Application of the Generalized Bessel Function to Two-Color Phase-of-the-Phase Spectroscopy

Abstract

In two-color strong field ionization of atoms, dynamical characteristics such as ionization rate and electron trajectory depend heavily on the relative phase of the strong fundamental field and its second harmonic. The phase-of-the-phase spectroscopy method reveals the oscillation of the photoelectron momentum distribution with the relative phase of the two-color field, and the relative phase contrast and its phase are usually obtained by the fitting process. Instead, we apply the generalized Bessel function to the strong field approximation and derive analytical expressions for the first- and second-order relative phase contrast and phase-of-the-phase spectra. Our analytical results are in good agreement with fitting-based ones, but calculating them is much less computationally intensive. We illustrate the advantage of using generalized Bessel functions in two-color phase-of-the-phase spectroscopy of argon.

1. Introduction

The photoelectron momentum distribution (PMD) generated by laser interaction with atoms and molecules contains a wealth of dynamical information such as ionization time and electron motion, and its interference pattern is extremely dependent on the specifics of laser pulses and targets [1,2,3,4,5,6,7,8,9,10]. In particular, accurate information about the amplitude and phase of the electron emission associated with the waveform can be obtained by changing the relative phase of the two-color field [11,12,13,14,15,16,17,18,19]. A simple and feasible method of analyzing the spatio-temporal information contained in the PMD is urgently needed. Recently, phase-of-the-phase (PP) spectroscopy has been proposed as a powerful tool to analyze the dynamical information [20,21,22]. The photoelectron yield $Y(\mathbf{p},\phi )$ is an observable depending on two variables: the momentum $\mathbf{p}$ and the the relative phase $\phi$ between the fundamental field with frequency $\omega$ and its second harmonic with frequency 2$\omega$ [23,24]. In this paper, we assume a linearly polarized two-color field $\mathbf{E}\left(t\right)=-{\partial }_t\mathbf{A}\left(t\right)$, where the vector potential has the form

$$\mathbf{A}\left(t\right)=A_0\left(t\right)[sin\omega t+\xi sin(2\omega t+\phi )]{\mathbf{e}}_z.$$

(1)

Here, $A_0\left(t\right)$ is the envelope of the laser, $\xi$ is the ratio between the 2$\omega$ and $\omega$ components. As the tunable parameter $\phi$ changes periodically, Fourier analysis can be used to find $Y(\mathbf{p},\phi )$ for each final momentum $\mathbf{p}$. Truncating the series at the second-order terms [19,25],

$$\begin{array}{cc}\hfill Y(\mathbf{p},\phi )=& Y_0\left(\mathbf{p}\right)+\left|Y_1\left(\mathbf{p}\right)\right|cos\left[\phi +{\Phi }_1\left(\mathbf{p}\right)\right]\hfill \\ \hfill & +\left|Y_2\left(\mathbf{p}\right)\right|cos\left[2\phi +{\Phi }_2\left(\mathbf{p}\right)\right]\hfill \end{array}$$

(2)

where $\left|Y_1\left(\mathbf{p}\right)\right|$ and $\left|Y_2\left(\mathbf{p}\right)\right|$ are the first- and second-order relative phase contrast (RPC) spectra, respectively. It can be seen that the contribution to the yield with momentum $\mathbf{p}$ at a given order follows a co-cosine curve as the frequency difference is modulated; its amplitude is the RPC at that order, and its phase is related to the periodic photoelectron signal modulation to $\phi$, denoted as a quantity ${\Phi }_1\left(\mathbf{p}\right)$, the PP at that order [26,27].

The details of RPC and PP spectra are extremely sensitive to the shape of the targets and laser fields. PP spectra contain a lot of ultrafast spatio-temporal dynamical information. For this reason, PP spectroscopy is a powerful and a widely investigated method of exploring the potential dynamics of strong fields, often used to distinguish different types of signal contributions, such as the weights of long and short orbitals or the contribution of different time windows to electron ionization [28]. Other important applications are the detection of attosecond time delay in Freeman resonance states [29] and the re-trapping resonance scattering mechanism to accurately interpret the time delays measured in non-perturbation above threshold ionization (ATI) regions [30]. In addition, recent studies have shown that PP spectroscopy can also be used to analyze the photoemission process induced by strong fields in bulk density media with atomic impurities [31].

The completion of the PP spectroscopy process requires as many PMDs in different relative phases as possible, which is very time-consuming. The use of semi-classical models offers an alternative. While these are usually solved by numerical fitting, some may be amenable to analytical solution [32,33]. However, there are still relatively few. In Ref. [34], a characteristic checkboard pattern in the ATI spectra of xenon has been demonstrated. In this paper, we focus on deriving the mathematical expressions of PP spectroscopy to the second order in the strong field approximation (SFA). In the framework of SFA, the first-order Fourier expansion is insufficiently accurate to identify the underlying dynamics. Therefore, an extension to the second order is necessary. For this purpose, we adopt the generalized Bessel function to SFA and extend the derivation of the Fourier component of PP spectroscopy to the second order. The use of generalized Bessel function in practice does not require so much data, which greatly saves the computational cost.

