Nonlinear Optical Materials: Theory and Modeling - American

Chapter 1 Nonlinear Optical Materials: Theory and Modeling Shashi P. K a r n a and Alan T. Yeates 1

2

Space Electronics Division, U.S. Air Force Phillips Laboratory, 3550 Aberdeen Avenue, Southeast, Kirtland Air Force Base, N M 87117-5776 Polymer Branch, Wright Laboratory, Wright-Patterson Air Force Base, O H 45433-7750

Downloaded by FLORIDA STATE UNIV on March 19, 2013 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0628.ch001

1

2

With a view to developing new materials for technological applica tions, the theory and mechanism of nonlinear optical phenomena in various dielectric media are reviewed. The role of theoretical mod eling in advancing our understanding of the structure-property rela tionships of nonlinear optical materials is discussed.

I. INTRODUCTION 1

Since the discovery by Franken et al of optical second harmonic generation (SHG) in quartz crystal, a number of optical frequency mixing phenomena have been observed in a variety of dielectric media. Detailed descriptions of these processes, collectively known as the nonlinear optical(NLO)processes, the dielectrics in which these are observed, and their practical applications have already appeared in a number of excellent texts and symposiaproceedings. In this chapter, we present a brief account of the theory and modeling ofNLOmaterials with a view of their applications in the current and future technology. 2-6

7-9

NLO phenomena encompass a broad range of light (electromagnetic radiation) mediated processes from the commonly observed Raman scattering to the less com monly observed two-photon absorption and optical harmonic generation. In this chap ter, we will limit our discussion to thoseNLOphenomena which are principally de termined by electronic polarizations alone. It should be noted, however, that even in these cases, there may be some contribution from nuclear motions in the molecule or unit cells. 5,6

0097-6156/96/0628-0001$15.50/0 © 1996 American Chemical Society

In Nonlinear Optical Materials; Karna, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

2

NONLINEAR OPTICAL MATERIALS

T h e l i n e a r a n d n o n l i n e a r p o l a r i z a t i o n s are generally c h a r a c t e r i z e d b y t h e i r re spective susceptibilities. T h e b u l k electronic susceptibilities are defined b y a p o w e r series e x p a n s i o n of the p o l a r i z a t i o n P as a f u n c t i o n of the a p p l i e d electric

P = x H e r e , x^

( 1 )

• E + x

( 2 )

: E E+ X

( 3 )

field,

2

: E E E + • • •.

(1)

is the rath-order s u s c e p t i b i l i t y of the m e d i u m a n d E represents the t o t a l

electric-filed experience b y the s y s t e m . T h e nth. order s u s c e p t i b i l i t y is a tensor q u a n t i t y of r a n k ( n - f 1) w h i c h has 3 ^

n+1

^ elements. T h u s , x ^ is

a

second-rank tensor w i t h


s i x elements, x ^ is a t h i r d - r a n k tensor w i t h twenty-seven elements, x ^ is a f o u r t h r a n k tensor w i t h eighty-one elements, a n d so o n . O f course, not a l l the elements of a s u s c e p t i b i l i t y tensor are l i n e a r l y independent a n d i n most p r a c t i c a l m a t e r i a l s m u c h fewer elements are r e q u i r e d t o describe the tensor. I n g e n e r a l , N L O p h e n o m e n a are c h a r a c t e r i z e d as second-order or t h i r d - o r d e r de p e n d i n g o n w h e t h e r they are described p r i m a r i l y t h r o u g h the x ^

or the x ^

terms.

