## Abstract

In this paper, a scaling law of photoionization of atoms irradiated by intense, few-cycle laser pulses is established. The scaling law sets a relation to the phase-dependent ionization with the kinetic energy of photoelectrons, the duration and peak intensity of short pulses, and the ionization potential of the target atoms. We find that it will be advantageous to manifest the phase-dependent photoionization by choosing the target atoms with larger ionization potential, using laser with smaller carrier-frequency, and increasing the pulse intensity.

©2007 Optical Society of America

## 1. Introduction

Recent developments in laser technology have made it possible to produce intense laser pluses with durations as short as only few optical cycles[1]. These ultrashort pulses have the advantage that higher intensity can be reached before the target being depleted, allowing much higher effective field strengths, thus provide a useful research tool widely used in various fields [2, 3]. For few-cycle pulses, the temporal shape of the electric field varies dramatically with the initial phase of the carrier wave with respect to the pulse envelope (the so-called carrier-envelope phase, CE phase), the physical process depending on the electric field is CE phase-dependent [4].

Measurements of the photoelectron angular distributions (PADs) manifested directly the CE phase of a short pulse [5, 6, 7, 8, 9]. The experiment performed by Paulus *et al*. detected the coincident photoelectrons emitted in opposite directions and showed the asymmetric ionization in few-cycle above-threshold ionization (ATI) [4]. From then on, the CE phase-dependence phenomena attract much more attentions in both theoretical and experimental studies [8, 10, 11, 12, 13].

Scaling is a powerful technique for analyzing many phenomena in intense laser fields [14, 15]. For relatively-long laser pulses, the scaling law established recently [16] reveals a remarkably simple physical mechanism: the main characters of PADs as well as of many strong-field phenomena are determined by three dimensionless numbers: (1) the ponderomotive number *u _{p}* =

*U*/

_{p}*ħ*

*ω*, the ponderomotive energy in the unit of laser-photon energy; (2) the binding number

*ε*=

_{b}*E*/

_{b}*ħ*

*ω*, the atomic binding energy in the unit of laser-photon energy; and (3) the absorbed-photon number

*q*[16]. The physical essences of processes with the same three parameters are just the same, even though their dynamical parameters look quite different. While for few-cycle pluses, all related physical processes may strongly depend on the CE phase. Is the scaling law still applicable in the short-pulse case?

Very recently, the asymmetry in the ionization of Rydberg atoms by radio-frequency few-cycle pulses were demonstrated [17]. Similar results were also found in the photoionization of atoms with intense few-cycle laser pulses [18]. The similarity strongly suggests a scaling relation in various few-cycle pulses. In this paper, a scaling law of photoionization in the few-cycle case is given, which determines the similarities of photoionization in various short pulses. The scaling law is proved theoretically and verified numerically. A discussion with the present experimental observations is given in the last part of this paper.

## 2. Theoretical proof of the scaling law

The scaling law in few-cycle case can be stated as: *For a given CE phase and the corresponding ATI peaks, the main characters of PADs for an atom of binding energy E _{b} irradiated by an ultrashort laser pulse (with fixed polarization) of duration τ, peak intensity I and carrier frequency ω are the same as that for an atom of binding energy rE_{b} irradiated by an ultrashort laser pulse of duration τ/r, peak intensity r^{3}I and carrier frequency rω, where the re-scaling parameter r is an arbitrarily positive number.* The scaling law says that all PADs admit an arbitrary scaling transformation.

The scaling law is indicated by the following fact: the main characters of PADs in the few-cycle case are determined by five dimensionless numbers, i.e., the CE phase ϕ_{0}, the cycle number *n*, the atomic binding number *ε _{b}*, the ponderomotive number

*u*, and the number of absorbed photons

_{p}*q*. The PADs are the same if the five parameters are the same. In contrast to the relatively-long-pulse case, the PADs in the few-cycle case are determined by two additional dimensionless numbers describing the shape of the electric field modulated by the pulse envelope, i.e. the CE phase ϕ

_{0}and the cycle number

*n*. Once the values of ϕ

_{0}and

*n*are given, the distribution of the electric field in the pulse envelope is fixed; thus the PADs of atoms irradiated by the short pulse satisfy the same scaling law as the one in the relatively-long-pulse case.

