Study on spin angular momentum balance in harmonics generated from counter-rotating two-color laser fields

Xiaolu Bai; Yingtong Su; Jingtao Zhang

doi:10.1364/OE.439695

1. Introduction

Transfer and conservation of angular momentum are frequently studied in modern physics, and recently, they drew much attention on high-order harmonics generation (HHG) in intense laser fields [1–7]. During HHG process, multiple photons of the driving laser field convert into one harmonic photon and the generated harmonics can span more than thousands of orders. How to balance the angular momentum is a basic question in the study of HHG.

In the past decades, HHG was studied extensively, mainly focusing on generating harmonics of higher order and/or higher conversion efficiency. The spin angular momentum (SAM) transfer during HHG process did not arouse much attention until recently. Using an $\omega -2\omega$ counter-rotating two-color circularly (CRTC) driving field, Fleischer et al. observed that the generated harmonics reach a conversion efficiency comparable to those in the linearly polarized (LP) laser fields [1]. The generated harmonics can be classified into two categories, including $3q+1$ series with the same helicity as and $3q-1$ series with the opposite helicity to that of the fundamental driving field. This feature, actually, is a consequence of the SAM conservation during HHG [8]. Together with the the electronic symmetry of the driven targets, the selection rule of harmonics was disclosed [9–13].

HHG in elliptically polarized (EP) driving laser fields includes rich processes. Early studies on this topic had shown that the harmonic rate changes greatly with the ellipticity of the driving fields [14–16]. Fleischer et al. also observed that, when one mode of the CRTC field is elliptically polarized, the harmonic rate varies with ellipticity in a complicated manner [1]. From the viewpoint of SAM, the circularly polarized (CP) photons have the SAM quanta $\hbar$, while that of EP photons is uncertain, thus how to balance the SAM during HHG from the EP driving fields is still ambiguous. In order to explain the intricate selection rules on the HHG observed in the EP driving fields, Fleischer et al. suggested a model using the expectation value of SAM of the EP field [1], while Pisanty et al. used a model in which the EP field is treated as a composition of two independent CP fields with the opposite helicity [8]. However, the classical-field treatment cannot describe accurately the amplitude of an electronic transition in which the number of involved photons is given, so the transition amplitude of a certain harmonic channel in these models should be obtained by other sources.

We used a quantum-field scattering theory [17] to treat the HHG from Ar atoms driven by the intense laser pulses. The laser field is fully quantized, and the analytical formula of the harmonic rate is obtained. This allows us to ascertain the contribution of each HHG channel and to identify the SAM balance process in a solid base. This treatment is self consistent, in the meaning that a generalized phased Bessel (GPB) function can accurately describe the amplitude of the electronic transition, both in the magnitude and in the quantum phase [18]. Using this feature, we obtain the phase of each HHG channel, by which the SAM conservation during the HHG process is derived in a solid base and in a straightforward manner. This, actually, presents a shortcut to study the SAM transfer in the HHG process.

In this paper, we extend our treatment to study the SAM transfer in the HHG process when one mode of the two-color driving field is elliptically polarized. We treat the EP field as a composition of two counter-rotating CP fields, and discuss how to balance the SAM during HHG from the EP fields. We derive the analytical formulas of the harmonic rate and the SAM conservation relations when one driving laser field is elliptical polarized. The observed V-type and $\Lambda$-type distributions of the HHG spectra with ellipticity are well recovered by numerical calculations and are discussed in details. The SAM balance mechanism from the EP field is confirmed substantially. Our conclusions hold for rare atoms and the numerical results are preformed on xenon targets.

2. Volkov state and HHG formula in three-mode fields

The EP field can be treated as a composition of two counter-rotating CP fields with the eletric-field strength being given by

(1)$$E_{R}=\frac{E_{0}}{\sqrt{2}}\left ( \cos \frac{\xi}{2}+ \sin \frac{\xi}{2}\right ), E_{L}=\frac{E_{0}}{\sqrt{2}}\left ( \cos \frac{\xi}{2}- \sin \frac{\xi}{2}\right ),$$

where $\xi$ and $E_{0}$ are the ellipticity and the maximal strength of the EP field, $R$ and $L$ represent right and left CP fields, respectively. Accordingly, the laser field of one CP mode plus one EP mode can be treated as a composition of three CP modes.

According to the three-step scenario [19–21], a bound electron first transits to the continuum state from the initial state, then recombines to the final state accompanying with the harmonic generation. In our quantum-field treatment, the continuum state of the photon-electron system is described by a quantum-field Volkov state [22–24]. For an electron moving in a three-mode laser field, it is given by [24] (atomic units $\hbar =m_{e}=\left | e \right |=1$ are used throughout this paper):

(2)$$\left|\Psi_{\mu}\right\rangle =\sum_{j_{1}, j_{2}, j_{3}} \exp \left \{i(\textbf{P}+\triangle\textbf{k} \cdot \textbf{r} \right \} \mathcal{X}_{{-}j_{1}, -j_{2}, -j_{3}}(\zeta)\left|n_{1}+j_{1}, n_{2}+j_{2}, n_{3}+j_{3}\right\rangle, \\$$

where P is the electron’s momentum in the Volkov state, and $\triangle \textbf {k}=(u_{p1}-j_{1})\textbf {k}_{1}+(u_{p2}-j_{2})\textbf {k}_{2}+(u_{p3}-j_{3})\textbf {k}_{2}$, and $\textbf {k}_{i}$ is the wave vector of the ith laser mode, respectively, $n_{i}\;(i=1,2,3)$ and $j_{i}$ denote, respectively, the number of the back-ground and the transferred photons in the $i$th mode, the sum over $j_{i}$ is performed from $-n_{i}$ to infinity. In Eq. (2), the quantity $u_{pi}$ is the ponderomotive parameter defined as $u_{pi}=E_{i}^{2}/(4\omega _{i}^{3})$, where $\omega _{i}$ and $E_{i}$ are, respectively, the frequency and the electric-field strength of the $i$th laser mode.

