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Two-colour three-photon ionization of hydrogen

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1997 J. Phys. B: At. Mol. Opt. Phys. 30 2599

(http://iopscience.iop.org/0953-4075/30/11/012)

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J. Phys. B: At. Mol. Opt. Phys.30 (1997) 2599–2608. Printed in the UK PII: S0953-4075(97)80069-7

Two-colour three-photon ionization of hydrogen

Magda Fifirig†, Aurelia Cionga‡ and Viorica Florescu§‖† Department of Chemistry, University of Bucharest, Bld Carol 13, Bucharest, R-70346 Romania‡ Institute of Space Sciences, Bucharest-Magurele, PO Box MG-36, R-76900 Romania§ Department of Physics, University of Bucharest, Bucharest-Magurele, R-76900 Romania

Received 4 December 1996, in final form 24 February 1997

Abstract. Based on analytical equations valid in perturbation theory, we study the process ofionization of the ground state of the hydrogen atom by the absorption of three photons, twoof them being of the same frequency. Results are given for the total ionization rate and forthe angular distribution of the ejected electrons, in a regime of frequencies in which two-colourtwo-photon ionization is not possible. The case of orthogonal photon polarizations, investigatedin some detail, displays a strong dependence on the electron direction.

1. Introduction

Interesting effects have been predicted and observed during the last few years in thesimultaneous interaction of an atomic system with two electromagnetic sources. We mentionthe possibility of revealing the details of the interaction between the system and a laser withthe help of a second electromagnetic field, used as a probe (Kruitet al 1983, Muller etal 1986) or the possibility of controlling chemical reactions (Shapiroet al 1988). Also,attention was given to the coherent control of photoionization (Kulander and Shafer 1992)and high harmonic generation (Zuoet al 1995).

Recently, the effect of an intense laser field on the Auger decay has been investigated(Schinset al 1994, 1995), and, very recently, the effect of the laser on the photoionizationof helium, caused by given harmonics (Schinset al 1996, Gloveret al 1996) has beenrecorded. Prior to experiment, the laser modification of photoionization was investigated bytheory (Freund 1973, Jain and Tzoar 1977, Cavaliereet al 1980, Ciongaet al 1993, Veniardet al 1995, Taiebet al 1996). The situation described is that of an atom which interactswith a relatively high-frequency field, able to ionize the atom alone, in the presence of alaser field, a source with a much stronger intensity than the first and with a lower frequency.At high laser field intensity the theory of such effects requires a non-perturbative treatmentof the electron–laser field interaction.

The examination of the literature of two-colour ionization processes shows that evensimpler situations than that mentioned before have not been investigated. To our knowledge,there are no data, based on LOPT or on any other method, for three-photon ionization ofground state hydrogen, in the case of photons of different frequencies, the only case studiedbeing that of three photons of the same frequency (Gontier and Trahin 1968, Karule 1975,1988, Laplancheet al 1976, Chang and Poe 1977, Maquet 1977, Gao and Starace 1988,Cormier and Lambropoulos 1995).

‖ Author to whom correspondence should be addressed.

0953-4075/97/112599+10$19.50c© 1997 IOP Publishing Ltd 2599

2600 M Fifirig et al

The purpose of our work is a study of the ionization of ground state atomic hydrogen,by simultaneous absorption of three photons, one of frequencyω1 and complex polarizations1, the other two of frequencyω2 and complex polarizations2. Each polarization vector isnormalized such thats · s∗ = 1. Energy conservation gives

E = E1+ hω1+ 2hω2 (1)

whereE1 is the ground state energy andE > 0 is the final electron energy. We investigatethe particular regimeω1 > ω2, hω1+hω2 < |E1|, i.e. the case in which the absorption of twophotons, one from one field, the other from the second field, is not sufficient for ionization.The conditions we impose do not exclude the possibility of ionization by absorption oftwo photons of frequencyω1, if 2hω1 > |E1|. The energy of the ejected electron in thistwo-photon process will beE = E1 + 2hω1, instead of that in equation (1). The dominantprocess will be decided by the values of the frequencies and intensities involved. For alaser frequencyω2, the corresponding field intensity could be thought to be much higherthan the intensity of the other field, which will favour the three-photon process. Also, forω1 ω2 the larger energy of the electron ejected in the two-photon process will reduce thevalue of the transition matrix element.

In section 2, we present the general equations describing the two-colour three-photonionization of the hydrogen atom initially in the ground state. Our numerical results for thetotal ionization rate and for the angular distribution of the ejected electrons are presented,as graphs, in section 3.

