Scaling law for energy-momentum spectra of atomic photoelectrons

Huiliang Ye; Yan Wu; Jingtao Zhang; D.-S. Guo

doi:10.1364/OE.19.020849

1. Introduction

Scaling and re-scaling are powerful tools for analyzing various physical phenomena. Searching for new scaling laws and applying these laws are interesting research topics and possess both fundamental importance and practical applications. For example, in designing a strong-laser experiment, if one can re-choose the sample atom with one-half binding energy and the laser beam with one-half frequency, also re-scale the laser beam intensity as one eighth, according to our previously published scaling law [1], one will observe the same physical phenomenon. If experiments can be performed in a re-scaled way, not only the experimental conditions will be greatly eased, but also the energy and the material consumptions will be greatly reduced. Many discussions have been devoted to scaling laws in treating atoms and molecules interacting with intense laser fields. Most studies of scaling relations of high-order harmonic generation in intense laser fields employed numerical methods [2,3]. Many scaling relations on the wavelength of the driving laser field were disclosed [4–6] by numerical methods while the deeper physical understandings still remain open. Thus, the theoretical study of scaling laws from basic physics relations is still needed with immediate applications.

Photoionization of atoms is a fundamental physical process which provides both photoelectron kinetic energy spectra and momentum spectra, or the four-momentum spectra. The magnitude of the photoelectron momentum is directly related to its corresponding kinetic energy, so the phrase “photoelectron momentum spectra” really means the photoelectron angular distributions (PADs) which, with the corresponding energy spectra, provide the full information of photoelectron four-momentum spectra. For above-threshold ionization (ATI) of atoms shined by laser fields, our analytical study has shown that the shapes of PADs depend only on three dimensionless parameters [1]: 1) the ponderomotive number u_p ≡ U_p/ω, i.e., the ponderomotive energy U_p in the unit of laser photon energy ω; 2) the binding number ɛ_b ≡ E_b/ω, i.e., the atomic binding energy E_b in the unit of ω; and 3) the absorbed-photon number. These three dimensionless parameters complete the dynamics for a single atomic photoelectron in the non-relativistic regime. Definitely these three parameters can be replaced by another set of three independent parameters to describe the same physics, but cannot be replaced by just one parameter, such as the Reiss parameter [7] or the Keldysh parameter [8]. Those physical interpretations just using a single parameter for a complicated physics phenomenon are, most likely, handicapped. Different physical processes for a physical phenomenon with the same values of the dimensionless parameters are essentially the same, while their measurable parameters may look quite different [9, 10].

Recently, the high-energy photoelectrons released from atoms and molecules were used to extract the structure information of targets [11–14]. When these laser-induced photoelectrons fall back to the vicinity of the target core, the Coulomb force dominates their subsequent motion which brings out the structure information of the target core. The harmonics generated during the recombination of photoelectron with its parent core were used to reconstruct the tomography of the outmost shell of target molecules [15]. The high-energy photoelectrons rescattered by the ionic parent core were used to retrieve the structure information of target atoms [16] and molecules [17, 18]. To reach this purpose, one needs to use the scaling law to classify all information from the irradiated targets. However, a question is, does the scaling relation still hold for rescattered photoelectrons, since the scaling law mentioned above is obtained for photoelectrons directly emitted?

Our previous work on the scaling law was mainly on PADs. A natural question is: can this scaling law be extended to the photoelectron kinetic energy spectra? The answer is yes.

2. The scaling law of photoelectrons

The extended scaling law states: the photoelectron production rate in one direction, as well as the total rate, for an atom with binding energy KE_b, shined by a laser beam of frequency Kω and intensity K³I is K times of that for an atom with binding energy E_b, shined by a laser beam of frequency ω and intensity I. According to this scaling law, different PADs linked by scaling transformations will have the same shape, subjected to the scaling ratio K; while their corresponding kinetic energy spectra will have the similar curve up or down, also subjected to the scaling ratio K. We can also say: Under a scaling transformation which maps (E_b, ω, I) to (KE_b, Kω, K³I), the photoelectron energy-momentum spectra in a logarithmic scale are invariant but up to a constant position difference logK.

