition no = - 1 as well as for the case no = + 1. In other words, we have two cases - two solutions: first, no = - 1 and, second, no = + 1. Consequently relations (21)-(24) do not depend on the quantity no. To see this, only look at the case for which the condition qp > 0 is satisfied. Substituting the soliton solution for P& in (8) and (1 l), we obtain for the quantity 0 the super pulsing soliton solution, 477 sin( 6, + S) OS’ = x
ch(2v(p)
(25)
’
where &=Wt-KZ,
6=
LtT-QZ+
cp,.
(26)
4. The appearance in (25) of sin( 6 1 + S > indicates the formation of periodic (slow in comparison with exp[i(wt - kz)]) beats in coordinates and time, with characteristic parameters w, 0, K and Q, as a result of which the soliton solution (23) for the quantity VnO is transformed into solution (25) for the quantity 0. The latter solution is a nonlinear wave of breather type with two oscillation frequencies (superbreather).
G.T. Adamashvili/
Physics Letters A 208 (1995) 323-327
The bond between (25) for 8sa and the “usual” breather (7) for 0, is especially clear from (25) when the inequality us =Z us is satisfied, 477 sin 6, cos 6 %a = F ch(2nrp) fcos
&@a*
Analyzing the whole process of the solution of the wave equation (4) we reach the conclusion that our solution is connected with a nonlinear wave motion characterized by the three different oscillation frequencies w, w and R. Herewith, the envelope of the pulse is characterized by the quantities w and Ll and their connection with the quantities K and Q is characterized by the two relations (17) and (18). In other words, the solution for the envelope of the optical pulse has a more complex interior structure than the breather (7) and is characterized by the interior parameters w, a, K and Q. Consequently, the resonance transparency of the medium under the condition of SIT is possible not only by soliton (4) or breather (7), but by the optical pulse (25) too. The obtained solution (25) is also valid in an ampli~ing medium under the condition that the connection between the quantities w and K instead of (18) has the form KC = w. In this case the quantity H n,,no > 0 and the condition pq > 0 canbe satisfied for c = - 1, which is valid for the amplifying medium. Con~uently, solution (25) for the optical pulse can exist for the resonance absorbing medium as
327
well as for the splicing one. The influence of the phase modulation and the inhomogeneous broadening of a spectral line can be realized by means of a standard procedure (see, for example, Ref. [5]). It should be noted that 6,) 6 and yl contain the “group velocities” us and ug and also the quantities a, Q, w and K, i.e. parameters which are more characterized by (w, K), and less by (0, Q) for rapid oscillation processes. The quantities w(,K) and n(Q) are connected only by the inequalities w Z+ 0 and K 2 Q. Here we obtain solution (25) by means of exp~sions (8) and (11) under condition (1). More interesting is the solution of Eq. (4) when condition (1) is not satisfied. It is possible to use the inverse scattering transform, which will be done in a future work.
References [ll S.L. McCall and E.L. Hahn, Phys. Rev. 183 (1%9) 457. 121 L. Allen and J.H. Eberly, Optical resonance and two-level atoms (Wiley, New York, 1975); G.L. Lamb, Rev. Mod. Whys.43 (1971) 99. [3] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon and H.C. Morris, eds., Solitons and linear wave equations (Academic Press, New York, 1982). [4] G.T. Adamashvili and S.V. Manakov, Solid Sta@ Commun. 48 (1983) 381; G.T. A~~hvili, L.V. Ckonia, 1.1. Chikaidze and G.G. Nozadze, Kvaut. Efektr. 18 (1991) 1251. I51G.T. Adamashvili, Teor. Mat. Fiz. 99 (1994) 92.
ELSEVIER
LETTERS
A
Physics Letters A 208 (1995) 323-327
Self-induced transparency and zero-area pulses G.T. Adamashvili Tbilisi State University, Tbilisi, Georgia
Received 4 January 1995; revised manuscript received 28 July 1995; accepted for publication 6 September 1995 Communicated by V.M. Agranovich
Abstract A theory for the self-induced transparency for a zero-area pulse of a new type is constructed. It is shown that a nonlinear wave has a complicated structure and is characterized by two different oscillation frequencies. The well-known breather solution in the limited case is obtained.
