Belenov, E. M.

Kryukov, P. G.

Nazarkin, A. V.

Prokopovich, I. P.

English

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Coherent interaction between phase-modulated light pulses and an active medium 
of two-level particles with inhomogeneous broadening 
E. M. Belenov, P. G. Kryukov, A. V. Nazarkin, and I. P. Prokopovich 
P. N. Lebedev Physics Institute, USSR Academy of Sciences 
(Submitted 5 November 1990) 
Zh. Eksp. Teor. Fiz. 100,105-1 12 (July 1991 ) 
It is shown that induced transitions of an ensemble of two-level particles with an 
inhomogeneously broadened line in the field of a phase-modulated pulse are analogous to 
Landau-Zener inelastic transitions in molecules when their terms cross. Even for T %  T:, where T 
and T: are, respectively, the pulse width and the inhomogeneous relaxation time of the medium, 
a transition of particles of the entire spectrum ( T:) - ' from an upper to a lower level is possible, 
signifying the complete removal by the pulse of the energy stored in the active medium. Analysis 
of the dynamics of a phase-modulated pulse by means of the Maxwell-Bloch equations shows, in 
addition, the possibility of amplifying it without the development of field-amplitude oscillations. 
1. INTRODUCTION 
One of the most interesting trends in modern laser phys- 
ics is the possibility of obtaining powerful ultrashort laser 
pulses (USP) of width < 1 psec and energies up to several 
J.' The power in the laser beam reaches a level of several 
terawatts. After focusing, the intensity can reach lo2' 
W/cm2, and the electric field strength of the light wave ex- 
ceeds the electric field of the Bohr atom. It is assumed that 
obtaining such pulses will make it possible to set up experi- 
ments on nonlinear quantum  electrodynamic^.^ 
At the present time, two main approaches are used to 
obtain powerful USP. The first consists in using wide-aper- 
ture excimer lasers to amplify femtosecond pulses in the UV 
range.3 The pulses being amplified are obtained from dye 
lasers, with subsequent conversion to the second harmonic. 
The amplification linewidth and pulsewidth are such that 
during amplification, coherent-propagation effects begin to 
play an appreciable role.4 The second approach, used both 
in excimer systems and in near-IR solid state l a ~ e r s , ~  con- 
sists in first producing phase modulation of the USP carrier 
frequency and stretching the pulse over time. After such a 
"chirped" pulse has been amplified, it is passed through an 
artificial medium with dispersion of the group velocity of 
such magnitude and sign that the pulse is compressed to a 
width < 1 psec. 
In both approaches, the amplification takes place either 
in a medium with an inhomogeneously broadened line, or in 
a medium whose properties are similar to those of an inho- 
mogeneous ensemble of radiators because of the complex 
vibration-rotation structure of the working  level^.^ This nat- 
urally gives rise to the problem of removal of the energy 
stored in the active medium under coherent interaction con- 
ditions, i.e., when the pulse width T is smaller than the time 
T2 of irreversible polarization relaxation of the substance." 
A remarkable manifestation of coherence effects in ac- 
tive two-level media with a homogeneously broadened line is 
the possibility of removal by the pulse of the entire energy 
stored in the medium. The energy is completely removed for 
pulses with a constant phase and an area equal to 
8, = ~ ( 2 m  + 1 ), m = 0, 1, .... Thus, in an extended amplifi- 
er, an initial pulse of small area 8'4 1 is shaped into a square- 
wave pulse dependent on self-similar variables, whose ener- 
gy increases linearly and whose width is reduced in 
proportion to the path traveled.'&l2 All of the above also 
applies in the presence of inhomogeneous broadening in the 
pulse-shaping stage where its width becomes shorter than 
the time T: of inhomogeneous dephasing of the radiators. 
However, the attainment by the electric field envelope of the 
self-similar amplification mode for an initial pulse width 
r0 % T f requires comparatively extended amplifying media. 
The question therefore arises, should the pulse field in the 
case of practical interest, 
be shaped so that the entire energy stored in the medium is 
removed at comparatively small (in the limit, infinitely 
small) amplification lengths? It will be shown below that the 
problem has a solution, and the determining pulse character- 
