Resonance effects on two photon absorption processes of
molecules
Y. Fujimura
Department of Chemistry, Faculty of Science, Tohoku University, Sendai 980, Japan
S.H.Lin
Department of Chemistry, Arizona State University, Tempe, Arizona 85281
(Received 4 June 1980; accepted 4 November 1980)
Effects of resonance on two photon absorption of molecules have been investigated. Using the expression for
the photon processes which was derived by using the density matrix method in the Markoff approximation,
the intensity of the two photon absorption can be separated into simultaneous, sequential, and their mixing
terms. The last term vanishes in the case of zero pure dephasing constant between the initial and final
electronic states. In order to study the resonance effects on the two photon absorption for a multilevel
molecular system, an expression for the transition probability of the two photon absorption processes is
derived in the Bom-Oppenheimer basis set. Analytical expressions for both simultaneous and sequential
processes are obtained for the displaced harmonic oscillator model. A temperature dependent simultaneous
two photon absorption probability for the multimode case is formulated. Some model calculations of the
vibrational structures in the low temperature limit for the molecular system are performed to illustrate the
resonance effect.
I. INTRODUCTION
It is frequently recognized that multiphoton processes
such as multiphoton absorption and multi photon ionization of atoms and molecules exhibit resonance effects.I - 7
The multiphoton process is said to be in resonance if
the energy of one or several quanta of the radiation field
is close to the transition energy from the initial to an
intermediate excited state. One of the resonance effects is revealed in the appearance of vibrational structures in the multiphoton spectra. This effect is related
to the difference among the potential energy surfaces involved in the multi photon transition, and may be explained by using the Franck-Condon principle. The potential energy differences between the initial and resonance states and/or between the resonance and final
states as well as those between the initial and final electronic states may reflect on the spectra. On the other
hand, it is necessary to carry out infinite summations
over the intermediate states in the nonresonant case.
The vibronic intensity of the spectra in this case is
analyzed in terms of the potential energy displacements
between the initial and final states.
Another interesting problem to be solved in the multi-
photon processes is concerned with the mechanism:
Terms "simultaneous process" and "sequential process"
are frequently used in describing the multi photon processes. 6,8-13 For the nonresonant case the Simultaneous
process in general makes the dominant contribution. On
the other hand, for the resonant case, the two mechanisms seem to operate on the process at the same time.
In this paper, we restrict ourselves to the two photon
absorption (TPA) of a molecular system. 14,15 We first
analyze the transition probability of the TPA, which was
derived by using the denSity matrix method in the Markoff approximation, and show that the transition probability is separated into three terms (simultaneous, sequential, and their mixing terms). The role of dephasing constants in the TPA processes which are expressed
by the sum of longitudinal and transverse relaxation constants is emphasized. Next, analytical expressions for
both simultaneous and sequential processes are obtained
for the displaced harmonic oscillator model. For the
simultaneous TPA process, a temperature dependent
transition probability in the multimode case is formulated. Finally, some model calculations of the vibrational structures appearing in the TPA spectra are performed.
II. MECHANISM OF TPA
We consider a system consisting of a molecule, heat bath, and radiation field. Using the density matrix method
in the Markoff approximation, the transition probability of the TPA at the steady state is expressed as 16 ,17
(2.1)
where suffix a and n denote the initial and final states, respectively, and k and m the intermediate states. P•• represents the initial population of the system. iI~ denotes the molecule-radiation field interaction Hamiltonian. r nk
denotes the dephasing constant related to the states (n and k), and is given by17-19
3726
J. Chem. Phys. 74(7),1 Apr. 1981
0021-9606/81/073726·09$01.00
© 1981 American Institute of Physics
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Y. Fujimura and S. H. Lin: Two photon absorption processes
3727
where r nn and r kk are population decay constants of the states nand k, respectively, and r~:nk denotes the pure dephasing constant originated from the elastic interaction process between molecules and heat bath.
Equation (2.1) can be written as
W == W. lm
+ W.oq + Wmlx
(2.2)
,
where W. lm represents the transition probability of the simultaneous TPA and is expressed as
Paa
\ " (H~)nm(H~)ma
ef,;-iwna+rna ~ iWma+rma
-.!..R"
W
.lm-If4
\2
(2.3)
•
W••Q represents the transition probability of the sequential TPA and is expressed as
W
~R
-
If 4
.eq -
and
Wmix
e
L: L:
an
(rnk + r mn - r mk)(H~)ak(H~)kn(H~)nm(H~)mg
mk (iw ma + r ma)(iw mn + r mn)(iw nk + r nk)(iw mk + r mk)
(2.4)
the transition probability of their mixing process, and is given by
Wmlx ==
bRe L:.
an
If,
.
