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Optimal component distribution for ultrashort pulse generation by two-component active media
22 September 1997
PHYSICS
ELSEXIER
LETTERS
A
Physics Letters A 234 ( 1997) 229-232
Optimal component distribution for ultrashort pulse generation by two-component active media A.V. Andreev, S.L. Sheetlin Physics Department, M.Y Lomonosov Moscow State University, Moscow I1 9899, Russian Federation
Received 12 May 1997; revised manuscript received 23 June 1997; accepted for publication 25 June 1997 Communicated by V.M. Agranovich
Abstract It is shown that by varying the density distribution of the components in a mixed two-component active medium we can increase the peak pulse intensity and shorten the temporal width of the generated pulse. This is of great interest for different schemes of superradiative generation as the peak pulse intensity can grow by ten times for the same pumping pulse energy and profile. @ 1997 Published by Elsevier Science B.V. PAC.? 42.5O.F~; 42.65.Re Keywords: Superradiance; Semiconductor laser; Superfluorescence
1. Introduction The dynamics of the generation by an active medium consisting of two species of quasiresonant atoms is essentially different from that for a onecomponent one. The inclusion of the second component drastically changes the threshold conditions of generation and generated pulse parameters. This is widely used in lasers with a saturable absorber, in semiconductor lasers, etc. Recently [l-4] it was shown that the adoption of a second component into the superradiative medium enables us to increase significantly the peak pulse intensity, shorten the temporal width of the pulse, and extend the delay time of superradiance (SR). As a result we can transform a broad pumping pulse into an ultrashort pulse in superradiative emission. There are two main types of two-component systems. In the first case the system consists of two spatially separated volumes of active
or passive media [5], as in the case of lasers with saturable absorbers or semiconductor lasers. In the second case the active medium is a mixture of two components [ l-41. We consider the latter case. Usually it is assumed that the components have a uniform density distribution. Here we show that a variation in the profile of the component density distribution results in a drastic change of the generated pulse parameters. This opens up new opportunities for pulse parameter control.
2. Main equations We consider an active medium consisting of two species of atoms, A and B, that have quasiresonant transition frequencies with WE = WA+ d and different dipole moments of the resonant transitions, dA < ds. The frequency of the Rabi oscillations is proportional to the dipole moment, 0,~ cx dA,B, and therefore we
037%9601/97/$17.00 @ 1997 Published by Elsevier Science B.V. All rights reserved. PI/ SO375-9601(97)00548-3
A.I! Andreev,S.L. SheetWPhysics LettersA 234 (1997) 229-232
230
can refer to A-type atoms as “slow” and B-type ones as “fast”. Let the active volume V contain NA slow and NB fast atoms distributed along the length L of the active volume with the densities PA(X) and PB (x) . The Maxwell-Bloch equations [ 6,7] for the slowly varying envelopes at ,z of the counterpropagating e.m. waves, the amplitudes of the polarization waves of the slow (~1.2) and fast (q1.2) atoms associated with these e.m. waves, and the population inversion densities of the slow (R) and fast (r) atoms are %+$=pt+q* Be(x) + B +dzTo 2
+vKpo aa
aa ---=
P2 +
ax
at
dP1 2
341
--$
2
+
a’=-“=, dr
NA
r(x, t) =
NA
The dimensionless density distribution of components Ro( x) , ro( x) does not depend on time as it is assumed that the lower levels of both components are metastable states or atomic ground levels,
NzA(X,t> + N1At-Gt> NA
= PLVO(X>~I,Z~,
2 (pnan* + ~,*a,) j
2
al,z(x,O) = 0,
(1)
Nx,O)
A = (‘?I,/,- mB)r,
(yA~B = +A,B) 2 2~~ldA,s12 ri
NA 2
(2)
7.
