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Four possible types of pulses for self-induced transparency
Volume
10, number
OPTICS
2
February
COMMUNICATIONS
FOUR POSSIBLE TYPES OF PULSES FOR SELF-INDUCED
1974
TRANSPARENCY*
C.T. LEE Department
of Physics,
Alabama
A. and M. University,
Received
26 November
kormal,
Alabama
35762.
USA
197 3
We have found four types of possible steady-state solutions to the coupled Maxwell-Bloc11 equations which describe extremely intense pulse propagation in a resonant medium. This is possible due to the replacement of the usual slowly varying envelope approximation by the assumption that (c/l’-l)/wr < 1, where V is the pulse speed and T is the pulse duration. We ignore possible non-resonant losses and set Tt = T, = m.
The self-induced transparency of a sech-pulse is well known [ 11. This is the steady-state solution to the coupled MaxwelllBloch equations under the assumption that 1/cn < 1, where o is the carrier frequency and r is the pulse duration. Recently, we [2] have obtained a sli.ght modification of the sech-pulse by assuming*
where c is the phase speed and V is the pulse speed. Assumption (1) will be better than the previous assumption when (c/V’-- 1) < 1. This will be true for a typical value of r 5 1O-lo set [4]. We are curious to know if there are other types of possible steady-state solutions. In this paper we will present four types of such solutions. We will assume exact resonance, ignore possible non-resonant losses, and put the relaxation times T, and T2 to be infinite. Then the coupled Maxwell& Bloch equations will give a set of five simultaneous equations of first order. They are derived in ref. [2] and are as follows: ; = -cY_lJu, 4 =
QWU/E
+ QKW,
Li =Ju. fi =
-$U
*Work supported 020. *E. Courtens
under
grant
made a similar assumption
No. NGR-Ol-OOl(ref.
[ 31).
In the above equations, E is the envelope of the pulse, $ is the phase. u and +J are dimensionless dispersion and absorption components of the dipole moments, and w is the atom energy divided by $fiw. These quantities depend on their arguments t and z only through the combination { = t z/V. The derivatives are with respect to <. The constants cyand K are defined as follows: K =
?p/i‘f,
(7)
where N is the number of interacting atoms per unit volume and p is the transition dipole moment. Dividing (6) by (2) and integrating we obtain W=(K/21YW)f2
-
1,
(8)
where the integration constant is chosen for the case that the atoms are in the ground state before the arrival of the pulse. Dividing (4) by (2) and using (3) and (8). we obtain du/de + (l/e)u
(3)
This is a typical first-order differential equation. By the standard way we can find the solution to be
(9
by NASA
(6)
(2)
(4) + KEW,
4 = -KEU.
u = -(K~/~,u~),~
= -(~~/2,,~)e~
+ (K/?,W)E
~
+ CJE.
K/O.
(9)
(10)
where the integration constant C must be zero; otherwise u will diverge when E = 0. 111
I~ehruary I Y 74
(8) and (1 0) into (3) WChave
Substituting
6 = (31&W)E2
~~;f_w.
which shows chirping Combining
the parameter
cqs. (4)
(1 I)
where the integration
variables
(0). we obtain
ot’( 16). followed
and a substitution
constant
is chosen
where r is intended
(8) and ( IO) into f 13). we obtain (i3)
I,‘w~,~)“~.
between
Then.
second 6 =
of the
the upper
sign is for
pulse and the lower sign for the
half.
modification
[ I t (I
as t’ollows:
1,1,2,+/2].
(18)
in this case are just a slight
of those obtained
by McCall and Hahn.
When we can neglect the term l/,2,2, eqs. ( 17) and ( 18) will reduce to those of McCall and Hahn. (b)
CtK/h
as sug-
eq. ( 14) will give the relation
We note that the results as well as in the following,
duration
the pulse speed and pulse duration I = (r;Ty
C/V
to be the pulse
[7] and we have to set 6 = I
gested by Eberlq
(I
the first half
oi gives
to be con
u = F (K2/XCY&
where.
by an inversion
of e2 for (8w”/u’).y,
(17)
with eq. (8).
