Coherently Enhanced Optical Kerr Effect

Case study in a medium of N2

A thesis submitted in conformity with the

requirements for the degree of Master of Science

Graduate Department of Chemistry

University- of Toronto

@ Copyright by Etienne McCullough 1997

National Library I*I of Canada Bibliothèque nationale du Canada

Acquisitions and Acquisitions et Bibliographie Services services bibliographiques

395 Wellington Street 395. nie Wellington Ottawa ON K I A ON4 OHawa ON K I A ON4 Canada Canada

The author has granted a non- L'auteur a accordé une licence non exclusive licence aIlowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sell reproduire, prêter, distribuer ou copies of this thesis in rnicroform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/6lm, de

reproduction sur papier ou sur format électronique.

The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantid extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation.

Coherently Enhanced Optical Kerr Effect

Case study in a medium of N2

by Etienrie .\lcCiillough

Master of Science

Graduate Depart ment of Chemistry

University of Toronto

1901

Abstract

Colierent ly enhanced optical Kerr effect ( C E 0 KE). introduced here, represents a novel

approach to quantum control of the incles of refraction. CVe present the theoretical de-

veloprnent of CEOKE. as well as a case study of CEOIiE in a medium cornposed of N2

molecules. Various conditions and parameters are investigated to ascertain the range of

control provided by CEOKE in this medium of ?i2. It is found that a substantial control

of the index of refraction is achieved in this medium. A brief survey of various other

schemes producing a variation of the index of refraction is provided.

Acknowledgments

1 would lilie to thank professor Paul Brumer and al1 t h e members of his group. Aliya.

Arjendu. Artiir. Danny. Joe. both Jeffs. Xue-Pei and Zhidang, for their guidance and

ericouragernents. 1 was lucky in meeting some incredible people in Toronto which are now

friencls. Mille mercis a tous. vous allez me manquer beaucoup.

Pour conclure. j'aimerais remercier mes parentso Robert et Pierrette. pour leur amour.

Contents

1 Introduction 1

2 Formalism 3

. . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction to the Formalism 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Review of Classical Optics 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 IndexofRefraction 4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Permittivity 4

. . . . . . . . . . . . . . . . . . . 2.3 Time-dependent Perturbation Expansion 6

. . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Interaction Harniltonian S

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Dipole Moment S

. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 First Principle Calculation 14

. . . . . . . . . . . . . . . . . 2 - 5 1 Evaluation of (9(')) :la(*)) and ( ~ L J ( ~ ) ) 15

. . . . . . . . . . . . . . . . . . . . . 2 . 2 Evaliiation of (j8')) and (PL3)) 18

. . . . . . . . . . . . . . . . . . . . . . . 2.6 Competing Effects, OKE and SFE 21

. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Optical Kerr Effect 21

. . . . . . . . . . . . . . . . 2.6.2 Self Focusing/Defocusing Effect (SFE) 23

. . . . . . . . . . . . . . . . . . . . . . 2.6.3 Magnitude of OKE and SFE 25

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Summary 26

3 Irnplementation 28

. . . . . . . . . . . . . . . . . . . . . . 3.1 Born-Oppenheimer Approximation 2S

. . . . . . . . . . . . . . . . . . . . 3.1.1 Spherical Coordinate Basis Set 29

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . 2 Selection Rules 33

3.3 Evaluation of the Dipole-Dipole Transition Matrix Elements . . . . . . . . 36

. . . . . . . . . . . . . . . . . . . . . . 3.3.1 Transition Dipole Moments 36

. . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Radial Functions Overlap 36

3.3.3 Angiilar Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .J.4 Parameters of CEOKE :3G

4 Results 40

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Simplified Mode1 40

. . . . . . . . . . . . . . . . . . . . . . 4.1.1 Ratio of the Incident Lasers 41

. . . . . . . . . . . . . . . . . 4.1.2 States Involved in the Superposition 4R

4.1.3 Phase of the Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

. . . . . . . . . . . . . . . . . . . . 4.1.4 XIagnitude of the Coefficient c? 50

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - 2 Thermal Distribution 51

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - 3 Near Resotiance 6 i

5 Discussion 71

. . . . . . . . . . . . . . . . . . . . . . . . 5.1 .Amplification without Inversion TI

5.1.1 Resonantly Enhanced Refractii-e Index without Absorption via Atomic

C'ol-ierence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

. . . . . . . . . . . . . . 5.1.2 Elect romagnctically Induced Transparency 81

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Siimmary 83

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Limitations of CEOIiE 54

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Applications 88

6 Conclusion 91

A Feynman Diagrams of Second Order Perturbation Expansion 96

B Feynman Diagrams of Third Order Perturbation Expansion 98

C Radial F'unctions Overlap 105

D Data of n' and n" Produced by CEOKE Near Resonance

List of Tables

2.1 Third-order nonlinear susceptibilities of various materials. . . . . . . . . . . 26

3.1 Transition dipole moments from Stahcl et al. between the ground electronic

state XX: and the other dipole allorvetl electronic States. . . . . . . . . . . 37

2 Ep needed to achieve I ~ ~ 1 ~ = 0 . 2 0 as a frinction of 7. 6 x IO-' sec is the

natural lifetime of the blII, level and 5 x 10-'O sec is the time between

collisions at 1 atm and :3001<. We assume for the next calculations tha t

only one collision is needed to produce a transition to the ground state.

Note that for a constant Ic2I2, a change in 7, due to different experimental

conditions. only implies a modification of Ep. . . . . . . . . . . . . . . . . . 39

4. L Magnitude of the electric field. the intensity and angular frequency of the

pump lascr involved in creating the siipeiposition between lul ?.Ji=O.Ml =O)

and 1-..J2=2.!v12=O). .A 7 of L x 10"-Iz and a negligible detuning from

resonance are assumed for the calculation of these values. . . . . . . . . . . 4.3

4 . Magnitude of the electric field and the associated intensity of the pump

laser needed to achieve the diffcrent values of lczl assuming a of 1 x 10'

Hz and a negligible detiining of the two-photon absorption Ilz,, « y. . . . 51

-1.3 Magnitude of the electric field. the associated intensity and the angular

Frequency of the pump laser involved in creating the superposition between

Iul=OJi) and Iv2=0.J2). A 7 OF 1 x IO9 Hz and a negligible detuning from

a two-photon resonance are assumed for the calculation of t hese values. . . 58

vii

C.1 Radial functions overlap between the v=O to 21 of XE: and v=O to 4 of

C.2 Radial functions overlap between the v=O to 21 of XE: and u=5 to 9 of

C.3 Radial functions overlap between the ~ = 0 to 21 of XCT and u=10 to 14 of

b'lz: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 S

C.4 Radial functions overlap between the v=O to 21 of XE$ and u= 1.j to 19 of

b'l 'P+ 109

C.5 Radial functions overlap between the u=0 to 21 of XE: and u=20 to 2.1 of

bfLS; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

C.6 Radial functions overlap between the v=O to 21 of XE: and u=25 to 2S of

b t l T I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I l l

C1.7 Radial functions overlap between the v=O to 21 of XCg and u=O to 4 of

C'.d Radial functions overlap between the v=O to 21 of XC: and v=5 to Y of

C'.Cl Radial functions overlap between the v=O to 21 of XX: and v=O to 2 of

et'S: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

C.10 Radial functions overlap between the u=O to 21 of XS: and v=O to -4 of

C.11 Radial functions overlap between the v=O to 21 of XS; and v=5 to 9 of

b' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

C.12 Radial functions overlap between the u=O to 21 of XE: and v=lO to 14 of

C.13 Radial functions overlap between the v=O to 21 of XC; and v=l5 to 19 of

C.14 Radial functions overlap between the v=O to 21 of XE: and v=O to 4 of

a . .

Vl l l

C.L.5 Radial functions overlap betwern the v=O to 21 of XE: and v=O to 4 of

o'n, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - 1 2 0

List of Figures

Interference of two lasers at and d 2 interacting with a system in a

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . superpositionstate. 9

Interference via two lasers at ZI and sl interacting with a system in a

superposition state created by the piimp laser a t w,, where A*,, is the

. . . . . . . . . . . . . . . . . . . . cletuning of the two photon absorption. 14

Feynman diagrams representing the creat ion of the wavefunction (a) IQ$)).

. . . . . . . . . ( b ) IU?:)). (c ) I@II(:)). ( d ) IQ??~). ( e ) 19!!12) and ( f ) lik!!L3). 16

. . . . . . Geometry of the interaction of ( a ) transverse KEE and (b) OIiE 21

(a) Feynman cliagram of OKE and ( b ) the corresponding energy-level de-

scription. Tlie laser a t wl experiences a tlifferent index of refraction due to

33 the presence of the second laser at + . . . . . . . . . . . . . . . . . . . . . -- (a) Feynman diagram of SFE and ( b ) t h e corresponding energy-level de-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s c r i p t i o n . . 23

(a) Feynman diagram and ( b ) the corresponding energy-level description

(3) of an off-resonance wavefunction 1 Q -,, ,-,, ,,, ) associated with SFE. . . . . 24

Positive lens created by the variation of the intensity, I , across the beam

. . . . . . . . . . . . . . . . profile in SFE where a positive n? is assumed. 25

Euler angles a, ,@ and y relating the laboratory coordinate frame (X,Y.Z)

. . . . . . . . . . . . . . . . . . with the molecular coordinate frame(s.y.z) 19

Electronic states of i\iz involved in the clipole allowed transitions from the

electronic ground stat.e XE:. . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Variation of the real component of the index of refraction from vacuum.

produced by CEOKE at (- ) ancl d2(- - -), as a function of the

ratio of the incident lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Control of the real part of the index of refraction b - CEOKE with different

y involved in the superposition state. The ground state XS~.lol=O.Ji=O.

M1=O) is in a superposition state with l ~ ~ = O ? . J ~ = 2 , ? ~ 1 ~ = 0 ) ( ) Ivz=l.

J2=2.M?=0) (- - -) lv2=2..J2=2.S12=0) ( - - - - ) I u ~ = ~ . . J ~ = ~ ~ & ~ ~ = O ) ( - .

- -) lv2=-4.J2=2,M2=O) (- - -). . . . . . . . . . . . . . . . . . . . . . . 45

The consequences of using a different rotational quantum number in the

excited state involvecl in the superposition on the control achieved b -

CEOIiE. The ground s t a t e XZ:.Ivi =O..JI =O,MI =O) is in a superposition

with the state Iu2=1..J2=O.?vI2=O ) ( ) and Iv2=1.J2=2.M2=0 ) (-

- - ) . . . . . * . . . . . . . . . . . . . . . . . . . * . . . . . . . . . . . .

Dependence of n' produced by C E 0 L<E on the relative phase of the lasers

"6, + - o1 =* (- . . . . . . ore l = - 2 ) . O ( ) . ( )

The irnaginary part of the index of refraction? nt'. produced by CEOKE

where the relative phase of the lasers OrCr = M8, + 82 - 0, = 7 ( 1- 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t- - -). ( - . . - ) . 49

Control of n' rvith Ic$= 0.20 (--- ). lc2I2= 0.10 (- - -). Ic212= 0.01

(. - - - - ) ? Icr12= 0.001 ( - - - -). T h e state involved in the superposition

are Iv1 =O..Ji=O,Mi=O) and =0..J2=2.M2=O). . . . . . . . . . . . . . . 52

Probability of finding a moIecuIe of Fi2 in a rotational state a t 29SIi. . . . 54

Control of n' by CEOKE with a superposition created between Iv1=O7J1 =O?

M1=O) and lv2=1,.Jz=0,Mz=0) of XE: ( ) where al1 the molecules

of nitrogen are initially in the ground s ta te and with a superposition be-

tween lvl=0..Ji=6,Ml=(-6 to 6) ) and l y=1,J2=6,M2=(-6 to 6)) where

the molecules of N2 are initially in a thermal distribution corresponding to

a temperature of 29SK (- - -). . . . . . . . . . . . . . . . . . . . . . . . 5.5

-4.9 Control of n' by CEOKE where the superposition is created between lul =O.

J1=6.M=(-6 to 6) ) + lu2=0,.J2=8..LI=(-6 to 6) ) (- - -) of X Z i in a

medium composed of Nz molecules initially in a thermal distribution of

states corresponcling to a teniperatiire of 29SK. For cornparison purposes.

the previous result involving the superposition between lul =O.J = O . M I =O)

and lu2=0..J2='2.h12=0) of XE: with a medium initially in a pure state. is

shown as citr\*e ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-1.10 Creation of the superposition ( a ) I~etwcen 1 v l = O , J =O, kll=O) -r (v2=0.

. . . . .J2=2.M2=O) ( b ) between (vl=0..Il=2..\ll=O) + Iu2=0..J2=0,hIZ=0).

4.11 Control of n' by CEOKE where t h e superposition is created between the

states jvl =O. J I =O.M1=O) and /y =0..J2=2.-Y12=0). The curve ( reP-

resents the use of a n initially pure state corresponding to a thermal distri-

bution of 01;. while the curve ( - - -) shows the result of using a medium

having an initial thermal distribution of the states corresponding to a tem-

perature of 2981i. The lower curve ( - ----•-) represents a fictitious medium

with the parameters of curve (- - -) but where the contribution from

. . the superposition lul =0..J1=2.X11 =O) ancl Iu2=0,J2=0.M2=O) is omitted.

-1.12 Interference of two lasers with angular frequencies wl and w2 interacting

with a svstern in a superposition statc. The two angular frequencies are

. . . . . . . . . . . . . . . . . . . . . . . . . . clet uned A,, from resonance.

-1.113 Real (- ) and imaginary (- - -) part of the index of refraction as

. . . . a function of the detuning il,, = RSL - LC'I of the angular frequency.

-4.14 nt- 1 (- ) and nu (- - -) where II ' and nu are respectively the real

and imaginary part of the index of refraction near resonance due to CEOKE. 65

4.15 71'- l(- ) and nu(- - -) as a function of the detuning. T h e negative

ni' implies that the electromagnetic wave a t wl could be amplified. The

propagation of the electromagnetic wave would have to be investigated

before we could say how much amplification is possible. . . . . . . . . . . . 67

xii

-1. L6 The vah~e of nt as a function of A,, ancl Brel where E2/ El is set to 1000.

nt= IO(- - - ) . 5 ( - - -1. l(- ) and 0.1(- -). . . . . . . . .

4-17 The value of n" as a function of A,, and Orcl where E2/ El is set to 1000.

Three le\-el configuration used in .-\KI. The quantum interference of the

atomic wavefunction can createcl a t rap state which inhibits any transfer

to the state Ion). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The field E experiences -4WI or. iinder certain conditions. a strong n' wit h-

out the normal accompanying absorption. L+, is the angular frequency. in

the microwave range. of the piimp laser which creates the superposition

between loi) and 14*). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Modification of n' and n" of a n electroniagnetic wave resonant between a

. . . . . superposition state. cl ldi)+c?loî). ancl the Iiigher lying s ta te Id3).

Modification of n' and n" due to the presence of a trapped state as a

function of the detuning A,, . Tlic cim-es (-- ) and (- - -) represent

n' and n" respcctivelv with a 02i = 3.0. C'urves ( - - - - ) and (- - - -) shows

the reduction of n' and n" when the superposition approaches the perfect

t rapped state. For the caiculation of t lie curves (. . .) and ( - - -). a

of :3.1 is assumecl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Creation of the superposition of different M states by a scheme involving

a two photon process. using circularly polarized light, represented in (a) is

not correct since the real nature of the electric fields requires the presence

of a counter superposition seen in (b) . As a result, the superposition used

in the calculations of Fig. 5.4 can not be created by this scheme if both M

. . . . . . . . . . . . . states Ipi) and 14*) are initially equally populated.

Four components of the left (a) and right ( b ) circularly polarized electro-

magnetic field, that are responsible for the creation of t h e superposition. .

... X l l l

The equal distribution of the population between the two states 14*) and

192) diminates the control of \ (&?) and ultimately of n(w ) via this quant uni

interference scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1

Modification of n1 and n" of a n electromagnetic wave resonant between

states 1 0 ~ ) and 142) via the quantum interference effects of the different

pathrvays due to the strong coupling between level Id3) and I Q 2 ) . . . . . .

( a ) Gain from a Raman laser invoived in AWI. Normally the inverse process

( b ) is the main loss Factor. In AWI. gain by stimulated Raman scattering

turns out to be possible even i f SI >N2+X3 due to the quantum interference

of the different pathwqs. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

( a ) First contribution to the absorption of a photon of the Raman laser.(b)

Another contribution to the absorption of the photon of t h e Raman laser

. . that can interfere destructively with the first contribution shown in (a).

