Spin-to-orbital angular momentum conversion in harmonic generation driven by intense circularly polarized laser

We first use a relativistic CP Gaussian laser with wavelength λ = 800 nm and amplitude ${a}_{0}=12$ to impinge normally on a solid density plasma target, and the corresponding laser intensity is ${I}_{0}=3.08\times {10}^{20}{\rm{W}}\,{{\rm{cm}}}^{-2}.$ Supposing that the driving laser pulse propagates along the z-axis, its transverse electric field is expressed by ${\boldsymbol{E}}={a}_{0}\exp \left(-{r}^{2}/{w}^{2}\right)\exp \left[-\,\mathrm{ln}\,2{\left(2\left(t-{k}_{0}z/{\omega }_{0}\right)/\tau \right)}^{2}\right]\left[\cos \left({\omega }_{0}t-{k}_{0}z\right){{\boldsymbol{e}}}_{x}+\sigma \,\sin \left({\omega }_{0}t-{k}_{0}z\right){{\boldsymbol{e}}}_{y}\right].$ The spot size of the laser pulse is w = 5λ and pulse duration τ = 5 T, where T is the laser period, and $\exp \left[-\,\mathrm{ln}\,2{\left(2\left(t-{k}_{0}z/{\omega }_{0}\right)/\tau \right)}^{2}\right]$ is the carrier envelope. The simulation box is 20λ(x)×20λ(y)×12λ(z) and is divided into 600 × 600 × 720 cells. The plasma target has a density of 10n_c and occupies the region of 10λ < z< 12λ, where ${n}_{c}={m}_{e}{\omega }_{0}^{2}/(4\pi {e}^{2})=1.74\times {10}^{21}{\rm{c}}{{\rm{m}}}^{-3}$ is the critical density for the wavelength of 800 nm. The critical density is defined as the plasma threshold density at which the plasma becomes opaque for an electromagnetic field with frequency ${\omega }_{0}$ [31].

When the driving CP Gaussian laser interacts with the target, it drives a vortex plasma surface to oscillate rapidly, and harmonics are generated in the reflected light. In our discussion, we will use the SAM along the z-axis but not the helicity, because the helicity is dependent on the direction of propagation of light. However, the propagation directions are opposite for the incident laser and the emitted harmonics. Therefore, it is not convenient to use the helicity when discussing the angular momentum features. If the driving laser is a CP light beam, the generated harmonics will still be circularly polarized, as shown in figure 1, where the SAM of the driving laser is $\sigma =+1.$ This figure shows the transverse electric fields (E_x and E_y) at the transverse position $\left(x,y\right)=\left(0.8\lambda ,\,\,0\right)$ for the first (figure 1(a)), second (figure 1(c)), and third (figure 1(e)) harmonics at t = 24 T. It is found that the harmonics have Gaussian envelopes in longitudinal direction, just as the driving laser pulse. The E_x and E_y fields of the harmonics have nearly identical amplitudes, and their phase difference is π/2 (which is more clearly shown in the embedded figures). This indicates the harmonics are circularly polarized. We also find that the SAM of the harmonic is $\sigma =+1,$ which is equal to the SAM of the incident driving laser. However, for the third harmonic shown in figure 1(e), there is a significant amplitude deviation between E_x and E_y after $z=7.5\lambda$ (marked by the black dashed line). To understand this large deviation, we plot the E_x field in the x–z plane for the harmonics, as shown in figures 1(b), (d) and (f), and find that the third harmonic becomes more and more turbulent after $z=7.5\lambda .$ ⁴ Figures 1(a) and (b) indicate that when the turbulent harmonic is being emitted, the main part of the driving laser has been reflected by the plasma surface and the laser intensity acting on the surface is gradually decreasing. Therefore, the current on the plasma surface becomes more and more turbulent at that moment, and the noise is produced to swamp the harmonics. The second harmonic is not affected because of the its higher intensity.

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Figure 2 shows the y-component of the harmonic electric field, E_y. The fundamental-frequency light is still Gaussian, as shown in figures 2(a) and (d). However, figures 2(b), (c), (e), and (f) indicate that the harmonics have vortex structures. In figures 2(a)–(c), where the driving laser carries the SAM $\sigma =+1$ and TAM ${j}_{0}=\sigma =+1,$ the OAMs of first to third harmonics are l₁ = 0, l₂ = +1, and l₃ = +2, respectively. In figures 2(d)–(f), where the driving laser beam carries the SAM $\sigma =-1$ and the TAM ${j}_{0}=\sigma =-1,$ the OAMs of first to third harmonics are l₁ = 0, l₂ = −1, and l₃ = −2. These results can be understood from the point of view of TAM conservation. In the HHG process, multiple angular momenta of fundamental-frequency photons are converted to the angular momentum of one harmonic photon. Therefore, the TAM carried by the n_th harmonic photon should be n times that of the driving laser, i.e. ${j}_{n}=n\sigma .$ Because the SAM of a photon can only be ±1, the extra SAMs would be converted to OAM so that TAM conservation is ensured, which leads to ${l}_{n}=(n-1)\sigma .$ ⁵

