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Theory of optical properties in nonstationary media created by femtosecond pulses: linear case
Chemical Physics North-Holland
16 1 ( 1992) 5 15-526
Theory of optical properties in nonstationary media created by femtosecond pulses: linear case B. Fain School of Chewstry, Sackler Faculty ofExact Sciences, TelAvrv Unrverslty, Ramat Avrv, 69978 Tel Awv, Israel
and S.H. Lin Department of Chemistry and Bzochemlstty and Centerfor the Study ofEarly Events m Photosynthesls, Ari:ona State Unwerslty, Tempe, AZ 85287-1604, USA Received
30 August
199 1; in final form 16 December
I99
1
In ultrashort ( femtosecond) pump-probe spectroscopy, the pump pulse creates a nonstationary medium, while the probe pulse propagates m such a medium. The constitutwe equations of the nonstationary media are different from those of the stationary media. Both constitutive equations and electrodynamic properties of the nonstationary media are analyzed. The conjecture that by dispersing the probe pulse, having a broad spectrum, in a spectrograph after it passed through the sample one can achieve simultaneously high resolution both in time and frequency, is examined. Such a conJecture IS based on the use of the stationary electrodynamics formula for the probe pulse absorption. Using the nonstationary electrodynamlc relations, derived in the paper, It is shown that probe pulses yield time-resolved spectra subjected to usual time-frequency hmitations.
1. Introduction The production of femtosecond (fs) optical pulses [ l-3 ] enabled researchers to investigate induced nuclear motion in molecules [4] and on the medium [ 5 1. In typical pump-probe pulse experiments the pump pulse creates a nonstationary state of the medium (molecule), while the probe pulse is used to measure the absorption in the medium. Nonstationary, coherent vibrational states have now been observed in a wide variety of systems including crystalline solids [ 7,8 1, molecular liquids [ 9- 111 and dye molecules [ 12- 16 1. Coherent wave packet evolution on excited transition states has been observed in the gas phase [ 4.17,18 1, solutions [ 191 and in biological systems [ 201. The novelty of these experiments is the use of pump and probe pulses whose duration is much shorter than the period of some vibrational mode of the molecule, liquid or solid. In most of these experiments only the energy of the probe pulse is measured. This necessi0301-0104/92/$
05.00 0 1992 Elsevier Science Publishers
tates averaging of the response of the sample over the spectral shape of the probe pulse. For very short (fs) pulses the spectral width is becoming very large AOPZ 1/T,
(1.1)
where Tis the mean duration of the probe pulse. One can obtain the time-resolved spectra of the molecules, or excitations in the solid, by performing series of the experiments with the probe pulse having different carrier frequencies. However, the spectral resolutions will be limited by the spectral width of the pulses. In a number of papers [ 2 l-281 a different approach has been proposed. It has been suggested to monitor the spectrum of the probe pulse. By dispersing the probe pulse, having a broad spectrum, in a spectrograph after it has passed through the sample, it is anticipated to achieve simultaneously high resolution both in time and frequency. A complete spectrum (with high resolution) corresponding to a given pump-probe delay is thus recorded in a single shot.
B.V. All rights reserved.
B. Fain, S.H. Lm /Optlcalpropertlesm nonstatronarymeddra
516
This approach has become the basis of an increasing number of experiments [ 26-321. This paper is aimed at analyzing electrodynamical properties of nonstationary media created by ultrashort (femtosecond) pulses. Generally speaking this is a nonlinear problem in which the electromagnetic fields of the pump and probe pulses are involved. Therefore, it is appropriate to describe such a system by third-order polarization PC3), which is proportional to the cube of the amplitudes of the pump and probe fields (see e.g. refs. [ 12,13,33] ). Such a description belongs to the domain of nonlinear optics characterized by third-order susceptibilities xC3) (wi, w2, c+). Essential simplification may be achieved in the case of non-overlapping pump and probe pulses [ 34,5 5 1. In this case we have linear optics of the nonstationary medium created by the pump pulse. In other words we are in the domain of linear nonstationary optics. Therefore, the suggestion to monitor the spectrum of the probe pulse (after it has passed through the sample) and to achieve high resolution both in time and frequency [ 251 has to be considered in the framework of nonstationary optics. It has become almost commonplace [21-29,361 to use in this situation the formula
o(w)=
yIm{P,,(w)/&(o)l
(1.2)
as a measure of the absorption of the probe signal E,, in the sample; while PPr(w) is the polarization induced by this probe field. Formula (1.2) has been derived as the photon absorption coefficient for stationayv media. The important assumption that the Fourier component of P,,( CO)is proportional to the Fourier component E has been introduced in the derivation of formula ( 1.2 ). In a nonstationary medium such an assumption is not valid any more. Therefore to find the equivalent of eq. ( 1.2 ), or more generally to interrogate the propagation of the probe signal in the nonstationary medium we have to perform an analysis germane to the nonstationary medium. The use of formula ( 1.2 ) leads, sometimes, to the conclusion that it is possible to simultaneously achieve high resolution of the photon absorption in both time and frequency. The analysis which is performed in the framework of nonstationary electro-
dynamics shows that there is no violation certainty relation AoAt>
of the un-
1
(1.3)
This means that we cannot achieve a resolution of the photon absorption spectrum Aw better than T - ‘, where T is the duration of the probe pulse. This paper distinguishes itself by the analysis of the electrodynamics of the propagation of a short pulse in a nonstationary medium. It does not take for granted formulae of the type of eq. ( 1.2) (derived for stationary media) but attempts to establish new theoretical concepts suitable for nonstationary media.
2. Constitutive equations in nonstationary media As a result of the ultrashort, femtosecond, pump laser pulses, space-time coherences are created in the medium [ 35 1. The probe pulses, or continuum electromagnetic field, will be affected by the coherences created by the pump pulses. In the linear approximation (for sufficiently weak probe pulses) electromagnetic properties of the medium activated by the pump pulse may be described by generalized susceptibilities. These susceptibilities connect the polarization and electric field (or magnetization and magnetic field). These connections are called constitutive equations for the medium. The general properties of the linear connection between the polarization (magnetization) and the electric (magnetic) field have been considered by Kubo [ 371 and others [ 38,391. In general, the linear connection between the polarization and the electric field has the form
P,(r, t)=
5 I d3rl 1 dti IV,b(r, t; ri, tl burl, --co
tl 1 . (2.1)
Here the field Eb( r, tl ) and the polarization P,( r, t ) are assumed to be classical quantities, we neglect their quantum fluctuations. From now on we will omit summation over repeating indices, as b in eq. (2.1). Response functions !Pab(r, t; rl, t, ) are characteristics of the medium (created by the pump pulse), determining its linear electrodynamic properties. It has to be stressed that relation (2.1) is not a result of a microscopic physical model but it is a general linear re-
B Fam, S.H. Lm /Opticalpropertles
lation taking into account the causality principle. Another representation of the connection between the polarization and the electric field, which is equivalent to eq. (2.1), can be obtained using the Fourier representation of the electric field E, (k, w ) E,(r. t)=
d3k
I
s
dwE,(k,
w) exp[-i(ot-k-r)]
Pa(r, t) =
(2.2) Developing the polarization in the Fourier integral and substituting eq. (2.2) into eq. (2.1) we get P,(r, t) = jd3kJdrr,jd3k, XEh(kl,wl)
{dw,Xa&,c&,W
exp[-i(wt-k-r)]
,
5
d3k
5
doxah(k,
(2.3)
where the generalized linear susceptibilities are expressed through the function You,,in the form
w) &(k, co)
.
Xexp[ -i(ot-k-r)]
.
517
in nonstatronar?, medra
(2.7)
Therefore, we see that the description of the electromagnetic properties of the medium by the susceptibility Xab(k, o) as function of k and w only (and in a majority of cases as function of o only) is valid for stationary and uniform media. As has been mentioned [34,35,40], a medium which was subjected to ultrashort pulses is neither stationary nor uniform. Properties of such media must be described by the generalized susceptibilities. To derive these susceptibilities we will consider a medium consisting of dipoles d, interacting with the electric field V= - 1 d;E(r,,
t) .
(2.8)
To find the linear response we will use the von Neumann equation in the interaction representation (2.9) dt,exp[i(ot-k-r)]
From these two equations, neglecting spatial dispersion and nonuniformity, one obtains [ 34,351
--oc
xexp[
-i(co,t,
-k,*rl
)] Y&r,
t; rl, t) .
(2.4)
These generated susceptibilities (with independent variables k, o and k,, co,) can describe electrodynamic properties in any linear media including nonstationary and inhomogeneous ones: i.e., the media whose properties depend on time t and coordinates r (even when an external field is not applied, or was applied in the past like in our example with the pump field). The bulk of physical literature deals with stationary and uniform media. In this case, the functions You,,characterizing such media will depend, as is known, only on the differences of their arguments Eab(r. t; rl, t, I= lu,,(p, 5) ,
(2.5)
where p=r-r, and 7=t-t,. Substituting eq. (2.5) in eq. (2.4) we find that for stationary and uniform media
xadk w k, w 1=d(k-k, 3
1 6(0-w
) xadk, w) >
(2.6) and relation
(2.3 ) takes the form
(2.10) and *
Pa=
I
dt, IV,tdt,tl) Edtl)
.
(2.11)
Here n is the density of dipole moments and the time dependence of &( t> is determined by the unperturbed Hamiltonian of the system, i.e. with I’= 0. The mean value in eq. (2.10) is taken over the initial in general, coherent state created by the pump pulse at time t,. This means that the initial density matrix at t = t, contains both diagonal elements and off-diagonal elements, coherences. These coherences p,,,,,, correspond to the low-frequency motion of the system [ 34,35,40] and particularly to the vibrational states with frequencies ; (E, -I?,)
=‘-4.