P~r na L:{(r
na + r ka mil
ZW na
-
)[.(
)
rnk)(iw mn + r mn)
r ]}
+ {ZWna +-rna -z Wmn+W ka -rmn+ ka
(.W
"rna
(H~)qk(H~)kn(H~)nm(H~»f
(2.5)
+r rna )(iw mn +r mn )(iw nk +r nk) -iw ka +r ka )
To qualitatively see which one is the dominant mechanism in the TPA, we consider a three level system in
which the effect of the geometrical structure of the molecule is neglected. The ratios of W.8<I to W. lm and that of
WmlX to W. lm are given by
n
(2.6)
~
and
YJ
m
f:
W1(sM
w2.(seq)
EIf!U
WI
(2.7)
0
respectively. From the above expressions, we can see
that the simultaneous and sequential processes in the
resonant TPA are independent of each other if the pure
dephasing constant between the initial and final states
vanishes or, if the condition r ~:t na « r mn is satisfied, the
two processes may be considered to be independent ones.
Which one of the two processes dominates depends on
the dephasing constants relevant to the states.
•
ttn
III. TPA FOR A MULTILEVEL SYSTEM
In this section in order to investigate the resonance
effect on the TPA intensity for a multilevel molecular
system, we derive an expression for the transition probability of the TPA in the Born-Oppenheimer basis set.
Analytical expressions for the transition probability of
the simultaneous and sequential processes are derived
in the displaced harmonic oscillator model. The contributions from the mixing between the simultaneous and
sequential processes are neglected.
A. Simultaneous TPA
We consider the transition probability for the simultaneous TPA process shown in Fig.!. We start with the
expression
q
FIG. 1. Model for a two photon absorption. a, m, and n
specify the electronic ground, resonant, and final states, respectively. €~ and €~ represent the energy gap between the final
and groWld states and that between the resonant and ground
states, respectively. ~a and ~nm denote the dimensionless displacements between the resonant and ground state and the final
and resonant states, respectively. In the one-dimensional displaced harmonic oscillator model, ~na = ~nm + ~ma. WI and Wz
represent the frequencies of the incident and probing lasers,
respectively. The figure inserted shows a model for the simultaneous and sequential two photon processes.
J. Chem. Phys., Vol. 74, No.7, 1 April 1981
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Y. Fujimura and S. H. Lin: Two photon absorption processes
3728
f'"
~
="21 _'" dtexp(-ixt-altl)
x a
(3.1)
In the Born-Oppenheimer approximation, the molecular
state la) is expressed as la)= liPa) lea)' where I<pa) and
I ea) represent electronic and vibrational eigenvectors of
the ath electronic state, respectively. Using the transformations
(3.2a)
and
ix 1+ b
=
1'" dT
0
exp[ - T(ix + b») •
(3.2b)
Equation (3.1) is expressed in the integral form as
(3.3)
where the Condon approximation was used to evaluate the transition moment Mnm = (<pn I er I cI>m)' Noting that wma
Wi and w n• = (En - E.)/Ff - Wi - w2, where Em represents the vibronic energy of the intermediate state,
and wi and w2 the frequencies of the incident and probing lasers, respectively, Eq. (3.3) is rewritten as
= (Em -E.)/Ff -
xL:L: exp[-rnaltl-it(Wi +w 2) -iWiT' +iWiT1(ealpaaexp[-iEa(t+,' -T)/Ff)
an
mm'
x exp(i Em T' In
- r ma ,') Iem) (em I exp(iEntlFf) Ien) (en I exp(- iE m• Tiff - r m'a T) Iem.)(e m.1 ea)
(3.4)
Using the closure relation of the vibrational states, and carrying out the summations in Eq. (3.4), we obtain
where
(3.6)
where Hm represents the vibronic Hamiltonian of the resonant electronic state. In deriving Eq. (3.5), the vibrational quantum number dependence of r was neglected. Equation (3.5) expresses the transition probability of the
simultaneous TPA using the Born-Oppenheimer basis set.