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The STOC not only exists in the HHG process driven by a CP Gaussian laser, but also occurs for vortex lasers. A typical vortex laser is the Laguerre-Gaussian (LG) mode, whose transverse electric field is expressed by

$$\begin{eqnarray}&&\begin{array}{l}{{\boldsymbol{E}}}_{{\rm{L}}{\rm{G}}}={a}_{0}{(\sqrt{2}r/w)}^{| {l}_{0}| }\exp (-{r}^{2}/{w}^{2}){L}_{p}^{| {l}_{0}| }(2{r}^{2}/{w}^{2})\exp [-\,\mathrm{ln}\,2{(2(t-{k}_{0}z/{\omega }_{0})/\tau )}^{2}],\\ \,\,\times \,[\cos ({\omega }_{0}t-{l}_{0}\theta -{k}_{0}z){{\boldsymbol{e}}}_{x}+\sigma \,\sin ({\omega }_{0}t-{l}_{0}\theta -{k}_{0}z){{\boldsymbol{e}}}_{y}]\end{array}\end{eqnarray} \tag{ 1 }$$

where ${L}_{p}^{l}\left(s\right)$ is the associated Laguerre polynomial, l₀ is the OAM number, and p is the number of radial nodes in the intensity distribution. The field of the LG mode has a hollow transverse structure (${{\boldsymbol{E}}}_{{\rm{L}}{\rm{G}}}=0$ at $r=0$), and its intensity has a ring-like maximum value. To ensure that the peak intensity of the LG mode is equal to the Gaussian mode above, we set the normalized parameter ${a}_{0}=20$ in the simulations here. The SAM of the incident laser is $\sigma =+1$ and other parameters are the same as the Gaussian pulse. Figure 3 shows the transverse electric fields, E_x and E_y, at the transverse position $\left(x,y\right)=\left(\lambda ,\,-2.5\,\lambda \right)$ for the first (figure 3(a)), second (figure 3(c)), and third (figure 3(e)) harmonics at t = 24 T, which are driven by a CP LG laser with $\sigma =+1$ and ${l}_{0}=+1.$ These figures indicate that the generated harmonics are still CP light carrying the same SAM as the driving laser. We also find the third harmonic becomes turbulent after $z=8\lambda .$

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Figure 4 illustrates the two groups of harmonics generated by the CP LG lasers with different angular momenta. The electric field components, E_y, of the harmonics plotted in figure 4 show clear vortex structures. In the first case, the driving laser pulse carries the OAM ${l}_{0}=+1,$ so its TAM is ${j}_{0}=\sigma +{l}_{0}=+2.$ The OAMs of first to third harmonics are ${l}_{1}=+1,$ ${l}_{2}=+3$ and ${l}_{3}=+5,$ as shown in figures 4(a)–(c). In the second case shown in figures 4(d)–(f), the driving laser pulse carries the OAM ${l}_{0}=-1,$ and its TAM is ${j}_{0}=\sigma +{l}_{0}=0,$ which leads the OAMs of all the harmonics to be $\left|{l}_{n}\right|=1.$ These results can be explained as follows: the driving laser carries the TAM ${j}_{0}=\sigma +{l}_{0}.$ According to the TAM conservation, the TAM of the n_th harmonic is given by ${j}_{n}=n{j}_{0}=n(\sigma +{l}_{0}).$ Since the SAM of the n_th harmonic photon is still $\sigma =+1,$ the OAM of the n_th harmonic photon is given by ${l}_{n}={j}_{n}-\sigma =\left(n-1\right)\sigma +n{l}_{0}.$ If the OAM of the driving laser is ${l}_{0}=+1,$ we have ${l}_{n}=2n-1;$ while in the second case the driving laser carries the OAM ${l}_{0}=-1,$ which leads to a constant OAM ${l}_{n}=-1$ for each order of harmonics.

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To clearly reveal the structure of the vector vortex of the harmonics, we plot the vector diagrams of the transverse electric fields for the first to third harmonics at the position $z=6\lambda$ at $t=24T$ in figure 5. The angular momenta of the driving laser are $\sigma =+1$ and ${l}_{0}=+1,$ which corresponds to the case shown in figures 4(a)–(c) respectively. The arrows represent the distribution of the electric field vector, and the background color shows the harmonic intensity. The intensity distribution shows a hollow structure just like a donut. The hollow phase derives from the evolution of the phase singularity at the center point on the focal spot plane.

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From the simulation results, it is concluded that since the SAM of a photon can only be +1 or −1, to ensure the TAM conservation, the OAM of the n_th harmonic photon should be

$$\begin{eqnarray}&&{l}_{n}=\left(n-1\right)\sigma +n{l}_{0}.\end{eqnarray} \tag{ 2 }$$

The extra SAMs of the fundamental-frequency photons will be converted to OAMs of the harmonic photons.

During the interaction of the intense laser with the plasma, the harmonics are emitted by the transverse current on the plasma surface. To clarify the mechanism of the harmonic generation, let us focus on the details of the evolution of the transverse current. Without loss of generality, we consider the driving laser to carry the angular momenta $\sigma =+1$ and ${l}_{0}=+1,$ which corresponds to the situation displayed in figures 4(a)–(c). Figure 6 shows the electron density evolution on the plasma surface at the time interval between 20T and 21T. This figure displays a section on the x–z plane. We can see the surface deformation within the skin depth. The electrons are pushed by the light pressure (ponderomotive force) to accumulate on the plasma surface, which forms thin electron shells. The ring-like intensity distribution of the vortex laser drives to form two cambered surfaces shown in figure 6(b).

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Figure 7 shows the formation and evolution of the transverse current in one laser period (from 16.2T to 17.2T) when the laser interacts with the plasma target. This figure displays the current distribution on the surface located at $z=10.1\,\lambda ,$ where is the position about 0.1λ depth inside the target. The arrows represent the transverse current vectors, and the background color denotes the electron density distribution. When irradiating the target, the intense laser drives the plasma electrons to produce a rotating and ring-like current first, as shown in figures 7(a)–(c), and then the current evolves into a vortex structure shown in figures 7(d)–(f). Figures 7(e) and (f) show the azimuth-dependent phase structures for the electron density, as well as the current, which are the typical features of the vortex current. This vortex current induces the emission of the vortex harmonics.