The different interfering pathways should. as in the resonant case. allow us

to create an interfering effect that cancels the absorption of the laser at the

transition Io1) - 1&) thus allowing us ta have gain in a non inverted system. 53

.A schernatic of t h e coherence length of C'EOIïE (a ) when t h e interference

is created between t.wo states with clifferent vibrational levels V I # 4 is

shorter than the coherence lengtli of CEOKE (b ) when the interference is

assumed between two different rotational quantum numbers .JI #d2 biit

having the same ul = uz. . . . . . . . . . . . . . . . . . . . . . . . . . . .

The incoming signal can be filtered and redirected to the appropriate de-

tectors by a hologram. thus essentially forming a massively parallel optical

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . switch.

Transmittance functions of a binary amplitude grating (a), its phase version

( b ) and a bleached grey grating ( c ) . . . . . . . . . . . . . . . . . . . . . . . 89

Simplified example of a kinoform whicli is basically a phaser version of a

. . . . . . . . . . . . . . . . . . . . . . . . . . binary arnplit~ide hologram. 90

xiv

D. 1 T h e value of n' as a function of A,, and Brel where E2/ El is extremely small.

making the contribution of the interference terms negligible. nr=1.5(- - - .

.. . . . . . . . . . . . . . . . . . . . . . -)4 )and0.85(-• - - ) . 123

D.2 T h e value of n' as a function of Ad, and Brel where E2/Ei is set to 1.

nt=1..5(- - - S . . . . . . . . . . . . . . . . - - ) , l ( ) and O.S.?(- -). I I4

D.3 The value of n' as a hnction of A,, and Orei where EÎ/Ei is set to 2.5.

n1=1.5(- - - - -).l( ) and O.S.?(. . . S . -). . . . . . . . . . . . . . 125

D.4 The value of n' as a function of A,, and O,,r where E2/E1 is set to 10.

. . . . . n1=13(- + - . -),l( ).O.S5(-. -. - - ) and 0.1(- - ). 126

D.3 The wlue of n' as a function of A,! and OrEl where E 2 / E 1 is set to LOO.

nt=,j(. ...... ). 1.q- - - - - ) J ( ).O.S5(.- - - - ) a n d 0.1(--).127

D.6 The value of n" as a funct ion of A,, ancl orel where E2 / El is extremely small

to make the contribution of the i nterference terms negligible. d ' = O . 1 (---• )

a n d O . O l ( - - - - - ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

D.7 T h e value of n" as a function of A,, and Orel where E 2 / E l is set to 1.

n"=O.l(....... ). 0.01(- - - --).O( ) and -0.01(. . . . . a). . . . . . . 129

D.8 T h e value of n" as a function of A,, and Orel where E2/ El is set to 2.5.

1 (. ...... ). 0.01(- - - - -).O( ), -O.Ol(- - - - - - ) and -O.l(-

-). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

D.9 The value of n" as a function of A,, and 8,,1 where E2/E1 is set to 10.

n"=l(- - -),0.1(*--)7 O.Ol(- - - . -),O( ), -0.01(. - . - . -).

. . . . . . . . . . . . . . . . . . . . . . -0.1 (- --) and -1(- - -- ).. 131

D.10 T h e value of n" as a function of A,, and Orcl where E 2 / E l is set to 100.

n"=l(- - -): O.l(-.-...), O( ). -O.l(- -) and -1(- - --). 132

Chapter 1

Introduction

Quantum mechanics gave birtli t.o a numher of riew concepts and ideas which are being

used increasingly to create novel mediums wi t h original propert ies. The superposition

state is one of these fundamental ideas that represents a new state of mat ter with prop-

erties mhich have no analogue in classical physics. As a result. some aut hors have labeled

siich a medium, composed of an ensemble of atoms in a well defined state of superposition.

a "p haseonium" [l].

Another concept hrought about by quaiititni mechanics is the constructive and de-

st riictive interference seen in wavefiinctions which are created by the use of simultaneous

independent pathways to the same final state. P. Brumer and M. Shapiro have since

the micl-1980s been working on the control of photodissociation products of molecules

tlirough the use of interfering pathways to iiltirnately control the cornplex outcome of

cliemical reactions. Tliey have called tliis novel concept Coherent Cont,rol[i?]. In t his the-

sis. we forther these concepts by trying to control the electrical properties of atoms and

molecules using coherent lasers. Specifically. we atternpt to control the electrical suscep-

tibility of a medium to increase the normal index of refraction that would be perceived

by an electromagnetic wave. Using two coherent lasers incident on a medium composed

of molecules in a superposition state, we are able to modify, through t h e constructive

and tlest ructive interference of the wavefunct ions descri bing the atoms, the propagation

of one of the lasers. LVe name this concept the coherently enhanced optical Kerr effect

(C'EOKE).

The organization of the thesis is as follows. As an introduction. we consider the

development of the general equations of CEOKE in Chapter 2. üsing time-dependent

perturbation theory. we calciilate the susceptibility of a general medium and s t u d - the

effecr s of the coherence between the states involred in the superposition. We also esamine.

in tliis chapter? ot her effects which rnodik tlic real part of the index of relraction (n t ) .

Narr-iely. ive invest igate the magnitude of the controi produced by the optical Kerr effect

(OIiE) and the self focusing effect (SFE).

In Chapter 3 , we provide the da ta necessary for the implementation of this theory in

a realistic medium cornposed of N2 molecules. brief overview of t he different approx-

imations used in this case study is given with the selection rules needed to evaluate the

varioris terms elaborated in the equations of C'liapter 2.

C'hapter 4 presents the results of CEOKE. The chapter is divided into three distinct

sections. The first section shows the results of implementing CEOKE far from resonance

in a medium initially in a single pure state. To make the medium more realistic, the

second section incorporates a thermal distribution of the initial eigenstates of N2. Finally,

we stiidy, in the last section. the consecluences of having the photon energy. of the two

coherent sources. close to a resonance.

\Ve introduce in Chapter 5 trvo recent developments which deal with the control of

t he indes of refract ion. Namely: we disciiss resonantly enhanced refract ive index wi t hout

absorption via atomic coherence and electrornagnetically induced transparency. .Mer

comparing CEOKE with these last theories. we introduce the limitations associated with

CEOIiE. This is followed by an overview of the advantages of CEOKE and some possible

applications involving the control of the propagation of electromagnetic waves.

Finally, a brief conclusion is given in Chapter 6, followed by a number of appendices

grouped at the enci of the thesis to complement the information given in some of the

previous chapters.

Chapter 2

Forrnalism

2.1 Introduction to the Formalism

The purpose of this cliapter is to theore t ica l i~ rlcrive the coherently enhanced optical Kerr

effect (CEOKE). 'Ilter a brief review of classical opt ics, the time-dependent perturbation

espansion, used for the quantum mechanical t rcatment of the medium, is introduced.

The different aspects of this new theory are then explained by calculating the inter-

ference terms produced by tivo lasers interacting with a medium composed of' elements in

a superposition state or mixture of suc11 states. Finaliy the theory is expanded to take

into account the preparation of the superposition. Specifically. rve introduce a thircl laser

into the equations to produce CEOKE on an initially unperturbed system.

2.2 Review of Classical Optics

The clerivation of C'EOKE requires a numher of concepts from classical optics and from

quantum mechanics. As an introduction. ive givc a brief overview of the concept of Light-

mat ter interaction.

2.2.1 Index ofRefraction

From classical optics. the index of refraction is clefined as a measure of the speed of an

electromagnetic wave in a medium ( v ) with respect to its speed in vacuum (c).

where Ir is the permeability (not to be confiised with the dipole moment) and c is the

permittivity of t h e material. The subscript O is to denote the properties of free space.

For most materials p z IL,; making the inclex of refraction a function of the permittivity

on l c The next section introduces the definitions needed to calculate e of a material.

2.2.2 Permitt ivity

The permittivity is a rneasure of t he response of a material to an applied electric field.

4 4 -. D = C E = c,crE (2.2)

4

where D is t h e displacement vector and ç, is the relative permittivity.

This equation can be expanded in the moltipolar component formulation [3]

wlwre P is t h e dipole mornent and V' - Q is t,lie qitadrupole moment. \Ve Focus only on the

first two terms becaiise the contribution from the Iiigher terms like the quadrupole moment

arc only observed at very high energy fields or very high frequencies. The polarization of

a nietliurn? P , is defined by the surn of the individual dipole moments per unit volumeo

Assurning a uniform density of p, the polarization is written as

cvhere we assumed a medium of con

4

P = pj i

ant density p. The pol arizat ion in the previous

equation can be decomposed into a permanent component and a transitory response due

4

to an applied electric field. To simplify the rest of the calculations we assume a medium

wit h no permanent polarization. The polarizat ion due to the transitory response is t hen

characterized by the susceptibility ( ,y)

:\[ter taking t lie Fourier t ransform. the susceptibility is defined in the frequency do-

main as

P(F. uj = c O [ \ f ( d ) + i Iu (w) lE(+) .

Tliese ecjuations can also be extendecl to take into account the nonlinear response of

t lie medium by e s panding the susceptibili ty as[4]

This expansion provides the basis for calcitlat ing the index of refraction involving the

miiltiple fields necessary to develop CEOKE.

By approximat ing the displacement ~vector as

ancl iising Eq. 2.1 and Eq. 1.2 we can write the index of refraction sirnply as

(2.10)

Xote that. since \ is complex, the index of refraction defined so far has a real and imag-

inary component. The real part is related io the speed of propagation of the wave front in

t h e iiiedium while the imaginary part describes the modification of the amplitude of the

elect romagnetic wave as it propagates t lirough the medium. After some macipulat ion[5]?

the real part of the index of refraction can be written as

and the imagina- component as

with s ign(xM) being the sign of but with sign(yM)=l for y"=O.

The formulation of CEOKE assumes the usual semiclassical formulation of nonlinear

op tics and laser physics w here the fields are described as classical electromagnet ic rvaves

ancl tvhere the interaction with the medium is described by quantum mechanics. Note -.

that the component E is cornplex. allorving us to introduce a phase in the calculations.

I f there are ntimerous fields E, incident on the system then the total field is

.-\ straightforward application of Eq. 2 3 and Eq. 2.7 gives

where we assumed y to be a tensor of rank 2. Thus, by evaluating the expectation value

of t h e dipole operator.

(p ) = w ) w w ) (2.16)

ive can calculate the susceptibility and tilt imately the index of refraction experienced

hy a n electromagnetic field passing through the medium. The wavefunction, 1 @ ( t ) ) , is

calculated using the t ime-dependent perturbation expansion described in the following

section.

2.3 Time-dependent Perturbation Expansion

Following Dirac's approach, we solve the Schrdinger equation

for the wavefunction using time-dependent perturbation theory.

The Harniltonian of an isolated system interacting with an electromagnetic wave is

given by,

H = H~ + ilr*,., + H'

wliere fîo. firad and fi' are the Harniltonian of t h e isolated system composing the medium.

of the radiation field. and of t lie interaction hetween the system and the radiation field-

respectively. Since the electromagnetic wave is considered classicaily. rve can eliminate

H r q d from the total Harniltonian.

The eigenfunctions and the eigenenergies of the isolated system are assumed to be

tvhere t h e eigenfunctions lm) for m=O. 1.T.. are orthonormal and form a complete basis

set. The wavefunction that we seek. 4(F. 1 ) . cari then be expanded in this basis set.

I-ysing this Last equation. we can write Eq. 2-17' as.

which can be simplified. by ~ising Eq. 2.19. t.o

hlult iplying by the bra (ml and remembering t liat the eigenfunctions are ort honormal,

Eq. 2.22 becomes

This differential ecluat ion can be solvetl iterat ively. .Assuming an initial condition of

cl(!) = 1. the first few iterations give

The wavefunction can then be reconstructeci from the coefficients c e ) and the eigen-

functions lm)

where the superscript refers to the level of tlie iterative process. If the interaction H f

is wrnk enough. tlicn ive can expect the expansion to converge and we can truncate the

scries at the desircd level of accirracy.

2.3.1 Interaction Harnilt onian

Performing the intcgration over time of the i terat ive process requires the definition of the

interaction Hamiltonian. ..\ssurning the usual electric dipole approximation. the interac-

tion Hamiltonian is [il 4 4 $1' = - p . 5 (2.26)

4 - where I; is the electric dipole operator and E 1s clefined in Eq. 2.13. The evolution of t he

coefficients c p ) is tlien given by

w h c r r /r' is the projection of j? dong the \-ector of polarization Ë of the electric field

witli nngular freqiicncy wl O,, is eqiial to ( En - Ei)/h and c.c. stands for the complex

conjugate.

The next section presents the theoretical cleveloprnent of CEOKE using the time-

dependent perturbation theory and the above clefinit ions from classical op tics.

2.4 Dipole Moment

To develop CEOIiE consider the interference of two incident lasers interacting with a

system in a superposition state. Figure 2.1 clisplays the two different pathways to the

same final virtual state that allow us to control the properties of this final state. In

our case, this resuits in a modification of the propagation of the incident lasers. To

introduce the different concepts behind tliis new effect, ive postpone the discussion of the

superposition statc preparation to section 2.5 and assume a medium with elernents in an

initialIy prepared superposition state. Specifically. the initial wavefunction is

Figure 2.1: Interference of t ~ o lasers at and d2 interacting with a system in a super-

position state.

l'sing first order perturbation theorj-. the wavefunction, in the presence of the fields.

is

wliicli leads to the following equation for the a\-erage value of the dipole moment'

Assuming a aeali interaction, we c m neglect the higher order term Ic!,!)12(q5,1~10,).

Furthermore, as we have stated earlier. ive assume a medium with no permanent dipole

moment, that is' (cD11P1(b1) and (#21$142) are equal to zero. Since we focus on N2: we set

(01 1p1@~) = (g21jllpl) = O because the dipole transition is forbidden when Idl) and 142)

are in the same electronic state (section 3.2).

C'onsequently. w e are left with

where pl, = (oi 1/710~) is the transition-clipole matrix element. Evaluating Eq. 2-27

assiiniing two incident C W lasers of angular frequency wl and Q gives

where the perturbation is assumed turneci on nt to + -m at which time the system is

in i ts initial superposition state[Eq. 2-28]. The iisual factor 7 has been added to force

the integrand of Ecl. 2.27 to go to zero as we take the limit of to -t -S. This factor

&O nlodels the finite lifetime or linewidth of the transitions [SI. For clarity in the rest of

t h e calculation. we eliminate the superscript ( O ) of the coefficients cl and cl , since al1 the

terms are now a function of the initial conditions. Substituting cm)(t) into Eq. 2.31 and

expanding the result gives

wliere the rn are sumrned over al1 the dipole al lowd transition. In this case. the y can be

clropped since ive assume an off-resonance transition, which implies that Rmi f Y; > y.

wliere i and j are 1 or 2 .

T h e first sixteen terms of Eq. 2.33 are the intluced di pole moments normally encoun-

teret1 in a system interacting wi t h lasers O C atigiilar Frequencies and W. The last sisteen

terms are a consecluence of the interference etfect due to the fact tha t the initial s ta te is

a siiperposition. Of the latter sixteen ternis. the last eight are "satellite terms". Tha t is.

they do not affect t h e observable dipole moment at the desired anguiar frequency ic.1 or

wz. Using the fact tha t wl - d 2 = [see Fig. 2.11, we can write the remaining eight

interference terms explicitly with an oscillating term a t frequency wl and LQ. Note that

a loss of coherence between the two initial states 141) and 142) due to collisions. sponta-

neous emission or other dephasing effects would cause the reduction in magnitude o r the

disappearance of the interference terms of Eq. 2.33.

Separating the average value of the inducecl dipole moment into it's various frequency

components, we are left with

and

Cysing Eq. 2.18 we finally get the sosceptibility y(wl ) and k(u2) of the medium:

The index of refraction is then given by Eq. 2.11. Assuming a negligible y"? however,

we can write the real part of the index of refraction a s a function of the real part of the

suscepti bility

which can further be simplified if we are in a regirne where y f ( w ) < L I

Thus. the difference between the speed of the ivavefront in the medium and the speed in

vacuum is given. as a first approximation. by

Looking back nt Eq. 2.36 and Eq. 2-37 riTe are now capable

between the creation of the interference terms and the control of

The first four ternis of Eq. 2.36 and Eq. X37 give the ordinary

(2.40)

of seeing a direct link

the index of refraction.

index of refract.ion. In

an off-resonance scenario, the contribution to the index of refraction [rom t hese terms is

relatively smail and can almost be considerecl as a constant independent of Y.. Control

is thus related to tlie modification of the last two terms of Eq. 2.36 and Eq. 2.37. The

paranieters that allow us to modify the index of refraction are then the magnitude of the

coefficients Icl/lczl. their phase. the ratio of tlie incident lasers' field E i / E j . their relative

phase and the choice of the states 1 & ) and Ioz) involved in the superposition. It should

be noted that the ratio E2/E1 for x(ul) is inverted relative to that of \(w)[Eq. 2-37].

