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ATOMS INTERACTING WITH
ELECTROMAGNETIC FIELDS, MULTIPHOTON IONIZATION
G. Mainfray
To cite this version:
G. Mainfray. ATOMS INTERACTING WITH ELECTROMAGNETIC FIELDS, MULTI- PHOTON IONIZATION. Journal de Physique Colloques, 1982, 43 (C2), pp.C2-367-C2-386.
�10.1051/jphyscol:1982228�. �jpa-00221839�
JOURNAL DE PHYSIQUE
Colloque C2, supplément au n°ll, Tome 43^ novembre 1982 page C2-367
A T O M S INTERACTING W I T H E L E C T R O M A G N E T I C F I E L D S , M U L T I P H O T O N IONIZATION
G. Mainfray
Service de Physique des Atomes et des Surfaces, Centre d'Etudes Nucléaires de Saclay, 91191 Gif-sur-Yvette Cedex, France
Résumé. - Des atomes soumis à un rayonnement laser intense peuvent être io- nisés en absorbant N photons par l'intermédiaire d'états "virtuels" induits par le champ laser. L'ionisation à N photons varie avec 1'éclairement laser I comme I . L'ionisation multiphotonique est surtout gouvernée par des effets de résonance et des effets de cohérence temporelle du laser. A des éclaire-
7 9 - 2
ments laser modérés (10 - 10 W cm ) correspondant à l'ionisation à deux, trois ou quatre photons, les effets de résonance jouent le rôle essentiel en comparaison des effets de cohérence. Au contraire, à des éclairements laser
1 ? 1 ? -?
très intenses (10 - 10 W cm ) les effets de résonance sont totalement amortis alors que les effets de cohérence conduisent â une augmentation de N! de l'ionisation. En conclusion, on présente quelques nouveaux domaines dans lesquels les processus d'absorption multiphotonique jouent un rôle im- portant.
Abstract. - The non linear interaction between an intense laser radiation and atoms leads to ionization through the absorption of N photons from the laser radiation via laser-induced virtual states. The multiphoton ionization rate varies as a function of the laser intensity I as IN. We discuss the two most important effects which govern multiphoton ionization processes : resonance effects and laser-coherence effects. In a moderate laser intensity
7 9 ?
range (10 - 10 W cm" ) corresponding to the two, three or four-photon io- nization of atoms, resonance effects play a dominant role while the photon statistics of the laser radiation are relatively unimportant. On the contra-
il 13 -2
ry, in the very high intensity range (10 - 10 W cm ) required to obser- ve large N - order ionization processes, coherence effects play a dominant role through N! enhancement in the N-photon ionization rate, while resonance effects are completely damped due to the high laser intensity. We conclude by reviewing the possible applications of multiphoton absorption processes in other fields.
1. Introduction.-
Multiphoton ionization of atoms is a typical example of one of the new fields of investigation in atomic physics that lasers have opened up. It is only over the past five years that considerable progress has been made in this field thanks to advances in sophisticated theoretical treatments, as well as to the possi- bility of conducting very accurate experiments through a better control of all the parameters of powerful pulsed lasers. As the physics of multiphoton excitation and ionization of atoms is now well understood, it is the right time to present a survey of this mature field.
It is essential to begin with the basic physics involved. Let us consider the ionization of an atom by optical radiation. The well-known photoionization of an atom takes place when the photon energy is higher than or equal the atom's ioni- zation energy. However, an atom with an ionization energy E^ can be ionized by pho-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982228
C2-368 JOURNAL DE PHYSIQUE
tons w i t h an energy hV much l e s s than Ei, i f t h e photon f l u x i s s t r o n g enough, which, from a p r a c t i c a l p o i n t o f view, can o n l y be achieved w i t h l a s e r r a d i a t i o n . I n t h i s case, t h e atom has t o absorb several photons from t h e l a s e r r a d i a t i o n i n o r - der t o be ionized. T h i s can be done u s i n g two d i f f e r e n t methods w i t h two v e r y d i f f e - r e n t i n t e n s i t y ranges.
F i g u r e 1 shows s c h e m a t i c a l l y t h e f i r s t method, u s i n g as an example t h e i o n i - z a t i o n of an atom through t h e a b s o r p t i o n o f t h r e e photons o f d i f f e r e n t energies El, E and E
.
Each absorbed photon matches t h e energy d i f f e r e n c e between two atomic sgates. ?or t h e d i f f e r e n t jumps, each photon has a p l a t f o r m t o step on, which enor- mously f a c i l i t a t e s t h e t r a n s i t i o n . The l i f e t i m e o f t h e i n t e r m e d i a t e atomic s t a t e si s t y p i c a l l y s. T h i s m u l t i - s t e p i o n i z a t i o n process can be performed using dye l a s e r s d e l i v e r i n g d i f f e r e n t l a s e r frequencies
vl,
)J2 anddl,
w i t h an i n t e n s i t y o fn
The second method designated m u l t i p h o t o n i o n i z a t i o n r e q u i r e s a much higher l a s e r i n t e n s i t y , and can be performed w i t h a s i n g l e l a s e r . Figure 2(a) shows schema- t i c a l l y t h e four-photon i o n i z a t i o n o f an atom. The v e r t i c a l arrows i n d i c a t e t h e photons absorbed i n t h e four-photon t r a n s i t i o n from t h e ground s t a t e t o t h e c o n t i - nuum. One o f the most e s s e n t i a l featBres of a m u l t i p h o t o n absorption process i s t h a t i t occurs through laser-induced v i r t u a l s t a t e s which are n o t eigen s t a t e s o f t h e
Ground State
Fig. 1 F i g . 2
atom. I n p r i n c i p l e , such a m u l t i p h o t o n i o n i z a t i o n process does n o t r e q u i r e any i n - termediate atomic s t a t e . The laser-induced v i r t u a l s t a t e s r e l a t e d t o t h e photon energy and i t s harmonics a c t as atomic states, t h e corresponding l i f e t i m e s a r e however much s h o r t e r . We may r o u g h l y regard t h e atom as spending a t i m e
z
i n a laser-induced v i r t u a l e x c i t e d s t a t e . T h i s t i m e r i s o f t h e order o f one o p t i c a l cycle, t y p i c a l l y 10-15s. Consequently, t h e absorption o f photons through laser-induced v i r t u a l s t a t e s must occur w i t h i n a t i m e < l ~ - ~ ~ s . Therefore, t h e photon f l u x has t o be strong enough f o r t h e r e t o be a l a r g e number o f photons w i t h i n 10-15 s. We thus understand why m u l t i p h o t o n i o n i z a t i o n processes can o n l y be achieved w i t h an i n t e n - se l a s e r r a d i a t i o n .We now consider a more r e a l i s t i c s i t u a t i o n because t h e l a s t b u t one photon i s absorbed i n t h e dense p a r t of t h e atomic energy spectrum. When an atomic s t a t e i s l o c a t e d n o t t o o f a r from a laser-induced v i r t u a l s t a t e , t h e a f o r e mentioned
time, 2, can be determined from l/dE, where
bE
i s t h e energy defect as shown in f i g u r e 2 ( b ) , i . e . r = 3x
10-l1 s f o r6~
= 1 cm-l. Such a quasi-resonant process requires a lower l a s e r intensity. Furthermore, the resonant multiphoton ionization of an atom, corresponding t o6
E = 0 leads t o very i n t e r e s t i n g e f f e c t s which w i l l be considered in d e t a i l in section 3.2. Non resonant multiphoton ionization of atoms.
Non resonant multiphoton ionization of atoms i s the subject of one of the chapters of volume 18, the most recent volume of the Avances in Atomic and Molecular Physics s e r i e s /1/ Consequently, we will only b r i e f l y survey t h i s topic.
The N-photon ionization r a t e W i s given by W
=rN
lIN, whereCN i s the genera- lized N-photon ionization cross section. W i s expressed in s - l u n i t s ,rN
i s expres- sed in cm 2Ns N-l units and t h e l a s e r i n t e n s i t y I in photons cm-2 s-'.