(2.12)
B. Farn, S.H. Lm /Opticalpropertles
518
Therefore, the generalized susceptibilities could display quantum beats with vibrational frequencies. The constitutive equation (2.11) is characterized by the susceptibility Yau,,(t, t, ) in time-time representation. In the frequency-frequency representation, we expand Eb( t, ) in the Fourier integral
and obtain P,(t)
= j do j dw,
xexp(
xdw WI1
Eb(wI)
-iot)
The functions Yab(t, t,), xab(w, wI) and xab(w, t) provide us with both information about the spectrum of the system and its time evolution. However, as we will see from the analysis of the electrodynamics of the system, only in exceptional cases can one derive these functions from the observational data. Therefore it is expedient to introduce “coarse-grained” susceptibilities which contain only partial information about the time-resolved spectra. Let us assume that the time dependence of the probe pulse may be presented in the form E,(t)=
(2.14)
,
m nonstationary medra
[&(a)
exp( -i&)
+E=( -0)
or
P,(u) =
dw, xab(~, WI) G(w,)
s
L(t)
.
(2.21)
Here L(t) is the dimensionless shape function. It is defined to be equal to one at the maximum t= t,, the probing time, and decaying when
(2.15)
.
exp(i&)]
Here
If-&,1 >T.
Here T may be called the effective time of pulse duration. We can present two simple forms of L (t)
t--11
X
s
dMab(t,
t-r)
exp(-io,z)
(2.16)
.
L(t)=l,
0
It is important to notice that in the nonstationary case, the two times t and t,. and two frequencies w and CO,, are independent variables. The frequency-time representation of the susceptibility xab( w, t) is obtained by substitution of eq. (2.13) in eq. (2.11) p,(t)
= 1
(2.22)
doxas(w,t) Eb(o)
exp( -iwt)
,
It-&I
=o,
It-t,I>$T,
(2.23)
or L(t)=exp(-2jt-t,I/T).
(2.24)
The integral of both of these functions
is
(2.17) L(t)
I
where
dt=T.
(2.25)
--m
,--11 x*l?(w. t) =
dr Yab(t, t-z)
s
exp(ior)
.
Now we define the coarse-grained
susceptibility as the response to the amplitudes ,!?a(0) in eq. (2.2 1); substituting the field, eq. (2.2 1) in eq. (2.17 ) we get
(2.18)
0
There are connections t): cc xAw,w~)=
between xab( o, or ) and X&( w,
t) &h(O) exp( --at)
where the coarse-grained bility takes the form
I exp[i(w-wM dtxadw,t)
k
P,(t)=&(W,
+c.c.,
frequency-time
(2.26)
suscepti-
I>
&(ti,
(2.19) and
xexp[ with
Xab(Wt
t) =
t)
=
s -cc
dw
xab(wr
0)
exp[-i(w,
-w)tl
(2.20)
.
t) L(W--0)
dfX&(W,
-i(o-G)t]
,
(2.27)
B. Fam, S.H. Lm /Optical properties m nonstatlonaty media co
i(o-S)
=
&
dtexp[i(w-tiQ)t]
L(t).
-m
(2.28) For a monochromatic field (L=l) i(w-a)= 6(0-CG) andjaab(W, t)=xab(ti ,I). It is expedient to present the above generalized susceptibilities using the dipole matrix elements and the eigenfrequencies of the unperturbed system (without the interaction with the probe field) [ 341
519
depend on time in this approach. However, the eigenfrequencies o,, form the continuous spectrum having a maximum near WE,, the eigenfrequencies which do not take into account the interaction with the thermal bath. In many situations connected with propagation of electromagnetic waves in a medium excited by ultrashort pulses one cannot neglect the nonuniformity of the medium. Therefore, we will generalize the above formula to the case when the ultrashort plainwave pulse generates a nonstationarity in the slab. Let tP be the time when the pump pulse creates the coherence p,,,,,, at the left edge of the slab. Then the same coherence will be created at point z at the time (2.34)
t,(z)=t;+Z/C. where p, is the initial pulse. Here
matrix
formed
by the pump
The formula
(2.29) will take the form
,&(W,&z)=~
f--Is
c
p,,,exp[-io,,(f-tP-z/c)]
m,n.k
i,(x) = -i
j
exp(irx)
dr
(2.30a)
,
x
0
P - -inS(x)
c,(x)=
X
[i,-z,r(@--O,k)
dbmkdakn
-ir-r,c(W--Ok,)
,
damkdbknl
(2.30b) =Xab(@
where P signifies the principal part of the corresponding integral. The frequency-frequency susceptibility can be obtained using relation (2.19). The function &b( w, t) may be split into two pXtS, one of which is time-independent and represents the part of X&( 0. t ) averaged over time
t-Z/C)
(2.35)
.
By the same token we have to interchange above formulae t, by t,(z). see eq. (2.34). formulae (2.35 ) and (2.19) one finds W
in all the From the
(2.31) -T
=xab(w
while the other part, f&,, is time dependent time average is equal to zero. Then Xab(wt
t)
=,%b(W)
Substituting
+&b(W,
t,
.
and its
WI)
exp[i(m-w
)z/c]
(2.36)
,
where Cc
(2.32)
Xab(W>
ml)=
&
this formula in eq. (2.19) one gets
s 6
d7exp[i(w-w,)r]
Xab(W,
7)
(2.37) does not depend on z. Formula It is worthwhile mentioning that formula (2.29) is an exact linear response function. This means that the thermal bath (like phonons), which gives rise to the decay of the density matrix elements, is taken into account in the eigenfrequencies and matrix elements of the system. Diagonal matrix elements pnn do not
Xab(@I,
w2,
+%b(w7
where &(wr,
z)=,%b(~2) 02)
exp[i(w
a(01
(2.3 3 ) takes the form -02)
-w2>z/cl
w2) is defined similarly
,
(2.38) to eq. (2.37).
B. Fain, S.H. Lin /Opticalpropertles
520
rn nonstatronary medra
tively small perturbation we will adopt the approximation of slowly varying envelopes E( w, z)
3. Major equations of linear nonstationary electrodynamics
a2E,(o, To analyze the propagation of an electromagnetic wave through an absorbing sample we have to use the general Maxwell equations for the electric and magnetic fields, E( r, t) and B(r, t)
f $ (E+4xP)
In this case we find from eq. (3.5) 8E,(W, z)
,
Cl(w, (3.1)
It follows from these equations that the wave equation governing optical wave propagation in a nonmagnetic medium is
-$+$$=_$$. 3-
(3.3)
This equation can be further simplified if we adopt certain approximations. Let us consider the Fourier transform of eq. (3.3). For this purpose we expand the electric field E and the polarization Pin the Fourier integral doE(w,z)
exp(-imt),
IdwP(ru,z)exp(-imt).
Substituting
(3.4)
a2E(o, z) to2 +7E(o,z)=-$w2P(ru_z). a9
(3.5)
We will look for a solution of this equation in the form
Assuming
=2rci T j dw, X&O, w,, 2) &(w .
--w)z/c]
(3.8)
qz) (3.9)
aZ
s) exp(ioz/c)
that the polarization
. P introduces
=2rriT
s
do,X(w,o,)E,(o,,z). (3.10)
Eqs. (3.9) and (3.10) are the basic equations for a linear nonstationary medium. To obtain the specific case of a stationary medium, we recall that eq. (2.6) in this case yields x(0, @),)=X(O)
S(w-0,)
(3.11)
.
Therefore we get from eq. (3.10) dE,(W, z)
aZ
(3.12)
=2xi:x(o)E,(w,z).
The absorption coefficient can be obtained from eq. (3.12) if we take into account that the energy density of the field U is proportional to 1E( w, z) 12. Thus we have
aw - =-4i+“(w)U. aZ
these integrals in eq. (3.3) we get
z)=E,(o,
.
We will further simplify this equation by considering an inhomogeneous, isotropic nonstationary medium which is characterized by the susceptibility x( o,o,, z)ofeq. (2.36).Inthiscaseweget aEo(O, z_)
(3.2)
P(z,~)=
P(0, z) exp( -iwz/c)
E(r,t)=-$$p(r,f). >
For the sake of simplicity we will consider the propagation of a plain wave, the electric field depending on the coordinate z only. In this case we get from eq.
E(o,
=2xiY
Xexp[i(o,
(3.2)
s
e)
az
V.B=O.
E(z,t)=
(3.7)
.
Substituting the constitutive equation (2.15) of the linear nonstationary medium in eq. (3.8) we get
V.(E+4nP)=O,
vx(vx)+$$
a-
C
aZ
WE=-;g,
VxB=
z) << g aEcl(w z)
aZ2
(3.6) a rela-
(3.13)
In this case of a stationary medium the Fourier transform of the polarization is proportional to E(o) p(w)
=x(o)
E(o)
(3.14)
.
Therefore the frequency dependent ficient CJ(o) can be presented as
absorption
coef-
B. Fam, S.H. LIII /Optrcaipropertres m nonstationary media
a(w)=47+“(w)=
FIm-
p (0) &Jr(W) .
(3.15)
Thus, we have obtained formula ( 1.2 ). However, as we see from its derivation, this formula is valid only for stationary media, and, therefore, it cannot describe ultrashort time-resolved spectra. Real time-resolved spectra in a nonstationary medium can be described by the susceptibilities ~(0, t), ~(w, wI) and f(w, t) for a nonstationary medium, introduced above. To complete the general electrodynamical consideration of a linear nonstationary medium we will write the electromagnetic energy balance, which follows from the Maxwell equations. The Poynting vector is given by the expression
521
Let 6w be a characteristic frequency scale of E( o, z) near some central frequency W. We assume that 60~~ o*, where w* is the characteristic frequency scale of x( w, t ). In this case one can show
P(t, z)=x(W, t-z/c) t-z/c)
=x(0,
s
E(o,
z) exp( -iwt)
E(O, t, z) exp( -iat)
,
Then, from the Maxwell equations E.~+H.:
divS=-&
(3.1) one gets
_E.$,
(3.17)
>
P(t, z)=x(o,
t-z/c)
E(o,
t, z) exp( -iwt)
U= ;
After we solve the Maxwell equation for E( w, z) we must verify that the time dependence of E satisfies the condition
au au
Therefore
,
t) exp[ -iw(t-z/c)] t-z/c)
along z-axis
8P
(3.19)
(4.4) We omit the letter w in the designation of the field though it is clear that E represents the time varying amplitude of the field with carrier frequency w. Neglecting higher derivatives of E and x, a2E, w aEo __ << a=2 c at
the expression
--)
Q= -Em g
(3.20)
represents a measure of the local (at a certain point z) energy dissipation.