The scaling relation between the laser frequency and the atomic binding energy is trivial, while the cubic relation pertinent to the laser intensity is the key of the scaling law and highly nontrivial.

Before using scaling to study the kinetic energy spectra of rescattered photoelectrons, we are going to examine the validity of the scaling law of PADs formed by these electrons. Applying the nonperturbative quantum electrodynamic theory of ATI [19], and using the quantum-field Volkov state as the intermediate state, we derive the differential ionization rate as follows (the unit h̄ = c = 1 is used throughout this paper) [20]

\frac{dW}{d Ω_{p_{f}}} = \frac{{(2 m_{e}^{3} ω^{5})}^{1 / 2}}{{(2 π)}^{2}} \sum_{q} {(q - ɛ_{b})}^{1 / 2} {| T_{fi}^{d} + T_{fi}^{r} |}^{2},

where m_e is the rest mass of electron and q is the number of photons absorbed during the overall ionization process and denotes the ATI order. The first transition matrix element is for the directly emitted photoelectrons

T_{fi}^{d} = (u_{p} - j_{i}) 𝒳_{- j_{i}} {(ζ_{f}, η)}^{*} 𝒳_{- j_{f}} (ζ_{f}, η) Φ_{i} (| P_{f} - q k |),

in which j_i and j_f are the numbers of absorbed photons in excitation and exit processes, respectively; P_f is the final momentum of photoelectrons; The second transition matrix element in Eq. (1) comes from the second term in Eq. (33) of Ref. [19], which is neglected in the previous studies, and can be described by

\begin{array}{l} T_{fi}^{r} \propto - i π \sum_{ℰ_{P, n} = ℰ_{f}} \sum_{ℰ_{P^{'}, n^{'}} = ℰ_{P, n}} 〈 ϕ_{f}, n_{f} | Ψ_{P, n} 〉 \\ \times 〈 Ψ_{P, n} | U | Ψ_{P^{'}, n^{'}} 〉 〈 Ψ_{P^{'}, n^{'}} | V | Φ_{i, n_{i}} 〉, \end{array}

where n_i and n_f are the numbers of photons before and after interaction, respectively, Φ_i denotes the initial wave function of the bound electron, and ϕ_f is the final plane wave of the photoelectron. V is the interaction operator between the electron and the laser field, U denotes the attraction of the ionic core to the electron; The quantity ℰ_P,n is the eigenenergy of the quantized-field Volkov state |Ψ_P,n〉, in which P is the momentum of the electron and n is the number of background photons [19]. The summations are performed over all the Volkov states with the same eigenenergy. The first factor of Eq. (3), from the right, describes the excitation of the initially bound electron to a Volkov state under the action of the laser field, and the third factor describes the exit process of the electron from the Volkov state to the final plane wave state, while the second factor describes the transition from a Volkov state to another on-energy-shell Volkov state under the attraction of the ionic core. Thus

T_{fi}^{r}

term describes the rescattering amplitude of on-energy-shell transitions [20]

\begin{array}{l} T_{fi}^{r} & = & \frac{i m_{e}}{4 π^{5 / 2}} \sum_{j_{i}} 𝒳_{- j_{f}} (ζ_{f}, η) (u_{p} - j_{i}) | P | Φ_{i} (| P |) \\ \times & \int d Ω_{P} 𝒳_{q - j_{i} + j_{f}} (ζ - ζ_{f}) 𝒳_{- j_{i}} {(ζ, η)}^{*} U (P_{f} - P - q k) \end{array}

with |P| = (2m_eω)^1/2 (j_i − u_p − ɛ_b)^1/2, and U(P) is the Fourier transform of the binding potential. In our calculations, we set u_p equal to j_f [19]. The phased Bessel functions and the generalized phased Bessel (GPB) functions are defined as [21]

\begin{array}{l} X_{n} (z) & \equiv & J_{n} (| z |) e^{in arg (z)}, \\ 𝒳_{j} (z, z^{'}) & \equiv & \sum_{m = - \infty}^{\infty} X_{j - 2 m} (z) X_{m} (z^{'}), \end{array}

where z and z′ are complex variables with the arguments in Eq. (2) and (4) given by