Thus. the control of n(wl) through this parameter occurs to the detriment of the control

of n ( 4 . The control's dependence on these parameters is studied in detail in Chapter 4.

T h e previous equations relied on an initially prepared superposition state a t time to.

\Vc cdescribe CEOIiE in the next section h m first principles by applying a third laser to

produce the desired superposition state frorn an initially unperturbed system. In order to

simplify the calculrttions involving the three lasers, we introduce in the next section, the

Feynman diagrams and some notations used in nonlinear optics.

2.5 First Principle Calculation

Herc we consider the creation of the superposition state from an initially unperturbed

state. This superposition could be created bu a number of different schemes. However.

since CEOIiE is a Function OF the coefficient tic? and not of the process that creates the

superposition. rve chose to focus on only one sclieme involving a two photon absorption

from a pump laser E(L+,) [see Fig. 2-21.

Figure 2.2: Interference via two lasers at and d 2 interacting with a system in a su-

perposition state created by the purnp laser at +. where A2, is the detuning of the two

phot.on absorption.

In order to accoiint for the interaction of the tliree lasers. ive extend the perturbation

expansion of the wavelunctions and calculate t lie higher order terms of the espectation

value of the dipolc moment.

In the next section, we evaluate the different wavefunctions

(2.41)

that are necessary to

calculate Eq. 2.4 1.

2 -5 -1 Evaluat ion of 19(')), 1 Q(*)) and 1

Assume that the niolecule is initially in the groiind state,

The contribution from first order perturbation using three CW lasers of angular frequency

q.4 and q, is

where we used Eq. 2-27 as before. The coefficients cg) are no longer present in Eq. 2.4-4

becaiise we startecl from a single pure state c g ) = hm,l. The 7 are also neglected since

we assume that none of the transitions are on-resonance with another eigenstate. Due t o

t h e number of terms involved in the nest calculations, we introduced in Eq. 2.43 a new

notat ion t hat speci fies the lasers involvecl in creating the perturbed wavefunction. The

(1) negative frequencies. e.g. 1 Q-,, ) are associated tri th the absorption of a photon while the

posi t ive frequencies. e.g. 1 @cl)), represent an eniission of a photon. Figure 2.3 explains

this notation by sclieniatically representing via Feynman diagrams the interaction of the

incident lasers witli the ground s ta te wavefunction.

il Feynman diagram is extrernely useful in describing the evolution of the wavefunction

as a function of time. A t time t o , the wavefunction of Fig. 2.3 ( a ) is in the initial ground

s t a t e 161). The interaction of t h e photon is depicted by a break in the initial straight line

a t t l a t which t ime the wavefunction transforms into the eigenstate lm). An oscillating

arrow pointing towards the break depicts the absorption of a photon while an oscillating

arrow pointing away from the break represents the emission of a photon. This notation

becomes useful when representing a nurnber of different events, as will be seen shortly

ivlien me espand the perturbation theory to take into account the second and third order

terms.

The second ortler perturbation expansior, of the wavefunction can be ivritten down as

where the coefficients cm) are

results in 36 terms which are

A number of these terms are

the result of the second iteration of Eq.

depicted in appenclix .A with the aid of

significantly bigger due to a resonance

2.24. This iteration

Feynman diagrams.

with an eigenstate.

The most interesting term is the two-photon absorption, since it creates the superposition

that is going to be needed to prodoce CEOIiE.

The terms

rvhere a= 1.2 or p. are also needed in the calculations. However, these terms contribute

to ot her effects than CEOKE and are described in greater detail in section 2.6.

Another important term represents a stimulated Raman transition induced by and

-. .-1s in the two-photon absorption. the final s tate is

Tlie contribution of this term is however negligible when compared to the two-photon

absorption because El and E2 are relatively w a k lasers when compared to the pump

laser E p . t hat is. 1 Ep 1 1 Epl > 1 Ei I I E2 1. -411 the other second order terms are negligible

sincc they do not possess a resonance with an eigenstate. Note that a limit. imposed by

the lise of the perturbation regime[9], restricts the power of the pump laser so that the

coefficient lczI2 5 0.20.

Taking the norm of Eq. 2.46.

ive can write the restriction on cz as

where A2up = - 2 4 , is the detuning of the two-photon absorption from resonance.

Assuming a negligible AZWp: we are left witli a restriction on only two parameters.

Tlierefore, knowing the y. we can maximize the coefficient Ic2 l 2 by applying the appropri-

ate field Ep. Table :3.2. found in the nest chapter, presents the magnitude of Ep needed

for various y to achieve I ~ ~ 1 ~ = 0 . 2 .

The third order perturbation contributes a niirnber of terms of interest. Since we are

dealing with six different frequencies f w l . &kt2 and f w p producing three events, we have

to consider 63 = 216 terms. Appendix B graphically represents the 216 terms. As with the

second order terms. we only consider terms with a resonance, which limits the 216 terms

to 5 - k . By discarding the --satellite terms*. ivhich do not possess an angular frequency of

dl or d 2 . ive furt. her reduce the number of ternis to consider to 30.

One of the most important terms encounteretl in the third order process is responsible

for part of the interference terms seen in Eq. 2-36

The other third order terms that contribute significantly to (p(3)(~1)) and ( p ( 3 ) ( ( 2 1 ) } are

i4;it.h the relevant wavefunctions defined. we can now calculate t he expectation value

of the dipole moment [Eq. 2.411.

2.5.2 Evaluation of ($I l) and (p("))

As was discussed earlier. (j8')) is equal to zero because we are assuming a medium without

a permanent dipole moment. The lack of inversion symmetry of our medium further

reduces the calculations since ,yL2)=0 [il. The first contribution by a nonlinear term is

consequently due to the terms of ($3)).

\Ve have classified the contribution to the espectation value of the dipole moment

according to the mual convention of nonlinear optics. At wl and ~ i 2 , we have respectively

for (p(')) the terms:

and

where the subscript nor is used t.o denote the normal terms which are the linear response

of a medium composed of elements in the groiincl state Idi) and are responsible for the

normal index of refraction and absorption of tlic ~Iectrornagnetic wave at wl and d 2 .

\#Vriting them espticitly with Eq. 2.44 rve get as the expectation value for the dipole

moment at i ~ < 1 and d z respectively.

and

The interference terms, responsible for the control achieved by C E 0 IiE. are respec-

whcre the subscript int is t o indicate the interference nature of the terms. LVe inserted

the off-resonant terms of t h e second and thircl order perturbation expansion in the Iast

two equations to show tha t the off-resonant terms are negligible when compared to the

resonant contribution. Note that. due to the possibility of interchanging events in the

Feynman diagranis. we have to rniiltiply t he terms involving two identical frequencies by

two. Csing Eq. 2.44. Eq. 2.46 and Eq. 7-51. the first two terms of Eq. 2.61. which are

resonantly enhanced. can be expanded to give

+ C.C. (2.63)

where we added the subscript res to specify the resonant nature of the terms.

The rest of the terms of Eq. 3.61. denoted by the subscript non - res, are not resonant

mith an eigenstate. This accounts for the extra sumrnation and the omission of y in the

folloming equation:

Thc ciifference between the denominators of Eq. 2.63 and Eq. 2.64 makes the off-

resonant terms negligible when compared to the resonant terms. That is,

For N2 molecules. the first di pole allowed transition is on the boundary between the UV

and the the far UV which makes the lowest Rab in the order of 1016 Hz. As a result. for a

laser with an ang~ilar frequency. G, in the visible light range, Qab f w, f wp is of the order

of at least 10" Hz. On the other hand. the 7 . is at most of the order of 10'' Hz. This

makes the off-resonant term at least five ordcrs of magnitude smaller than the resonant

term which justifies the elimination of such terms in ($3)).

Tliere exist 10s terms for ( F ( ~ ) ) that are resonantly enhanced. We have so iar classified

onlj- four as interference terms. Of the 104 left. 4S terms contribute to satellite terms. some

of which were encountered previously in Eq. 2.33. The 56 remaining terms are responsi ble

for a number of different effects that compete with CEOKE. In the next section Ive classifv

the r>6 terms as either the optical Kerr effect or the self-focusing/defocusing effect and

present their consequences on the index of refraction.

2.6 Competing Effects, OKE and SFE

i v e present in this section effects that compete with CEOIiE. After giving a brief overview

of t h e opt ical Kerr effect and t lie self-focusing/defocusing effect. we present the condi-

tions tinder which these effects become coniparable to CEOKE. In order to simplily the

ecluations. we describe these effects only for d l .

2.6.1 OpticalKerr Effect

For a nurnber of years. the Kerr electrooptic effect ([CEE) has been used to dynamically

control the index of refraction of an electromagnetic wave. It is present in a number of

devices. ranging from the fast optical switch ncetled in an optical mernory storage device

to t, he multiplexing and dernultiplexing of a signals t hrough optical fibers.

IïEEI also knomn as the DC Kerr effect. is realized by applying a static field on a

niediiim. Figure 2.4 (a ) presents the effect schematically. The effect is normally modeled

Figure 2.4: Ckometry of the interaction of (a) transverse KEE and (b ) OKE

where no is ordinary index of refraction and nir,,E is the effect produced by the appli-

cation of the static field E. .As an example. for X=lpm, GaAs has a coefficient nir of

approximately 60 x 10-"ni/V.

-1s in IiEE. the optical Kerr effect (OKE) modifies the index of refraction that a probe

laser experiences by modifying the suscept ibili ty of the material. Hoivever: in this case the

control over t h e susceptibility is achieved by the use of a strong optical electromagnetic

field. as shown schematically in Fig. 2.4 (b). Although OKE permits a faster response

time. t h e modification tends to be q~i i te srnaller than the one produced by IiEE. The

Fe!-nman diagram and the energy-level description of OIiE are seen in Fig. 2.5.

Figure 2.5: (a) Fe>-nman diagram of OIiE and (b ) the corresponding energy-level descrip-

tion. The laser a t J I experiences a different inclex of refraction due to the presence of the

second laser at wz

The terms (~(~)(q)) associated with OIiE. are

+ C.C. (2.67)

CiTe have ornittecl in this section the non-resonant contribution to OKE since. as mention

al~ove. t hey do not coritri butc significant 1'; to ( i i (3 ) )0KE, or to our insight..

Al1 three lasers involve in CEOKE? esperience OIiE. However. we prove in section 2.6.3

tha t OKE can be negiected when we consider the application of CEOKE to a medium

composed of N2 moleciiles, if the three lasers are beiow a certain intensity threshold.

2.6.2 Self Focusing/Defocusing Effect (SFE)

SFE is ver- similar to OKE. Figure 2.6 presents the Feynman diagram and the Energy-

level description associated with SFE. The clifFerence between OKE and SFE lies in the

Figure 2.6: ( a ) Feynman diagram of SFE and ( b ) the corresponding energy-level descrip-

tion

substitution of the controlling laser of angular frequency wz of OKE by the first laser at

Hence. SFE is the first nonlinear correction to the linear susceptibility when only one

laser is present. In ottier morcls. a powerfiil lascr can change the index of refraction that

it experiences. This can be modeled by the lollowing equation

wliere n21(d i ) is ttie nonlinear correction t,o tlic linear index of relract,ion n o ( q ).

The terms of ($3) associated with ÇFE arc

Again. me have omit ted the off-resonance contribution. Figure 2.7 shows schemat ically

(3) t liis off-resonance character of a t hird order wavefunction 14-,, ,-,, ,,, ) t hat would be

associated with SFE

Figure 2.7: ( a ) Feynman diagram and (b ) the corresponding energy-level description of

(3) an off-resonance wavefunct ion 10 -,, ,-,, ,,, ) associated wit h SFE.

Thus far, the only consequence of SFE is a modification of the index of refraction.

However, the name of this effect is derived from the creation of a lens in the medium due

to the variation of the intensity across the beam profile. If we assume n2 to be positive.

the center of the beam which has a higher intensity will experience a stronger index of

refraction. consequently creating a positive lens. as shown in Fig. 2.S. SFE has been

Figure 2.8: Positive lens created by the variation of the intensity. 1. across the beam

profile in SFE where a positive nz is assurnecl.

observcd in a number of experiments[S] where t h e effect created conditions under mliich

tlie beam distorted and formed filaments. This ultimately can impair the propagation of

the laser and even cause physical damage to tlie medium.

As can be seen lrom Fig. 2.6 and appendis B. the three lasers present in the study of

C E O i X can experience SFE.

\Ve have shown in this section that the terms of OKE [Eq. 2.671 and SFE [Eq. 2-69] are

prcsent in an experiment involving CEOICE. The next section considers the magnitude of

t hese t tvo effec t S.

2.6.3 Magnitude of OKE and SFE

Modeling OKE by a sirnilar equation as Eq. Z.(iS.

makes the evaluation of the magnitude of these effects quite simple. The intensity depen-

dent index of refraction has been measured in a number of different experiments. Table

15

2.1 presents the values of n~ for various materials[ï].

Xir(20° C)

Carbon disulfide

Semiconductor-doped glass

C;aAs(room temperat tire)

Ga.As/Gar\l.As

Table 2.1 : Third-order nonlinear susceptibilities of various materials.

.As already mentioned, we implement . in the following chapter. CEOIiE in a medium

composed of nitrogen molecules. We chose tliis medium for a number of reasons. One is

the availability of theoretical and spectroscopic data in the literature. The other reasons

are Iiased on the possible applications which arc cIiscussed further in chapter 5 . Here we

just provide calculations to eliminate any concern about the possibility of having OliE or

SFE compete with CEOKE.

-4 medium composed of nitrogen molecule at standard temperature and pressure has

a n index of refraction of 1.000Z'i [IO]. Thus. for a value nî of 5 x 10-Lgcm'/W for air? a

change in the index of refraction of IO-" is only possible by OKE or SFE with a laser in

tlic range of '2 x 101' W/cm? The laser field needed to create stich an intensity woiild

have to be bigger tlian 2 x 10' V/m. Below me show that CEOKE can produce such

changes with far less power.

2.7 Summary

Neglecting the OIiE and SFE. (jZ(wl)) is just a sum of Eq. 2.59 and Eq. 2 - 6 3 . The

siisceptihility at srl is then given by,

1,L-e noow need to evaluate \ (dl) in orcler to quantify the control achieved by CEOKE.

The equations ancl data necessary to evaluate the terms in Eq. 2-71 are

0 the transition-dipole matrix elements. pisd

a the eigenstates of N2 involved in the dipole ailowed transition, In)

O the eigenenergy of the different eigenstatcs. ( E, - E û ) / h =

a the linewidth of the transition. 7 .

These are presented in the next chapter.

To demonstrate the power of CEOKE. we then calculate. in chapter 4. the index of

refraction achieved by modifying the various parameters in Eq. 2-71:

a the magnitude of the ratio of the two incident lasers. E2/ El

r the choice of the two states involvecl in the superposition, 1 & ) and Id2)

m the relative phase of the lasers, e'(2'p+"2-Jl )

m the magnitude of the ptirnp laser. Ep.

Chapter 3

Implement at ion

The Born-Oppenheimer approximation great ly simplifies the calculation of the transition-

dipole matris elements introduced in the previoiis chapter. After a brief review of this

approximation we derive the select ion rules applicable to nit rogen molecules and present

data necessary for t h e numerical implementatiori of CEOKE.

3.1 Born- Oppenheimer Approximation

The calculation of [(dl) requires the evaluation of the transition-di pole mat ris elernents

pk,. For reasons tliat will become apparent in section 3.1.1. a significant reduction in the

numbcr of calculations results i f we carry ou t the evaluation of these matrix elements via

a procluct p i l p , k in t he case where the intermetliate state l j ) is virtual. This is the case

4

in CEOKE since. as seen in Eq. 2.71. WC recpire &&, and p i n ~ i 2 ~ 2 m / ~ & L wliere lm)

and 1 1 2 ) are virtual intermediate states.