T h e S H G is a n e x a m p l e of a second-order p h e n o m e n o n i n w h i c h t w o p h o t o n s , each of a n a n g u l a r frequency a;, c o m b i n e t o produce a t h i r d p h o t o n w i t h a n g u l a r fre quency 2u. Some of the i n o r g a n i c m a t e r i a l s such as d i h y d r o g e n p h o s p h a t e ( K D P ) , l i t h i u m n i o b a t e , a n d b a r i u m t i t a n a t e h a v i n g large x ^ value have f o u n d a p p l i c a t i o n s as frequency doublers for the p o w e r f u l lasers used i n laser f u s i o n . F o r l o w p o w e r a p p l i c a t i o n s , o r g a n i c dyes w i t h s u i t a b l e x ^ upconverting semiconductor l a s e r s . ' ' 5

9

1 0

values have been f o u n d t o be useful for

These dyes m u s t be i n c o r p o r a t e d i n t o p o l y

m e r host for m e c h a n i c a l s t a b i l i t y a n d p r o c e s s i b i l i t y , w h i c h presents the m a i n c h a l lenge t o the development of S H G m a t e r i a l s . Since the dyes are i n h e r e n t l y disordered i n the host s y s t e m , there is a center of s y m m e t r y w h i c h destroys the

second-order

response. I n order t o overcome t h i s difficulty, the dye molecules are a l i g n e d i n a n intense s t a t i c electric field at a t e m p e r a t u r e above glass t r a n s i t i o n of the p o l y m e r . W h e n the p o l y m e r is cooled, the dye molecules are a l i g n e d p r e f e r e n t i a l l y g i v i n g rise t o a second-order

response. U n f o r t u n a t e l y , such systems are u n s t a b l e w i t h respect

t o t i m e a n d t e m p e r a t u r e , t e n d i n g t o r e t u r n t o t h e i r o r i g i n a l disordered state. F o r a m a t e r i a l t o be c o m m e r c i a l l y v i a b l e as a component i n a n " o n - c h i p " o p t i c a l device, i t m u s t m a i n t a i n a g o o d second-order

response w h e n exposed t o t e m p e r a t u r e s as

h i g h as 320 ° C . F o r m i l i t a r y a p p l i c a t i o n s i t m u s t also m a i n t a i n 9 5 % of i t s o r i g i n a l response after 10 years at 125 ° C .

1 1

These are very stringent requirements w h i c h w i l l

necessitate m u c h f u r t h e r research. T h e second-order N L O phenomenon of most interest for m i l i t a r y a p p l i c a t i o n s is the e l e c t r o o p t i c ( E O ) or P o c k e l s effect. T h e E O effect, m e d i a t e d b y a x ^ t e r m , arises f r o m the change i n the i n d e x of r e f r a c t i o n of a dielectric i n the presence of a s t a t i c or low-frequency

o p t i c a l field. T h e refractive i n d e x of the m a t e r i a l varies l i n e a r l y

w i t h the s t r e n g t h of the a p p l i e d electric field. T h e E O p h e n o m e n o n finds a p p l i c a t i o n s


1.

Theory and

KARNA & YEATES

3

Modeling

i n the development of active o p t i c a l interconnects a n d switches, w h i c h w i l l be used i n f u t u r e d a t a processing a n d c o m m u n i c a t i o n systems i n a w i d e range of a i r c r a f t a n d s a t e l l i t e systems. Here organic p o l y m e r s have a great advantage over i n o r g a n i c m a t e r i a l s . W h i l e the E O response of organic m a t e r i a l s is c o m p a r a b l e t o or b e t t e r t h a n the best i n o r g a n i c m a t e r i a l s , the low dielectric constants of the former result i n a 5

m u c h faster response. T h i s is because the p o l a r i z a t i o n i n organics results p r i m a r i l y f r o m electronic m o t i o n s a n d have m u c h s m a l l e r nuclear ( v i b r a t i o n a l )

component.

T h u s E O switches c a n operate at m u c h greater speeds (40 G H z ) . I n a d d i t i o n , o r g a n i c p o l y m e r s c a n be easily processed i n t o films a n d waveguides, m a k i n g o n - c h i p a p p l i c a t i o n s m u c h easier.