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If the plane plasma target is replaced with a spiral phase plate (SPP) target, more interesting phenomena occur. The SPP is an optical element with a helical surface. The thickness of the component increases smoothly with azimuthal position according to ${l}_{s}\lambda \theta /\left(4\pi \right),$ shown in figure 8. When a plane wave laser is reflected on the SPP, it can be converted into a vortex mode with the helical phase $\exp \left(i{l}_{s}\theta \right)$ in azimuth [18, 32], where l_s is the OAM parameter induced by the SPP. Now, let us use a vortex laser beam to normally irradiate a SPP plasma target. The TAM of the fundamental-frequency light becomes ${j}_{0}=\sigma +{l}_{0}+{l}_{s}.$ Based on a similar analysis of the above discussion, the OAM of the n_th harmonic photon would be

$$\begin{eqnarray}&&{l}_{n}=\left(n-1\right)\sigma +n\left({l}_{0}+{l}_{s}\right).\end{eqnarray} \tag{ 3 }$$

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Table 1 shows the simulation results under different conditions. The plane target and Gaussian pulse correspond to ${l}_{s}=0$ and ${l}_{0}=0,$ respectively. Obviously, the OAM of the harmonic photon can be tuned and controlled by changing the three parameters: σ, l₀ and l_s. The simulation results indicate that the OAM of the n_th harmonic photon follows equation (3). These results demonstrate that the STOC occurs during the HHG process driven by a CP laser beam. Besides, it should be noted that the increasing target thickness with the azimuthal angle forms a slope structure in the azimuthal direction on the target surface. Then the laser irradiation on the target is equivalent to an oblique incidence at a small angle. It is well known that obliquely incident CP laser can drive the plasma surface to emit the harmonics [28]. So, both the oblique incidence mechanism and longitudinal electric field work to generate harmonics on the SPP target, which enhances the harmonic intensity.

Physically, STOC is required for the conservation of TAM. The details of how the harmonics are generated are not important when studying the above-mentioned phenomenon. Here, to understand the mechanism how the SAM is converted to OAM in our condition, let us first analyze the HHG induced by the CP laser pulse in theory. Suppose that the target surface is located at $z=0.$ When the CP laser is reflected on the target, a standing wave field is formed outside the plasma target at $z\lt 0.$ In the Coulomb gauge, the vector potential of the standing wave field can be expressed by

$$\begin{eqnarray}&&{{\boldsymbol{A}}}_{{\rm{o}}{\rm{u}}{\rm{t}}}(x,y,z,t)={A}_{0}\left(\begin{array}{c}u(r)\sin ({k}_{0}z)\sin ({\omega }_{0}t)\\ -\sigma u(r)\sin ({k}_{0}z)\cos ({\omega }_{0}t)\\ {k}_{0}^{-1}{\partial }_{r}u(r)\cos ({k}_{0}z)\sin ({\omega }_{0}t-\sigma \theta )\end{array}\right),\end{eqnarray} \tag{ 4a }$$

where A₀ is the amplitude, $r=\sqrt{{x}^{2}+{y}^{2}},$ u(r) is the transverse profile and $u\left(r\right)=\exp (-{r}^{2}/{w}^{2})$ describes a Gaussian beam, $\theta =\arctan (y/x)$ is the azimuth angle, $\sigma =\pm 1$ is the SAM, and k₀ and ω₀ are the wave vector and frequency of the laser, respectively. The corresponding electric field can be derived using ${\boldsymbol{E}}\left(x,y,z,t\right)=-\left(1/c\right){\partial }_{t}{\boldsymbol{A}},$

$$\begin{eqnarray}&&{{\boldsymbol{E}}}_{{\rm{o}}{\rm{u}}{\rm{t}}}\left(x,y,z,{\rm{t}}\right)=-{A}_{0}\left(\begin{array}{c}{k}_{0}{\rm{u}}\left(r\right)\sin \left({k}_{0}z\right)\cos \left({\omega }_{0}{\rm{t}}\right)\\ \sigma {k}_{0}{\rm{u}}\left(r\right)\sin \left({k}_{0}z\right)\sin \left({\omega }_{0}{\rm{t}}\right)\\ {\partial }_{r}{\rm{u}}\left(r\right)\cos \left({k}_{0}z\right)\cos \left({\omega }_{0}{\rm{t}}-\sigma \theta \right)\end{array}\right).\end{eqnarray} \tag{ 4b }$$

Inside the plasma target ($z\gt 0$), the laser field decays rapidly within a skin layer, and the vector potential is approximated by

$$\begin{eqnarray}&&{{\boldsymbol{A}}}_{{\rm{i}}{\rm{n}}}(x,y,z,t)={{A}}_{d}\left(\begin{array}{c}u(r)\exp (-z/d)\sin ({\omega }_{0}t)\\ -\sigma u(r)\exp (-z/d)\cos ({\omega }_{0}t)\\ d{\partial }_{r}u(r)\exp (-z/d)\sin ({\omega }_{0}t-\sigma \theta )\end{array}\right),\end{eqnarray} \tag{ 5 }$$

with the skin depth $d\ll {\lambda }_{0}$ and the amplitude at the surface A_d.

It is well known that the HHG is caused by the transverse current formed owing to the collective oscillating motion of electrons. In the existing HHG theory [27–29], the ponderomotive force of the laser pulse drives the plasma surface oscillation in the longitudinal direction, and the harmonic is emitted by the current on the plasma surface. For linear polarization, the ponderomotive force oscillates at twice the frequency of the driving laser; thus, only odd harmonics are generated when the driving laser impinges normally on the target. However, for circular polarization, the ponderomotive force is constant, and it cannot excite an oscillating plasma surface; therefore, no harmonic is generated.