ζ_{f} = ζ_{0} P_{f} \cdot ɛ, ζ = ζ_{0} P \cdot ɛ, η = u_{p} ɛ \cdot ɛ / 2,

where

ζ_{0} = 2 \sqrt{u_{p} / (m_{e} ω)}

and ɛ is the polarization vector of the laser beam. For directly emitted photoelectrons, the order and the arguments of the GPB functions in

T_{fi}^{d}

are determined by the three dimensionless parameters mentioned above, and the wave function Φ_i(|P_f|) does not affect the PADs in the long-wavelength limit [22], thus the PADs are the same if those three parameters are kept unchanged. For rescattered photoelectrons, the ionization amplitude

T_{fi}^{r}

includes transitions between a Volkov state to another on-energy shell Volkov state. By keeping the three parameters unchanged, we find the value of 𝒳_{−j_f}(ζ_f,η ) is still the same, while the values of other two Bessel functions are not. The arguments and the order of the rest two Bessel functions vary with the number of photons absorbed in the ionization process, say j_i. Since this is a dummy index of the summation, it does not result in any change in the summation of Bessel functions, thus the dynamic part of the formula keeps the same under the scaling change. The form of U(P_f – P) depends on the Coulomb potential. For the widely used screened Coulomb potential

U (r) = \frac{- Z e^{2}}{4 π r} exp (- λ r),

we obtain

U (P_{f} - P) = \frac{- Z e^{2}}{{| P_{f} - P |}^{2} + λ^{2}} .

Since the momentum P is also a dummy variable, we expect the value of U(P_f – P) does not depend on the direction of P_f. Thus, the PADs of rescattered photoelectrons obey the scaling law.

3. Numerical verification by the analytical formula

Then, we numerically verify the scaling law using the derived formula through the following steps: (1) choosing the PAD of one ATI peak as a reference, (2) enlarging the atomic binding energy and the laser frequency as K times, and (3) changing the laser intensity as K², K³ and K⁴ times of the original value respectively, then (4) comparing the main features of the calculated PADs with the original one. One comparison is depicted in Fig. 1, in which (a) is the PAD served as the reference. In these PADs, we see clearly the main lobes along the laser polarization and the jets striking from the waist of the main lobes. Figure 1(a) depicts the PAD of the 23rd ATI peak of the hydrogen atom irradiated by the laser field of wavelength 800nm and intensity 1.04 × 10¹⁴ W/cm², while the others depict the PADs of the 23rd ATI peak of a model atom with binding energy 27.20 eV irradiated by the laser field of wavelength 400nm and intensity 4.16 (b), 8.32 (c) and 16.64 (d) × 10¹⁴ W/cm², respectively. We see the PAD in (c) agrees well with the referenced PAD in (a), while the other two PADs show more or less differences from the referenced PAD, in the size and the number of jets and the breadth of main lobes. The fact that the PADs formed by rescattered photoelectrons are invariant under the scaling transformation confirms the scaling law of PADs of rescattered photoelectrons.

Fig. 1 Polar plots of the PADs of the 23rd ATI peaks. (a) serves as a reference which is for hydrogen atoms irradiated by laser field of intensity 1.04 × 10¹⁴ W/cm² and of wavelength 800 nm. (b), (c) and (d) are from a model atoms with binding energy 27.20 eV, irradiated by laser field of wavelength 400 nm and of intensity 4.16, 8.32, and 16.64 × 10¹⁴ W/cm², respectively. The similarity between the PADs in (a) and (c) verifies the scaling law. We set λ = 0.1 and change it as $\sqrt{K}$ times in the scaling transformation.