In the displaced harmonic oscillator model, the vibronic Hamiltonians of the initial, resonant, and final electronic
states are written as
(3. 7a)
(3.7b)
and
(3.7c)
respectively, where N represents the number of the vibrational modes, wI is the frequency, and ql and PI are the
dimensionless nuclear coordinate and the conjugate momentum of the ith vibrational mode, respectively. Amal represents the dimensionless displacement between the equilibrium points of the ith vibrational mode in the resonant
and ground electronic states, and E~ the electronic energy gap between the bottoms in the two electr"onic states (Fig.
1). Equation (3.5) can be expressed in the analytical form as
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Y. Fujimura and S. H. Lin: Two photon absorption processes
X
(c. mal 112) <PI
SI!)
h!
+Ol'l
1
{.lLn-Wt+bPI-q/+rl-slwl
rE~
~
+
) ] r}
(
2
3729
(3.8)
ma
where C. nmi represents the displacement between the equilibrium points of the ith vibrational mode in the final and
resonant states. The detail of the derivation of Eq. (3.8) is given in the Appendix.
B. Sequential TPA
In the Born-Oppenheimer approximation the transition probability of the sequential TPA is expressed as
(3.9)
where the approximation (iw mk + r mk)"1 "" 0mklr mm was used, i. e., the contribution from the off -diagonal terms in the
intermediate states was neglected. This approximation is valid for the system without overlapping between the vibronic levels. If the vibrational quantum number dependence of the dephasing constants may be neglected, the transition probability is expressed as
W.eq (wI'
2( .
_ 2 IMnmMma r 2r mn - r mm) "'I ( )1 ( )
Jf4 r r
4...J a-m Wt m-n w2
W2 ) -
mm
mn
,
(3.10)
m
where Ia_m(wt) and I m_n(W2) represent the line shape functions of the absorption from the electronic ground state to a
single vibronic level m in the resonant state and from this level to the final electronic state, and are given by
(3.11)
and
(3. 12)
respectively.
In the displaced harmonic oscillator model for the single mode, an analytical expression for W.eq in the low temperature limit can be obtained as
(3. 13)
IV. DISCUSSION
In the preceding sections, the mechanism of the resonance TPA in the presence of the heat bath has been investigated, the role of the dephasing constants has been clarified, and the analytical expressions for both simultaneous
and sequential processes have been derived in the displaced harmonic oscillator model.
To illustrate the resonance effect on the TPA, model calculatiOns of the vibronic structures appearing in the TPA
are performed by using the analytical expressions derived in Sec. III. In the model calculatiOns, we consider the
TPA spectra for a single mode in the low temperature limit for the molecular system. In this case, the simultaneous TPA spectra, Eq. (3.8) reduces to
(4.1)
J. Chern. Phys., Vol. 14, No.1, 1 April 1981
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3730
Y. Fujimura and S. H. Lin: Two photon absorption processes
...' 0
.
oCD
oN
-1.0
1
0
I
J.
2.0
1.0
3.0
5.0
40
l.I..2 -(~- £~ )/11
kk
-1.0
FIG. 2. The calculated TPA spectrwn as a function of
w2 - k~ -f!I/1i. The spectrwn is shown without the constants
21 M"",M"., 12/n4. The parameters used are w = 1000 cm-I ,
Arm '" 1. 0, A"" = 1. 6, the detuning frequency WI - f !/1i - 200 cm-I,
rma '" 5. 0 cm- I , r,.,"'5.0 cm- I , r"",=lO.O cm- I , rmm'" 10 cm- I ,
and f!/1i = 39 200 cm-I . The bands whose peak locates at the integral multiple on the abscissa correspond to those of the sequential TPA, and the other bands to those of the simultaneous
TPA.
To calculate the sequential TPA spectra, Eq. (3.13)
was used. In Figs. 2 and 3, the vibronic intensity distribution of the TPA in the case of the excitation of the
detuning frequency Wt - E:~/1f = - 200 cm- t is shown as a
function of Wz - (E:~ - E:~)/1f. The dimensionless displacement between the resonant and final states Anm was taken
to be 1. 0 and 2.0 for Figs. 2 and 3, respectively, and
Ama = 1. 6 was used for both figures. The dephasing constants r rna = 5. 0, rna = 5. 0, and r mn = 10.0, and the population decay constant r mm = 10. 0 were used through the
model calculations. The doublet structure in Figs. 2
and 3 reflects the simultaneous and sequential mechanism of the TPA; the bands whose peak locates at the
integral multiple on the abscissa correspond to those
originated from the sequential TPA mechanism, and the
other bands correspond to those originated from the
simultaneous TPA mechanism. The vibronic intensity
distribution of the sequential TPA depends on the sum
of the transition probability from the vibronic levels in
.