In our scheme the mechanism of harmonic generation is completely different. When a CP laser pulse is tightly focused, a longitudinal electric field, E_z, is induced due to the finite transverse size and profile of the laser field, as expressed by equations (4) and (5). This longitudinal electric field rapidly oscillates at the frequency equaling the laser frequency, and this drives the plasma surface to oscillate at the same frequency. Consequently, both odd and even harmonics are generated. In addition, E_z has a vortex phase $\exp ({\rm{i}}\sigma \theta )$ similar to an LG beam with a topological charge of 1 as shown in figure 9. This vortex phase is associated with the SAM of the laser pulse, which produces the angular momentum transfer from the driving laser to plasma surface during the laser-plasma interaction. We notice that E_z is proportional to ${\partial }_{r}u(r).$ If the focal spot size is much larger than the wavelength, we find ${\partial }_{r}u(r)\ll {k}_{0}u(r),$ which implies the longitudinal field strength is much smaller than the transverse field, and the intensity of the harmonic is low. However, when the spot size is small and comparable with the laser wavelength, the factor ${\partial }_{r}u(r)\leqslant {k}_{0}u(r),$ that means E_z can be comparable with the transverse electric field of the driving laser, and the harmonic intensity can be increased. In addition, E_z can also induce longitudinal ponderomotive force. But this ponderomotive force is much smaller than the electric field force, ${F}_{z}=e{E}_{z},$ and can be neglected.

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We know that the harmonics are generated and emitted from the oscillating transverse current, ${{\boldsymbol{J}}}_{\perp },$ on the plasma surface. The transverse current is generated via the collective transverse quiver of the electrons driven by the transverse laser field [33]. The longitudinal component of the laser field drives the current to oscillate in the longitudinal direction, and then, the harmonics are emitted. The circular polarization makes the current acquire a vortex structure with the phase $\exp ({\rm{i}}\sigma \theta ),$ which plays a key role in the angular momentum conversion in the HHG process. To study the details of the harmonic characteristics, we need to solve the nonlinear wave equation

$$\begin{eqnarray}&&\displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{{\partial }^{2}\pmb{\mathscr{A}}}{\partial {t}^{2}}-{{\rm{\nabla }}}^{2}\pmb{\mathscr{A}}=\displaystyle \frac{4\pi {{\rm{J}}}_{\perp }}{c}.\end{eqnarray} \tag{ 6 }$$

Using the Green function method, the solution to equation (6) can be expressed in an integral form

$$\begin{eqnarray}&&\pmb{\mathscr{A}}({\boldsymbol{x}},t)=\displaystyle \frac{1}{c}\displaystyle \int {{\rm{d}}}^{3}{\boldsymbol{x}}^{\prime} {\rm{d}}t^{\prime} \displaystyle \frac{1}{| {\boldsymbol{x}}-{\boldsymbol{x}}^{\prime} | }\delta \left(t^{\prime} -t+\displaystyle \frac{| {\boldsymbol{x}}-{\boldsymbol{x}}^{\prime} | }{c}\right){{\boldsymbol{J}}}_{\perp }({\boldsymbol{x}}^{\prime} ,t^{\prime} ).\end{eqnarray} \tag{ 7 }$$

The ${\rm{d}}t^{\prime}$ integral can be worked out easily by using the delta-function. If we consider that the observation point z_obs is far away from the plasma surface in the z-direction and use the paraxial approximation, we get ${\left(z-z^{\prime} \right)}^{2}\gg {\left({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} \right)}^{2}.$ Then, the transverse contribution from ${\left({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} \right)}^{2}$ can be neglected, implying that the r-dependent transverse profile of the current is not considered in detail. In the longitudinal direction, the current is sharply localized, say at the position Z(t), with a thin shell on the plasma surface; then, the ${\rm{d}}z^{\prime}$ integral can be approximated

$$\begin{eqnarray}&&\pmb{\mathscr{A}}({\boldsymbol{x}},t)\approx \displaystyle \frac{1}{c}\displaystyle \int {\rm{d}}\theta ^{\prime} {\rm{d}}r^{\prime} r^{\prime} \displaystyle \frac{1}{| z-Z({t}_{{\rm{r}}{\rm{e}}{\rm{t}}},\theta )| }{{\boldsymbol{J}}}_{\perp }(Z({t}_{{\rm{r}}{\rm{e}}{\rm{t}}},\theta ),\theta ^{\prime} ,r^{\prime} ,{t}_{{\rm{r}}{\rm{e}}{\rm{t}}}),\end{eqnarray} \tag{ 8 }$$

where

$$\begin{eqnarray}&&{t}_{{\rm{r}}{\rm{e}}{\rm{t}}}=t-(Z({t}_{{\rm{r}}{\rm{e}}{\rm{t}}},\theta )-z)/c\end{eqnarray} \tag{ 9 }$$

is the retarded time at the observation point $z={z}_{{\rm{o}}{\rm{b}}{\rm{s}}}$ and $Z\left({{\rm{t}}}_{{\rm{r}}{\rm{e}}{\rm{t}}}\right)$ describes the longitudinal oscillation of the current. The transverse current driven by the laser field ${\boldsymbol{A}}\left({\boldsymbol{x}},t\right)$ is highly nonlinear and should be described by the nonlinear fluid theory. For simplicity, we neglect some higher order relativistic terms here, and approximate this current as [28]:

$$\begin{eqnarray}&&{{\boldsymbol{J}}}_{\perp }(Z({t}_{{\rm{r}}{\rm{e}}{\rm{t}}},\theta ),\theta ,r,{t}_{{\rm{r}}{\rm{e}}{\rm{t}}})\sim {{\boldsymbol{a}}}_{\perp }(Z({t}_{{\rm{r}}{\rm{e}}{\rm{t}}},\theta ),\theta ,r,{t}_{{\rm{r}}{\rm{e}}{\rm{t}}}),\end{eqnarray} \tag{ 10 }$$

where ${{\boldsymbol{a}}}_{\perp }=e{\boldsymbol{A}}/\left({m}_{e}{c}^{2}\right)$ is the normalized transverse vector potential of the CP laser field. The ${\rm{d}}\theta ^{\prime}$ and ${\rm{d}}r^{\prime}$ integrals are not important here, because we have neglected the transverse profile of the current in the above approximation. By inserting equations (4) and (10) into (8) we obtain an approximate solution to equation (6):

$$\begin{eqnarray}&&\pmb{\mathscr{A}}({\boldsymbol{x}},t)\sim \displaystyle \frac{{A}_{0}}{| z-Z({t}_{{\rm{r}}{\rm{e}}{\rm{t}}},\theta )| }\left(\begin{array}{c}\sin ({\omega }_{0}{t}_{{\rm{r}}{\rm{e}}{\rm{t}}})\\ -\sigma \,\cos ({\omega }_{0}{t}_{{\rm{r}}{\rm{e}}{\rm{t}}})\end{array}\right).\end{eqnarray} \tag{ 11 }$$

Equation (11) is a quasi-one-dimensional solution, and the transverse profile of the current has been neglected when solving the wave equation. Therefore, this solution is insufficient for explaining all the details of the harmonic features, especially the harmonic intensity and its transverse distribution. However, this theory is sufficient for describing the TAM conservation and the STOC in the HHG process, which is the focus of this work. In addition, the solution (11) indicates that the reflected light has the same polarization as the incident laser.