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Next, we prove that the scaling law which applies well for the PADs also applies for the energy spectra of photoelectrons. The ionization rate of a given ATI peak is the angle-integrated PAD. Thus the energy spectra of photoelectrons are expected to obey the same scaling law. Furthermore, in the unit of laser photon energy, the plateau in the energy spectra cuts off at 10u_p. Since keeping u_p unchanged is the most important features of the scaling law, one may anticipate that the energy spectra obey the same scaling law. On the other hand, since the PADs disclose only the relative variation of the ionization rate with the azimuthal angle, the difference in the absolute value of PADs may change the energy spectra of photoelectrons. The question is to what extent this difference changes the photoelectron energy spectra.

We first analysis the change of energy spectra under the scaling transformation with a ratio constant K. For directly emitted photoelectrons, as we discussed above, the GPB functions keep unchanged under the scaling transformation, then any change in the energy spectra is caused by the initial wave function Φ_i(|P_f|). For hydrogen-like 1s wave function, Φ_i(|P_f|) is given by

Φ_{i} | P_{f} | = \frac{8 {(π β^{5})}^{1 / 2}}{{(β^{2} + {| P_{f} |}^{2})}^{2}},

where

β = \sqrt{2 m_{e} E_{b}}

. In the denominator both β² and

P_{f}^{2} = 2 m_{e} (q ω - E_{b} - u_{p} ω)

scale as constant K. The wave function, as well as the value of

T_{fi}^{d}

, scales as K^−3/4 under the scaling transformation. For rescattered photoelectrons, the changes in the energy spectra depend also on the value of U(P_f – P) which varies with the charge number Z and the screen parameter λ. For analytical simplicity, we set λ = 0. According to the relation E_b ∝ – Z²/n^*2 where n^* is the principle quantum defect number, the value of Z changes as

\sqrt{K}

times when the binding energy is enlarged by K times. Under the scaling transformation, the value of |P_f – P|² is enlarged by K times, thus the value of U(P_f–P) is enlarged by K^−1/2 times. Considering the momentum |P| in front of the integral, we find the value of

T_{fi}^{r}

changes as that of Φ_i(|P_f|) and is also enlarged by K^−3/4 times in the rescattering case. Considering the factor ω^5/2 in front of the summation, we find the ionization yield is enlarged by K times under the scaling transformation. This means the energy spectra obey the scaling law, because the overall ionization rate scales as K times under the scaling transformation.

To show numerically, we calculate the energy spectra changed according to the scaling law, then compare them with the original one. The comparison is shown in Fig. 2 for two different scaling ratios. The re-scaled photoelectron energy spectra are almost the same as the original one, and the only difference is that the rescaled spectra are overall shifted by a ratio of K for both directly emitted photoelectrons and re-scattered photoelectrons. Thus we conclude that the kinetic energy spectra obey the same scaling law as the PADs do. In a logarithmic scale, the energy spectra of the re-scaled case are overall shifted by a constant logK.

Fig. 2 (Color online) The calculated energy spectra for different scaling ratios. The original one is for the hydrogen atoms irradiated by laser field of wavelength 800 nm and intensity 1.04 × 10¹⁴ W/cm². The rest two spectra are these calculated according to the scaling law with different scaling ratios. The value of λ is chosen as 0.1.

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4. Verification by the numerical method

Verification by independent methods provides a strong check to the scaling law. In the following we verify the scaling law of photoelectron energy spectra by the time-dependent Schrödinger equation (TDSE) method, which is widely used in the study of the interaction process of atoms and molecules with intense laser fields. Under the single active electron approximation, the TDSE is solved by the QPROP code [23] in the velocity gauge, in which the spherical coordinate system is employed and the wave function is expanded on the radial-position angular-moment grid. The detailed procedure is presented in Refs. [23,24]. The simulation is done with 60 angular momenta involved. The laser field is chosen as that a 16-cycle flat-top laser pulse with a half-cycle in both the raising and the tailing edges. The target is chosen as the argon atom in its ground state with binding energy 15.76 eV. The kinetic energy spectrum of photoelectrons is obtained by employing a window operator on the time-resolved wave function. The window operator is taken as [25]

W_{γ} (ℰ) = \frac{γ^{2^{n}}}{{(H_{0} - ℰ)}^{2^{n}} + γ^{2^{n}}},

where 2γ is the width of the energy bin centered at energy ℰ. In our calculations, we set γ = 5.44 × 10⁻² eV and n = 3.

