On the cutoff law of laser induced high harmonic spectra

Abstract

The currently well accepted cutoff law for laser induced high harmonic spectra predicts the cutoff energy as a linear combination of two interaction energies, the ponderomotive energy U_p and the atomic biding energy I_p, with coefficients 3.17 and 1.32, respectively. Even though, this law has been there for twenty years or so, the background information for these two constants, such as how they relate to fundamental physics and mathematics constants, is still unknown. This simple fact, keeps this cutoff law remaining as an empirical one. Based on the cutoff property of Bessel functions and the Einstein photoelectric law in the multiphoton case, we show these two coefficients are algebraic constants, 9-4\sqrt{2}\approx 3.34 and 2\sqrt{2}-1\approx 1.83, respectively. A recent spectra calculation and an experimental measurement support the new cutoff law.

Graphical abstract

Keywords

high harmonic generation / cutoff law / strong laser physics / nonperturbative quantum electrodynamics / Bessel functions / Einstein photoelectric effect

1 Introduction

High harmonic generation (HHG) is one of the most important phenomena in strong laser physics. Holding great promises on new types of stimulated emission of short wavelengths, it has attracted many experimental and theoretical attentions. Spectra of high harmonics show common features such as plateau and cutoff frequency. To describe the width of the plateau of harmonic energy spectra and the frequency or the energy of the highest valuable harmonic, a cutoff law q_c\mathrm{\hslash }\omega =3U_p+I_p, as an empirical one, was initially suggested by Krause et al.[1], where ω is the incident laser frequency, q_c the cutoff order of high harmonics, U_p the ponderomotive energy, and I_p the ionization potential energy. Since then, many discussions have been devoted to the cutoff law. A commonly accepted version of the cutoff law is q_c\mathrm{\hslash }\omega =3.17U_p+I_p[2], which was derived as a three-step process: a semiclassical ionization, a classical trajectory motion, and an electron recombination process. There are also some improved versions of the cutoff law, such as q_c\mathrm{\hslash }\omega =3.17U_p+1.32I_p[3].

Even though this law is well accepted, it still remains as an empirical law from the point of physics. The reason is as follows. In this law, there are two distinct physical constants, 3.17 and 1.32, as the coefficients of the ponderomotive energy U_p and the atomic binding energy I_p, respectively. The background information about these two physical constants is unclear. As long as these two constants remain unclear, the empirical nature of the cutoff law also remains. When these two constants were given, some questions naturally arose. Such as, Q.1. Are they fundamental physics constants independent from other fundamental physics constants? A.1. No, they cannot. Because HHG is of the electric-magnetic interaction, the relevant fundamental physics constants can only be few, such as the Planck constant \mathrm{\hslash }, the Bohr radius a₀, the fine structure constant α, the speed of light c, the rest mass of electron m_e, and the electron charge e. These fundamental ones rule out the possibility for other physics constants to be independent. For example, the electron classical radius r_0=a_0{\alpha }^2 is not an independent one. Q.2. If the two coefficients are not fundamental physics constants, how do they relate to the fundamental physics constants, or do not relate at all? Q.3. Are they mathematical constants? The makers of the original cutoff law did not provide the answers to these questions. If these questions cannot be theoretically answered, the physical mechanism underneath the HHG phenomena still needs to be discovered.

Recent developments in laser techniques allowed experimentalists to observe ultrahigh harmonics up to orders greater than 5 000 [4]. The currently accepted cutoff law and its improved version do not quite predict the harmonics of orders as that high. Pursuing better interpretation to the experimental result, also trying to answer above basic addressed physics questions, we intend to re-derive the cutoff law from fundamental theories with less assumptions.

In the nonperturbative quantum electrodynamics (NPQED) theory [5], the electron transition amplitudes due to photon fields are expressed as Bessel functions with the index denoting the photon number emitted during the transition. In our previous work, the transition rate formulas for above-threshold ionization [5, 6], Kapitza–Dirac effect [7, 8], Freeman resonance [9, 10], and HHG [11–13], are all expressed in terms of Bessel functions and wave functions of initial atomic bound state. If a physical effect is not specially related to the wave function of the atomic bound state, it must be entirely determined by the property of Bessel functions. We showed that the photoelectron angular distributions (PADs) were so [14–16].

In this paper, without using the whole NPQED theory and any HHG rate calculation, we provide a self-contained theoretical proof of the cutoff frequency of high harmonic spectra. In this proof, we only use the cutoff property of Bessel functions and the Einstein photoelectric law in the multiphoton case. Since the Einstein photoelectric law is the energy conservation law and we have exact photo-number counting in the derivations, the phase-matching processes caused by the energy-momentum and photon number changes are taken care of automatically. From a direct inspection, we find that Bessel functions for a fixed argument do have a cutoff order. The example in Fig. 1 is made with the argument x=1472.88, while the cutoff order n_c\approx x and , as expected. A natural question arising here is: Can one determine the cutoff order of HHG only from that of Bessel functions and the dynamic condition specified by the Einstein photoelectric law in the multiphoton case? To answer this question, we have to define what is cutoff order for Bessel functions. The cutoff order of Bessel functions, at a fixed argument, is defined as the order after that the value of the Bessel functions is monotonically decreasing as the order goes to the positive infinity. The reasonability and rigorous description of the definition are presented in Appendix 1.