Substituting
I >6>0.
( a1
The integration
effect.
I,2 + UZ t ,L.? = 11
sistent
~3.
I!>&>
The integration
I. of eq. ( 16) gives
(14) and where
7c = [h/3-&$w]
time” defined Substitution
‘E is the “cooperation
by Arecchi and Courtens of( 13) into (2) gives
IS 1.
; = qKQ30)E
where (15)
we have to set 6 = I + (I l/w*~~)~/~. This is a completely new pulse shape and should. perhaps. be the most interesting result of this work. To assess its physical realizability. we find its minimum peak value at 6 = I (OI- wr = I) to be
EC = &&/K After
a multiplication
K~E~/~L_?,
by E and a substitution
the integration
of S =
of eq. ( IS) can be written
2lS
By definition S must be positive. For the expression under the square root sign to be positive. we must also have 6 > S > 6 2. So we put 6 to be the upper limit of the integral. The integration constant is so chosen that the peak of E = ( %J/K)~ occurs at < = 0 The integration in eq. (I 6) can be carried out easily with the help of integration table [6]. The result will be different for four different ranges of values of 112
= &Q./p.
(10)
We see that low frequency and high transition dipole moment will lower this critical value. As an example, we consider the I .06 p laser pulse. We have w = I .78 X 1015 set-l, Assuming p - 10P’* esu, we have E - 8 X I O*V/cm or a peak power density - 2 X c II W/cm2. This power range has certainly been IO achieved by amplifying single pulses from a modelocked Nd+3-doped glass laser 181. In general, the peak value of E is E(O)=(2fiK7)[1
-(I
~+L?7*)1/2]P1’* (21)
= (4W/K) [l ~~ l/8U2,’ It is interesting creases.
-~ 5/1 28W474 ~~ . ..I.
to note that as E(O) increases,
This is just the opposite
Y-also in-
of the previous
case.
Volume
10. number
The relation corresponding c/v-
1 =(;/rc)2(l
February
OPTLCS COMMUNICATIONS
2
-(I--
1974
where we have to set 6 = 1 + (I + 1/4~‘r~)‘/~. Eq.( 25) represents a pulse train. The pulse speed and pulse duration are related by
to eq. ( 18) is
l/,V)“2] (22)
;_ Here we see that the factor (c/V- 1) continues to decrease as the intensity. and hence the pulse duration, increases. (c)
6 = 7.
+ 1]-1'2,
6 > 2.
Eq. ( 16) gives di-) =
(26)
of these pulses is still an open ques-
(13)
which is of lorentzian form. The peak value is e(0) = 2tiw/p, which is very close to cc‘ of eq. (20). The area as defined by McCall and Hahn [ &] will be infinite in this case. The pulse duration is not defined here. The pulse speed is
(d)
The stability tion.
~~1 ~~~ ~, +(1+1/4c.W)*‘2]
The author is grateful to Dr. R.L. Kurtz and Dr. L.M. Narducci for valuable discussions.
Eq. (16) gives e(c)= (~CA/K)[~W'+~
1 =_---__ &A;[1
$(I t &)l’l ~c+)]-1’2.(25)
References 111S.L. McCall and E.L. Hahn, Phys. Rev. Lett. 18 (1967) 908; Phys. Rev. 183 (1969)
457.
I21 C.T. Lee, Opt. Commun. 9 (1973)
1. paper presented at the Sixth Intern. Own[31 E. Courtens, tum Electron. Conf., Kyoto, Japan, 1970. and quantum optics, ed. L. Man141 J.11. Eberly, Coherence de1 and E. Wolf (Plenum Press. 1973). p. 652. 151 I:.T. Arccchi and E. Courtens, Phys. Rev. 2A (1970) 1730. 161 I.S. Gradshteyn and I.M. Ryzhik, Table of integrals. series and products (Academic Press, New York. 1965). p. 84. I71 J.H. Eberly. private communication. [81 See, for example. G.W. Gobeli, J.C. Bushnell, P.S. Peercy E.D. Jones, Phys. Rev. 188 (1969) 300.