H o w e v e r , as w i t h the S H G a p p l i c a t i o n s , t h e r m a l a n d t e m p o r a l s t a b i l i t y of the o r g a n i c m a t e r i a l s r e m a i n s a n issue. T h e o p t i c a l q u a l i t y of the m a t e r i a l s is also a n issue t o be t a k e n i n t o a c c o u n t . O p t i c a l losses o n the order of l d B / c m or less m u s t be o b t a i n e d , m a k i n g the phase s t a b i l i t y a n d p u r i t y of the m a t e r i a l s a c r i t i c a l issue. M a t e r i a l s w i t h a g o o d c o m b i n a t i o n of properties for second-order on-chip device a p p l i c a t i o n s h a v e yet t o be developed. W h i l e second-order m a t e r i a l s are t a n t a l i z i n g l y close t o f i n d i n g a p p l i c a t i o n , t h i r d order m a t e r i a l s are m u c h f u r t h e r f r o m real a p p l i c a t i o n s . H o w e v e r , the possible a p p l i c a t i o n s for t h i r d - o r d e r m a t e r i a l s are j u s t as p r o f o u n d . T h e u l t i m a t e g o a l for the a p p l i c a t i o n of t h i r d - o r d e r m a t e r i a l s is the development of " a l l - o p t i c a l " ( A O ) c o m p u t i n g a n d s i g n a l processing. T h i s w i l l require the development of m a t e r i a l s t h a t h a v e a large nonresonant

value of x ^ • S u c h m a t e r i a l s m u s t be able t o undergo o p t i c a l l y

i n d u c e d changes i n the refractive i n d e x . So far the largest values of x ^ f r o m h i g h l y conjugated organic p o l y m e r s .

5 - 9

have come

W h i l e a large n u m b e r of such c o m

p o u n d s have been i n v e s t i g a t e d , none have d e m o n s t r a t e d the necessary m a g n i t u d e of X ^

t o be of p r a c t i c a l , or even l a b o r a t o r y , use i n a p r o t o t y p e A O s w i t c h . O n e m a j o r

p r o b l e m t h a t exists i n the development of l i n e a r (or quasi-linear)

conjugated

poly

mers for A O a p p l i c a t i o n s is the interference f r o m l o w - l y i n g t w o - p h o t o n a b s o r p t i o n s .

5

C u r r e n t l y i t appears l i k e l y t h a t because of h i g h o p t i c a l losses f r o m such a b s o r p t i o n s , l o n g - c h a i n conjugated p o l y m e r s w i l l p r o b a b l y never be usable for A O s w i t c h i n g a p p l i cations. A l t e r n a t i v e materials currently being investigated include two-dimensionally c o n j u g a t e d molecules, w h i c h have higher l y i n g t w o - p h o t o n s t a t e , a n d molecules w h i c h p r o d u c e a l i g h t i n d u c e d refractive i n d e x change as a result of cascaded optical nonlinearities.

second-order

12

T h e o u t l o o k for the t h i r d - o r d e r m a t e r i a l s is not e n t i r e l y b l e a k . Uses have been f o u n d recently for m a t e r i a l s w i t h a large resonant value of x ^ • T h e first is the t w o p h o t o n u p c o n v e r t e d l a s i n g . I n this a p p l i c a t i o n , a t w o - p h o t o n a b s o r p t i o n i n the 600 - 900 n m region of the s p e c t r u m can be used t o produce a p o p u l a t i o n i n v e r s i o n a n d l a s i n g i n the blue region (400 - 480 n m ) . H o w e v e r , as of t h i s w r i t i n g , o n l y superr a d i a n c e (single-pass a m p l i f i c a t i o n ) has been d e m o n s t r a t e d .