The quantum-field Volkov state is an overlapping of the electronic plane waves with weights being determined by the generalized phased Bessel (GPB) functions. We have shown that the electronic transition dynamics is determined by the GPB function [25,26]. The three-mode GPB function $\mathcal {X}_{j_{1},j_{2},j_{3}}(\zeta )$ is defined as

(3)$$\begin{aligned} \mathcal{X}_{j_{1},j_{2},j_{3}}(\zeta) & =\sum_{m_{i}={-}\infty}^{\infty} X_{j_{1}-2m_{1}-m_{4}-m_{5}-m_{6}-m_{7}} (\zeta_{1}) X_{j_{2}-2m_{2}-m_{4}+m_{5}-m_{8}-m_{9}}(\zeta_{2})\\ & \times X_{j_{3}-2m_{3}-m_{6}+m_{7}-m_{8}+m_{9}}(\zeta_{3}) X_{m_{1}}(z_{1})\cdots X_{m_{9}}(z_{9}), \end{aligned}$$

where the sum index $m_{i}$ includes $m_{1},m_{2}\cdots m_{9}$ and $X_{n}(z)$ is the phased Bessel function which relates to the ordinary Bessel function as [27]

(4)$$X_{n}(z)=J_{n}(|z|) e^{i n \arg (z)},$$

and $arg(z)$ denotes the principle angle of a complex argument $z$. The twelve arguments of the GPB function stem from the interaction among the electron and the laser modes, which is obtained by the mini-coupling principle as $\textbf {P} \cdot \textbf {A}_{i} + \textbf {A}_{i} \cdot \textbf {A}_{j}$ and A$_{i}$ denotes the vector potential of the $i$th laser mode. In general case, they are given by

(5)$$\begin{aligned} \zeta_{1} & =\frac{E_{1}}{\omega_{1}^{2}} \mathbf{P}\cdot \mathbf{\epsilon}_{1},z_{1}=\frac{1}{2}{u_{p1}}{\epsilon}_{1}\cdot{\epsilon}_{1},\\ \zeta_{2} & =\frac{E_{2}}{\omega_{2}^{2}} \mathbf{P} \cdot \mathbf{\epsilon}_{2}, z_{2}=\frac{1}{2}{u_{p2}}{\epsilon}_{2}\cdot{\epsilon}_{2},\\ \zeta_{3} & = \frac{E_{3}}{\omega_{3}^{2}} \mathbf{P}\cdot \mathbf{\epsilon}_{3},z_{3}=\frac{1}{2}{u_{p3}}{\epsilon}_{3}\cdot{\epsilon}_{3},\\ z_{4} & =\frac{E_{1} E_{2} \mathbf{\epsilon}_{1} \cdot \mathbf{\epsilon}_{2}}{2 \omega_{1} \omega_{2}\left(\omega_{1}+\omega_{2}\right)},z_{5}=\frac{E_{1} E_{2} \mathbf{\epsilon}_{1} \cdot \mathbf{\epsilon}_{2}^{{\ast}}}{2 \omega_{1} \omega_{2}\left(\omega_{2}-\omega_{1}\right)},\\ z_{6} & =\frac{E_{1} E_{3} \mathbf{\epsilon}_{1} \cdot \mathbf{\epsilon}_{3}}{2 \omega_{1} \omega_{3}\left(\omega_{1}+\omega_{3}\right)},z_{7}=\frac{E_{1} E_{3} \mathbf{\epsilon}_{1} \cdot \mathbf{\epsilon}_{3}^{{\ast}}}{2 \omega_{1} \omega_{3}\left(\omega_{3}-\omega_{1}\right)},\\ z_{8} & =\frac{E_{2} E_{3} \mathbf{\epsilon}_{2} \cdot \mathbf{\epsilon}_{3}}{2 \omega_{2} \omega_{3}\left(\omega_{2}+\omega_{3}\right)},z_{9}=\frac{E_{2} E_{3} \mathbf{\epsilon}_{2} \cdot \mathbf{\epsilon}_{3}^{{\ast}}}{2 \omega_{2} \omega_{3}\left(\omega_{3}-\omega_{2}\right)}. \end{aligned}$$

where $\zeta _{i }$ and $z_{i}$ ($i$=1,2,3) variables come from $\textbf {P} \cdot \textbf {A}_{i}$ and $\textbf {A}_{i} \cdot \textbf {A}_{i}$ terms, and the rest $z_{i}$ variable comes from the coupling between laser modes, say $\textbf { A}_{i} \cdot \textbf { A}_{j}$. In Eq. (5), the quantity ${\epsilon }_{i}$ denotes the polarization vector of the $i$th laser mode and can be written as