Multiphoton ionizatioh cross sections have mainly been measured f o r alkaline atoms and r a r e ga- ses with currently available s o l i d - s t a t e l a s e r , a t a few selected wavelengths, and - with l a s e r i n t e n s i t i e s ranging from 10 - / W cm-' (two-photon ionization of a l k a l i n e atans) t o 1015 W cm-' (twenty-two-photon ionization of helium). From 1975 onwards, i t has been possiblet o
obtain a c c u r a t e r values with the a v a i l a b i l i t y of l a s e r s which have good s p a t i a l and temporal cohevence.Before giving absolute values f o r the non resonant multiphoton ionization cross sections of d i f f e r e n t atoms, we will present a very simple argument which leads t o an order of magnitude estimate of the N-photon ionization cross section.
Let us consider the simplest case, the two-photon ionization of an atom with a laser frequencywand an i n t e n s i t y I /2/. If an energy-conserving f i r s t - o r d e r t r a n s i t i o n were possible, the one-photon t r a n s i t i o n would take olace a t a r a t e
w
= G7.
I . where -r1,
the one-photon absorption cross section, i s t y p i c a l l y 10-l7 cm2. A second pho- ton can be absorbed i f i t i s incident within the time T which i s of the order of W - l , i . e . s. Again, the r a t e of t h e second event i sa;
I , so t h a t the ovtra.11 r a t e f o r t h e two-photon ionization will beW Z ~ I W - ' ~ ; J Therefore 0- =
X z
cm 4 s .I'
I f there i s an atomic s t a t e not too f a r from the laser-induced v i r t u a l s t a t e , we must replace 7 =w-' by
l / d ~ ,
where dE i s the energy defect as shown in f i g . Z ( b ) .This estimate might be of i n t e r e s t t o some of t h e readers of t h i s paper, due t o t h e continued i n t e r e s t multiphoton processes have aroused in new f i e l d s . Even though the argument leading t o t h i s crude estimate should not be taken l i t e r a l l y , the numerical value obtained i s in good agreement with experimental and theoretical data. For example, the two-photon ionization cross section of cesium atoms a t 528
nm
has been measured t o be r 2 = (6.7 f 1.9)x cm
4 s 3 I t i s in very good agreement w i t h d i f f e r e n t calculations /4-7/ as shown i n the following t a b l e . TheTwo-photon ionization cross section of Cs Measurement
(6.7
'
l o g ) x10-50 cm4s a th
= 528 nmNormand and Morellec (1980)
Calculations
~ x I o - ~ O
un
4 s Bebb (1966)I . ~ X I O - ~ ~ cm4s Teague e t a1
(1976)
1 . 2 x 1 0 - ~ ~ c m ~ s Crance and Aymar (1980)
6 . 6 ~ 1 0 -5OCm4 Rachman e t a l . (1979)
C2-370 JOURNAL DE PHYSIQUE
two-photon i o n i z a t i o n cross s e c t i o n s o f s i n g l e t and t r i p l e t metastable helium atoms a t 347 nm are r e s p e c t i v e l y (2.7 t 2)x and (1.5 f 1.4) x cm4s. /8/. The agreement between experimental d a t a and c a l c u l a t i o n s i s v e r y s a t i s f y i n g .
As f a r as three-photon i o n i z a t i o n processes are concerned, l e t u s r e t u r n f l r s t l y t o t h e simple argument which l e d t o an order o f magnitude estimate f o r C 2 . The o v e r a l l r a t e f o r t h e three-photon i o n i z a t i o n i s :
w =
cq
Iro-,
I Tq
I Theref o r ev3
=+.c
5 x cm6 s2T h i s estimate i s i n good agreement w i t h experimental and t h e o r e t i c a l d a t a on t h r e e - photon i o n i z a t i o n cross s e c t i o n s o f t r i p l e t and s i n g l e t metastable helium atoms
6 2
which a r e r e s p e c t i v e l y , a t 6943.5 nm., (3.0
*
2.2) x 1 0 - ~ ' cm s and (3.3 f 1.9) x an6 s2 /8/. The higher v a l u e obtained f o r s i n g l e t s t a t e s i s explained by t h e f a c t t h a t t h e 6% s t a t e i s o n l y 40.5 cm-I away from t h e two-photon resonance.Two general remarks have t o be made. F i r s t l y , t h e m u l t i p h o t o n i o n i z a t i o n cross s e c t i o n i s v e r y s e n s i t i v e t o t h e p r o x i m i t y o f a resonance. For example, two o f t h e values f o r r 4 published i n t h e l i t e r a t u r e f o r cesium d i f f e r by an order o f magnitude a t almost t h e same l a s e r wavelength: C4 = 7 . 5 x 1 0 - ~ ~ ~ cm8 s3 a t 1056 nm/3/
a n d r 4 = 1 . 0 ~ 1 0 ~ ~ ~ ~ cm8 s3 a t 1060 nm /9/. These values were obtained on e i t h e r s i d e o f t h e resonance on t h e 6F s t a t e a t 1059 nm. Secondly, some o f t h e m u l t i p h o t o n i o n i - z a t i o n cross s e c t i o n s o f a l k a l i n e atoms published i n t h e l i t e r a t u r e may be somewhat misleading due t o t h e c o n t r i b u t i o n t h a t dimers make t o t h e atomic i o n signal. T h i s
i s due t o t h e la.rge i o n i z a t i o n cross s e c t i o n o f dimers r e l a t i v e t o atoms. T h i s p r o - blem w i l l be considered i n d e t a i l i n s e c t i o n 4, and i s e s p e c i a l l y important i n t h e two-photon i o n i z a t i o n o f a l k a l i n e atans, and t o a l e s s e r e x t e n t i n t h r e e and f o u r - photon i o n i z a t i o n . From t h i s p o i n t o f view, t h e m u l t i p h o t o n i o n i z a t i o n of r a r e gases a t low d e n s i t y avoids t h e molecular problem completely.
Much experimental work has been undertaken on t h e non resonant m u l t i p h o t o n i o n i z a t i o n of r a r e gases. However, experimental i o n i z a t i o n cross s e c t i o n s can r a r e l y be checked a g a i n s t t h e o r y due t o t h e lacR o f r e l e v a n t c a l c u l a t i o n s .
I n N-photon i o n i z a t i o n w i t h l a r g e N values (as i n t h e case f o r r a r e gases), t h e N-photon i o n i z a t i o n r a t e W which v a r i e s w i t h t h e l a s e r i n t e n s i t y I, as I ~ , c h a r a c t e r i z e s much more p r e c i s e l y t h e non l i n e a r process than t h e g e n e r a l i z e d i o n i - z a t i o n cross s e c t i o n r N d e f i n e d as being independent o f I. N-photon i o n i z a t i o n o f r a r e gases has been i n v e s t i g a t e d i n h i g h l a s e r i n t e n s i t y ranges, up t o 1015 W cm-2 corresponding t o a l a s e r e l e c t r i c f i e l d o f t h e order o f i n t r a a t o m i c f i e l d s . I n these u l t r a s t r o n g l a s e r f i e l d s , no departure from t h e law was observed. For example, accurate measurements i n 22-photon i o n i z a t i o n o f helium w i t h a bandwidth-
l i m i t e d 15 ps l a s e r p u l s e a t 1.06pm show a I
'' ' '.'
i n t e n s i t y dependence/lo/.
Such an accuracy i s o n l y p o s s i b l e by measuring t h e s p a t i a l d i s t r i b u t i o n o f t h e focused l a s e r i n t e n s i t y /11/. None o f t h e experiments confirmed t h e t u n n e l i n g i o n i - z a t i o n hypothesis /12/ which was b e l i e v e d t o dominate over m u l t i p h o t o n i o n i z a t i o n i n u l t r a s t r o n g f i e l d s .