Next, we focus on the physical mechanism of STOC in the HHG process. The oscillation of the plasma surface is driven by the longitudinal laser field, which is a nonlinear process. Here, for the sake of simplicity, we approximate this surface motion as a harmonic oscillation

$$\begin{eqnarray}&&Z({t}_{{\rm{r}}{\rm{e}}{\rm{t}}},\theta )=-{Z}_{s}\,\sin ({\omega }_{0}{t}_{{\rm{r}}{\rm{e}}{\rm{t}}}-\sigma \theta )\end{eqnarray} \tag{ 12 }$$

with the oscillation amplitude Z_s. Inserting equations (9) into (11) the transverse vector potential of the reflected light takes the form

$$\begin{eqnarray}&&\pmb{\mathscr{A}}({z}_{{\rm{o}}{\rm{b}}{\rm{s}}},t)\sim \displaystyle \frac{{A}_{0}}{| z-Z({t}_{{\rm{r}}{\rm{e}}{\rm{t}}},\theta )| }\left(\begin{array}{c}\sin ({\omega }_{0}t-{k}_{0}Z({t}_{{\rm{r}}{\rm{e}}{\rm{t}}},\theta )+{k}_{0}z)\\ -\sigma \,\cos ({\omega }_{0}t-{k}_{0}Z({t}_{{\rm{r}}{\rm{e}}{\rm{t}}},\theta )+{k}_{0}z)\end{array}\right).\end{eqnarray} \tag{ 13 }$$

The terms $\left({k}_{0}z\right)$ within sine and cosine are not important and can be removed, because we can set the observation point $z={z}_{{\rm{o}}{\rm{b}}{\rm{s}}}=N{\lambda }_{0}$ with the integer N. Inserting equations (12) into (13) the vector potential of the reflected light at the observation point is obtained. If the integer $\left|N\right|\gt 1,$ one finds $\left|{k}_{0}{Z}_{s}\right|=2\pi \left|{Z}_{s}/{\lambda }_{0}\right|\gg \left|{Z}_{s}/\left(N{\lambda }_{0}\right)\right|,$ the term $Z({t}_{{\rm{r}}{\rm{e}}{\rm{t}}},\theta )$ in the denominator of equation (13) is neglectable. For relativistic intensities ${a}_{0}\gt 1,$ the amplitude of the plasma surface motion is very small compared with the wavelength of the laser: ${Z}_{s}\lt {\lambda }_{0}/\left(4\pi \right),$ so we have ${k}_{0}{Z}_{s}\ll 1.$ Then the vector potential of the reflected light is approximately derived

$$\begin{eqnarray}&&\pmb{\mathscr{A}}\left({z}_{{\rm{o}}{\rm{b}}{\rm{s}}},t\right)\sim \displaystyle \frac{{A}_{0}}{\left|N{\lambda }_{0}\right|}\left(\begin{array}{c}\sin \left({\omega }_{0}t+{k}_{0}{Z}_{{\rm{s}}}\,\sin \left({\omega }_{0}t-\sigma \theta \right)\right)\\ -\sigma \,\cos \left({\omega }_{0}t+{k}_{0}{Z}_{{\rm{s}}}\,\sin \left({\omega }_{0}t-\sigma \theta \right)\right)\end{array}\right).\end{eqnarray} \tag{ 14 }$$

Fourier expansion of equation (14) gives two angular momentum modes of harmonic components,$\pmb{\mathscr{A}}\left({z}_{{\rm{o}}{\rm{b}}{\rm{s}}},t\right)={\pmb{\mathscr{A}}}^{\left(\sigma \right)}\left({z}_{{\rm{o}}{\rm{b}}{\rm{s}}},t\right)+{\pmb{\mathscr{A}}}^{\left(-\sigma \right)}\left({z}_{{\rm{o}}{\rm{b}}{\rm{s}}},t\right)$

$$\begin{eqnarray}&&{\pmb{\mathscr{A}}}^{\left(\sigma \right)}\left({z}_{{\rm{o}}{\rm{b}}{\rm{s}}},t\right)\sim \displaystyle \sum _{n=1}^{\infty }{J}_{n-1}\left({k}_{0}{Z}_{{\rm{s}}}\right)\left(\begin{array}{c}\sin \left[n{\omega }_{0}t-\left(n-1\right)\sigma \theta \right]\\ -\sigma \,\cos \left[n{\omega }_{0}t-\left(n-1\right)\sigma \theta \right]\end{array}\right)=\displaystyle \sum _{n=1}^{\infty }{{\rm{\varepsilon }}}_{\perp }^{\left(\sigma \right)}\left(n{\omega }_{0}\right),\end{eqnarray} \tag{ 15a }$$