40
5.0
kk
FIG. 4. The calculated TPA spectrum as a function of w2
-(f~-E!I/1i. The same parameters as in Fig. 2 are used except A"". = O.
the resonance state to the final vibronic levels. On the
other hand, the vibronic intensity distribution of the
simultaneous TPA is not expressed by a simple combination of the optical transition such as that of the sequential TPA. Figure 4 shows the vibronic intensity distribution as a special case in which Anrn = 0 and the other
parameters were taken to be the same as in Fig. 2. In
Fig. 4, the singlet structure originated from the simultaneous TPA mechanism forms a progression; only the
line corresponding to the 0-0 optical transition is observed for the sequential TPA. The different behavior
involved in the vibronic structure between the simultaneous and sequential TPA spectra can simply be explained as follows: For Anm = 0, by using Eq. (4.1), the
transition probability of the simultaneous TPA is expressed as
W
(
) _ 2 1MnmMma 12 rna
1f4
r
slm Wt, W2 -
ma
(4.2a)
or
':-0
where Ama = Ana has been used, and Iao.nl(Wt +wz) represents the line shape function of the optical transition
from the lowest vibronic level in the initial state to that
in the final state, and is given by
0
to
0
~
~
Iao.nl(Wt
0
N
-1.0
o
1.0
20
Wr(f~- ("N'h
3.0
4.0
50
6.0
kk
FIG. 3. The calculated TPA spectrwn as a function of
w2
- k~ -E!I/1i. The same parameters as in Fig. 2 are used except
A"". '" 2. O.
r na(A~/2)I exp[ - (A~/2)]
+ wz) = lI[(E:~I1f + lw -Wt -WZ)Z + r~] .
(4.3)
From the above expressions it can be seen that only the
band corresponding to the 0-0 optical transition from
the resonant to the final state is observed for the sequential TPA, and the vibronic structure of the simultaneous TPA forms its progression.
The intensity ratio between the simultaneous and sequential TPA spectra depends on the parameters involved such as dephasing constants, displacements, the
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Y. Fujimura and S. H. Lin: Two photon absorption processes
detuning energy, etc. However, in the case of the exact
resonance Wt =E!/Ii, the intensity (peak height) ratio of
the sequential to simultaneous TPA bands is approximately determined by the dephasing constants and the
population decay constant r mm and is independent of the
3731
position of the peak as can be seen in Fig. 5, where the
same parameters as in Fig. 2 were used except WI =E!I
Ii. In fact, in the case of the zero detuning, in Eq. (4.1)
the term with k = 0 and j = l makes a dominant contribution;
(4.4)
where w» r was used. On the other hand, for the sequential TPA in the case of the zero detuning Eq. (3.13) is
written as
_OIl<
) '" 2 1MnmMma 12 (2r mn - r mm) exp (_ ~~a _ ~~m) f-Wseq (Wt-Em
", W2
l<4
r mm r na
2
2 L.i
f£
1=0 l'
.
(~2 /2) I
(4.5)
[(EOn- 10 m) +lw-w J2 +r2
Ii
mn
2
By using Eqs. (4.4) and (4.5), the intensity ratio of the sequential and simultaneous TPA at lth band can be estimated as
WSeq ( WI
=
0
0
)/
(
0
0
)
Em0
En-Em
Em0
_En-Em
-If
' w2 = --li+ lw W s1m w t = If ' w2 - --fi- + lw = (2r mn - l)r mar na ,
(4.6)
where f mn = r mn/r mm' Using the parameter set for the
dephasing and population decay constants adopted in the
calculation, we find that the intensity ratio appearing in
Fig. 5 gives W ••q[Wl=E~/fi,W2=(E~-E~)/Ii+lwJlW.lm(WI
=10 !/fi, w 2= (E~ -10 ~)/Ii + lw) = 0.25.
state to the lowest level in the final state makes a significant contribution to the sequential TPA process, and
the transitions from the higher vibronic levels than the
level with one quantum excitation may be neglected in
the model calculations.