2. Basic equations

We consider a hydrogen atom in interaction with two monochromatic electromagnetic fields.We work in the velocity gauge, in the dipole approximation, and treat the interaction withboth fields as a perturbation. Three Feynman diagrams of third order in the fine-structureconstantα are associated with the process in which three photons, two of them identical, are

diagram 3diagram 2diagram 1

ω2

ω2

ω1

ω2

ω1

ω2

ω1

ω2

ω2

Figure 1. Feynman diagrams associated with the process of ionization of the hydrogen atominitially in the ground state, by three-photon absorption, two of the same frequency.

Two-colour three-photon ionization of hydrogen 2601

absorbed (see figure 1). The amplitude of the process is constructed from a basic third-ordertensor with Cartesian components

5ijk(′, ) = 〈E − |PiGc(

′)PjGc()Pk|E1〉 (2)

and from the Cartesian components of the polarization vectorss1 ands2 as follows:

M =∑i,j,k

s2i s2j s1k[5ijk(

′1, 1)+5jki(′2, 2)+5kij (

′3, 3)

](3)

whereP is the momentum operator,Gc the Coulomb resolvent operator,|E−〉 and |E1〉are, respectively, the final continuum (incoming) and the initial ground state electron energyeigenvectors. The energy eigenvector|E−〉 is normalized in the energy and solid anglescales. The parameters and′ corresponding to the three Feynman diagrams in figure 1are:

1 = E1+ hω1 ′1 = E1+ hω1+ hω2

2 = E1+ hω2 ′2 = E1+ hω2+ hω1

3 = E1+ hω2 ′3 = E1+ 2hω2 .

(4)

We have recently derived (Ciongaet al 1994) an analytical compact expression for thetensor5ijk in (2) using the analytical expression of the linear response of the hydrogenatom to an electromagnetic monochromatic plane wave for the ground state (Florescu andMarian 1986) and for the continuum state (Florescu 1986). We have also obtained (Fifiriget al 1995) equivalent analytical results, using the second-order correction (Florescuet al1993) to the wavefunction of a hydrogenic system, initially in the ground state, due to thepresence of a weak uniform harmonic electric field. According to Ciongaet al (1994) andFifirig et al (1995), this tensor can be expressed with three rotationally invariant amplitudesA,B andC:

5ijk = ϒ [Aniδjk + B(njδki + nkδij )+ Cninjnk] (5)

where

ϒ = √me(αZ)−2c−2 n = p/p

with A,B andC depending only on,′ andE. We have used the notation:Z for theatomic number,me for the electron mass andc for the velocity of light. The expressionin equation (5) is symmetric with respect to the indicesj and k due to the symmetriccharacter of the second-order correction to the ground state of a Coulomb electron in aweak electromagnetic field (see equation (21) in Florescuet al (1993)). The analyticalequations obtained previously (Ciongaet al 1994, Fifirig et al 1995) express the invariantamplitudes as a series of terms which are products of a hypergeometric Gauss function2F1 with a sum of several Appell functionsF1 (Appell and Kampe de Feriet 1926). Forillustration, we reproduce the analytical expression of the amplitudeB:

B = β∞∑m=0

(6)mm!

[(iη − τ ′)(τ − τ ′)(iη + τ ′)(τ + τ ′)

]m2F1

(−m, 2− iη, 6; −4iητ ′

(iη − τ ′)2)f (m) . (6)

The following notations have been used:

β = −212√

2

πτ ′η9/2eπη/20(2− iη)

[ττ ′

(1+ τ)(τ + τ ′)]6(iη − τ ′)−2+iη

(iη + τ ′)4+iη

f (m) = 1

3− τ ′ +m[(1+ τ)2

2− τ F1(2− τ, 6+m,−m, 3− τ ; ξ, ζ )

− (1− τ)2

4− τ F1(4− τ, 6+m,−m, 5− τ ; ξ, ζ )]

2602 M Fifirig et al

where the two variables of the Appell functions are

ξ = (1− τ)(τ ′ − τ)(1+ τ)(τ ′ + τ) ζ = ξ

(τ ′ + ττ ′ − τ

)2

.