The Born-Oppcnheirner approximation. which is the assumption that the nuclear

and electronic motion are independent due to their considerable differences in time scale

and mass. perrnits us to separate the wavefunction into an electronic and a vibrational-

rotational component. As a ftirther approximation. we assume that the vibrational and

rotationai components of the wavefunction can also be separated. The product of matrix

elernents can then be written as

where I'Bel1). 18v1 ) and 1 qA, jh) are respcctivcly the electronic. the vibrational and the

rot ational cornponent of the ground wavefunction I&).

3.1.1 Spherical Coordinate Basis Set

To further simplify the evaluation of Eq. :j.l. we introduce the spherical coordinate b a i s

set and the D functions. The transformation I~etween two rotated coordinate frames.

sliorvn in Fig. 3.1. cari be simplified by iising tlic spherical coordinate basis set ancl the D

fonctions? also knoivn as the Wigner functions or the generalized spherical functions.

Figure 3.1: Euler angles a. /? and y relating the laboratory coordinate frame (X,Y.Z) with

the niolecular coordinate frame(x,y,z)

Tliese concepts a re needed in order to simplify the evaluation of the transition-dipole

matrix elements which involve the polarization of a laser, described in the laboratory

frame, and the wavefunction. which is nrrturally represented in the molecular frarne[ll].

The derivation below is based on t h e work of Lin e t al.[l2].

The spherical coordinate basis set is definecl accordiog to Zare[l3] as'

To describe the interaction of the laser and the molecule. we define the laboratory

frame R& by

and the molecular fised frarne Rzol by

fi::[ = = F I -(S* if)

The D function is defined by

where D(0) is the operator of the successive rotations that are associated with the Euler

angles defined with respect to the laboratory franie. see Fig. 3.1, and IJI<) is the eigenstate

of the total ongular mornenturn operators j2 and &. ( 1 ( a A ) ( y )

D ( û . 3 . 7 ) = exp -Jz e s p - J y e ~ p -Jz

The operation of the D functions on the spherical harrnonics is given by

Since the spherical coordinates basis set are proportional to the spherical harrnonics

ive can use the D function of order one to transform between frames,

Mul t i plying by the complex conjugate and applvi ng the uni tary relation.

we find t be transformation between t h e laborator- and the molecular fised coordinate

The clipole moment in the lab frame is thus

Substitiiting this last equation into Eq. 3.1 gii-es

where the polarization of t he laser E~ is describe in the spherical basis set [Eq. 3.31.

The sum. x,,,. over the dipole allowed transitions in Eq. 1.71 involve a considerable

niimber of calculations. To recloce the nurnbrr of calculations, m e approxirnate the de-

nominators of Ecl. 2.71 to be independent of the rotational quantum numbers. Tha t is,

can be approsimated to be independent of the rotational quantum numbers. J and

M . This allows us to sum the elements j72m/th over the rotational quantum numbers.

Rernembering that 1 KPJm,icr,) Eorm a complete subspace

we can approximate the Cm,, in Eq. 2.71 to a stim over the electronic states, el,: el, and

the vibrational quantum numbers, u,. v,.

Equation 3.14 can be simplified further by using the following property of the D

ftinct ion

iihere ( ) is the Wigner 3j ~ y d i 0 l ~ ( l 2 ] . The restrictions on the surni oie. Ml :b12

./..II and f" are givcn by the following properties of the Wigner :3j symbols:

Eqiiation 3.1.5 is the simplification mentioned earlier that arises when we evaluate

proclucts of dipole-transition matrix elements that possess a virtual intermediate state.

Eq~iation 3-14 simplifies to

1 1

Note tliat, on resotiance with a real intermediate state. Eq. 3.15 is no longer valid. In such

a case. the number of Wigner :3j symbols to e d u a t e doubles. However, the resonance

allows us to eliminate the contribution from the ot lier di pole allowed transit ions, t hus

rernoving C m and m, n from Eq. 2.7 1.

(4 In order to evaliiate (@ j, ,,kI2 1 D M K ( f i ) lqJl ). we write the normalized rotational wave-

function in terms of the D function and integrate over the angles. From Zare[13]

ivhere Ji. h', and .\fi represent the total angular mornenturn. the component of .Il along

t h e inter-nuclear z axis and the component of .Ji dong the Z-axis of the Laboratorp frarne.

TIic iritegration of t h e tliree D functions is simplified by the following relation

The final form of the sum of the pair of transition-dipole ~na t r ix elements is

To fiirther reduce the amount of calculations needed to study CEOKE? ive introduce a

number of selection riiles that apply to t he nitrogen molecule[l LI.

Since the parity operator cornmutes with the total Hamiltonian of a diatomic molecule.

we can characterize the total ivavefiinction by its parity.

where Q,,, is composed of qv and qJ,,bl. The nuclear wavefunction (@,,,), composed of

the vi brat ional and rotational wavefunction. possesses a parity (- 1)'[14]. Assuming a

certain parity for the electronic wavefunction. which we denote as a superscript + or -:

we get the following eigenvalues for the parity operator:

O ( - 1 l J for the even electronic states I+

O ( - L ) ~ + ' for the odd electronic states Se.

Homonuclear molecules possess an e'tt ra classification due to the invariance of the

elect ronic 1-Iamiltonian with respect to the molecular frarne. The electronic states of the

honionuclear molecoles are Iabeled with a sulxcript g or u to denote its even (gerade)

or otld (ungerade) nature with respect to inversion of the electronic coordinates in the

rnolccular frarne. Yote that the previoiis notation. Cf. denotes the parity due to the

inversion of the electronic and nuclear coordinates rvith respect to the laboratory frarne.

The first selection rule stems from the use of the electric dipole operator. which enters

the di pole-t ransition matrix elernents. This operator. being an odd function, forbids the

transitions between states of the same parity. Note that. this agrees with the well known

selectioii rule for an electric dipole alloivecl transition l .J=f 1 when there is no change

in $JCr This ultimately restricts the electric di pole transition between electronic states

of different parity. Hence, for transitions between two odd or even electronic states. the

restriction is

O A J is odd

and for transitions between an odd or even electronic states, the restriction is

r i1J is even

Additionally, a number of simplifications t,o Eq. 3.20 are possible by considering the

resrriction imposed by the Wigner 3j symbols[lZ] :

Assurning that the dipole is dong the internuclear axis of the rnolecule. # O ,

eliminates the sum over a and b in Eq. 3.20. This in turn eliminates the surn over li

because of the restriction Ml + AI2 + &=O on the Wigner 3j terrn, thus.

Eq~iation 3.20 can be simplified further by appiying the following relation for the Wigner

3j s ~ m b o l s of Eq. 3.20:

if J I + .J2 + J3 is odd. .As a result. we are forcecl to consider only the terms with J=0 and

M .

In addition.

eiectronic state

since the projection of .J, dong the internuclear axis. I\;, is O for the

XS:! and l i = O , the fourth Wigner 3j symbol

is also reduced to a term of the form

I-sing the second restriction on the Wigner 3j symbols. IJi - .J21 5 .J3 5 lJ1 + .J2[. forces

t h e use of only specific values of J,

The final step in calculating CEOKE is the evaluation of the elements of Eq. 3.20.

3.3 Evaluation of the Dipole-Dipole Transition Ma-

t rix Element s

C'alculating CEOIi E requires the evaluat ion of t liree distinct compooents of equation 3.20.

The following three sections describe the method used to evaluate them.

3.3.1 Transition Dipole Moments

?'O evaluate the transit ion di pole moments. we assume the Condon approximati on. whi

allows us to treat the transition dipole moment connecting two electronic states as

constant. independent of the internuclear distance.

The electronic states that allow a di pole transition moment frorn the ground electronic

state of Nz, are b ' i S ; . d l E ~ ~ e f l Y ~ . bln,,.clIlu,olII,. These electronic states are

shown in Fig. 3.2 as a function of the interniiclear distance, plotted using the data in the

review article of Lolthiis et al. [15].

The transit ion di pole moments for t hese calculations,

are based on the theoretical numbers of Stahel et a1.[16] between the electronic ground

state. XE:: and the states mentioned above.

3.3.2 Radial Functions Overlap

The radial functions. 1 Qu,) , were caiciilated using Dr. G. Campoliet i's uniform WIiB

program[l'i]. The eigenenergies calculated by the program were very close to those com-

pileci hy Lofthus et al. [Ml. A separate program was written to calculate the overlap of

the radial eigenfunctions

C(% I Q ' J ('Ln I%J). (3.27) m

Th- were found to be in escellent agreement with the values calculated by Stahel et

al. [16] for the vibrational ground state. -411 the relevant data is reproduced in appendix

. -

Elec. State (el,)

Table 3.1: Transition dipole moments from Çtaliel et al. between the ground electronic

state X S a and the other dipole allowed electronic states.

3.3.3 Angular Functions

LVe evaluate the angular functions, after the siniplifications mentioned in section 3 . 2 .

directlu. The tabulation of the possible values. as was done with the radial functions in

appendix C7 is prohibitive due to the large niimber of possible cases.

3.4 Parameters of CEOKE

Equation 2.71 has shown that the control parameters in CEOKE are the ratio of the

interfering lasers E2/EI , the overall pliase of the interference terms. the choice of the

states loi) and Id2) and the magnitude of Ep.

The addition of 7 to the list of parameters is required because the experimental setup

will determine the linewidth of the transition between the states It$i) and 142). Since the

Figiire 3.2: Electronic states of !L; involveci in the dipole allorved transitions from t,he

clectronic ground state XE:.

transition is dipole forbidden between t liese two states, the collisions arc going to be the

limitirig factor in t h e lifetime of the stiperposition. The temperature. the pressure and

the experimental setiip are t hus going to dictat,e this factor.

Assuming a negligible detuning A2, [seen in Fig. 2.21, Eq. 2-50 states t hat 7 and 1 Epl

are related by the coefficient cz

Hcnce: we are going use t h e parameter /cz12 to investigate the effect of modifying the

magnitude of Ep or 7. This also allows us to t w i f - that the calculations remain in a valid

perturbation theory regime. Note that the parameter Ic2I2 is still calculated using the

equations of section 2.5 and the values of 1 EpI and 7 are provided for al1 the calculations.

Table 3.2 sumrnarizes some values of the pump field needed to attain the value of

Ic212 = 0.20 as a function of 7. The pump laser is set a t the angular frequency needed

to have a resonant two-photon absorption between the levels Iol) = (v=OJ=O.M=O) and

Io2) = I Y = O . . J = ~ . M = O ) of the ground electronic state ?(Y:.

Table 9.2: Ep needed to achieve lc212=0.20 as a Iitnction of -1. 6 x 10-%sec is the natural

Iiktime of the 6'Il,, lcvel and 5 x 1 0 - ' O sec is the time between collisions a t 1 atni and

: )OOIi . CCé assume for the next calciilations t hat only one collision is needed to produce

a transition to the ground state. Note that for a constant lc2I2. a change in y. due to

different experimental conditions? only implies a modification of Ep.

( H z )

In surnrnary. the next chapter calculates CEOIiE for the folloiving parameters:

a the magnitucle of the ratio of the two inciclcnt lasers. &JEi

T = 1/27 (sec)

a the choice of the two states involvctl in ilie superposition, lbi) and 1 & )

Ep (VIm)

the relative phase of the lasers, e ' ( 2 B p C " ~ o l )

a the magnitude of the coefficient of t he superposition state, Ic21.

In orcler to compare the varioiis consecluences of these parameters, we assume. in the

calculation of Ecl. 2.71. a constant density of p=2.68845 x 1 0 ~ ~ molecules/m3. which is

the value of p a t standard temperature and pressure.

Chapter 4

Result s

Tliis chapter shows the extent of the control o w r the index of refraction achieved by- the

colierently enhanccd optical Kerr effect.

The first calculations involve an ideal medium composed of nitrogen molecules in a

single pure state. allowing us to explore the advantages and limitations of this new effect.

Tliis is followed by calcitlations which incorporate a thermal distribution of initial Nz

eigenstates to study the consequences of a rnorc- rralistic medium on CEOKE. Finall-, we

esamine the consequences of using CEOKE ncar a resonance.

4.1 Simplified Mode1

A nitmher of characteristics of CEOKE can be stiidied by analyzing (di) [Eq. 2-71]. T h e

relei-ant control parameters in this study. as mciitioned in section 3.4, are

r the ratio of the amplitudes of the two incident lasers & / E l ,

a the choice of the vibrational-rotational states, 1 & ) and Id2) of the ground electronic

state XX:, involved in the superposition.

r t he relative phase of the lasers 20, + O2 - 01,

a the magnitude of the coefficient in t he superposition state Iczl.

The next four sections describe the influence of these factors on the control obtained

by CEOKE in a medium consisting of X2 molecules initially in the ground state Iol)=

Iu=O..J=O.M=O) of the ground electronic s ta te X Y i .

4.1.1 Ratio of the Incident Lasers

The calculations in this section examine the consequences on CEOKE of varying the

ratio E2/ El. In doing so. the three otlier parameters are optimized (ivithin a limited

parameter range) to achieve the greatest effect on the real part of the index of refrac-

tion. To rneet these conditions. a superposition is created between the ground state

/ol)=lv=OJ=O..\.I=O) and the excited state Io2)=lu=0..J=2,M=0) of the ground electronic

state .YE: using a linearly polarized laser witli a n angular frequency of 1.12-1294 x 1O1'Hz.

The tmo-photon absorption is assumed to b e exactly on resonance and the natural linewidth

of the state is 1 x 10' Hz. The magnitude of E,, is set to 1.1 x 108 V/m. maximizing the

coefficient lc2I2 to 0.20. The relative phase of the lasers 26, + O2 - is set to 7 to

insure. as shown below that there is no modification in the amplitude of eitlier El or

&. s, and w2 arc 3 x 1015 HZ and 2.99775 x 10'" Hz. respectively. The choice of t hese

optimized parameters is explained in greater cletail in the next three sections. Results of

tliis calculation arc shown in Fie. 4.1.

The log-log plot of Fig 4.1 shows. as is espected [see Eq. 2-71]. that n ' ( u t ) (-- 1

grows linearly, in the regime where the approximation of Eq. 2.40 is valid. with the ratio

of the incident lasers E2/E1 once the interference terms dominate in Eq. 2.71. Neglecting

higher order nonlinear processes such as SFE. r l ' ( ~ c : ~ ) (- - -) does not experience tliis

growth and remains constant at the ordinary nt(d2). since the interference terms vanish

in proportion to the ratio m. Hence. as in OIiE, the more powerful laser Ez controls

the nt(w1) experienced by the weaker probe laser El.

Llnlike OKE, however, the control achieved by CEOKE, as seen in Fig. 4.1, is sub-

stantial. For example. a quick calculation wit h the parameter n2 for air from section 2.6.3

shows that a laser in the range of the TW/cm2 is needed to observe a variation of 0.01%

Figure 4.1: Variation of t he real component of the index of refraction from vacuum,

produced by CEOKE at wl ( ) and d l ( - - -): as a function of the ratio of the

incident lasers.

in n'(ai ) experienced by the probe laser in an O1iE experiment in air. By cornparison, in

CEOIiE a probe laser of a few pW/crn' esperiences a substantial change of a few percent

in n'(dl ) when the controlling laser is in the range of the W/cm2.

4.1.2 States Involved in the Superposition

The calculations disciissed in the previous section involved the superposition of t h e states

loi) = (v=O.J=O.hl=O) and 102) = Iu=O.J=Z.?vI=O) of the ground electronic state sr:. CVe

now examine the consequences on CEOKE of rcplacing the excited state 14,) with other

higher energetic states. To simplify the notation. hence forth the eigenstates involved in

the superposition are taken to be in the ground electronic state XE: and a subscript of 1

or 2 is appended to the quantum numbers to show the relationship wit h 14') or Io2). The

selection rules. mcntioned in section 3.2. liiiiit tlir possible higher Iying states Iv2.Jz.hI2).

of a two photon absorption in the same electronic state. to states with J2=.J1 or J2=Ji f 2.

As cliscussed in the previous section. in al1 cases the linearly polarized pump laser was

acljusted to achieve the maximum allowecl coefficient lc2 l 2 = 0.2 and the relative phase of

the three lasers 2B, + B2 - el is set to 7. Table -L.L presents the values of Ep used in the

following calculations.

Table 4.1: Magnitiide of the electric field' the intensity and angular frequency of t h e pump

laser involved in creating the superposition between lul ,J1=O,M1 =O) and Iu2,J2=2,MZ=0).

.A 7 of 1 x 1 0 W z and a negligible detuning from resonance are assumed for the calculation

of these values.

Figure 4.2 presents the variation of n'(wl) mlien the superposition is created between

the ground vibrational state and the first four escited vibrational states. u2=l to 4. with

.12=2.k12=0. The real part of the index of refraction at wl ( ) presented in Fig. 4.1

has been reproduced in Fig. 4.2 for comparison purposes.