.
Eo(z, t) exp[-iw(t-z/c)]
(3.18)
we get for the propagation c~+~=-E.~.
(4.3)
where At is a characteristic time scale of change of E( o, t, z). We again start from the propagation equation (3.3). The solution of this equation we will be looking for is in the quasi-monochromatic form
P(z. t)=x(w,
(E2+H2),
. (4.2b)
E=E,(z.
we recall that B= H. Introducing the energy density
(4.2a)
or we can put it in another way
u*At>> 1 , (3.16)
S=;ExH.
dw
a3
at’ =w2x.
ax -
at
<<
wx,
!!kJ at <
we obtain after substituting
(4.5)
eq. (4.4) in eq. (3.3) (4.6)
4. Propagation of quasi-monochromatic
probe pulses
First we have to properly define the notion of a quasi-monochromatic probe pulse. We will use relations (2.17) and (2.35) P(t, z)=
s
x(w, t-z/c)
E(w, z) exp( -iot)
do. (4.1)
Now, let us consider the propagation of the signal (probe pulse) from left to right through a slab of thickness 1. We assume that at z= 0 (left boundary of the slab) the amplitude of the incoming wave has the form E,(z=O,
t)=E(t)
.
(4.7)
B. Fatn, S.H. Lm /Optlcalproperties
522
Then a solution
of eq. (4.6) inside the slab (z< 1) is
(4.8)
E,(t,
has the form
(4.12)
t) ,
provided the envelope E( r) is a smooth enough function of its argument.
P(z,t)=j(w,t-z/c)E(@z,t)exp(-iGt)+c.c. (5.1) In contrast to x(w, T), which contains all the information about time-resolved spectra, details of the spectrum at Aw 5 1/T are lost in the coarse-grained susceptibility 2, where T is the pulse duration. For a short pulse the energy dissipation per unit time is ]341 I&S)\2
L(t-z/c)
. (5.2)
The total loss of the energy during the pulse is mI
-Cc
dtE$=-Ztij”(w,t,)l&@I’T,
au
(5.3)
(5.5)
2
The bar takes into account the averaging over a period of the oscillation. In the coarse-grained approach E(z, t)={E(ti,
z. t) exp[ -iQ(t-z/c)]
+El*(O, z, t) exp[iti(t-z/c)]}L(t-z/c),
(5.6)
and the averaged (over a period of the oscillation) field energy density takes the form cr= &E.E*=
The equation nonstationary
au
at +7 z
We will consider now the propagation of the energy of an ultrashort pulse. Short pulses may be characterized by the “coarse-grained” constitutive equation (2.26)
t-z/c)
au
r+ca=-E-$
au
5. Propagation of energy in a nonstationary medium
-
(5.4)
is incorporated in the energy propagation equation (3.19). We rewrite this equation in the form
-21.
It is easy to see that the imaginary part of x: x” = Im x at the moment t-z/c determines the attenuation (absorption) of the amplitude. Thus, in this case one can find from the experimental data the time-resolved spectrum x”( w, t ). Of course, the procedure consists of sending many probe pulses with different frequencies 0. To satisfy condition (4.3) of quasi-monochromaticity of the signal one has to interrogate solution (4.9). The result of such inspection is
= -2c3j”(0,
L(t) dt=T.
Now we will show how the local energy loss eq. (5.3)
xexp[(2nio/c)X(w,t-z/c)/],
-Em g
I
-m
z)=E(t-z/c)
w*>>2sc(o/c)l~(w,
where cu
E,(z,t)=E(t-z/c)exp[(2rcio/c)X(w,t-z/c)z].
Outside the slab (z> l) the solution
m nonstationary media
lF(“;,‘y
‘)12Lz(t-z/c)
.
(5.7)
for the propagation of the energy in a medium takes the form = -2f3j”(O,
xlE(o,z,t)12L(t-z/c).
t-z/c) (5.8)
Now we will make two assumptions about the propagation of the pulse in the medium. Let at z=O the maximum of the pulse shape L( t-z/c) be at t= t,. We assume that at arbitrary z#O inside the sample the maximum is at (5.9)
t-z/c=t,.
This means that we neglect the shape change during the pulse propagation in the sample. We also assume that L (t - z/c) is a rather narrow function so that its width is much smaller than the change of the amplitude ]E(w, z, t) I during the time Tor at the distance CT due to the absorption in the medium Az=cT<<
(Y--I ,
(5.10)
where (Yis the effective absorption coefficient. Now let us find the equation of the propagation of the energy density integrated over the time of the pulse. We find from eq. (5.7)
B. Fam, S.H. Lm /Optlcalproperties
523
in nonstationary media
cc
s
,rJ&=
I(z)=
Ina z, t&l+z/c)
I2
F
2x
(5.11)
g+2Ca~“(m,lp)E(t)&0.
(5.18)
-cc
Thus the decay of the total energy of the laser pulse is determined by the quantity
while eq. (5.8) yields c~=-2~~“(~,t,)l~(a,z,t,+z/C)lT,
(5.12)
co I -cc
L2(t-z/c)
(5.19)
It is assumed here that the duration T of the pulse is much smaller than the characteristic time of change of;y(w, 1):
where
F=
fP) g .
y=20cj”(C&
dt,
1
L(t-z/c)
dt.
(5.13)
where w, is a characteristic
vibrational
frequency.
---co Neglecting the shape change of the pulse we can ignore the z-dependence of T and F (which have the same order of the magnitude). Thus we obtain cg
+47ru$f(w,
(5.14)
tP) $I=O.
Therefore the coarse-grained tion coefficient has the form
z-dependent
absorp-
(5.15) It is worthwhile to compare eq. (3.13) for the propagation of the energy density in a stationary medium with eq. (5.14) describing the propagation in a nonstationary medium. Formula (5.15 ) provides a timeresolved spectrum of the nonstationary medium but with the accuracy Aoe
(5.16)
l/T.
The latter limitation is an implication of the coarsegraining procedure, see eq. (2.27). We may also introduce another quantity: the total energy of the pulse at time t
6. Time-resolved spectrum information which might be obtained by dispersing one probe pulse. Conclusions Spectral information about nonstationary, timedependent media is contained in the above introducedcharacteristics Yab(t, t,), eq. (2.10),xnb(c0, t), eq. (2.18) and xab(m, w,), eq. (2.19). These characteristics of nonstationary media are not independent of each other. They can be expressed one through another. Therefore, in principle, it is enough to know one of them. We have seen that a quasi-monochromatic probe signal propagating through the nonstationary medium can provide information about the time-dependent susceptibility xab( o, t). This means that to obtain the time- and frequency-resolved information about the molecule, one has to perform a series of experiments using probe pulses over a range of frequencies. On the other hand, if the measurements are performed using very short, nonmonochromatic pulses, one obtains information about the coarse-grained susceptibilities f( o, t) [ 34 1, eq. (2.27). However, in this case the spectral resolution Ao is limited by the spectral width of the pulse T-l AozT-’
E(t)=
j: U(z,t) -CC
dz
.
(6.1)
(5.17)
(to be exact, SE(t) is the total energy, where S is the area of the pulse in the transverse direction ). We get from eq. (5.8)
The latter relation conforms to the time-frequency uncertainty relation. However, as it was mentioned in the introduction, a number of authors propose .an alternative approach to dynamic absorption spectroscopy [ 2 l-28,36]. A
524
B. Fain, S.H. Lm /Optical properties rn nonstationary media
short pulse contains a broad Fourier spectrum. Therefore, it is rather attractive to obtain the entire information about a nonstationary medium, just analyzing a signal passing through the medium. Thus, Walkup et al. [ 361 consider the transmission through an optical thin sample. The total field after the transmission is (6.2)
E,,, = ~5, + Cad >
where E,, is the incident field and Eradis induced by the radiation field. The spectral power density of the total output field is L(m)
x I&(W)
+&ad(~)
I2 .
(6.3)
The absorption coefficient (Y(0) is defined in terms of the change in spectral power density due to the transmission through the sample AZ(w)=-a(w)
Z,,(o)
AZ,
(6.4)
or a(w)
= -
(6.5)
It is easy to verify that this formula leads to eq. However, in this case the nonstationary media tion (6.5 ) does not describe exponential decay photon energy, as it is expressed by eq. (3.13). in the case when Erad(w ) is proportional to E,,
( 11). relaof the Only
Erad(~)~Eu,(o)
(6.6)
>
(6.7) We recall from eq. (3.6) that the electric field Fourier component is given by E(w, ~)=&(a,
z) exp[i(o/c)-]
.
(6.8)
A general inspection of eq. (6.7) shows that a short incident pulse with a broad spectrum Aw= l/T
(6.9)
is “coarse-graining” the susceptibility function X( 0, 0, ) by the Fourier component of the electric field envelope Eo(w,. z). Assuming that the sample (slab) is very thin we can neglect the depletion of the incident field. In this case we find the radiation field at the right side of the slab (z= 1) E,,(w,Z)=2ai:Z
I
dw,X(o,o,)&(o,). (6.10)
Using relation (2.33) we can split the radiation into two components
field
( w ),
the change of the spectral density of the very thin sample determines the absorption coefficient in the bulk. As we learn from eq. (3.9), a relation like (6.6) does not occur for a nonstationary medium characterized by the two-frequency susceptibility x( w, or ). Therefore, relation ( 1.2) does not describe the phonon absorption in the bulk of a nonstationary medium. On the other hand, energy propagation of very short pulses in the bulk of nonstationary media can be described by eq. ( 5.14)) which is similar to eq. (3.13 ). The difference is that for nonstationary media the coarse grained susceptibility x( w, tP) at time t = t, determines the electromagnetic field absorption. To analyze the information which can be obtained by dispersing one short pulse after its interaction with a nonstationary medium, we, again, turn to eq. (3.10).
+2niyI
s
dw,i(w,w,)
E,,(q).
