Relativistic ionization of hydrogen by

linearly polarized light

D. P. Crawford and H. R. Reiss

Physics Department, American University, Washington, DC 20016-8058, USA

reiss@american.edu

Abstract: Relativistic ionization of hydrogen by intense, linearly po-larized light is treated by the Strong Field Approximation (SFA). Bothbound and ionized states are described by the Dirac equation, withspin effects fully included. The applied laser field is also treated rel-ativistically. There is no recourse to the dipole approximation nor tolarge-component, small-component approximations. Examples are cal-culated for the long-pulse limit of a uniformly distributed laser field.A prediction is verified that relativistic effects will appear with linearpolarization of the laser at lower intensities than with circular polar-ization. Strong-field atomic stabilization is found to be enhanced byrelativistic effects.c©1998 Optical Society of AmericaOCIS codes: (270.6620) Strong-field processes; (350.5720) Relativity; (020.4180)Multiphoton processes

References

1. H. R. Reiss, “Relativistic strong-field ionization”, J. Opt. Soc. Am. B 7, 574-586 (1990).

2. D. P. Crawford and H. R. Reiss, “Stabilization in relativistic photoionization with circularlypolarized light”, Phys. Rev. A 50, 1844-1850 (1994).

3. L. D. Landau and E. M. Lifshitz, Classical Theory of Fields (Pergamon, Oxford, 1959).

4. E. S. Sarachik and G. T. Schappert, “Classical theory of the scattering of intense laser radiationby free electrons”, Phys. Rev. D 1, 2738-2753 (1970).

5. H. R. Reiss,“Theoretical methods in quantum optics: S-matrix and Keldysh techniques forstrong-field processes”, Prog. Quantum Electron. 16, 1-71 (1992).

6. D. P. Crawford, “Relativistic ionization with intense linearly polarized light”, doctoral disserta-tion, American University, 1994.

7. H. R. Reiss, “Energetic electrons in strong-field ionization”, Phys. Rev. A 54, R1765-R1768(1996).

8. H. R. Reiss,“Effect of an intense electromagnetic field on a weakly bound system”, Phys. Rev.A 22, 1786-1813 (1980).

9. L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave”, Sov Phys. JETP 20,1307-1314 (1965).

10. H. R. Reiss, “High-frequency, high-intensity photoionization”, J. Opt. Soc. Am. B 13, 355-362(1966).

11. H. R. Reiss, “Frequency and polarization effects in stabilization”, Phys. Rev. A 46, 391-394(1992).

12. H. R. Reiss and V. P. Krainov, “Approximation for a Coulomb-Volkov solution in strong fields”,Phys. Rev. A 50, R910-R912 (1994).

13. U. Mohideen, M. H. Sher, and H. W. K. Tom, “High intensity above-threshold ionization of He”,Phys. Rev. Lett. 71, 509-512 (1993).

14. B. Walker, B. Sheehy, and L. F. DeMauro, “Precision measurement of strong-field double ion-ization of helium”, Phys. Rev. Lett. 73, 1227-1230 (1994).

15. J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964).

(C) 1998 OSA 30 March 1998 / Vol. 2, No. 7 / OPTICS EXPRESS 289#2871 - $15.00 US Received October 16, 1997; Revised February 13, 1998

16. H. R. Reiss, “Physical basis for strong-field stabilization of atoms against ionization”, Laser

Phys. 7, 543-550 (1997).

1. Introduction

When an atom is ionized by an intense laser field, the final-state electron will have anenergy of interaction with the field measured by the ponderomotive energy Up. Thisquantity represents the energy of the oscillatory motion of the free electron immersedin the strong field. When Up approaches the rest energy mc2 of the electron, then thevelocity of the electron must be regarded as relativistic, and theoretical treatments ofthe ionization process must be done relativistically. This conclusion is borne out by fullyrelativistic calculations[1],[2] done for ionization by circularly polarized light. In thosecalculations it is found that the inequality

Up mc2 (1)

is satisfied in that Up ≈ .01mc2 leads to observable relativistic effects. The criterion inEq. (1) is strongly frequency dependent because of the inverse quadratic dependence ofUp on the frequency. A factor 10−2 inserted into Eq. (1) would correspond to about1018W/cm2 laser intensity for an excimer laser wavelength of 248nm, 1017W/cm2

for 800nm, and 5 × 1014W/cm2 for the CO2 laser at 10.6µm. These intensities areachievable.

