EXCITING HIGHER ANGULAR MOMENTUM STATES IN ROTATIONAL
WAVE PACKETS
by
Omid Masihzadeh
B.S, University of Colorado at Denver, 2000
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineering
2005
This thesis for the Master of Science
degree by
Omid Masihzadeh
has been approved
by
otfr/zr/or
Date
Mike Radenkovic
Masihzadeh, Omid (M.S., Electrical Engineering)
EXCITING HIGHER ANGULAR MOMENTUM STATES IN ROTATIONAL
WAVE PACKETS
Thesis directed by Assistant Professor Mark Baertschy
ABSTRACT
Rotational wave packets in a gas of linear molecules have been used as opti
cal modulators capable of shaping the phase and spectral content of ultrashort
light pulses. The wave packet is initially formed by a pump pulse that rotation
ally aligns the molecules. The alignment induced in the molecules periodically
dephases and rephases, leading to an ultrafast transient in the index of refrac
tion. If a second, probe, pulse is timed to coincide with particular features of
the periodic revivals of the rotational wave packet one can broaden and/or shift
the spectrum of the probe pulse. The limit to how much the spectrum of the
probe pulse can be manipulated depends upon the temporal width of the revival
features as well as the degree of the rotational alignment itself. By using two ap
propriately spaced pump pulses to form the rotational wave packet it is possible
to achieve the same degree of alignment as from a single, more intense pulse but
without exceeding the dissociation threshold of the molecules. Furthermore, if
m
!
(
the directions of polarization of the two pump pulses are rotated by 45 degrees
it should be possible to excite higher angular momentum states compared with
the case of coincident polarization. This would lead to more rapid fluctuations
in the optical properties of the gas allowing for greater spectral modulation of
the probe pulse. I have developed a set of computer codes that are capable of
modelling two pump pulses, with arbitrary linear polarization rotation, inter
acting with a gas of linear molecules. Initial results from these codes indicate
that this scheme is indeed capable of exciting higher angular momentum states.
Future work will be centered around better understanding and optimizing this
procedure as well as experimentally verifying its feasibility and practical utility.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
IV
DEDICATION
To my dearest friends, Chris and Racheal Flynn.
CONTENTS
Figures ......................................................... viii
1. Introduction..................................................... 1
2. Quantum Mechanics and Molecular Dynamics......................... 8
2.1 Molecular Interaction............................................ 8
2.2 Molecular Dynamics.............................................. 16
2.3 Periodicity in the Wavepacket................................... 18
2.4 Alignment Factor ............................................. 19
2.5 Computational Model........................................... 24
3. Controlling Alignment of Molecules........................... 31
3.1 Optimization of ^ (cos2(6)}................................... 32
3.1.1 Maximizing the Applied Torque on a Rotating Body........... 37
3.2 The Rotation Matrix............................................. 42
3.2.1 Construction of the Rotation Matrix........................ 46
4. Excited Higher Angular Momentum................................. 55
4.1 Spectral Representation of (cos2(0)).......................... 55
4.2 Results......................................................... 58
5. Conclusion...................................................... 68
Appendix
vi
A. Derivation of Kick Strength
B. Derivation of ((cos2{9)))t
References....................
FIGURES
Figure
1.1 Adiabatic Alignment................................................... 2
1.2 Alignment of a Molecule With a Short Laser Pulse...................... 4
2.1 Interaction of a Linear Molecule and External Field.................. 13
2.2 Alignment of an Ensumble of Molecules ............................... 21
2.3 Allowed Transitions ................................................. 27
2.4 Effect of Temperture on the Maximum Value of (cos2(6)) 28
2.5 Effect of Temperture on the (cos2(9)) 29
2.6 Effect of Temperture on the Alignment {cos2(6)} 30
3.1 Phase Modulator...................................................... 34
3.2 Sinusoidal Phase Modulator........................................... 35
3.3 Periodic Index of Refraction......................................... 36
3.4 Temporal Behavior of Index of Refraction as a Nonperiodic Phase
Modulator............................................................ 37
3.5 Structure of the Rotational Revivals................................. 38
3.6 Effect of Different Polarizability: IV2 Molecule vs. CO2 Molecule . 40
3.7 Torque Induced on a Anisotropic Molecule............................. 41
3.8 Vector Model for Rotation of  JM) States............................ 44
viii
1
3.9 Euler Angles.................................................... 45
3.10 Error for J 10.................................................. 49
3.11 Error for J = 30.................................................. 50
3.12 Error for J = 50.................................................. 51
3.13 Error for J = 60.................................................. 52
3.14 Error for J = 70.................................................. 53
3.15 Error for Different J States vs. Degree of Rotation 6.......... 54
4.1 Spectral Representation of (cos2(0)) ........................... 57
4.2 Spectral Representation of (cos2(0)): 10 Rotation................. 60
4.3 Spectral Representation of (cos2(0)): 20 Rotation................. 61
4.4 Spectral Representation of (cos2(0)): 30 Rotation................. 62
4.5 Spectral Representation of (cos2(0)): 40 Rotation................. 63
4.6 Spectral Representation of (cos2(0)): 45 Rotation................. 64
4.7 Spectral Representation of (cos2(0)): 50 Rotation................. 65
4.8 Spectral Representation of (cos2(0)): 60 Rotation................. 66
4.9 Comparing 45 Rotation and 50 Rotation......................... 67
IX
1. Introduction
In this thesis, we report on our investigation in the field of Rotational Re
vivals The aim of this presentation is to develop and further understand the
mechanism of quantum mechanics of molecular alignment, and investigate fur
ther properties of this phenomenon. By doing that, we hope to implement these
results to optimize the effects of induced dipole moment interaction between
electromagnetic field and an anisotropic molecule like CO2 .
Different methods for alignment of molecules have been employed using
techniques like focusing by an electric hexapole field [7], optical pumping [19],
supersonic nozzel expansion and collisions! alignment in the seeded beam [36],
and alignment in a strong dc field [26, 15] The latest one, the alignment in
dc field, has been extensively analyzed theoretically and experimentally. In this
procedure, alignment is due to the mixing of large amount of angular momenta,
coherently, by the electric field. However, this alignment vanishes as the field is
removed ( See Figl.l).
The notion of alignment with an short, strong laser field is a direct con
sequence of alignment with a dc field. When molecules are hit by an intense
laser, they align themselves along the polarization axis of the field. The signif
icance of using a short and strong laser pulse for alignment is due to the fact
1
0.336
time
Figure 1.1: Adiabatic alignment of a molecule: Here the pulse length is long
compared with the rotational period of the molecule. We can see that the
alignment factor follows the field. This arises from the tendency of the molecular
axis spends a more time in the direction of the field.
2
that the alignment takes place not only during the laser interaction, but also
occurs periodically when the field is gone ( See figure 1.2). Basically, at the
time of laser interaction, the laser developes a wavepacket consisting of coherent
superposition of rotational states, the aligned states. After the laser pulse is
gone, the wavepacket dephases and the alignment is gone. After some time Tr
the revival time, the wavepacket rephases itself and the alignment takes place
again. This is a direct consequence of the phenomenon of quantum revivals,
and would not happen if angular momentum was not quantized. Generation of
these states requires an intese and ultrashort laser pulse. Due to wide spectrum
of ultrashort laser pulse, r < picosecond they are essential tool for generation
of these coherent wave packets consisting of many different energy levels [44].
The apprehension that intense laser field of nonresonant frequencies can in
duce controllable alignment [31, 8, 11, 12] has raised a lot of interest during the
last decade. However, due to the complexity of internal molecule structure, our
understanding of dynamics of molecular interaction is less than our comprehen
sion of the atomic one. Therefore, in order to understand the problems involved
with the alignment of molecules, one has to study the quantum mechanics of the
dynamics between the laser field and the molecule. This is the main discussion
in chapter 2.