(6.11)
It is clear from eq. (6.11) that the spectral content of the emerging pulse differs from that of the incident pulse. The unshifted field is due to the absorption and dispersion connected with the averaged (or background) susceptibility x( 0). This component has nothing to do with the time evolution of the nonstationary medium. The rest of the part of the emerging field arises due to the scattering on the vibrational (or other low-frequency) movements excited by the pump pulse. One can easily recognize that the emerging field contains red- and blue-shifted fields E( w f co,,) which arise due to Stokes and anti-Stokes scattering, respectively. Frequencies o,, = + o0 are connected with coherences pmn. To exemplify this situation in the context of impulsive stimulated scattering, we can refer to the paper of Nelson and co-work-
525
B. Fam. S.H. Lin /Optrcal propertres tn nonstationary media
ers. (Experimental and theoretical results of this group have been reviewed in ref. [ 5 1. ) For short pulses, the spectral width of each field exceeds the frequency separation between them and interference occurs among unshifted and scattered fields. This means that for the ultrashort probe pulse, with very broad frequency content, the dispersing of such a pulse after its transition through the sample provides information about the integral on the right-hand side of eq. (6.11). The function Jdw, x(w, w,) E,,(o,) describes a combined property of the medium and the electromagnetic field E,, (co). This is the price that one has to pay for dealing with nonstationary media. In stationary cases spectral information about the media is determined by the function x(w) (see eqs. (3.13), (3.14)) which is a property of the medium and does not depend on the laser pulse shape. To extend the discussion about the spectral information provided by one short probe pulse we consider the differential transmission function 6T( w) defined in ref. [ 421
(6.12) where “pump on/off ” designates the electromagnetic probe field transmitted through the optically thin slab with and without the pump laser pulse, respectively. For a weak probe pulse we write E, (1, ~)purnpon = E,&)+&,(oM
>
&(A
.
0) mm=&
(w) +Eraci (~4
(6.13)
Substituting these relations in eq. (6.12) and assuming Erad(w, 1) to be small one obtains 67-(o)
(6.14) Using relations 6T(o)=
(2.15 ) and (6.10) one gets
4niol Irn P(o) -P(w) - c & (0)
4lrwl =--1m C
(6.15)
where p(o) and P(w) are the polarizations induced by the probe pulse with and without pump pulse, respectively, and GP(o)=&o)-P(w)
(6.16)
is the part of the polarization which is connected with the density matrix jump [ 35 ] induced by the pump pulse. Using eq. (6.11) the differential transmission function takes the form 6T(w, t,,) 4nwl =--Im C
doi K(w 01) E,,(W) (6.17)
One can see that formula ( 1.2) essentially coincides withformulae (6.15), (6.17). Formula (6.17) isthe transmission function for optically thin samples at time delay tP. In general, this function does not describe the photon absorption at frequency o. The observable presented by formula (6.17 ) is a combined property of both the medium and the probe field. Therefore the fact that one can obtain both high spectral and time resolution from the function 6T(co, tP) does not contradict the uncertainty relation ( 1.4). In conclusion we wish to emphasize the following. In ultrashort pump-probe spectroscopy, the pump pulse creates a nonstationary coherent state of the molecule (medium). The probe pulse propagates in the nonstationary medium. The constitutive equations in such a medium are different from those of a stationary medium. Therefore the electrodynamics of the nonstationary medium is also different. There are new rules of the game. Usage of the conventional formula for the stationary medium, as eq. ( 1.2) can lead to the erroneous conclusion that one can get both infinite spectral and time resolution in photon absorption. The analysis in this paper shows that in a nonstationary medium one can get time-resolved spectra subject to the usual time-frequency uncertainty relations. It is quite possible that ultrashort (fs) pulses experiments can open a new field of nonstationary optics.
526
B. Fain, S.H. Lm /Optical propertles rn nonstatronary medra
Acknowledgement The Arizona State University Center for the Study of Early Events in Photosynthesis is funded by the US Department of Energy (US DOE) Grant No. DEFG02-88ER1969 as part of the US Department of Agriculture-US DOE-National Science Foundation (NSF) Place Science Center Program. This work was supported in part by NATO and ASU.
References [ 1] R.L. Fork, C.H. Brito Cruz, P.C. Becker and C.V. Shank, Opt. Letters 12 (1987) 483. [2] C.V. Shank, Science 233 (1986) 1276; W. Domcke and H. Koppl, Chem. Phys. Letters 140 (1987) 133: A.L. Hams, J.K. Brown and C.B. Harris, Ann. Rev. Phys. Chem. 39 (1988) 341. [3] G.R. Fleming, Ann. Rev. Phys. Chem. 37 (1986) 81. [ 41 M. Dantus, M.Y. Rosker and A.M. Zewail, J. Chem. Phys. 87 (1987) 2395; L.R. Khundkar and A.H. Zewail, Ann. Res. Phys. Chem. 44 (1990) 15. [ 51 K.A. Nelson and E.P. Ippen, Advan. Chem. Phys. 75 (1989) 1 [ 6 ] K.P. Cheung and D.H. Auston, Phys. Rev. Letters 55 (1985) 2152. [ 7 ] L.R. Williams and K.A. Nelson, J. Chem. Phys. 87 ( 1987) 7396. [ 81 L.T. Cheng and K.A. Nelson, Phys. Rev. B 37 ( 1988) 3603. [ 91 R. Leonhardt, W. Holzapfel, W. Zinth and W. Kaiser. Rev. Phys. Appl. 22 (1987) 1735. [lo] D. McMorrow, W.T. Lotshaw and G.A. Kenney-Wallace, Chem. Phys. Letters 145 (1988) 309. [ 111 S. Ruhman, A.G. Joly and K.A. Nelson, IEEE J. Quantum Electron. QE-24 ( 1988) 460. [ 12 ] C.L. Tang, F.W. Wise and LA. Walmsley, Rev. Phys. Appl. 22 (1987) 1695. [ 131 J. Chesnoy and A. Mokhtary, Phys. Rev. A 38 (1988) 3566. [ 141 HI. Fragnito, J.Y. Bigot, P.C. BeckerandCV. Shank,Chem. Phys. Letters 160 (1989) 101. [ 15 1P.C. Becker, H.L. Fragnito, J.Y. Bigot, C.H. Brito Cruz and C.V. Shank, Phys. Rev. Letters 63 (1989) 505. [ 161 S. Sarkan, Phys. Rev. A 38 (1988) 4669. [ 17 ] M.J. Rosker, T.S. Rose and A.H. Zewail, Chem. Phys. Letters 146 (1988) 175. [ 181 R.M. Bowman, M. Dautus and A.H. Zewail, Chem. Phys. Letters 156 (1989) 131.
[ 191 A.G. Joly and K.A. Nelson, J. Phys. Chem. 93 (1989) 2876. [20]R.A. Mathtes, C.H. Brito Cruz, W.T. Pollard and C.V. Shank, Science 240 ( 1988) 777; R.G. Alden, Z.T. Chu, S.H. Lin, R. Islampour, J.P. Allen and J.C. Williams, Advan.‘Multiphoton processes spectry. 7 (1991) 237. [21] R.F. Lonng, Y.J. Yan and S. Mukamel, J. Chem. Phys. 87 (1987) 5840. [22] W.T. Pollard, C.H.B. Cruz, C.V. Shank and R.A. Mathies. J. Chem. Phys. 90 (1989) 199. [23] S.Y. Lee, W.T. Pollard and R.A. Mathies, Chem. Phys. Letters 160 (1989) 531. [ 241 W.T. Pollard, C.H.B. Cruz and C.V. Shank, J. Chem. Phys. 90 (1989) 199. [ 251 W.T. Pollard, S.Y. Lee and R.A. Mathies, J. Chem. Phys. 92 (1990) 4012. [26] J. Misewich, J.H. Glownia. J.E. Rothenberg and P.P. Sorokin, Chem. Phys. Letters 150 ( 1988) 374. [27] J.H. Glownia, J. Misewich and P.P. Sorokin, in: Supercontmuum laser source, ed. R.R. Alfano (Springer, Berlin, 1989) p. 337. [ 28 ] R.E. Walkup, J.A. Misewich, J.H. Glownia and P.P. Sorokin, J. Chem. Phys. 94 ( 199 1) 3389. 1291 J.H. Glownia, Y.A. Misewich and P.P. Sorokin, J. Chem. Phys. 92 (1990) 3335. [30] H.L. Fragnito, J.Y. Bigot, P.C. Becker and C.V. Shank, Chem. Phys. Letters 160 (1989) 101. [31] W.H. Knox, R.L. Fork, M.C. Downer, D.A. Miller, D.S. Chemla. C.V. Shank, A.S. Gossard and W. Wiegman, Phys. Rev. Letters 59 (1985) 1306. [ 321 C.H. Bnto Cruz, R.L. Fork, W.H. Knox and C.V. Shank, Chem. Phys. Letters 132 ( 1986) 34 1. [ 331 I.A. Walmsley and C.L. Tang, J. Chem. Phys. 92 (1990) 1568; S. Mukamel, Ann. Rev. Phys. Chem. 41 ( 1990) 697. 1341 B. Fain, S.H. Lin and N. Hamer, J. Chem. Phys. 91 ( 1989) 4485; S.H. Lin and B. Fam, Chem. Phys. Letters 155 (1989) 216. [35 B. Fain and S.H. Lin, J. Chem. Phys. 93 (1990) 6387. t36 R.E. Walkup, J.A. Misewmh, J.H. Glownia and P.P. Sorokm, Phys. Rev. Letters 65 (1990) 2336. and E. Hanamura, Solid State [37 R. Kubo, T. Takagahara Commun. 32 ( 1979) 1. [38 W. Bernard and H.B. Callen, Rev. Mod. Phys. 3 1 (1959) 1017. electronics 139 V.M. (B.) Fam and Y.I. Khanin, Quantum (Pergamon Press, Oxford, 1969); V.M. (B.) Fain, Photons and non-linear media (Soviet Radio, Moscow, 1972) [in Russian]. [ 401 S.H. Lin, B. Fain and N. Hamer, Advan. Chem. Phys. 119 (1990) 133. [ 411 Y.X. Yan, E.B. Gamble and K.A. Nelson, J. Chem. Phys. 83 (1985) 5391. [42] M. Lindbergand SW. Koch, Phys. Rev. B 38 ( 1988) 7607.