As can be seen in Fig. 4.2. the control of n ' ( d l ) by CEOKE is greatly dependent on

the choice of u2 of the excited state used in the superposition. It should be noted that the

resuit of the calculat,ion for v2=2 ( - - - . - ) and v2=3 ( - . - - -) are almost su~erirnposed

which makes it difficult to discern the two ciirvcs. The variation of the four lower curves

of Fig. 4.3 results from a reduction in the radial function overlap terms (QablU!zb) in Eq.

3-27 which reduces the magnitude of the interference terms. The use of the same radial

function, VI =Y. in Io1) and 1 @ 2 ) . as is the case for the uppermost curve ( ). reduces

Eq. 3.27 to the sum of the Franck-Condon factors 1 (4:, 11~~) 1 2 . This maximizes Eq. 3.27

The substantial reduction in the dope and in tlie onset of the linear regime in the last 4

curves can be attrihuted to tlie fact that (I[lzibllP::o)(Qzbl@Yib) unlike the F.C. factor. cari

be negatire? thus retliicing the total sum of Ecl. :3.27 and consequently, the magnitude of

the interference terms in Eq. 2-71.

The use of J2=0. instead of .J2=2' with il2= 1 to -1 cloesn't noticeably change the control

on n' obtained by CEOKE. Figure 4.3 sliows the difference in n1(w1 ) when Iul=O..Ji=O.

411=0) and lu2=1..J2=0.iC12=0 ) ( ) and IV~=O..J~=O,M~=O ) and Iu2=1,J2=2.k12=0

) k - -) are used as states in the superposition. The same parameters mentioned at

the beginning of this section have been used for this calculation. The difference between

the two curves of Fig. 4.3 stems from Eq. 3.28 and is a term dependent only on the

quantum numbers of the rotationa1 wavefiinction J i , M l , J2 and Mq. The reduction in

the onset and the dope of curve (- - -): is attributable to the reduction of Eq. 3.38

which consequently reduces the interference terms in Eq. 2.71. The three lower curves in

Fig. -1.2 will experience the same modification if we change the rotational number J2 to

2, since Eq. 3.25 is independent of vz.

Figure 4.2: Control of the real part of the index of refraction by CEOKE with different

112 involved in the superposition state. The groiind state XS;, lui=O..Ji =O. Mi =O) is in

a superposition state with Iu2=O1J2=2-M2=O) [--- ) Iu2=l, J2=2.hl2=O) (- - -)

I I ~ ~ = ~ . J ~ = ~ , A ~ ~ = O ) ( - - - . - ) 1-=:3.J2=2..\[2=0) ( - - - - - ) ~ W ~ = L L , J ~ = ~ , & I ~ = O ) (- - -).

100 EUE 1

Figure 4.:;: The consequences of using a different rotational quantum number in the

excited state involved in the superposition on the control achieved by CEOKE. The ground

state XCQ,IvI=O,.Ji =O,MI=O) is in a superposition with the state Iu2=1,.J2=O,M2=0 ) (-

-) and lv2=1,J2=2,Mz=0 ) (- - -).

4.1.3 PhaseoftheLasers

The previous calcdat ions were done wi t h lasers having opt imized phase components to

masimize the control over Figiirc -4.4 prcsents the dependence of the control on

the relative pliase orel = 28, + O2 - el of the t hree lasers involved in C E 0 KE. Figure -1.4

s1iov;s clearly the loss of control produced by v a r ~ i n g Orel from -1 to $.

Once again. the interference terms of Eq. 2.7 1 decrease from t h e maximum previously

clisplayed ( ) in Fig. 4.1. From Eq. 2-11 and Eq. 2.71, we know that in order to

masimize nt while having a negligible n". we need to rnaximize y t ( w i ) and thus the real

part of the interference terms. The resonance nature of terms like those îound in Eq. 2.51

give a contribution of to the total phase of the interference terms of ~(q) . Thus. the

relative phase of the lasers 20, + O2 - 81 necessary to maximize the real part of \.(diri) is

-5 + 2n . n . where n=0.1.'2 .....

1 1 remarkable leattire of Fie. 4.4 is the clraniatic reduction of nt at Orel=" 2 ( - - - - - 1.

This reduction is easily explainecl by noticing t hat for O < Orci < rr: the interference terms

of \(dl ) are negative. which in turn reduces nt of the material. The dramatic drop is due

to t lie logarit hmic nature of Fig. 4.4.

Ciirve O r e r = O (- - -) presents an interesting feature when the ratio E2/& ap-

proaclies 500. Since O r f i = 0. the real part of tlie interference terms. yt(wl) [Eq. S.'il],

is negligible and nt [Eq. 2.1 11 becornes only a function of yM(ul). Although the control

of n' is possible under these conditions. the propagation of the incident beam Et is al-

tered dramatically by the absorption/arnplification produced by the imaginary part of the

refractive index, n". when a high ratio E2/Ei is used.

The linear-log plot of Fig. 4.5 presents the modification of nu as a function of the ratio

E2/ El for O r e l = - $ ( - - - ). O(- - -) and i; ( S . . a ) . Figure 4.5 shows that the increase

of nt seen in curve (- - -) of Fig. 4.4 is concurrent with a non negligible nu. shown as

curve (- - -) of Fig. 4.5. This is not the case for Brel = -' since a strong n' is not 2

procluced at the detriment of the variation of t h e amplitude of El. Curve ( ) of Fig.

4.5 shows that tlie n" for O,,, = -: is negligible.

Figure 4.4: Dependence of n' produced by CEOKE on the relative phase of the lasers

O,,/ = 3Op + O2 - OL =y (- ) ? O (- - -1. : (- - * -).

100 Ratio E2E1

Figure 4.5: The imaginary part of the index of relraction, n": produced by CEOKE where

the relative phase of the lasers OrCr = 20, + O2 - B i =y ( ), 0 (- - -): T (.-• .).

The n" associated with the nt ( - - - sa-) in Fig. 4.4 is the same as the curve (- )

of Fig. 4.5. Changing O,,/ = 7 to Orci = only changes the sign of r f ( w l ) . .As long as

l ' ( d l ) > - 1, the effect is to reduce n" to zero. However. the result of n" for OrCr =

increases when the ratio E 2 / E i is big enough to make yr(wl) < -1. With the parameters

mentioned above. this condition is fulfilled only when E2/E1 > 20000.

C'urve (. . . . - ) of Fig. 4.5 shows that witli firpr = r, El could be amplified as it travels

through the medium. .A change over n' is also associated with this amplification. The

ciirr-c (- - -) of Fig. 4.4 represents esactly the change in n' that would be experienced

I ~ J - tlic electromag~wtic waïe when O,,! = r . The same answer of nt for Orci = a and

O,,r = O is expected since under these conditions. nt is only dependent on ) I and is

not a function, unlike nt', of sign(,yf'(wl)).

;\lthough the control of nt with OrCr = n is possible. it should be noted that a inodifi-

cation of El is procl~iced as the lasers propagate through the medium. feedback effect

is created where the growth of El would reduce E 2 / E l , which in turn would reduce the

control over n' and n". For this reason. we cliosc to control the propagation of the elec-

tromagnetic mave hy moclifying nt while having a negligible nt'. Consequently. we use

O r e l = 7 in the calculations.

Note also. that the phase control observed in CEOKE represents a sigriificant difference

from OKE since OIiE is not a function of the relative phase of the lasers.

Possible applications of t his control over the index of refraction through the modifica-

tion of the phase of the lasers is discussrcl in tlic next chapter. along with a detail analysis

of t lie disadvantagcs imposed by the phase inatcliing conditions.

4.1.4 Magnitude of the Coefficient c2

The magnitude of Ep has a drastic effect on the controi achieved by CEOKE. As was

discussed in section 3.4. we study the effect of varying the parameter Ic21 of Eq. 2.50

instead of lEp',l. Figure 4.6 shows that the initial preparation of the superposition by

the pump laser Ep is extremely important since the interference terms of Eq. 2.71 are

proportional to 1 Ep 1 2 . We assume for t hese calculations the usual y of 1 x 10' Hz and a

negligible detuning of the two-photon absorption A2+ « 7. Table 4.2 lists the different values of Ep needed to achieve the values of lczl iised in

Fip. -1.6.

Table 4.2: Magnitude of the electric fielcl and the associated intensity of the pump laser

needed t,o achieve the different values of lez 1 assuming a 7 of 1 x log Hz and a negligible

cletuning of the two-photon absorption &,, < 7 .

The usual lul =O.J =O~?vI l=O) and Iv2=0..J2=L>.M2=O) eigenstates were involved in the

superposition and the relative pliase of t he lasers 20, + & - O1 was set to F. LVe see in

Fig. -1.6 a gradua1 decline of the dope going Ironi the curve (- ) to the curve(-----)

due to the reduction of Ic21. This de1aj.s the oriset of the different curves to t h e point

mliere E2/E1 compensates for the reduction of ( E,J2 in the interference terms.

The next section considers a more realistic medium by including the effect on CEOIiE

of the thermal distribution of the initial states.

4.2 Thermal Distribution

.As in section 4.1 .2 we consider here, the consequences on CEOKE of the choice of the

states involved in the superposition. Here. hoivever, the pure initial s tate Ivi=O,Jl =O,

M1=O)? is replaced hy a thermal distribution of the initial states. Ultimately. we will see

that the effect of this distribution over the initial rotational states is to reduce the control

due to the decrease in the number of moleciiles involved in the superposition state.

Figure 4.6: Control of n' with Ic21Z= 0.20 (-- ). Ic2I2= 0.10 (- - -)_ Ic2I2= 0.01 (---) ,

Icz12= 0.001 ( - - - -). The state involvecl in tlic superposition are lui = O,J i=O, iLL1=O)

and luÎ = 0,J2=2.M2=0).

At roorn temperature: over 99.99% of the molecules are in the vibrational ground

state v=O. The thermal distribution of the rotational states at 29SK, however. involves a

large nurnber of states. The distribution is calculated using the Boltzmann factor and the

stat ist ical weight due to the degeneracies of the rotational state and the nuclear spin[18].

The number of molecules ni in the state with energy Ei is

iV is the number of molecules. Q is the partition function: g; = ( 2 J + l)g,,, is the

s ta te clegmeracy of the rotational s tate and the nuclear spin.

In the case of N2. g,., is 6 and 3 for the even and odd rotational wavefunctions

respectively[lS]. Figure 4.7 presents the probnhility of finding a molecule of N2 in a

certain rotational s tate a t 29SEi.

One can see in Fig. 4.7 that the degeneracy of the nuclear spin has a great effect on

the probability of finding a molecule in ari ocid or even rotational state. This is what gives

the illusion of having tivo sets of data.

T h e present calculation uses a linearly polarized pump laser of angular freqtiency

4.3901 x 10'" Hz ii-liere we assumed a riegligihle cletuning ol the two-photon absorption

hctween lul =O. J i =6) and Ii= 1 ..J2=6): lierc ; = 1 x lOgHz. This creates a superposi-

tion between lvl=O..Jl =6,Mi=(-6 to 6 ) ) ancl l ~ ~ = l . . J ~ = 6 . k 1 ~ = ( - 6 to 6)). We chose

Ivi =O.Ji =6.MÎ=(-6 to 6)) since it is the state t hat displays the maximum population

at 29SK. Figure 4.8 shows the results of tliis calculation (- - -) compared to the pre-

vioiis results for Ivl =O.Jl =O,iLII=O) superposccl with I Y ~ = I , J ~ = O $ I ~ = O ) ( ) [Fi&

-1.31. The decrease in the population involved in the superposition accounts for the reduc-

tion of almost an ordcr of niagnitude in the control provided by CEOKE. Note that the

curve ( ) in Fig. 4.S is not the usual result of Ivl=O,Jl=O,MI=O) superposed with

lv2=0.Jz=2,M2=0) seen in most of the previous graphs, since we wanted to isolate the

effect of modifying the initial population frorn the effect of varying uz.

1t should also be noted that Fig. 4.8 shows good agreement between the calculated

O 5 1 O 15 20 25 30 35 Rotational state

Figure 4.7: Probability of finding a molecule of N2 in a rotational state at 29SK.

100 Ratio

Figure 4.S: Control of n' by CEOKE with a superposition created between lul =O..Jl=O.

kI1=0) and (u2=1..J2=0,M2=0) of XCQ (-- ) where a11 the molecules of nitrogen a r e

initially in the ground state and with a superposition between lui=0,Ji=6,M1 =(-6 to

6 ) ) and lvz=1,Jz=6.Mz=(-6 to 6)) where the niolecules of N2 are initially in a thermal

distribulion corresponding to a temperature of 29SK (- - -).

index of refraction of '12 at STP (1.00010) ancl the experimental index of refraction of

clv air. l.OOO'Li[lO]. at STP for a wavelength of 600nm in the E2/EL regime where the

i nterference terms are negligi ble. The small discrepancy could be a t t ributed to the values

of the electronic t ransitioo moments calculated by Stahel et al. [l6] mentioned in the

previous chapter and the difference in the composition of the mediums since N2 forms

only 78.084 Mole percent of dry air.

In section 4.1.2. ive saw in Fig. 4.2 that the use of Ivl=O.Jl=OIMI=O) and Iv2=0.J2=2.

.L12=O) in the superposition produced the greatest control over n'. The thermal distri-

bution of the initial states now gives us t h e possibility to create superpositions between

different values of J and J 2 with ul = 4 = O. .As stated before. the selection rules impose

a A J = f 2. Table 4.3 lists the field strengt h and the angular frequency of the pump laser

iised to create the superposition between the different J states. These calculated values

assumed a y of 1 x 10' Hz. a negligible detiining from resonance and are optimized to

achieve l ~ ~ 1 ~ = 0 . 2 0 . d l is still 13 x IOL5 HZ and d2 is adj~isted to satisfy the usual condition.

flli = izi - d?. For al1 of these calculations. O,,! is set to 7. Figure 4.9 presents only the results for a superposition between lui =0.J1=6.Mi =(-6

to 6 ) ) and lu2=0,J2=d.M2=(-6 to 6 ) ) , iihich prodiiced the strongest control over n'. since

the rest of the curves are relatively close and rnake the figure unreadable. The previous

resiilts of Fig. 4.1 was also inserted in orcler to show the reduction in the control over the

real part of the inclex of refraction.

.-\s in Fig. 4.8. t lie evident reduction of the intcrference term can partially be accounted

for bj* the reduction of the number of moleciiles involved in the superposition. The

decrease in the population of 1&), however. should result in a stronger reduction then

that seen in Fig. 4.9. In order to account for the discrepancy, we need to consider the

superposition between ~ v ~ = O ~ . J ~ = S , M ~ = ( - ~ to 6 ) ) and lu2=0,J2=6,Mz=(-6 to 6)). This

plienomena was not encountered in the previous calculations since the second state used

in t h e superposition was always devoid of an- population.

To isolate tliis effect from the influence of the various quantum numbers, we explain

the discrepancy by comparing the same superposition between Ivl=O,Jt =O,M1=O) and

Figure 4.9: Control of n' by CEOIïE where t lie superposition is created between lu1 =O.

J I =6.M=(-6 to 6 ) ) + / y=0,J2=S,M=(-6 to 6) ) ( - - -) of XE: in a medium composed

of '12 molecules ini t ially in a thermal distri but ion of states corresponding to a temperature

of 29SK. For comparison purposes, the previous result invoiving the superposition between

Ivi =O1.Ji =O,MI=O) and Iv2=0,.J2=2;M2=O) of XE: with a medium initially in a pure state,

is shown as curve ( ) -

Tahle 4.3: Magnitude of the electric field. the associated intensity and the angular fre-

quency of the punip laser involved in creating the superposition between lul=O..JL) and

Iv2=0..J2). A 7 O € I x 10g HZ and a negligible rlet~ining from a two-photon resonance are

assunied for the calculation of these values.

1-=0..J2=2.M2=0) in a medium which has initially a pure single state with a medium

whicli has initiaily a thermal distribution of the states corresponding to a temperature of

29SIi. Figure 4.10 represents schematically t h e creation of the two superpositions involved

in a resonant two photon absorption of the pump laser when both ~vl=O,J1=O,M1=0) and

I V ? =0,.J2=2,M2=0) are popdatecl.

To show the role of the second superposition [Fig. 4.10 (b)] , an artificial medium was

created in which this last superposition WLS oniitted. Figure 4.11 shows the reduction of

n' when the second superposition is eliminated (. - .). The curve (- - -) represents

Figure 4. IO: Creation of the superposition (a) betwveen Ivl =O.Jl=O.MI =O) + Ivz=O.