113
10, number
OPTICS
2
February
COMMUNICATIONS
FOUR POSSIBLE TYPES OF PULSES FOR SELF-INDUCED
1974
TRANSPARENCY*
C.T. LEE Department
of Physics,
Alabama
A. and M. University,
Received
26 November
kormal,
Alabama
35762.
USA
197 3
We have found four types of possible steady-state solutions to the coupled Maxwell-Bloc11 equations which describe extremely intense pulse propagation in a resonant medium. This is possible due to the replacement of the usual slowly varying envelope approximation by the assumption that (c/l’-l)/wr < 1, where V is the pulse speed and T is the pulse duration. We ignore possible non-resonant losses and set Tt = T, = m.
The self-induced transparency of a sech-pulse is well known [ 11. This is the steady-state solution to the coupled MaxwelllBloch equations under the assumption that 1/cn < 1, where o is the carrier frequency and r is the pulse duration. Recently, we [2] have obtained a sli.ght modification of the sech-pulse by assuming*
where c is the phase speed and V is the pulse speed. Assumption (1) will be better than the previous assumption when (c/V’-- 1) < 1. This will be true for a typical value of r 5 1O-lo set [4]. We are curious to know if there are other types of possible steady-state solutions. In this paper we will present four types of such solutions. We will assume exact resonance, ignore possible non-resonant losses, and put the relaxation times T, and T2 to be infinite. Then the coupled Maxwell& Bloch equations will give a set of five simultaneous equations of first order. They are derived in ref. [2] and are as follows: ; = -cY_lJu, 4 =
QWU/E
+ QKW,
Li =Ju. fi =
-$U
*Work supported 020. *E. Courtens
under
grant
made a similar assumption
No. NGR-Ol-OOl(ref.
[ 31).
In the above equations, E is the envelope of the pulse, $ is the phase. u and +J are dimensionless dispersion and absorption components of the dipole moments, and w is the atom energy divided by $fiw. These quantities depend on their arguments t and z only through the combination { = t z/V. The derivatives are with respect to <. The constants cyand K are defined as follows: K =
?p/i‘f,
(7)
where N is the number of interacting atoms per unit volume and p is the transition dipole moment. Dividing (6) by (2) and integrating we obtain W=(K/21YW)f2
-
1,
(8)
where the integration constant is chosen for the case that the atoms are in the ground state before the arrival of the pulse. Dividing (4) by (2) and using (3) and (8). we obtain du/de + (l/e)u
(3)
This is a typical first-order differential equation. By the standard way we can find the solution to be
(9
by NASA
(6)
(2)
(4) + KEW,
4 = -KEU.
u = -(K~/~,u~),~
= -(~~/2,,~)e~
+ (K/?,W)E
~
+ CJE.
K/O.
(9)
(10)
where the integration constant C must be zero; otherwise u will diverge when E = 0. 111
I~ehruary I Y 74
(8) and (1 0) into (3) WChave
Substituting
6 = (31&W)E2
~~;f_w.
which shows chirping Combining
the parameter
cqs. (4)
(1 I)
where the integration
variables
(0). we obtain
ot’( 16). followed
and a substitution
constant
is chosen
where r is intended
(8) and ( IO) into f 13). we obtain (i3)
I,‘w~,~)“~.
between
Then.
second 6 =
of the
the upper
sign is for
pulse and the lower sign for the
half.
modification
[ I t (I
as t’ollows:
1,1,2,+/2].
(18)
in this case are just a slight
of those obtained
by McCall and Hahn.
When we can neglect the term l/,2,2, eqs. ( 17) and ( 18) will reduce to those of McCall and Hahn. (b)
CtK/h
as sug-
eq. ( 14) will give the relation
We note that the results as well as in the following,
duration
the pulse speed and pulse duration I = (r;Ty
C/V
to be the pulse
[7] and we have to set 6 = I
gested by Eberlq
(I
the first half
oi gives
to be con
u = F (K2/XCY&
where.
by an inversion
of e2 for (8w”/u’).y,
(17)
with eq. (8).