12

It is possible, at least


4


t h e o r e t i c a l l y , to place these m a t e r i a l s i n a c a v i t y to o b t a i n t r u e l a s i n g . T h i s process, w h e n r e a l i z e d , w o u l d p r o v i d e a n a l t e r n a t i v e t o the S H G for the p r o d u c t i o n of coher ent b l u e l i g h t . O t h e r p o t e n t i a l a p p l i c a t i o n s of such m a t e r i a l s i n c l u d e higher d e n s i t y o p t i c a l d a t a storage a n d high-frequency o p t i c a l c o m m u n i c a t i o n s . A successful m a t e r i a l w o u l d have t o floresce i n the b l u e , w h i l e h a v i n g a s t r o n g t w o - p h o t o n

absorption

i n the n e a r - I R . Since no s t r u c t u r e - p r o p e r t y r e l a t i o n s h i p s have been e s t a b l i s h e d for t w o - p h o t o n a b s o r p t i o n , c o m p u t a t i o n a l m o d e l i n g can have a m a j o r influence o n t h e development a n d successful a p p l i c a t i o n s of these m a t e r i a l s . These same m a t e r i a l s also show p r o m i s e for a n o t h e r a p p l i c a t i o n , n a m e l y , o p t i c a l l i m i t i n g i n the v i s i b l e a n d near-


I R . O p t i c a l l i m i t i n g is a m a t e r i a l p r o p e r t y by w h i c h l o w - i n t e n s i t y l i g h t is passed a n d h i g h - i n t e n s i t y l i g h t is a b s o r b e d . T h e m e c h a n i s m is governed a g a i n b y the t w o - p h o t o n a b s o r p t i o n process because, w h i l e the f r a c t i o n of l i g h t absorbed is i n d e p e n d e n t i n t e n s i t y for the l i n e a r region (Beer's l a w ) , the f r a c t i o n of l i g h t absorbed

of

through

t w o - p h o t o n process depends l i n e a r l y o n the i n t e n s i t y of l i g h t . T h u s intense l i g h t w i l l be m o r e s t r o n g l y a b s o r b e d

due t o the t w o - p h o t o n a b s o r p t i o n s l e a d i n g t o a l i m i t

i n g of the passed i n t e n s i t y . A s is clear, t h i s p r o p e r t y has i m p o r t a n t c o m m e r c i a l a n d m i l i t a r y a p p l i c a t i o n s i n h i g h - i n t e n s i t y r a d i a t i o n p r o t e c t i v e devices. T h e m i l i t a r y a p p l i c a t i o n s of the o p t i c a l l i m i t i n g process center a r o u n d p r o t e c t i o n of sensors a n d eyes f r o m h o s t i l e laser t h r e a t s . A g a i n , t h e o r e t i c a l m o d e l i n g d i r e c t e d t o w a r d a d v a n c i n g o u r u n d e r s t a n d i n g of the t w o - p h o t o n a b s o r p t i o n process a n d i t s r e l a t i o n s h i p s w i t h t h e s t r u c t u r a l features of m a t e r i a l s is of p a r a m o u n t i m p o r t a n c e for a speedy

development

of o p t i c a l l i m i t i n g m a t e r i a l s . I n w h a t follows, a b r i e f survey of the general t h e o r y of N L O processes is g i v e n i n S e c t i o n I I . I n S e c t i o n I I I , t h e o r e t i c a l models used t o describe the o r i g i n a n d m e c h a n i s m s of the second- a n d the t h i r d - o r d e r N L O m a t e r i a l s are r e v i e w e d . T h e recent o b s e r v a t i o n s of S H G i n s i l i c a glass based m a t e r i a l s is discussed i n section I V . F i n a l l y , the role of t h e o r y a n d m o d e l i n g i n developing new N L O m a t e r i a l s is s u m m a r i z e d i n Section V . II. Q U A N T U M

MECHANICAL THEORY

OF NLO

P H E N O M E N A

T h e first successful t h e o r e t i c a l e x p l a n a t i o n of N L O processes i n a n i n f i n i t e , ho mogeneous, n o n l i n e a r dielectric m e d i u m was p r o v i d e d by A r m s t r o n g et a l .