(6)$$\mathbf{\epsilon}_{i}=\mathbf{\epsilon}_{x} \cos \frac{\xi_{i}}{2}+i \mathbf{\epsilon}_{y} \sin \frac{\xi_{i}}{2},\quad (i=1,2,3)$$

in which $\mathbf {\epsilon }_{x,y}$ are the unit vectors along $x/y$ directions, and $\xi _{i}$ varies from $-\pi$ to $\pi$. This quantity monitors the ellipticity degree of the $i$th mode such that $\xi _{i}$=0 is for linear polarization and $\xi _{i}=\pm \pi /2$ are for right/left circular polarization, respectively. The Volkov state can be used to describe the state of an electron moving in a three-mode mulit-cycle laser pulse where the pulse duration is not a key parameter.

By means of the transition matrix given in Ref. [17], the generation rate of a harmonic at frequency $q\omega$ in a three-mode laser field can be expressed by

(7)$$w _{q}=\frac{1}{q(2\pi)^{6}}\left | \sum_{q_{1},q_{2},q_{3}}\Delta \mathcal{E}\left | \textbf{P} \right | T_{f_{i}}^{(q_{1},q_{2},q_{3})}\right |^2,$$

where $\Delta \mathcal {E} = (j_{1}-u_{p1})\omega _{1}+(j_{2}-u_{p2})\omega _{2}+(j_{3}-u_{p3})\omega _{3}$, in which $j_{i}$ and $j'_{i}$ denote the number of photons absorbed in the electron ionization and emitted in the recombination processes, respectively, and $q_{i}=j_{i}-j'_{i}\;(i=1,2,3)$ denotes the number of net photons converted into the harmonics in one laser mode. In Eq. (7) the sum over $q_{i}$ is performed over the range $\left [- \infty,\infty \right ]$ under the energy conservation condition that $q\omega =q_{1}\omega _{1}+q_{1}\omega _{2}+q_{3}\omega _{3}$, and a set of $(q_{1},q_{2},q_{3})$ that satisfies the previous relation is termed as an HHG channel. Many channels contribute to one harmonic, and the harmonic rate is a coherent sum over all possible channels. The transition amplitude for the $q$ th harmonic in a certain channel is given by

(8)$$\begin{aligned} & T_{f i}^{\left(q_{1}, q_{2}, q_{3}\right)} =\sum_{j_{1},j_{2},j_{3}}\int d\Omega\Phi_{i}(\mathbf{P}) \Phi_{f}^{*}(\mathbf{P}_{q}) \epsilon^{\prime *} \cdot\left \{ \mathbf{P} \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j_{1}', -j_{2}',-j'_{3}}(\zeta) \right. \\ \qquad\qquad & +\sqrt{U_{p 1}} \left [ \epsilon_{1} \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1}-1, -j'_{2},-j'_{3}}(\zeta) +\epsilon_{1}^{*}\mathcal{X}_{{-}j_{1},-j_{2},-j_{3}}(\zeta)^{*}\mathcal{X}_{{-}j_{1}'+1,-j_{2}',-j'_{3}}(\zeta) \right ] \\ \qquad \qquad & +\sqrt{U_{p 2}} \left [ \epsilon_{2} \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1}, -j'_{2}-1,-j'_{3}}(\zeta) +\epsilon_{2}^{*} \mathcal{X}_{{-}j_{1},-j_{2},-j_{3}}(\zeta)^{*}\mathcal{X}_{{-}j_{1}',-j_{2}'+1,-j_{3}'}(\zeta) \right ] \\ \qquad \qquad & +\sqrt{U_{p 3}} \left [ \epsilon_{3} \mathcal{X}_{{-}j_{1}, -j_{2}.-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j_{1}', -j_{2}',-j_{3}'-1}(\zeta) +\epsilon_{3}^{*} \left. \mathcal{X}_{{-}j_{1},-j_{2},-j_{3}}(\zeta)^{*}\mathcal{X}_{{-}j_{1}',-j_{2}',-j_{3}'+1}(\zeta) \right ] \right\}, \end{aligned}$$

in which $\left |\textbf {P}\right |=\sqrt {2 \Delta \mathcal {E}}$ and $\left |\textbf {P}_{q}\right |=\sqrt {2q\omega }$ , and $\epsilon ' =\epsilon _{x}$cos$(\xi '/2) + i\epsilon _{y}$sin $(\xi '/2)$ denotes the polarization vector of the generated harmonics. The quantities $\Phi _{i}(\textbf {P})$ and $\Phi _{f}(\textbf {P}_{q})$ are the wave functions of the initial and the final states in the momentum space, respectively. In HHG process, one-active-electron approximation is used and the final state of the electron is generally treated as the same as the initial state.

The above formula holds for three-mode laser fields of arbitrary polarization. In the following, we will study the harmonic generation when three modes are circularly polarized. We term the first laser mode as the fundamental driver field and set $\omega _{1}=\omega$.