Substituting
I >6>0.
( a1
The integration
effect.
I,2 + UZ t ,L.? = 11
sistent
~3.
I!>&>
The integration
I. of eq. ( 16) gives
(14) and where
7c = [h/3-&$w]
time” defined Substitution
‘E is the “cooperation
by Arecchi and Courtens of( 13) into (2) gives
IS 1.
; = qKQ30)E
where (15)
we have to set 6 = I + (I l/w*~~)~/~. This is a completely new pulse shape and should. perhaps. be the most interesting result of this work. To assess its physical realizability. we find its minimum peak value at 6 = I (OI- wr = I) to be
EC = &&/K After
a multiplication
K~E~/~L_?,
by E and a substitution
the integration
of S =
of eq. ( IS) can be written
2lS
By definition S must be positive. For the expression under the square root sign to be positive. we must also have 6 > S > 6 2. So we put 6 to be the upper limit of the integral. The integration constant is so chosen that the peak of E = ( %J/K)~ occurs at < = 0 The integration in eq. (I 6) can be carried out easily with the help of integration table [6]. The result will be different for four different ranges of values of 112
= &Q./p.
(10)
We see that low frequency and high transition dipole moment will lower this critical value. As an example, we consider the I .06 p laser pulse. We have w = I .78 X 1015 set-l, Assuming p - 10P’* esu, we have E - 8 X I O*V/cm or a peak power density - 2 X c II W/cm2. This power range has certainly been IO achieved by amplifying single pulses from a modelocked Nd+3-doped glass laser 181. In general, the peak value of E is E(O)=(2fiK7)[1
-(I
~+L?7*)1/2]P1’* (21)
= (4W/K) [l ~~ l/8U2,’ It is interesting creases.
-~ 5/1 28W474 ~~ . ..I.
to note that as E(O) increases,
This is just the opposite
Y-also in-
of the previous
case.
Volume
10. number
The relation corresponding c/v-
1 =(;/rc)2(l
February
OPTLCS COMMUNICATIONS
2
-(I--
1974
where we have to set 6 = 1 + (I + 1/4~‘r~)‘/~. Eq.( 25) represents a pulse train. The pulse speed and pulse duration are related by
to eq. ( 18) is
l/,V)“2] (22)
;_ Here we see that the factor (c/V- 1) continues to decrease as the intensity. and hence the pulse duration, increases. (c)
6 = 7.
+ 1]-1'2,
6 > 2.
Eq. ( 16) gives di-) =
(26)
of these pulses is still an open ques-
(13)
which is of lorentzian form. The peak value is e(0) = 2tiw/p, which is very close to cc‘ of eq. (20). The area as defined by McCall and Hahn [ &] will be infinite in this case. The pulse duration is not defined here. The pulse speed is
(d)
The stability tion.
~~1 ~~~ ~, +(1+1/4c.W)*‘2]
The author is grateful to Dr. R.L. Kurtz and Dr. L.M. Narducci for valuable discussions.
Eq. (16) gives e(c)= (~CA/K)[~W'+~
1 =_---__ &A;[1
$(I t &)l’l ~c+)]-1’2.(25)
References 111S.L. McCall and E.L. Hahn, Phys. Rev. Lett. 18 (1967) 908; Phys. Rev. 183 (1969)
457.
I21 C.T. Lee, Opt. Commun. 9 (1973)
1. paper presented at the Sixth Intern. Own[31 E. Courtens, tum Electron. Conf., Kyoto, Japan, 1970. and quantum optics, ed. L. Man141 J.11. Eberly, Coherence de1 and E. Wolf (Plenum Press. 1973). p. 652. 151 I:.T. Arccchi and E. Courtens, Phys. Rev. 2A (1970) 1730. 161 I.S. Gradshteyn and I.M. Ryzhik, Table of integrals. series and products (Academic Press, New York. 1965). p. 84. I71 J.H. Eberly. private communication. [81 See, for example. G.W. Gobeli, J.C. Bushnell, P.S. Peercy E.D. Jones, Phys. Rev. 188 (1969) 300.
113