1 3

These

a u t h o r s were able t o describe various o p t i c a l frequency m i x i n g p h e n o m e n a , e.g. S H G a n d t h i r d - h a r m o n i c - g e n e r a t i o n ( T H G ) , b y i n t r o d u c i n g a generalized m a c r o s c o p i c n o n linear polarization, P

N

L

a n d s o l v i n g the M a x w e l l e q u a t i o n i n a n i n f i n i t e , a n i s o t r o p i c

d i e l e c t r i c m e d i u m . T h e i r theory was subsequently extended Pershan

1 4

by Bloembergen

and

t o finite b o u n d a r y c o n d i t i o n s w h i c h described the o p t i c a l h a r m o n i c gen

e r a t i o n i n e x p e r i m e n t a l s i t u a t i o n . A c c o r d i n g to t h i s theory, i n the presence of a n e x t e r n a l o p t i c a l electric field E , the t o t a l p o l a r i z a t i o n P of a d i e l e c t r i c m e d i u m c a n


1.

Theory and

KARNA & YEATES

5

Modeling

be expressed as t h e s u m of a l i n e a r p o l a r i z a t i o n P

L

and a nonlinear polarization P

N

L

defined, respectively, as P

= X

L

(

1

)

(2)

'E

and pNL

T h e s u s c e p t i b i l i t y tensors x^

^(2) .

=

E

E

+

x

(3) :

E

E

E

. . .

+

^

(3)

i n the above equations are r e l a t e d t o the p o l a r i z a b i l i t i e s

of the m i c r o s c o p i c u n i t s of the d i e l e c t r i c . C o n s i d e r i n g the a n g u l a r frequency of t h e i n c o m i n g l i g h t b e a m t o be ( j , the l i n e a r s u s c e p t i b i l i t y tensor,

of the m e d i u m c a n


be w r i t t e n as

xi>) = E E

() 4

I n t h e a b o v e e q u a t i o n , a\^ (w) is the element of the l i n e a r p o l a r i z a b i l i t y tensor at t h e pth. m i c r o s c o p i c site a n d N^^u)

is r e l a t e d t o the l o c a l field c r e a t e d b y t h e o p t i c a l

b e a m of frequency u at the same site. If we consider t w o o p t i c a l beams of a n g u l a r frequencies u\ a n d u

2

Lj , a

i n t e r a c t i n g i n a m e d i u m t o creat a t h i r d b e a m of frequency

t h e n t h e second-order s u s c e p t i b i l i t y tensor x^

1S

r e l a t e d t o the m i c r o s c o p i c

p o l a r i z a b i l i t y tensor, /3, as

xl'l.eK

= "2

+

=E E

(

W

» ) < * ( ^ ) ^ , * K

= "2 + «,)•

(5)

P

T h e t h i r d - o r d e r s u s c e p t i b i h t y tensor, x^ K c o r r e s p o n d i n g t o the g e n e r a t i o n of a l i g h t 3

b e a m of frequency u

a

f r o m the m i x i n g of three i n p u t beams of frequencies u\,

u, 2

a n d us is r e l a t e d t o t h e m i c r o s c o p i c p o l a r i z a b i l i t y tensor, 7 , as

Xa\,c,d( * U

= L J

= E E P

3

+^2

»i1{»*)*l ]w^

()

p

6

t,j,Ar,I

I n a c e n t r o s y m m e t r i c m e d i u m , the o d d - r a n k tensor (e.g., x^

a

n

(

l

P)

vanishes,

whereas t h e even-order tensors have n o n v a n i s h i n g values regardless of t h e s y m m e t r y of t h e m e d i u m . It is i m p o r t a n t t o note t h a t the m a c r o s c o p i c s u s c e p t i b i l i t y t e n s o r , has t h e p o i n t s y m m e t r y properties of the m e d i u m , s u c h as a c r y s t a l l a t t i c e

