Influence of light statistics on the many photon resonance ionization

Volume 56A, number 2

PHYSICS LETTERS

8 March 1976

INFLUENCE OF LIGHT STATISTICS ON THE MANY PHOTON RESONANCE IONIZATION J. MOSTOWSKI Institute of Physics, Polish Academy of Sciences, Ale/a Lotników 32/46, 02-668 Warsaw, Poland Received 24 November 1975 We study the influence of laser temporal coherence on the width and shape of induced resonances in the many photon ionization.

Many photon resonance ionization was recently intensively studied both experimentally and theoretically. Experiments show that the ionization probability as a function of light frequency has resonance behavior, where the resonance width is proportional to the light intensity. Theories of these processes [e.g. 5, 8] assume that the laser beam can be represented as an Nphoton state in one single frequency. This assumption simplifies the calculations but is often not well justifled. It is well known that the laser may operate with many excited modesshown and that these modes [7, are10, incoherent. It has been experimentally 111 and theoretically [1, 3, 8] that the multi-mode laser action leads to essential modifications of many photon transition probabilities in the non-resonant case. In the present paper we study the influence of the multi-mode laser action on the many photon resonance ionization. This problem was stated also in ref. [6] however no explicit formulas were given. We shall show that the multi-mode laser action leads to an essential broadening of the resonance line, It follows from theories given in e.g. refs. [5,81 that the probability for the resonant s-photon ionization when the laser beam is represented by the N-photon state is given by: ,

2

p0, = IfI2Ial2~ eXP~ a 2i1 ol 2 Im Li/a

r

x

1~’1(AIu* + 1, ~

[

~+

+ 2;

I al 2)

*

~

—

1 c.c.j

ía

where 1F1 is the confluent hypergeometric function and Im denotes the imaginary part. Since the mean number of photons in the laser beam is large we may expand in al 2: 2L2 +~O(1a12). (3) = + a1a1 Comparing eq. (3) with (1) one can see that the ionization probabilities are the same in the N-photon state as in the coherent state a), where al 2 (the mean number of photons) is equal to N. We shall discuss now the most interesting case when the laser operates with many excited modes. We shall assume, that the number of excited modes is large but the spectral width of the beam is so small that the transition probability can be taken as constant within the width. It is known that in this case one can describe the beam as monochromatic but with the Gaussian statistics [2, 121.

/

(1) where i~is the detuning,f is a factor depending on the oscillator strengths and the density of final states, aN beam, We shall discuss now the case when the laser beam is represented by a coherent state cs). Following the same lines of reasoning as in refs. [5, 8] one can show that the ionization probability is given by:

2

15

//

\~

)‘

_—~-‘~

~_—~i” °~

-----

_____________

2

0

2

4

Fig. 1. Shape of resonance lines for Re E/Im ~

C =

0.

87

PHYSICS FETTERS

Volume 56A, number 2

8 March 1976

E

Fig. 2. Shape of resonance lines for Re ~/Im ~

=

6

1.

Hg. 3. Shape of resonance lines for Re ~/Ini ~

In order to calculate the transition probability in this case we should integrate p~from eq. (2) with the 2—I

2

2

1

weight w~(a)= (7n1131 ) exp(—IaI /1131 ) where ill is the mean number of photons [4] Integrating in polar coordinates one obtains: 2s! l1312s [ 2) (4) 1 Ifl Im ~/aL2Fi(l,s+1,2+~/a*;_j13I 2iIaI~ c.c.J .

-

8

- ~~~—--—-

2.

Figs. l—3 present plots of the functions p0/l fl 21131 2s versus = ~‘/Im ~ for s 3, 4 and vanous values of Re ~/Im The dashed line represents the same resonances in the coherent field. Note essential broadening of the resonances as well as their asymmetrical shape. -,

~.

The author wishes to thank Professor I. BialynickiBirula and Dr. J. Zawistowski for helpful discussions.

where 2F1 denotes the hypergeometric function. Once more we shall make use of the fact that 1131 2 is large. The asymptotic expansion for p0 is found to be: 2

2s

fl 1131 S. ~ 2iIm~/~I~I~ =

(5)

s—i

x

~

—~--

[ex~(__)(~) 2~ ~

W~1 ~ (~)_c.c.~

---

--

where ~= a1131 2 and W denotes the Whittaker function. For large detuning, i.e. when the frequency is far from the resonance, p0 reduces to P0

=

Li ~ j~

2101~

2s/~2~~~ I

(6)

and is s! times larger than in the coherent field with the same intensity the well known result, [e.g. 1].

88

References [1] G.S. Agarwal, Phys. Rev. Al (1970) 1445. [2] S. Carusotto and C. Strati, Nuovo Cimento B 15 (1973) 159. [3] J.L. Debethune, Nuovo Cimento B 12 (1972)101. [41-R. Glauber, Phys. Rev. 131 (1963) 2766. 15] Y. Gontier and M. Trahin. Phys. Rev. A7 (1973) 1899. [6] W.A. Kovarski and N.F. Perelman, Zh. Eksp. Teor. Fiz. 68 (1975) 465.

[71 Ct al., Optics Comm. (1974) 304.Phvs. 81 J. P. Krasitiski Lambropoulos, C. Kikuchi and 12 R.K. Osborn, Rev. 144 (1966) 1081. [9] 1101

P. (1974) 1992. 265; C. Lambropoulos, Lecompte et al.,Phys. Phys.Rev. Rev.A9 Lett. 32 (1974)

Phys. Rev. Al 1(1975)1009. [11] Van et al., Phys. (1973) 91. 112] M. B.R.EuMollow, Phys. Rev.Rev. 175 A7 (1968)1555.