istics are not amplitudinal, but phase characteristics. 
2. INTERACTION OFTWO-LEVEL PARTICLES WITH THE 
FIELD OF A PHASE-MODULATED PULSE ASTHE LANDAU- 
ZENER PROBLEM OF THE PREDlSSOClATlON OF A 
DIATOMIC MOLECULE 
Let us consider an inhomogeneous ensemble of two- 
level particles which interacts resonantly with a pulse field 
8 ( t )  = E  ( t )cos [wtf cp ( t )  1, (2) 
where w is the carrier frequency, and E and q7 are slowly 
changing amplitude and phase. Considering that the pulse is 
shorter than the homogeneous dephasing time T2, we can 
describe the dynamics of an individual two-level particle 
with the aid of the probability amplitudes aj of the upper 
(j = 2) and lower = 1 ) levels: 
Herep,, is the transition dipole moment; w2, = w:, + Aw is 
the eigenfrequency of the individual particle, detuned from 
the amplification line center w:, by Aw. Before the interac- 
tion with the field ( t -  - CQ ), all the particles are in the 
57 Sov. Phys. JETP 73 (I), July 1991 0038-5646/91/070057-04$03.00 @ 1991 American Institute of Physics 57 
upperlevel,i.e.,a,(Aw, - W )  = 1, a,(Aw, - W )  =O. 
The energy removal efficiency in the ensemble of parti- 
cles after passage of the pulse ( t - w ) is characterized by the 
quantity 
m 
q = 5 n i ( A o ,  - ) g ( A m ) d A o ,  (4) 
- m 
where n, (Aw, t) = 1 a, (Aw, t) 1 is the population of the 
particle levels, and g(Aw) is the particle distribution func- 
tion as a function of the detuning ($g( Aw)dAw = 1 ). 
Below we consider phase-modulated pulses. First, how- 
ever, it is helpful to discuss the case of a pulse with a constant 
phase. 
For @ = 0, the effects of the phase memory of the medi- 
um will be manifested only for particles whose eigenfrequen- 
cies lie close to the pulse carrier frequency w = w;, within 
the confines of the spectral pulsewidth: 1 Aw 1 5 7; I. From 
Eqs. ( 3 ) ,  we have an approximate expression for the energy 
level populations of these particles: 
Particles that have detunings from resonance that ex- 
ceed the spectral pulsewidth, I Aw 1 > T; ', interact with the 
field in a quasistationary manner; the polarization and popu- 
lations in this case are functions of the instantaneous value of 
the pulse field:13 
It follows from expressions (5)  and (6)  that after pas- 
sage of the pulse (E( 00 ) +0), only those particles of the 
ensemble that have memory will remain in the lower level 
and will thereby contribute to the energy removal of the me- 
dium. From the expressions (4)  and (5)  we obtain the fol- 
lowing estimate for the energy removal efficiency: 
where 
OD 
is the pulse area, and Ami is the inhomogeneous linewidth 
Awiz (TT)  -I. 
We note that the above estimate can be rigorously sub- 
stantiated when the pulse shape is of the hyperbolic secant 
form, for which parameter-dependent exact solutions ex- 
ist.I4 An important consequence of the expression (7 )  is that 
for a fixed pulsewidth ro,  an increase in pulse amplitude 
(which can be obtained, for example, by focusing the radi- 
ation into the active medium) does not lead to an increase in 
energy removal, so that the efficiency of the latter for 
Awir0 ) 1 remains small, v <  1. It is pertinent at this point to 
emphasize the qualitative difference of this coherent interac- 
tion mode ( 1 ) from the incoherent, when the pulsewidth is 
substantially longer than the time of irreversible relaxation 
of polarization T,. In the incoherent mode, the energy re- 
moval can be increased by means of field broadening, and in 
principle, it is possible to remove one-half of the total energy 
stored in an inhomogeneous ensemble.I5 
We now turn to phase-modulated pulses, for which 
@ #O. The efficiency with which the particle ensemble inter- 
acts with a field can be increased by continuously varying the 
instantaneous pulse frequency w ( t )  = w + @(  t)  with time. 
In the simplest case of phase modulation linearly dependent 
on time, we have 
and in order for the instantaneous pulse frequency to pass 
through resonance with all the eigenfrequencies of the en- 
semble in a time r0,  the magnitude of the chirp 0 = d+ /at 
must satisfy the condition 
By virtue of the assumption ( 1 ), this condition signifies that 
when the particles interact with the pulse field, what is sig- 
nificant is not the change in its amplitude, but the rapid 
phase change at the instant t ' when it passes through reso- 
nance w(t ') = w,, . Hence, one can neglect the change in 