When the ionizing laser radiation is linearly polarized, an additional relativisticeffect emerges. In contrast to the simple straight-line oscillatory motion of a free electronin a plane-wave field in the non-relativistic case, a free electron in a strong plane-wavefield executes a figure-8 motion[3],[4] with the long axis along the polarization directionof the electric field, and the plane of the figure defined by that direction and the directionof propagation of the plane wave. This figure-8 motion arises from the magnetic fieldof the plane wave coupled to the electric field, and the usual non-relativistic dipoleapproximation neglects the magnetic field. A failure of the dipole approximation shouldhave physical consequences when the short axis of the figure-8 is of the order of theatomic radius. On these grounds, the dipole approximation is expected to fail[5] when

Up < 2~ω/α, (2)

where α is the fine-structure constant. The limit (2) is even more strongly frequencydependent than the condition (1) for most frequencies, and is reached at lower intensities.It is found herein that a factor of about 10−2 is appropriate in Eq. (2), just as it was inEq. (1), at least for the high frequencies examined here. However, the effects of this lossof the dipole approximation on total ion yields, angular distributions, or photoelectronspectra may not be of equal significance. All of this remains to be explored at the lowfrequencies (ω 1 a.u.) for which very intense lasers are available.

The theoretical method to be employed here is the Strong Field Approximation(SFA). This has already been applied to the problem of atomic ionization under rela-tivistic conditions, but results have been published[1, 2] only for circular polarizationof the laser radiation, in the framework of the Dirac theory. We report here analogousDirac results for linear polarization[6] of the ionizing laser.

We note that the SFA, by employing Volkov solutions for the final state, auto-matically includes all field-induced oscillations of the ionized electron. Rescattering inthe final state is neglected. The neglect of rescattering is unimportant in the circularpolarization case, which explains the great accuracy attainable with the SFA[7] for thiscase. With relativistic Volkov solutions employed, the emitted photoelectrons acquire

(C) 1998 OSA 30 March 1998 / Vol. 2, No. 7 / OPTICS EXPRESS 290#2871 - $15.00 US Received October 16, 1997; Revised February 13, 1998

the expected relativistic projection in the direction of propagation of the field[1], aneffect that can be interpreted as a result of the momentum of the absorbed photons.The resulting increase in the distance from the atom of the field-induced trajectory ofthe photoelectron can be viewed as the cause of the relativistic strengthening of thestabilization effect that was found[2].

A brief overview of the SFA is helpful to understand why it is useful for thedescription of relativistic ionization in strong fields. In its original non-relativistic form[8,5], the SFA approximates the exact time-reversed transition amplitude

(S − 1)fi = −i

∫ ∞−∞

dt(

Ψ(−)f , HIΦi

)(3)

i∂tΦ = H0Φ

i∂tΨ = (H0 + HI) Ψ

HI = −1

cA · p +

1

2c2A2 (4)

by replacing the complete final-state wave function Ψ by the Volkov solution. Equation(3) is written in atomic units. These units are employed in the remainder of the paper.The Volkov solution ΨV olk is an exact solution for a free charged particle in an elec-tromagnetic field, and so the nature of the SFA is that it assumes that the final-statemotion of the ionized electron is dominated by the laser field, and effects on the detachedelectron of the binding potential are neglected.

We remark on the differences between the SFA and the well-known Keldyshapproximation[9]. The Keldysh approximation can be formulated from Eq.(3), thoughit was not so expressed by Keldysh. The differences are: (a) The HI used by Keldysh isin the “length” gauge, rather than the “velocity” gauge of Eq. (4); (b) Keldysh makesa low-frequency approximation from the outset; and (c) Keldysh assumes that the finalstate electron is produced with zero momentum. Extension to the relativistic case is notpossible because of limitation (a); limitation (b) precludes extension to high frequencies,unlike the SFA, which works well[10] at high frequency; and (c) taken together with (a)make the Keldysh approximation into a tunneling method, which cannot be used toexplore the stabilization phenomenon for which the SFA is well suited[11, 2].

The limitations of SFA have been explored for circular polarization of the elec-tromagnetic field by finding the principal Coulomb correction[12] to the SFA. When socorrected, the SFA is valid when either α0 & 10 or Up/EB & 10, where α0 is the radiusof motion of a classical free electron in the circularly polarized field, and EB is thebinding energy of the field-free atom. Since either or both of these conditions are satis-fied in recent strong-field experiments, a comparison of the predictions of the SFA withexperimental measurements[13] of the photoelectron spectrum shows essentially perfectagreement[7] for circular polarization. For conditions leading to relativistic ionizationby circularly polarized fields, the SFA has negligible error.