3. SAM balance and harmonic spectra when the fundamental driver changes ellipticity

When the fundamental driver is elliptically polarized, it can be treated as a composition of two counter-rotating fields of frequency $\omega$, and the strength of each field is given by Eq. (1). We set $\omega _{1}=\omega _{2}=\omega$ and $\xi _{1}=-\xi _{2}=-\xi _{3}=\pi /2$, that is, the $\omega _{1}$ mode is right CP while the other two modes are left CP. In this case, the GPB function reduces to

(9)$$\begin{aligned} \mathcal{X}_{j_{1},j_{2},j_{3}}(\zeta) & =\sum_{m_{i}={-}\infty}^{\infty} X_{j_{1}-m_{4}-m_{6}}(\zeta_{1}) X_{j_{2}-m_{4}-m_{9}}(\zeta_{2})\\ & \times X_{j_{3}-m_{6}+m_{9}}(\zeta_{3}) X_{m_{4}}(z_{4})X_{m_{6}}(z_{6})X_{m_{9}}(z_{9}), \end{aligned}$$

where the sum over $m_{i}$ is performed over $i=4,6,$ and $9$, respectively, and $\zeta _{i}=E_{i}\textbf {P}\cdot \epsilon _{i}/\omega _{i}^{2} \ (i=1,2,3).$

According to Eqs. (5) and (6), we find

(10)$$\arg \left(\zeta_{1}\right)= \phi, \quad \arg \left(\zeta_{2}\right)=\arg \left(\zeta_{3}\right) ={-}\phi,$$

where we have set $\textbf {P}=( | \textbf {P}|,\theta,\phi )$. Hence, by the definition of the phased Bessel function in Eq. (4), we find

(11)$$\arg \left(X_{{-}j}\left(\zeta_{1}\right)\right)={-}j \phi, \quad \arg \left(X_{{-}j}\left(\zeta_{2}\right)\right)=\arg \left(X_{{-}j}\left(\zeta_{3}\right)\right)=j \phi.$$

The phased Bessel function $X_{-j}(\zeta _{i})$ describes the electron transition amplitude of absorbing $j-$photons in the CP fields. The above relations indicate that the phase depends on the momentum of freed electron and the number of photons involved in the electronic transition. Noting that the eigenstate of orbital angular momentum (OAM) projection operator $l_{z}$ is exp$(im\phi )/\sqrt {2\pi }$, and that the SAM of the photons in the CP fields is $\pm 1$, the relations in Eq. (12) suggest that the phased Bessel functions bring the information of the OAM of the freed electron and the SAM of transferred photons. The freed electron moves around a circle in the CP field, with the direction determined by the field helicity, and thus has an OAM that depends on the number of photons it absorbed during ionization. This feature is reflected by the quantity $\textbf {P}\cdot \mathbf {\epsilon }_{i}$ in the $\zeta _{i}-$arguments of the phased Bessel functions and leads to the phase of the transition amplitude depending on the OAM of the freed electron and the SAM of photons. According to Eq. (4), we find

(12)$$\arg \left(\mathcal{X}_{j_{1},j_{2},j_{3}}(\zeta)\right) =\left(j_{1}-j_{2}-j_{3}\right) \phi,$$

which indicates that the GPB function automatically brings the SAM of the CRTC field and the OAM of the freed electron. This relation holds in both the ionization and the recombination processes, which brings many advantages in analyzing the electronic transition and leads to the selection rule of harmonic order in the CRTC fields.

For a given HHG channel, say for one combination of (${q_{1},q_{2},q_{3}}$), the transition matrix element includes four terms. The first term describes the transition caused by coupling of the electron momentum with the potential vector of harmonic field, while the rest three terms correspond to the transition caused by the coupling of the vector potential of the harmonic field with the three laser modes, respectively. The transition amplitude in each term is described by a coherent sum over two pairs of the GPB functions, so its value depends on many factors, but its phase depends only on the value of $q_{i}$ and $\phi.$ For example, according to Eq. (12), we have

(13)$$\begin{aligned} \quad\arg \left ( \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*}\mathcal{X}_{{-}j'_{1}\pm 1,-j'_{2},-j'_{3}}(\zeta) \right ) & =\left ( q_{1}-q_{2}-q_{3}\pm 1 \right )\phi,\\ \arg \left ( \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*}\mathcal{X}_{{-}j'_{1} ,-j'_{2}\pm 1,-j'_{3}}(\zeta) \right ) & =\left( q_{1}-q_{2}-q_{3}\mp 1 \right )\phi. \end{aligned}$$

where $q_{i}=j_{i}-{j}'_{i} \ (i=1,2,3)$ denotes the number of photons transferred from each mode into harmonics. The integration of each term over the azimuthal angle produces the nonzero condition as

(14)$$q_{1}-q_{2}=q_{3}\mp 1$$

Note that $q_{1}$ and $q_{2}$ are the numbers of photons transferred into harmonics from the two component fields of the $\omega -$field, hence the quantity $q_{1}-q_{2}$ means the net SAM quanta of the electron obtained from the fundamental driving field. This quantity differs from that obtained from the harmonic driver, say $q_{3}$, by $\pm 1$, accordingly, the generated harmonics are right/left CP, respectively. Thus we see, when the fundamental driver is elliptically polarized, the SAM is balanced via the manner given in Eq. (15), and the $\mp 1$ term denotes the SAM of generated harmonics. The relation, actually, denotes the SAM conservation during the HHG process, which is set as a predetermined condition in other treatments. In addition to the fact that the SAM conservation relation is derived in a solid base and in a straigtforward manner, our treatment has another advantage that the HHG amplitude of an arbitrary channel can be accurately calculated, and thus is advantageous to determine the dominant balance mechanism of the SAM.