pulse amplitude, assuming it to be constant in the course of 
the interaction and equal to E ( t  '). Substituting Eq. (8)  in 
the system ( 3 )  and considering the foregoing remark, we can 
readily see that the equations which we have analyzed are 
analogous to those considered in the theory of Landau- 
Zener inelastic transitions in the crossing of molecular 
termsI6 (see also the adiabatic passage through reso- 
nance15 ) . 
Using the asymptotic expression for the transition 
probability,16 we obtain the following estimate for the ener- 
gy removal efficiency: 
where i? is the characteristic magnitude of the pulse field 
strength. The applicability of this equation is determined by 
the condition that the Landau-Zener interaction time 
rint = IP 1 - be small in comparison with the pulse width 
rO and is ensured by the conditions ( 1 ) and (9).  According 
to Eq. ( lo) ,  the energy removal reaches unity when 
The minimum value of the pulse field at which the entire 
energy stored in the ensemble is removed obviously corre- 
sponds to the minimum possible rate of frequency change 
10 I ,in -.Awir; I ,  whence we have 
Thus, control of the pulse phase has the most significant 
effect on the energy removal in an inhomogeneous ensemble 
of two-level particles: In the absence of phase modulation at 
any field amplitude, the removal efficiency is less than 
7 - (rO Ami ) - 4 1, and in the presence of a chirp and under 
the condition E> Emin, the removal efficiency is 17- 1. It 
58 Sov. Phys. JETP 73 (1). July 1991 Belenov et a/. 58 
should be emphasized that this effect is essentially nonlinear 
in the pulse field. Indeed, estimating from the expressions 
( 5 ) and ( 8 the energy removal for pulses of small area 
,ox!%- Ezo<l, 
f i  
cne can readily ascertain that when the excitation of the 
transition is weak, chirping of the pulse does not produce a 
gain in energy removal, and at best (when P z  = Ao~T; I ) ,  
the energy removal efficiencies are found to be equal. 
3. NUMERICAL MODELLING OFTHE AMPLIFICATION OF A 
CHIRPED PULSE 
Presented below are some results obtained by consider- 
ing the propagation dynamics of a field pulse with initial 
phase modulation, specified by the law (8  ), in the halfspace 
z  > 0 filled with a medium of two-level particles with an inho- 
mogeneously broadened Gaussian shape of the amplifica- 
tion line. 
The change in the reactive and active components P,,, 
of polarization of the particle and in the difference of the 
populations and levels was described by the equations 
with the initial conditions 
Pi (z, -m) =Pz(z, - 0 0 )  =0, n(z, -m) =I. (14) 
and the field amplitude and phase were described by the 
equations 
where N is the particle number density and x is the nonre- 
sonant index of refraction of the medium. 
It is assumed that a pulse with a Gaussian amplitude 
profile 
1 E (I) =Eo eap[ - - (t/s) '1 . 2 (16) 
whose carrier frequency w coincides with the amplification 
line center o:, , is supplied to the input of the medium 
( Z  = 0).  The results of the numerical integration of the sys- 
tem ( 13), ( 15) are presented in dimensionless quantities: 
the z  coordinate is measured in units of reciprocal linear gain 
a - ' = &icx/2mNp:, TT, time t is measured in units of ini- 
tial pulsewidth T,, and the electric field E ( t )  is normalized 
to the quantity (Wp,, T, ) . 
It is convenient to introduce the average efficiency of 
energy removal on the length L of the active medium accord- 
ing to the relation 
Figure 1 shows ?j as a function of the input-pulse amplitude 
at the maximum E, over relatively short lengths of the active 
medium ( L  = 0.3). It  is evident that when the chirp is ab- 
sent ( P  = O), an increase in pulse amplitude does not result 
in an increase in energy removal efficiency. A characteristic 
feature is the oscillating form of the dependence .fj(E, ), 
caused by the effects of coherent saturation of the particles 
interacting with the field. As the chirp size increases for 
P >  1, first, an increasingly large number of particles enter 
into the interaction with the field, and second, a change to 
the mode of Landau-Zener interaction of the particles with 
the field takes place (Fig. 1). In this mode, an increase in 
field amplitude leads to a complete removal of energy. In 
contrast to the case p < 1, the particle ensemble behaves in 
an irreversible manner. 
Figure 2 shows the dynamics of the field envelope E( t )  
and of the function @(t)  of the chirped pulse as the latter 