The parametersτ, τ ′ andη are given by

τ = λ/√−2me τ ′ = λ/

√−2me′ η = λ/p (7)

with λ = αZmec.From equations (3)–(5), we obtain the general structure of the amplitude of the process

of two-colour three-photon ionization, showing the dependence on photon polarizationsexplicitly,

M = (n · s2)(s2 · s1)a + (n · s1)(s2 · s2)b + (n · s1)(n · s2)2c (8)

where

a = A1+ A2+ B1+ B2+ 2B3

b = A3+ B1+ B2

c = C1+ C2+ C3 .

(9)

The subscript of each amplitude represents the number of the corresponding Feynmandiagram.

The differential rate of two-colour ionization is

d0 = (2π)4α3 h2

m6e

|M|2ω2

1ω42

I1I22 d (10)

whereI1 andI2 are the intensities of the two fields.Dividing the rate by the incident flux of photons of the first field (I1/hω1) we obtain a

quantity which is a generalized ‘ionization cross section’, with the dimensions of an area.We present two particular cases: the case of identical polarizations (s1 ≡ s2),

dσ‖d= KI 2

2 |s1 · n|2|s21|2|a + b|2+ |s1 · n|4|c|2+ 2 Re[(n · s∗1)2(s1)

2(a + b)c∗] (11)

and the case of orthogonal linear polarizations (s1 · s2 = 0)

dσ⊥d= KI 2

2 |s1 · n|2|b + (n · s2)2 c|2 . (12)

In these equations Re denotes the real part andK is the quantity

K = 8π2α

Z14

(a0

I0

)2 1

k1k42

(13)

with kj ≡ hωj/|E1|, j = 1, 2, the energies of the two kinds of photon measured inZ2× Rydberg,I0 the atomic unit for field intensity anda0 the Bohr radius.

The integration on the ejected electron directions leads to the result

σ‖ ≡∫

dσ‖ = 43πKI

22

γ |a + b|2+ 3

35(2+ 3γ )|c|2+ 65γ Re[(a + b)c∗] (14)

for identical polarizations (s1 ≡ s2), with γ = |s22|2, and to

σ⊥ ≡∫

dσ⊥ = 43πKI

22 |b|2+ 3

35|c|2+ 25 Re(bc∗) (15)

for orthogonal linear polarizations (s1 · s2 = 0).

Two-colour three-photon ionization of hydrogen 2603

3. Results

First, we mention that in order to check our numerical codes for the evaluation of theinvariant amplitudeA,B andC, we have made extensive comparisons over a large rangeof photon energy with different numerical results, given as tables in the literature (Laplancheet al 1976, Chang and Poe 1977, Karule 1988, Gao and Starace 1988), for the total crosssection in the case of ionization by three identical photons, for both the linear and circularpolarizations. Good agreement was obtained in all the cases, but the best agreement (relativeerrors under 0.2%) was found with the results of Gao and Starace (1988).

In the case of two electromagnetic fields, we consider here only the regime in which theabsorption of two photons with different frequencies is not sufficient for ionization. Moreprecisely, we have investigated the domain

k2 < k1 1− 2k2 < k1 < 1− k2 . (16)

In this domain, we evaluate the Appell hypergeometric functions in the amplitudesA,B

andC (equation (6) here, equations (15)–(23) of Ciongaet al (1994)), through their seriesexpansions. Nontrivial changes have to be made in our codes in order to account for othercases.

The process we study presents resonances in the following cases:

k1 = 1− 1/n2 k1+ k2 = 1− 1/n2 2k2 = 1− 1/n2 (17)

wheren is the principal quantum number. The first and the third type of resonance mayappear only for the first and the third diagram, respectively, while the second one mayappear in each of the first two Feynman diagrams.

In the case of three-photon ionization of the ground state, there are three possiblechannels for the quantum numberl of the involved states (Laplancheet al 1976):

0→ 1→ 0→ 1 0→ 1→ 2→ 1 0→ 1→ 2→ 3 . (18)

The amplitudeA has contributions from all these channels,B from the last two, whilefor the amplitudeC we have a contribution only from the channel leading tol = 3. Adetailed discussion will be presented elsewhere (Fifirig 1996). In terms of these channels,one understands the resonance structure of the different cross sections. When the two fieldshave parallel linear polarizations all three channels are allowed by all the Feynman diagrams,as one can see from equations (14) and (9). For identical circular polarizations, only thechannel 0→ 1→ 2→ 3 is allowed, since only the|c|2 term is present in equation (14) forγ = 0. When the fields have linear orthogonal polarizations, the first channel is allowedonly by the third diagram as is obvious from equations (15) and (9).