The interaction between the molecule and radiative field introduces temporal
changes in the index of refraction which have dependence upon the field ampli
3
t/Tr
Figure 1.2: Alignment of a molecule with a short laser pulse : Here we are
using the same system as Figure ( 1.1 ), but this time the laser pulse length is
much smaller compared to the rotational period. Hence, the transitions among
the rotational state occures instantanously and a coherent wave packet gets
created which dephases and rephases itself at some time Tr .
I
4
i
tude, polarizability, and the angle between the field and the molecular axis [25].
For short pulses (short compared to the rotational period of the molecule), tran
sitions among the rotational quantum states occurs instantaneously, in which
case the phase relationship among the final superposed states will be well de
fined. As mentioned before, this superposed quantum states occurs even after
the excitation. This opens up a broad field of research on the effect of these tran
sitions, change of index of refraction and hence the temporal phase, on a pulse.
Production of extremely short pulses and shifting the entire spectrum is a result
of this realization. At present time, there are several methods for generation of
short pulses. The most common one is selfphase modulation (SPM). Here ,one
uses the electronic Kerr nonlinearity to produce SPM that is proportional to
the pulse intesity, and gives rise to a positive chirp on a pulse. However, there
are some limiting factor on this method. Since, SPM process is a thirdorder
nonlinear process, it is very sensetive to variation in the input pulse shape and
could amplify any noise in the input signal. Recently, however, it was demon
trated by Bartels et al [3] the use of the molecular rotational wavepacket for
phase modulation of ultrashort pulses. Here, they demonstrated that rotational
wave packet revivals in a molecule (C02) can be used to phase modulate and
selfcompress single, ultrashort, fight pulse. It has also been shown that one
could shift the entire spectrum of such pulse to create pulses in some region of
Electromagnetic Spectrum that can not be achieved by conventional lasers.
5
The idea would be to exite a rotational wave packet in the gas that would
go through periodic full revivals at some time Tr = h/2B and then introduce a
second pulse, the probe pulse, after excitation of the molecule with the first pulse,
the pump pulse, at a time delay that matches the revivals, and study the changes
that the probe pulse experiences during its interaction with the temporal index
of refraction variation of a revival. This kind of phase modulation is very general
and should allow for selfcompression of light over the entire transparency range
of the molecular gas from IR into the UV region of spectrum [18, 3]. A measure
of changes in the index of refraction ^ja, introducing a frequency shift due to
the temporal phase modulation, of a linear molecule is governed by {cos2(9(t))),
expectation value of the alignment factor cos2(6(t))) which is proportional to
the induced dipole potential [33].
It is the control and optimization of this quantity that will mainly be dis
cussed in chapter 3 and chapter 4. Here, we will introduce a novel idea on
maximizing the torque exerted on the molecule which will increases the angular
momentum of the rotating molecule. This increase in the angular momentum,
will populate or exite the higher J levels, the angualr momentum quantum num
ber, which in its turn will introduce a shift in the angualr frequency. In the first
part of chapter 3, we will be showing how this maximization leads to an en
hanced change of index of the refraction and hence an enhanced spectral shift.
In the second part of chapter 3, we show the algorithm and implimentation of
6
this algorithm. Finally, in chapter 4 we will show our results.
2. Quantum Mechanics and Molecular Dynamics
In this chapter I am going to present the mathematical description and
computation for rotational revivals. The majority of the analysis done in this
chapter minus the computational description could be found in any literture on
quantum physics and is presented here just for an introductory point of view.
Here, first I am going to talk about the interaction between the molecule and
the field, then briefly talk about the dynamics of the molecules upon this inter
action, and finally give a brief description on the computational model that has
been used to implement the theory.
2.1 Molecular Interaction
The time evolution of the wavefunction in quantum mechanics is described
by the timedependent Schrodinger equation [16],
ihV = HV. (2.1)
at
where h is Plancks constant, H is the Hamiltonian operator, and ^(r, 0, , t)
is particles wavefunction in spherical coordinate. The wavefunction is in general
seperable and could be written as,
^(r, 6, , t) = ^>(r, 6, 4>) exp(iEt/h) (2.2)
8
with E being the energy of the state. This gives rise to Schrodingers time
indepent equation,
6,4>) = Â£ii'(r, 6, )
(2.3)
where the total Hamiltonian,
(2.4)
The solution of this eigenvalue problem is a set of wavefunctions {ipi} and
corresponding set of eigenvalues! 2?,}, and is a description of the molecular struc
ture [4]. In general, the Hamiltonian for an unperturb molecule consist of elec
tronic Hamiltonian and nuclear Hamiltonian. The electronic Hamiltonian con
sist of the electronic kinetic energy, Coulomb interaction between the electrons
and the Coulomb interaction between the electrons and the nuclei. The nuclear
Hamiltonian consist of the nuclear kinetic energy and the Coulomb interaction
between the nuclei. Using the BornOppenheimer approximation, one could
seperate the nuclear motion from the electron motion [4]. By doing that one
could write the total wave function as a sum of the electron wave function and
the nuclei wave function, and the Hamiltonian would look like,
Heil>e(r, 6, (j>) = Eei)e(r, 0, )
(2.5)
9
o, 4>) = r, 6, )
(2.6)
where the subscripts stands for electron and nuclei. The kinetic energy
Hamiltonian for an unperturbed molecule, in the groundstate, is [21]
h2 d2
Hmb + Hrot = [
2M'dIP + RdR + 21
(2.7)
where M is the reduced mass of the nuclei, I = MR2 is the principle moment
of inertia at internuclear distance R, and J2 is the angular momentum operator.
In the present work, we are only interested in the mechanism of a purely
rotational excitation. The motivation for this is that the vibrational states are
assumed not to be excited during the interaction of molecules with nonresonant
electric field. Therefore, neglecting the electronic excitation and the rovibronic
coupling the unperturbed Hamiltonian looks like
= Yr (28)
In driving Eq. (2.8) one has made the so called Rigid Rotor approxima
tion. This model is the simplest approximation for molecular rotation and is
10
similar to classical model of energy for a rotating rigid body [10]. The Rigid
Rotor approximation is due to the fact that nonspherical molecules vibrational
displacement is only a small fraction of the equilibrium bond length [4].
Eq. (2.8) assumes that, molecules have quantized energy levels for rotation,
and individual molecules can be considered as free rotors without no interaction
with neighboring molecules. By inserting Eq. (2.8) into Eq. (2.6), and solving
the eigenvalue problem, the rotational eigenvalues are given by
Erot = Bj(j + 1), j = 0,1,2,.... (2.9)
and the eigenfunction are spherical harmonics,
W = e/(2fe(J1 + H)?0 ew^lflcose) (2.10)
where j is the rotational quantum number, m is the magnetic quantum
number, B = 2A}Ri is the rotational constant, and Ra the interatomic distance.
The P are the associated Legendre functions.
Next let us consider perturbing our system with an applied electric field
e(t).
When a molecule is exposed to an external field, the molecule is going to
align in the direction of the field. In general, one could associate the potential
energy of the molecule with the orientation of the dipole in the field, where the
11
dipole has its least potential energy when it is in its equilibrium, i.e when it is
lined up with the filed. The work done to orient a dipole to a specific orientation
is,
r6 2
w= Tdd = fiÂ£{t) (2.11)
JOi
where jl is the molecular dipole moment vector and i{t) the electric field
vector. The response of the dipole moment in each molecule to an applied
electric field can be written as
V Vo + aiJÂ£J + qPijkÂ£jÂ£k +....... (212)
where fi0 is the permanent dipole moment, ocjj is the linear polarizability tensor,
0ijk the first hyperpolarizability tensor, the indices (/, J, K,...) are the principle
coordinate in molecular frame, and the Einstein summation convention assumed
[35].
Then our total perturbed Hamiltonian is
Hrot = ^ + v^e,t) + vi(e,t) (2.13)
where
Vp(d, t) = hqÂ£cos{6) (2.14)
12
s
e /*
Figure 2.1: Interaction betweent a linear molecule and external field: Here e
represents the external electric field, /i the induced dipole moment, 9 the angle
between the molecular axis and the field polarization, Q represents the parallel
polarizability and a is the perpendicular polarizability. The torque r = fi x e
exerted on the molecule rotates the molecule counterclockwise into alignment
with the applied field.