The rest of the paper is organized as follows: In Section 2, the SFA method is briefly introduced. In Section 3, applying the generalized Bessel function to SFA, we derive analytical expressions for first- and second-order RPC and PP spectra. In Section 4, the change of strong field ionization of Ar atom with relative phase in a two-color linearly polarized laser field has been analyzed by the SFA method. It sounds odd to say PP spectra are obtained by PP spectroscopy that would seem to be the obvious way one would obtain them. Meanwhile, we present the result of analytical expressions. Finally, some conclusions from our work are presented in Section 5. We also include some technical details in Appendix A. (Atomic units are used unless noted otherwise.)

2. Theoretical Method

In this section, we apply the SFA [35,36,37,38] to photoelectron interference in the case of the vector potential in Equation (1). The simplest form of SFA accounts only for the so-called “direct” electrons, which are bound to the atom up to the detachment time and at later times just move in the laser field without any further interaction with the parent atom. Thus, the probability amplitude of transition from the bound state ${\Psi }_0$ to a continuum state ${\Psi }_{\mathbf{p}}$ with asymptotic momentum $\mathbf{p}$ can be written as [39]

$$\begin{array}{cc}\hfill M_{\mathbf{p}}& =-i{\int }_{-\infty }^{\infty }dt〈{\Psi }_{\mathbf{p}}^{GV}\left(t\right)\mid \mathbf{r}·\mathbf{E}\left(t\right)\mid {\Psi }_0\left(t\right)〉\hfill \\ \hfill & =-i{\int }_{-\infty }^{\infty }dt{e}^{i{S}_{\mathbf{p}}\left(t\right)}〈\mathbf{p}+\mathbf{A}\left(t\right)\mid \mathbf{r}·\mathbf{E}\left(t\right)\mid {\Psi }_0\left(t\right)〉.\hfill \end{array}$$

(3)

Here,

$$\left|{\Psi }_{\mathrm{p}}^{GV}\left(t\right)\right.〉=|\mathbf{p}+\mathbf{A}\left(t\right)\rangle e^{-i{S}_{\mathbf{p}}\left(t\right)}$$

(4)

is the Gordon–Volkov state, i.e., a solution to the time-dependent Schrödinger equation [40,41,42] $i\left|{\dot{\Psi }}_{\mathbf{p}}^{\mathrm{GV}}\left(t\right)\right.〉=\left[\frac{{\mathbf{p}}^2}{2}+\mathbf{r}·\mathbf{E}\left(t\right)\right]\left|{\Psi }_{\mathbf{p}}^{\mathrm{GV}}\left(t\right)\right.〉$ with no Coulomb potential. The action during the transition process is written as

$$S_{\mathbf{p}}\left(t\right)={\int }_{-\infty }^t\left\{\frac{{\left[\mathbf{p}+\mathbf{A}\left(t^{\prime }\right)\right]}^2}{2}+I_p\right\}d{t}^{\prime },$$

(5)

where $I_p$ is the ionization potential. Using the saddle-point approximation (SPA), the transition amplitudes in Equation (3) can be approximated by [43]

$$M_{\mathbf{p}}=\sum _s\frac{2^{-1/2}{\left(2I_p\right)}^{5/4}}{\mathbf{E}\left(t_s\right)·\left[\mathbf{p}+\mathbf{A}\left(t_s\right)\right]}{e}^{i{S}_{\mathbf{p}}\left(t_s\right)},$$

(6)

where $t_s$ satisfies the saddle point equation

$${\left.\frac{\partial S_{\mathbf{p}}\left(t\right)}{\partial t}\right|}_{t=t_s}=\frac{{\left[\mathbf{p}+\mathbf{A}\left(t_s\right)\right]}^2}{2}+I_p=0.$$

(7)

3. Derivation of RPC and PP

The generalized Bessel function is very useful in the analytical treatment of strong-field processes, especially in the case of linearly polarized laser fields [44,45,46,47,48,49,50,51,52,53,54]. In this section, we will elaborate on the PP spectroscopy method in the framework of SFA theory. Applying the generalized Bessel function in the SFA, analytical expressions for the first- and second-order RPC and PP can be obtained.

3.1. First Order RPC and PP

In the SFA framework, with the 1s state chosen to be the initial state, the transition matrix element from the initial state to the final state can be written as [46]

$$M_{\mathbf{p}}=i{\Psi }_0\left(\mathbf{p}\right)\left(\frac{p^2}{2}+I_p\right){\int }_{-\infty }^{\infty }dt{e}^{i{S}_{I_{p,\mathbf{p}}}\left(t\right)}.$$

(8)