.J2=2..L12=O) ( b ) between Ivl =0.J1=2.iL11 =O) -+ lv2=0.J2=0,M2=0).

the correct value of the control over n' ivhicti takes into account the second superposition.

The modifications to the equations to produce these results are minimal. Using Eq.

?.-KI we can write the tivo wavefunctions associated to Fig. 4.10(a) and Fig. -L.lO!b). as

where ]cl l 2 and Ic-1' are the magnitude of the population of the states ( d l ) and Io?). The

perturbeci wavefunction' I @ ( ' ) ) and 1 @ ( 3 ) ) also have to be calculated, however. for clarity,

WC omit the elaboration of al1 the details. Thc iesonant interference terms of Eq. 2.61

now have to be espanded to take into accoiint the extra interference terms,

+ C.C.

Expanding t his last equation gives,

Figure 4.1 1: Control of n' by CEOKE where the superposition is created between the

states jul=O,.Ji =O.Mi=O) and I V ~ = O , . J ~ = ~ . ? V I ~ = O ) The curve (-- ) represents the use

of an initially pure s t a t e corresponding t o a thermal distribution of O I i l while the curve

(- - -) shows the result o l using a medium having an initial thermal distribution of

the s tates corresponding to a temperature of 29SE;. T h e lower curve (. . . - ) represents

a fictitious medium with t h e parameters of curve (- - -) but where t h e contribution

frorn the superposition 1 vl = O . J i =%?M1=O) and lv2=0,J2=0,Mz=0) is ornitted.

where we have used R Z i - ;ip = w, and Rzl = - ~9~ t o simplify some of the eyuations.

Tlie interference terrn of the susceptibility is tlicri written as

I L should he notecl that altbough t lie distrib~ition of the initial states decreases the

cont rol provided bj- CEOIiE. the overall range ol t h e real part of t h e index of refraction

is still sribstantiai.

4.3 Near Resonance

This section presents the results of irnplementirig CEOIiE with El a n d E2 capable of cre-

ating a transition hetween the superposition state and a higher lying rnolecular eigenstate.

Hence we are considering the near resonant case. In order to simplify the calculations.

cve rnodel the medium of N2 in this section by a simple three level system as shown

in Fig. 4.12 where t h e eigenstate, 1 # 1 ) and 1 & ) . involved in the superposition s t a t e are

Ivi=O.Ji=O,M1=O) and (u2=O:J2=%,M2=O) of X S g and where (44 is Iu3=0,J3=l?>l3=0)

of bllI,, the lowest energy state allowed 114' a dipole transition from Idi) and 1&).

Figure 4.12: Interference of two lasers with aiigular frequencies wi and w~ interacting

witli a syst,em in a superposition state. The tii-O angular frequencies are detuned A,,

from resonance.

An increase of the real and imaginary part of the index of refraction is expected when

the photons can transfer the populations hetireen different eigenstates in an ordinary

medium. Figure -4.~3 presents the change in the index of refraction as a function of the

clet iining from resonance t hat the laser EL esperiences near the resonance between ( p l )

and Io3) Note tha t n' - I was plotted ancl not n' t,o follow the previous convention to plot

the clifference between the speed of the electromagnetic wave in the mediuni from that

in vacuum. For these calculations, we assumecl a -1 of 1 x logHz and an ordinary medium

witlioiit a preparecl superposition state. I t shoulcl be noted that these calculations do not

talic into account the effect of the other higlier lying eigenstates. The overall effect of

inclucling these other states would be a slight increase in the real and imaginary part of

t Iie index of refract ion.

The results show that a high n' is possible without the use of CEOKE, but that this

is only achievable i f it is accompanied by absorption due t o a non negligible nu. Thus,

under ordinary conditions, the modification of n' is accompanied by the attenuation of

t lie incident beam. Although Fig. 4.13 seems t o show that the attenuation is negligible. in

fact it increases by more than an order of magnitude between A,, =30 GHz and A,, =10

C X z . growing from 0.000SS to 0.0043. T'herefore, even for 5 GHz< A,, <20 GHz, we

have to consider the attenuation of the electromagnetic wave if it has a significantly long

-50 -40 -30 -20 -10 O 10 20 30 40 50 A,, (GHz)

Figure 4.13: Real ( ) and irnaginary (- - -) part of the index of refraction as a

lunction of the detuning A,, = n31 - wl of the angular frequency.

pat h lengt h t hrough the medium.

As shown below. the presence of an interference term. however, permits us to create

conditions under which a substantial n' is experienced by the electromagnetic wave with-

out the associated absorption. The calculations which follow are again based on Eq 2.36.

1-Iowever, unlike the previous calculat ions [section 4.1 and section 4 2 1 . the resonance wi t h

Io3) recluires that we keep y in Eq. 3.l3-L. The terms which possess a denominator of the

form Rua + w + in, are labeled antiresonant ternis and are at least five orders of magnitude

smaller than the rest of the terms which are clefined as resonant terms. By dropping the

ant iresonant contri bution. Eq 2.36 simplifies to

Bu expanding the result , separating the real ancl imaginary parts and remembering t hat

we are dealing witli a three level system. ive get

P Plrn/lml P2mPm2 AyI + y + I c ~ I ~ ( A ~ + 0 1 2 ) (A,, + R i 2 ) 2 +

and

Figure 4.14 prcsents the result of iising Eq. 4.9 and Eq. 4.10 in conjunction with

Ecl. 2.11 and Eq. 2.12 to obtain the real and imaginary parts of the index of refrac-

tion. The preparation of the superposition state is identical to the one done in section

1.1.1. This creates a coefficient lc2I2 of 0.20. and u2 are respectively, on resonance,

l.9OOSS4Tï"i' x 10i6 Hz and l.gOO6.jg420 x 10L6 Hz. The ratio of the two lasers, E2/ El is

taken to be 1000, O,,/ = 7 and the y=l xlOgHz.

\\;c see in Fig. 4-14 that a considerablc n' is achievable without associated absorption

when the detuning A,, is bigger than 20 GHz. Note that this increase is huge by the

Figiire 4.14: n'-l (--- ) and n" (- - -) where n' and n" are respectively the real

ancl irnaginary part of the index of refraction near resonance due to CEOKE.

standards set in Fig. -1.13. Once the interference term dominates in Eq. 4.10, the resultant

n' and nt' can easily be explained by noting tlmt Bref = -- and A,, » make \"(dl) 2

negligible. This. in turn. simplifies n' [Eq. 2. L 11 to

and n" [Eq. 2.121 to

\Ve see thât assiirning a negligible \''(dl ). nt tends to 41 + i1(ul) and to O when \ ' ( d l ) 2

O and i'((zi) 5 -1. respectil-ely. This is the reverse of the case of n" where t ' ( & y i ) 2 O

and y t ( ~ , ) 5 - 1 malies n" tend to O and 41 + i;'(wl) respectively. This explains the

asymmetry and the mirror images of n' and n" in Fig. 4.14.

The interference term in Eq. 4.10 allorvs. as seen in Fig. 4.5, the possibility of inducing

negative n"; which implies the amplification of tlie electromagnetic wave at wl. Figure

-1.15 presents the data on n' and n" whiclr restilts from modifying the Iast calciilations

[Fig. -~.14] by chanjing Brel to a/- + 1 IO-". The only differences from the previous

calculations are the creation of a negligible negative y"(wi) and the inversion of the sign

of as a function of A,,. The negative \''(di) inverts the results of n" seen in Fig.

-4.14 and creates the mirror image in Fig. 4-13 mit11 respect to the n" = O a i s . Similarly.

tlie inverse y'(wl) creates a mirror image iritli rrspect to the line A,, = O of both n' and

n".

Since there is 3 parameters of intercst in tlrese calculations.

a Ratio of the lasers, E z / E i

a relative phase. Bref

detuning from resonance, A,,

the representation of a11 the results of the calctilations requires a large number of figures.

Figures 4.16,4.17 represent the isosample lines of the calculation of n' and n" respectively,

as a function of A,, and The remaining parameter, E2/E1, is set to 1000. Figures

4.l-~.-!. 15 are just cross-section of these 9 dimensional results. Appendix D contains the

rest of the graphics ivit h different parameters Ei/ E l . A brief description is also given in

appendix D to describe the variation of nt and n" as a function of A,, . Orci and E2/ El .

The substantial ratio E2/ El allows us to treat t h e interference terms as the dominant

contribution to Eq. 4.9 and Eq. 4.10. The region around A,, = O still remains the area

wit h the most drastic change in the real part of the index of refraction. We see that it

is possible to exploit tliis region of higli n' II! making the appropriate choice of n". It

slioiilcl be noted tliat Fig. 4-17 shows aii interesling jagged feature which represents a cut

wliere the extrema of n" join. This espiains tlia presence of al1 t he different isosample

lines in t his area. The strong variation in the values of nu is clue to the variation of t t ' (wl )

between positive and negative values. The consequences of such a change was depicted

in t h e curve (- - -) of Fig. 4.14 and Fig. 4.15. The form of the feature is easily

esplained by noticing that t''(di ) becomes O i f A,, sin(&) = -7 COS(&^). AS a result.

t h e iariation of sign(\"(wl)) which is necessarily close to the region where i t t ( d l ) = O .

prodiices a substantial change in n" and creates a feature ivith an arctan($) shape.

In surnmary. CEOIiE represents a powerful iiirans of controlling the index of refraction

of a iveak probe laser. The cont rol has been shown to be su bstantial in a medium composed

of N2 molecules wit h a range of initial conclitions. The similarities between OKE and this

sceriario and the sul~stantial control over nt have lead us to name t his effect the coherently

enhanced optical Kerr effect.

Figure 4.16: The value of n' as a function of A,, and Brel where E2/EI is set to 1000.

n'=IO(- - - ) , 5 ( - - - -1: l(- ) and 0.1(--- -).

Figure 4.17: The value of n" as a function of A,, and Brel where E2/Ei is set to 1000.

= 5(--- )A- - - ) ,O( . - . - -1,- 1 ( - - - --)and 4- - -1.

Chapter 5

Discussion

This chapter will discuss some of the implications of CEOKE. We also compare CEOKE to

other schernes that modify the susceptibility via quantum interference. In order t o discuss

the ciifferences. we give a brief overview of the work on amplification without inversion

(AWI) that has appeared in the last few years in the literature. Specifically, we focus on

the work of Scully on the resonantly enhancecl refractive index without absorption via

atomic coherence[l9] and Harris' work on Elect romagnetically Induced Transparency [-O].

This is followed by a discussion on the limitations and the possible applications of CEOKE.

5.1 Amplification wit hout Inversion

A\VI is defined by the amplification of a weak probe field through the transfer of energy

From a strong and coherent driving field wit h a smaller frequency where the energy is

extracted from the medium \vit hout the requirement for a population inversion between

the lasing levels[21]. AWI is intimately linked to the quantum interference of different

pathways seen in CEOKE, which creates, in these schemes, a "hidden inversion". This

approach provides the required control over thc susceptibility which leads to changes in

the index of refract ion of the material.

The creation of the hidden inversion by q~ian tum interference is easily understood if

we calculate the transition [rom the superposition state, formed by Id1) and )42), to state

Id3) due to the tivo fields El and Ez seen in Fig. 5.1. Starting from the usual superposition

Figure 5.1: Three level configuration usecl in .+\\VI. The quantum interference of the atomic

waveftinction can created a trap state which inhibits any transfer to the state I&).

state [Eq. 2.231, the transition probability to Io3) is given by,

.-\ssuming that the transition matrix elements are both equal to p ~ .

Hence. it is possible to have an absence of population transfer to 1 & ) i f cl = - c z The

atorns or molecules wit h a corresponding wavefunction are essentially trapped in the

superposition state. Moreover, if lms) is popiilated, it cannot transfer its population to

IQi ) and Id2) which are involved in the trapped state characterized by the coefficients

cl = -c2. [Note that we have not considered in this simple calculation the effect of

collisions and other dephasing effects that would contribute to the destruction of the

trapped state.]

The hidden inversion mentioned earlier is based on the creation of a sufficiently large

trapped state to render the population of 1&) bigger than the remaining population in

lot) and Id2) that are not involved in the trapped state,

Hence. the extraction of the energy ironi the sniall population of the higher lying level is

possible. This can lead to AW'I. Alzetta e t a l . [Z] reported the first experimental evidence

of coherent population t rapping. They foiinci t liat the fluorescence spect rum disappeared

(dark lines) from a sodium vapor medium escited by a multimode laser beam, when the

proper conditions were met for the interference process to occur.

It is convenient to begin the cornparison of CEOKE with Scully's work[l9] on the

enhancement o i the index of refraction via quantum coherence because of the similarity

of tincierlying principles. This discussion is followed by a brief description of Harris' work

or1 EIT.

5.1.1 Resonant ly Enhanced Refract ive Index wit hout Absorp-

tion via Atomic Coherence

Resonant ly enhancecl refract ive index rvi t hoii t absorption via atomic coherence originates

from the work of Sctilly et a1.[23] on AM'I. Thc core of their work is based on the trappecl

s tn tc and t he hidclen inversion tliat it creates. Iloivever. unlilie Fig. 5.1. t h e superposition

is created between two energetically close states. T h e difference in the eigenenergies of

Iol) and Idz) is of the order of the lineividth of /on). This allows the use of only one laser

having a frequencj- as opposed to the two laser configuration of Fig. 5.1. This, as will be

seen below. creates an index of refraction that is independent of the field strength E(ir.).

T h c system studicd is rnodeled by a simplifiecl tliree level atom.

As in CEOKE. Sciilly e t al. work \vit11 a pliase-coherent medium which the- term

"phaseonium" . The two-photon absorption used in CEOKE is replaced by the absorption

of two strong coherent photons in the microwave range. However, unlike CEOKE, Scully's

work is based on the resonant transition with an eigenstate which possesses a residual

population. The trapped s ta te allows AWI of the field E il the residual population in

Id3) satisfies Eq. .5.3. More importantly. the moclification of the suscep ti bility allows the

control of n'. However, in order to get a large n'. resonantly enhanced refractive index via

quantum interference[lg] requires the high n' that is experienced near a resonance. AS

Figure 5.2: The field E experiences AWI or. uncler certain conditions. a strong nt without

the normal accompanying absorption. dp is the arigular frequency, in the microwave range.

of t h e purnp laser which creates the superposition between 1#*) and IO2).

iras seen in section -1.3. in an ordinary unperturbeci medium, the high n' is accompanied

by a non negligible n". The consequence. in an ordinary unperturbed medium. can be

the loss of t h e incident laser due to absorption. The novelty of Scully's scheme lies in

the use of t h e quantum interference to modify the absorption of the material and hence

allow t lie laser to experience tlie strong n' wi t liout the associated at tenuation. Shus. the

interference effect leads to a t ransparency or a negligi ble absorption close to the resonance.

Figurc 1 of [19] presents the results of 110th :\\+:I and the enhancement of the index of

refraction via this q~iantum coherence scheme.

In the case of zero population in the upper level 163) a t the point of vanishing ab-

sorption, Scully et al. report that the effect of tlie disappearance of the absorption also

leads to a vanishing of the real component of tlie refractive index. Modifying CEOKE, so

that the superposition is created between tivo states with the same rotational quantum

nuniher Ji = J2 but having different iCI states. Ml # hl2, we can reproduce the results

of vanishing n' and n". Figure 5.3 presents schematically the interaction of such a su-

perposition s ta te with a laser field, E l , resonant with the first dipole allowed transition,

I#3)=lv3=0,J3=07S13=0) of blII,. It will become apparent below that the use of a single

laser makes, as is the case in the resonantly enhanced refractive index via quantum inter-

Figure 5.3: Modification of nt and n" of an eiectrornagnetic wave resonant between a

siiperposition state. cl l@l)+~2j&). and the higher lying s t a t e Id3).

ference, t h e indes of refraction independent of the laser field. This reduces the control of

nr since the ratio & / E l is no longer a parameter.

Following the same development as in section 2.4. we calculate the expectation value

of the induced dipole moment produced by the laser El.

Remembering that we are dealing with a superposition of a degenerate s tate? Rzl = 0,

the susceptibility 1 (ul ) is written as,

Note that this equation is not dependent on field El or, as is the case in CEOKE. on a

ratio E j / Ej. .As in section 4.3. the resonant nature of this scheme allows us to neglect the

ûnt iresonant contribution. Hence. the susceptibility ) can be simplified to

wtiere A,, is the cletiining from resonance. i-e.. Qmi - d i . By assuming pml z /lmz and

71 - 7, ive can further sirnplib Eq. 5.6 to

which can be separated into the real and imaginary y" components.