!
and
Vi(9, t) = \E2(a\\cos2(d) + aj_sm2(0)) (215)
Â£
Equations (2.14) and (2.15) are due to the permanent and induced dipole
interaction with the electric field, respectively. Here, 9 is the angle between the
electric polarizability and the molecular axis, and ay and qi are the parallel
and perpendicular polarizability. The origion of Eq. (2.14) is that the torque r
, on an electric dipole with permanent dipole moment vector /Iq in an electric
!
I
j
i
i
i
13
filed is,
T = /iOÂ£sin(0) (216)
which upon integration in Eq. ( 2.11 ) give us Eq. ( 2.14 ). The potential
energy due to induced dipole moment, however, is a two step process. First the
field has to induce the dipole, then interact with it. In the lab frame, we can
represent the field in the space fixed z direction,
Ziab = Ezz (2.17)
and,
l^lab QlabElab
(2.18)
where a/06 is the molecular suceptibility in the lab frame and is given by rotation
of the linear polarizability tensor, in the molecular frame,
f aL 0 0 ^
0 q_l 0
0 0 QT
V )
(2.19)
with a rotation matrix
14
^ cos (f> cos ip sin
sin 4> cos ip cos 8 cos cp sin ip cos
sin
V /
(2.20)
The orientational averaging for bulk of molecules in spherical coardinate
with polarizibility specified by Eq. (2.19),
p2ir /7r r2ir
(au)ij = {(tfj) / d
Jo Jo Jo
where the capital indice indicate the molecular frame and small letters indicate
the lab frame. A is the directional cosines and G(6, (p, ip) is the probability of
finding a molecule in a direction define by Euler angles. Evaluating the integral,
we find an experession for /2
fiiab Ez (a + (a ocj) cos(0) sin(9)cos(
+ Ez (an + (ax a) cos(^) sin(0) sin(0)) fby+ (2.22)
+ Ez (an + (ax aj)) cos2(0)) fiz
Now, using Eq. (2.11), and realizing the field is assumed to be only in the z
direction, we find the Hamiltonian for induced dipole.
15
In this work, we axe interested with molecules with no or negligable per
manent dipole moment. Therefore, the total Hamiltonian for a single linear
molecule exposed to an electric field is,
Hrot = ^e2t(all a)cos2(6) + a]. (2.23)
Next, we are going to solve the eigenvalue problem, Eq. (2.53) for the above
Hamiltonian.
2.2 Molecular Dynamics
As the timeindependent Schrodinger equation is a description of the molec
ular structure, the timedependent Schrodinger equation describes the dynamics
of the molecular structure. The timedependent Scrodinger equation for an iso
lated molecule, under the rigid rotor approximation, exposed to an external
potential field is described by,
= ii\mt)). (2.24)
where the Hamiltonian operator is,
H = {H0 + Hi) = ^ + V(d,t) (2.25)
and
V(9,t) = ~e2[a.&cos2(6(t)) + ax\
(2.26)
16
is the interaction potential with ct a = a a. The time independent
eigenfunctions of H0 ,Ho^j = Ej\pj describe the unperturbed system which
eigenvalues are Ej = BJ(J + 1). The eigenfunctions of a rotating body are
expanded on the well known spherical hermonics basis Yjm(@, 4>) =  JM). These
eigenfunctions have the timedependence, ^j{t) = ipj exp(iEjt) = JM)exp(
iEjt). Because these functions constitute a complete system, we could expand
any arbitary function in terms of them. Hence,
=^C^M"yM)expHÂ£Jt/R] (2.27)
JM
where, are complex expansion coefficients with the superscripts denoting
the initial values. Inserting this equation in the timedependent Shrodinger
equation, we get,
^qJq,Mq _.
\JM)*M~iEJm = ^HiYJCJJ^\JM)eMiEjt/h] (2.28)
JM JM
Multipljdng the above by,  f M') exp [iEyt/h] and integrating, we get,
o.Afo
dCJo i
exp HBjsW (229)
dt
JM
which constitutes a set of linear differential equations, with coupling terms
rising from the perturbation Hi with nonzero offdiagonal terms. Equations
(2.25) and (2.26) tell us that the change of energy of the rotational states is a
function of anisotropy a a, and d(t) the angle between the molecular axis and
17
the electric field polarizability [24]. The polarizability of the field is chosen in
such way that it commutes with J2.
2.3 Periodicity in the Wavepacket
Lets consider time zero of the wave function described by Eq. (2.27) .
OO
l**j,(0)> = Â£ JM) (2.30)
J,M
with, Mo(0)) being the time zero. From Eq. (2.27) one could easily
observe the reason for the rotational revivals. Consider the time Tr = irk/B for
example. Then the exponential part of Eq. (2.27) simplifies to,
EjTr/h=J(J + 1)tt (2.31)
where J(J + 1) is an even integer. Therefore, EjTr/h is an even multiple of
7r and for all J,
exp \iE{J)t/h) = 1 (2.32)
Inserting Tr into Eq. (2.27) we get,
l**o(rr)> = Â£ exp \iE(J)Tr!h} = (2.33)
JM
Y^Cj^\JM) = (2.34)
JM
IVJoMo(O)) (2.35)
18
Therefore, the wavefunction at some t = Tr returns to the initial state,
IVjm(O)). In most literature, the time Tr is refered to the full revival time.Its
worth noticing that this phenomenon is a consequence of angular momentum
being quantized and would not have occured in a classical sense.
2.4 Alignment Factor
As outlined in the previous sections, after a strong nonresonant short pulse
excites the molecules, a periodic rotational wavepacket with period Tr forms in
the medium. The interaction between the field and the molecule is characterized
by the potential energy given by,
V{d,t) = ~Â£2[a^cos2(d(t)) + a_ij (2.36)
z
where,
Â£ = [~E(r) exp[i(a;ot fc0r)] + c.c.]a2 (2.37)
is the timevarying field oscillating at optical center frequency ujq. Here, E(r)
is the complex amplitude of the field and az is the unit vector in the direction
of the polarization of the field, which is the projection axis for the commuting
component of the angular momentum. The square of e ,
e2 = 2\E\2[cos(2u>0t) + 1] (2.38)
19
contains frequencies near zero and twice the optical frequency u>o [5]. How
ever, the molecular orientation can only response to near zero frequencies only.
Hence, one could replace E2 with its average value over many optical cycles.
The physical explaination for that is that the angular response of molecules to
the torque exposed to the molecule at 2ujq is very weak [24]. Therefore, the
mean potential energy (V) is given by,
(V) = \\E\2[aA(cos2(9(t))) + ax]. (2.39)
Here, (cos2(9)) denotes the expectation value of cos2(6) and is computed as,
(cos2 (6)) = (# JM\cos2(9)\% JM) (2.40)
and is a measure of average molecular alignment with respect to the field po
larizability and strength of the induced alignment. Hence, if (cos2(6)) = 1 then
we have a perfect alignment and in the case of (cos2(9)) = 0, the molecules are
perfectly perpendicular to the fields polarization or are said to be antialigned.
The time average (cos2(9))mean over a rivaval period, as well as the maximum
value of (cos2(9))max are both used to characterize the rotational wavepacket.
For small or no excitation of the wavepacket (isotropic medium) (cos2(9))mean =
. After excitation of molecule, each thermal rotational state creates an inde
20
Figure 2.2: The alignment
of the molecules is charac
terized by the value of the
(cos2(9)). When (cos2(6)) =
1, then the molecules are per
fectly alligned with the fields
polarizability and lower poten
tial energy is obtained, and
(cos2(9)) = 0 indicates that
the molecules are perfectly
perpendicular to the field,
hence, higher potential energy
[5],
pendent coherent wave function.