Here, ${\Psi }_0\left(\mathbf{p}\right)$ is the initial state. Substituting Equation (1) with $A_0\left(t\right)=A_0$ into Equation (5), $S_{I_{p,\mathbf{p}}}\left(t\right)$ becomes

$$\begin{array}{cc}\hfill S_{I_p,\mathbf{p}}=& \left(\frac{p^2}{2}+I_p+U_p\right)t-\frac{A_0{p}_z}{\omega }sin\left(\omega t+\frac{\pi }{2}\right)+\frac{A_0^2}{8\omega }sin\left[2\left(\omega t+\frac{\pi }{2}\right)\right]\hfill \\ \hfill & -\frac{A_0\xi }{\omega }\left[\frac{p_z}{2}cos(2\omega t+\phi )+\frac{A_0}{6}(3\omega t+\phi )-\frac{A_0}{2}(\omega t+\phi )\right]+\mathrm{O}\left({\xi }^2\right).\hfill \end{array}$$

(9)

Neglecting the term $\mathrm{O}\left({\xi }^2\right)$ in Equation (9) and substituting it into Equation (8), then using the generalized Bessel function [55,56,57,58,59,60] $e^{i\left[\right(usin\phi +vsin\left(2\phi \right)]}={\sum }_{n=-\infty }^{\infty }{e}^{in\phi }{J}_n(u,v)$ and its recursive relation $J_n(-u,v)={(-)}^n{J}_n(u,v)$ (see Appendix A for more details), Equation (8) can be rewritten as

$$\begin{array}{cc}\hfill M_{\mathbf{p}}=& i{\Psi }_0\left(\mathbf{p}\right)\left(\frac{p^2}{2}+I_p\right)\times {\int }_{-\infty }^{\infty }dt{e}^{i\left(\frac{p^2}{2}+I_p+U_p\right)t}\hfill \\ \hfill & \sum _{n=-\infty }^{\infty }{(-i)}^n{J}_n(u,v)e^{-in\omega t}{e}^{-\frac{i{A}_0\xi }{\omega }\left[\frac{p_z}{2}cos(2\omega t+\phi )+\frac{A_0}{6}sin(3\omega t+\phi )-\frac{A_0}{2}sin(\omega t+\phi )\right]},\hfill \end{array}$$

(10)

where the two parameters of the generalized Bessel function have values of

$$u=\frac{p_z{A}_0}{\omega },v=-\frac{U_p}{2\omega }$$

(11)

with the ponderomotive energy $U_{\mathrm{p}}=A_0^2/4.$

For the term $exp\left\{-\frac{i{A}_0\xi }{\omega }\left[\frac{p_z}{2}cos(2\omega t+\phi )+\frac{A_0}{6}sin(3\omega t+\phi )-\frac{A_0}{2}sin(\omega t+\phi )\right]\right\}$ in the integral of Equation (10), one can use its Taylor series expansion, and the first two terms can be rewritten as $1-\frac{i{A}_0\xi }{\omega }\left[\frac{p_z}{2}cos(2\omega t+\phi )+\frac{A_0}{6}sin(3\omega t+\phi )-\frac{A_0}{2}sin(\omega t+\phi )\right]$. Then, we have $M_{\mathbf{p}}$ split in two terms,

$$\begin{array}{cc}\hfill M_{\mathbf{p}}=& i{\Psi }_0\left(\mathbf{p}\right)\left(\frac{p^2}{2}+I_p\right){\int }_{-\infty }^{\infty }dt{e}^{i\left(\frac{p^2}{2}+I_p+U_p\right)t}\sum _{n=-\infty }^{\infty }{(-i)}^n{J}_n(u,v)e^{-in\omega t}\hfill \\ \hfill & +i{\Psi }_0\left(\mathbf{p}\right)\left(\frac{p^2}{2}+I_p\right){\int }_{-\infty }^{\infty }dt{e}^{i\left(\frac{p^2}{2}+I_p+U_p\right)t}\sum _{n=-\infty }^{\infty }{(-i)}^n{J}_n(u,v)e^{-in\omega t}\hfill \\ \hfill & \times \left\{-\frac{i{A}_0\xi }{\omega }\left[\frac{p_z}{2}cos(2\omega t+\phi )+\frac{A_0}{6}sin(3\omega t+\phi )-\frac{A_0}{2}sin(\omega t+\phi )\right]\right\}.\hfill \end{array}$$

(12)

The first part of Equation (12) is denoted by

$$\begin{array}{cc}\hfill M_{\mathbf{p}}^{\left(0\right)}& =i{\Psi }_0\left(\mathbf{p}\right)\left(\frac{p^2}{2}+I_p\right){\int }_{-\infty }^{\infty }dt{e}^{i\left(\frac{p^2}{2}+I_p+U_p\right)t}\sum _{n=-\infty }^{\infty }{(-i)}^n{J}_n(u,v)e^{-in\omega t}\hfill \\ \hfill & =2\pi \mathrm{i}{\Psi }_0\left(\mathbf{p}\right)\left(\frac{p^2}{2}+I_p\right)\sum _{n=-\infty }^{\infty }{(-i)}^n\delta \left(\frac{p^2}{2}+I_p+U_p-n\omega \right)J_n(u,v).\hfill \end{array}$$

(13)