The theoretical proof of the scaling is based on our earlier works [19, 20]. We use a non-perturbative scattering theory of ATI [21], which treats the Volkov states as intermediate states while the final state of ATI is an electron-photon plane wave. The electric field of *n*-cycle pulses with carrier frequency *ω* is written as

where *n* is the cycle number defined by *n* =*τω*/(2*π*) through the pulse duration *τ*, and ϕ_{0} the CE phase. The defined CE phase is always zero when the maxima of the field and the envelope coincide [12]. A three-mode laser field is used to mimic an infinite sequence of *n*-cycle pulses:

with the intensity of the sideband modes being a quarter of that of the central mode. The advantage of this treatment is that the ionization rate can be obtained analytically [19]. Our treatment reproduces the asymmetric PADs, and also shows the reversal of the asymmetry with shifting the CE phase by π[19, 20]. The scaling law preserves these properties.

The differential ionization rate for a given ATI peak is (*ħ* = 1, *c* = 1) [20]

$$\times {\mid \sum _{{q}_{i},{j}_{i}}{\chi}_{{j}_{1}-{q}_{1},{j}_{2}-{q}_{2},{j}_{3}-{q}_{3}}\left({z}_{f}\right){\chi}_{{j}_{1},{j}_{2},{j}_{3}}{\left({z}_{f}\right)}^{*}\mid}^{2},$$

where **P**
* _{f}* is the final momentum of the photoelectron, and k the central wave-vector of the laser pulse. The integer

*q*relates the final kinetic energy of the photoelectron as

Here *q* denotes the order of an ATI peak and gives the net absorbed-photon number. According to Eq. (2), the number of transferred photons in the *i*th mode in overall process, say *q _{i}*(

*i*= 1,2,3), satisfies the following integer equation

and the sum over *q _{i}* is performed over all possible sets of {

*q*} satisfying above relation. The sum over

_{i}*j*is performed on the energy shell;

_{i}*u*is the ponderomotive parameter defined as the ponderomotive energy in the unit of

_{p}*ω*. The generalized phased Bessel (GPB) functions are given by

$$\times {X}_{-{j}_{3}+{2m}_{3}+{m}_{6}+{m}_{7}+{m}_{8}-{m}_{9}}\left({\zeta}_{3f}\right){X}_{-{m}_{1}}\left({z}_{1}\right)\cdots {X}_{-{m}_{9}}\left({z}_{9}\right),$$

where the sum is performed over integer indices *m _{i}*(

*i*= 1,2,⋯9): -∞ <

*m*< ∞, and

_{i}*X*(

_{n}*z*) is phased Bessel function defined in terms of the ordinary Bessel function as

The arguments of the GPB function are defined as

where *ε _{j}*(

*j*= 1,2,3) is the polarization vector of the

*j*th mode defined by

where *ε _{x}* and

*ε*are unit vectors vertical to each other;

_{y}*ξ*determines the degree of polarization, such that

*ξ*=

*π*/2 denotes circular polarization and

*ξ*= 0 and

*π*linear polarization. The phase angle

*ϕ*(

_{j}*j*= 1,2,3) relate to the CE phase

*ϕ*

_{0}as

In Eq. (8), Λ* _{i}*(

*i*= 1,2,3) is the classical-field amplitude of the ith mode and given by

The photoelectron rate of a given ATI peak is obtained by integrating over the solid angle *d*Ω*P _{f}* = sin

*θ*of the final photoelectron, where

_{f}dθ_{f}dϕ_{f}*θ*is the scattering angle and

_{f}*ϕ*is the azimuthal angle. The PAD denotes the ionization rate for different azimuths at a fixed scattering angle

_{f}*θ*=

_{f}*π*/2.