1. When intensive optical radiation propagates in a condensed medium, various nonlinear phenomena occur. Among the nonlinear phenomena, of particular interest are those that lead to the formation of nonlinear waves with a stable shape. They can transfer energy without essential loss over considerable distances. Solitons or their various modifications, like breathers, kinks, etc., are basic among such nonlinear waves. In nonlinear physical theory, they play as fundamental a role as harmonic oscillations do in linear wave theory. Every above mentioned soliton has its conditions of existence and specific regularity of propagation. Revealing new types of solitons and the investigation of their properties is one of the basic problems of nonlinear wave physics. For the investigation of such waves one should look at the mechanisms for the formation of solitons. There are several different mechanisms for the formation of a soliton. The reverse statement seems to be true as well - by means of one and the same mechanism two or more different kinds of solitons can be formed. Depending on the physical situation, resonant solitons can be formed by means of the Elsevier Science B.V. SSDI 0375-9601(95)00734-2
mechanism of McCall and Hahn, i.e. under the conditions of a nonlinear coherent interaction of optical waves with impurity resonance atoms contained in the medium when the conditions of self-induced transparency (SIT) are satisfied [ 1,2], oT a 1, T. .K T,,z, where T, and o are the width and the frequency of the pulse, the T,,, are the longitudinal and transverse relaxation times of impurity atoms. For this soliton (27r-pulse) the condition can be realised when the area of the optical pulse envelope is O( z, t) = ~~1’ ,??(z, t’) dt’ > n. --co (where E = ge-‘@- 0)) is the x-component of the strength of the electrical field, ~a = 2 d/h, d is the dipole matrix element, h is the Planck constant, k is the wave number), and the breather (Orr-pulse) solution when o=
1.
(1)
The question whether the breather is the only possible Or-pulse or not has been an open one till now. It
324
G.T. A~~s~v~l~/
Physics Letters A 208 (199Si 323-327
is precisely this question we shall consider in this paper. 2. For the consideration of the SIT phenomenon it is sufficient to use the one-dimensional wave equation, a2E;
---~ az2
q2 a2E
c2 at2
=--
4~
a2p
2
89 )
12)
where c and 77are the speed of light in vacuum and the refractive index, P = nods, is the macroscopic polarization of the impurity non-interacting two-level atoms under the conditions of homogeneous broadening of the spectral line and in the absence of phase modulation, n, is the number of atoms per unit. The dimensionless amplitudes of the electric dipoles of impurity atoms s, and s2, together with the population difference between the ground and excited states sf are governed by the optical Bloch equations, a%( t> =
-q$,(t), at as2tt> = WoS,(t) + KoE( t, Z)Sg( t), at as, ( t > = -K&t, Z)s,(t), at
a20
a20
-CX~ sin 0,
law w = Ck, where CX~= is the speed of light in the medium. We can solve this equation by means of the inverse scattering transform and obtain the solution in the form of a 27r-pulse or a soliton [3] (see also Refs. [ 1,2]), 2rrn,d2u/fiq2,
i=
2 -
t-z/u
T,
’
where u is the speed of the pulse. Quite another kind of SIT was investigated by Lamb [2], who considered the propagation of pulses in a medium without any loss but with a change of shape. The pulse areas which are equal to 2nn belong to such pulses (where n = 0, 1, 2,. . . >. Zeroarea pulses play an essential role among such pulses and we can consider them as boundary conditions for pulses whose total area is equal to zero. We can obtain the analytical expression for such a pulse from Eq. (4) under condition (1). Then (4) leads to the following equation,
a20
(4)
C = c/q
sech ~
%Tu
T+C