We now focus on the second part of Equation (12) and write it further as the sum of the six terms, i.e.,

$$\begin{array}{cc}\hfill M_{\mathbf{p}}^{\left(1\right)}=& i{\Psi }_0\left(\mathbf{p}\right)\left(\frac{p^2}{2}+I_p\right){\int }_{-\infty }^{\infty }dt{e}^{i\left(\frac{p^2}{2}+I_p+U_p\right)t}\sum _{n=-\infty }^{\infty }{(-i)}^n{e}^{-in\omega t}{J}_n(u,v)\left(-\frac{i{A}_0\xi }{4\omega }\right)\hfill \\ \hfill & \times \left[p_z{e}^{i(2\omega t+\phi )}+p_z{e}^{-i(2\omega t+\phi )}+\left(-\frac{i{A}_0}{3}\right)e^{i(3\omega t+\phi )}\right.\hfill \\ \hfill & +\left.\left(\frac{i{A}_0}{3}\right)e^{-i(3\omega t+\phi )}+\left(i{A}_0\right)e^{i(\omega t+\phi )}+\left(-i{A}_0\right)e^{-i(\omega t+\phi )}\right].\hfill \end{array}$$

(14)

As a brief illustration for the simplification of Equation (14), we take the first term. Denoting this term by $M_{\mathbf{p}}^{\left(11\right)}$ and setting $n=n+2$, we obtain

$$\begin{array}{cc}\hfill M_{\mathbf{p}}^{\left(11\right)}& =i{\Psi }_0\left(\mathbf{p}\right)\left(\frac{p^2}{2}+I_p\right){\int }_{-\infty }^{\infty }dt{e}^{i\left(\frac{p^2}{2}+I_p+U_p\right)t}\sum _{n=-\infty }^{\infty }{(-i)}^n{e}^{-in\omega t}{J}_n(u,v)\left(-\frac{i{A}_0\xi }{4\omega }\right)\left[p_z{e}^{i(2\omega t+\phi )}\right]\hfill \\ \hfill & =2\pi i{\Psi }_0\left(\mathbf{p}\right)\left(\frac{p^2}{2}+I_p\right)\sum _{n=-\infty }^{\infty }{(-i)}^n\delta \left(\frac{p^2}{2}+I_p+U_p-n\omega \right)\left(\frac{i{A}_0\xi }{4\omega }\right)p_z{J}_{n+2}(u,v)e^{i\phi }.\hfill \end{array}$$

(15)

Using the same method for the remaining five terms of Equation (14), we have

$$\begin{array}{cc}\hfill M_{\mathbf{p}}^{\left(1\right)}=& 2\pi i{\Psi }_0\left(\mathbf{p}\right)\left(\frac{p^2}{2}+I_p\right)\sum _{n=-\infty }^{\infty }\left\{{(-i)}^n\delta \left(\frac{p^2}{2}+I_p+U_p-n\omega \right)\left(\frac{i{A}_0\xi }{4\omega }\right)\right.\hfill \\ \hfill & \times \left[\left(-A_0{J}_{n+1}(u,v)+p_z{J}_{n+2}(u,v)-\frac{A_0}{3}{J}_{n+3}(u,v)\right)e^{i\phi }\right.\hfill \\ \hfill & \left.\left.+\left(-A_0{J}_{n-1}(u,v)+p_z{J}_{n-2}(u,v)-\frac{A_0}{3}{J}_{n-3}(u,v)\right)e^{-i\phi }\right]\right\},\hfill \end{array}$$

(16)

Adding Equations (13) and (16), Equation (12) can be written as

$$\begin{array}{cc}\hfill M_{\mathbf{p}}=& 2\pi i{\Psi }_0\left(\mathbf{p}\right)\left(\frac{p^2}{2}+I_p\right)\sum _{n=-\infty }^{\infty }{(-i)}^n\delta \left(\frac{p^2}{2}+I_p+U_p-n\omega \right)\hfill \\ \hfill & \times \left\{J_n(u,v)-\frac{i{A}_0\xi }{4\omega }\left[\left(A_0{J}_{n+1}(u,v)-p_z{J}_{n+2}(u,v)+\frac{A_0}{3}{J}_{n+3}(u,v)\right)e^{i\phi }\right.\right.\hfill \\ \hfill & \left.\left.+\left(A_0{J}_{n-1}(u,v)-p_z{J}_{n-2}(u,v)+\frac{A_0}{3}{J}_{n-3}(u,v)\right)e^{-i\phi }\right]\right\}.\hfill \end{array}$$

(17)

The probability is given by

$$Y(\mathbf{p},\phi )={\left|M_{\mathbf{p}}\right|}^2.$$

(18)