In Figs. 6 and 7, the calculated vibronic intensity distribution of the TPA as a function of w2 - (E:~ -E:~)/fi in
the case of the detuning w t -E~/fi=800 cm- t is shown.
The same values of the constants as in Figs. 2 and 3
were used in Figs. 6 and 7, respectively. The features
in Figs. 6 and 7, compared with other figures, show the
appearance of the high intensity band at the position w 2
-(E~-E!)/fi=-1000 cm- t for the sequential TPA. This
result indicates that the optical transition from the vibronic level with one quantum excitation in the resonance
In the model calculation of the TPA spectra, the low
temperature limit for the molecular system was assumed; the heat bath with a certain temperature was
implicitly assumed. These assumptions are valid for
the case in which fiw> kT "'fiw B, where w B represents
the high frequency of the bath mode. The heat bath may
supply energy of 200 cm- 1 responsible for the appearance of the sequential peaks in Figs. 2-4 in which the
detuned frequency of 200 cm- 1 under the lowest vibronic
level in the intermediate state was used, and for the
-1000 cm- 1 peak in Figs. 6 and 7 in which the detuned
frequency of 800 cm- 1 above the vibronic level was used.
.
To
q
N
0
rD
~
0
..:;
o
N
-1.0
.0
1..0
2..0
U7-(E~-~m)lh
3.0
4 ..0
5 ..0
kk
-1.0
FIG. 5. The calculated TPA spectrum as a function of ""2
- (~~ -~~)/1f in the case of the exact resonance (w 2 =E~/1fl. The
broken and straight lines correspond to the spectrum of the simultaneous and sequential TPA. The same parameters as in
Fig. 2 are used.
a
to
2.0
3..0
40
wr (~- ~"fl
50
6..0
7..0
80
kk
The calculated TPA spectrum as a function of w2
The detuning Wl-€~/1f=800 cm- 1 is used. For
the other parameters, the same values as in Fig. 2 are used.
FIG. 6.
- (€~-€~)/1f.
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3732
Y. Fujimura and S. H. Lin: Two photon absorption processes
o
ui
o
N
8.0
9.0
FIG. 7. The calculated TPA spectrum as a function of w2
The same parameters as in Fig. 6 are used
except ~"'" =2. O.
- «(~ -(~)/1f.
stants for separating the TPA process into the simultaneous and sequential processes has been emphasized.
The TPA spectra originating from both the sequential
and simultaneous processes have been calculated for the
displaced harmonic oscillator model. Effects of the
resonance on the TPA spectra have been discussed in
this model system. The effect of saturation on the TPA
was neglected. This effect may become important only
for the molecular system to which strong incident laser
is applied. It is important to note that the expressions
derived for the simultaneous TPA [see Eq. (3.8) and the
Appendix] includes not only the multimode effect but
also the temperature effect for the molecular system.
In this paper, although we only consider the case of the
allowed electronic transitions for both a- m and m - n
(and the Condon approximation has been used), the present theory can be extended to treat the resonant TPA of
the symmetry -forbidden transitions.
ACKNOWLEDGMENTS
Finally, it should be noted that the analytical expression for the simultaneous TPA [Eq. (3.8)] is applicable
to resonance Raman scattering from molecules: If we
put ~ma = - ~"m' E ~ = 0, and Wj - w 2 instead of Wj + w2 in
Eq. (3.8), we obtain the expression for the resonance
Raman scattering [Eq. (3.16)] appearing in a previous
paper. 20
In summary, in this paper, the mechanism of the res0nant TPA process of molecules interacting with a heat
bath has been clarified; the role of the de phasing con-
Acknowledgment is made to The Donors of the Petroleum Research Fund, administered by the American
Chemical Society, for support of this research. S. H.
L. wishes to thank Professor E. W. Schlag and Professor G. L. Hofacker for their gracious hospitality,
and Y. F. wishes to thank Professor T. Nakajima for
his encouragement during the course of this work. We
are grateful to Professor M. Ito and Professor K. Kaya
for stimulating discussions and for having sent us the
copy of their manuscripts before publication.