In a given HHG channel $(q_{1},q_{2},q_{3})$, when $q_{1}-q_{2}=q_{3}-1$, the generated harmonics are of the same helicity as that of the $2\omega$-field. In the present case, the harmonics are left CP with the helicity $-1$, and the generation amplitude is given by

(15)$$\begin{aligned} T_{{-}1} & =2 \pi \sum_{j_{1} ,j_{2},j_{3}} \int_{0}^{\pi} \sin \theta d \theta \Phi_{i}(\textbf{P}) \Phi_{f}^{*}\left(\textbf{P}_{q}\right)\\ & \times\left[ \left|\textbf{P}^{\prime}\right| e^{i\phi} \mathcal{X}_{{-}j_{1},-j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1},-j'_{2},-j'_{3}}(\zeta)\right.\\ & +\sqrt{U_{p1}} \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1}+1, -j'_{2},-j'_{3}}(\zeta)\\ & +\sqrt{U_{p2}} \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1}, -j'_{2}-1,-j'_{3}}(\zeta)\\ & +\sqrt{U_{p3}} \left. \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1}, -j'_{2},-j'_{3}-1}(\zeta) \right], \end{aligned}$$

while when $q_{1}-q_{2}=q_{3}+1$, the generated harmonics are of the same helicity as that of the fundamental field. In the present case, the harmonics are right CP with the helicity $+1$, and the generation amplitude is given by

(16)$$\begin{aligned} T_{{+}1} & =2 \pi \sum_{j_{1} ,j_{2},j_{3}} \int_{0}^{\pi} \sin \theta d \theta \Phi_{i}(\textbf{P}) \Phi_{f}^{*}\left(\textbf{P}_{q}\right)\\ & \times\left[ \left|\textbf{P}^{\prime}\right| e^{{-}i\phi} \mathcal{X}_{{-}j_{1},-j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1},-j'_{2},-j'_{3}}(\zeta)\right.\\ & +\sqrt{U_{p1}} \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1}-1, -j'_{2},-j'_{3}}(\zeta)\\ & +\sqrt{U_{p2}} \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1}, -j'_{2}+1,-j'_{3}}(\zeta)\\ & +\sqrt{U_{p3}} \left. \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1}, -j'_{2},-j'_{3}+1}(\zeta) \right]. \end{aligned}$$

We have set $\left |\mathbf {P}^{\prime }\right |=|\mathbf {P}|$sin$\theta /\sqrt {2}$ and replaced the integration over $\phi$ by $2\pi$, and the value of $\phi$ is arbitrary.

The analytical formulae allows us to obtain the channel-resloved harmonic rate independently. In our calculations, we set the final state being the same as the initial state and the wave function is chosen as the hydrogen-like one. Figure 1 depicts the channel-resolved rate of the 19th harmonic when the ellipticity of the $\omega -$field is changed. Many bright island-like structures are highlighted from the blue background in each panel. These bright islands distribute symmetrically about $\xi =\pi /2$, reflecting a fact that the combined field is also symmetrically distributed. A bright island-like structure represents a higher harmonic rate, and the area corresponds to the range of ellipticity, which is formed mainly by two factors. One is the variation of the GPB function with the ellpicitiy, the other is the intra-channel interference, say the coherent sum over $(j_{1},j_{2},j_{3})$ in a given channel. The island-like structures in one horizontal line denote the higher harmonic rates in one channel, and their location in the panel varies with the channel index. We define a central channel that $|q_{2}|$ is the smallest one among all the possible channels. The central channel contributes most to the harmonic yield, especially in the region around $\xi =\pi /2$. For the 19th harmonic, the central channel is $(7,0,6)$ channel for the right-hand harmonic and is the $(6,1,6)$ channel for the left-hand harmonic. Other dominated channels distribute around the central channel, and the main contribution shifts to high ellipticity as $| q_{2}|$ deviates from its minimum. As a result, the bright islands distribute mainly in a region enveloped by two intersecting lines in each panel. As the laser intensity increases, more channels become notably contributed to the harmonics. Generally, although many channels are possible, the harmonic generation is dominated only by several channels of smaller channel index, and more channels become dominated in the large ellipticity case and when the laser intensity increases.

Fig. 1. Channel-resolved rate of the 19th harmonic varying with the ellipticity of $\omega$-field. The panels in left/right columns are for right and left CP component of the generated harmonics, respectively. The maximal intensity of the EP field is the same as that of the EP field. In (a) and (b), the intensity of the CP field is taken as $1.0 \times 10^{14}W/cm^{2}$ , while in (c) and (d), it is taken as $2.0 \times 10^{14}W/cm^{2}$ . The fundamental driving field is of 800nm wavelength and the target atom is xenon.