Setting $B=\frac{A_0\xi }{4\omega },a=A_0{J}_{n+1}(u,v)-p_z{J}_{n+2}(u,v)+\frac{A_0}{3}{J}_{n+3}(u,v),a_1=A_0{J}_{n-1}(u,v)-p_z{J}_{n-2}(u,v)$$+\frac{A_0}{3}{J}_{n-3}(u,v)$, substituting Equation (17) into Equation (18) and ignoring the higher order term $B^2\left(a{e}^{i\phi }+a_1{e}^{-i\phi }\right)\left(a{e}^{-i\phi }+a_1{e}^{i\phi }\right)$ that contributes very little to the sum, we have

$$\begin{array}{cc}\hfill Y(\mathbf{p},\phi )& =2\pi {\left|{\Psi }_0\left(\mathbf{p}\right)\right|}^2{\left(\frac{p^2}{2}+I_p\right)}^2\sum _{n=-\infty }^{\infty }\delta \left(\frac{p^2}{2}+I_p+U_p-n\omega \right)\hfill \\ \hfill & \times \left\{J_n{(u,v)}^2-J_n(u,v)\xi sin\phi \left[4v\left(J_{n+1}(u,v)-J_{n-1}(u,v)\right)\right.\right.\hfill \\ \hfill & \left.\left.+\frac{u}{2}\left(J_{n+2}(u,v)-J_{n-2}(u,v)\right)+\frac{4v}{3}\left(J_{n+3}(u,v)-J_{n-3}(u,v)\right)\right]\right\}.\hfill \end{array}$$

(19)

Next, we use the property of the generalized Bessel function $2n{J}_n(u,v)=u(J_{n-1}(u,v)$$+J_{n+1}(u,v)\left)\right)+2v(J_{n-2}(u,v)+J_{n+2}(u,v))$ to eliminate the terms of $J_{n+3}(u,v)$ and $J_{n-3}(u,v)$ and to rewrite Equation (19) as

$$\begin{array}{cc}\hfill Y(\mathbf{p},\phi )=& 2\pi {\left|{\Psi }_0\left(\mathbf{p}\right)\right|}^2{\left(\frac{p^2}{2}+I_p\right)}^2\sum _{n=-\infty }^{\infty }\delta \left(\frac{p^2}{2}+I_p+U_p-n\omega \right)\hfill \\ \hfill & \times \left\{J_n{(u,v)}^2-\frac{4J_n(u,v)\xi }{3}sin\phi \left[(4v+n)\left(J_{n+1}(u,v)-J_{n-1}(u,v)\right)\right.\right.\hfill \\ \hfill & \left.\left.+\left(J_{n+1}(u,v)+J_{n-1}(u,v)\right)-\frac{u}{8}\left(J_{n+2}(u,v)-J_{n-2}(u,v)\right)\right]\right\}.\hfill \end{array}$$

(20)

Here, we can see that Equation (20) has the same form as Equation (2) truncated to the first order, which is expressed as

$$Y(\mathbf{p},\phi )=Y_0\left(\mathbf{p}\right)+\left|Y_1\left(\mathbf{p}\right)\right|cos\left[\phi +{\Phi }_1\left(\mathbf{p}\right)\right],$$

(21)

with

$$\begin{array}{cc}\hfill Y_0\left(\mathbf{p}\right)=& 2\pi {\left|{\Psi }_0\left(\mathbf{p}\right)\right|}^2{\left(\frac{p^2}{2}+I_p\right)}^2\sum _{n=-\infty }^{\infty }\delta \left(\frac{p^2}{2}+I_p+U_p-n\omega \right)J_n(u,v){}^2\hfill \\ \hfill Y_1\left(\mathbf{p}\right)=& 2\pi {\left|{\Psi }_0\left(\mathbf{p}\right)\right|}^2{\left(\frac{p^2}{2}+I_p\right)}^2\sum _{n=-\infty }^{\infty }\delta \left(\frac{p^2}{2}+I_p+U_p-n\omega \right)\hfill \\ \hfill & \times \left\{-\frac{4J_n(u,v)\xi }{3}\left[(4v+n)\left(J_{n+1}(u,v)-J_{n-1}(u,v)\right)\right.\right.\hfill \\ \hfill & \left.\left.+\left(J_{n+1}(u,v)+J_{n-1}(u,v)\right)-\frac{u}{8}\left(J_{n+2}(u,v)-J_{n-2}(u,v)\right)\right]\right\},\hfill \end{array}$$

(22)

and

$${\Phi }_1\left(\mathbf{p}\right)=arg\left[Y_1\left(\mathbf{p}\right)\right].$$

(23)

Therefore, one obtains analytic expressions for $\left|Y_1\left(\mathbf{p}\right)\right|$ and ${\Phi }_1\left(\mathbf{p}\right)$ as the first order RPC and PP derived from the generalized Bessel function, respectively.