X^ \ n

as a w h o l e , whereas the p o l a r i z a b i l i t i e s ( a ,

7 , etc.) h a v e t h e s y m m e t r y p r o p e r

ties of i n d i v i d u a l m i c r o s c o p i c u n i t s , for e x a m p l e a n a t o m o n the p t h l a t t i c e s i t e , a u n i t cell c o n t a i n i n g several a t o m s , or a d i a t o m i c b o n d , t h a t c o n s t i t u t e t h e m e d i u m . B o t h , t h e m a c r o s c o p i c a n d the m i c r o s c o p i c tensors of a g i v e n order n,

however,

o b e y some c o m m o n p e r m u t a t i o n a l s y m m e t r y . F o r e x a m p l e , i n t h e second-order case, Pijk(v

a

= u

2

+ u>i) = Pkij{u\

= u

a

- u) 2

= Pjki(u2

= u

a

- ui),

w h i c h is also


6


t r u e for t h e elements of x ^ • S i m i l a r r e l a t i o n holds for the t h i r d - o r d e r s u s c e p t i b i l i t y ( p o l a r i z a b i l i t y ) tensor. I n other w o r d s , the frequency a r g u m e n t s m a y be p e r m u t e d at w i l l p r o v i d e d the C a r t e s i a n indices i, j, k, etc. are s i m u l t a n e o u s l y p e r m u t e d so t h a t a g i v e n frequency is always associated w i t h the same i n d e x . I n the l i m i t of l o w o p t i c a l frequencies, however, one c a n freely p e r m u t e the indices i, j, k, etc., w i t h o u t m u c h loss o f generality. T h i s a p p r o x i m a t i o n , w h i c h considerably reduces t h e n u m b e r of i n dependent elements t o describe the s u s c e p t i b i l i t y tensor, is k n o w n as the K l e i n m a n symmetry.

1 5

T h e s u s c e p t i b i l i t y tensors x ^ a n d x ^ i n eq. (3) are a measure of t h e N L O response o f a m e d i u m . T h a t i s , a large value for x^ - of a m e d i u m is associated w i t h


n

large o p t i c a l n o n l i n e a r i t y a n d a fast N L O response. F r o m eqs. (5) a n d ( 6 ) , one c a n easily infer t h a t t h e large o p t i c a l n o n l i n e a r i t y of a m e d i u m is d i r e c t l y r e l a t e d t o t h e N L O response o f i t s m i c r o s c o p i c u n i t s . W e w i l l see i n the next section h o w t h i s basic concept has p l a y e d a c e n t r a l role i n t h e development of new N L O m a t e r i a l s . A r m s t r o n g et a l

1 3

also derived q u a n t u m m e c h a n i c a l expressions t o c a l c u l a t e t h e

elements of t h e m i c r o s c o p i c p o l a r i z a b i l i t i e s j3 a n d 7 i n terms of t h e m a t r i x elements of t h e d i p o l e m o m e n t o p e r a t o r a n d energy of the e x c i t e d states. A d e t a i l e d q u a n t u m m e c h a n i c a l t r e a t m e n t of m i c r o s c o p i c p o l a r i z a b i l i t i e s i n a n o n a b s o r b i n g m e d i u m w a s later given by Franken and W a r d by O r r and W a r d

1 8

and W a r d

1 6

1 7

w h i c h was subsequently generalized

t o i n c l u d e resonance cases. T h e i r expressions for t h e elements of

j3 a n d 7 , respectively, i n a nonresonant case c a n be w r i t t e n as

Pabc(u*; ^ 1 , ^ 2 ) = ^K(u> ,LJi,U ) a

2

• p(w ,U>l,U> ) • a

2

y> 1 - hu )

(AE

mg

m

m

(AE

ng

.

- hi*) y

U

E^

l

a

g

E ^ g

In t h e a b o v e e q u a t i o n s , a,b,c,d

(AE

lg

- hu ) a

{AE

mg

- ^ 1 - hu )(AE 2

(AE

mg

- hu )

= x,y,z;

a

(AE

ng

- hwtUAEn,

ftwi)

-

ng

) + hu )J' 2

{

}

g represents the g r o u n d s t a t e , / , m , n are

the e x c i t e d states, a n d AE represents t h e energy difference between t w o states. T h e m a t r i x element w i t h b a r r e d operator is defined as =

— .


(9)

1.