$$\begin{eqnarray}&&{\pmb{\mathscr{A}}}^{\left(-\sigma \right)}\left({z}_{{\rm{o}}{\rm{b}}{\rm{s}}},t\right)\sim \displaystyle \sum _{n=1}^{\infty }{\left(-1\right)}^{n}{J}_{n+1}\left({k}_{0}{Z}_{{\rm{s}}}\right)\left(\begin{array}{c}\sin \left[n{\omega }_{0}t-\left(n+1\right)\sigma \theta \right]\\ \sigma \,\cos \left[n{\omega }_{0}t-\left(n+1\right)\sigma \theta \right]\end{array}\right)=\displaystyle \sum _{n=1}^{\infty }{{\rm{\varepsilon }}}_{\perp }^{\left(-\sigma \right)}\left(n{\omega }_{0}\right),\end{eqnarray} \tag{ 15b }$$

where ${J}_{\nu }(s)$ denotes the Bessel function of first kind. Equations (15a) and (15b) represent two different angular momentum modes of harmonics with OAM ${l}_{n}=\left(n\pm 1\right)\sigma .$ These two modes are closely related to the SAM of the generated harmonics, which can be explained by TAM conservation. Since the driving laser does not carry OAM, its total angular momentum is only determined by SAM, ${j}_{0}=\sigma .$ During the HHG process, the n_th harmonic photon carries the TAM ${j}_{n}=n\sigma .$ If the harmonic photon carries the same SAM as the incident photons in driving laser, the extra SAMs would be converted to OAM due to the TAM conservation, which leads to ${l}_{n}=\left(n-1\right)\sigma$ as formulated by mode ${{\boldsymbol{\varepsilon }}}_{\perp }^{\left(\sigma \right)}(n{\omega }_{0})$ in equation (15a). This result is consistent with equation (2) for ${l}_{0}=0.$ However, there is another case. If the SAM of the harmonic photon flips to $-\sigma ,$ the OAM must be ${l}_{n}=\left(n+1\right)\sigma ,$ which is expressed by mode ${{\boldsymbol{\varepsilon }}}_{\perp }^{\left(-\sigma \right)}(n{\omega }_{0})$ in equation (15b).

Since the amplitude of the plasma surface motion is small, ${k}_{0}{Z}_{s}\ll 1,$ the Bessel function in equation (15) can be expressed by the limiting form: ${J}_{n}\left({k}_{0}{Z}_{s}\right)\approx {\left({k}_{0}{Z}_{s}/2\right)}^{n}/{\rm{\Gamma }}\left(n+1\right),$ where ${\rm{\Gamma }}\left(s\right)$ is the Gamma function. Thus, the electric field intensity of the n_th harmonic mode ${{\boldsymbol{\varepsilon }}}_{\perp }^{\left(-\sigma \right)}(n{\omega }_{0})$ is much lower than the one of the mode ${{\boldsymbol{\varepsilon }}}_{\perp }^{\left(\sigma \right)}(n{\omega }_{0}),$ with the ratio $\left|{{\rm{\varepsilon }}}_{\perp }^{\left(-\sigma \right)}\left(n{\omega }_{0}\right)/{{\rm{\varepsilon }}}_{\perp }^{\left(\sigma \right)}\left(n{\omega }_{0}\right)\right|\sim {\left({k}_{0}{Z}_{s}\right)}^{2}/\left(4n\left(n+1\right)\right)$ $\ll 1.$ In principle, modes ${{\boldsymbol{\varepsilon }}}_{\perp }^{\left(\sigma \right)}(n{\omega }_{0})$ and ${{\boldsymbol{\varepsilon }}}_{\perp }^{\left(-\sigma \right)}(n{\omega }_{0})$ can be distinguished via the OAM spectrum. However, we cannot extract the ${{\boldsymbol{\varepsilon }}}_{\perp }^{\left(-\sigma \right)}(n{\omega }_{0})$ component from the simulations because its intensity is too weak. This is just the reason why the SAM of the harmonic is always the same as the one of the driving laser in our simulations. On the other hand, from figures 1 and 3 we find some small amplitude deviations between E_x and E_y of the second and third harmonics, especially for the LG driving laser. This small amplitude deviation may be a potential evidence that there exists the counter-polarized mode ${{\boldsymbol{\varepsilon }}}_{\perp }^{\left(-\sigma \right)}(n{\omega }_{0}).$ ⁶ Physically, the superposition of modes ${{\boldsymbol{\varepsilon }}}_{\perp }^{\left(\sigma \right)}(n{\omega }_{0})$ and ${{\boldsymbol{\varepsilon }}}_{\perp }^{\left(-\sigma \right)}(n{\omega }_{0})$ will lead to an elliptic polarization vector, so that E_x and E_y components have different amplitudes.

If the driving laser pulse is the CP LG beam, the vector potential outside the plasma surface is written as

$$\begin{eqnarray}&&{{\boldsymbol{A}}}_{{\rm{L}}{\rm{G}}}={A}_{0}\left(\begin{array}{c}{u}_{{\rm{L}}{\rm{G}}}(r)\sin ({k}_{0}z)\sin ({\omega }_{0}t-{l}_{0}\theta )\\ -\sigma {u}_{{\rm{L}}{\rm{G}}}(r)\sin ({k}_{0}z)\cos ({\omega }_{0}t-{l}_{0}\theta )\\ {k}_{0}^{-1}({\partial }_{r}{u}_{{\rm{L}}{\rm{G}}}(r)-(\sigma {l}_{0}/r){u}_{{\rm{L}}{\rm{G}}}(r))\cos ({k}_{0}z)\sin ({\omega }_{0}t-{j}_{0}\theta )\end{array}\right)\end{eqnarray} \tag{ 16 }$$

in the Coulomb gauge, where ${j}_{0}=\sigma +{l}_{0}$ is the TAM, ${u}_{{\rm{L}}{\rm{G}}}(r)={(\sqrt{2}r/w)}^{| {l}_{0}| }\exp (-{r}^{2}/{w}^{2}){L}_{p}^{| {l}_{0}| }(2{r}^{2}/{w}^{2}).$ Based on the similar theoretical analysis discussed above, the two angular momentum modes of harmonic components are obtained, with the results