For carrying out the summation on Eq. (2.40), we start with the wave
packet For each initial Jo, Mo, we have,
^ Jo,mo = J2CtMVM)exp(iEjt/H) (2.41)
J,M
Then the expactation value of cos2(0) is,
(*cos2()*)= ]T CjfyMo(J'M'\cos2(0)\Jexp(i(EjEj')t/h)
(2.42)
Where the selection rule for a Ramon pulse allows j' = J, j' = J I 2, J' =
21
J 2 Carrying out the summation over the primes,
(*l cos2(fl)Â¥) = Y, CiM"20i)+
J,M
E P(iM0+ (2.43)
E C^e^Cj^ exp(i(u,,,)t)
J=2,M
Where fuvj = (J+2)[(j+2)+l]J(J+1) = 4J+6, and 0^ = (JMcos2(0) JM)
are the stationary states, 0JJ+2 = (JMcos2(0)J + 2, M) are the transitions
between J to J + 2 and j,j_2 = (JMcos2(0)J 2,M) are the transitions
between the J to J 2 states. As one can see, due to the selection rule and the
fact that, at frequencies well bellow resonant frequency the rotational excitation
is a two photon process, the only allowed transition states are J = 0, 2, 4 .
The matrix elements for cos2 (6) are found in literture and are as follows [32],
(JM\cos2(d)\JM) = + l(
(2J + l)[J(J + 1) 3M2])
(2J + 3)(2J1)(2J+1))
(2.44)
{JM\cos2(9)\J + 2, M)
V(2J + 1)(2J + 5)( J + 1 M))
(2J + 1)(2J + 3)(2J + 5)
V(J + 2 M){J + 1 + M)(J + 2 + M)
(2J + l)(2J + 3)(2J + 5)
(2.45)
22
(2.46)
(JM\cos2(6)\J 2, M)
yj (2J 3)(2J + 1)( J 1 M))
(2J3)(2J 1)(2J+1)
V(JM)(J1 + M)(J + M)
(2J 3)(2J 1)(27 + 1)
Next we consider the effect of temperature of ambient on an ensemble of
molecules. Due to thermal agitation which tend to randomize the orientation of
the molecules, the tendency of the molecule to become aligned with the electric
field is disturbed [5]. Previous studies have shown that due to the rapid increase
of rotational population, the degree of alignment has a rapid decrease with
increasing temperture [32, 28, 27].
The ensemble average of molecular alignment is calculated by thermal av
erage over occupation probability of each initial rotational states, and is given
by [28],
where,
(cos2(9))T = Qr1^Tp{Jo)
Jq0 Mq=Jq
(2.47)
m,) = exp[MJ \ )R/kT\ (2.48)
is the Boltzmann distribution associated with the rotational states, and Qr
is the rotational partition function [9],
23
(2.49)
Qr 'y "(2Jp + 1) exp[Jo(Jo + 1)J5/A:T].
Jo
Basically, each initial rotational state J0 is being redistributed into (2J0 +1)
states with a statistical weight given by the Boltzmann factor [13, 14]. Finally,
one could find the orientation probability density at some time t as,
P(6, ; t) = Qr1 Â£ ItfjcMo (t)l2 exp[Ejt/kT] (2.50)
jo,Mo
where, is defined as Eq. (2.27).
2.5 Computational Model
The computation code for propagation ( solving timedependent Schrodinger
equation ) and calculation of (cos2(8))t, follows closely what we have discussed
so far. The molecule of interest here is C02 gas with ax = 12.87(a.u) and
0 = 27.14(a.it). The molecule, initially assumed to be in a pure  JqMq) state,
at thermal equilibrium with temperture T, will be excited by a 20 fs laser pulse
( pump pulse ), into a coherent superposition of different  JM) states, where the
population of each state are given by the Boltzmann distribution. The electric
field of the laser pump pulse is described as,
Spump(^) = E(t')pumpCos(ut') (2.51)
24
where the field envelope E(t) is given by,
Efypump^le At2
= e
2
(2.52)
with a = 20fs being the FWHM, A = 800nm the optical wave
length, ojpump the carrier frequency, and X = 1013W/cm the intensity of the
envelope. The structure of the code is seperated into two part; Solving the
timedependent Schrodinger equation (TDSE)given by Eq. (2.53), and compu
tation of (cos2(6))t The numerical challenge with the solving the ( TDSE ) had
conviniently been done by Dr. Beartschy ( My Advisor ) for different purposes.
In general for one dimentional TDSE, with a timedependent hamiltonian H(t),
we could write,
p\
ihy{t) = H*(t). (2.53)
at
which a formal solution of ^(t) for any t > 0 could be expressed by,
*(t + At) = Â¥(<) (2.54)
There are several ways to approximate the exponential function. Chebyshev
polynominal expansion has been used in our case. First of all, we are going to
use stair step approximation for H{t). so,
25
V(tk) = V(tki)exp[
iAtH(tki + t)
h
]
(2.55)
Then, we introduce a new matix W j + bl, where I is the identity
matrix. This matrix has a characteristic where its eigenvalues are between 0
and 1 Solving for ^ we get, ^ ^ K Then, the wave function can be
written as,
T.. T r . ,iW At, ,ibAt,
y(tk) = exp[:] exp[]
a
(2.56)
Then, we expand the exp[ tT^At] in terms of Chebyshev polynominal as,
exp[] = W (257)
CL
n=0
where Tn(W) is the nth order chebyshev polynominal. The adventage of this
methode is that it converges very rapidaly, and there exist a recursion formula
for Tn.
For computation of (cos2(6))t refer to Appendix 2.
26
Figure 2.3: Allowed transition at nonresonant frequencies : Rotational exci
tation takes place via twophoton cycles. A system initially in a pure \JqMq)
state will be excited by the field into coherent superposition of different  J' Mq)
states. Selection rule requires J' = J0, Jo 2, J0 4 .
27
Figure 2.4: Effect of temperture on alignment: As temperture goes up, ro
tational population of rotational states increases and the degree of allignment
decreases. Here, we are plotting the maximum value of (cos2(6)) versus kick
strength.
28
t/Tr
Figure 2.5: Effect of temperture on alignment: Although the alignment factor
(cos2(6)) decreases with temperture, the shape of the (cos2(6)) stays almost the
same. Her we are showing (cos2(6)) versus, Note that the revival time does
not change with temperture.
29
t/Tr
Figure 2.6: Effect of temperture on alignment: This figure is a close up of
Figure (2.5). As temperture goes up, average alignment is close to thermal
equilibrium value of . Due to thermal agitation, the wavepacket dephases itslef
more rapidaly.
30
3. Controlling Alignment of Molecules
The ability to align samples of molecules with a laser has been studied in
tensively in the past years and is shown to be crucial for many applications in
ultrafast optics and molecular dynamics (see e.g [37]). It has been shown that
with high intensity nonresonant laser fields, one could yield periodic molecular
alignment along the polarizability of the field [22, 30, 20, 46]. For short strong
laser pulse, it has been shown that one could obtain rotational wave packets
which reshape and reform their initial shape and phase after the interaction
[38, 39, 24, 25, 2]. This fieldfree transient alignment has been shown to be
mostly beneficial in generation of ultrashort light pulses [3, 18], and control
of high harmonics generation [40]. In both cases, the limit and rate to which
one could align the molecules is a crucial factor. Many techniques have been
employed to maximize the amount of alignment. Recently, it has been shown
that there is a limit to the degree of molecular alignment that can be achieved
with only a single laser pulse [23], and more complicated pulse shapes should be
used to achieve a bigger alignment.
Here, I introduce a novel strategy for controlling the shape and dynamics of the
temporal behavior of these alignments by controling the spectral component of
it. Using this, I propose an enhanced shift in the central frequancy. In this
31
chapter, I am going to first, introduce the motivation of such task and then
introduce a novel strategy for a solution to this idea.