3.2. Second Order RPC and PP

We will now extend the first-order derivation to second order. Truncating the term $exp\left\{-\frac{i{A}_0\xi }{\omega }\left[\frac{p_z}{2}cos(2\omega t+\phi )+\frac{A_0}{6}sin(3\omega t+\phi )-\frac{A_0}{2}sin(\omega t+\phi )\right]\right\}$ of the integral in Equation (10) to the third term after Taylor expansion splits $M_{\mathbf{p}}$ into three terms. The first two terms are the same as those of Equation (12). The third term can be expressed as

$$\begin{array}{cc}\hfill M_{\mathbf{p}}^{\left(2\right)}& =i{\Psi }_0\left(\mathbf{p}\right)\left(\frac{p^2}{2}+I_p\right){\int }_{-\infty }^{\infty }dt{e}^{i\left(\frac{p^2}{2}+I_p+U_p\right)t}\sum _{n=-\infty }^{\infty }{(-i)}^n{J}_n(u,v)e^{-in\omega t}\hfill \\ \hfill & \times \left\{-\frac{A_0^2{\xi }^2}{2{\omega }^2}{\left[\frac{p_z}{2}cos(2\omega t+\phi )+\frac{A_0}{6}sin(3\omega t+\phi )-\frac{A_0}{2}sin(\omega t+\phi )\right]}^2\right\}.\hfill \end{array}$$

(24)

Using the same method as for obtaining Equation (16), Equation (24) can be rewritten as

$$\begin{array}{cc}\hfill M_{\mathbf{p}}=& 2\pi i{\Psi }_0\left(\mathbf{p}\right)\left(\frac{p^2}{2}+I_p\right)\sum _{n=-\infty }^{\infty }\left\{{(-i)}^n\delta \left(\frac{p^2}{2}+I_p+U_p-n\omega \right)\left(-\frac{A_0^2{\xi }^2}{16{\omega }^2}\right)\right.\hfill \\ \hfill & \times \left[\left(\frac{A_0^2}{2}{J}_{n+2}(u,v)-A_0{p}_z{J}_{n+3}(u,v)+\frac{A_0^2}{3}{J}_{n+4}(u,v)\right.\right.\hfill \\ \hfill & \left.-\frac{A_0{p}_z}{3}{J}_{n+5}(u,v)+\frac{A_0^2}{18}{J}_{n+6}(u,v)\right)e^{2i\phi }\hfill \\ \hfill & +\left(\frac{A_0^2}{2}{J}_{n-2}(u,v)-A_0{p}_z{J}_{n-3}(u,v)+\frac{A_0^2}{3}{J}_{n-4}(u,v)\right.\hfill \\ \hfill & \left.\left.\left.-\frac{A_0{p}_z}{3}{J}_{n-5}(u,v)+\frac{A_0^2}{18}{J}_{n-6}(u,v)\right)e^{-2i\phi }\right]\right\}.\hfill \end{array}$$

(25)

Considering $M_{\mathbf{p}}=M_{\mathbf{p}}^{\left(0\right)}+M_{\mathbf{p}}^{\left(1\right)}+M_{\mathbf{p}}^{\left(2\right)}$ and inserting Equation (13), Equations (16) and (25), we can obtain the expression of the $M_{\mathbf{p}}$ expanded to the second order as follows:

$$\begin{array}{cc}\hfill M_{\mathrm{p}}=& 2\pi i{\Psi }_0\left(\mathbf{p}\right)\left(\frac{p^2}{2}+I_p\right)\sum _{n=-\infty }^{\infty }{(-i)}^n\left\{\delta \left(\frac{p^2}{2}+I_p+U_p-n\omega \right)\right.\hfill \\ \hfill & \times \left\{J_n(u,v)-\frac{i{A}_0\xi }{4\omega }\left[\left(A_0{J}_{n+1}(u,v)-p_z{J}_{n+2}(u,v)+\frac{A_0}{3}{J}_{n+3}(u,v)\right)e^{i\phi }\right.\right.\hfill \\ \hfill & \left.+\left(A_0{J}_{n-1}(u,v)-p_z{J}_{n-2}(u,v)+\frac{A_0}{3}{J}_{n-3}(u,v)\right)e^{-i\phi }\right]-\frac{A_0^2{\xi }^2}{16{\omega }^2}\hfill \\ \hfill & \times \left[\left(\frac{A_0^2}{2}{J}_{n+2}(u,v)-A_0{p}_z{J}_{n+3}(u,v)+\frac{A_0^2}{3}{J}_{n+4}(u,v)\right.\right.\hfill \\ \hfill & \left.-\frac{A_0{p}_z}{3}{J}_{n+5}(u,v)+\frac{A_0^2}{18}{J}_{n+6}(u,v)\right)e^{2i\phi }\hfill \\ \hfill & +\left(\frac{A_0^2}{2}{J}_{n-2}(u,v)-A_0{p}_z{J}_{n-3}(u,v)+\frac{A_0^2}{3}{J}_{n-4}(u,v)\right.\hfill \\ \hfill & \left.\left.\left.-\frac{A_0{p}_z}{3}{J}_{n-5}(u,v)+\frac{A_0^2}{18}{J}_{n-6}(u,v)\right)e^{-2i\phi }\right]\right\}.\hfill \end{array}$$

(26)