16 1 ( 1992) 5 15-526
Theory of optical properties in nonstationary media created by femtosecond pulses: linear case B. Fain School of Chewstry, Sackler Faculty ofExact Sciences, TelAvrv Unrverslty, Ramat Avrv, 69978 Tel Awv, Israel
and S.H. Lin Department of Chemistry and Bzochemlstty and Centerfor the Study ofEarly Events m Photosynthesls, Ari:ona State Unwerslty, Tempe, AZ 85287-1604, USA Received
30 August
199 1; in final form 16 December
I99
1
In ultrashort ( femtosecond) pump-probe spectroscopy, the pump pulse creates a nonstationary medium, while the probe pulse propagates m such a medium. The constitutwe equations of the nonstationary media are different from those of the stationary media. Both constitutive equations and electrodynamic properties of the nonstationary media are analyzed. The conjecture that by dispersing the probe pulse, having a broad spectrum, in a spectrograph after it passed through the sample one can achieve simultaneously high resolution both in time and frequency, is examined. Such a conJecture IS based on the use of the stationary electrodynamics formula for the probe pulse absorption. Using the nonstationary electrodynamlc relations, derived in the paper, It is shown that probe pulses yield time-resolved spectra subjected to usual time-frequency hmitations.
1. Introduction The production of femtosecond (fs) optical pulses [ l-3 ] enabled researchers to investigate induced nuclear motion in molecules [4] and on the medium [ 5 1. In typical pump-probe pulse experiments the pump pulse creates a nonstationary state of the medium (molecule), while the probe pulse is used to measure the absorption in the medium. Nonstationary, coherent vibrational states have now been observed in a wide variety of systems including crystalline solids [ 7,8 1, molecular liquids [ 9- 111 and dye molecules [ 12- 16 1. Coherent wave packet evolution on excited transition states has been observed in the gas phase [ 4.17,18 1, solutions [ 191 and in biological systems [ 201. The novelty of these experiments is the use of pump and probe pulses whose duration is much shorter than the period of some vibrational mode of the molecule, liquid or solid. In most of these experiments only the energy of the probe pulse is measured. This necessi0301-0104/92/$
05.00 0 1992 Elsevier Science Publishers
tates averaging of the response of the sample over the spectral shape of the probe pulse. For very short (fs) pulses the spectral width is becoming very large AOPZ 1/T,
(1.1)
where Tis the mean duration of the probe pulse. One can obtain the time-resolved spectra of the molecules, or excitations in the solid, by performing series of the experiments with the probe pulse having different carrier frequencies. However, the spectral resolutions will be limited by the spectral width of the pulses. In a number of papers [ 2 l-281 a different approach has been proposed. It has been suggested to monitor the spectrum of the probe pulse. By dispersing the probe pulse, having a broad spectrum, in a spectrograph after it has passed through the sample, it is anticipated to achieve simultaneously high resolution both in time and frequency. A complete spectrum (with high resolution) corresponding to a given pump-probe delay is thus recorded in a single shot.
B.V. All rights reserved.
B. Fain, S.H. Lm /Optlcalpropertlesm nonstatronarymeddra
516
This approach has become the basis of an increasing number of experiments [ 26-321. This paper is aimed at analyzing electrodynamical properties of nonstationary media created by ultrashort (femtosecond) pulses. Generally speaking this is a nonlinear problem in which the electromagnetic fields of the pump and probe pulses are involved. Therefore, it is appropriate to describe such a system by third-order polarization PC3), which is proportional to the cube of the amplitudes of the pump and probe fields (see e.g. refs. [ 12,13,33] ). Such a description belongs to the domain of nonlinear optics characterized by third-order susceptibilities xC3) (wi, w2, c+). Essential simplification may be achieved in the case of non-overlapping pump and probe pulses [ 34,5 5 1. In this case we have linear optics of the nonstationary medium created by the pump pulse. In other words we are in the domain of linear nonstationary optics. Therefore, the suggestion to monitor the spectrum of the probe pulse (after it has passed through the sample) and to achieve high resolution both in time and frequency [ 251 has to be considered in the framework of nonstationary optics. It has become almost commonplace [21-29,361 to use in this situation the formula
o(w)=
yIm{P,,(w)/&(o)l
(1.2)
as a measure of the absorption of the probe signal E,, in the sample; while PPr(w) is the polarization induced by this probe field. Formula (1.2) has been derived as the photon absorption coefficient for stationayv media. The important assumption that the Fourier component of P,,( CO)is proportional to the Fourier component E has been introduced in the derivation of formula ( 1.2 ). In a nonstationary medium such an assumption is not valid any more. Therefore to find the equivalent of eq. ( 1.2 ), or more generally to interrogate the propagation of the probe signal in the nonstationary medium we have to perform an analysis germane to the nonstationary medium. The use of formula ( 1.2 ) leads, sometimes, to the conclusion that it is possible to simultaneously achieve high resolution of the photon absorption in both time and frequency. The analysis which is performed in the framework of nonstationary electro-
dynamics shows that there is no violation certainty relation AoAt>
of the un-
1
(1.3)
This means that we cannot achieve a resolution of the photon absorption spectrum Aw better than T - ‘, where T is the duration of the probe pulse. This paper distinguishes itself by the analysis of the electrodynamics of the propagation of a short pulse in a nonstationary medium. It does not take for granted formulae of the type of eq. ( 1.2) (derived for stationary media) but attempts to establish new theoretical concepts suitable for nonstationary media.
2. Constitutive equations in nonstationary media As a result of the ultrashort, femtosecond, pump laser pulses, space-time coherences are created in the medium [ 35 1. The probe pulses, or continuum electromagnetic field, will be affected by the coherences created by the pump pulses. In the linear approximation (for sufficiently weak probe pulses) electromagnetic properties of the medium activated by the pump pulse may be described by generalized susceptibilities. These susceptibilities connect the polarization and electric field (or magnetization and magnetic field). These connections are called constitutive equations for the medium. The general properties of the linear connection between the polarization (magnetization) and the electric (magnetic) field have been considered by Kubo [ 371 and others [ 38,391. In general, the linear connection between the polarization and the electric field has the form
P,(r, t)=
5 I d3rl 1 dti IV,b(r, t; ri, tl burl, --co
tl 1 . (2.1)
Here the field Eb( r, tl ) and the polarization P,( r, t ) are assumed to be classical quantities, we neglect their quantum fluctuations. From now on we will omit summation over repeating indices, as b in eq. (2.1). Response functions !Pab(r, t; rl, t, ) are characteristics of the medium (created by the pump pulse), determining its linear electrodynamic properties. It has to be stressed that relation (2.1) is not a result of a microscopic physical model but it is a general linear re-
B Fam, S.H. Lm /Opticalpropertles
lation taking into account the causality principle. Another representation of the connection between the polarization and the electric field, which is equivalent to eq. (2.1), can be obtained using the Fourier representation of the electric field E, (k, w ) E,(r. t)=
d3k
I
s
dwE,(k,
w) exp[-i(ot-k-r)]
Pa(r, t) =
(2.2) Developing the polarization in the Fourier integral and substituting eq. (2.2) into eq. (2.1) we get P,(r, t) = jd3kJdrr,jd3k, XEh(kl,wl)
{dw,Xa&,c&,W
exp[-i(wt-k-r)]
,
5
d3k
5
doxah(k,
(2.3)
where the generalized linear susceptibilities are expressed through the function You,,in the form
w) &(k, co)
.
Xexp[ -i(ot-k-r)]
.
517
in nonstatronar?, medra
(2.7)
Therefore, we see that the description of the electromagnetic properties of the medium by the susceptibility Xab(k, o) as function of k and w only (and in a majority of cases as function of o only) is valid for stationary and uniform media. As has been mentioned [34,35,40], a medium which was subjected to ultrashort pulses is neither stationary nor uniform. Properties of such media must be described by the generalized susceptibilities. To derive these susceptibilities we will consider a medium consisting of dipoles d, interacting with the electric field V= - 1 d;E(r,,
t) .
(2.8)
To find the linear response we will use the von Neumann equation in the interaction representation (2.9) dt,exp[i(ot-k-r)]
From these two equations, neglecting spatial dispersion and nonuniformity, one obtains [ 34,351
--oc
xexp[
-i(co,t,
-k,*rl
)] Y&r,
t; rl, t) .
(2.4)
These generated susceptibilities (with independent variables k, o and k,, co,) can describe electrodynamic properties in any linear media including nonstationary and inhomogeneous ones: i.e., the media whose properties depend on time t and coordinates r (even when an external field is not applied, or was applied in the past like in our example with the pump field). The bulk of physical literature deals with stationary and uniform media. In this case, the functions You,,characterizing such media will depend, as is known, only on the differences of their arguments Eab(r. t; rl, t, I= lu,,(p, 5) ,
(2.5)
where p=r-r, and 7=t-t,. Substituting eq. (2.5) in eq. (2.4) we find that for stationary and uniform media
xadk w k, w 1=d(k-k, 3
1 6(0-w
) xadk, w) >
(2.6) and relation
(2.3 ) takes the form
(2.10) and *
Pa=
I
dt, IV,tdt,tl) Edtl)
.
(2.11)
Here n is the density of dipole moments and the time dependence of &( t> is determined by the unperturbed Hamiltonian of the system, i.e. with I’= 0. The mean value in eq. (2.10) is taken over the initial in general, coherent state created by the pump pulse at time t,. This means that the initial density matrix at t = t, contains both diagonal elements and off-diagonal elements, coherences. These coherences p,,,,,, correspond to the low-frequency motion of the system [ 34,35,40] and particularly to the vibrational states with frequencies ; (E, -I?,)
=‘-4.