3.1 Optimization of ft(cs2(9)).
The generation of short pulses and higher fluence has been amoung the pri
mary directions in the field of optics. Concentration of optical energy in time
has many applications in ultrafast optics and communication. In the begin
ning of the sexties the high power sources of 10100 ns Qswitch laser pulses
became available, generating powers up to 10100 MW. The penetration into
picosecond time scale laser was available by the beginning of the seventies, gen
erating powers up to 10GW. In the early eighties, using the selfmodelocked
dye laser, several groups passed the mark of 100 fs getting into the new era of
femtosecond laser technology. Eventually, by the end of the eighties, 6 fs long
pulses in the visible range was possible. This was the beginning of the realiza
tion of optical methods in studies of ultrafast optics involving phase matching
in nonlinear material, since the transition to femtosecond pulses was accompa
nied with a jump into higher Intensities. Due to the science of nonlinear optics,
we have been able to produce field with temporal duration as low as lfs, and
intesities in order of 1017 1018W/cm?.
The general principle of creating short pulses is basically achieved by modulation
of intensity and phase of a continous optical source. The intensity modulation
i
32
of such source has been extensively studied and achieved up to a nanosecond
range. Generation of shorter pulses though requires manipulation of the phase of
such pulse. In general, the phasematching of various spectral components of the
pulse makes the task of generation of ultrashort pulse possible. The idea would
be, by controlling the temporal behavior of index of refraction of the medium,
controlling the phase of the propagating pulse ( i.e the prob pulse). This con
troll over the temporal phase, not only allows for broadening the spectrum, but
also is a good source for shifting the entire spectrum of pulse into regions not
easily achieved by conventional lasers. Its been shown that rotational revivals
could be a alternative source for this task [17]. Its been shown that due to
the large number of coherently exited levels, wellseperated temporal structure,
magnetude can be controlled through the relative delay of the pump and the
prob pulse. Here, I am going to show how one could use this temporal behavior
to shift the central frequency of a short pulse.
The set up is pictorially demonstrated by Figure (3.1)
An input signal Ein(t) = f(t) exp[i(uot)] with a center frequency u0 will
enter the phase modulator with length L, and timedependent index of refraction
n(t). On the output a phase modulated signal E^it) = Ein exp[will
come out, where (t) is the temporal phase relating the real and imaginary
part of the complex amplitude. This phase, in general, could be complicated
33
L
Figure 3.1: After the Gaussian wave with a constant phase goes through a
phase modulator, its phase gets modified. This modification is a function of
length of the medium L and temporal behavior of index of refraction n(t).
deterministic or random function. For a deterministic sinusoidal phase,i.e,(f>(t) =
A^costhe input field is going to experience a peridic index of refraction
with period of The change of the output signal is shown on Figure (3.2).
In this case, the periodic structure has give rise to sidebands where the distance
between the sidebands and the central frequency is equal to the ujm.
However, if one could adjust an ultrashort pulse in such way that the prob
will experience the linear region of the periodic struction of the index of refrac
tion, then the phase takes a form of 4>{t) = fit ( see Figure(3.3)).
Then, the spectrum of the output signal is not going
to have sidebands anymore, but will be shifted in an amount depending on the
slope of the line and in a direction depending on the sign of the slope. Since,
the slope of line /3 = ^ is dependent on how fast the temporal phase changes,
the maximum value of this quantity would be achieved if ^ is maximized,
which in turn requires faster structure on the sinusoidal index of refraction.
34
Figure 3.2: A sinusoidal phase modulator creates sidebands.
This is the central idea behind my attempt to maximze the shift upon a signal
that goes through an periodic index of refraction, for example, those of the
rotational revivals shown on Figure(3.4). My task would be to by manupulating
the spectral behaviour of such revivals, maximize the slope of the structures seen
by the prob pulse. Figure (3.5) shows a zoomed version of one of the revivals.
35
Periodic Index of Refraction
Figure 3.3: The probe pulse experiences a linear change of index of refraction
in the linear regim of the periodic index of refraction. The more the slope of the
line is, the more of a frequency shift the pulse will experience.
Representing this revival in terms (j){i) = e(t) cos(cujt), we can write,
= wje(t) sin(Qjt) + cos(ojjt) (3.1)
at at
a>je(t)moa; (3.2)
Equation (3.2) tells us that by increasing the angular momentum frequency
u>j related to each angular momentum J, and also the central frequency of the
36
4
Â£ 2
V
A o
+*
'S'
2
4
0 5 10 15 20 25 30 35 40 45
Time Delay, t (ps)
Figure 3.4: Temporal behavior of index of refraction as a nonperiodic phase
modulation.
revivals, we could increase the slope of the structure and consequently increase
the amount of the spectrum shift the prob pulse experiences. Next, we are go
ing to consider how we can increase the angular frequency of a rotating single
molecule, by increasing the torque exposed on it.
3.1.1 Maximizing the Applied Torque on a Rotating Body
Due to the anisotropic property of the molecules, the component of the
induced dipole moment along the molecular axis is bigger than then component
perpendicular to the molecular axis. Therefore, upon an interaction between
electric field and induced dipole moment, a torque
x 10"
37
Figure 3.5: Structure of the rotational revivals, allows for design of smooth
spectral shift and broadening.
t = fix E (3.3)
will be exerted on the molecule. This induced torque, tends to rotate the
molecule into alignment with the applied field. The more torque is exerted on
the molecule, the more rotational energy is being transfered to the molecule. On
way of maximizing the torque is to choose a molecule with a large anisotropy
Aa, since
38
(3.4)
dV(9,t)
T~ dd
d[Â£2[Aacos2(0(Â£)) + ajJ]
dt
= ^AaÂ£2sin(20)
However, anisotropy is a property of the molecule and can not be opti
mized. Another option is to apply a more intense electric field to the molecule.
Here, once again, we are going to be restricted to the physical properties of the
molecule, since there is limit to how much energy one can put into the molecule
before molecules start to ionize. Figure (3.6) shows the maximum alignment
(cos2(9))max as a function of kick strength for two different molecules. One can
see that for both molecules the maximum alignment increases with kick strength
(see Appendix 1 for derivation of kickstrength). Also, the molecule with bigger
anisotropy has a bigger alignment factor than the molecule with less anisotropy.
Here, we introduce a novel thought for optimizing the molecular alignment with
out putting any restriction upon the structure of the molecule; maximizing
sin(20), which has a maximum value at 6 = 45. Therefore, one could max
imize the torque exerted on one molecule by applying the field at an angle of
45 with respect to the molecular axis.
We will implement this strategy in four parts:
39
Maximum Alignment For Co2 and N2 Molecules
Figure 3.6: Maximum allignment for different molecules: Molecules with larger
anisotropy Aa, experience a greater torque and therefore have a larger alignment
factor. Here, we are plotting the maximum allignment versus different kick
strength for two different molecules. At smaller kickstrength, The allignment
for C02 gas with = 14.6[a.it] have a larger value at a specific kick strenght
with respect to N2 molecule with a a = 5.6[a.u], For larger kickstrength though,
the allignment, is almost the same. That is due to the quadratic dependence of
the energy of the field.
40
E
A
0
Figure 3.7: Torque induced on a anisotropic molecule attempting to align the
molecule to the direction of polarization.
1. Apply a first pump pulse to align an ensemble of molecules initially at a
pure state  JM), into a superposition of different states  J' M).
2. Rotate the obseravable plane an amount of 6 = 45, meaning a rotation
of the coefficient calculated in step 1.
3. Apply a second pump pulse to the rotated molecules.
4. Apply a probe pulse to the rotated molecules and calculate the alignment
factor.
41
I
i

3.2 The Rotation Matrix
A general computational problem associated with the transformation prop
erty of angular momentum under rotation is that, upon rotation one can trans
form  JM) into a linear combination of other M' [45]. A mentioned in previous
section, if the polarization of the pulse is chosen in the z direction, the magnetic
quantum number M becomes a good quantum number and does not hybridize.