Theory and

KARNA & YEATES

7

Modeling

T h e s u m m a t i o n s i n eqs. (7) a n d (8) are p e r f o r m e d over a l l t e r m s generated b y t h e p e r m u t a t i o n of t h e frequency a r g u m e n t s as i n d i c a t e d b y t h e o p e r a t o r p. T h i s i n eq. (7) leads t o s i x t e r m s a n d i n eq. (8) t o t w e n t y - f o u r t e r m s . T h e values of t h e n u m e r i c a l coefficients K(u ;ui,U2)

a n d K(u ;ui,u> ,U3)

a

a

depend on p a r t i c u l a r N L O

2

processes a n d have been t a b u l a t e d b y O r r a n d W a r d .

1 8

T h e O r r - W a r d t h e o r y of t h e

m i c r o s c o p i c n o n l i n e a r p o l a r i z a b i l i t i e s , also k n o w n as h y p e r p o l a r i z a b i l i t i e s , has p l a y e d a p i v o t a l role i n a d v a n c i n g o u r u n d e r s t a n d i n g of t h e m e c h a n i s m a n d o r i g i n o f N L O p h e n o m e n a i n a t o m s a n d molecules. T h i s t h e o r y has also p l a y e d a c r u c i a l r o l e i n t h e d e v e l o p m e n t of t h e " i n t e r m e d i a t e - s t a t e m o d e l " w h i c h has p r o v e d v e r y successful i n


e x p l a i n i n g t h e m e c h a n i s m of N L O processes i n m e t a l s a n d c r y s t a l s .

6

E q u a t i o n s (7) a n d (8) p r o v i d e a s t r a i g h t - f o r w a r d means for a q u a n t i t a t i v e p r e d i c t i o n of t h e N L O properties of a t o m s a n d molecules. A l l t h a t is r e q u i r e d is t h e m a t r i x elements of t h e dipole o p e r a t o r a n d t h e energies. A l t h o u g h t h e s u m m a t i o n i n eqs. (7) a n d (8) r u n over t h e e n t i r e e x c i t e d state space l e a d i n g u p t h e c o n t i n u u m , i n r e a l i t y there are f a r fewer m a t r i x elements i n t h e n u m e r a t o r w h i c h have sufficient m a g n i t u d e t o be of i m p o r t a n c e . T h e r e f o r e , i n q u a n t u m m e c h a n i c a l c a l c u l a t i o n s , i t often suffices t o p e r f o r m t h e s u m m a t i o n over a t r u n c a t e d space c o m p r i s i n g of t h e m o s t i m p o r t a n t states i n t h e excited-state m a n i f o l d . A n o t h e r q u a n t u m m e c h a n i c a l m e t h o d t h a t has p r o v e d t o be q u i t e useful i n the recent years for t h e o r e t i c a l p r e d i c t i o n of t h e l i n e a r a n d N L O p o l a r i z a b i l i t i e s is the d e n s i t y m a t r i x a p p r o a c h of S e k i n o a n d B a r t l e t t .

1 9

In Sekino-Bartlett theory, the

p o l a r i z a b i l i t i e s , w h i c h are p e r t u r b a t i o n enrgies of various o r d e r , t h e e x p e c t a t i o n value of t h e o p e r a t o r H ( r , 2 ) ( =

2 0

are c a l c u l a t e d as

+ A*' E ( r , 2 ) ) a s

2 1

p=. where,

( ) 1 0

is t h e t o t a l w a v e f u n c t i o n of t h e s y s t e m i n t h e presence of t h e e x t e r

n a l o p t i c a l f i e l d , E ( r , J). A T a y l o r series e x p a n s i o n of t h e a b o v e e q u a t i o n y i e l d s t h e following expressions:

19

a

a 6

(«) = - T r [ h ^ D ^ M ] ,

(11)

/W-w,;«i,a>2) = - T r [ h ^ D ^ ^ x , ^ ) ] ,

(12)

7a6cd(-w/3

Nonlinear Optical Materials: Theory and Modeling - American

Recommend Documents