The analysis of second harmonic generation in sodium vapour presented here follows th a t of Uchiki et al [ 9 7 ] , but t h e i r t h e o r e t i c a l treatment is
extended by d is tin g u is h in g between homogeneous and inhomogeneous broadening processes, and by consid eration of the special case of single frequency fundamental r a d i a t i o n . The use of a s in g le frequency fundamental beam permits examination of many i n t e r e s t i n g e f f e c t s which would have been "smudged out" by a large lin e w id th la s e r .
The absorption of two-photons from the fundamental beam by an atom can induce a coherence between i t s 4D and 3S s ta te s . Under c e r ta in conditions t h i s can cause second harmonic generation in the forward d i r e c t i o n , by means of an e l e c t r i c quadrupoLe t r a n s i t i o n . The aim of the t h e o r e t i c a l modelling described here is to c a lc u la t e the value of the 3S - 4D
quadrupole moments under the conditions of i n t e r e s t . Expressions are thus obtained fo r the power and p o l a r i s a t i o n of the second harmonic beam as
functions of the laser frequency, the magnetic f i e l d strength and the power and p o l a r i s a t i o n angle of the fundamental l i g h t .
The atomic system is considered to consist of the three groups of states l i s t e d in the previous s e c tio n , th a t i s , the 3S ground s t a t e s , the 3P interm ediate states and the resonantly enhancing 4D states of the sodium atom. These are la b e lle d s , p and d r e s p e c t iv e ly . The Hamiltonian, Hg, of the atomic system includes f i n e , hyperfin e and Zeeman s p l i t t i n g terms as described above. The homogeneous lin ew id th enters the analy sis as a
damping constant in the two-photon-induced coherence between the upper and 3.12
lower s t a t e s . The inhomogeneous (Doppler) linew idth is taken in t o account by i n t e g r a t i n g the equations f o r the induced quadrupole moments across the Maxwellian v e l o c i t y d i s t r i b u t i o n of the sodium atoms, with a p p ro p ria t e ly s h i f t e d atomic resonant frequencies.
The expectation values of the quadrupole moment o s c i l l a t i n g at the second harmonic frequency can be deriv ed by consid eratio n of the time e v o lu tio n of the density matrix operator p [ 1 2 ] . The quantum mechanical system is
considered to be in a s ta te
(j)(r,t) = ^ c ^ ( t ) ' f ^ ( r ) , ( 3 . 2 . 1 ) n
where c ^ ( t ) = i|)(r,t) d t , and Y ^ ( r ) form a complete orthonormal set of e ig e n fu n c tio n s . The average value of an expectatio n value of an operator A is
Amn'
nrti
where A_ = [ Y * ( r ) A Y ( r ) dr. mn J m n ( 3 . 2 . 3 ) Density matrix elements are defined as the ensemble average
Pnm = <3.2.4)
and the m atrix form of these elements is known as the density matrix [ 1 2 ] . The time e v o lu tio n of t h i s matrix may be determined from the time-dependent Schroedinger equation
K <i>(r,t) = ib ^ ( r , t ) , ( 3 . 2 . 5 ) ôt
as each system in the ensemble must s a t i s f y t h i s equation. This leads to the L i o u v i l l e equation fo r the time e v o lu tio n of the density matrix
= 1 [ p , K ] , ( 3 . 2 . 6 )
ô t fi
The e f f e c t of c o l l i s i o n s may be introduced by modifying t h i s eqyation to = - 1 % p ] .j - %ij(p - p ) . j ( 3 . 2 . 7 )
where K = Kq + ( 3 . 2 . 8 )
= - p .E = V e x p [ - i w t ] + C . C . , ( 3 . 2 . 9 )
and
p -
expC-Kg/kgTlI ( 3 . 2 . 1 0 )t r T e x p C - H g / k g T ] )
is the thermal e q u ilib r iu m value of the density matrix [ 1 2 ] . Here is defined to be the r e l a x a t i o n r a te f o r the population in level i , which depends on spontaneous emission as well as c o l l i s i o n processes, and
j ( i j ) is the r a te at which p_.j relaxes to zero. This q u a n tity
describes the phase coherence between le v e ls i and j , which w i l l decay due to c o l l i s i o n a l dephasing. H/j is the p e rtu rb a tio n caused by the e l e c t r i c dipole i n t e r a c t i o n energy between the fundamental Laser f i e l d
E^= 8 e x p [ - i w t ] + c . c . and the atomic system, where e and w are the
amplitude and angular frequency of the fundamental f i e l d r e s p e c t iv e ly . A rep resen tatio n is chosen in which ( K n ) ■• = E . 6 . . , and w.. is defined asTJ 1 ] 1 1] 1J w_.j = (E^ - E j ) / b . The thermal e q u ilib r iu m values of the density matrix elements then take the form
j = 0 fo r i ^ j ( 3 . 2 . 1 1 ) p . . = e x p [ - E . / k T ] / ( £ exp[-E /kp.T]j 11 1 D J ( 3 . 2 . 1 2 )
states
The subscripts i and j correspond to the s, p and d states of the sodium atom. The equations of motion of the density matrix are then
Ôp.. = - i w . j P . j - ^ C V , p ] . j - %%j(p - p > i j ( 3 . 2 . 1 3 ) where [V ,p ] = 2 ( p . ^ y ^ . - jj .^p^. ) E ( t ) , ( 3 . 2 . 1 4 ) and ju_.j is the i j element of the dip ole moment operator. An i n t e g r a l
p e rtu rb ic s o lu tio n may be found f o r p [12] as
(t) = ( - i / b ) J e xp[i(wUj-i%%j)(t'-t)]
X [ V ( t ' ) , p ( " " 1 ) ( t ' ) ] . j d f ( 3 . 2 . 1 5 ) (0) ^ (1) ^ (2) ^
where pUj = p^j P ij + + . . . . Co.2.16) and p^9^ = p . . 6 . . . r ij r 1] 1] ( 3 . 2 . 1 7 ) Equation 3 . 2 . 1 7 describes the populations of the states when no e l e c t r o magnetic f i e l d is present. In the magnetic f i e l d s used in t h i s study the' maximum s p l i t t i n g of the 3S sta te s is less than 0.1% of k^T, so
the e q u ilib riu m populations of these states are e s s e n t i a l l y equal. The energy of the 3S - 3P t r a n s i t i o n is about t h i r t y times greater than kgT, so the populations of the 3P and 4D states are e s s e n t i a l l y zero in thermal e q u ilib riu m . The zero value fo r o ff-d ia g o n a l matrix elements p.jj (i ^ j ) is due to the lack of coherence betweeen states when no
electromagnetic f i e l d is present; any coherenece th at had been generated by an electromagnetic f i e l d p revio u sly is assumed to have been lo st due to damping processes.
Equation 3 .2 .1 5 may now be evaluated to give the higher order terms in