7 .5 M ultiph ot on S pe c tro s c opy
Ef - Ei =
j j : multiphot on ex cit ation
7 .5 .1 T w o - P h ot on A b s orption
* v irtu al le v e l , V (F i g . (b ) ) : re p re s e n t e d b y a lin e ar c o m b in a ti o n o f th e w av e f u n c t io n s o f a ll re al m o le c u l ar le v e l s |kn>
T he prob ability for a t w o- photon tr an sition bet w een the groun d st at e Ei and an ex cit ed st ate Ef :
A if
ifI1I2
[ if - 1 - 2 - v ( k1 + k2) ]2 + ( if / 2)2
|
k Rk iik- e11 -Rkfv ke12 + Rk iik- e22 -Rkfv ke21|
2
(7.53)
# fir st fact or : line profile of the tw o- phot on tr an sition
# second fact or : tr an sition prob ability for the tw o- phot on tr an sition
< adv ant ag es >
* can reach st at es w hich cann ot be populat ed by single photon tr an sition .
* can select the acces sible upper st at e by proper choice of the las er polarization .
* can obt ain Doppler - fr ee spectrum w ithout v elocity - selection of atom s by a counter - pr opag ating s chem e
< requirem ent >
* Each sin gle- photon tr an sition lines : opposite parity
7 .5 .2 D opple r - F re e M ulti - P h ot on S pe c tro s c opy
does not n eed v elocity selection becau se all m olecules in the ab sorbing st ates , reg ardles s of their v elocities , can contribute t o the Doppler - free tr an sition .
< principle >
r eson ance condition for the tw o- phot on ab sorption of a m olecule w ith a v elocity v :
( Ef - Ei) / = 1 + 2 - v ( k1 + k2) (7.55)
# If 1 = 2 and k1 = - k2 => all m olecules ab sorb , independent of thier v elocities , at the s am e fr equency 2 = 1 + 2
# Although the pr obability of a t w o- photon tr an sition is g ener ally much low er th an th at of a single- phot on tr an sition , the fact th at all m olecules in the abs orbing st at es can contribut e t o the sign al m ay out w eight the low er tr an sition prob ability .
# tw o- phot on ab s orption pr obability I2 => focu sin g !!
T w o- phot on ab sorption prob ability :
Wif
|
m{
R-im eim1 -Rm fv ek21 + R-ik eim2 -Rkfv e1k2}|
2
I2
[
( if - 2 )42 if+ ( if / 2)2 + ( if - 2 - 2 v if k )2 + ( if / 2)2+ if
( if - 2 + 2 v k )2 + ( if / 2)2
]
(7.56)
# fir st fact or : tr an sition pr ob ability for the tw o- phot on tr an sition
# fir st term in [] : Doppler - fr ee ab s orption sign al
# second, third term s in [] : Doppler - broadened b ackgroun d
* firs t t erm ≡ 2 × (s e c on d + th ird t erm s )
∵ Let a : incident photon, b : reflected photon ,
=> fir st t erm ≡ (ab + ba )2 : ab , ba ar e indistinguish able s econd + third term s ≡ (aa )2 + (bb )2
==> the area under the Doppler - profile is h alf of th at under the n arr ow Lorent zian profile w ith linew idth if = i + f
* Doppler b ackgr ound peak relativ e t o the t w o- photon sign al peak :
= i f
2 k vp
i f
2 D
ex ample)
7 .5 .3 Influ e n c e of F oc u s s in g on th e M ag nitu de of th e T w o - P h ot on S i g n al
t w o- photon ab sorption prob ability I2 => focu sing !!
T w o- Photon sign al :
S ( 1
2 if) Ni
|
m Rim e1- Rimmf e2|
2
z rI2( r , z ) 2 r d r dz (7.59) w here, I ( r , z ) = 2 P0
w2 e - 2 r2/ w2 (7.60)
w ith w ( z ) = w0 1 + z
w0 : beam size (7. 61)
=> sign al becomes m ax imum if
z
- z
dz
1 + ( z / w20)2 = 2 L t an - 1
(
2 Lz)
h as m aximum !!7 .5 .5 M ultiph ot on S pe ctro s c opy condition :
k pk =
k kk = 0
ex ample)