APPENDIX: DERIVATION OF EO. (3.8)
In this Appendix, a brief derivation of Eq. (3.8) is presented. We start with Eq. (3.5):
W.1m(Wj, w2) = IM"nAfma 12
f~ dt
-~
J'" dT J~ dT' exp[ - rna It I - r matT + T') - it(W j + W
2) -
0
iWj(T' - T) ]G(T, T', t) ,
(A1)
0
where
G(T, T', t)
=Av{exp[ -
iHa(t + T'
-
T)/1f] exp(iHm T' In) exp[iHntln] exp(- iHm TIn)} ,
(A2)
whereAv'oo =L:.<Bal··· PaaIBa)'
In the displaced harmonic oscillator model, the vibronic Hamiltonians of the initial, resonant, and final electronic
states are expressed as
(A3a)
(A3b)
and
(A3c)
The vibrational wave vector IBa) is expressed as IBa)=TI j IB al ). The density operator of the initial state Paa is assumed to be given by the Boltzmann distribution
(A4)
where {3 = (kT)"j •
Substituting Eqs. (A3) into Eq. (A2), we obtain
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3733
Y. Fujimura and S. H. Lin: Two photon absorption processes
,
[i€O(r-r')
if.Ot]II
'
G(r,T,t)=exp
m
+ - , t - , G,(r,T,t) ,
Fi
(A5)
where
(A6)
G, (T, T', t) =A v, {exp( - iii., (t + T' - r)/Fi] eXp(u1mf T' /Fi) exp(iHn,t/Fi) exp(- iii m, T/m} ,
,
where Av denotes the average for the ith vibrational mode. Using the relation
exp(iHm, r' /m =
exp[~~.' (e'W,T'
2t
(A7)
-1)] exp(iH., T' /Fi) exp( - Amaf (T')bJ] exP(A:.(r')b,] ,
where Am.I(T') =(~m.,N2}[(1_e-'W,T·)], Eq. (A6) is rewritten as
G, (T, r', t)
= exp( (~~,/2)(efWII -1) + (~~.,/2)(elwIT· -
1) + (~~af/2)(e-'W,T -1) ]L I ( T, T'. t) ,
(AS)
where
L, (T, ,', t) =A vj {exp[ - iH.f(t + T' - T)/Fi] exp(iH.,T' /Fi) exp( - Am4f (T')bll exp[,\;.,(T')b j
]
x exp(iH.,t/Fi) exp[ - Anaf(t)bi] exp[A:.,(t)bf] exp(- iiia,dFi)exp[ - ,\:.,( T)bi] exp{Am.l( T)b f ]} •
(A9)
The evaluation of Eq. (A9) is accomplished by using the relation22
AVf[J(b" bj)]= (0,0,1/("1 +11, bf +,rn;al, "'I +nf
bi +~a,) 10/0,),
(AID)
where 11, =[exp(f:lFiwf) _l]-t. Equation (A9) is expressed as
= (Of 01 Iexp[ -
L,(T, T', t)
x exp[ - Ana' (t)("'1 +
n, 'bl + ~ af)] exp( A:.f (t)(v't + n, b, + ~ am
xexp[Amaf(T)(v't +n,
hi + ~ a,)]exp[ -
By using the Boson operator algebra,
= exp[ (~;'f/2)(eIW, 1 _
L, ('T, T', t)
bi + ~ al)] exp(A:',(,') e'w, '("'1 +11, h, + ~ an]
Ama,(T') e- fw , '("'1 +n,
22
A:",(T)("1 +n,
b, + v'1i;ii1)J! 010,) •
(All)
Eq. (All) is easily evaluated as
I) + (~~.,!2)(e'W'T· -I)
+ (~~a,/2)(e-fW,T -1) + (1 +n,)[A:af(T')A:'f(t) + A:'f(t}Ama,(T} + A:",(T')A,nal(T) efWII ]
I
I
- n,[ lAma' (T') 12 + Amaf(T} 12 + Ana,(t) 12 - Ama, (T')Anof (t) - Ana' (t)A:a,(T) - Amaf( T'}A:a,(T) e-
IW
,,]} •
(A12)
Equation (A12) is rewritten as follows:
+ 1)(~rj +
= exp [- (2n,
Li(T, T', t}
~imf) - ~nm2~mq, \nf + 1)(efW fT"
+ e-fWIT ) _
~nm2~mqf
ni(e-lW,T' + eIWIT )
l)(~JfI + ~e'WjT") (~I + ~al e-IWIT) elWjl +11f(~vr + ~Hf e-IWjT') ( ~Jf
+\nl +
+
Jt'
e'W,T)e-IWI1] ,
(A13)
where ~nm'=~n.' -~ma"
Wstm(Wt> W2)
Substituting Eqs. (A2), (A5), (AS), and (A11) into Eq. (AI), W.tm(Wt>W2} is expressed as
fMn",Mma 12
Fi'
=
f" dt f"° dT f'"° dT
x exp [- L(2n, +
I
+
_00
l)(~~a'
+
2
[ r no It I - r ma(T + T,) -.zt(Wt + W2) - tWt(T
. , - T) + if.°T'
i€Ot
i€~Tl
~ + if- - -Fi-]
exp -
~~ml) _ 2: ~nml~maf \nf + 1)(elwIT' + e-IWIT} _ L ~nm,~mqfn, (e-IW,T' + efWfT )
2
I
2
-
I
2
~\nf + 1)(~Jif + ~efWfT') (~+ ~e-fWfT) efwf' + ~n'(~JfI + -1re-IWfT")(~Jff
+ Jtl e'WI T) e- IWII] .