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The island-like structures in different channels for fixed $q_{1}$ form regular distributions. Figure 2(a) and (b) depict two distributions for the harmonic spectra calculated at $2.0 \times 10^{14}W/cm^{2}$, in which a V-shaped distribution of the highlighted island-like structures is evident, both for $q_{1}-q_{2}-q_{3}=\pm 1$. The V-type distribution for $q_{1}-q_{2}-q_{3}=1$ case is a little narrower than that for $q_{1}-q_{2}-q_{3}=-1$ case, and thus the two legs of the V-type structure cross with each other at the 22nd harmonics, while at 23rd harmonic for $q_{1}-q_{2}-q_{3}=-1$, both at around $\xi =\pi /2$. Note that the fundamental field is circularly polarized when $\xi =\pi /2$, these two crossing points denotes the $3q\pm 1$ harmonics in the CRTC fields. These harmonics are of opposite helicity. In Fig. 2(a), the 17th harmonic is the highest when the ellipticity angle tends to zero, indicating that the harmonic rate in the two-color field is comparable to that in the LP fields. Moreover, the 19th harmonic is the strongest harmonic among its neighboring harmonics, for both $q_{1}-q_{2}-q_{3}=\pm 1$ cases. All features agree perfectly with these observed by Fleischer et al.[1].

In above V-shaped distributions, the number of the transferred photons from the first mode, say $q_{1}$, is kept unchanged. As the harmonic order increases by one, the value of $q_{2}$ decreases but that of $q_{3}$ increases by one, and thus $q_{2}+q_{3}$ is kept unchanged. This feature holds for $q_{1}-q_{2}-q_{3}=\pm 1$ cases. This feature puts a deep insight into the SAM balance pattern in the EP field. Both $q_{1}$ and $q_{2}+q_{3}$ are kept unchanged is a requirement of the SAM balance in physics. This is ensured by the strong-coupling between two left CP modes. In Eq. (5), the $z_{i}-$ arguments determine the coupling of laser modes. In the present case, the argument $z_{9}\propto \epsilon _{2} \cdot \epsilon _{3}^{*}$ describes the coupling between the two left CP fields and is the largest one among three $z_{i}-$ arguments. This strong coupling means a large probability to annihilate a $q_{2}-$photon and create a $q_{3}-$photon. Since $\omega _{3}=2\omega _{2}$, this process accompanies the generated harmonic increasing one order.

In Ref. [1], the SAM balance for the harmonics generated from the $(10,5)-$channel is not fixed substantially. This channel generates the 20th harmonics and corresponds to two channels in our treatment. One channel is marked by $(7,3,5)$ which denotes that among ten photons absorbing from the $\omega -$field, seven photons are left CP while the rest three photons are right CP. Adding the five right CP photons from the $2\omega -$field, we find that seven left CP photons and eight right CP photons are transferred to generate the 20th harmonic. The generated harmonic is right CP field, so the SAM is balanced. The other channel is marked by $(8,2,5)$ which denotes that eight left CP photons and seven right CP photons are transferred. The generated harmonics is left circularly polarized, and the SAM is also balanced. Our study show that the harmonic spectra involving these two channels from evident V-shaped distributions as the harmonic order increases with the ellipticity, as is shown in Fig. 2. From the brightness of the island areas, we judge that these two kinds of oppositely rotated harmonics are of comparable strength but appear at different ellipticity, thus the helicity of the overall harmonic changes dramatically. Similarly, on the $(14,3)-$channel in Ref. [1], it also corresponds to two channels, marked by $(8,6,3)$ and $(9,5,3)$, respectively. The harmonic spectra involving these two channels form evident V-shaped distributions as both the harmonic order and the ellipticity increase, as is shown in Fig. 2(c) and Fig. 2(d).

Fig. 2. Calculated harmonic spectra versus ellipticity of the fundamental driving field. The harmonic driving field is left-circularly polarized. The intensity of each laser mode is $1.0 \times 10^{14}W/cm^{2}$for (a,b) and $2.0 \times 10^{14}W/cm^{2}$for (c,d), respectively. The panels in left/right columns are for right and left CP component of the generated harmonics, respectively.

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4. SAM balance and harmonic spectra when the harmonic driver changes ellipticity

When the harmonic driver is the EP field but the fundamental driver is the CP field, we set $\omega _{2}=\omega _{3}=2\omega$ and $\xi _{1}=\xi _{2}=-\xi _{3}=\pi /2$, and the electric-field strength of modes 2 and 3 can be obtained by Eq. (1). In this case, the GPB function reduces to

(17)$$\begin{aligned} \mathcal{X}_{j_{1},j_{2},j_{3}}(\zeta) &=\sum_{m_{i}={-}\infty}^{\infty} X_{j_{1}-m_{5}-m_{6}}(\zeta_{1}) X_{j_{2}+m_{5}-m_{8}}(\zeta_{2}) \\ & \times X_{j_{3}-m_{6}-m_{8}}(\zeta_{3}) X_{m_{5}}(z_{5})X_{m_{6}}(z_{6})X_{m_{8}}(z_{8}), \end{aligned}$$

where the sum over $m_{i}$ is performed over $i=5,6,$ and $8$, respectively. Because $\zeta _{i}=E_{i}\textbf {P}\cdot \epsilon _{i}/\omega _{i}^{2} \ (i=1,2,3)$, we find $\arg (\zeta _{1})= \arg (\zeta _{2})=\phi$, $\arg (\zeta _{3})=-\phi$, respectively. According to Eq. (4), we have