The spontaneous relaxation in the resonant transitions is simulated by the choice of a small, but non-zero [ 7 ] , spontaneous polarisation po and qo. The rate of spontaneous emission is different for slow and fast atoms and it is proportional to the dipole moments of the transitions fi and the population of the excited states (Ro + R) /2 and (ro + r) /2. In computer
PI,z(x,~) = q1,2(x,O) = 0,
=Ro(x),
al(x=O,t) We have introduced in Eq. ( 1) the dimensionless coordinate x’ = x/L and time t’ = t/r, where 7 = L/c, and omitted their primes. The dimensionless relaxation rates LYA,B, detuning of the component frequencies A, and parameters BA,Bare determined by 7
t
We assume that initially the slow atoms are in the excited state while the fast atoms are in the lower level of the resonant transition. We also assume that there is no reflection at the active volume boundaries. The effect of boundary reflection on the two-component SR dynamics has been investigated earlier [4] in the case of a uniform density distribution. Therefore we use the following initial and boundary conditions,
,
= PARo(x
(qnan* + qian>. at = - cn=,
PA,B =
Nm(X, t) - NIB(X,t>
NA 2
c
t) - NIA(x, t>
N~E(x, t) + NIB(X, t)
+ (0~ + i&ql,2
C?R
hA(x,
ro(x) =
q2
R”(x;+R+~qo
ffAp1.2
R(x, t) =
Ro(x>=
ro(x) + r
+ &PO -I$-
ro(x) + r 2 3
simulations we assume po = qo N 10e3. The population inversion is normalised to the mean density of the slow atoms,
=u~(x=
rtx,O)
=+-0(x>,
1,t) =O.
(34
(3b)
3. Results of numerical simulations The case when the components are distributed uniformly along the active medium length has been considered in Refs. [ I-41. In this case Ro( x) , r-g(x) are constants and there is an optimal ratio ro/Ro when the peak pulse intensity has an absolute maximum. This ratio depends on the dipole moments, line widths and frequency detuning of the resonant transitions of the slow and fast components. In the case when the components are distributed non-uniformly the peak pulse intensity is a functional of the functions Ro( x), ro( x). The problem of interest is whether we can get a further increase in the peak pulse intensity by optimizing the functional mentioned above.
A.V Andreev, S.L. Sheetlin/Physics Letters A 234 (1997) 229-232
23
(a)
2.0 7 x=0
x=1
x=1
Fig. I. Scheme of the optimized two-component medium: the slow component is distributed uniformly in the interval x E [ 0, 1 1, and the fast component is uniformly distributed in the interval x E [I,ll.
Let the dimensionless
parameters
a,JJ, d, and P4.n
be (YA=O.Ol,
PA = 1,
LYE=0.04,
pB=4,
A=O.
In the case of a uniform distribution of the component densities the maximum of the peak pulse intensity occurs when Ro = 1 and ra = 0.75. For these values the intensity of the two-component SR is approximately three times higher [ 3,4] than that for a one-component medium ( Ro = 1, ra = 0). With the help of the algorithm for the optimal control for the functional ]at (x = 1, t = fmax) I* = F( Ro( x) , ro( x) ), which determines the peak pulse intensity as a function of the component density distribution, we have found the two optimal distributions. The maximal benefit in the peak pulse intensity follows from the solution illustrated in Fig. 1. The slow component is distributed uniformly in the interval x E [ 0, 11, while the fast component is uniformly distributed in the interval x E [Z, 11. The optimal value for 1 is equal to 0.1. Notice that when 1 = 0 we have a two-component medium with a uniform distribution of component concentration, while the case [ = 1 corresponds to a one-component medium of slow atoms. Fig. 2 shows the peak pulse intensity I max = lai(x = 1,t = to)]* (a), the temporal width of the pulse ~-0 (b) , and the delay time ta (c) versus the length 1. It is seen from Fig. 2 that ‘variation in the length of the volume filled by the fast atoms can result in an increase of the peak pulse intensity by an order of magnitude. The increase in the intensity is accompanied by a reduction in the temporal width of the pulse and of the delay time. However, the ratio of delay time to temporal width of the pulse is in excess of 100 at 1 = 0.1, when we have a maximum for the peak pulse intensity. As was shown in Refs. [l-3] the pumping process does not affect the SR pulse parameters and shape if its duration is smaller than or equal to the delay time. Therefore for I = 0.1 we still
0.0
, 1.0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8 , 1.0
0.6
0.8
(b)
8-
6-
1
00
0.0 600 ,‘“’
0.0
0.2
0.4
Fig. 2. Peak pulse intensity (a), temporal and delay time (c) versus the length 1.
, 1.0
width of the pulse (b),
232
A.V Andreev. S.L. SheetlWPhysics
Letters A 234 (1997) 229-232
0 < x < 1, plays the role of the seed pulse. As a result we have a sharp asymmetry in the spatio-temporal evolution of the counterpropagating waves at the initial stage of generation. By varying the length 1 we can completely suppress the generation in the left direction as is seen from comparison of Figs. 2 and 3. In this case we have uni-directional emission instead of the bi-directional emission that occurs for 1 = 0 or 1= 1. 5. Conclusions
0.0
0.2
0.4
0.6
0.6
,
1.0
Fig. 3. Peak pulse intensity of the opposite wave as a function of the length 1.
have the possibility to transform the broad pumping pulse into an ultrashort pulse of SR emission.