where s,(t) = (6,(t)), (i = 1, 2, 3) B,(t) are the Pauli operators, o0 is the resonant frequency of the impurity atoms. All the relaxation times have been assumed to be infinite. The most signi~c~t effects at the interaction of an optical wave with a reson~tly absorbing medium are usually observed at exact resonance. Therefore, for simplicity, we consider Eq. (3) at exact resonance w = we. Then s2 = -@ sin 0 e-i(kz-or), where ci = rt 1, The plus sign corresponds to the initial condition, when all the impurity atoms are initially in the ground states, i.e. t + - w, s3 = - 1. The minus sign corresponds to the case when t -+ - ~0,s3 = + 1, i.e. all the impurity atoms for the initial condition are initially in the excited states. Substituting this solution in (2) and using the slowly-vying-envelop ap~oximation for the quantities E and P, Eq. (2) reduces to the well-mown sine-Gordon equation,
-p+e -=ataz
and the dispersion
a20 -= ata
-a,ZO+p203+0(05),
(6)
where p = f o,“. For the analysis of the equation we make use of the perturbation reduction method by obtaining the solution of Eq. (6) in the following form [41,
where the quantities q, 6 and cp are determined below by formulas (22) and (26) for ug = 0. Expression (7) is an analytical expression of the zero-area pulse, which is often called a pulsing solition or breather. We can see from (7) that the breather has an interior structure - in the process of the propagation in a medium, it oscillates according to the law exp[i(Qz - 0 r)]. Here the oscillation parameters R and Q are interior parameters of the breather. Additional conditions are necessary for the existence of this solution: R Ta > 1, QLB z+ 1, where La and Ta are the length and the width of the breather. These conditions are satisfied for many optical waves, for example, in solid dielectrics containing a small concentration of impurity atoms. For this the quantity
325
G.T. Adamashvili/ Physics Letters A 208 (1995) 323-327
oTa exceeds 106. This circumstance suggests to us the idea that the realization of a zero-area pulse with a more complicated interior structure, with two different ~h~~~~stic o~illation frequencies, is possible for the optical wave. In other words, we wish to consider a solution of Eq. (4) under condition (1) of the type of a zero-area pulse with two different characteristic oscillation frequencies.
where
3. To this end we will search the solution of (6) in the form
w-0,
O( z, r) =
Substituting (11) in (10) we obtain the equation
i X,F,(Z m= --m
T),
The use of this expansion is possible following conditions are satisfied,
the
KBQ,
1afifl,,/a7 I ~3: n I f:y;
(8)
beCaUse
I,
I af:ll,,/az
I c f2 I fLYi) I.
where im(rn-
X,=e
wr)
,
Z=z-u,t,
T=t,
ug=;.
Herewith the following inequalities are satisfied,
a’,a”mv
k>> K,
WSW,
Xf(l) m m Substi~ting (8) in 6) we obtain
cm
%I
k
(_
s,n
J m,n =Q[qz----4
(10)
n
,_
r
f(P'?f("% in ,n s,r
en’+a” -co
I
=o,
(12)
-n(fl+
- fiq,
+ nQ2q, - naQq,
+ nfl*),
3ug + 2nQq, Qu,)qs + 2nRu,],
H m.n =Q2(qcq~Ug+$),
q1 =t?Z2W(KC-W),
h m.m = q3 + nQq5 - 2nfi.
m(2Wl.4, - WC - CKtl,),
q5 = c -
_
n’,r=
$n.*= 41 + WPI,“~
qr + iq,% + i&G
q,=RZ(KC-2W),
ppos= --m
-cc
where
W “,n = -n(Qq2
C& =
c
c
z
-p2
To determine the values of fA:nf we set the terms corresponding to the same powers of E equal to zero. As a result, we obtain a chain of equations: in the first order in E
q4=Ug(Ug-c),
2u,.
For the solution of Eq. (10) we use the perturbative reduction method according to which the quantity F,,, can be represented in the following form,
(%., f LYf)f;‘J! = 0, in the second order in E
F,( 2, T) = g ? &“Ynti;:( 5, T), n-1 n---m
(131
(11)
(14
326
G.T. Aabzashuili/Physics
and in the third order in E
Letters A 208 f1995) 323-327
where the quantities p, the expressions
and y are determined from
q
2 ,
q=2p
p
=
H%Al
nOQ2$,,n, ’
nohn 0.n”
Z= \/3;y+ugT.
(22) = 0.
(16)
This is the well-known nonlinear Schriidinger equation (NSE), which under the condition PQ > 0 has the soliton solution
To begin with we consider the situation when m = n = n, = - 1. Then it follows from Eq. (14) that
Pno= no X 2in
“n,,., +(X,2=0.
where
(17)
w-4 -icp,no) ch(2r)q)
(23)
’
From this relation follows the connection between w and K, (CK-W)W+Ct;=o.
(18)
and the relation between 0 and Q is determined from
Qq2-@+ni,(Q*cr
flQqs +n*)
=O.