(2.12)
B. Farn, S.H. Lm /Opticalpropertles
518
Therefore, the generalized susceptibilities could display quantum beats with vibrational frequencies. The constitutive equation (2.11) is characterized by the susceptibility Yau,,(t, t, ) in time-time representation. In the frequency-frequency representation, we expand Eb( t, ) in the Fourier integral
and obtain P,(t)
= j do j dw,
xexp(
xdw WI1
Eb(wI)
-iot)
The functions Yab(t, t,), xab(w, wI) and xab(w, t) provide us with both information about the spectrum of the system and its time evolution. However, as we will see from the analysis of the electrodynamics of the system, only in exceptional cases can one derive these functions from the observational data. Therefore it is expedient to introduce “coarse-grained” susceptibilities which contain only partial information about the time-resolved spectra. Let us assume that the time dependence of the probe pulse may be presented in the form E,(t)=
(2.14)
,
m nonstationary medra
[&(a)
exp( -i&)
+E=( -0)
or
P,(u) =
dw, xab(~, WI) G(w,)
s
L(t)
.
(2.21)
Here L(t) is the dimensionless shape function. It is defined to be equal to one at the maximum t= t,, the probing time, and decaying when
(2.15)
.
exp(i&)]
Here
If-&,1 >T.
Here T may be called the effective time of pulse duration. We can present two simple forms of L (t)
t--11
X
s
dMab(t,
t-r)
exp(-io,z)
(2.16)
.
L(t)=l,
0
It is important to notice that in the nonstationary case, the two times t and t,. and two frequencies w and CO,, are independent variables. The frequency-time representation of the susceptibility xab( w, t) is obtained by substitution of eq. (2.13) in eq. (2.11) p,(t)
= 1
(2.22)
doxas(w,t) Eb(o)
exp( -iwt)
,
It-&I
=o,
It-t,I>$T,
(2.23)
or L(t)=exp(-2jt-t,I/T).
(2.24)
The integral of both of these functions
is
(2.17) L(t)
I
where
dt=T.
(2.25)
--m
,--11 x*l?(w. t) =
dr Yab(t, t-z)
s
exp(ior)
.
Now we define the coarse-grained
susceptibility as the response to the amplitudes ,!?a(0) in eq. (2.2 1); substituting the field, eq. (2.2 1) in eq. (2.17 ) we get
(2.18)
0
There are connections t): cc xAw,w~)=
between xab( o, or ) and X&( w,
t) &h(O) exp( --at)
where the coarse-grained bility takes the form
I exp[i(w-wM dtxadw,t)
k
P,(t)=&(W,
+c.c.,
frequency-time
(2.26)
suscepti-
I>
&(ti,
(2.19) and
xexp[ with
Xab(Wt
t) =
t)
=
s -cc
dw
xab(wr
0)
exp[-i(w,
-w)tl
(2.20)
.
t) L(W--0)
dfX&(W,
-i(o-G)t]
,
(2.27)
B. Fam, S.H. Lm /Optical properties m nonstatlonaty media co
i(o-S)
=
&
dtexp[i(w-tiQ)t]
L(t).
-m
(2.28) For a monochromatic field (L=l) i(w-a)= 6(0-CG) andjaab(W, t)=xab(ti ,I). It is expedient to present the above generalized susceptibilities using the dipole matrix elements and the eigenfrequencies of the unperturbed system (without the interaction with the probe field) [ 341
519
depend on time in this approach. However, the eigenfrequencies o,, form the continuous spectrum having a maximum near WE,, the eigenfrequencies which do not take into account the interaction with the thermal bath. In many situations connected with propagation of electromagnetic waves in a medium excited by ultrashort pulses one cannot neglect the nonuniformity of the medium. Therefore, we will generalize the above formula to the case when the ultrashort plainwave pulse generates a nonstationarity in the slab. Let tP be the time when the pump pulse creates the coherence p,,,,,, at the left edge of the slab. Then the same coherence will be created at point z at the time (2.34)
t,(z)=t;+Z/C. where p, is the initial pulse. Here
matrix
formed
by the pump
The formula
(2.29) will take the form
,&(W,&z)=~
f--Is
c
p,,,exp[-io,,(f-tP-z/c)]
m,n.k
i,(x) = -i
j
exp(irx)
dr
(2.30a)
,
x
0
P - -inS(x)
c,(x)=
X
[i,-z,r(@--O,k)
dbmkdakn
-ir-r,c(W--Ok,)
,
damkdbknl
(2.30b) =Xab(@
where P signifies the principal part of the corresponding integral. The frequency-frequency susceptibility can be obtained using relation (2.19). The function &b( w, t) may be split into two pXtS, one of which is time-independent and represents the part of X&( 0. t ) averaged over time
t-Z/C)
(2.35)
.
By the same token we have to interchange above formulae t, by t,(z). see eq. (2.34). formulae (2.35 ) and (2.19) one finds W
in all the From the
(2.31) -T
=xab(w
while the other part, f&,, is time dependent time average is equal to zero. Then Xab(wt
t)
=,%b(W)
Substituting
+&b(W,
t,
.
and its
WI)
exp[i(m-w
)z/c]
(2.36)
,
where Cc
(2.32)
Xab(W>
ml)=
&
this formula in eq. (2.19) one gets
s 6
d7exp[i(w-w,)r]
Xab(W,
7)
(2.37) does not depend on z. Formula It is worthwhile mentioning that formula (2.29) is an exact linear response function. This means that the thermal bath (like phonons), which gives rise to the decay of the density matrix elements, is taken into account in the eigenfrequencies and matrix elements of the system. Diagonal matrix elements pnn do not
Xab(@I,
w2,
+%b(w7
where &(wr,
z)=,%b(~2) 02)
exp[i(w
a(01
(2.3 3 ) takes the form -02)
-w2>z/cl
w2) is defined similarly
,
(2.38) to eq. (2.37).
B. Fain, S.H. Lin /Opticalpropertles
520
rn nonstatronary medra
tively small perturbation we will adopt the approximation of slowly varying envelopes E( w, z)
3. Major equations of linear nonstationary electrodynamics
a2E,(o, To analyze the propagation of an electromagnetic wave through an absorbing sample we have to use the general Maxwell equations for the electric and magnetic fields, E( r, t) and B(r, t)
f $ (E+4xP)
In this case we find from eq. (3.5) 8E,(W, z)
,
Cl(w, (3.1)
It follows from these equations that the wave equation governing optical wave propagation in a nonmagnetic medium is
-$+$$=_$$. 3-
(3.3)
This equation can be further simplified if we adopt certain approximations. Let us consider the Fourier transform of eq. (3.3). For this purpose we expand the electric field E and the polarization Pin the Fourier integral doE(w,z)
exp(-imt),
IdwP(ru,z)exp(-imt).
Substituting
(3.4)
a2E(o, z) to2 +7E(o,z)=-$w2P(ru_z). a9
(3.5)
We will look for a solution of this equation in the form
Assuming
=2rci T j dw, X&O, w,, 2) &(w .
--w)z/c]
(3.8)
qz) (3.9)
aZ
s) exp(ioz/c)
that the polarization
. P introduces
=2rriT
s
do,X(w,o,)E,(o,,z). (3.10)
Eqs. (3.9) and (3.10) are the basic equations for a linear nonstationary medium. To obtain the specific case of a stationary medium, we recall that eq. (2.6) in this case yields x(0, @),)=X(O)
S(w-0,)
(3.11)
.
Therefore we get from eq. (3.10) dE,(W, z)
aZ
(3.12)
=2xi:x(o)E,(w,z).
The absorption coefficient can be obtained from eq. (3.12) if we take into account that the energy density of the field U is proportional to 1E( w, z) 12. Thus we have
aw - =-4i+“(w)U. aZ
these integrals in eq. (3.3) we get
z)=E,(o,
.
We will further simplify this equation by considering an inhomogeneous, isotropic nonstationary medium which is characterized by the susceptibility x( o,o,, z)ofeq. (2.36).Inthiscaseweget aEo(O, z_)
(3.2)
P(z,~)=
P(0, z) exp( -iwz/c)
E(r,t)=-$$p(r,f). >
For the sake of simplicity we will consider the propagation of a plain wave, the electric field depending on the coordinate z only. In this case we get from eq.
E(o,
=2xiY
Xexp[i(o,
(3.2)
s
e)
az
V.B=O.
E(z,t)=
(3.7)
.
Substituting the constitutive equation (2.15) of the linear nonstationary medium in eq. (3.8) we get
V.(E+4nP)=O,
vx(vx)+$$
a-
C
aZ
WE=-;g,
VxB=
z) << g aEcl(w z)
aZ2
(3.6) a rela-
(3.13)
In this case of a stationary medium the Fourier transform of the polarization is proportional to E(o) p(w)
=x(o)
E(o)
(3.14)
.
Therefore the frequency dependent ficient CJ(o) can be presented as
absorption
coef-
B. Fam, S.H. LIII /Optrcaipropertres m nonstationary media
a(w)=47+“(w)=
FIm-
p (0) &Jr(W) .
(3.15)
Thus, we have obtained formula ( 1.2 ). However, as we see from its derivation, this formula is valid only for stationary media, and, therefore, it cannot describe ultrashort time-resolved spectra. Real time-resolved spectra in a nonstationary medium can be described by the susceptibilities ~(0, t), ~(w, wI) and f(w, t) for a nonstationary medium, introduced above. To complete the general electrodynamical consideration of a linear nonstationary medium we will write the electromagnetic energy balance, which follows from the Maxwell equations. The Poynting vector is given by the expression
521
Let 6w be a characteristic frequency scale of E( o, z) near some central frequency W. We assume that 60~~ o*, where w* is the characteristic frequency scale of x( w, t ). In this case one can show
P(t, z)=x(W, t-z/c) t-z/c)
=x(0,
s
E(o,
z) exp( -iwt)
E(O, t, z) exp( -iat)
,
Then, from the Maxwell equations E.~+H.:
divS=-&
(3.1) one gets
_E.$,
(3.17)
>
P(t, z)=x(o,
t-z/c)
E(o,
t, z) exp( -iwt)
U= ;
After we solve the Maxwell equation for E( w, z) we must verify that the time dependence of E satisfies the condition
au au
Therefore
,
t) exp[ -iw(t-z/c)] t-z/c)
along z-axis
8P
(3.19)
(4.4) We omit the letter w in the designation of the field though it is clear that E represents the time varying amplitude of the field with carrier frequency w. Neglecting higher derivatives of E and x, a2E, w aEo __ << a=2 c at
the expression
--)
Q= -Em g
(3.20)
represents a measure of the local (at a certain point z) energy dissipation.