The real component of the index of relraction is reduced to zero while the absorption

vanislies when the following condition is met:

where Bz1 = Oz - O1 is the relative phase of the coefficients ci and cz and where we assume

the entire population to be in the ground state. Ici l 2 + Equation 5.11 is solved

by letting cl = -c2 with Icll = a. Figure 5.4 presents xt(w1) and f"'wl) as a. function

of the detuning of the angular frequency dl of laser El from resonance with 1&& When

the condition of Eq. 5.11 is met: n' and n" vanish for al1 frequencies. To illustrate the

redoction in the index of refraction. ive plotted. in Fig. 5.4. nr (- ) ancl n" (- -

-) as a fiinction of the detuning il,, where we assumed a relative phase. 021. of 3.0.

C'urves (. . - . .) antl ( - - - - - ) show the reduct,ion of nr and nt'. respectively, when Eq.

5.1 1 approaches - 1. For these last calculatioiis. hl is assumed to be 3.1.

The following parameters have been used in the calculations:

0 tlensity of N2 at STP. psTp=--688-I.5 x 10" rno le~u ies / rn~~

0 angular frequency on resonance, dl = 1.9OOT:H.5 x 10L6 HZ,

El is a lineariy polarized field.

first eigenstate involved in the superposition. I@l)=lvi=O:J, = l:M,=- i ) of XZY,

second eigenstate involved in the superposition, 1&)=I v2=O?J2= 1 .MZ=+ 1) of ?<ET.

third eigenstate. Id3)= iu3=OJ3=0.M3=O) of blIIu,

Note that . the results of Fig. 5.4 do not involve the previous assumption, pml z pm2

iisecl in Eq. 5.S antl Eq. 5.9. The equations usecl for the calculation of Fig. 5.4 are

and

I t should also be noted that under different conditions,

CC)

I I

-1 0 -8 -6 -4 -2 O 2 4 6 8 10 a,, (GHz)

Figure 5.1: Modification of nt and n" due to the presence of a trapped state as a function

of the detuning il,, . The curves ( ) and (- - -) represent n' and n" respectively

witli a Bzl = 3.0. Curves (. - ) and ( - . - - - ) shows the reduction of nt am! n" when

the superposition approaches the perfect trappecl state. For the calculation of the ciirves

(. - . . a ) and ( - . - . -), a 021 of 3.1 is assumed.

it is possible to increase n' and n". Hoivever. to control ni adequately? there has to be a

rcsidiial population in Id3). This use of t he residual population in Id3) with a *phaseo-

niuni" are the foundations of t h e resonantly enlianced refractive index via quantum in-

terference t heory.

The creation of the superposition of differcnt 31 s ta tes was omitted in the above

calctilations because it presents some difficu1t~- in a thermally populated medium. T h e

superposition between 1 4i) and 1 oz) can be creat,ed using two circularly polarizecl lasers.

T h e left circuiarly polarized photon pumps ~ i p Iol) to a virtual intermediate state which

is t.hen piirnped down by a right circiilarly polarized photon t o lm2), as shown in Fig.

5..5 (a). Hoivever. t lie two photon process depicted in Fig. 5.5 ( a ) does not allow us to

create the required initial superposition if both s tate 1 & ) and 142) are initially equally

popiilated. This is due to the presence of the inverse process seen in Fig. 5.5 (b) .

Figure 5.5: Creation of the superposition of cliffcrent 31 states by a scheme involving a

tmo photon process. iising circiilarly polarized light. represented in ( a ) is not correct since

the real nature of tlie electric fields requires t h e presence of a counter superposition seen

in (1)). As a result. t he superposition iised in tlie calculations of Fig. 5.4 can not be

created by t his scheme if bot h M states Idi) and 142) a re initially equally populated.

To see this note tha t left and right circularly polarized light. E+I and E-i, are defined

b y

where q = I l . T h e definition of t he spherical coordinate basis set, ëq, is given in Eq. 3.2.

There exist four distinct terrns in Fig. 5.5 which are shown in Fig. 5.6 as left and right

circularly polarized light.

iot

Figure 5.6: Four components of the left ( a ) and right (b) circularly polarized electromag-

netic field. that are responsible for the creation of the superposition.

The coefficient cl>) and c r ) . where the superscript (a) and ( b ) refer to Fig. 5..j(a) and

( b ) respectively. are. using Eq. 2.45.

( 6 ) B - assurning an q u a 1 population in hoth states. 1cY)l = Ic, 1, the magnitude of the

coefficients cp' and cib' becorne equal. The terni clc; + c ; ~ of Eq. 5.8. 5.9 becornes

where 19 is the relative phase that results from the fields E t E,R' and the extra phase factor

of 7ï/2 is due to t h e denominator -iy of Eqs. 5.16,5.17. Consequently, the control over

the susceptibility vanishes. -4s a result. the irnplementation of a scheme such as the one

depicted in Fig. 5.7 is not possible if t h e initial population of Idi) and 142) are equal, such

as in degenerate thermally populated M states.

Figure 5.7: The equal distribution of t h e population betiveen the two states loi) and

1p2) eliminates the control of y(w) ancl iiltirnatcly of n ( w ) via this quantum interference

scheme.

5.1.2 Electromagnetically Incluced Transparency

Harris and cowor kers also suggest ed a t hree- lewl scheme of lasing wi t hou t inversion[24]

depicted in Fig. 5.S. A stroiig, coherent fielcl drives the transition 1 & ) - 1q3) while

lasing is experienced on the transition Ion) - loi) by a weaker probe field. The systern

is assumed to be in a state where the population of state Idi) is bigger than the sum

of t h e population in state Id2) and Ids)li.e.. > N2 + iV3 They presuppose a system

Figure 5.8: Modification of n' and n" of an electromagnetic wave resonant between states

lot) and 142) via the quantum interference effects of the different pathways due to the

strong coupling between Ievel Id3) and Id2).

witli RLy2 >R2+yI where RI and Rz are the purnping rates from 14,) and 1 9 2 ) to 143)

respectively. The stimulated Raman scattering, seen in Fig. 5.9(a), is responsible for the

cain experienced by the probe field El (nt' < 0 ) while the inverse process [Fig. 5.9(bj] is 3

Figure 5.9: (a) Chin from a Raman laser involred in AWI. Normally the inverse process

( h ) is t he main loss factor. In ACVI, gain by stirnulated Raman scattering turns out to be

possible even if X I >Xz+N3 due to the quantum interference of the different pathways.

responsible for the loss(nU > O)[%]. With an inversion of population between Ioz) and

In,). i t is possible t,o achieve gain at the liigher frcquency of laser El . However. Harris and

coworkers where capable of generating a gain even under conditions where !VI > rV2+N3 by

creating a hidden inversion. The absorption of a photon of El is canceled by a destructive

quantum interference of the two transition patlimays. shown in Fig. 5.10 ( a ) and (b) . and

procliices a cohererit population trapping. This scheme has considerable experimental

Figure 5.10: (a) First contribution to the absorption of a photon of the Raman laser.@)

Another contribution to the absorption of the photon of the Raman laser that can interlere

destructively with the first contribution shown in (a).

difficulties because of the stringent constraints on the system[25]. However, Harris and

CO-workers have observed significant reduction of absorption via quantum interference

which they termed electromagnetically induced transparency. To our knowledge, t hey

did not attempt. as in Scully's case to modify n'.

I t tvould seem natural to extend the work to the off-resonance scenario as in Fig. 5.1 1

since. ~inlike Scull'-'s scheme. the idea of Harris and CO-worliers does not need a population

Figure 5-11: The diffcrent interfering pathirays should. as in the resonant case. alloiv us

to create an interfering effect that cancels t h e absorption of the laser at the transition

19,) - (&) thus allowing us to have gain in a non inverted system.

in Io3) in order to create the desired effect. Harris has published some results of such

a. scheme[26]. Harris found that n' coold be recluced to ~inity. The reasons behind the

absence, in Harris' scherne, of an increase in n'. as is observed in OKE, is unclear.

5.1.3 Summary

Unlike C:E,OE;E? resonantly enhanced refract ive index without absorption via atomic co-

herence recluires tlie presence of some population in 1 @ 3 ) in order to achieve tlie desired

conti-01. Furthermore. it does involve a second controlling laser which in CEOKE creates

the parameter E 2 / E I . It would be intercsting to study further the equations to see the

effect of the various components of an electromagnetic field elliptically polarized. CVe have

also shown that the preparation of the "phaseonium" in a system consisting of degenerate

levels is more corn plex t han it first appears.

Flirt her considerat ion of the propagation equat ions of the elect romagnet ic waves has

to be done in order to study Harris' worli in detail. This should elucidate the reasons why

AWI or a strong absorption are not observed in an off-resonance scheme and why 0143

is not considered.

5.2 Limitations of CEOKE

A process in which the scattering events l e a w t lie states of the atoms or molecules un-

changed is defined as a parametric process. Cliaracteristically, these processes are con-

strained by phase matching conditions[-l]. CEOIiE is a parametric process and its major

limitation is due to its phase matching condition.

Çect ion 4.1.3 shows the dramatic result of rarying the relative phase between t h e

varioils components of the lasers in a C E 0 LE scenario. T h e major limitation of CEOKE

stems from the effect of the spatial propagation of the different laser beams on the relative

phase.OrCr. Assuming the medium is created wi t h a uniform superposition. we are st il1 left

with the task of phase matching the two lasers. n-hich couple Idl) and l&) t o the virtual

state. to a constant Orci = 7. The phase of t h e lasers. ciel. seen in Eq. 2.14 varies as

c l ( ~ ~ + ~ t ) where ic is the propagation constant or the wave number defined by

where n is t h e index of refraction, X is the ivavelcngth of the electromagnetic wave in the

mecliiim and the sobscript O is to denote tlie property in vacuum.

Since me are dealing with trvo Lasers ivit h clifferent wavelengths? we can expect some

difficulty in achieving a constant relative phase matching condition over a n extended in-

teraction lengt h. Hoirever. using lasers ivi t h siinilar wavelengths greatly increases the

interaction lengt h where the relative phase is acceptable. Figure 5.12 presents schemati-

cally the spatial variation of the two different wavelengths which gives rise to the spatial

variation of B r e l . As a result. we expect that t he interference between two states having

the same vibrational quantum number, vi = UL. hut having different rotational quantum

numbers. J i #J2' as seen in section 4.1.1. is going to have a longer coherence length than

a systcm where tlie vibrational quantum iiunibers. ul a n d 14, are not equal.

For a medium composed of Pi2 moleciiles. a qiiick calculation has tha t a change of the

relative phase of %. in Fig. 5.12 (a)? is going to take place in, at most, 2.2pm while this

same relative phase variation would occur for Fig. 5.12 (b ) on a length scale of, at most,

2.5nim.

F i e 5 . 1 A schematic of the coherence length of CEOKE ( a ) when the interference

is created between two states with different vibrational levels V I # v2 is shorter than the

coherence length of CEOKE (b ) when the interference is assumed between tmo different

rotational quantum numbers J l #J2 but tiaving the same V I = v2.

This interaction length (or coherence length) I~ecomes crucial since a large n' on a very

small length scale may not alter the propagation of the electromagnetic wave sufficiently

to allow CEOKE to be useful in practical applications. Consequently. the control of the

overall phase of thc rvavefront of El is what ive are seeking,

irliere 1~~ is the modification of the wavc nutiil->er produced by CEOKE on E l . Note

tliat the choice of 2 r as the modification is esplained below when we discuss the kinoform

[section 5.31. As was mentioned in section 2.4. the more powerful controlling laser is

not affected appreciably by the presence of the probe laser. Thus we can treat the wave

nuniber of the controlling laser' i i (wz), as a constant. The modification of rc(wl ) can be

written with Eq. .5.19 as

where nc is the indes of refraction produced hy CEOKE and n,,, is the normal index

of refraction. In the case of nitrogen n,,, can be considered negiigible if we are far from

cesonance.

Section 4.1 -3 has shown that the index of refraction is intimateiy linked to the relative

phase. Consequeritlc it is conceivable to invert this relation to make OrCr independent

of the spatial propagation by creating a specific nt . In other words. the increase in n'

procliiced by CEOKE is going to facilitate the phase matching condition since a bigger

n' decrease the wavenuniber of the probe field Ei. Assuming a ratio Ez/Ei ~ 1 0 . we can

achieve a n l ( u I ) of 1.00034 [see Fig. 4.11. This produces a perfect phase matching between

the two lasers since &(dl) = K ( I J ~ ) . AS a result. El and E2 in Fig. 5.12 (b) become esactly

superposed. Under such conditions, the length. z . needed to achieve Eq. 5.20 is about 4

mm.

I-sing a superposition s ta te with v l f -. would at a minimum require a change in

n ' ( d l ) of 0.14 to achieve K ( w ~ ) = ~ ( w < ~ ) . This woi~ld reducegreatly the length of interaction

needed to achieve Eq. 5-20, however. the ratio &/El would have to be at least bigger

than 106.

One of the advantages of OIiE and IiEE o w r CEOIiE is evident from t his discussion.

Both of these effects clo not require any phase iiiatching conditions. Tlius. even though

the effect of OKE ancl KEE on the n' is cpite small. it is possible to observe the effect

of OIiE and KEE since the interaction length can be arbitrarily long. Nevertheless. the

strong control over n' produced by CEOIiE m a - allow us to have a shorter interaction

lengt h. This may allow us to. for example. miriiat urize some electro-optic devices such as

rnult ipiexers and demoltiplexers.

Hoivever, a number of other factors have to be considered before thinking of imple-

menting CEOKE. The inclusion in the previous theory of the coherence degrading process

sucli as Doppler broadening and collisions have to Ile considered in order to understand the

limits of CEOKE under different conditions. LVe expect that t he influence of both factors

will degrade the overall control provided by CEOKE. The decoherence due to collisions

rvould seem to forbid CEOKE in a CW rcgime. However, under certain conditions, it has

heen proven that the coherence between the states involved in t h e superposition can b e

maintained[27]. Thus. it is espected that C'EOICE is still valid in certain regimes where

the clccoherence effects would only part ially recluce the number of molecules involved in

the superposition.

Furtherrnore. we have incorporated the damping phenornena into the theory in a

crude way, by including the factor in Eq. 2.32 and the subsequent equations. This

procedure is not totally acceptable, because it cannot describe dephasing processes that

arc not accompanied by the transfer of population. Therefore. in order to study further

the effects of decohcrence. it would be appropriate to pursue this work with the density

mat rix formalism[2ll.

The Doppler broadening can also be seen as a factor that will reduce the number of

molecules involved in the superposition state since not al1 the moleciiles will satisfy the

conditions of resonance, wl - 122 = of the two lasers with the molecular system.

\+'e have seen in section 4.1.4 that such a reduct ion in the number of molecules involved

in the superposition can have some drarnat ic consequences. Scully and coworkers have

investigated the effect of Doppler broadening on the resonantly enhanced refractive index

without absorption via atomic coherence. The>- found that under certain conditions. the

inhomogeneous broadening can wash out the coherence effect.

Further ivork is needed to fully understancl the propagation of the different lasers in-

volvccl in CEOIiE ancl to test the stability of the index of refraction to perturbation due

to. for esample, noise in the phase of t h e lasers or noise in the amplitude of the laser.

/\net lier possible area of further researcli is the investigation of the pulse or multiple fre-

quencies regime. The applications t hat are cliscussed in section 5.3 would ~ r o b a b l y have to

be implemented in such a regime. The developnient of the equation of propagation. would

be necessary to evaluate the group velocity which would become the crucial parameter to

control in that context.

5.3 Applications

The potential applications of coritrolling nt via qiiantum interference are numerous. Scully

ment ions a number of interest ing examples[ll)] :

laser part icle acceleration.

optical microscopy.

atomic tests of electroweak physics

0 and magnetornetr.

b-e can increase this the list by focusing on the rapidly erpanding field of communi-

cations. -4s was previously mentioned. the modification of nt by the Kerr electro-optic

effect is used in a nrimber of devices to modify the branching of an incoming pulse of light

or to encode information in an electromagnetic wave by modulating its phase and/or am-

plit~ide. Today. wi t h the increasing demancl on processing speed. more emphasis is being

placecl on the idea of using light and liolograrns to process information. The idea is built

on the use of holograms to filter and redirect tlw incorning information. I t can be easily

understood if we ji~st perceire the hologram as a cornplex lens. This idea is illustrated in

Fig. 5.13.