The situation is more complicated for linear polarization. The essential differ-ence between the two polarization states is that the photoelectron spectrum for strong-field circular polarization is relatively narrow, and centered approximately at the Upcorresponding to the peak laser intensity laser in the laser pulse. All emitted electronsare thus energetic, and the Coulomb field has little effect. Furthermore, there is no “re-visiting” of the ionized atom by the photoelectron. For linear polarization, a substantialportion of the spectrum is at low energy, and the ionized electron does oscillate backto the vicinity of the ionized atom. Although calculated linear polarization spectra[7]show all the features of the experimental spectra[13, 14], the agreement is not unquali-fied, as it is for circular polarization. However, as the laser intensity increases into therelativistic domain, field dominance of the photoelectron motion will be more strongly

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asserted, and a decreasing proportion of the spectrum is at low energy. Stated in otherterms, the rescattering effects neglected in the SFA are of decreasing significance as theintensity increases. Therefore, the SFA should work well for linear polarization in therelativistic domain.

2. Relativistic Dirac calculation

The transition amplitude in a Dirac formulation of the ionization problem starts fromthe exact expression

(S − 1)fi = −i

∫d4xΨ

(−)

f

1

cAµγµΦi (5)(

iγµ∂u − γ0V/c− c

)Φ = 0 (6)(

iγµ∂u − γµAµ/c− γ

0V/c− c)

Ψ = 0 (7)

replacing the non-relativistic Eq. (3). In place of the “natural units” with ~ = 1, c = 1customarily employed in relativistic work, atomic units are retained here to be consistentwith the non-relativistic expressions. The states Φ and Ψ are Dirac spinor states for theatom bound by the central potential V for, respectively, the system without and withthe laser field present. The laser field is defined by the four-vector potential Aµ, and theγµ are the Dirac matrices. Conventions for quantities in the Dirac space are as describedin the book of Bjorken and Drell[15].

As in the non-relativistic SFA, the Dirac form of the relativistic SFA comes fromthe replacement of Ψ representing an exact solution of Eq. (7) with the correspondingDirac Volkov solution

Ψ(−)V olk =

(c2

EV

)1/2(1 +

1

2cp · kkµAνγ

µγν)u (8)

× exp

[−ip · x+ i

∫ ∞k·x

d (k · x)′(A · p

cp · k−

A2

2c2p · k

)],

where m is the mass of the electron, E is the relativistic electron energy, V is the volumeof a box normalization, pµ and kµ are four-momentum vectors for the electron and theapplied field of frequency ω, and scalar products like p · k, k · x, and A · p are relativisticscalar products such that, for example,

p · k = p0k0 − p · k

= Eω/c2 − p · k.

The quantity u in Eq. (8) is a Dirac spinor that satisfies the equation (γµpµ − c)u = 0.The field-free solution Φi in Eq. (5) is taken to be the solution of the Dirac equation forthe hydrogen atom as given, for example, in [15].

The procedure followed in proceeding from the transition amplitude in Eq.(5) to the expression for the transition rate follows closely that given for the circularpolarization calculation in Ref. [1]. The differential transition rate with respect to thesolid angle Ω is

dW

dΩ=

2a2

πc2Z3

∑n

p (UA + UB + UC)[1 + (ρ/Z)

2]4 , (9)

where Z is the magnitude of the charge on the hydrogen-like atom, a is the amplitudeof the vector potential, and ρ is the magnitude of the three-vector

ρ = p/c− (n− η) k/c,

(C) 1998 OSA 30 March 1998 / Vol. 2, No. 7 / OPTICS EXPRESS 292#2871 - $15.00 US Received October 16, 1997; Revised February 13, 1998

where p and k are the three-vector parts of pµ and kµ, and η = a2/4c2p · k. Thesummation index n can be identified as the number of photons participating in theprocess. Considerable physical content is represented by the U parameters (that is, UA,UB , and UC) stated below.

The non-relativistic limit of Eq. (9) is equivalent to the expression in Ref. [8]

dW

dΩ=

8ω

π

(EB

ω

)5/2 ∞∑n=n0

(n − z −EB/ω)1/2

(n− z)2 (Jn)2, (10)

where the angular dependence is contained in the Bessel function, Jn. For linear polar-ization Jn is the generalized Bessel function described in previous work[8]

Jn

(z1/2χ,−

z

2

), χ = 81/2

(n− z −

EBω

)1/2

cosθ. (11)

In the polar coordinate system chosen for the linear polarization case, θ is the anglebetween the velocity vector of the emitted electron and the polarization vector. Thequantity z is the intensity parameter defined by

z ≡ Up/ω.