Emphasis should be g i v e n t o t h e f a c t t h a t N-photon i o n i z a t i o n w i t h l a r g e N values can be observed by simply i n c r e a s i n g t h e l a s e r i n t e n s i t y . As an example, f i g u r e 3 shows t h a t , by u s i n g a coherent l a s e r p u l s e a t 1.06,~m, t h e 4-photon i o n i - z a t i o n o f cesium occurs a t 10" W t h e 11-photon i o n i z a t i o n o f xenon a t 10 13 W an-2 and t h e 22-photon i o n i z a t i o n o f helium a t 1015 W cm-', f o r t h e same i o n i z a -
8 -1 t i o n r a t e W = 10 s
.
Multiphoton ionization rate IS-')
Laser intensity IW.
~ r n - ~ )
Fig. 3
3. Resonance e f f e c t s i n m u l t i p h o t o n i o n i z a t i o n o f atoms.
gy t u n i n g t h e l a s e r frequency, t h e m u l t i p h o t o n i o n i z a t i o n r a t e of atoms can be made t o e x h i b i t a t y p i c a l resonant c h a r a c t e r when t h e energy o f an i n t e g r a l num- ber o f photons approaches t h e energy o f an atomic s t a t e . T h i s occurs i n t h e W ex- pression when we sum a l l t h e i n t e r m e d i a t e s t a t e s w i t h resonance detunings i n t h e denominator. W increases d r a m a t i c a l l y w h i l e r e t a i n i n g a f i n i t e v a l u e due t o dam- ping terms a r i s i n g from t h e coupling o f t h e resonant s t a t e w i t h t h e ground s t a t e and t h e cont4 nuum.
7 9 2
3.1. Resonance e f f e c t s i n a moderate l a s e r i n t e n s i t y range (10 -10 W cm- ) The c h a r a c t e r i s t i c s o f t h e resonance e f f e c t s have been e x t e n s i v e l y i n v e s t i g a t e d i n t h e four-photon i o n i z a t i o n o f Cs atans w i t h a Nd-glass l a s e r p u l s e and t h e t u n i n g o f t h e frequency through t h e resonant three-photon t r a n s i t i o n 6S+6F.
/13-16/. T h i s example has been chosen f o r two reasons. F i r s t l y , c a l c u l a t i o n s can be c a r r i e d o u t w i t h a h i g h degree o f accuracy f o r Cs atans /17-19/. Secondly, t h e Nd- g l a s s l a s e r i s v e r y w e l l s u i t e d t o perform t h i s experiment a t 1059 nm because i t can d e l i v e r a coherent p u l s e and t h e wavelength can be tuned w i t h i n t h e 1052-1065 nm range. It i s of i n t e r e s t t o use a coherent p u l s e because resonance e f f e c t s can be i n v e s t i g a t e d w i t h o u t any m i x i n g w i t h l a s e r coherence e f f e c t s and l a s e r band- w i d t h e f f e c t s so t h a t a d i r e c t ccmparison w i t h t h e o r y can e a s i l y be made. F u r t h e r - more, i t should be p o i n t e d o u t t h a t a few years ago coherent pulses from high-power dye l a s e r were n o t a v a i l a b l e .
As we have seen p r e v i o u s l y , t h e i o n i z a t i o n t i m e o f an off-resonance multiphc- t o n i o n i z a t i o n processis g i v e n by 1/
6~ ,
i.e., 10-l1 secford^
= 3 cm-l. Here we a r e concerned w i t h resonance processes a t a l a s e r i n t e n s i t y of 10 w.cm-'; 8 t h e i o n i z a t i o n t i m e i s much longer and i s g i v e n by t h e l i f e t i m e o f t h e resonant s t a t e induced by t h e l a s e r f i e l d . With our experimental c o n d i t i o n s , t h e 6F resonant s t a t e i s much more s t r o n g l y coupled t o t h e continuum than t o t h e ground s t a t e . T h i s means t h a t t h e p h o t o i o n i z a t i o n r a t e from t h e 6F l e v e l i s much g r e a t e r t h a n t h e decay r a t e t o t h e ground s t a t e due t o s t i m u l a t e d emission. At t h e exact resonance, t h e i o n i z a t i o n t i m e T = 1/ ( T I ) , w h e r e c = 1.4x10-'~ cm2 i s t h e p h o t o i o n i z a t i o n8 2
cross s e c t i o n o f t h e 6F l e v e l . T z
lo-'
sec f o r I = 10 Wan- .
Thus t h e resonant m u l t i p h o t o n i o n i z a t i o n process r e q u i r e s a l a s e r i n t e n s i t y , r o u g h l y 100 times lower than f o r t h e off-resonance process, i n agreement w i t h t h e r a t i o o f t h e aforementio- ned t i m e s ( ( 1 0 - ~ / 10-'I). T h i s p o i n t i s v e r i f i e d by t h e r e s u l t s shown i n Fig.4, which shows t h e l a s e r i n t e n s i t y I r e q u i r e d t o forml o 4
ions as a f u n c t i o n o f r e s o -JOURNAL DE PHYSIQUE
Resonance detuning (cm-'1 F i g . 4
Laser frequency (cm-l) F i g . 5
nance detuning. The f u l l curve i s d e r i - ved f r m a t h e o r e t i c a l c a l c u l a t i o n /20/
and t h e experimental p o i n t s have been obtained using a single-mode l a s e r p u l - se /14/.
Conversely, i f t h e l a s e r i n t e n s i t y i s kept f i x e d , resonance e f f e c t s w i l l be expressed as an enhancement i n t h e number o f i o n s a t t h e resonance f r e - quency, as shown i n F i g . 5. T h i s r e s u l t was obtained using a bandwidth-limited 15-psec l a s e r p u l s e /16/. I t c l e a r l y demonstrates a s h i f t i n t h e resonance p r o f i l e s when t h e l a s e r i n t e n s i t y i s increased from 10 t o 10' w.cm-'. 8 As i s w e l l known, t h e exchange o f photons between l a s e r r a d i a t i o n and atoms s h i f t s and broadens atomic l e v e l s . These e f
-
f e c t s have been w e l l described w i t h i n t h e framework o f t h e dressed atom theo- r y /21/. Atcmic l e v e l s h i f t s induce a resonance s h i f t as shown i n Fig.5. T h i s resonance s h i f t i s l i n e a r w i t h r e s p e c t t o t h e l a s e r i n t e n s i t y
I,O=
4 I w i t h 4 = 2 c m - l / ~ ~ i n e x c e l l e n t agree- ment w i t h c a l c u l a t i o n s /20, 23/. As t h e resonance s h i f t i s l i n e a r as a func t i o n o f t h e l a s e r i n t e n s i t y ; t h i s means t h a t a photon i s absorbed and r e e m i t t e d i n a d d i t i o n t o t h e four-photon absorp t i o n leading t o t h e i o n i z a t i o n of t h e atom.The importance o f atomic l e v e l s h i f t s induced by t h e l a s e r f i e l d i s a l s o emphasized through t h e law d e s c r i - b i n g t h e v a r i a t i o n i n t h e number o f i o n s Ni as a f u n c t i o n o f t h e l a s e r i n - t e n s i t y
I
v e r y near t o t h e resonance.L e t us consider a l a s e r frequency t h a t g i v e s r i s e t o a small s t a t i c resonance detuning AE = E6F
-
E6S-
3Ep, where E i s t h e photon energy. ~ t o m i c l e v e lP
s h i f t s b r i n g about a dynamic resonance detuning
SE
= A E + d I, which can tune t h e resonance t o be c l o s e r o r f a r t h e r away, depending upon whetherAE i s p o s i t i v e o r negative. Consequently, t h e v a r i a t i o n i n t h e number o f i o n s formed w i l l be f a s t e r o r slower than t h e simple I 4 law. We observed indeed an law. F i u r e 6 shows t h e v a r i a -9b