(18)$$\arg \left(\mathcal{X}_{j_{1}, j_{2},j_{3}}(\zeta)\right)=\left(j_{1} +j_{2}-j_{3}\right)\phi,$$

and thus the phase of an HHG channel is worked out to be

(19)$$\begin{aligned}\quad\arg \left ( \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*}\mathcal{X}_{{-}j'_{1}\pm 1,-j'_{2},-j'_{3}}(\zeta) \right ) &=\left ( q_{1}+q_{2}-q_{3}\pm 1 \right )\phi, \\ \arg \left ( \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*}\mathcal{X}_{{-}j'_{1} ,-j'_{2},-j'_{3}\pm 1}(\zeta) \right ) &=\left( q_{1}+q_{2}-q_{3}\mp 1 \right )\phi. \end{aligned}$$

Similarly, we obtain the existence condition of the harmonic order as

(20)$$q_{3}-q_{2}=q_{1}\mp1$$

for four terms in Eq. (8) after the integration over $\phi$. This relation suggests a SAM conservation mechanism that the net SAM quanta transferred from the $2\omega -$field differs one from that from the $\omega -$field, and the extra SAM quanta is taken by the generated harmonics. This relation, together with the energy conservation relation as $q_{1}\omega +2(q_{2}+q_{3})\omega =q\omega$, defines a generation channel when the harmonic field is elliptically polarized.

In a given HHG channel $(q_{1},q_{2},q_{3})$, when $q_{3}-q_{2}=q_{1}-1$, the generated harmonics are of the same helicity as that of the fundamental driver. In the present case, the harmonics are right CP with helicity $+1$, and the generation amplitude is given by

(21)$$\begin{aligned}T_{{-}1} &=2 \pi \sum_{j_{1} ,j_{2},j_{3}} \int_{0}^{\pi} \sin \theta d \theta \Phi_{i}(\textbf{P}) \Phi_{f}^{*}\left(\textbf{P}_{q}\right) \\ & \times \left [ \left|\textbf{P}^{\prime}\right|e^{i\phi} \mathcal{X}_{{-}j_{1},-j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1},-j'_{2},-j'_{3}}(\zeta)\right. \\ & +\sqrt{U_{p1}} \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1}-1, -j'_{2},-j'_{3}}(\zeta) \\ & +\sqrt{U_{p2}} \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1}, -j'_{2}-1,-j'_{3}}(\zeta) \\ & +\sqrt{U_{p3}}\left. \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1}, -j'_{2},-j'_{3}+1}(\zeta)\right], \end{aligned}$$

while when $q_{3}-q_{2}=q_{1}+1$, the generated harmonics are of the same helicity as that of the $2\omega -$field. The harmonics are left CP with helicity $-1$, and the generation amplitude is given by

(22)$$\begin{aligned} T_{{+}1} &=2 \pi \sum_{j_{1} ,j_{2},j_{3}} \int_{0}^{\pi} \sin \theta d \theta \Phi_{i}(\textbf{P}) \Phi_{f}^{*}\left(\textbf{P}_{q}\right) \\ & \times\left[ \left|\textbf{P}^{\prime}\right| e^{{-}i\phi} \mathcal{X}_{{-}j_{1},-j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1},-j'_{2},-j'_{3}}(\zeta)\right. \\ & +\sqrt{U_{p1}} \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1}+1, -j'_{2},-j'_{3}}(\zeta) \\ & +\sqrt{U_{p2}} \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1}, -j'_{2}+1,-j'_{3}}(\zeta) \\ & +\sqrt{U_{p3}}\left. \mathcal{X}_{{-}j_{1}, -j_{2},-j_{3}}(\zeta)^{*} \mathcal{X}_{{-}j'_{1}, -j'_{2},-j'_{3}-1}(\zeta)\right]. \end{aligned}$$

We have set $\left |\mathbf {P}^{\prime }\right |=|\mathbf {P}|$sin$\theta /\sqrt {2}$ and replaced the integration over $\phi$ by $2\pi$, and the value of $\phi$ is arbitrary.

According to the formulas given above, we calculate the harmonic spectra in given channels for different ellipticity. Figure 3 depicts the channel-resolved rate of the 20th harmonic generated from Xe atoms when the ellipticity of the $2\omega -$field changes. When the ellipticity of the $2\omega -$field changes, the HHG rate in a given channel varies distinctively, and many island-like structures become highlighted. The location of the island-like structures changes with the laser intensity and the ellipticity. These features are similar to those shown in Fig. 1. One difference is that the dominated channels contributing to a given harmonic are less, because the possible combinations of $q_{i}$ that satisfy $q_{1}+2(q_{2}+q_{3})=q$ are less for a given $q$. Another difference is that the central channel becomes more dominated than that in the previous case.

Fig. 3. Channel-resolved rate of the 20th harmonic varying with ellipticity of the 2$\omega$ -field. The panels in left/right columns are for right and left CP component of the generated harmonics, respectively. The maximal intensity of the EP field is the same as that of the EP field. In (a) and (b), the intensity of the CP field is taken as $1.50 \times 10^{14}W/cm^{2}$ , while in (c) and (d), it is taken as $2.0 \times 10^{14}W/cm^{2}$. The fundamental driving field is of 800 nm wavelength and the target is Xe atoms.