(A14)
Performing the expansition of the exponentials involving exp(iw,t) and exp(-iw,t}, and integrating over t, Eq. (4.14)
is expressed as
Waim (Wt>W 2)=
21M
~ma
12
exp
[
2 2)]" 2: ...
-L(2n,+I}(~t +~2ml
00
I
X
X
eif,2 +
...
(~;;N
:tr2e-IW2TY2 ...
+
'7fN
(~JiN
eIWNT)'N exp [-
+
0
00..
..
L
kt'O k2'O
kN'O 't'O '2,0
\nl + !)kl\n? + l)k2 ••• \nN + 1}kNnftn:2nl!f" d
kt !k 2 ! ... kN 1ll!l21 ... IN!
..
L
TeXp
(.
ZWIT-
2:L'"
i€~T
Fi
-
~JfN e-IWNTrN(~1 + ~e'WIT
L
,N,O
r
)(~ + ~ -IW T)kl
maT
teJf-
v'2
2+
v'2 e
~Jf2 elw2T
~ ~nmf2~mq, \n, + 1) e-'WfT - ~ ~n!/l2Aml!# 11, e'WI T]
r
t
I
(A15)
J. Chem. Phys.• Vol. 74, No.7, 1 April 1981
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Y. Fujimura and S. H. Lin: Two photon absorption processes
3734
Using the binomial and Tayler expansions in Eq. (A15), and integrating over T, the analytical expression for the
transition probability of the simultaneous TPA for a multimode case is evaluated as
W. 1m (Wt>W 2 )=
21M M
nm 4 ma
fi
12 exp [~
(~2
-LJ(2n,+l)
rna;
1=1
2
+l)kln'IJ
L: LJ'" L......
L L'" L. [N
IT en k,.l;.
~2)]" ~
+----1l.!1li..
2
k1=Ok 2=O
I
k N=O/1=O'2=o
'N=O
,
,
I
1=1
(A16)
Ip. Lambropoulos, Phys. Rev. A 9, 1992 (1974).
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Teil A 33, 1546 (1978).
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(1979).
6M. Kawasaki, K. Tsukiyama, M. Kuwana, K. Obi, and I.
Tanaka, Chern. Phys. Lett. 67, 365 (1979).
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IOD. Huppert, J. Jortner, and P. M. Rentzepis, J. Chern.
Phys. 56, 4826 (1972).
liN. Mikami and M. Ito, Chern. Phys. Lett. 31, 472 (1975).
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(1975).
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Lett. 42, 494 (1976).
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16 y . Fujirnura, H. Kono, T. Nakajima, and S. H. Lin (to be
published) .
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3623, 3105 (1977).
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(1979) .
19D• J. Diestler and A. H. Zewail, J. Chern. Phys. 71, 3103
(1979).
20 y . Fujirnura and S. H. Lin, J. Chern. Phys. 71, 3733 (1979),
where the exponential factor exp [- Lf.l (2n; + 1)(~;;2)1 should
be rewritten as exp[ - Lf=l (2n , + 1) ~;].
21y. Fujirnura and S. H. Lin, J. Chern. Phys. 70, 247 (1979).
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(Wiley, New York, 1973), p. 132.
J. Chern. Phys., Vol. 74, No.7, 1 April 1981
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