t i o n o f K = ( l o g N i ) / ( b l o g l ) as a f u n c t i o n o f A E . T h i s r e s u l t has been obtained using a single-mode Nd:glass l a s e r p u l s e /14/, I n t h i s representa- t i o n , a resonance appears as a sharp phenomenon marked by a dramatic change
i n t h e n o n l i n e a r order K,which
-
6Log Ni no longer corresponds t o t h e6Log I number o f photons absorbed by
t h e atom. F o r A E values l a r g e r than 10 cm-l, atomic l e v e l s h i f t s become i n s i g n i f i c a n t 4 compared t o .A E, and t h e I law i s again v a l i d and charac- t e r i z e s t h e off-resonance four-photon i o n i z a t i o n process
I t should be p o i n t e d o u t t h a t a v e r y good agreement between t h e o r y and experiment i s obtained as f a r as t h e pre- ceding r e s u l t (Fig.6) i s con- , cerned, provided a l l t h e ex-
-
-30 -20 -10 0 10 20 perimental parameters a r e
brought i n t o t h e c a l c u l a t i o n s . L e t us g i v e an example con- Resonance detuning
bE
lcm") cerning t h e l a s e r i n t e n s i t y .K i s c a l c u l a t e d from K = ( h l o g N i ) / ( b l o g ~ ) , and t h e number o f ions, Ni, i s d e r i v e d from t h e v e r y simple expression
Let us consider a f i x e d l a s e r i n t e n s i t y t h a t i s assumed t o be homogeneous i n t h e i n t e r a c t i o n volume. I n t h e v i c i n i t y o f t h e resonance, t h e v a r i a t i o n o f K e x h i b i t s p o s i t i v e and negative values, which are i n c o n s i s t e n t w i t h t h e experimental r e s u l t s shown i n Fig.6. The assumption o f a homogeneous l a s e r i n t e n s i t y i n t h e i n - t e r a c t i o n volume i s u n r e a l i s t i c , because t h i s i d e a l c o n d i t i o n i s never f u l f i l l e d i n experiments. The experimental s p a t i a l d i s t r i b u t i o n f u n c t i o n o f t h e l a s e r i n t e n s i t y i s Gaussian i n most cases, so t h a t t h e l a s e r i n t e n s i t y I can be w r i t t e n
"
I = IM exp ( - ( r ~ r ~ ) ~ )
When t h i s Gaussian d i s t r i b u t i o n i s introduced,negative values o f K are no longer observed, which i s i n good agreement w i t h a1 1 experimental r e s u l t s on reso- nance effects i n t h e m u l t i p h o t o n i o n i z a t i o n o f atoms.
Such c o n s i d e r a t i o n s must be extended, because t h e experimental r e s u l t s shown i n Fig.6 were n o t obtained a t a f i x e d l a s e r i n t e n s i t y b u t f o r a f i x e d number o f ions.
By f u l f i l l i n g t h i s a d d i t i o n a l requirement, G o n t i e r and T r a h i n /23/ have c a l c u l a t e d t h e v a r i a t i o n o f K i n t h e v i c i n i t y o f t h e resonance, which i s i n v e r y good agree- ment w i t h experimental r e s u l t s .
As f a r as t h e w i d t h (FWHM) o f t h e resonance p r o f i l e i s concerned. Fig.5 s h w s a w i d t h o f 1 cm-I governed by t h e broad l a s e r bandwidth (1.4 cm-l) o f t h e 1 . 5 ~
lo-''
sec pulse. The s i t u a t i o n i s d i f f e r e n t when a single-mode l a s e r p u l s e w i t h a dura- t i o n o f 3 x sec and a narrow bandwidth of 20 MHz i s used. F i g u r e 7 obtained w i t h such a l a s e r p u l s e shows f o u r resonance peaks due t o a 0.3 cm-I h y p e r f i n e s t r u c t u r e of t h e 6s ground s t a t e o f t h e Cs atom and t o a 0.1
an-'
f i n e s t r u c t u r e o f t h e 6F resonant s t a t e /15/. The w i d t h o f each o f t h e f o u r resonance peaks i s 600MHz, much broader than t h e 20-MHz l a s e r bandwidth. T h i s i s due t o b o t h an i n t e n s i t y inde-pendent Doppler broadening, which c o n t r i b u t e s 300 MHz, and an i n t e n s i t y dependent broadening due t o t h e f a c t t h a t t h e l a s e r i n t e n s i t y i s inhomogeneous i n t h e i o n i z a - t i o n region. There i s a d i s t r i b u t i o n o f i n t e n s i t y between z e r o and a maximum value.
As t h e resonance s h i f t i s l i n e a r w i t h respect t o t h e l a s e r i n t e n s i t y , we a l s o have
JOURNAL DE PHYSIQUE
Laser Wavelength (nm)
a l a r g e d i s t r i b u t i o n i n t h e s h i f t s , which appears as a broadening i n t h e resonance p r o f i l e s and c o n t r i b u t e s 300 MHz f o r I = 2 x 1 0 ~ W cm-2. T h i s i n t e n s i t y dependent broadening i n - creases when t h e l a s e r i n t e n s i t y i s increased, as shown i n Fig.7, i n good agreement w i t h a c a l c u l a t i o n performed by G o n t i e r and T r a h i n /17/.
3.2. Temporal e f f e c t s i n resonant m u l t i p h o t o n i o n i z a t i o n processes.
Temporal aspects induced by t h e l a s e r p u l s e d u r a t i o n p l a y an important r o l e i n resonance e f f e c t s . Figure 8 ( a ) shows schematically t h e t i m e e v o l u t i o n o f t h e p o p u l a t i o n o f t h e resonant s t a t e nr
/lo/.
F i g u r e 8 ( b ) shows t h e t i m e e v o l u t i o n o f t h e corresponding m u l t i p h o t o n i o n i z a t i o n p r o b a b i l i t y P. Both f i g u r e s e x h i b i t t h r e e successive temporal regions.The p o p u l a t i o n o f t h e resonant s t a t e nr increases as t i n r e g i o n 2 I , r e a - ches a s t a t i o n a r y regime i n r e g i o n 11, and decays exponential l y t o z e r o i n r e g i o n 111. As f a r as t h e i o n i z a - t i o n p r o b a b i l i t y P i s concerned, t h e
Fjg. 7 temporal e v o l u t i o n obeys a t3 law
i n t h e f i r s t r e g i o n and a l i n e a r law i n t h e second r e g i o n before reaching an i o n i c s a t u r a t i o n (due t o t h e t o t a l i o n i - z a t i o n o f a l l t h e atoms) i n r e g i o n 111. It should be p o i n t e d o u t t h a t a time-inde- pendent i o n i z a t i o n r a t e , i.e., an i o n i z a t i o n p r o b a b i l i t y p e r u n i t time, can be de- f i n e d o n l y i n t h e second region.
The separation between r e g i o n s I and I 1 allows us t o d e f i n e a c h a r a c t e r i s t i c t i m e :
T = a I
( 6 ~ ) 2 + ( ~ 1 ) ~
3.3 Resonance e f f e c t s i n a h i g h l a s e r i n t e n s i t y (1013w cm-')
Resonance e f f e c t s i n t h e 1013 W cm-2 i n t e n s i t y range can be i n v e s t i g a - t e d i n t h e m u l t i p h o t o n i o n i z a t i o n o f r a r e gases u s i n g a tunable-wavelength coherent Nd:glass l a s e r pulse. We consider t h e twelve-photon i o n i z a t i o n o f k r y p t o n when a resonance occurs on a 5d o r 4d' s t a t e w i t h t h e penultimate photon absorbed. When t h e l a s e r i n t e n s i t y used i s h i g h enough, no s i g n i f i c a n t enhancement i n t h e number o f i o n s N Z i s observed, although t h e resonance e f f e c t s e x h i b i t a l a r g e v a r i a t i o n
~ n t n e e f f e c t i v e order o f n o n l i n e a r i t y K = ( logNi) / (
b
l o g I ) v e r y near t o t h e resonance/lo/.