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The island-like structures in different channels for fixed $q_{3}$ form regular distributions. Figure 4(a) and (b) depict two distributions for the harmonic spectra calculated at $2.0 \times 10^{14}W/cm^{2}$, in which a $\Lambda -$shaped distribution of the highlighted island-like structures is evident, both for $q_{1}-q_{2}-q_{3}=\pm 1$. The $\Lambda -$type distribution for $q_{1}-q_{2}-q_{3}=1$ case is a little broad than that for $q_{1}-q_{2}-q_{3}=-1$ case, and thus the two legs of the V-type structure cross with each other at the 16th harmonics, while at 17th harmonic for $q_{1}-q_{2}-q_{3}=-1$, both at around $\xi =\pi /2$. Note that the driving field is CP when $\xi =\pi /2$, these two crossing points denotes the $3q\pm 1$ harmonics in the CRTC fields. This agrees that in the CRTC field, the harmonics are of $3q\pm 1$ orders and the adjacent harmonics are of opposite helicity. These features agree well with the experimental observation by Fleischer et al.[1].

Fig. 4. Harmonic spectra versus ellipticity of the 2$\omega$-field. The fundamental driving field is right circularly polarized. The intesity of the both laser modes is $2.0 \times 10^{14}W/cm^{2}$. The panels in left/right columns are for right and left CP component of the generated harmonics, respectively.

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Next, we identify the observed harmonics generated from the $(3,8)-$channel when the $2\omega -$field is elliptically polarized. In our treatment, this channel includes two combinations marked by $(3,3,5)$ and $(3,2,6)$ channels. The harmonic generated from the $(3,3,5)$ channel is right CP while that from the $(3,2,6)$ channel is left CP. The harmonic spectra including these two channels are depicted in Fig. 4. Along with the ellipticity change, many islands appear, and all are distributed symmetrically about $\xi =\pi /2$. The $\Lambda$ distribution is distinct in each panel and multiple lags are clear. Figure 4(a) is for the right CP harmonics, and the $\Lambda$ distribution is a little broad than that in the Fig. 4(b). Moreover, the harmonic rate is higher than the counterpart in the right panel. The harmonic is dominated by the right CP component.

Finally, we show in Fig. 5 the ellipticity-resolved spectra for the harmonics generated from several dominated channels, for a comparison with the observation of Fleischer et al. [1]. Figures 5(a) and (b) are for the ellipticity of the $\omega -$field being changed, corresponding to Fig. 2(a) in Ref. [1]. The highlighted islands linked by white dotted lines form a V-shaped structure, for both the right and the left CP harmonics. This V-shaped distribution agrees qualitatively with that observed by Fleischer et al.. In addition to this main structure, other islands outside the V-shaped structure form another big V-shaped distributions, similar to that disclosed by the numerical results shown in the Fig. 2(c) of Ref. [1] . Similar agreement is also obtained when the ellipticity of the $2\omega -$field is changed. The $\Lambda -$shaped distributions of the highlighted islands are evident in Figs. 5(c) and (d), respectively, for the harmonic series of opposite helicity.

Fig. 5. The calculated harmonic spectra versus ellipticity of the EP mode. (a) and (b) are for the $\omega$-field being elliptically polarized, while (c) and (d) are for the 2$\omega$ -field being elliptically polarized. The intensity of the CP mode and the maximal intensity of the EP are $1.0 \times 10^{14}W/cm^{2}$ in (a) and (b), and are $2.0 \times 10^{14}W/cm^{2}$ in (c) and (d) respectively. The fundamental driving field is of 800nm wavelength and the target is Xe atoms. The panels in left/right columns are for right and left CP component of the generated harmonics, respectively.

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5. Conclusions

In this paper, the harmonics generated from Xe targets irradiated by the bichromatic laser field with one mode being elliptically polarized is studied using a fully quantum treatment. The EP laser field is treated as a combination of two counter-rotating CP fields and an analytical formula of the HHG rate is derived. The HHG amplitude can be accurately described in both the magnitudes and the phase, by which the SAM conservation conditions are derived in a solid base and in a straightforward manner. We find that the SAM of harmonics is balanced in a manner that the number of photons converted into the harmonics from one mode differs one from other modes of opposite helicity, thus the net SAM which is obtained by the harmonic photon is $\pm 1$, and the target atom recovers its initial state after harmonic emission. We find that in a given HHG channel, as the ellipticity changes, the harmonic rate varies distinctively, so many island-like structures appear in the harmonic spectra. When the fundamental laser field is elliptically polarized, the island-like structures form a V-shaped distribution, while when the harmonic laser field is elliptically polarized, the island-like structure form a $\Lambda -$shaped distribution. These features agree well with the experimental observations. We further show that in these distributions, the number of the transferred photons of the same helicity keeps fixed. Our treatment can be extended into other targets and our conclusions holds for other rare atoms.

Funding

Natural Science Foundation of Shanghai (20ZR1441600); National Natural Science Foundation of China (11674231, 12074261).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Study on spin angular momentum balance in harmonics generated from counter-rotating two-color laser fields

Abstract

1. Introduction

2. Volkov state and HHG formula in three-mode fields

3. SAM balance and harmonic spectra when the fundamental driver changes ellipticity

4. SAM balance and harmonic spectra when the harmonic driver changes ellipticity

5. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (22)

Optics Express