By substituting Equation (26) into Equation (18) and ignoring the higher order perturbation terms, the PMD $Y(\mathbf{p},\phi )$ truncated to second order expansion can be expressed as

$$\begin{array}{cc}\hfill Y(\mathbf{p},\phi )& =2\pi {\left|{\Psi }_0\left(\mathbf{p}\right)\right|}^2{\left(\frac{p^2}{2}+I_p\right)}^2\sum _{n=-\infty }^{\infty }\delta \left(\frac{p^2}{2}+I_p+U_p-n\omega \right)\hfill \\ \hfill & \times \left\{J_n{(u,v)}^2-\frac{4J_n(u,v)\xi }{3}sin\phi \left[(4v+n)\left(J_{n+1}(u,v)-J_{n-1}(u,v)\right)\right.\right.\hfill \\ \hfill & \left.+\left(J_{n+1}(u,v)+J_{n-1}(u,v)\right)-\frac{u}{8}\left(J_{n+2}(u,v)-J_{n-2}(u,v)\right)\right].\hfill \\ \hfill & -\frac{J_n(u,v){\xi }^2}{9}cos\left(2\phi \right)\left[-uv\left(J_{n+1}(u,v)+J_{n-1}(u,v)\right)\right.\hfill \\ \hfill & +\left(32v^2-\frac{u^2}{2}\right)\left(J_{n+2}(u,v)+J_{n-2}(u,v)\right)\hfill \\ \hfill & +(un+7uv)\left(J_{n+3}(u,v)+J_{n-3}(u,v)\right)+3\left(J_{n+3}(u,v)-J_{n-3}(u,v)\right)\hfill \\ \hfill & \left.\left.+\left(4nv-\frac{u^2}{2}+24v^2\right)\left(J_{n+4}(u,v)+J_{n-4}(u,v)\right)+16v\left(J_{n+4}(u,v)-J_{n-4}(u,v)\right)\right]\right\}.\hfill \end{array}$$

(27)

Here, we define $Y_2\left(\mathbf{p}\right)$ as

$$\begin{array}{cc}\hfill Y_2\left(\mathbf{p}\right)=& 2\pi {\left|{\Psi }_0\left(\mathbf{p}\right)\right|}^2{\left(\frac{p^2}{2}+I_p\right)}^2\sum _{n=-\infty }^{\infty }\delta \left(\frac{p^2}{2}+I_p+U_p-n\omega \right)\hfill \\ \hfill & \times \left\{-\frac{J_n(u,v){\xi }^2}{9}\left[-uv\left(J_{n+1}(u,v)+J_{n-1}(u,v)\right)+\left(32v^2-\frac{u^2}{2}\right)\left(J_{n+2}(u,v)+J_{n-2}(u,v)\right)\right.\right.\hfill \\ \hfill & +(un+7uv)\left(J_{n+3}(u,v)+J_{n-3}(u,v)\right)+3u\left(J_{n+3}(u,v)-J_{n-3}(u,v)\right)\hfill \\ \hfill & \left.\left.+\left(4nv-\frac{u^2}{2}+24v^2\right)\left(J_{n+4}(u,v)+J_{n-4}(u,v)\right)+16v\left(J_{n+4}(u,v)-J_{n-4}(u,v)\right)\right]\right\},\hfill \end{array}$$

(28)

and

$${\Phi }_2\left(\mathbf{p}\right)=arg\left[Y_2\left(\mathbf{p}\right)\right].$$

(29)

Equations (28) and (29) provide the second-order RPC and PP derived from Bessel functions, respectively.

4. Results and Discussion

In our work, we study the ionization of Ar atom ($I_p=0.579$) in a two-color field consisting of a strong 800-nm ($\omega =0.057$) and a weak 400-nm (2$\omega$) laser field with different phases. The strong laser intensity is $I=5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, and $\xi =0.1$ in Equation (1), where $A\left(t\right)$ is the vector potential of the laser pulse. The envelope rises linearly for two periods, then remains constant for four periods, and falls linearly for the last two periods.

For the wave function ${\Psi }_0\left(\mathbf{p}\right)$ in Equation (22) and (28), we choose the $1s$ as the initial state. For hydrogen-like atoms, this can be expressed as ${\Psi }_0\left(\mathbf{p}\right)=\frac{4A\sqrt{\pi }\kappa }{{\left(p^2+{\kappa }^2\right)}^2}$, where $\kappa =\sqrt{2I_p}$ is the characteristic momentum of a bound electron, and $A={\kappa }^{(1/4)}/\left(\sqrt{2\pi }\right)$ is the normalization coefficient. The Gaussian distribution $exp\left[-{\left(p^2/2+I_p+U_p-n\omega \right)}^2/a^2\right]/a\sqrt{\pi }$ with $a=1/200$ approximates the delta function $\delta \left(p^2/2+I_p+U_p-n\omega \right)$ in Equation (22) and (28) [34].

In Figure 1, we show the PMDs calculated by SFA of the Ar atom at relative phase $\phi =0,\pi /2$, and $3\pi /2,$ respectively. The ATI circles are clearly visible with different $\phi$. The PMDs show an obvious left-right symmetry regardless of the relative phase. Two of the PMDs, namely those with a relative phase difference of $\pi$, exhibit up-down reversal because the addition of the weak component 2$\omega$ destroys the symmetry of the laser pulse, making the PMD asymmetric along the laser polarization direction $P_z$. In particular, the ATI circle moves periodically in the direction of the laser’s polarization as the relative phase changes.