4. Discussion Thus the results of the computer simulations show that there is an optimal ratio of lengths of volumes filled by the slow, Lt = 1, and fast, LZ = 1 - 1, atoms. This ratio depends on the dipole moments and widths of the resonant transitions, as well as the component densities. There are two main reasons that explain such a drastic reconstruction of the pulse shape in the optimized two-component media. Firstly, the optimized media exhibits selectivity in the direction of generation. The intensity of the opposite wave at the left side of the active medium, la;?(x = 0, to) I23 is negligibly small in comparison with jar (X = 1, to) 12.Fig. 3 shows the peak pulse intensity of the opposite wave la = laa(x = 0, t = to) I2 as a function of the length 1. Secondly, the spontaneous decay in the region of the active medium I < x < 1 filled by both components is suppressed because the fast component is initially in the ground state. Therefore the spontaneous emission of the one-component part of the active medium,
The results presented here clearly demonstrate the strict dependence of the two-component SR pulse shape on the distribution of the component densities. It opens up new opportunities for pulse parameter control. By varying the length of the volumes filled by the components we can vary the SR peak pulse intensity and temporal width in a wide range. The intensity of emission by an optimized medium can exceed that for a medium with uniform density distribution by an order of magnitude. Such a sharp dependence is due to the selection of the spatial modes evolved in formation of an SR avalanche.
Acknowledgement This work was supported by the Russian Foundation for Basic Research (project no. 96-02-19285).
References [l] A.V. Andreev and P.V. Polevoy, Quantum Optics 6 ( 1994) 57. [2] A.V. Andreev and PV. Polevoy, Infrared Phys. Tech. 36 (1995) 15. [3] A.V. Andreev and P.V. Polevoy, Quantum Electr. 26 ( 1996) 724. [4] A.V. Andreev and D.Yu. Kobelev, Laser Phys. 6 (1996) 744. [5 ] 0. Svelte, Principles of lasers (Plenum, New York, 1976). [6] L. Allen and J.H. Eberly, Optical resonance and two-level atoms (Wiley, New York, 1975). [ 71 A.V. Andreev, V.I. Emelyanov and Yu.A. Ilinskii, Cooperative effects in optics (IOP, Bristol, Philadelphia, 1993).
PHYSICS
ELSEXIER
LETTERS
A
Physics Letters A 234 ( 1997) 229-232
Optimal component distribution for ultrashort pulse generation by two-component active media A.V. Andreev, S.L. Sheetlin Physics Department, M.Y Lomonosov Moscow State University, Moscow I1 9899, Russian Federation
Received 12 May 1997; revised manuscript received 23 June 1997; accepted for publication 25 June 1997 Communicated by V.M. Agranovich
Abstract It is shown that by varying the density distribution of the components in a mixed two-component active medium we can increase the peak pulse intensity and shorten the temporal width of the generated pulse. This is of great interest for different schemes of superradiative generation as the peak pulse intensity can grow by ten times for the same pumping pulse energy and profile. @ 1997 Published by Elsevier Science B.V. PAC.? 42.5O.F~; 42.65.Re Keywords: Superradiance; Semiconductor laser; Superfluorescence
1. Introduction The dynamics of the generation by an active medium consisting of two species of quasiresonant atoms is essentially different from that for a onecomponent one. The inclusion of the second component drastically changes the threshold conditions of generation and generated pulse parameters. This is widely used in lasers with a saturable absorber, in semiconductor lasers, etc. Recently [l-4] it was shown that the adoption of a second component into the superradiative medium enables us to increase significantly the peak pulse intensity, shorten the temporal width of the pulse, and extend the delay time of superradiance (SR). As a result we can transform a broad pumping pulse into an ultrashort pulse in superradiative emission. There are two main types of two-component systems. In the first case the system consists of two spatially separated volumes of active
or passive media [5], as in the case of lasers with saturable absorbers or semiconductor lasers. In the second case the active medium is a mixture of two components [ l-41. We consider the latter case. Usually it is assumed that the components have a uniform density distribution. Here we show that a variation in the profile of the component density distribution results in a drastic change of the generated pulse parameters. This opens up new opportunities for pulse parameter control.