I n c o n t r a s t w i t h t h e v e r y good agreement between t h e o r e t i c a l and experimentd r e s u l t s on resonance e f f e c t s i n t h e four-ohoton i o n i z a t i o n o f Cs i n t h e i n t e n s i t y ranqe o f
- l o 7 -
10 W an-2, t h e experimental r e s u l t s on resonance e f f e c t s f o r t h e 9 twelve-photon i o n i z a t i o n o f k r y p t o n a t 1013 W cannot, as y e t , be q u a n t i t a t i v e -l y explained due t o t h e l a c k o f r e l e v a n t c a l c u l a t i o n s . The absence o f such calcu- l a t i o n s i s m a i n l y due t o t h e l a c k o f a t a n i c d a t a and e s p e c i a l l y o s c i l l a t o r s t r e n - g t h s f o r r a r e gases. However, we can g i v e a q u a l i t a t i v e e x p l a n a t i o n i n terms of t h e
A
c h a r a c t e r i s t i c i o n i z a t i o n t i m e TP
i n t h e close v i c i n i t y of t h e reso-nance. The p h o t o i o n i z a t i o n cross s e c t i o n o f t h e resonant s t a t e ,
,
i s n o t known w i t h any accuracy f o r 5d o r 4d' s t a t e s i n krypton
.
I ti s estimated t o be between and 10-I' cm2. As t h e resonant m u l t i p h o t o n i o n i z a t i o n r a t e i s pro- p o r t i o n a l t o T, t h e v a r i a t i o n of T as a f u n c t i o n o f b
E
should g i v eb,
v a l u a b l e i n f o r m a t i o n on t h e r e s o - nance p r o f i l e . I n Fig. 9, T i s p l o t t e d f o r 1012 and 1013 W cmq2, assu- ming a p h o t o i o n i z a t i o n cross sec- t i o n 0- = cm2. ForI
= 10 13 W cm-', no s i g n i f i c a n t enhancement of T i s observed, which e x p l a i n s(1) I (1) i (1)
why no s i g n i f i c a n t resonance p r o - I f i l e i s e x p e r i m e n t a l l y observedunder t h e same c o n d i t i o n s . T h i s f l a t t e n i n g o f t h e resonance pro- f i l e a r i s e s from t h e d m i n a n t con-
I
2I
t r i b u t i o n o f t h e ( T I ) term due t oa)
t h e h i g h I v a l u e over t h e( F E ) ~
1
term throughout a v e r y broad reso-I nance detuning. Conversely, i f we
1
consider examples w i t h a s i g n i f i c a r tI
weaker value o f l a s e r i n t e n s i t y o rI a resonant s t a t e i n which t h e pho-
t o i o n i z a t i o n cross s e c t i o n i s
UI Time
smaller, a resonance p r o f i l e w i t h aT= ca~?+con2
small amplitude m i g h t s t i l l be ob- served, as i n a previous experimentF i g . 8 on t h e eleven-photon i o n i z a t i o n o f
Xe performed a t t h e Lebedev I n s t i - t u t e by Alimov and Delone /24/. I n t h i s experiment, a resonance p r o f i l e was obser- ved w i t h an enhancement o f t e n a t t h e most. The measurement of such a resonance p r o - f i l e , together w i t h a knowledge o f t h e l a s e r i n t e n s i t y as an absolute value, would a l l o w us t o determine w i t h accuracy t h e p h o t o i o n i z a t i o n cross s e c t i o n f o r a g i v e n a t a n i c s t a t e . Such d a t a a r e o f t e n missinq i n t h e l i t e r a t u r e .
The dramatic change i n t h e e f f e c t i v e order o f non l i n e a r i t y K = ( blogNi)/
(
b
1 o g I ) i s a1 so observed very near t o a resonance i n k r y p t o n a t 1013 W cm-', even when t h e resonance p r o f i l e i s t o o f l a t t o be measured. The v a r i a t i o n o f K i s e x p l a i - ned, as f o r Cs a t 10 8 W un-', i n terms o f atomic l e v e l s h i f t s induced by t h e l a s e r f i e l d . However a t 1013 W cm-', c a l c u l a t i o n s would have t o take i n t o account n o t o n l y t h e l i n e a r t e r m a I b u t a l s o higher-order terms 612,d,. . . ,
which are expected t o l i m i t s h i f t values.4. Ant iresonance e f f e c t s fn two-photon i o n i z a t i o n o f Cs atoms.
As i s w e l l known, t h e two-photon i o n i z a t i o n cross s e c t i o n ? a t l a s e r f r e - quency w i s g i v e n by second-order p e r t u r b a t i o n theory
JOURNAL DE PHYSIQUE
and i t i s necessary t o sum a l l t h e i n t e r m e d i a t e s t a t e s I n r o f energy
bun.
F i g u r e 10 shows t h e v a r i a t i o n of t h e two-photon i o n i z a t i o n cross s e c t i o n f o r Cs atoms, c a l c u l a t e d by Bebb /4/, as a f u n c t i o n o f t h e photon energy. The v a r i a t i o n o f 5 ex- h i b i t s b o t h successive resonance p r o f i l e s a r i s i n g from minima i n t h e denominator of t h e expression f o r ? , and deep minima due t o d e s t r u c t i v e i n t e r f e r e n c e among t h e terms of t h e sum. Second-order p e r t u r b a t i o n t h e o r y p r e d i c t s a f i r s t minimum f o r a 2.6 eV photon energy, i . e . about 480 nm, a r i s i n g from c a n c e l l a t i o n o f t h e 6P and t h e o t h e r nP s t a t e s w i t h a main c o n t r i b u t i o n from t h e 7P states.t
Characteristic time (s)Resonance detuning i E icm-')
Fig. 9 --
I t was expected t h a t t h i s minimum i n t h e two-photon i o n i z a t i o n cross s e c t i o n f o r Cs atoms would be d i f f i c u l t t o i n v e s t i g a t e e x p e r i m e n t a l l y due t o t h e c o n t r i b u - t i o n dimers make t o t h e Cs+ s i g n a l through processes such as
cs2
+
2 h ~ -CS;+
e f 01 lowed by CS;+
h)l+
CS'+
Cst o g e t h e r w i t h t h e two-photon d i s s o c i a t i o n o f Cs
.
As t h e d e n s i t y o f resonant states i s h i g h,
such processes can be enhanced resonangly, e s p e c i a l l y a t l a s e r wavelengths 1 1 3 - i n t h e 480 nm r e g i o n where t h e r e i s a s t r o n g Cs2 a b s o r p t i o n band : Cn r
X Z g.
Atomic CS+ ions r e s u l t i n g from these molecular processes are i n d i s t i n g u i s h a b l e
Photon energy lev) .Fig. 10 --
from CS' i o n s o r i g i n a t i n g from t h e two-photon i o n i z a t i o n o f Cs atoms.
The molecular d e n s i t y n i s much lower than t h e atomic d e n s i t y ncs, t y p i c a l - l y
2
= 5 x However, t h e two-photon i o n i z a t i o n and d i s s o c i a t i o n r a t e s f o r Cs2n r ,
Cs2 2:e much h i g h e r than t h e two-phobon i o n i z a t i o n r a t e f o r Cs atoms. Consequently, i t i s necessary t o minimize t h e dimer d e n s i t y . T h i s can be p a r t i a l l y achieved thrcu- gh a thermal d i s s o c i a t i o n i n a superheater. The molecular component r e d u c t i o n f a c - t o r i s r o u g h l y t e n /25/. There i s no p o s s i b i l i t y whatsoever o f d i s s o c i a t i n g a l l t h e dimers by t h i s process.
The most important parameter which makes t h i s experiment f e a s i b l e i s t h e l a s e r i n t e n s i t y . I t has been c l e a r l y demonstrated i n a previous experiment on four- photon i o n i z a t i o n o f Cs atoms a t 1.06 M m, t h a t t h e c o n t r i b u t i o n dimers make t o t h e
.