First, the PAD does not depend on the initial angular state of the atom. Even though the transition matrix element does contain the initial-state momentum wave function, the final-state momentum may still directly or indirectly incorporate with the initial-state momentum. This momentum wave function is squared and averaged over the initial orientations. Thus, the factor of the momentum wave function reduces to its radial part. Since the photon momentum is very small compared with the electron momentum, this factor does not affect the PADs [16].

Second, we show that the sum over the GPB functions is invariant under the scaling transformation by a constant *r*. First we show that the arguments of the GPB function are unchanged under this scaling change. By means of Eqs. (2), (4) and (9-11), we obtain the following relations

which varies with *q*, *n* and *ε _{b}*;

which depend only on *u _{p}* and

*n*; and

which vary with *ϕ*
_{0} and *n* for a fixed polarization. Thus, the arguments given in Eq. (8) are completely determined by the five dimensionless parameters, that is, the CE phase *ϕ*
_{0}, the cycle number *n*, the atomic binding number *ε _{b}*, the ponderomotive number

*u*, and the absorbed-photon number

_{p}*q*. This means that the arguments of the GPB function are kept unchanged under the scaling change. Here, the scaling change means that the five parameters mentioned above are kept unchanged but other quantities, such as the laser frequency and the binding energy, changed according to the scaling law. Next, we show the orders of the GPB functions, say

*j*and

_{i}*q*, do not change under the scaling change. In Eq. (3), the sum over

_{i}*j*(

_{i}*i*= 1,2,3) is performed on the energy shell for a given value of

*q*. If

*q*doesn’t change,

*j*in the sum doesn’t change, too. The possible value of

_{i}*q*also determined by

_{i}*q*, (see Eq. (5)). If

*q*doesn’t change, all the possible value of qi are kept unchanged. Thus, for a fixed

*q*, the orders of the GPB functions keep invariant under the scaling change. Finally, we show the number of transition channels is fixed in the scaling change. A set of

*q*satisfying Eq. (5) for a given integer q defines a transition channel. Since

_{i}*q*and

*n*are fixed numbers in the scaling change, all the possible value of qi as well as their combination are fixed.

Then, we conclude that the PADs in few-cycle case satisfy the scaling law. In the following we describe the tests to the scaling law using numerical methods.

## 3. Numerical verification of the scaling law

In few-cycle case, the photoelectron rates in a pair of opposite directions are not always equal to each other, which is termed as inversion asymmetry [19]. The inversion asymmetry varies with the CE phase, as shown in Fig. 1. The laser pulse is of five-cycle duration, and the pulse is linearly polarized and of peak intensity 4 × 10^{13} W/cm^{2}. A direct comparison can be made between our calculation and the measurements of Linder *et al*. [8]. For a revealing calculation to verify the scaling law, we study the PADs of given ATI peaks of xenon in five-cycle pulses for various CE phases and compare them with the calculated PADs using doubled (*r* = 2) binding energy, doubled central frequency, and eight times of the laser intensity of the former one in five-cycle pulses. The PADs of the same ATI order *q* for the same CE phase show identical feature, thus the comparison verifies the scaling law.

Direct experimental tests to the scaling law is somewhat difficult, since it requires the frequency, peak intensity and pulse duration of the short pulse as well as the atomic binding energy varying in a large scale at the same time. A favorite candidate will be the PADs of argon target irradiated by 616 nm laser pulses and that of xenon target irradiated by 800 nm laser pulses. Keeping the cycle number and the CE phase unchanged, choosing the peak intensity of the latter as 2.21 times that of the former, the PADs of the corresponding ATI peaks (with same *q*) will be the same, for linear polarization and for circular polarization, respectively. A numerical verification according to Eq. (3) is presented in Fig. 2.