.
Eo(z, t) exp[-iw(t-z/c)]
(3.18)
we get for the propagation c~+~=-E.~.
(4.3)
where At is a characteristic time scale of change of E( o, t, z). We again start from the propagation equation (3.3). The solution of this equation we will be looking for is in the quasi-monochromatic form
P(z. t)=x(w,
(E2+H2),
. (4.2b)
E=E,(z.
we recall that B= H. Introducing the energy density
(4.2a)
or we can put it in another way
u*At>> 1 , (3.16)
S=;ExH.
dw
a3
at’ =w2x.
ax -
at
<<
wx,
!!kJ at <
we obtain after substituting
(4.5)
eq. (4.4) in eq. (3.3) (4.6)
4. Propagation of quasi-monochromatic
probe pulses
First we have to properly define the notion of a quasi-monochromatic probe pulse. We will use relations (2.17) and (2.35) P(t, z)=
s
x(w, t-z/c)
E(w, z) exp( -iot)
do. (4.1)
Now, let us consider the propagation of the signal (probe pulse) from left to right through a slab of thickness 1. We assume that at z= 0 (left boundary of the slab) the amplitude of the incoming wave has the form E,(z=O,
t)=E(t)
.
(4.7)
B. Fatn, S.H. Lm /Optlcalproperties
522
Then a solution
of eq. (4.6) inside the slab (z< 1) is
(4.8)
E,(t,
has the form
(4.12)
t) ,
provided the envelope E( r) is a smooth enough function of its argument.
P(z,t)=j(w,t-z/c)E(@z,t)exp(-iGt)+c.c. (5.1) In contrast to x(w, T), which contains all the information about time-resolved spectra, details of the spectrum at Aw 5 1/T are lost in the coarse-grained susceptibility 2, where T is the pulse duration. For a short pulse the energy dissipation per unit time is ]341 I&S)\2
L(t-z/c)
. (5.2)
The total loss of the energy during the pulse is mI
-Cc
dtE$=-Ztij”(w,t,)l&@I’T,
au
(5.3)
(5.5)
2
The bar takes into account the averaging over a period of the oscillation. In the coarse-grained approach E(z, t)={E(ti,
z. t) exp[ -iQ(t-z/c)]
+El*(O, z, t) exp[iti(t-z/c)]}L(t-z/c),
(5.6)
and the averaged (over a period of the oscillation) field energy density takes the form cr= &E.E*=
The equation nonstationary
au
at +7 z
We will consider now the propagation of the energy of an ultrashort pulse. Short pulses may be characterized by the “coarse-grained” constitutive equation (2.26)
t-z/c)
au
r+ca=-E-$
au
5. Propagation of energy in a nonstationary medium
-
(5.4)
is incorporated in the energy propagation equation (3.19). We rewrite this equation in the form
-21.
It is easy to see that the imaginary part of x: x” = Im x at the moment t-z/c determines the attenuation (absorption) of the amplitude. Thus, in this case one can find from the experimental data the time-resolved spectrum x”( w, t ). Of course, the procedure consists of sending many probe pulses with different frequencies 0. To satisfy condition (4.3) of quasi-monochromaticity of the signal one has to interrogate solution (4.9). The result of such inspection is
= -2c3j”(0,
L(t) dt=T.
Now we will show how the local energy loss eq. (5.3)
xexp[(2nio/c)X(w,t-z/c)/],
-Em g
I
-m
z)=E(t-z/c)
w*>>2sc(o/c)l~(w,
where cu
E,(z,t)=E(t-z/c)exp[(2rcio/c)X(w,t-z/c)z].
Outside the slab (z> l) the solution
m nonstationary media
lF(“;,‘y
‘)12Lz(t-z/c)
.
(5.7)
for the propagation of the energy in a medium takes the form = -2f3j”(O,
xlE(o,z,t)12L(t-z/c).
t-z/c) (5.8)
Now we will make two assumptions about the propagation of the pulse in the medium. Let at z=O the maximum of the pulse shape L( t-z/c) be at t= t,. We assume that at arbitrary z#O inside the sample the maximum is at (5.9)
t-z/c=t,.
This means that we neglect the shape change during the pulse propagation in the sample. We also assume that L (t - z/c) is a rather narrow function so that its width is much smaller than the change of the amplitude ]E(w, z, t) I during the time Tor at the distance CT due to the absorption in the medium Az=cT<<
(Y--I ,
(5.10)
where (Yis the effective absorption coefficient. Now let us find the equation of the propagation of the energy density integrated over the time of the pulse. We find from eq. (5.7)
B. Fam, S.H. Lm /Optlcalproperties
523
in nonstationary media
cc
s
,rJ&=
I(z)=
Ina z, t&l+z/c)
I2
F
2x
(5.11)
g+2Ca~“(m,lp)E(t)&0.
(5.18)
-cc
Thus the decay of the total energy of the laser pulse is determined by the quantity
while eq. (5.8) yields c~=-2~~“(~,t,)l~(a,z,t,+z/C)lT,
(5.12)
co I -cc
L2(t-z/c)
(5.19)
It is assumed here that the duration T of the pulse is much smaller than the characteristic time of change of;y(w, 1):
where
F=
fP) g .
y=20cj”(C&
dt,
1
L(t-z/c)
dt.
(5.13)
where w, is a characteristic
vibrational
frequency.
---co Neglecting the shape change of the pulse we can ignore the z-dependence of T and F (which have the same order of the magnitude). Thus we obtain cg
+47ru$f(w,
(5.14)
tP) $I=O.
Therefore the coarse-grained tion coefficient has the form
z-dependent
absorp-
(5.15) It is worthwhile to compare eq. (3.13) for the propagation of the energy density in a stationary medium with eq. (5.14) describing the propagation in a nonstationary medium. Formula (5.15 ) provides a timeresolved spectrum of the nonstationary medium but with the accuracy Aoe
(5.16)
l/T.
The latter limitation is an implication of the coarsegraining procedure, see eq. (2.27). We may also introduce another quantity: the total energy of the pulse at time t
6. Time-resolved spectrum information which might be obtained by dispersing one probe pulse. Conclusions Spectral information about nonstationary, timedependent media is contained in the above introducedcharacteristics Yab(t, t,), eq. (2.10),xnb(c0, t), eq. (2.18) and xab(m, w,), eq. (2.19). These characteristics of nonstationary media are not independent of each other. They can be expressed one through another. Therefore, in principle, it is enough to know one of them. We have seen that a quasi-monochromatic probe signal propagating through the nonstationary medium can provide information about the time-dependent susceptibility xab( o, t). This means that to obtain the time- and frequency-resolved information about the molecule, one has to perform a series of experiments using probe pulses over a range of frequencies. On the other hand, if the measurements are performed using very short, nonmonochromatic pulses, one obtains information about the coarse-grained susceptibilities f( o, t) [ 34 1, eq. (2.27). However, in this case the spectral resolution Ao is limited by the spectral width of the pulse T-l AozT-’
E(t)=
j: U(z,t) -CC
dz
.
(6.1)
(5.17)
(to be exact, SE(t) is the total energy, where S is the area of the pulse in the transverse direction ). We get from eq. (5.8)
The latter relation conforms to the time-frequency uncertainty relation. However, as it was mentioned in the introduction, a number of authors propose .an alternative approach to dynamic absorption spectroscopy [ 2 l-28,36]. A
524
B. Fain, S.H. Lm /Optical properties rn nonstationary media
short pulse contains a broad Fourier spectrum. Therefore, it is rather attractive to obtain the entire information about a nonstationary medium, just analyzing a signal passing through the medium. Thus, Walkup et al. [ 361 consider the transmission through an optical thin sample. The total field after the transmission is (6.2)
E,,, = ~5, + Cad >
where E,, is the incident field and Eradis induced by the radiation field. The spectral power density of the total output field is L(m)
x I&(W)
+&ad(~)
I2 .
(6.3)
The absorption coefficient (Y(0) is defined in terms of the change in spectral power density due to the transmission through the sample AZ(w)=-a(w)
Z,,(o)
AZ,
(6.4)
or a(w)
= -
(6.5)
It is easy to verify that this formula leads to eq. However, in this case the nonstationary media tion (6.5 ) does not describe exponential decay photon energy, as it is expressed by eq. (3.13). in the case when Erad(w ) is proportional to E,,
( 11). relaof the Only
Erad(~)~Eu,(o)
(6.6)
>
(6.7) We recall from eq. (3.6) that the electric field Fourier component is given by E(w, ~)=&(a,
z) exp[i(o/c)-]
.
(6.8)
A general inspection of eq. (6.7) shows that a short incident pulse with a broad spectrum Aw= l/T
(6.9)
is “coarse-graining” the susceptibility function X( 0, 0, ) by the Fourier component of the electric field envelope Eo(w,. z). Assuming that the sample (slab) is very thin we can neglect the depletion of the incident field. In this case we find the radiation field at the right side of the slab (z= 1) E,,(w,Z)=2ai:Z
I
dw,X(o,o,)&(o,). (6.10)
Using relation (2.33) we can split the radiation into two components
field
( w ),
the change of the spectral density of the very thin sample determines the absorption coefficient in the bulk. As we learn from eq. (3.9), a relation like (6.6) does not occur for a nonstationary medium characterized by the two-frequency susceptibility x( w, or ). Therefore, relation ( 1.2) does not describe the phonon absorption in the bulk of a nonstationary medium. On the other hand, energy propagation of very short pulses in the bulk of nonstationary media can be described by eq. ( 5.14)) which is similar to eq. (3.13 ). The difference is that for nonstationary media the coarse grained susceptibility x( w, tP) at time t = t, determines the electromagnetic field absorption. To analyze the information which can be obtained by dispersing one short pulse after its interaction with a nonstationary medium, we, again, turn to eq. (3.10).
+2niyI
s
dw,i(w,w,)
E,,(q).