The photoelectron spectrum of the argon atom in the laser pulse of wavelength 800 nm and intensity 6.80×10¹³ W/cm² is chosen as the reference. Same as the aforementioned procedure, we enlarge the atomic binding energy and the laser frequency as 2 times, and vary the laser intensity to 4, 8 and 16 times the original intensity, respectively. Figure 3(a) is the kinetic energy spectrum served as the reference, which exhibits a falloff in the low-energy region and a plateau followed by a cutoff as the kinetic energy increases. Owing to resonant enhancement caused by the level shift in strong laser field [26], the 5th to 8th ATI peaks become higher than their neighboring peaks. Figure 3(b) depicts the spectrum for a model atom with binding energy 31.52 eV irradiated by laser field of wavelength 400 nm and intensity 2.72 × 10¹⁴ W/cm². The spectrum exhibits a rapid decrease as the ATI order increases, but the plateau is not remarkable which indicates that the rescattering effect is not notable. Figure 3(d) depicts the calculated spectrum for the model atom for the laser intensity 1.09 × 10¹⁵ W/cm² (2⁴ times the original). The spectrum shows a striking plateau that is longer than that in Fig. 3(a), and the resonance structure becomes significantly shifted to high-energy part. Figure 3(c) is the spectrum for the model atom for the laser intensity enlarged by 2³ times the original one. The spectrum resembles the referenced one in many aspects. The shape of the two spectra is quite similar to each other and both the spectra begin to ramp down at the same order. A detailed similarity is that the resonance structure starts after the fourth peak. We thus conclude that the spectrum for laser intensity changed by K³ times shows the best agreement with the original one, which verifies the scaling law.

Fig. 3 The kinetic energy spectra of photoelectrons calculated by the TDSE method: (a) is for argon atoms in laser field of wavelength 800nm and intensity 6.80 × 10¹³ W/cm²; (b)–(d) are calculated spectra for a model atom with binding energy 31.52 eV irradiated by laser field of wavelength 400 nm and intensity 2.72 × 10¹⁴ W/cm² (b); 5.44 × 10¹⁴ W/cm² (c); 1.09 × 10¹⁵ W/cm² (d). For convenience of comparison, the abscissa is plotted in the unit of ω.

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5. Conclusions and discussions

In this paper we establish the scaling law of photoelectron energy spectra including the rescattered photoelectrons. We present the detailed verifications of the scaling law by analytical and numerical methods. We show that the scaling law working for photoelectron momentum spectra can be well extended to the photoelectron energy spectra. The well-extended scaling relation for photoelectron four-momentum spectra applies for both directly ionized and rescattered photoelectrons.

The scaling law can be used to obtain the necessary reference to extract detailed structure information of the targets. This has potential applications in the study of molecules. Since the scaling law states the same dynamics behavior while the wave function determines the initial electronic state before interaction, the scaling relation depends less on the wave function. The numerical calculations also support this statement. This result implies that the scaling law holds for the molecular case, although the single-active-electron wave function of molecules is quite different from that of atoms.

Acknowledgments

This work was supported by the Chinese National Natural Science Foundation under Grant Nos. 10774153, 61078080, and 11174304, and by the 973 Program of China under Grant Nos. 2010CB923203 and 2011CB808103.

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Scaling law for energy-momentum spectra of atomic photoelectrons

Abstract

1. Introduction

2. The scaling law of photoelectrons

3. Numerical verification by the analytical formula

4. Verification by the numerical method

5. Conclusions and discussions

Acknowledgments

References and links

Cited By

Figures (3)

Equations (10)

Optics Express