2. Main equations We consider an active medium consisting of two species of atoms, A and B, that have quasiresonant transition frequencies with WE = WA+ d and different dipole moments of the resonant transitions, dA < ds. The frequency of the Rabi oscillations is proportional to the dipole moment, 0,~ cx dA,B, and therefore we
037%9601/97/$17.00 @ 1997 Published by Elsevier Science B.V. All rights reserved. PI/ SO375-9601(97)00548-3
A.I! Andreev,S.L. SheetWPhysics LettersA 234 (1997) 229-232
230
can refer to A-type atoms as “slow” and B-type ones as “fast”. Let the active volume V contain NA slow and NB fast atoms distributed along the length L of the active volume with the densities PA(X) and PB (x) . The Maxwell-Bloch equations [ 6,7] for the slowly varying envelopes at ,z of the counterpropagating e.m. waves, the amplitudes of the polarization waves of the slow (~1.2) and fast (q1.2) atoms associated with these e.m. waves, and the population inversion densities of the slow (R) and fast (r) atoms are %+$=pt+q* Be(x) + B +dzTo 2
+vKpo aa
aa ---=
P2 +
ax
at
dP1 2
341
--$
2
+
a’=-“=, dr
NA
r(x, t) =
NA
The dimensionless density distribution of components Ro( x) , ro( x) does not depend on time as it is assumed that the lower levels of both components are metastable states or atomic ground levels,
NzA(X,t> + N1At-Gt> NA
= PLVO(X>~I,Z~,
2 (pnan* + ~,*a,) j
2
al,z(x,O) = 0,
(1)
Nx,O)
A = (‘?I,/,- mB)r,
(yA~B = +A,B) 2 2~~ldA,s12 ri
NA 2
(2)
7.
The spontaneous relaxation in the resonant transitions is simulated by the choice of a small, but non-zero [ 7 ] , spontaneous polarisation po and qo. The rate of spontaneous emission is different for slow and fast atoms and it is proportional to the dipole moments of the transitions fi and the population of the excited states (Ro + R) /2 and (ro + r) /2. In computer
PI,z(x,~) = q1,2(x,O) = 0,
=Ro(x),
al(x=O,t) We have introduced in Eq. ( 1) the dimensionless coordinate x’ = x/L and time t’ = t/r, where 7 = L/c, and omitted their primes. The dimensionless relaxation rates LYA,B, detuning of the component frequencies A, and parameters BA,Bare determined by 7
t
We assume that initially the slow atoms are in the excited state while the fast atoms are in the lower level of the resonant transition. We also assume that there is no reflection at the active volume boundaries. The effect of boundary reflection on the two-component SR dynamics has been investigated earlier [4] in the case of a uniform density distribution. Therefore we use the following initial and boundary conditions,
,
= PARo(x
(qnan* + qian>. at = - cn=,
PA,B =
Nm(X, t) - NIB(X,t>
NA 2
c
t) - NIA(x, t>
N~E(x, t) + NIB(X, t)
+ (0~ + i&ql,2
C?R
hA(x,
ro(x) =
q2
R”(x;+R+~qo
ffAp1.2
R(x, t) =
Ro(x>=
ro(x) + r
+ &PO -I$-
ro(x) + r 2 3
simulations we assume po = qo N 10e3. The population inversion is normalised to the mean density of the slow atoms,
=u~(x=
rtx,O)
=+-0(x>,
1,t) =O.
(34
(3b)
3. Results of numerical simulations The case when the components are distributed uniformly along the active medium length has been considered in Refs. [ I-41. In this case Ro( x) , r-g(x) are constants and there is an optimal ratio ro/Ro when the peak pulse intensity has an absolute maximum. This ratio depends on the dipole moments, line widths and frequency detuning of the resonant transitions of the slow and fast components. In the case when the components are distributed non-uniformly the peak pulse intensity is a functional of the functions Ro( x), ro( x). The problem of interest is whether we can get a further increase in the peak pulse intensity by optimizing the functional mentioned above.
A.V Andreev, S.L. Sheetlin/Physics Letters A 234 (1997) 229-232
23
(a)
2.0 7 x=0
x=1
x=1
Fig. I. Scheme of the optimized two-component medium: the slow component is distributed uniformly in the interval x E [ 0, 1 1, and the fast component is uniformly distributed in the interval x E [I,ll.
Let the dimensionless
parameters
a,JJ, d, and P4.n
be (YA=O.Ol,
PA = 1,
LYE=0.04,
pB=4,
A=O.