-CS' s i q n a l p l a y s a dominant r o l e w i t h l a s e r i n t e n s i t i e s l e s s than 10 . - 7 W cm-2 /26/.
I t i s t h e r e f o r e necessary t o use a h i g h i n t e n s i t y i n t h e 10'
- lo1'
W range.The problem can be solved by t a k i n g advantage o f t h e d i f f e r e n t i n t e n s i t y dependen- c i e s o f t h e number o f atomic Cs ions produced by t h e two-photon i o n i z a t i o n o f Cs atoms t h a t obey a law, and t h e number o f a t a n i c Cs i o n s o r i g i n a t i n g from t h e d i f - f e r e n t molecular channels t h a t become s a t u r a t e d a t h i b h l a s e r i n t e n s i t y and obey an law, w i t h K d l . The c o n t r i b u t i o n dimers make t o t h e CS' s i g n a l t h e r e f o r e do- minates completely i n t h e low l a s e r i n t e n s i t y range, w h i l e t h e cs+ s i g n a l o r i g i n a - t i n g from t h e two-photon i o n i z a t i o n o f Cs atoms dominates t h e dimer c o n t r i b u t i o n when t h e l a s e r i n t e n s i t y i s h i g h e r than 10 W 9
The two-photon i o n i z a t i o n cross s e c t i o n o f atomic cesium was measured i n t h e 460-540 nm ranqe i n an experiment t h a t f u l f i l l s t h e f o l l o w i n q two requirements : a l a s e r i n t e n s i t y o f
l o l o
W cm-', and a p a r t i a l thermal d i s s o c i a t i o n o f dimers i n a superheater /25/. Figure 11 shows t h a t C 2 e x h i b i t s a deep minimum, as p r e d i c t e d by second-order p e r t u r b a t i o n theory. T h i s r e s u l t has two important consequences:-
F i r s t l y , i t confirms t h e v a l i d i t y o f second-order p e r t u r b a t i o n theory, and c l a r i f i e s a p u z z l i n g s i t u a t i o n which arose from a p r e v i o u s experiment i n which t h e r e was no minimum i n t h e two-photon i o n i z a t i o n cross s e c t i o n o f cesium and i n which i o n i z a t i o n cross s e c t i o n values up t o f o u r orders o f magnitude higher than t h e o r e t i c a l values occured /27, 28/. T h i s r e s u l t can be explained because t h e ex- periment was performed i n a v e r y low i n t e n s i t y range ( l o 4- l o 5
W cm-2) where t h eJOURNAL DE PHYSIQUE
Laser Wavelength trim)
c o n t r i b u t i o n dimers make t o t h e
540 520 480 lbO CS' i i g n a l dominates completely
( )
I I
t h e CS+ s i r n a l o r i r i n a t i n q from5. Laser temporal coherence e f f e c t s
5.1.
-
Coherence e f f e c t s i n non resonant m u l t i p h o t o n i o n i z a t i o nlo-"
l o - 5 z -
2.3 2.4 2.5 2.6 2.7 resonance w i t h an i n t e r m e d i a t e s t a t e , t h e two-photon i o n i z a t i o n Photon energy (eV) cross s e c t i o n i s determined from o n l y a few m a t r i x elements. For
Fig. 11 example, i n t h e resonant two-
photon i o n i z a t i o n o f Cs atoms on t h e 7P s t a t e , i t has been shown t h a t t h e s i n g l e i n t e r m e d i a t e 7P s t a t e approximation y i e l d s r e s u l t s w i t h i n a f a c t o r o f 2 of t h e completely converged summations over
several hundred wavenumbers on e i t h e r s i d e o f t h e resonance /31/. At, o r v e r y near t o a minimum, t h e two-photon i o n i z a t i o n cross s e c t i o n i s determined f r a n a l a r g e number o f m a t r i x elements, as a l a r g e number o f terms m a n i f e s t c a n c e l l a t i o n prodd- c i n g t h e minimum. The exact p o s i t i o n and depth o f t h e minimum can o n l y be determi- ned by an exact i n f i n i t e summation of terms w i t h exact m a t r i x elements. A l l t h e c a l c u l a t i o n s published i n t h e l i t e r a t u r e p r e d i c t a minimum i n t h e two-photon i o n i z a - t i o n cross s e c t i o n o f Cs atoms, b u t t h e y disagree on t h e p o s i t i o n and t h e depth of t h e minimum. Thereflore, t h e experimental r e s u l t can be a v e r y s e n s i t i v e t e s t of t h e accuracy o f t h e d i f f e r e n t c a l c u l a t i o n models. The comparison o f t h e experimental r e s u l t s ( d o t t e d 1ine)with c a l c u l a t i o n s /5, 31-33/, i s shown i n F i g . 12. T h i s B i g u r e demonstrates t h a t t h e model p o t e n t i a l t h e o r y (curve C / 5 / and D 1 3 3 1 i n f u l l l i n e ) g i v e s much more accurate r e s u l t s than t h e quantum d e f e c t t h e o r y (curve A 1 3 1 1 and B /32/ i n dashed l i n e ) . T h i s i s because t h e f i r s t P l e v e l s g i v e t h e main c o n t r i b u - t i o n t o r2 a t t h e minimum, w h i l e t h e quantum d e f e c t method g i v e s m a t r i x elements o f poor accuracy f o r t h e f i r s t
P
l e v e l s . Furthermore, a good agreement between experimental r e s u l t and t h e model p o t e n t i a l t h e o r y i s obtained by t a k i n g i n t o account t h e continuum c o n t r i b u t i o n .t h e t ~ o - ~ h o t o n i o n j z a t i o n - o f Cs atoms. However, t h i s r e s u l t brought about a confused s i t u a t i o n /29/
and even l e d some t h e o r e t i c a l p h y s i c i s t s t o q u e s t i o n t h e v a l i d i - t y o f p e r t u r b a t i o n theory. F i n a l - l y , t h e observation o f t h e m i n i - mum, i n t h e two-photon i o n i z a t i o n cross s e c t i o n o f Cs atoms shown i n Fig.11, as w e l l as i n t h e three-photon i o n i z a t i o n o f K atans /30/ makes t h e s i t u a t i o n c l e a r and confirms once and f o r a l l t h e v a l i d i t y of p e r t u r b a t i o n theory.
-
Secondly, t h e accuracy o f t h e r e s u l t s shown i n Fig.11 a l l w s v e r y p r o f i t a b l e comparisons t o be made w i t h several c a l c u l a t i o n s u s i n g second-order p e r t u r b a t io n t h e o r y . It should be p o i n t e d out t h a t i n t h e c l o s e v i c i n i t y o f a- - -
- - - -
- -
-
1 t
- - - - - - - -
- - - - -
- -
-
-
-
- -
7P1/2- -
-
1 I I I I I
Photon energy (crn-'1 Fig. 12
M u l t i p h o t o n i o n i z a t i o n o f atoms i s an i n h e r e n t l y non l i n e a r process and as such depends n o t simply on t h e l a s e r i n t e n s i t y b u t a l s o on i t s coherence p r o p e r t i e s . Many m u l t i p h o t o n i o n i z a t i o n experiments r e p o r t e d i n t h e l i t e r a t u r e have been performed w i t h incoherent 1 aser pulses generated by mu l t i m o d e Q-switched l a s e r s which had s p e c t r a l bandwidths o f about 1 cm-1 and strong temporal f l u c t u a - t i o n s . The d u r a t i o n o f these peak i n t e n s i t i e s i s g i v e n by l / b where b i s t h e spec- t r a l bandwidth o f t h e l a s e r pulse, i. e. 30ps f o r b = 1 cm-'. As was shown i n t h e i n t r o d u c t i o n , t h e c h a r a c t e r i s t i c i o n i z a t i o n t i m e of non resonant m u l t i p h o t o n i o n i z a -
- -
t i o n o t a t m s can be as s h o r t as
lo-''
s. As a r e s u l t , atoms "see" t h e f l u c t u a t i o n s i n t h e " a r r i v a l " o f photons and respond n o t o n l y t o t h e average number o f photons p e r u n i t time b u t a l s o t o t h e way t h i s number f l u c t u a t e s . Photons a r r i v e bunched i n an incoherent l a s e r p u l s e Fig. 13(a), w h i l e t h e y a r r i v e i n s i n g l e f i l e i n a cohe- r e n t l a s e r p u l s e as shown i n F i g u r e 13(b). T h i s i s r e f e r r e d t o as t h e e f f e c t of c o r r e l a t i o n s o r photon s t a t i s t i c s .The study o f t h e effects o f i n t e n s i t y f l u c t u a t i o n s o r photon s t a t i s t i c s began as e a r l y as 1964 i n non l i n e a r o p t i c a l processes /34/ and 1966 on two-photon absorp- t i o n /35/. S p e c i f i c c a l c u l a t i o n s on coherence e f f e c t s i n m u l t i p h o t o n i o n i z a t i o n haw been performed l a t e r /36, 37/. The fundamental r e s u l t of these c a l c u l h t i o n s i s t h a t t h e r a t e o f non resonant N-photon i o n i z a t i o n w i t h c h a o t i c l i g h t i s l a r g e r by a f a c t o r N! t h a n w i t h p u r e l y coherent l i g h t .