Mathematically this is shown below:
 J0, M0) ^ ^2 c^ol J Mj) (3.5)
j
which means that, different J states couple but M states dont. However,
upon rotation
J, Mp) > YjMMo\J,M) (3.6)
M
only the different M states couple and the J states dont.
When a rotation operator acts on some eigenstate J, M) of J2, it transforms
 J, M) to a linear combination of other M values. That is,
R(0,M)J,M) = (3.7)
m'
42
where the rotation coefficients are,
Du\u = (JM'\R(cf>AnJM). (3.8)
The rotation coefficient elements D Ju, M are elements of a unitary (2 J +1) x
(2J + 1) rotation matrix R, where the rotation R(<Â£, 9, y) is defined as [45],
R($, 0, = exp(iJz)exp(idJ^)exp(ixJz) (3.9)
Here, the quantum numbers and Jz indicate rotation about the Â£* and
z axis, where the angles 4> and 9 are familiar spherical polar coordinates, and
X measures the angle from the fine of nodes Â£, defined to be the intersection of
the x'rj and xy planes, to the x axis ( see Figure (3.9) ).
Upon substitution of Eq. (3.9) into Eq. (3.8) one gets,
Dm'M = exP(.~i(}>M')dJM>M(e)exP(iXM') (310)
where
dl'M = (JM'\exp(i6JY)\JM). (3.11)
In our case, however, the angles
the elements of the rotation matrix simplifies to dJM, M only. Figure (3.8) shows
43
z
Figure 3.8: A vector model for the rotation of state functions \JM). The
projection is from the Z axis, the observable plane with magnetic quantum
numbers M,to Z by an angle of 6. Here, a single M state will be represented
by a superposition of set of M' states [45].
a vector model for the rotation of state function  JM) from Z to Z by an angle
of 6.
It has been shown that [43] the dJM, M(0) can be expressed as a finite poly
nomial in arguments of the angle () :
= [(/ + M)KJ M)KJ + M')\(J M')!]5
(JMv)\
(i r
)!(J + M + t>)!(v + M M)\v\
(3.12)
XfcosC^j^^^lsinC^)]^^,
A Z
where the summation is over v for which the factorial arguments are non
44
Figure 3.9: Courtesy of MathWorld ( mathworld.wolfram.com )
negative. The dJM, M satisfies a number of symmetry relationship which yield
tremendous simplifications for the construction of the rotation matrix [45]. First
of all, Eq. (3.12) is not going to change under the substitution M M' and
M' * M. Also, for all 9, the replacement of 6 by 6 yields an alternating
change of sign. Moreover, a rotation by negative 6 is the inverse of rotation by
positive 6. However, dJM, are real and elements of a unitary matrix. Therefore,
di',M = dM,M' 0I
dM',M W ~ dM,M' (
Combining all the results above [45],
(3.13)
= (1)" = (1)M .() (314)
M'M'
45
Also, it can be shown that, dJM> M(0) = Vm and dM>MW = (1)J+M Vm>
with 5 being the Kronecker delta function, and for a full J ( Fullintegral J, spins
excluded ), rotation by 0 = 27T is the same as by 6 = 0, which is with agreement
with our physical intuition.
3.2.1 Construction of the Rotation Matrix
Although unexpected, the computation of of Eq. (3.12) was not trivial.
The problems encountered were mostly due to evaluation of the factorial and
summation of big numbers. It turned out that the summation in the Eq. (3.12)
created an underflow upon computation. Therefore, we evaluated our factorial
and summation in LOG space. Rewriting Eq. (3.12) in the form,
(3.15)
V
where
K= (J + M)\(J M)\(J + M')\(J M')\ 5 (3.16)
B = (J M v)\(J + M + u)!(u + M' M)\v\ (3.17)
/ ~\ 2J+M~M Zv f / a\\ZV
D = cos ( ) sin ( ) (3.18)
Letting Pv = ^ and rewriting Eq. (3.15),
46
(3.19)
di '*() =
V
Then, the summation could be evaluated as the sum of the LOGs,
Qv = log K + log D + log B (3.20)
and ,
= (ir'Â£(l)*e (3.21)
V
Despite all the effort, we still experienced some error accumulation during
the computation of the rotation matrix. Figure (3.10 3.14) show the error
accumulation for 6 = 45 for different J. For computation of the error, we have
used the fact that upon rotation of an angle 6 and a sucsesive rotation of ()0
one should get the identity matrix. Therefore letting the ordinate represent the
error, we can write,
Error = R(0)R(0) 1 (3.22)
where I is the Identity matrix.
After some trial and error and educated guess, we realized that the error was
much 0 sensetive. We realized that the error for smaller angles were significantly
47
less than for larger angles. The solution to this problem was rather easy. Since,
successive rotation correspond to multiplication of the appropriate rotation ma
trices, we concluded that an easy solution to our problem would be to rotate a
small angle 6\, and then by squaring the matrix get an angle d2 = 2d\. This
procedure, could of course be used succesively to get the the desired rotation
matrix. For example, to get an rotation of 6 = 45, one could first construct a
rotation matrix for 9 = 5, and then raise the matrix to the power of 36. Hence,
the rotation matricies 045 = (0s)36
48
Figure 3.10: Error for J = 10.
i
49
I
1
I
I
1
!
50
Figure 3.12: Error for J = 50.
51
s
1
53
Figure 3.15: Error due to rotation by 6 for different J states.
54
4. Excited Higher Angular Momentum
In this chapter I am going to present the result of my computation. First,
I am going to show the format of the presentation of the results, then show my
results, and last talk about the future work.
4.1 Spectral Representation of (cos2(0))
In general, one could represent any function in terms of its Fourier series.
Therefore, we could represent (cos2(0))in terms of its spectral component as,
where ao is the average or the dc value of the spectrum, Re represent the
in great detail on Appendix B. Basically, starting with the wave packet ipj0tMo
we have,
(4.1)
J
real part and (ujj) is the angular frequency of the Jth angualr momentum. In
general, (cos2(0)) is a real quantity and therefore could be represented by its real
part only. Figure(4.1) shows the temporal behavior of (cos2(0)) along with its
spectral representation. The derivation of the above expression has been done
tfjbjfo = Â£ exptiBjl/S)
(4.2)
55
Then the expactation value of cos(0)2 is,
Mcos2(0M= E C;y
Using the selection rule for a Raman pulse,
Jmax
>l cos2()V.> = E
Jmax2
E exp(i(Ej EJ+2)t/h)+
J,M
Jmax
E 0 p(i(Ej Ej,)t/n)
J=2,M
Where,
Ej EJ+2 =J(J + 1) (J + 2)[(J + 2) +1]
= 4 J 6
^ Ej2 =J(J + 1 )~(J~ 2)[(J 2) + 1]
= 4 J 2
Introducing variables huij = 4J 4 6,and W, = 4 J 2,by inspection,
see that cjj = Wj_2 Then Eq. (4.4) looks like,
r >)t/H)
(4.3)
(4.4)
(4.5)
(4.6)
3 can
56
(Vcos2()=^ciM2eJ,J+
J,M
2* E exp(i(Wj)()
J,M
(4.7)
which could be written in the form of, a0 + 2Re J2j ajexp(i(cjj)t).
0.8
0.6
0.4
0.2
o
0 0.5 1 1.5 2 2.5 3
Time Delay (ps)
Figure 4.1: Spectral Representation of (cos2(0)).
T11r
57
As discussed in the previous chapter, the goal of our investigation is to by
changing the spectral behavior of the (cos2(0)) change its temporal shape. This,
as explained in the previous chapter, will be achieved by maximizing the torque
on the molecules. The set up has already been discussed on chapter3:
Align the molecules with a first pulse.
Rotate the molecules.
Kick the molecules with a second pulse.
Probe the change in the spectrum and temporal index of refraction.