(19)
Such a “separation” of Eq. (17) into two relations is not arbitrary. The connection between the quantities R and Q should be such as to guarantee the carrying out of Eq. (15), i.e. for m = n = n, the quantity J no,“o should be equal to zero. Indeed, taking into account the definition of the quantity us one can easily observe that the following relation occurs,
Q
J %.“O = ---=
no
a%,.,,
o
aQ
*
Consequently, if relations (18) and (19) occur for cn) then Eq. (15) is satisfied. Accordany value of L,.,,, ing to Eq. (14), only the following terms of all the quantities fi’,! differ from zero: fn($O. We have to note that relations (17)-(20) occur also when the condition m = n = no = + 1 is satisfied. TO guamntee the reality of the quantity 0, we set 0 = 0 * and consenquently we obtain f
(21)
The quantities 5, r), ‘p. and y. are the scattering data that arise when the NSE is solved by the inverse scattering transform [3]. Relations (21)-(24) are valid under the condition no = - 1 as well as for the case no = + 1. In other words, we have two cases - two solutions: first, no = - 1 and, second, no = + 1. Consequently relations (21)-(24) do not depend on the quantity no. To see this, only look at the case for which the condition qp > 0 is satisfied. Substituting the soliton solution for P& in (8) and (1 l), we obtain for the quantity 0 the super pulsing soliton solution, 477 sin( 6, + S) OS’ = x
ch(2v(p)
(25)
’
where &=Wt-KZ,
6=
LtT-QZ+
cp,.
(26)
4. The appearance in (25) of sin( 6 1 + S > indicates the formation of periodic (slow in comparison with exp[i(wt - kz)]) beats in coordinates and time, with characteristic parameters w, 0, K and Q, as a result of which the soliton solution (23) for the quantity VnO is transformed into solution (25) for the quantity 0. The latter solution is a nonlinear wave of breather type with two oscillation frequencies (superbreather).
G.T. Adamashvili/
Physics Letters A 208 (1995) 323-327
The bond between (25) for 8sa and the “usual” breather (7) for 0, is especially clear from (25) when the inequality us =Z us is satisfied, 477 sin 6, cos 6 %a = F ch(2nrp) fcos
&@a*
Analyzing the whole process of the solution of the wave equation (4) we reach the conclusion that our solution is connected with a nonlinear wave motion characterized by the three different oscillation frequencies w, w and R. Herewith, the envelope of the pulse is characterized by the quantities w and Ll and their connection with the quantities K and Q is characterized by the two relations (17) and (18). In other words, the solution for the envelope of the optical pulse has a more complex interior structure than the breather (7) and is characterized by the interior parameters w, a, K and Q. Consequently, the resonance transparency of the medium under the condition of SIT is possible not only by soliton (4) or breather (7), but by the optical pulse (25) too. The obtained solution (25) is also valid in an ampli~ing medium under the condition that the connection between the quantities w and K instead of (18) has the form KC = w. In this case the quantity H n,,no > 0 and the condition pq > 0 canbe satisfied for c = - 1, which is valid for the amplifying medium. Con~uently, solution (25) for the optical pulse can exist for the resonance absorbing medium as
327
well as for the splicing one. The influence of the phase modulation and the inhomogeneous broadening of a spectral line can be realized by means of a standard procedure (see, for example, Ref. [5]). It should be noted that 6,) 6 and yl contain the “group velocities” us and ug and also the quantities a, Q, w and K, i.e. parameters which are more characterized by (w, K), and less by (0, Q) for rapid oscillation processes. The quantities w(,K) and n(Q) are connected only by the inequalities w Z+ 0 and K 2 Q. Here we obtain solution (25) by means of exp~sions (8) and (11) under condition (1). More interesting is the solution of Eq. (4) when condition (1) is not satisfied. It is possible to use the inverse scattering transform, which will be done in a future work.
References [ll S.L. McCall and E.L. Hahn, Phys. Rev. 183 (1%9) 457. 121 L. Allen and J.H. Eberly, Optical resonance and two-level atoms (Wiley, New York, 1975); G.L. Lamb, Rev. Mod. Whys.43 (1971) 99. [3] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon and H.C. Morris, eds., Solitons and linear wave equations (Academic Press, New York, 1982). [4] G.T. Adamashvili and S.V. Manakov, Solid Sta@ Commun. 48 (1983) 381; G.T. A~~hvili, L.V. Ckonia, 1.1. Chikaidze and G.G. Nozadze, Kvaut. Efektr. 18 (1991) 1251. I51G.T. Adamashvili, Teor. Mat. Fiz. 99 (1994) 92.