$$\begin{eqnarray}&&{\pmb{\mathscr{A}}}_{{\rm{L}}{\rm{G}}}^{\left(+\sigma \right)}\left({z}_{{\rm{o}}{\rm{b}}{\rm{s}}},t\right)\sim \displaystyle \sum _{n=1}^{\infty }{J}_{n-1}\left({k}_{0}{Z}_{{\rm{s}}}\right)\left(\begin{array}{c}\sin \left(n{\omega }_{0}t-{l}_{n}^{\left(\sigma \right)}\theta \right)\\ -\sigma \,\cos \left(n{\omega }_{0}t-{l}_{n}^{\left(\sigma \right)}\theta \right)\end{array}\right),\end{eqnarray} \tag{ 17a }$$

$$\begin{eqnarray}&&{\pmb{\mathscr{A}}}_{{\rm{L}}{\rm{G}}}^{\left(-\sigma \right)}\left({z}_{{\rm{o}}{\rm{b}}{\rm{s}}},t\right)\sim \displaystyle \sum _{n=1}^{\infty }{\left(-1\right)}^{n}{J}_{n+1}\left({k}_{0}{Z}_{{\rm{s}}}\right)\left(\begin{array}{c}\sin \left(n{\omega }_{0}t-{l}_{n}^{\left(-\sigma \right)}\theta \right)\\ \sigma \,\cos \left(n{\omega }_{0}t-{l}_{n}^{\left(-\sigma \right)}\theta \right)\end{array}\right),\end{eqnarray} \tag{ 17b }$$

where

$$\begin{eqnarray}&&{l}_{n}^{\left(\pm \sigma \right)}=n{l}_{0}+\left(n\mp 1\right)\sigma \end{eqnarray} \tag{ 18 }$$

are the OAM of the n_th harmonic photons for modes ${\pmb{\mathscr{A}}}_{{\rm{L}}{\rm{G}}}^{\left(+\sigma \right)}$ and ${\pmb{\mathscr{A}}}_{{\rm{L}}{\rm{G}}}^{\left(-\sigma \right)},$ respectively, and they obey the TAM conservation. Equation (18) is well consistent with equation (2) for the mode ${\pmb{\mathscr{A}}}_{{\rm{L}}{\rm{G}}}^{\left(+\sigma \right)}.$ By considering the same reason as the situation of CP Gaussian laser, the harmonic with mode ${\pmb{\mathscr{A}}}_{{\rm{L}}{\rm{G}}}^{\left(-\sigma \right)}$ is much weaker than mode ${\pmb{\mathscr{A}}}_{{\rm{L}}{\rm{G}}}^{\left(\sigma \right)}.$ In addition, for the LG laser beam, there is an additional contribution to the longitudinal electric field: $\sim \left(\sigma {l}_{0}/r\right){u}_{{\rm{L}}{\rm{G}}}\left(r\right)\cos \left({k}_{0}z\right)\sin \left({\omega }_{0}t-{j}_{0}\theta \right),$ which leads to a higher field strength than the strength of Gaussian laser, especially for large l₀. So, the harmonic intensity is higher with LG laser beams than Gaussian laser.

Physically, the conservation laws are closely associated with the symmetries. Regardless of any detailed considerations of the laser-plasma interaction, there are overriding symmetry constraints on the HHG process, and then the selection rules are derived [34, 35]. The electromagnetic field is a vector field, so there are two individual dynamical symmetries (DS): the polarization-dependent DS (related to the SAM conservation) and the spatially dependent DS (related to the OAM conservation). We have known that the harmonics are emitted by the transverse current on the plasma surface. Since the current is driven by the incident laser fields, it obviously has the same symmetry as the driving laser, and then the harmonic response also needs to be invariant under the same symmetry transformations.

We first consider a linearly polarized driving laser. Without loss of generality, we suppose its polarization is along the x axis, and express the vector potential as ${\boldsymbol{A}}\left(\theta ,z,t\right)={a}_{0}\,\sin \left({k}_{0}z\right)\sin \left({\omega }_{0}t-{l}_{0}\theta \right){{\boldsymbol{e}}}_{x},$ where k₀, ω₀ and l₀ are the wave vector, frequency and OAM, respectively. This linearly polarized laser possesses a polarization-dependent DS, ${\hat{P}}_{S}=\left({{\boldsymbol{e}}}_{x}\to -{{\boldsymbol{e}}}_{x},t\to t+\pi /{\omega }_{0}\right),$ where ${\hat{P}}_{S}$ is the DS operator with eigenvalue +1 when acting on the driving laser field: ${\hat{P}}_{S}{\boldsymbol{A}}\left(\theta ,z,t\right)={\boldsymbol{A}}\left(\theta ,z,t\right).$ Then the harmonic ${{\boldsymbol{A}}}_{n}={{\rm{e}}}^{{\rm{i}}n{\omega }_{0}t-{\rm{i}}{l}_{n}\theta }{{\boldsymbol{e}}}_{x}$ should possess the same symmetry and be invariant under the ${\hat{P}}_{S}$ transformation: $-{{\rm{e}}}^{{\rm{i}}n{\omega }_{0}t+{\rm{i}}n\pi -{\rm{i}}{l}_{n}\theta }{{\boldsymbol{e}}}_{x}={{\rm{e}}}^{{\rm{i}}n{\omega }_{0}t-{\rm{i}}{l}_{n}\theta }{{\boldsymbol{e}}}_{x}.$ It is satisfied if and only if the harmonic order $n=2N-1$ with the integer N, which is the selection rule for the HHG in the linearly polarized laser field. The ${\hat{P}}_{S}$ symmetry implies that the SAM is conserved in the HHG process (here the SAM can be regarded as zero for the linear polarization), and only the odd harmonics are generated. The linearly polarized laser field also possesses a spatially dependent DS, ${\hat{P}}_{O}=\left(\theta \to \theta +\varphi ,t\to t+{l}_{0}\varphi /{\omega }_{0}\right),$ with the eigenvalue +1: ${\hat{P}}_{O}{\boldsymbol{A}}\left(\theta ,z,t\right)={\boldsymbol{A}}\left(\theta ,z,t\right),$ where φ is an arbitrary rotation angle. The invariance of the harmonics under ${\hat{P}}_{O}$ transformation leads to ${e}^{{\rm{i}}n{\omega }_{0}t+{\rm{i}}n{l}_{0}\varphi -{\rm{i}}{l}_{n}\theta -{\rm{i}}{l}_{n}\varphi }{{\boldsymbol{e}}}_{x}={{\boldsymbol{e}}}^{{\rm{i}}n{\omega }_{0}t-{\rm{i}}{l}_{n}\theta }{{\boldsymbol{e}}}_{x}.$ φ is an arbitrary angle, and thus the OAM of the harmonics is derived, ${l}_{n}=n{l}_{0}=\left(2N-1\right){l}_{0}.$ The ${\hat{P}}_{O}$ symmetry implies the OAM conservation in the HHG. If the driving laser is a plane wave with ${l}_{0}=0,$ the generated harmonics also do not carry the OAM.