propagates in an extended amplifier. Shown for comparison 
are calculated data for a pulse with the same initial profile of 
the envelope, but a constant phase 4, = 0. On entering the 
medium, the pulse is chirped in such a way that the frequen- 
cy tuning during the pulse width T, is approximately equal 
to the width Aw, of the inhomogeneous line, and the magni- 
tude of the field strength is chosen from the condition ( 12) 
of complete energy removal. It is interesting to note that 
amplification in this case is not only more efficient, but sub- 
stantially more uniform than in the case 4, = 0 (cf. Fig. 2a 
and 2b). This is because a chirped pulse at each fixed point of 
the active medium at different instants interacts with differ- 
ent particles of the inhomogeneous ensemble. As a result, at 
considerable amplification lengths, there is an absence of 
response accumulation effects, which are characteristic of 
the amplification of pulses without phase modulation and 
which usually lead to the development of an oscillating pro- 
file and to a stretching of the pulse trailing edge. 
4. CONCLUSION 
It has been shown that the process of coherent amplifi- 
cation of powerful light pulses in a medium with an inhomo- 
FIG. 1 .  Mean efficiency i j  of energy removal by a pulse in an active medi- 
um with an inhomogeneously broadened line vs pulse amplitude E, at the 
input for different values of 8: 1-P= 0; 2-8 = 0.3; 3-B = 0.7; 4- 
p= 2; 5-8 = 4. Calculated parameters: L = 0.3; A ~ , T ~  = 4; T, = 10, 
T, = m. 
59 SOV. Phys. JETP 73 (I), July 1991 Belenov et a/. 59 
FIG. 2. Evolution of envelope E and of phase modulation @ of the pulse 
undergoing amplification in a medium with an inhomogeneously broad- 
ened line: a-input pulse is phase-modulated (/3 = 3), b--without phase 
modulation. Calculated parameters: A0,rO = 3; T2 = 10, TI  = W .  
geneously broadened line can be optimized by modulating 
the carrier-frequency phase. When the field strength of the 
amplified phase-modulated pulse exceeds a certain critical 
value, complete exchange of the energy of the active medium 
takes place. 
To estimate the parameters of the chirped pulse, we ex- 
amine the amplification in the active medium of an XeCl 
excimer laser on the most intense vibronic transition 
B(v = 0) +X(v = 2), corresponding to the wavelength 
A = 308 nm. At a pulse width T, > 1 psec, the contribution of 
other vibronic transitions can be negle~ted.~ The contribu- 
tion to the amplification on this transition is due to - 100 
lines in the P and R branches of the rotational structure of 
the transition; the width of each of the lines is determined by 
the rotational relaxation time T, = TR = 10 psec, and the 
total linewidth of the transition is Am, z 30 cm - ' (which 
corresponds to Tf =: 1 psec). Assuming that the pulsewidth 
is r0 5 10 psec and that the transition dipole moment is 
p,+, = 0.5 X 10- l 8  e ~ u , ~  we find from the expression (12) 
that the minimal chirped-pulse intensity providing for com- 
plete energy removal is I=: lo7 W/cm2 and substantially be- 
low the maximum permissible value for the active medium in 
question. 
In conclusion, we note the following. If dispersive ele- 
ments are used to compress the shaped chirped pulse in an 
optimal manner, the pulsewidth obviously becomes of order 
T 5 (Ami ) - I ,  and the amplitude reaches E > f i ~ m ~ / p , ~ .  
Then the area of the pulse is of order 6, -r, and this pulse 
can also remove the entire energy stored in the medium. 
However, amplification of a frequency-modulated pulse is 
much more advantageous, since self-focusing and break- 
down in the active medium limit the peak value of the inten- 
sity of the radiation being amplified. 
l 1  Note that such aspects of the amplification of chirped pulses as ampli- 
tude-phase modulation and its influence on further compression of the 
pulse (but not on energy removal in the medium) were discussed in 
Refs. 7-9. 
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Translated by Adam Peiperl 
60 Sov. Phys. JETP 73 (I), July 1991 Belenov et al. 60 


Coherent interaction between phase-modulated light pulses and an active medium of two-level particles with inhomogeneous broadening

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Coherent interaction between phase-modulated light pulses and an active medium of two-level particles with inhomogeneous broadening

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