Next, we focus on the use of the generalized Bessel function to represent PP spectroscopy results directly, which we do in two ways. In the first method, we use its infinite series expansion, i.e., $J_n(u,v)={\sum }_{k=-\infty }^{\infty }{J}_{n-2k}\left(u\right)J_k\left(v\right)$. In the second method, we use its tunneling-approximation form, which frequently arises in the analytical processing of strong-field processes [46]:

$$J_n(u,v)=\frac{exp\left(\frac{1}{2}\sqrt{n^2-z^2}-\frac{n}{2}\left|{cosh}^{-1}\frac{n}{z}\right|\right)}{\sqrt{\pi }{\left(n^2-z^2\right)}^{1/4}}cos\left(u\sqrt{\frac{n+z}{2z}}-\frac{n\pi }{2}\right),$$

(30)

Here, $z=U_p/\omega$.

We use the above two methods to obtain the analytical results. Figure 2a,b and Figure 2e,f are the analytical first and second order RPC spectra obtained by the above different generalized Bessel functions, respectively. Figure 2c,d and Figure 2g,h are the corresponding first- and second-order PP spectra. One can see that the choice of generalized Bessel function makes almost no difference. The PMDs vary significantly with the relative phase, as shown in Figure 1. Physically, this means that the electron motion changes. The first- and second-order components of the RPC and PP spectra decouple the physical quantities of the ionization process and visualize them with attosecond accuracy. The phase value of the PP spectra can reflect the electron–ionization time delay. In general, the second-order RPC spectra is weaker than the first-order, but both show reflection symmetry around the direction of laser polarization $P_z$. As the modulation of the relative phase is periodic, the RPC contains the amplitude information of the PMDs for each relative phase. The PP spectra range from −$\pi$ to $\pi$. Structurally, there is a phase shift of $\pi$ at the origin of $P_z$ in the first-order PP spectra that corresponds to two electron emissions within half a laser cycle. The second-order PP spectra exhibit symmetry along $P_z$.

To verify the correctness of the analytical results, we show the first- and second-order RPC and PP spectra fitted by Equation (2), which are determined by a series of PMDs for various relative phases (see Figure 3). We scan the relative phases from 0 to $2\pi$ in steps of 0.1$\pi$. The fitting RPC spectra have a clear ATI ring structure, as do the analytical spectra. The analytical results contain the amplitude information of yield oscillations with phases. The phase flip characteristic of the first-order PP spectra and the phase symmetry characteristic of the second-order PP spectra on the positive and negative axis of $P_z$ are reflected in Figure 3c,d, respectively. However, compared with the analytical PP spectra, the fitted results have a rich high-precision structure. Two reasons explain these differences: (1) there are many parameters that determine the structure of PMDs needed for fitting, among them laser intensity I, wavelength $\lambda$, laser period T, and relative phase $\phi$. However, T is not included in the analytical derivation, resulting in the loss of spectral information. (2) the values of $Y_1\left(\mathbf{p}\right)$, $Y_2\left(\mathbf{p}\right)$ obtained by the generalized Bessel function calculation are real, and the PP cannot be represented by Equations (22) and (28). To reflect its symmetry characteristics, we set ${\Phi }_n\left(\mathbf{p}\right)=\pm \pi /2(n=1,2)$, depending on the sign of $Y_n\left(\mathbf{p}\right)$. Therefore, this treatment leads to the loss of information, and accurate extraction of physical information is very difficult to achieve. Our goal in the future will be to achieve an accurate separation of real and imaginary parts and analytically express the PP spectra of the fine structure.

In terms of computational time, using the analytical expression is much faster than fitting. In the SFA, calculating the PMD of 20 million trajectories in a single relative phase takes nearly two days, a computation which must then be repeated for all the relative phases. By contrast, the first generalized Bessel function method takes less than 20 h, and the second only a few minutes. This proves the feasibility and superiority of solving for the RPC and PP analytically. However, many parameters in the SFA method are not reflected in the generalized Bessel function method, such as laser period and envelope shape. It is necessary to include these parameters in the analysis method in the future.

5. Conclusions

To conclude, we have used the SFA theory to study photoemission of Ar atoms in a two-color linearly polarized laser. The photoelectric yield oscillates periodically with the relative phase of the two-color field, and the underlying physical mechanism of the oscillation can be revealed by the PP spectroscopy method. Two important quantities, RPC and PP, are used to characterize the oscillations of the fitted curve. Applying the generalized Bessel function to SFA, we derive first- and second-order analytical expressions for both RPC and PP, respectively. The analytical RPC and PP spectra are consistent with the fitted results in terms of structure and strength. The analytical expressions found in this work avoid the requirement of large amounts of data for fitting, and provide a new approach to the investigation of strong field dynamics by PP spectroscopy. It is worth noting that, compared with the continuous scanning of PMD in a series of relative phases through SFA, RPC and PP can be obtained through fitting, and it is easier to directly solve the analytic expression. The time of analytical solution can be said to be negligible compared to the fitting calculation. At the same time, it is possible to obtain analytical spectra with attosecond time resolution for real-time detection and control of electronic dynamic processes.