2 The dynamic condition to limit the transferred photon numbers

In the NPQED theory, HHG is derived as a two-step process [17]. The first process is the ionization process during which the bound atomic electron ionizes into real Volkov states by absorbing j photons. The second process is the recombination process during which Volkov electrons absorb extra photons then transit back to the original bound state with an emission of a large photon of energy q\mathrm{\hslash }\omega. The dynamics condition governing the first process is

\begin{array}{}\text{(1)}& {\displaystyle \frac{P^2}{2m_e}}=\mathrm{\hslash }\omega (j-u_p-e_b)\end{array}

which is the Einstein photoelectric law in the multiphoton case, where u_p\equiv U_p/(\mathrm{\hslash }\omega ) and e_b\equiv I_p/(\mathrm{\hslash }\omega ). The Bessel functions which describe the amplitude of the electron transition induced by absorbing j photons look J_{-j}(x), where

\begin{array}{}\text{(2)}& x=\sqrt{8u_p(j-u_p-e_b)}\end{array}

as the maximized argument, is derived when the photoelectron emits in the polarization direction. In Fig. 2, the entire square-root line x=\pm \sqrt{8u_p(j-u_p-e_b)} describes the dynamics condition specified by the Einstein photoelectric law in the multiphoton case, say Eq. (1). The line x₁ intersects with the positive square-root line at two points which are the cutoff orders of Bessel functions of positive and negative even indices. The line x₃ intersects with the negative square-root line at two points which show the cutoff orders of Bessel functions of negative odd indices, without any extra photon absorption. The lowest line x₄ denotes the maximum photon absorption due to ionization and the extra photon absorption. The line x₂ denotes the relation between the argument of the Bessel function as a function of positive index –j+q, which gives the maximum photon number q_c converted to the harmonic photon; since the index is positive, the argument-index relation has to satisfy the same condition for x₁ and the dynamic condition specified by the square root curve. The maximum photon conversion number q_c, i.e., the cutoff high-harmonic order, is the distance between the intersects of line x₄ and line x₂ with the vertical axis.

3 The cutoff order of high harmonics

The complete lengthy mathematical proof is presented in the Appendix. The exact expression of the cutoff law in the U_p>I_p case reads

\begin{array}{}\text{(3)}& q_c=2(4u_p-2\sqrt{2u_p^2-2u_p{e}_b})+u_p-e_b.\end{array}

In the limit of U_p\gg I_p, i.e., u_p\gg e_b, we obtain the following version of the cutoff law

\begin{array}{}\text{(4)}& \begin{array}{cc}{q}_c\hfill & =(9-4\sqrt{2})u_p+(2\sqrt{2}-1)e_b\hfill \\ \hfill & \approx 3.3431457u_p+1.8284271e_b.\hfill \end{array}\end{array}

In the recent experiment [4], ultrahigh harmonics in keV X-ray regime from mid-infrared femtosecond lasers were observed. The orders of these harmonics are greater than 5000. The data from this experiment can be used to test different cutoff laws. In this experiment, one of the gas media used was helium. The laser wavelength was 3900nm, beam intensity was 3.3\times {10}^{14} W/{\mathrm{cm}}^2, the highest harmonics observed were of photon energy 1.6\mathrm{keV}=5033\mathrm{\hslash }\omega. With I_p=24.58741\mathrm{eV}, U_p=468.30\mathrm{eV}, and \mathrm{\hslash }\omega =0.3179\mathrm{eV}, we have u_p=1472.88, e_b=77.3432. With the laser photon energy 0.3179 eV, the cutoff order converted from the experimental cutoff energy 1.6 keV is 5033. The cutoff order predicted by the 3.17 law is q_c=3.17u_p+e_b=4746. The cutoff order predicted by the improved 3.17 law is q_c=3.17u_p+1.32e_b=4771. The cutoff order predicted by this theory is q_c=3.34u_p+1.83e_b=5060, which is very closed to the photon number 5033 observed by the experiment, also consistent with our earlier declaration as the upper bound of the cutoff order of high harmonics. The geometric relation between the cutoff order q_c and other parameters can be seen in Fig. 2, where both the horizontal axis and the vertical axis are labeled according to the experimental conditions. The exact cutoff order can also be obtained from Fig. 2. Indeed the graphical method provides an alternative proof to the cutoff order, in addition to the regular mathematical proof presented in Appendix 1.

Why the cutoff orders predicted by the expression q_c=3.34u_p+1.83e_b are greater than those predicted by the classical-field or semiclassical-field theories? The reason is as follows. In the Bessel function method, the laser photons to participate the photon-mode up-conversion are not only those ionization photons which show up in the Einstein’s photoelectric law, but also some non-ionization photons. In a full quantum mechanical treatment, all possible transition channels must occur if they are not ruled out by mathematical restrictions which embody all kinds of conservation laws in physics. The process for absorbing non-ionization photons to participate the photon-mode up-conversion may be called accompanying Raman effect. The absorbed laser photons to participate the photon-mode up-conversion are from the both, photoelectric effect and accompanying Raman effect.

4 Conclusion

The cutoff order for laser induced high harmonics can be completely determined by the Einstein photoelectric law in the multiphoton case and the properties of Bessel functions of first kind. The new cutoff law

\begin{array}{}\text{(4)}& q_c\mathrm{\hslash }\omega =(9-4\sqrt{2})U_p+(2\sqrt{2}-1)I_p\end{array}

extends the cutoff orders predicted by the traditional one and offers a new reference to future experimental measurements of cutoff orders. It also answers the questions addressed in the Introduction section. The two coefficients in the cutoff law do not depend on any fundamental physics constants which are already contained in the two interaction energies, U_p and I_p. The two coefficients are of algebraic constants. To the new cutoff law, the current paper offers two different theoretical proofs, the graphical method and the mathematical deduction method; the calculation using the formulas derived from the NPQED theory of HHG [17] provides a fine numerical test; and the recent experimental measurement [4] shows a good agreement with. The four independent methods constitute a firm basis of the new cutoff law.

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