Verifications from other sources can provide additional strong supports for the scaling law. The following are two. One is the calculation of the asymmetry degree, the ratio of the difference to the sum of the ionization rates in the two opposite directions, using the numerical solutions of the one-dimension time-dependent Schrodinger equation (TDSE). One result is shown in Fig. 3 (For detailed calculations, see Ref. [22]). The likeness of the asymmetry degree in (a) and (d), where the latter relates to the former with a re-scaling parameter *r* = 2, verifies the scaling law. The other lies in the data published by Chelkowski *et al*. [13], where the asymmetry degree in few-cycle laser pulses is calculated by three-dimension TDSE method. In Fig. 3 of Ref. [13], the asymmetry in photoionization of H atoms by a 788 nm pulse shows similar structure to that in a 800 nm pulse, but shifts to higher intensities. This likeness verifies the scaling law qualitatively.

## 4. Discussions

The scaling law in few-cycle regime sets a relation to the CE phase-dependent asymmetric ionization with the kinetic energy of photoelectrons, the duration and peak intensity of short pulses, and the ionization potential of the target atom. The CE phase is manifested by the inversion asymmetry that is caused by the interference among ionization channels. More channels can be formed in shorter pulses while less channels in longer pulses [19, 20]. If a pulse is not short enough that only one ionization channel may be formed, the PADs in such a pulse will be inversion symmetric then the CE phase plays no role. Thus a question is: How short should a pulse be, in which the CE phase as an important role comes into play? The answer to this question lies in the number of channels to form an ATI peak. The number of channels is counted as all possible combinations for different *q _{i}* satisfying Eq. (5) for a given

*q*and the channels with

*q*

_{2}=

*q*

_{3}are regarded as one because they are indistinguishable. The number of the ionization channels is determined by the kinetic energy and the pulse duration given in Eqs. (4) and (5). We find that when the cycle number is bigger than the order of an ATI peak, i.e.,

there is only one transition channel, thus the PAD is inversion symmetric and CE phase-independent. Since *q* relates with the kinetic energy of photoelectrons and the binding energy of the atom (see Eq. (4)), this relation indicates that the role of the CE phase varies with the kinetic energy of photoelectron and the binding energy of the target atom From Eq. (15) we find that more channels are available for the higher-energy electrons. Thus, it is reasonable that the PADs for low-energy electrons have inversion symmetry and that for high-energy electrons have inversion asymmetry in relatively longer few-cycle pulses [19]. The scaling law is also suggestive to the experimental observations of the CE phase-dependence. According to Eq. (15), it will be advantageous to choose the target atoms with larger binding energy or to use laser with smaller carrier-frequency, or to do the both. Moreover, increasing the laser-pulse intensity will help to manifest the CE phase in relative long few-cycle pulses, because the low ATI peaks (with small *q*) will be suppressed by the increased ponderomotive energy and the total ATI rate is contributed mainly by the higher ATI peaks, where the multi-channel ionization is easy to be formed in longer few-cycle pulses. This analysis agrees qualitatively with the observation of Gurtler *et al*. [ 17].

In the experiment performed by Paulus et al., the dependence of photoionization on the CE phase is very slight for 6-cycle laser pulses. For Kr atoms of binding energy 14*eV* irradiated by 800*nm* laser shot, 10 or more photons are to be absorbed for ionization. In energy spectra, photoelectrons distribute mostly to the 4*th* ∼ 6*th*order ATI peaks (*q* ≃ 13 ∼ 15). Thus, it is expected that the CE phase makes an appreciable effect on those ATI peaks even for the pulses with more than 10 cycles.

In our analytical treatment, the atomic structure was neglected which may lead to some uncertainties in the scaling relation. For example, if the carrier frequency is close to resonance with some bound states, the PADs will be different and the scaling relation will be poor. This implies that the scaling law becomes a better one when the resonant transition plays a negligible role. Generally, it is expected that the scaling law is more effective for rare gases, hydrogen and hydrogen-like atoms, as well as negative ions.

## Acknowledgments

This work is supported by Shanghai Rising-Star Program, the China National NSF under Grant No. 60408008 and 973 Program of China.

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