A similar derivation was also performed[6] in the context of a scalar electron,where solutions to the Klein-Gordon equation were used for the Volkov solution and forthe hydrogen-atom wave functions. Spin-averaged results for Dirac spin-1/2 electronswere found[6] to be very similar to those for Klein-Gordon scalar electrons. These resultswill not be presented here. Both derivations rely upon the SFA which is expected tobecome very accurate at high energies. That is, when UP EB and p2/2 EB thenthe solution using the SFA method should be very accurate.

The differential transition rate resulting from the derivation reported here fol-lows from a Dirac treatment of all elements of the problem, including bound states,unbound states, and interaction terms. Hence, spin effects are included in full. As de-rived, the differential transition rate is a result of averaging over initial spin states ofthe electron, and summing over final spin states.

Since fully relativistic Dirac formalisms are used, many physical processes areimplicitly accounted for, including negative energy state processes such as pair produc-tion. With suitably stated S matrices, the rates for any of the processes can be derived.The process of interest in this treatment is the ionization of an electron from a groundstate.

The polar coordinate system used in this derivation is defined such that Θ is theangle between p, the spatial part of the momentum four-vector for the emitted electron,and k, the spatial part of the em field propagation four-vector. A coordinate system isadopted where the direction of propagation of the laser field is along the x3 axis, andthe field is polarized along the x1 axis. Furthermore, Φ is defined as the angle betweenthe projection of the momentum in the spatial plane transverse to k and the x1 axis.

The U quantities are

UA =1

4P (ρ/Z)

2[Jn+1 (u, v) + Jn−1 (u, v)]

2

×

(E

c2− 1

)(ξρ/Z)

2 U2 +

(E

c2+ 1

)β2V2

+2ξ (β/Z)p

c

[pc

(1− 2 sin2 Θ cos2 Φ

)− (n− η)

ω

mcos Θ

]UV, (12)

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UB = −P

4(ρa0/Z)

2 (zω/m)1/2(

Em −

pm cos Θ

) pm

sin Θ cos Φ

×(n

v+ 2)Jn (u, v)−

u

2v[Jn−1 (u, v) + Jn+1 (u, v)]

× [Jn−1 (u, v) + Jn+1 (u, v)]

×

[(ξρa0/Z)

2 U2 + 2 (βa0/Z)mξ

(2p

mcos Θ −

E

m−bω

m

)UV + β2V2

],(13)

and

UC =ω

m

z

8(Em −

pm cos Θ

)P (ρa0/Z)2

×[(

2 +n

v

)Jn (u, v)−

u

2v[Jn−1 (u, v) + Jn+1 (u, v)]

]2

×

[(ξρa0/Z)

2 U2 + 2 (βma0/Z) ξ

(p

mcos Θ −

bω

m

)UV + β2V2

]. (14)

Auxiliary quantities used in these expressions are

ξ ≡(1− Z2α2

),

β ≡ (1− ξ) /Zα (15)

P ≡(1 + ξ) [Γ (ξ)]

222(ξ−1)

Γ (1 + 2ξ)

[1 + (ρ/Z)

2]2−ξ

(ρ/Z)6 .

Upon reduction to the non-relativistic limit, the differential transition rate be-comes

dW

dΩ→

NRL

8ωε5/2B

π

∞∑n=n0

(n− z − εB )1/2

(n− z)2J2n

(uNRL,−

1

2z

), (16)

which is identical to the expression derived using the non-relativistic state vectors inthe SFA (see Reiss[8]).

3. Numerical examples

Results for the dependence of ionization rate on field intensity are given here for twofrequencies, both large. The computation of low frequency rates is very computer inten-sive, and will be presented in a later publication. Angular distributions require extensivegraphics, and will not be reported here. Sample results for angular distributions andphotoelectron spectra may be found in Ref. [6].

Figures 1 and 2 show ionization rates as a function of intensity for frequenciesof ω = 8 a.u. and ω = 2 a.u., respectively. Some common features are exhibited in bothfigures:

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102 103 104

Intensity (au)

1014

1015

1016T

rans

ition

Rat

e (

1/s)

RelativisticNon-relativistic

Figure 1. Transition rate as a function of intensity for a frequency ω = 8 a.u.Both scales are logarithmic. One hundred points were computed for each decade inintensity.