Now, we present our numerical results. In figure 2 we plot total cross sections(equations (14) and (15) divided byI 2

2 ) for linearly polarized fields as a function of thelarger wavelengthλ2. The polarization vectors are parallel in figure 2(a) and orthogonal infigure 2(b). We choose the fifth harmonic of the Nd:YAG laser (¯hω1 = 5.85 eV) for theother field; therefore only the last two types of resonances in equation (17), correspondingto n = 2, are present in the spectral distribution. All the resonances in figure 2 are dueto an intermediate 2s state, allowed by the first channel in equation (18). To understandthe resonance structure, we note that this channel only has a contribution to the amplitudewe have denoted byA. According to equations (15)–(22) derived by Ciongaet al (1994),the analytical form of this amplitude has an infinite number of poles, which correspond tointeger values of the parametersτ andτ ′ that satisfy the relationsτ > 2 andτ ′ > 1. In thefrequency domain represented in figure 2 we always haveτ < 2. In what follows we shall

2604 M Fifirig et al

225 250 275 300λ2 (nm)

0.01

0.1

1

10

100

0.01

0.1

1

10

100σ

/I 22 (c

m6 W

-2)

(a)

225 250 275 300λ2 (nm)

0.01

(b)

Figure 2. (a) Three-photon ionization cross sectionσ/I22 (cm6 W−2), multiplied by a factor of

1044, as a function of the larger wavelengthλ2, for hω1 = 5.85 eV, in the case of parallel linearpolarizations. (b) Same as (a), but for orthogonal linear polarizations.

attach toτ andτ ′ a subscript which represents the number of the corresponding diagram infigure 1.

In the case of parallel polarizations, shown in figure 2(a), for λ2 = 243 nm, onlyτ ′3is an integer (τ ′3 = 2), therefore the resonance is connected to the third diagram. The 2sstate is reached by the absorption of two identical photons of energy ¯hω2 = 5.1 eV. Incontrast, forλ2 = 285 nm,τ ′1 = τ ′2 = 2 while τ ′3 is not an integer, therefore the resonanceis connected with the first two diagrams in figure 1. The 2s state is reached in this case bythe absorption of two photons of different energies: ¯hω1 = 5.85 eV andhω2 = 4.35 eV.

As mentioned before, in the case of orthogonal polarizations that is shown in figure 2(b),we have to consider only the third Feynman diagram. Forλ2 = 243 nm, whereτ ′3 = 2,the amplitudeA(τ3, τ

′3) has a pole. The 2s state is reached, as before, by absorption of two

identical photons (¯hω2 = 5.1 eV). There is no resonance atλ2 = 285 nm becauseτ ′3 is notan integer at this wavelength.

We have also investigated the angular distribution of the ejected electron in severalcases. Figure 3 displays angular distributions for two pairs of frequencies. The value of thefrequencyω2 was chosen to correspond to a laser frequency ¯hω2 = 1.17 eV (Nd:YAG laser)for the full curves and ¯hω2 = 1.55 eV (Ti:sapphire laser) for the broken curves. The otherfrequency is, respectively, ¯hω1 = 11.5 eV andhω1 = 10.75 eV, for the two types of curve.The two fields have identical polarizations, linear in figure 3(a) and circular in figure 3(b).Thez-axis is chosen to lie along the common polarization direction in figure 3(a) and alongthe common direction of the three photons in figure 3(b). The angular distribution isφ-independent in both cases. We see in figure 3(a) that the cross section vanishes forθ = 90

Two-colour three-photon ionization of hydrogen 2605

0 60 120 180θ (degrees)

0

I 2-2 d

σ /d

Ω (c

m6 W

-2)

(a)

0 60 120 180θ (degrees)

0

10

20

30

40

50

0

10

20

30

40

50 (b)

Figure 3. (a) Electron angular distributions in equation (11), multiplied by a factor of 1044,for hω1 = 11.5 eV, hω2 = 1.17 eV (full curve) and ¯hω1 = 10.75 eV, hω2 = 1.55 eV (brokencurve) in the case of identical linear polarizations.θ is the angle between the electron momentumand the polarization vector. (b) Same as (a), but for identical circular polarizations, withθ theangle between the electron momentum and the common direction of the three photons.

because of the overall cos2 θ factor in the expression (11). The curves of figure 3(b) havea maximum atθ = 90, because now the cross section is proportional to sin6 θ .

