The second harmonic p o l a r i s a t i o n produced in a quantised medium of atomic - ^ 0
density N by an o p t i c a l e l e c t r i c f i e l d E = ( 1 / 2 ) E „ e^^^ + c . c . is
P(2w) = N < l | e r | 3 > < 3 | e r | 2 > < 2 | e r | l > E^q e''^'^^. ( 2 . 2 . 1 )
4 ti^ (Wg^ - 2w) (Wg^ - w)
For media with d e f i n i t e p a r i t y , such as atomic vapours, at least one of the < i | e r | j > dipole matrix elements in equation 2 .2.1 must be zero. However, the a p p lic a t io n of an e x t e r n a l s t a t i c e l e c t r i c f i e l d can mix le v e ls of opposite p a r i t y , thus allo w in g a l l three dip ole matrix elements to be non-zero. The s t a t i c f i e l d imposes a p re fe rre d d i r e c t i o n on the vapour, and i t is no longer p erm issib le to quantise the angular momentum of the atom along the propagation d i r e c t i o n ; the momentum must be quantised along the d i r e c t i o n of the s t a t i c e l e c t r i c f i e l d . Conservation of angular
momentum among the th re e waves is possible in t h i s d i r e c t i o n , so second harmonic generation becomes allowed by both p a r i t y and angular momentum arguments.
For atoms and symmetrical molecules th ere is u sually no l i n e a r s h i f t in energy le v e ls with an e l e c t r i c f i e l d E^ ( l i n e a r Stark e f f e c t ) , as they do not possess permanent d ip o le moments which would i n t e r a c t with the f i e l d . However, an e l e c t r i c f i e l d can induce such a dip ole moment in the atom, p roportional to E^, and thus give r i s e to energy le v e l s h i f t s p ro p o rtio n a l to Eg. This is c a lle d the quadratic Stark e f f e c t . I f the states of
opposite p a r i t y are well separated in energy, second order p e rt u rb a tio n theory may be used to determine the e f f e c t of a weak e l e c t r i c f i e l d on the atom [203:
W(o(JM) = WgOxj) + | E g | ^ < f |« * 'J 'M |e r|K J M > |Z ( 2 . 2 . 2 ) ^ W(«J) - W ( a 'J ')
V(o^JM) = Y(«JM) - |Eg| <^Y((X'J'M) <w'J'M|erk%JM> ( 2 . 2 . 3 ) W(cKJ) - W (K 'J')
where W is the energy Level of the s t a t e . Eg is the s t a t i c e l e c t r i c f i e l d , and
y
and Y are the perturbed and unperturbed eigenfu nctions r e s p e c t iv e ly . These re la tio n s h ip s break down at f i e l d s which are s u f f i c i e n t l y strong that the Stark s p l i t t i n g s are comparable to the energy d iffe r e n c e s betweenstates of opposite p a r i t y . The same is tr u e when the basis s ta te s are almost degenerate, as in the hydrogen atom, or in high le v e l Rydberg s t a t e s , which have hydrogenic eig e n fu n c tio n s . In these cases the Stark e f f e c t can be shown to produce energy le v e l s h i f t s p ro p o rtio n a l to the e l e c t r i c f i e l d strength [ 2 0 ,2 1 3 .
As a s p e c if ic example of an e l e c t r i c - f i e l d - i n d u c e d mixing process in which only the quadratic Stark e f f e c t is in v o lv e d , consider the scheme reported by Bethune et al [223 where sodium vapour was used as the nonlinear medium. The angular frequencies and w^ of the o p t i c a l e l e c t r i c f i e l d s were chosen to be close to the 3S - 3P and 3P - 4D resonances. With no s t a t i c e l e c t r i c f i e l d a p p l i e d , the d ip o le matrix elements involved were
<3s t er 14d><4d 1 er 13pX3p I er 13s>, the f i r s t of which is zero. When a s t a t i c e l e c t r i c f i e l d was ap p lied to the vapour, eigenfunctions of opposite p a r i t y were mixed as described by eq. 2 . 2 . 3 . Equation 2 . 2 . 1 , modified fo r mixing ra ther than second harmonic g e n e ra tio n , then becomes
P(2w) = N / - < 3 s 1 er 1 n pX npl er 14d> + <np 1 er 14 d X 3 s | e£| np>\ . Eg + "2 - "np)
X < 4 d |e r|3 p > < 3 p |e r|3 s > E^Eg ( 2 . 2 . 4 ) ("4d - "1 - "2 - '^^ ("3p ■" Wf)
Thus the sum frequency p o l a r i s a t i o n is p roportional to the s t a t i c e l e c t r i c f i e l d Eg, and to the two o p t i c a l e l e c t r i c f i e l d s . This dependence on three
(3)