Let us d e s c r i b e t h e temporal f l u c t u a t i o n s o f a l a s e r p u l s e used f o r m u l t i - photon i o n i z a t i o n experiments. The instantaneous l a s e r i n t e n s i t y seen by a t m s can be expressed i n t h e form :
I ( t ) =
.
G ( t ) i ( t )C2-380 JOURNAL DE PHYSIQUE
where
$,
i s t h e maximum time- averaged i n t e n s i t y . G ( t ) i s t h e normalized temporal d i s t r i b u t i o n f u n c t i o n envelope o f t h e l a s e r i n t e n s i t y , i t s d u r a t i o n i s about sec f o r a Q-switched l a - ser pulse. i ( t ) i s a p e r i o d i c f u n c t i o n , which w i l l p l a y a v e r y fundamental r o l e i n t h i s study. I t has a s t o c h a s t i c p a t t e r n which depends on t h e number o f modes and on b o t h r e l a t i v e pha- ses and amplitudes o f modes. We g e n e r a l l y measure o n l y t h e t i - me-averaqed i n t e n s i t y-
2 0 ns -w
w i t h o u t t a k i n g i n t o account peak i n t e n s i t y f u n c t i o n i ( t ) . Whe- I ( t ) =T?, .
G ( t ) reas m u l t i p h o t o n i o n i z a t i o n o f atoms i s a h i g h l y non l i n e a r process which i s v e r y s e n s i t i v e t o peak i n t e n s i t y f u n c t i o n s i n - ce t h e N-photon i o n i z a t i o n r a t e i s p r o p o r t i o n a l t o t h e Nth po- wer o f t h e instantaneous i n t e n s i t y - i ( t ) s h a l l be c a l l e d t h e F i g . 13 peak i n t e n s i t y f u n c t i o n .L e t us d e f i n e t h e N~~ order time-independent peak i n - t e n s i t y moment, by
-
N fN = ( i>
Multimode laser pulse GHz bandwidth]
Single mode laser pulse
where t h e b r a c k e t stands f o r t h e ensemble average. fN which r e a d i l y c h a r a c t e r i z e s t h e s t a t i s t i c s o f t h e l a s e r pulse can be e a s i l y r e l a t e d w i t h m u l t i p h o t o n i o n i z a t i o n . The number o f i o n s induced by a multimode l a s e r p u l s e i s :
Nm = pJ~N(t)dt =
p
J?(t). i N ( t ) d t The number o f ions induced by a single-mo- de l a s e r p u l s e having t h e same average i n - t e n s i t y i s :N1 =
fjN
( t ) d t =p
J ; N ( t ) d t Hence,Thus fN may be r e l a t e d t o m u l t i p h o t o n i o n i - z a t i o n v e r y simply. I t i s t h e enhancement o f t h e i o n s i g n a l due t o t h e multimode ope- r a t i o n o f t h e l a s e r . L e t us mention t h a t two o t h e r moments have been considered i n t h e l i t e r a t u r e gN and bN /37/.
Laser intensity (arb. units) Fig. 14
The t h e o r e t i c a l p r e d i c t i o n s o f a N! enhancement i n t h e non resonant N-photon i o n i z a t i o n r a t e have been e x p e r i m e n t a l l y corroborated. For example, F i g u r e 14 shows t h a t t h e non resonant four-photon i o n i z a t i o n r a t e o f Cs atoms induced by an incohe- r e n t 3GHz bandwidth l a s e r p u l s e i s enhanced by 4! compared t o t h e r a t e induced by a single-mode l a s e r pulse o f t h e same average i n t e n s i t y /38/. Likewise, t h e non r e - sonant five-photon i o n i z a t i o n r a t e o f Na atoms w i t h an incoherent l a s e r p u l s e i s more e f f i c i e n t by a f a c t o r 5! than t h a t encountered w i t h a single-mode l a s e r p u l s e o f t h e same average i n t e n s i t y /39/.
The most dramatic experimental demonstration o f these e f f e c t s has been p e r f o r - med i n t h e 11-photon i o n i z a t i o n o f xenon atoms by v a r y i n g t h e mode-structure o f a Nd-glass l a s e r p u l s e from a single-mode t o about one hundred modes /40/. F i g u r e 15
shows, i n s o l i d l i n e , t h e enhance-
b
ment o f t h e number o f ions, i . e . t h e10'-
f11 experimental v a l u e o f the fll moment
when t h e number o f modes i s increased If! = 4x107
---
0----
from one t o one hundred f o r t h e same average l a s e r i n t e n s i t y . The number o f i o n s i s enhanced by n e a r l y 10 7 when t h e number o f modes i s increased from one (coherence t i m e 40 ns) t o one hundred (coherence time 8 ps).As a comparison, t h i s f i g u r e a l s o shows i n dashed l i n e , t h e fll moment c a l - c u l a t e d i n assuming t h a t phases o f t h e modes are independent. The d i f - ference between t h e experimental and t h e c a l c u l a t e d fll moment m a i n l y comes from t h e f a c t t h a t t h e l a - s e r spectrum c o n s i s t e d o f several bands w i t h t e n modes i n each band.Therefore, t h e s t a t i s t i c a l properties o f t h i s l a s e r r a d i a t i o n could be d i f - f e r e n t from t h a t used i n c a l c u l a t i n g t h e fll moment assuming t h a t phases o f t h e modes are independent. As shown i n f i g u r e 15, experimental and c a l c u l a t e d f values tend towards each o t h e r aAi a r e expected t o be- become i d e n t i c a l a t an asymptotic value ll! f o r a v e r y l a r g e number o f modes.
As a conclusion, t h e o f f - r e s o n a n t N-photon i o n i z a t i o n r a t e can be
,
Number o f modes as1 10
lo2
,Coherence ti&4 1
10-8 10"O w = C T N fN I"'
1 0 " ~ (sec.)