4.2 Results
Figures (4.3 4.8) show the spectrum and the temporal behavior of the
(cos2(0)). Here, we have used the same molecule as before C02 with polariz
abilities mentioned in chapter 2. It is very clear from these plots that the we
have a maximum shift in the spectrum for rotation at 45. Figure (4.2) shows
that upon 10 rotation, we are exciting the same angular momentum numbers
as the first pulse. The temporal behavior of the (cos2(0)) has slightly changed
though. Figure (4.2) shows that upon 20 rotation, we are still exciting the
same angular momentum as the first pulse. At 30 rotation, we start seeing
58
some changes in the spectrum. Although same quantum number are being ex
cited, the magnitude of excitation has changed. It is not until 45 rotation that
we see a shift on the entire spectrum. We can see that, not only we are exciting
additional angular momentum, we are also seeing a shift on the central frequency
and consequently the whole spectrum. The temporal behavior has also changed
dramatically. A closeup picture on these structures are shown on Figure (4.9).
It is seen in Figure (4.9) that it is not quite clear whether 45 or 50 has the
most shift on the spectrum. At 60 rotation, the spectrum is shifting back to
its original position and the temporal structures axe vanishing again.
The results shown above is from a point of view of the probe pulse at an angle of
80. There is no reason to believe that this is the optimal angle for the polariza
tion of the probe pulse. These figures shown, are just for a sake of showing the
results of my computation and does not conclude in any shape or form, what
the optimal probe polarization or pump polarization are. For further discussion
on these results and future work refer to the conclusion section.
59
Spectrum at 9*10
Fust Pulse Second Pulse
Figure 4.2: Spectral Representation of (cos2(0)): 10 Rotation.
60
Spectrum at 9 20
First Pulse
Second Pulse
Figure 4.3: Spectral Representation of (cos2(0)): 20 Rotation.
61
Spectrum at 6 = 30
First Pulse Second Pulse
Figure 4.4: Spectral Representation of (cos2(0)): 30 Rotation.
62
A
V
First Pulse
Second Pulse
Figure 4.5: Spectral Representation of (cos2(0)): 40 Rotation.
63
First Pulse
Prob Delay
Figure 4.6: Spectral Representation of {cos2(6)): 45 Rotation.
64
Spectrum at 6 50
First Pulse
Second Pulse
Figure 4.7: Spectral Representation of (cos2(0)): 50 Rotation.
65
Spectrum at 6 60
First Pulse
Second Pulse
Prob Delay
Prod Delay
Figure 4.8: Spectral Representation of (cos2(0)): 60 Rotation.
66
Figure 4.9: Comparing 45 Rotation and 50 Rotation.
67
5. Conclusion
As it is clear to the reader, this work is not conclusive. Although the results
are in perfect agreement with the theory, it still remains to see what is the op
timal frequency shift for a specific rotation. In general there are three questions
to be answered for the future work. Firstly, we have to find out whether or
not a 45 rotation is the optimal rotation for exiting higher angular momen
tum. As it was seen on chapter 4, the comparison between the 45 and 50
rotation showed us that it is not clear if we indeed achieve a maximum shift
on the spectrum for the 45 rotation. Secondly, we have to find out the exact
point where the molecules are perfectly aligned. Although due to the thermal
agitation the molecules are never going to be perfectly aligned, there is a point
where the maximum alignmet occurs and it is on this perticular time we have
to introduce the rotation and the second pump pulse. The difficulty of finding
this point is the enormous amount of data to be examined. For each calculation
of (cos2(0)) I had about 4Gigabite worth of data to investigate. That is for a
specific rotation, let us say 50 rotation, and at a specific time, lets say 42ps
after the first pulse. For accuratly understand the behavior of these revivals
one need to approximatly investigate about 500Gigabite of data, and this is not
including the complication with the probe pulse, which is the third aspect to be
68
understood in the future. It is not yet clear to me what the optimal polarization
of the probe pulse is supposed to be for which the most spectral shift is being
introduced. The data I have introduced on chapter 4 is taken for an arbitrary
probe polarization of
for all range of probe
80. In reality, on should look at the data, the alignment,
polarization and choose the one that give the best result.
i
!
I
I
I
j
i
i
I
i
i
I
69
Appendix A. Derivation of Kick Strength
For a nonresonant laser field, the interaction between the laser field and
the induced dipole moment, averaged over fast optical oscillation, is [5]
V(6,t) = E2 [Aa cos2 (0(t)) + a_ij.
(A.l)
Here aj and Aa axe the parallel and perpendicular components of the polar
izability, and E is the envelope of the laser pulse which in our case is a Gaussian.
Here we introduce the dimensionless time, r = y and the interaction strength
e =
(ail Aa)E2{t)I
4n
where I the moment of inertia is,
1 =
2B
K2
(A2)
(A.3)
and B is the rotational constant. We define the kickstrength as
= e{r)dr.
J 00
(A.4)
Using equations (A.2A.4) we get,
(anaA)/0 f At2
^ 32hc?Bneo *
f
(A.5)
70
where I0 is the intensity of the field, h is the planks constant, c is the speed
of fight n is the index of refraction, and eQ is the permitivity of the material. The
integrand is gaussian pulse with A = p^^ji Assuming that the revivals occurs
at r = 27r, and realizing that the revival time Tr = we find r = 47TBet and
consequently, t2 = (j^)2 Introducing the new t in Eq. (A.5) and using the
fact that,
/
eu2du = \[t\
we get,
(A.6)
P =
7T (OT a^)I0TTTp
(A.7)
In 2 16/icneo
where rp = FWHM. Basically, the kickstrength P oc rpI0 represents the energy
of the pulse.
Appendix B. Derivation of ((cos2(8)))T
Starting with the wave packet ipj0,mo which is a description of superposition
of different wavefunction, for each initial Jo, Mo, we have,
tPjoMo = Y, C?mVM) exp(iEjt/h) (B.l)
Then the expactation value of cos(0)2 is,
,coS2(0M = Y, {J' M'\ cos2{S)\J M)Cj^ exp(i(EjEj<)t/ti)
(B.2)
Where the selection rule for a Raman pulse allows j' J,J' = J + 2, J' =
J 2 For a linearly polarized pulse, there is no coupling between the Mstates,
so M = M'. Then, carrying out the summation over the primes,
Jmax
Mcos2(0M = Â£cX"l2e,>J+
J,M
Jmax2
E C^""eJ+2JC^Mexp(i(Â£J&+2)t/)+ (B.3)
J,M
Jmax
E exp(i(Ej Ej2)t/h)
J=2,M
72
Where,
Ej Ej+2 J(J + 1) (J + 2)[(J 2) + 1]
= 4 J 6
(B.4)
 Ej.2 =J(J + 1) (J 2)[(J 2) + 1]
= 4J2
(B.5)
Introducing variables hajj = 4J + 6,and /kjj = 4J 2,by inspection, we can
see that uj = Then Eq. (B.3) looks Hke,
M cos2()V) = Â£ \ci*0f&j,j+
J,M
2Re Â£ Cj^ewC^ exp(iMl)
J,M
(B.6)
Thermal averaged (cos2{6)), namely ((cos2(0)))r would be obtained by,
((cos2(0)))t = ]>^Q 1 exp(E^/fcT) (cos2(0)) (B.7)
Jo Mq=Jq
Where, Q = (2J0+l)exp(Ejo/kT) is the rotational partition function.
Then,
((cos2{6{t))))T = + 2i2e^ajexp(ia;jt) (B.8)
73
Where,
Jo
dj =05, Y1B* icÂ£i+2 E icitfIs +
Jo \ Mo /
lCl! + I+ 2 E (C?i,"2 + C*l5f a)
Mo=l
Ee"Efl*
Jo
M=1
(B.9)
and
Jo
__a0 \ d [ /o*Jo,0 , q \ *~t*Jo,M0/~tJo,Mo 1 i
J WJ,J+2 I
Jo
M0=l
E E +c?:tMcJj%+ (b.io)
M=1 Jo
Jo
O \ //^*Jo,Mo/mJo,Mo I /"iJo,Mo /^Jo,Mo\i
Z 2^ ^J+2,MUJ,M + J+2,MJ,M JJ
M0=l
with, BJo Q 1 exp(EjQ/kT), the probability of occupation of Jth rota
tional level, and 6^, = J YJM cos2(6)Yy Mdfl.