Figure 5.13: The incoming signal can be filtered and redirected to the appropriate detec-

tors by a hologram. thus essentially forming a massively parallel optical switch.

By having the source act as the clock, the dynamic holographic lens can process in

parallel a two dimensional array of information in one clock cycle which can rcpresent an

enormous amount of information. Furtbermorr. the processing rate tvould be increased

since the speed achieved by an optical logic element is intrinsically faster than the speed

achieved by semiconductor logic[%]. This lens could also prove to be an interesting

optoelectronic component which could manipulate the information while making the link

bet~veen various electronic and optical devices.

In essence? a hologram is just a comples "diffraction grating". A simple object, such

as a point. can illustrate this concept. Shining coherent source on this point creates

sptierical wave fronts. L'sing the same coherent source. it is possible to interfere the

plane wavefront coming from the coherent source and the spherical wavefronts emanating

Irorii the point source. With a n emulsion. we can record the image of the interference of

these t.i-t.0 wave fronts which is a number of concentric circles. These concentric circles

form basically a "diffraction grating'. Sliining t lie coherent source through the diffraction

grating recreates the wave fronts that wvere produced by the object. Modeiing more

coniples objects is simply done by increasing the number of points. Figure 5.14 presents

different types of gratings which in essence. represent simple types of holograms.

Amplitude Amplitude

- O L N 1.01- Amplitude

l - O F

Phrise f hase Phase

7 -;p 7

............................ ...................... ...................... 0 0 0 8 &

-7r X X X

(4 (b) ( c )

Figure 5.14: Transmit tance functions of a binary amplitude grating (a), its phase version

( b ) and a bleached grey grating (c).

Due to the possible applications in optical computing, a number of different computer-

generated holograms have been investigated [30]. The kinoform[29] represents one of the

simplest implementation of a hologram which could be created by CEOKE. The idea.

as in the binary phase grating [Fig. 5 . i ~ ( b ) ] is hased on modifying the plane cvave front

of the coherent source by modulating the phase at various points in a plane to recreaie

the wave fronts that would be emanating from the object we want to create. Figure 5-15

prescrits schematically an example of a binary kinoform. Thus, using Eq. 5.20 and the

Figure 5.15: Simplified example of a kinoform which is basically a phaser version of a

binary a.mplitude hologram.

subsequent discussion. me know that it rvoitld bc possible to create a dynamic kinoform

in air by using CEOIiE.

The optoelectronic components. in Fig. 5.1:1 could also be replaced by the human eye.

The formation of real-tirne dynamic holograms could produce a 3 dimensional display

terminal. The concept of 3 dimensional television has been around for a number of years.

.A group at MIT have produced a prototype which uses an acousto-optical rnodulator to

create the variation in n' needed to produce the phase variation discussed above[3 11. Such

devices could be invaluable in domains such as medical imaging, cornputer-aided design

and navigation. The advantage of using CEOKE would be to create the holographic Lens

in air. This is one of the major reasons which justifies the present study of CEOIiE in a

medium of N2 molecules.

Chapter 6

Conclusion

This thesis has shown that it is possible to create. through the coherently enhanced

optical Kerr effect (CEOIiE). a substantial variation of the real component of the index

of refraction ( n t ) while maintaining a negligible imaginary component of the index of

refraction (nt'). In ot her words. CEOIiE is capable of controlling the propagation of an

elect rornagnetic wave wit hout modifying its ampli tude. and hence its energy. as it. t ravels

throiigh a medium. We applied CEOIiE to a medium composed of molecules of N2 to

stiidu the implementation of such a scheme in a realistic medium. Of the case that we

esamined, the superposition with the states Ivl =O.Ji=O,Mi=O) and 1u2=0,J2=2.M2=0)

created the largest variation of nt. CEOKE lias shown to be a more powerful technique

then the optical Kerr effect (OKE) or the self foçusing effect (SFE) in controlling n' of the

elcctromagnetic ivaïe as it propagates througli air. However. unlike OKE and SFE. we

have also shown that CEOKE has some possible shortcomings due to its phase matching

conditions. Xe~ert1ieles.s~ we have shown that it is possible to create a medium in which

the phase matching conditions are met.

Further study is needed to investigate the stability of the different parameters as

the electromagnetic wave propagates through the medium since spatial propagation and

the temporal variations have direct effects o n t lie parameters, consequently producing

a feeclback mechanism on CEOKE. This coiild prove to be an advantage under certain

conditions. GVe could imagine that the substantial change between the absorption and

the amplification of an electromagnetic wave produced by CEOKE near resonance could

be iised t o create a binary optical switch. In addition. the decoherence of the medium

due to colIisions and other factors needs to be hetter addressed by introducing the density

matris formalism.

In conclusion. CEOKE has shown to provide a substantial control over the index of

refraction. The possi bili ties of applications are numerous:

a laser particle acceleratioci.

optical microscopy,

rn at-omic tests of electroweak physics.

magnet omet ry.

a clynamic holograms to create a 3 dimensional terminal,

a rnultiplexing and demultiplexing of signals.

a electroptic elements.

Tliese implementations of CEOKE require more work to fiirther our understanding of

certain basic principles such as the propagation of the electromagnetic wave witli C E 0 KE.

The underlying principle of quantuni interference involve in calculating the wavefunc-

tions should also be investigated further. The use of different schemes wi th the same

underlying principle could prove to have other interesting properties. This could prove to

be a fertile area for innovative devices.

Bibliography

[ I I 11. Fleischlauer. C H . Keitel. M.O. Scully. C'Su. B.T. Ulrich and Shi-Yao Zliii. Phys.

Reo. A. 46 (19%) 1468.

[2] P. Brurner and A I . Shapiro. Act-of Ckeni. Rcs.. 22 (1994) 407.

[ 3 ] Ci. .Arken, !Ilathernatical Methods j'or Phgsicists (Academic Press. Inc., San Diego.

19S.5)

[-LI D. C. Hanna..LL. A. Yiiratich and D. C'otter. .Vonlinenr Optics oJ Free -4torn.s and

.Ilolecules (Springer-Verlag. Berlin. 1979)

[.il 11. Fleischhauer. C H . Iieitel, hI.O.Sco1ly. C 'Su. B.T. Ulrich and S.Y. Zhu P h p . Rev.

-4. 46 (1992) 1-168.

[6] R. L. Liboff, Introd uctory Quantum .Clechnnics ( Holden-Dav. San Francisco. 1980)

[il R. W. Boyd. :\.onlinear Optics (.AcacIemic Press. Inc.. San Diego. 1902)

[Y] A. Yariv. Quant urn Electronics (.John CVilcy k Sons, Inc., Toronto: 1989)

[9] P. Brumer, private communication.

[IO] D.R. Lide and H.P.R. Frederikse Handboob of Chemistry and Physics (CRC Press,

Inc.. New York, 1996)

[ I I ] I .N. Levine. Quantum Chemistry uolume II: iI.loleculnr Spectroscopy (Allyn and Ba-

con. Inc.. Boston, 1970)

[l?] S. K. Lin, Y. Fujimura. H. J. Neusser. E. W. Schlag, Multiphoton Spectmscopy O/

:CIolecules (Academic Press- Inc., Orlando. 1984)

[1:3] R. N. Zare. .-lngular !blomentun (John Wiley & Sons, Inc., New York. 198s)

[l4] G.G. Balint-Iiurti and M. Shapiro. C'hem. Phgs.. 61 (1981) 137.

[15] .A-Lofthus and P.H. Iirupenie. J. Pfiys. C%cin. Ref. Data, 6 (1977) 113.

[ IG] D. Stahel.M.Leoni a n 9 I<.Dressler. .J. C ' h e m . P h p . 79 (1983) 2541.

[l'il G. Campolet i. private communication.

[ l S] Ci. Herzberg. Spectra of diatomic molecules ( Van Nostrand Rein hold. Inc., New York.

1 %O)

[20] S.E. Harris. .J. E. Field and A.Imamoglu Phys. Rev. Lett. 64 (1990) 1107.

[21] P. Mandel. Conternporary Physics 34 ( 1993) 235.

[22] G. Alzetta, :\. Gozzini, L. Moi and Ci. Orriols, Nuouo Cim B. 36, (1976) 5.

[ 2 3 ] .\~I.O.Scully. :\.Y .Zhu and A. Gavrielides Ph 9s. Reu. Lett., 62 ( 1989) 2813.

2 - .-1. Imamoglu. .J.E. Field and S.E. Harris. Phys. Reu. Lett. 66 (1991) 1154.

[-.il W. Gawlik, Comments At. Mol. P h p . 29 ( 1993) 189.

[2û] S.E. Harris, Optics Letters 19 (1994) 201s.

[27] M. Shapiro and P. Brumer, J.Chem. Phys.. 90 (1989) 6179.

['ZS] P. Das. Lasers and Optical Engineering (Spring-Verlag, Inc., New York, 1991)

[29] L.B. Lesem, P.M. Hirsch and J.A. Jordan. Jr.? IBM J. Res. Deuelop. 13 (1969) 150.

[30] Sing H . Lee, C'omputer-Generated Holograms and Diffractive Optics ( S P I E Optical

Engineering Press , Bellingham , l g E ) .

[31] P. St-Hilaire. S. Benton and M. Lucente. J . Opt. Soc. Am. A 9 (1992) 1969.

Appendix A

Feynman Diagrams of Second Order

Perturbation Expansion

The :36 possible terms which results from the second order perturbation expansion of the

wavefunction, [Eq. 2-45], interacting with 3 lasers of distinct angular frequencies di: ~2

and u,.

Appendix B

Feynman Diagrams of Third Order

Perturbation Expansion

The 216 possible t,erms which results from the tliird order perturbation expansion of the

wavcfunction interacting witli 3 lasers of dist,inct angular frequencies c.1' ~412 and up-

Appendix C

Radial Funct ions Overlap

The radial eigenfunctions overlap,

between the US of the various dipole allowed electronic states and XX9 are compiled in

the following tables:

0 Table C. L to C.6 represent the radial functions overlap between XCJ and b"S;,

0 Table C.7 to C.8 represent the radial functions overlap between XCJ and ct1CZ,

0 Table C.9 represents the radial functions overlap between XC,f and eflC$,

0 Table C.10 to C.13 represent the radial functions overlap between XE: and bllI,,

0 Table C.14 represents the radial functions overlap between XE: and c'II,,

Table C.15 represents the radial functions overlap between XE: and o1 II,.

Table C.1: Radial functions overlap between the u=O t o 21 of XCO and v=O to 4 of b"C;

Table C.2: Radial functions overlap between the v = O to 21 of XE: and v=5 to 9 of b"C:

Table C.3: Radial functions overlap between the v=O to 21 of XE,+ and v=10 to 14 of

Table C.4: Radial functions overlap between the v=0 to 21 of ?<Cg+ and u=15 to 19 of

b" x;

Table C.5: Radial functions overlap between the v=O to 21 of XC; and v=20 to 24 of

Table C.6: Radial iunctions overlap between the v=O to 21 of XC; and v=25 to 28 of

Table C.7: Radial functions overlap between the v=0 to 21 of x C , ~ and v = O to 4 of c"C;

Table C.8: Radial functions overlap between the v=O to 21 of XC; and v=5 to S of c"C;

Table C.9: Radial functions overlap between the v=O to 21 of XC; and v=O to 2 of e"C;

Table C.10: Radial fiinctions overlap between t h e v=O to 21 of XCI and v=O to 4 of

bln,

- --

vbl il,

vxrg

O

i

3 d

3

4

95

6 - 1

s 9

10

I l

12

1 3

14

1 5

16

17

1s

19

20

'21

Table C. l l : Radial functions overlap between the v=O to 21 of XE: and v=5 to 9 of

b1nU

Table C.11: Radial functions overlap between the v = 0 to 21 of XE,+ and u=10 t o 14 of

b 1 n U

Vbl il,

US s+

Table C.13: Radial functions overlap between the u=0 to 21 of XC; and v=15 to 19 of

b1 IIu

Table C. 14: Radial functions overlap between the v=O to 21 of XE$ and u=0 to 4 of c'Il,

Table C.15: Radial functions overlap between the u=0 to 21 of XE: and v=O to 4 of ollI,

Appendix D

Data of n' and n" Produced by

CEOKE Near Resonance

Due to the number of parameters involveci in the full calculation of CEOIiE near reso-

nance. we have compiled the data that was not presented in section 4.3 in this appendix.

The parameters used for these calculations are iclentical to the parameters used to calcu-

late Fig. 4.16 and Fig. 4.17. Xamely, they are

Figure D.1 to Fig. D.5 represent the variation of nt as a function of A,, and O,,[. We

incrernent the parameter E 2 / E 1 from 0.0001 to 100 to allow us to see the influence of

the interference tcrms [Eqs. 4.9, 4-10] on the overall value of nt. Figure D.6 to Fig. D.10

represent nt' for the same conditions.

To help in understanding the evolution of the different values of nt in Fig. D.1 to Fig.

D.5, we have assign a specific line pattern for eacli value of nt:

Figure D.l represents the isosample lines of the values of n', mentioned above. seen nor-

mally in the three state system discussed in section 4.3 without the influence of CEOKE.

LVe see very cleariy that nt is not a function of Q r e l . Figure 4.13 is basically a cross-section

of this :3 dimensional surface. The curve ( ): which represents n l = l , separates the

region of stronger nt from the region of lower values. The same curve is used in Fig. D.6

to Fig. D.10 to separate the regions of absorption from the regions of amplification. This

aIlows 11s to view nt a quick glance the effect o l the different regions on the propagation

of the electromagnet ic wave El.

To show the evolution of the different isosample lines, we chose to represent the dif-

Serent values of n" wi t 11 the same curves in al1 t h e figures. The following values of n" are

symbolized by

Figure D.l: T h e \.alue of n' as a function of A,, and Brel where E2/Ei is extremely smali,

making the contri bu t ion of the interference ternis negligible. nf=l.5(- - -) , l( )

and O.85(- . a ) .

-50 -40 -30 -20 -10 O 10 20 30 40 50

41, (GHz)

Figure D.2: The value of n' as a function of A,, and Brel where E2/E1 is set to 1.

n'=1.5(-. - . -):l( ) and O.S5(. . - . .).

I I I l ' i l I I I 1

-50 -40 -30 -20 -10 O 10 20 30 40 50

% (GHz)

Figure D.3: The value of n' as a function of A,, and Brel where E2/Ei is set to 2.5.

n f = 1 . 5 ( - - - -).l( ) a n d 0 . 8 5 ( . . - . - - ) .

-50 -40 -30 -20 -10 O 10 20 30 40 50

h (GHz)

Figure D.4: The value of n' as a function of A,, and Brcr where E Î / E 1 is set to 10.

. - . - ) Y 1 ( ) O ( - - - . - ) and 0.1(- -).

% (GHz)

Figure D.5: The value of n' as a function of &, and OrCr where E2/EI is set to LOO.

n f = 5 ( . . . . . . . ). 1.5(- - - - -)A ).O.S5(- - . . . .) and 0.1(- -).

Figure D.6: The value of n" as a function of A,, and O,,/ where E2/EI is extremely

small to make the contribution of the interference terms negligible. n''=O. 1 ( - . O - - - ) and

O . O l ( - - - - -).

Figure D.7: The value of n" as a function of A,, and Brel where E2/Ei is set t,o 1.

1 ( . . . . . . . ), O . O l ( - - - - -).O( ) ancl -0.01(-. . - m.).

Figure D.S: The value of n" as a function of A,, and Orel where E2/Ei is set to 2.5.

1 ( . . . . . . . ), 0.01(- * - . - 1 :O( ). - O . O l ( - -) and -0.1(--).

Figure D.9: The value of nu as a function of A,, and Brel where E Z / E L is set to 10.

nn=l(- - -),O. i ( ....... ), O.Ol(- . - . -):O( ) , -O.O1(* . a . - ) ? -O.l(- -)

and -L(- - --).

Figure D.lO: The value of n" as a function of A,, and Br,/ where E2/ El is set to 100.

n f f = l ( - - -), O . l ( - - a ) , O( ). -0.1(-- -) and -1(- - --).

C V e see t hat the choice of t h e regions Brel = :. A,, < O and Brel zz 7, A,, > O offer,

once - /El >2.5 the possibility of creating a high n' with negligible n".

IMAGE EVALUATION TEST TARGET (QA-3)

A P P L I E D IWGE. lnc 1653 East Main Street -

,=-; Rochester. NY 14609 USA I I -- - - Phone: 71 6/W-0300 -- -- - - Fax: 716/288-5989

0 1993. Appiied Image. Inc.. All Rights Reseived