• A “stabilization peak” (i.e., a maximum in the transition rate) precedes a sharpdrop in transition rate, followed by large-amplitude oscillations in rate thereafter.

• The oscillations in rate are strongly suppressed in the relativistic as compared tothe non-relativistic case for the first several oscillations after the maximum rate,with later (albeit more subdued) resumption of oscillations in relativistic vis-a-visnon-relativistic rates.

• A consequence of the above property is that the relativistic rates have a much moreisolated and prominent pre-stabilization maximum than do the non-relativisticrates.

• The onset of significant departure between relativistic and non-relativistic rates isnot far beyond the occurrence of the rate maximum.

• The stabilization effect is more strongly manifested in the relativistic rates thanin the non-relativistic case.

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101 102 103

Intensity (au)

1015

1016T

rans

ition

Rat

e (

1/s)

RelativisticNon-relativistic

Figure 2. Transition rate as a function of intensity for a frequency ω = 2 a.u.Both scales are logarithmic. Twenty five points were computed for each decade inintensity.

4. Discussion and conclusions

The linear polarization problem has no azimuthal symmetry about the propagation vec-tor axis as in the circular polarization case. Furthermore, with relativistic effects, thelinear polarization differential transition rate loses the symmetry about the polarizationaxis that exists in the non-relativistic calculation. Also, the relativistic differential tran-sition rate is shifted in the direction of the propagation vector due to the momentum ofthe absorbed photons, thereby destroying symmetry about this axis.

The numerical predictions, appropriately, show that the relativistic ionizationrate approaches the non-relativistic ionization rate at low intensities. At intensitiessomewhat beyond the pre-stabilization maximum in the rate, the relativistic ionizationrate is depressed by an order of magnitude or more when compared to the non-relativisticrate, and its oscillations are suppressed. At still higher intensities, the relativistic ratebegins to oscillate in similar fashion to the non-relativistic case, but always retains asmaller value - that is, the stabilization phenomenon remains more pronounced in therelativistic domain.

From Figs. 1 and 2, one can appraise the applicability of Eq.(2). If the factor10−2 is incorporated in Eq.(2) as was found to be true for Eq. (1), then the dipoleapproximation (which is a type of non-relativistic approximation) should fail when theintensity is about I ≈ 10ω3. In fact, the figures show major differences between rela-tivistic and non-relativistic results when

I ≈ 5ω3, (17)

which can be regarded as good agreement.An interesting correspondence between the dipole limit in Eq. (17) and the

predicted onset of stabilization can be found. In Ref. [16] it was found that, at high fre-

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quencies, a maximum in the transition rate as a function of intensity (i.e., stabilization)will occur when

I ≈ 4ω3 (1− EB/ω) = 4ω3 − 2ω2 ≈ 4ω3 (18)

for the case of hydrogen, where the binding energy EB is EB = .5 a.u. The last elementof Eq. (18) follows from the presence of high frequencies. In fact, Eq. (18) is found to bewell satisfied in both figures. Furthermore, the ω3 behavior found for both the onset ofrelativistic effects as found in Eq. (17) and for the onset of stabilization as in Eq. (18)means that the near concurrence of the two effects found in Figs. 1 and 2 is a generalfeature of strong-field ionization at high frequency.

We note that, in the long-pulse, fixed-intensity problem treated here, there is nofield-induced drift of the photoelectron beyond its simple circular (circular polarization)or figure-8 (linear polarization) motion. This motion occurs in a frame of referencefixed with respect to the atom. “Revisiting” of the atom as described by the Volkovsolution replicates itself on each cycle of the field. Quantum calculations involve nostatement of initial conditions as in the case of classical problems. (One may say thatquantum calculations represent an average over the possible initial conditions arising inthe corresponding classical calculations.) These statements must be modified to somedegree when the temporal intensity profile of the laser pulse is included, so that somenet drift of the classical orbit with respect to the atom might occur. This effect shouldappear in a full laser-pulse calculation (as in Ref. [7]), where the spatial and temporalprofile of a laser brought to a Gaussian focus in included, and where even the acceptanceangle of the electron spectrometer is contained in the calculation. This type of completelydetailed calculation has not yet been performed with relativistic expressions.

(C) 1998 OSA 30 March 1998 / Vol. 2, No. 7 / OPTICS EXPRESS 297#2871 - $15.00 US Received October 16, 1997; Revised February 13, 1998