For the case of the CP plane wave, there is a rotation symmetry for the polarization vector. When a polarization-dependent DS operator, ${\hat{P}}_{S}=\left(\vartheta \to \vartheta -\sigma \varphi ,t\to t+\varphi /{\omega }_{0}\right),$ acts on the CP light ${\boldsymbol{A}}\left(z,t\right)={a}_{0}\,\sin \left({k}_{0}z\right)\left(\sin \left({\omega }_{0}t\right){{\boldsymbol{e}}}_{x}-\sigma \,\cos \left({\omega }_{0}t\right){{\boldsymbol{e}}}_{y}\right),$ the light field is invariant. Here $\vartheta$ is the azimuthal angle of the polarization vector. It implies that the SAM is conserved in the HHG process, and the harmonics must also be circularly polarized. The invariance of the harmonic, ${{\boldsymbol{A}}}_{n}={{\rm{e}}}^{{\rm{i}}n{\omega }_{0}t}({{e}}_{x}-{\rm{i}}\bar{\sigma }{{\boldsymbol{e}}}_{y}),$ under the same ${\hat{P}}_{S}$ transformation requires ${{\rm{e}}}^{{\rm{i}}n{\omega }_{0}t+{\rm{i}}n\varphi -{\rm{i}}\sigma \bar{\sigma }\varphi }={{\rm{e}}}^{{\rm{i}}n{\omega }_{0}t},$ where $\bar{\sigma }$ is the SAM of the harmonic photon. It is satisfied if and only if $\bar{\sigma }=\sigma$ and $n=1,$ which indicates the harmonics are forbidden by the DS at CP plane wave.

The situation is completely different when the focal spot of the intense CP laser is focused to a small size. The induced longitudinal electric field expressed in equation (4) breaks the dynamical symmetries ${\hat{P}}_{S}$ and ${\hat{P}}_{O}$ individually, and both the SAM and OAM are not conserved in HHG. On the other hand, if the combined DS operator, ${\hat{P}}_{T}=\left(\vartheta \to \vartheta -\sigma \varphi ,\theta \to \theta +\sigma \varphi ,t\to t+\varphi /{\omega }_{0}\right),$ acts on equation (4), the driving laser field is invariant. Therefore, the combined symmetry still holds in this case, and the TAM is conserved in HHG. Under this DS constraint, the harmonic, ${{\boldsymbol{A}}}_{n}={{\rm{e}}}^{{\rm{i}}n{\omega }_{0}t-{\rm{i}}{l}_{n}\theta }({{\boldsymbol{e}}}_{x}-{\rm{i}}\bar{\sigma }{{\boldsymbol{e}}}_{y}),$ needs to be invariant, which leads to ${{\rm{e}}}^{{\rm{i}}n{\omega }_{0}t+{\rm{i}}n\varphi -{\rm{i}}\sigma \bar{\sigma }\varphi -{\rm{i}}{l}_{n}\theta -{\rm{i}}{l}_{n}\sigma \varphi }={{\rm{e}}}^{{\rm{i}}n{\omega }_{0}t-{\rm{i}}{l}_{n}\theta }.$ Consequently, both the odd and even harmonics are allowed, and the harmonic photon carries the OAM of ${l}_{n}=n\sigma -\bar{\sigma },$ which is consistent with equation (18) for ${l}_{0}=0.$

If the driving laser is the CP LG beam expressed in equation (16), the longitudinal field also breaks each individual DS, but the combined DS, ${\hat{P}}_{T}=\left(\vartheta \to \vartheta -\sigma \varphi ,\theta \to \theta +\sigma \varphi ,t\to t+\left(1+\sigma {l}_{0}\right)\varphi /{\omega }_{0}\right),$ still holds, which ensures the TAM conservation in the HHG process. Then the harmonic ${\hat{{\boldsymbol{A}}}}_{n}={{\rm{e}}}^{{\rm{i}}n{\omega }_{0}t-{\rm{i}}{l}_{n}\theta }({{\boldsymbol{e}}}_{x}-{\rm{i}}\bar{\sigma }{{\boldsymbol{e}}}_{y})$ is invariant under this ${\hat{P}}_{T}$ transformation, and we obtain ${{\rm{e}}}^{{\rm{i}}n{\omega }_{0}t+{\rm{i}}n\varphi +{\rm{i}}n\sigma {l}_{0}\varphi -{\rm{i}}\sigma \bar{\sigma }\varphi -{\rm{i}}{l}_{n}\theta -{\rm{i}}{l}_{n}\sigma \varphi }={{\rm{e}}}^{{\rm{i}}n{\omega }_{0}t-{\rm{i}}{l}_{n}\theta }.$ Under this DS constraint, the selection rules for the HHG process is derived, ${l}_{n}=n{l}_{0}+\left(n-\sigma \bar{\sigma }\right)\sigma ,$ which is also consistent with equation (18).