In the case of a uniform distribution of the component densities the maximum of the peak pulse intensity occurs when Ro = 1 and ra = 0.75. For these values the intensity of the two-component SR is approximately three times higher [ 3,4] than that for a one-component medium ( Ro = 1, ra = 0). With the help of the algorithm for the optimal control for the functional ]at (x = 1, t = fmax) I* = F( Ro( x) , ro( x) ), which determines the peak pulse intensity as a function of the component density distribution, we have found the two optimal distributions. The maximal benefit in the peak pulse intensity follows from the solution illustrated in Fig. 1. The slow component is distributed uniformly in the interval x E [ 0, 11, while the fast component is uniformly distributed in the interval x E [Z, 11. The optimal value for 1 is equal to 0.1. Notice that when 1 = 0 we have a two-component medium with a uniform distribution of component concentration, while the case [ = 1 corresponds to a one-component medium of slow atoms. Fig. 2 shows the peak pulse intensity I max = lai(x = 1,t = to)]* (a), the temporal width of the pulse ~-0 (b) , and the delay time ta (c) versus the length 1. It is seen from Fig. 2 that ‘variation in the length of the volume filled by the fast atoms can result in an increase of the peak pulse intensity by an order of magnitude. The increase in the intensity is accompanied by a reduction in the temporal width of the pulse and of the delay time. However, the ratio of delay time to temporal width of the pulse is in excess of 100 at 1 = 0.1, when we have a maximum for the peak pulse intensity. As was shown in Refs. [l-3] the pumping process does not affect the SR pulse parameters and shape if its duration is smaller than or equal to the delay time. Therefore for I = 0.1 we still
0.0
, 1.0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8 , 1.0
0.6
0.8
(b)
8-
6-
1
00
0.0 600 ,‘“’
0.0
0.2
0.4
Fig. 2. Peak pulse intensity (a), temporal and delay time (c) versus the length 1.
, 1.0
width of the pulse (b),
232
A.V Andreev. S.L. SheetlWPhysics
Letters A 234 (1997) 229-232
0 < x < 1, plays the role of the seed pulse. As a result we have a sharp asymmetry in the spatio-temporal evolution of the counterpropagating waves at the initial stage of generation. By varying the length 1 we can completely suppress the generation in the left direction as is seen from comparison of Figs. 2 and 3. In this case we have uni-directional emission instead of the bi-directional emission that occurs for 1 = 0 or 1= 1. 5. Conclusions
0.0
0.2
0.4
0.6
0.6
,
1.0
Fig. 3. Peak pulse intensity of the opposite wave as a function of the length 1.
have the possibility to transform the broad pumping pulse into an ultrashort pulse of SR emission.
4. Discussion Thus the results of the computer simulations show that there is an optimal ratio of lengths of volumes filled by the slow, Lt = 1, and fast, LZ = 1 - 1, atoms. This ratio depends on the dipole moments and widths of the resonant transitions, as well as the component densities. There are two main reasons that explain such a drastic reconstruction of the pulse shape in the optimized two-component media. Firstly, the optimized media exhibits selectivity in the direction of generation. The intensity of the opposite wave at the left side of the active medium, la;?(x = 0, to) I23 is negligibly small in comparison with jar (X = 1, to) 12.Fig. 3 shows the peak pulse intensity of the opposite wave la = laa(x = 0, t = to) I2 as a function of the length 1. Secondly, the spontaneous decay in the region of the active medium I < x < 1 filled by both components is suppressed because the fast component is initially in the ground state. Therefore the spontaneous emission of the one-component part of the active medium,
The results presented here clearly demonstrate the strict dependence of the two-component SR pulse shape on the distribution of the component densities. It opens up new opportunities for pulse parameter control. By varying the length of the volumes filled by the components we can vary the SR peak pulse intensity and temporal width in a wide range. The intensity of emission by an optimized medium can exceed that for a medium with uniform density distribution by an order of magnitude. Such a sharp dependence is due to the selection of the spatial modes evolved in formation of an SR avalanche.
Acknowledgement This work was supported by the Russian Foundation for Basic Research (project no. 96-02-19285).
References [l] A.V. Andreev and P.V. Polevoy, Quantum Optics 6 ( 1994) 57. [2] A.V. Andreev and PV. Polevoy, Infrared Phys. Tech. 36 (1995) 15. [3] A.V. Andreev and P.V. Polevoy, Quantum Electr. 26 ( 1996) 724. [4] A.V. Andreev and D.Yu. Kobelev, Laser Phys. 6 (1996) 744. [5 ] 0. Svelte, Principles of lasers (Plenum, New York, 1976). [6] L. Allen and J.H. Eberly, Optical resonance and two-level atoms (Wiley, New York, 1975). [ 71 A.V. Andreev, V.I. Emelyanov and Yu.A. Ilinskii, Cooperative effects in optics (IOP, Bristol, Philadelphia, 1993).