In figure 4 we present angular distributions for orthogonal linear polarizations. Nowthe z-axis is chosen along the polarization vectors1 and thex-axis along the vectors2.According to equation (12), we have

I−22

dσ⊥d= K cos2 θ(α1+ α2 sin2 θ cos2 φ + α3 sin4 θ cos4 φ) (19)

where

α1 = |b|2 α2 = 2[Re(b)Re(c)+ Im(b) Im(c)] α3 = |c|2 . (20)

The electron angular distribution depends on both electron momentum polar angles,θ andφ, but it is invariant at each of the changesθ → π − θ, φ→−φ andφ→ π ± φ. Due tothe first invariance, the angular distribution is symmetric with respect to the reflection in theplane orthogonal tos1. The other two properties are responsible for the fourfold symmetryof the angular distribution. We focus our attention on a frequency domain situated betweentwo resonances, located at 12.09 and 12.75 eV. Figures 4(a)–(c) display the azimuthalangular distribution as polar diagrams for the second harmonic of the Nd:YAG laser at thelower frequency, and the high frequency chosen such that the energy of the ejected electronis 0.95 eV in figure 4(a), 1.05 eV in figure 4(b) and 1.15 eV in figure 4(c). In these figures

2606 M Fifirig et al

1000

500

0

500

1000(a)

200

150

100

50

0

50

100

150

200(c)

100

50

0

50

100(b)

Figure 4. (a) Azimuthal angular distributions for an ejected electron in (19), multiplied bythe factor of 1046, for orthogonal linear polarizationss1 and s2, chosen asz-axis andx-axis, respectively, in the case of the second harmonic of Nd:YAG (¯hω2 = 2.34 eV), forhω1 = 9.87 eV. Full curve,θ = 0; long-broken curve,θ = 20; short-broken curve,θ = 35;dotted curve,θ = 60; chain curve,θ = 75. The azimuthal angleφ is measured in thecounterclockwise sense from the horizontal line. (b) Same as (a), for hω1 = 9.97 eV, (c) sameas (a), for hω1 = 10.07 eV.

the plotted quantity,I−22 dσ⊥/d, is multiplied by the factor 1046. Bearing in mind the

mentioned symmetry with respect to the reflection in the planexOy, we restrict ourselvesto four values ofθ between 0 and 90. For θ = 0 the polar diagrams reduce obviously toa circle and to a point forθ = 90. There is a strong dependence of the angular distributionswith respect to the photon frequencies, illustrated in figures 4(a)–(c). The change in theshape of the curves can be understood in terms of the corresponding functionsα1, α2 andα3, given in table 1. For small values ofθ the curve is deformed along they-axis (φ = 90)becauseα2+α3 sin2 θ < 0. With the increasing ofθ , we see a second pair of lobes alignedalong thex-axis (φ = 0). The magnitude of these lobes increases due to an increasing

Two-colour three-photon ionization of hydrogen 2607

Table 1. The dimensionless functionsα1, α2 andα3, defined in equation (20), for several photonenergies.

hω1 (eV) hω2 (eV) α1 α2 α3

9.87 2.34 1771.868 −17 652.377 52 987.1839.97 2.34 259.693 −743.768 3 596.317

10.07 2.34 793.414 −3 432.602 6 187.801

positive contribution, coming from(α2 + α3 sin2 θ) sin2 θ . When θ approaches 90, theentire curve decreases because of the overall cos2 θ factor.

In conclusion, in this work we have investigated two-colour three-photon ionizationof hydrogen based on the compact analytical results we had derived previously(Ciongaet al 1994) for the relevant rotationally invariant amplitudes. We have focusedour attention on the evaluation of transition rates and angular distributions for the case inwhich two photons of different frequencies are not enough for ionization. In our evaluationswe have used frequencies of two available laser sources (Ti:sapphire and Nd:YAG). Ouranalytical expressions allowed us to understand the effect of the polarization state of thephotons on the spectral distribution of the total cross sections. Electron angular distributionshave been studied for different combinations of photon polarizations. In the case of linearorthogonal polarizations they display a strong dependence on the ejected electron directionand photon energies.

Acknowledgments

This work is part of the EC contract CIPD CT940025 and of the contract 5009/1996 of theRomanian CNCSU. The numerical calculations were performed on the workstation HP-715of the Computer Center of the Quantum and Statistical Physics Group (Bucharest-Magurele),a donation of the SOROS Foundation.

The authors are grateful to A Maquet for useful discussions, encouragements andcriticism of this work. The competent assistance of M Dondera for numerical problemsis warmly acknowledged.

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