The N~~ order peak i n t e n s i t y moment, Fig. 15
o r N~~ order a u t o c o r r e l a t i o n function, fN, i s equal t o u n i t y f o r a s i n g l e -
C2-382 JOURNAL DE PHYSIQUE
mode.laser pulse, o r a bandwidth-limited p u l s e t h a t i s , a pulse completely devoid o f i n t e n s i t y modulation. I n t h e l i m i t o f an i n f i n i t e number o f independent modes, fN equals N!. fN depends b o t h on t h e l a s e r s p e c t r a l bandwidth and on t h e order N o f t h e non l i n e a r process. It i s a small c o r r e c t i o n f a c t o r ( < 2 ) f o r a two-photon process, w h i l e i t has a dramatic e f f e c t i n h i g h order non l i n e a r processes such as m u l t i p h o t o n i o n i z a t i o n o f r a r e gases. T h i s e f f e c t can give, i n many cases, t h e key t o t h e e x p l a n a t i o n o f discrepancies between
r N
c a l c u l a t e d i n assuming coherent l a s e r r a d i a t i o n a n d l r measured w i t h incoherent l a s e r pulses. Conversely, m u l t i p h o - t o n i o n i z a t i o n procesges a l l o w us t o consider an atom i r r a d i a t e d by a l a s e r pulse, as an i d e a l photon d e t e c t o r concerning t h e s t a t i s t i c a l p r o p e r t i e s o f t h e l a s e r p u l - se. The d e t e r m i n a t i o n o f t h e ~ t h - o r d e r a u t o c o r r e l a t i o n f u n c t i o n o f t h e l a s e r peak i n t e n s i t y i s o f s p e c i a l i n t e r e s t t o f u l l y c h a r a c t e r i z e a l a s e r r a d i a t i o n .5.2. Coherence e f f e c t s i n resonant m u l t i p h o t o n i o n i z a t i o n
The c h a r a c t e r i s t i c i o n i z a t i o n t i m e i n non resonant m u l t i p h o t o n i o n i z a - t i o n o f atans can be as s h o r t as 1 0 - l 5 s. As a r e s u l t , atoms "see" t h e f l u c t u a t i o n s i n t h e l a s e r i n t e n s i t y and are v e r y s e n s i t i v e t o t h e s t a t i s t i c a l p r o p e r t i e s o f l a - ser r a d i a t i o n . I n resonant m u l t i photon i o n i z a t i o n o f a1 k a l i ne atoms i n moderate l a - s e r i n t e n s i t y range (
l o 7 -
10 9 W i n s t e a d , t h e c h a r a c t e r i s t i c i o n i z a t i o n t i m e can be as long aslob9
s. Consequently t h e s t a t i s t i c a l p r o p e r t i e s o f l a s e r r a d i a - t i o n are n o t expected t o enhance d r a m a t i c a l l y t h e resonant m u l t i p h o t o n i o n i z a t i o n r a t e . However, t h e resonance s h i f t due t o laser-induced atomic l e v e l s h i f t s , i s l i n e a r as a f u n c t i o n of t h e l a s e r i n t e n s i t y and can be s i g n i f i c a n t l y enhanced by l a s e r i n t e n s i t y f l u c t u a t i o n s o f an incoherent l a s e r pulse. I n a d d i t i o n , t h e l a s e r bandwidth begins t o p l a y a r o l e as soon as i t becomes ccmparable t o t h e resonance detuning. Therefore, laser-temporal coherence e f f e c t s cannot be i n v e s t i g a t e d inde- pendently of l a s e r bandwidth e f f e c t s i n resonant m u l t i p h o t o n i o n i z a t i o n .lo3 I n t h e p a s t few years, a number o f
authors have i n v e s t i g a t e d t h i s p r o - blem t h e o r e t i c a l l y /41-43/. It i s only v e r y r e c e n t l y t h a t s p e c i f i c c a l c u l a - t i o n s have been performed /44-46/. Re- c e n t l y , l a s e r coherence e f f e c t s and bandwidth e f f e c t s have been i n v e s t i - gated e x p e r i m e n t a l l y i n four-photon
- lo2
i o n i z a t i o n o f Cs, w i t h a three-photonV)
C resonance on t h e 6F l e v e l /38/. The
.-
c s o p h i s t i c a t e d l a s e r used i n t h i s ex-=
periment a l l o w s t o change both wave-f
m-
l e n g t h and temporal coherence by vary-V) i n g t h e number o f l o n g i t u d i n a l modes.
E 0 I n c r e a s i n g t h e number o f modes leads
-
t o stronger i n t e n s i t y f l u c t u a t i o n s10 and a l a r g e r bandwidth. The resonance
curves obtained w i t h incoherent l a s e r pulses a r e observed t o be s h i f t e d and broadened w i t h regard t o those indu- ced by coherent pulses w i t h t h e same average i n t e n s i t y . The s t a t i s t i c a l enhancement o f t h e resonance s h i f t i s shown d i r e c t l y i n f i g u r e 16. T h i s f i -
1 gure g i v e s two resonance p r o f i l e s
9443.2 9443,4 9443,6 obtained a t t h e same l a s e r i n t e n s i t y
-
I = 5.6 xl o 7
W cm-'.with a coherent Laser frequency (cm-'1 p u l s e f i g . 16(a) and an incoherent~ u l s e f i g . 16(b) which has a band-
Fig. 16 w i d t h b = 0.1 cm-l. I n t h e resonance
p r o f i l e obtained w i t h t h e multimode 0.4
-
+ L
$
0
I I I I I , , ,
l a s e r pulse, t h e 6F f i n e s t r u c t u r e i s n o t r e s o l v e d because t h e l a s e r band- w i d t h i s l a r g e r . The important r e s u l t o f t h i s f i g u r e i s t h e a d d i t i o n a l shiR induced by t h e incoherent l a s e r pulse T h i s a d d i t i o n a l s h i f t i s 0.2 cm-l ex- pressed i n terms o f t h e energy o f t h e three-photon t r a n s i t i o n 6S96F. The s t a t i s t i c a l enhancement o f t h e reso- nance s h i f t i s c l e a r l y demonstrated i n f i g u r e 17 which shows t h e two laws o f v a r i a t i o n o f t h e resonance s h i f t as a f u n c t i o n o f t h e l a s e r i n t e n s i t y , w i t h a coherent pulse(dashed lineland an incoherent l a s e r p u l s e ( f u l l line).
The d i f f e r e n c e between t h e two l i n e s g i v e s t h e a d d i t i o n a l s h i f t due t o t h e l a s e r i n t e n s i t y f l u c t u a t i o n s , i.
e. (3.6
*
0.3) cm-'/ 6W T h i s experimental r e s u l t i s i n e x c e l l e n t agreement w i t h a previous c a l c u l a - 0.5 1 t i o n /44/. F i g u r e 17 shows t h a t t h eresonance s h i f t induced w i t h an i n - Laser intensity (x108 w . c ~ - ~ ) coherent l a s e r p u l s e i s enhanced by a f a c t o r 2.8
*
0.2 compared w i t h t h e s h i f t induced w i t h a coherent Fig. 17 pulse. As shown i n a r e c e n t c a l c u l a -t i o n /47/, i t i s remarkable t h a t two l a s e r modes o f equal i n t e n s i t y a l - ready account f o r h a l f o f t h e en-
-
hancement o f t h e s h i f t obtained i na c h a o t i c f i e l d .
#
0.3
.-
The w i d t h o f t h e resonance pro-
- - '
f i l e s induced by multimode l a s e r- -
- -
p u l s e s depends on t h e l a s e r band- width, t h e i n t e n s i ty-dependent broadenSng and s t a t i s t i c a l broade- ning. F i g u r e 18 shows t h e v a r i a -
0.2 -
t i o n o f t h e resonance w i d t h s as a-
CI f u n c t i o n o f t h e l a s e r i n t e n s i t y ,f o r single-mode l a s e r pulses ( a )
- b - fc)
and mu1 timode l a s e r pulses w i t hbandwidths 3 x
lo-'
cm-l ( b ) ~ 8 xl o m 2
c m - l j c ) and 0.15 cm-'(d).At low i n t e n s i t i e s , t h e resonance
(b)
w i d t h i s governed by t h e l a s e r bandwidth w h i l e a t h i g h i n t e n s i t i e s i t dependson s t a t i s t i c a l broadening4' (3)
~ h e s e r e s u l t s a r e i n good agree- ment w i t h c a l c u l a t i a a s .0 0.5 1
Laser intensity (x108 W . C ~ - * )
F i g . 18