(6.11)
It is clear from eq. (6.11) that the spectral content of the emerging pulse differs from that of the incident pulse. The unshifted field is due to the absorption and dispersion connected with the averaged (or background) susceptibility x( 0). This component has nothing to do with the time evolution of the nonstationary medium. The rest of the part of the emerging field arises due to the scattering on the vibrational (or other low-frequency) movements excited by the pump pulse. One can easily recognize that the emerging field contains red- and blue-shifted fields E( w f co,,) which arise due to Stokes and anti-Stokes scattering, respectively. Frequencies o,, = + o0 are connected with coherences pmn. To exemplify this situation in the context of impulsive stimulated scattering, we can refer to the paper of Nelson and co-work-
525
B. Fam. S.H. Lin /Optrcal propertres tn nonstationary media
ers. (Experimental and theoretical results of this group have been reviewed in ref. [ 5 1. ) For short pulses, the spectral width of each field exceeds the frequency separation between them and interference occurs among unshifted and scattered fields. This means that for the ultrashort probe pulse, with very broad frequency content, the dispersing of such a pulse after its transition through the sample provides information about the integral on the right-hand side of eq. (6.11). The function Jdw, x(w, w,) E,,(o,) describes a combined property of the medium and the electromagnetic field E,, (co). This is the price that one has to pay for dealing with nonstationary media. In stationary cases spectral information about the media is determined by the function x(w) (see eqs. (3.13), (3.14)) which is a property of the medium and does not depend on the laser pulse shape. To extend the discussion about the spectral information provided by one short probe pulse we consider the differential transmission function 6T( w) defined in ref. [ 421
(6.12) where “pump on/off ” designates the electromagnetic probe field transmitted through the optically thin slab with and without the pump laser pulse, respectively. For a weak probe pulse we write E, (1, ~)purnpon = E,&)+&,(oM
>
&(A
.
0) mm=&
(w) +Eraci (~4
(6.13)
Substituting these relations in eq. (6.12) and assuming Erad(w, 1) to be small one obtains 67-(o)
(6.14) Using relations 6T(o)=
(2.15 ) and (6.10) one gets
4niol Irn P(o) -P(w) - c & (0)
4lrwl =--1m C
(6.15)
where p(o) and P(w) are the polarizations induced by the probe pulse with and without pump pulse, respectively, and GP(o)=&o)-P(w)
(6.16)
is the part of the polarization which is connected with the density matrix jump [ 35 ] induced by the pump pulse. Using eq. (6.11) the differential transmission function takes the form 6T(w, t,,) 4nwl =--Im C
doi K(w 01) E,,(W) (6.17)
One can see that formula ( 1.2) essentially coincides withformulae (6.15), (6.17). Formula (6.17) isthe transmission function for optically thin samples at time delay tP. In general, this function does not describe the photon absorption at frequency o. The observable presented by formula (6.17 ) is a combined property of both the medium and the probe field. Therefore the fact that one can obtain both high spectral and time resolution from the function 6T(co, tP) does not contradict the uncertainty relation ( 1.4). In conclusion we wish to emphasize the following. In ultrashort pump-probe spectroscopy, the pump pulse creates a nonstationary coherent state of the molecule (medium). The probe pulse propagates in the nonstationary medium. The constitutive equations in such a medium are different from those of a stationary medium. Therefore the electrodynamics of the nonstationary medium is also different. There are new rules of the game. Usage of the conventional formula for the stationary medium, as eq. ( 1.2) can lead to the erroneous conclusion that one can get both infinite spectral and time resolution in photon absorption. The analysis in this paper shows that in a nonstationary medium one can get time-resolved spectra subject to the usual time-frequency uncertainty relations. It is quite possible that ultrashort (fs) pulses experiments can open a new field of nonstationary optics.
526
B. Fain, S.H. Lm /Optical propertles rn nonstatronary medra
Acknowledgement The Arizona State University Center for the Study of Early Events in Photosynthesis is funded by the US Department of Energy (US DOE) Grant No. DEFG02-88ER1969 as part of the US Department of Agriculture-US DOE-National Science Foundation (NSF) Place Science Center Program. This work was supported in part by NATO and ASU.
References [ 1] R.L. Fork, C.H. Brito Cruz, P.C. Becker and C.V. Shank, Opt. Letters 12 (1987) 483. [2] C.V. Shank, Science 233 (1986) 1276; W. Domcke and H. Koppl, Chem. Phys. Letters 140 (1987) 133: A.L. Hams, J.K. Brown and C.B. Harris, Ann. Rev. Phys. Chem. 39 (1988) 341. [3] G.R. Fleming, Ann. Rev. Phys. Chem. 37 (1986) 81. [ 41 M. Dantus, M.Y. Rosker and A.M. Zewail, J. Chem. Phys. 87 (1987) 2395; L.R. Khundkar and A.H. Zewail, Ann. Res. Phys. Chem. 44 (1990) 15. [ 51 K.A. Nelson and E.P. Ippen, Advan. Chem. Phys. 75 (1989) 1 [ 6 ] K.P. Cheung and D.H. Auston, Phys. Rev. Letters 55 (1985) 2152. [ 7 ] L.R. Williams and K.A. Nelson, J. Chem. Phys. 87 ( 1987) 7396. [ 81 L.T. Cheng and K.A. Nelson, Phys. Rev. B 37 ( 1988) 3603. [ 91 R. Leonhardt, W. Holzapfel, W. Zinth and W. Kaiser. Rev. Phys. Appl. 22 (1987) 1735. [lo] D. McMorrow, W.T. Lotshaw and G.A. Kenney-Wallace, Chem. Phys. Letters 145 (1988) 309. [ 111 S. Ruhman, A.G. Joly and K.A. Nelson, IEEE J. Quantum Electron. QE-24 ( 1988) 460. [ 12 ] C.L. Tang, F.W. Wise and LA. Walmsley, Rev. Phys. Appl. 22 (1987) 1695. [ 131 J. Chesnoy and A. Mokhtary, Phys. Rev. A 38 (1988) 3566. [ 141 HI. Fragnito, J.Y. Bigot, P.C. BeckerandCV. Shank,Chem. Phys. Letters 160 (1989) 101. [ 15 1P.C. Becker, H.L. Fragnito, J.Y. Bigot, C.H. Brito Cruz and C.V. Shank, Phys. Rev. Letters 63 (1989) 505. [ 161 S. Sarkan, Phys. Rev. A 38 (1988) 4669. [ 17 ] M.J. Rosker, T.S. Rose and A.H. Zewail, Chem. Phys. Letters 146 (1988) 175. [ 181 R.M. Bowman, M. Dautus and A.H. Zewail, Chem. Phys. Letters 156 (1989) 131.
[ 191 A.G. Joly and K.A. Nelson, J. Phys. Chem. 93 (1989) 2876. [20]R.A. Mathtes, C.H. Brito Cruz, W.T. Pollard and C.V. Shank, Science 240 ( 1988) 777; R.G. Alden, Z.T. Chu, S.H. Lin, R. Islampour, J.P. Allen and J.C. Williams, Advan.‘Multiphoton processes spectry. 7 (1991) 237. [21] R.F. Lonng, Y.J. Yan and S. Mukamel, J. Chem. Phys. 87 (1987) 5840. [22] W.T. Pollard, C.H.B. Cruz, C.V. Shank and R.A. Mathies. J. Chem. Phys. 90 (1989) 199. [23] S.Y. Lee, W.T. Pollard and R.A. Mathies, Chem. Phys. Letters 160 (1989) 531. [ 241 W.T. Pollard, C.H.B. Cruz and C.V. Shank, J. Chem. Phys. 90 (1989) 199. [ 251 W.T. Pollard, S.Y. Lee and R.A. Mathies, J. Chem. Phys. 92 (1990) 4012. [26] J. Misewich, J.H. Glownia. J.E. Rothenberg and P.P. Sorokin, Chem. Phys. Letters 150 ( 1988) 374. [27] J.H. Glownia, J. Misewich and P.P. Sorokin, in: Supercontmuum laser source, ed. R.R. Alfano (Springer, Berlin, 1989) p. 337. [ 28 ] R.E. Walkup, J.A. Misewich, J.H. Glownia and P.P. Sorokin, J. Chem. Phys. 94 ( 199 1) 3389. 1291 J.H. Glownia, Y.A. Misewich and P.P. Sorokin, J. Chem. Phys. 92 (1990) 3335. [30] H.L. Fragnito, J.Y. Bigot, P.C. Becker and C.V. Shank, Chem. Phys. Letters 160 (1989) 101. [31] W.H. Knox, R.L. Fork, M.C. Downer, D.A. Miller, D.S. Chemla. C.V. Shank, A.S. Gossard and W. Wiegman, Phys. Rev. Letters 59 (1985) 1306. [ 321 C.H. Bnto Cruz, R.L. Fork, W.H. Knox and C.V. Shank, Chem. Phys. Letters 132 ( 1986) 34 1. [ 331 I.A. Walmsley and C.L. Tang, J. Chem. Phys. 92 (1990) 1568; S. Mukamel, Ann. Rev. Phys. Chem. 41 ( 1990) 697. 1341 B. Fain, S.H. Lin and N. Hamer, J. Chem. Phys. 91 ( 1989) 4485; S.H. Lin and B. Fam, Chem. Phys. Letters 155 (1989) 216. [35 B. Fain and S.H. Lin, J. Chem. Phys. 93 (1990) 6387. t36 R.E. Walkup, J.A. Misewmh, J.H. Glownia and P.P. Sorokm, Phys. Rev. Letters 65 (1990) 2336. and E. Hanamura, Solid State [37 R. Kubo, T. Takagahara Commun. 32 ( 1979) 1. [38 W. Bernard and H.B. Callen, Rev. Mod. Phys. 3 1 (1959) 1017. electronics 139 V.M. (B.) Fam and Y.I. Khanin, Quantum (Pergamon Press, Oxford, 1969); V.M. (B.) Fain, Photons and non-linear media (Soviet Radio, Moscow, 1972) [in Russian]. [ 401 S.H. Lin, B. Fain and N. Hamer, Advan. Chem. Phys. 119 (1990) 133. [ 411 Y.X. Yan, E.B. Gamble and K.A. Nelson, J. Chem. Phys. 83 (1985) 5391. [42] M. Lindbergand SW. Koch, Phys. Rev. B 38 ( 1988) 7607.