74
REFERENCES
[1] I. Sh. Averbukh and R. Arvieu. Anguar focusing, squeezing, and rainbow
formation in a strongly driven quantum rotor. Physical Review Letters,
87:163601, 2001.
[2] A. D Bandrauk and L. Claveau. Molecular rotational coherence at high
intensities. J. chem. Phys, 93:107, 1989.
[3] R.A. Bartels, Weinacht T.C, N. Wagner, M. Baertschy, C.H. Green, Mur
nane M.M, and Kapeyn H.H. Phase modulation of ultrashort light pulses
using molecular rotational wave packets. Phy. Rev. Lett., 88(1):013903,
2002.
[4] Peter F. Bemath. Spectra of Atoms and molecules. Oxford University
Press, Inc, New York, 1995.
[5] Robert W. Boyd. Nonlinear Optics. Academic Press, San Diego, 2003.
[6] A.D. Buckingham. Adv. Chem. Phys., 12:107, 1967.
[7] Victoria A. Cho and Richard. B. Bernstein. Tight focusing of beams of
polar polyatomic molecules via the electrostatic hexapole lens. J. Phys.
Chem., 95:8129, 1991.
[8] P. Dietrich, L.A. Strickland, M. Laberge, and P.B. Corkum. Molecular
reorientation during dissociative multiphoton ionization. Phys. Rev. A,
47:2305, 1993.
[9] C. M. Dion, A. Ben HajYedder, E. Cances, C. Le Bris, A. Keller, and
O.Atabek. Optimal laser control of orientation: The kicked molecule. Phys
ical Review A, 65:063408, 2002.
75
[10] Claude M. Dion, Arne Keller, Osman Atabek, and Andre D. Bandrauk.
Laserinduced alignment dynamics of hen: Roles of permanent dipole mo
ment and the polarizability. Physical Review A, 59:13821391, 1999.
[11] G.R. Kunmar et al. Phys. Rev. A, 53:3098, 1996.
[12] J.J. Larsen et al. J. Chem. Phys., 111:7774, 1999.
[13] B. Friedrich and D. Herschbach. Alignment and trapping of molecules in
intense laser fields. Physical Review Letters, 74(23) :46234626, 1995.
[14] B. Friedrich and D. Herschbach. Polarization of molecules induced by in
tense nonresonant laser fields. J. Phys. Chem, 99:1568615693, 1995.
[15] B. Friedrich and D.R Herschback. Modification of the minimum degree
algorithm by multiple elimination. Phys. D, 18:153, 1991.
[16] David J Griffiths. Introduction to Quantum Mechanics. Prentice Hall,
NewJersey, 1995.
[17] E. Herz, O.Faucher, B. Lavorel, F. Dalla via, and Chaux. Quantum control
of the groundstate rotational coherence in a linear molecule. Phys. Rev.
A, 61:033816, 1999.
[18] V.P. Kalosha and M. Ivanov. Generation of single dispersion precompen
sated 1fe pulses by shapedpulse optimized highorder stimulated raman
scattering. Phy. Rev. Lett, 88(10): 103901, 2002.
[19] Ziv Kamy, Ron C. Estler, and Richard N. Zare. Tight focusing of beams of
polar polyatomic molecules via the electrostatic. J. Chem Phys., 69:5199
5201, 1978.
[20] G. R. Kumar, P. Gross, C. P. Safvan, F. A. Rajgara, and D. Mathur.
molecular pendular states in intense laser fields. Phys. Rev. A, 53:3096,
1996.
[21] L.D. Landau and E. M. Lifshitz. Quantum Maechanics. Pergamon Press,
Moscow, 1989.
76
[22] J.J. Larsen, H. Sakai, C. P. Safvan, I. WendtLarsen, and H. Stapelfeldt.
Aligning molecules with intense nonresonant fields. J. Chem. Phys.,
111:7774, 1999.
[23] M. Leibscher, I. Sh. Averbukh, and H. Rabitz. Molecular alignment by
trian of short laser pulses. Physical Review Letters, 90:213001, 2003.
[24] C.H. Lin, R.Y. Chian, and J.P. McTague. Birefringence arising from the
reorientation of the polarizability anisotropy of molecules in collisionless
gases. Phy. Rev. A, 13(2) :813, 1976.
[25] C.H. Lin, J. P. Heritage, and T.K. Gustafson. Susceptibility echos in linear
molecular gases. Applied Physics Letters, 19(10) :379400, 1971.
[26] H.J. Loesch and A. Remscheid. Chem. Phys, 93:4779, 1990.
[27] Mette Machholm. Postpulse alignment of molecules robust to thermal av
eraging. J. Chem. Phys., 115:10724, 2001.
[28] Mette Machholm and Niels E. Henriksen. Fieldfree orientation of
molecules. Physical Review Letters, 87(10):193001, 2001.
[29] H.P. Neitzke and R. Terlutter. Molecular alignment in gaseous flow
detected by molecular fragmentation following electron impact double ion
ization. Phys. B: At. Mol. Opt, 25:19311947, 1992.
[30] D. Normand, L. A. Lompre, and C. Cornaggia. Laserinduced molecular
alignment probed by a doublepulse experiment. J. Phys. B, 25:L497,1992.
[31] D. Normand, L.A. Lompre, and C. Cornaggia. J. Phys. B, 25:L497, 1992.
[32] Juan. Ortigoso, Mirta. Rodriguez, Manish. Gupta, and Bretislav. Friedrich.
Time evolution of pendular states created by the interaction of molecular
polarizability with a pulsed nonresonant laser field. Journal of Chemical
Physics, 110(8):38703875, 1999.
[33] Govind P.Agrawal. NonLinear Fiber Optics. Academic Press, New York,
2001.
77
[34] T. Seideman. Revival structure of aligned rotational wave packets. Physical
Review Letter, 83:4971, 1999.
[35] Hideo Sekino and Rodney J. Baxtlett. Molecular hyperpolarizabilities. J.
Chem. Phys., 98:30223037, 1992.
[36] M.P. Sinha, C.D. Caldwell, and R.N. Zare. Chem. Phys, 61:491, 1974.
[37] H. Stapelfeldt and T.Seideman. Colloquium: Aligning molecules with
strong laser pulses. Rev. Mod. Phys., 75:534, 2003.
[38] T.Seideman. Rotational excitation and molecular alignment in intense laser
fields. J. chem. Phys, 103:7887, 1995.
[39] T.Seideman. Manipulated external degree of freedom wih intense light:
laser focusing and rapping of molecules. J. chem. Phys, 106:2881, 1997.
[40] R. Vellota, N. Hay, M. B. Mason, M. Castillejo, and J. P Marangos. High
order harmonic generation in aligned molecules. Phys. Rev. Lett, 87:183901,
2001.
[41] Q.Z. Wang, P.P. Ho, and R.R. Alfano. Degenerate crossphase modulation
for pulse compression and amplification of ultrashort laser pulses, opt. lett.,
15(18):1023, 1990.
[42] A. M. Weiner. Femtosecond pulse shaping using spatial fight modulators.
Review of Scientific Instruments., 71:19291960, 2000.
[43] E. P. Wigner. Groupe Theory. Academic Press, New York, 1959.
[44] J. A. Yeazell and C. R. Stroud Jr. Observation of fractional revivals in the
evolution of a rydberg atomic wave packet. Phys. Rev. A, 43:5153, 1991.
[45] Richard N. Zare. Angular Momentum: Understanding spatial aspects in
chemistry and physics. Wiley interscience, Canada, 1988.
[46] B. A. Zon and B. G. Katsnelson. Nonresonant scattering of intense fight
by a molecule. Sov. Phys. JETP, 69:1166, 1975.
78
