High-harmonic spin-orbit angular momentum generation in crystalline solids preserving multiscale dynamical symmetry

Abstract

Symmetries essentially provide conservation rules in nonlinear light-matter interactions and facilitate control and understanding of photon conversion processes or electron dynamics. Since anisotropic solids have rich symmetries, they are strong candidates for controlling both optical micro- and macroscale structures, namely, spin angular momentum (circular polarization) and orbital angular momentum (spiral wavefront), respectively. Here, we show structured high-harmonic generation linked to the anisotropic symmetry of a solid. By strategically preserving a dynamical symmetry arising from the spin-orbit interaction of light, we generate multiple orbital angular momentum states in high-order harmonics. The experimental results exhibit the total angular momentum conservation rule of light even in the extreme nonlinear region, which is evidence that the mechanism originates from a dynamical symmetry. Our study provides a deeper understanding of multiscale nonlinear optical phenomena and a general guideline for using electronic structures to control structured light, such as through Floquet engineering.

Crystal symmetry offers a versatile design approach for using electronic systems to manipulate high-harmonic structured light.

INTRODUCTION

Coherent interactions between intense light fields and matter give rise to a variety of intriguing phenomena (1, 2), including high-harmonic generation (HHG) and generation of attosecond pulses (3–5), coherent driving of electrons (6, 7), and dynamical modulation of light-dressed electronic structures (8, 9). The overarching framework governing these phenomena is characterized by a spatiotemporal symmetry called dynamical symmetry (DS) (10–16). DS serves as a powerful tool for finding universal rules in seemingly elusive and intricate phenomena arising from perturbative and nonperturbative light-matter interactions. In particular, the application of DS to microscopic light-matter interactions has provided general insights into the selection rules for polarization of HHG (11, 13, 16), as well as into symmetry breaking spectroscopy (14), molecular symmetry sensing (12, 15), and light-induced symmetry-breaking phenomena (8, 9, 17). For a Hamiltonian H representing an electron system interacting with an external periodic light field, called a Floquet system, the DS operator $\widehat{G}$ works as follows:

$$\widehat{G}H(\overrightarrow{r},t){\widehat{G}}^{-1}=H({\widehat{\mathrm{\gamma }}}_G\overrightarrow{r},{\widehat{\mathrm{\delta }}}_{\widehat{G}}t)=H(\overrightarrow{r},t)$$ (1)

where ${\widehat{\mathrm{\gamma }}}_G$ and ${\widehat{\mathrm{\delta }}}_{\widehat{G}}$ are, respectively, the microscopic spatial part and temporal part of the DS operator (13). The distinctive feature of the DS operation is that it operates not only within spatial dimensions but also simultaneously in the temporal domain. This feature is important for describing nonperturbative phenomena, where light behaves more as a temporary oscillating electric field than individual photons.

The applicability of DS has been extended to multiscale spatial light structures, and, thereby, it allows for comprehensive predictions to be made on a wide range of nonperturbative optical phenomena induced by spatially nonuniform driving fields (18, 19). Micro- and macroscale light structures are respectively characterized by spin angular momentum, corresponding to the helicity of the polarization, and orbital angular momentum (OAM), which corresponds to the twist in the wavefront of light (20). This extension was motivated by the recognition of the potential significance of using both of these fundamental degrees of freedom to control gas-phase HHG (21–25). It has allowed us to develop control strategies for spin angular momentum and OAM states in extreme ultraviolet light pulses (18, 19, 22–27), for nonlinear beam propagation (19), and for generation of topological light (18, 26, 27). However, atomic gasses have only isotropic symmetry, making them poor choices for spatial design of structured light. As a result, the current strategies for designing structured light rely only on controlling the driving field to date (18, 19, 21–27).

Establishing a framework of DS for multiscale interactions with solid systems is of utmost importance for improving the design of structured light. Crystalline solids have anisotropic symmetry, which not only expands the design possibilities of symmetry in light-matter interactions but also has potential for using electronic structure to control in spatial structures of light, such as through Fermi-level control (28), photoinduced phase transitions (29), moiré engineering (30), and Floquet engineering (31, 32). An additional advantage is the capability of nano/microfabrication, including metasurfaces and photonic crystals, which may lead to arbitrary control of structured harmonics (19, 33, 34). One difficulty with solid systems is that, unlike isolated gas-phase atoms, they have inherently complex nonlinear interactions with light due to their dense atomic arrangements, and this complexity makes an understanding of their nature elusive. Strong light fields can induce a range of microscopic phenomena in solids including tunneling and intraband acceleration of electrons in multiple bands, leading to nonlinear emission (35–38). In addition, macroscopic propagation of light in bulk solids involves complex effects, such as self-phase modulation (39), cascade processes (40), and reabsorption (39). Thus, solid systems require a predictive framework for understanding behaviors that are universal to solids, but efforts to develop such a framework have remained within the bounds of theoretical research so far (19, 41).

Here, we experimentally demonstrate the generation of vectorially structured harmonics linked to the discrete crystal symmetry, starting from the concept of DS. As a platform for creating a situation characterized by multiscale DS, we made use of the spin-orbit interaction (SOI) of light in uniaxial crystals. In this situation, even from a single circularly polarized Gaussian driving beam, we observed structured harmonics composed of multiple OAM modes and found that the conservation of total angular momentum governs our observations. Our findings demonstrate that DS provides a robust framework to comprehensively explain nonlinear processes, including the complex propagation processes of nonlinear spin-orbit angular momentum cascades.

RESULTS

Total angular momentum conservation rule of light

The SOI of light (42–46) is a notable effect that essentially creates particular multiscale symmetries. Spin angular momentum and OAM behave independently in isotropic media, e.g., atomic gasses, for paraxial beams (20). However, they can be entangled in structured materials and anisotropic media, such as solid crystals, optical fibers, metasurfaces, and liquid crystals (44). An important example of this occurs in uniaxial crystals, which have unique different refractive indices along their single optical axis. When a circularly polarized beam is tightly focused onto a thick uniaxial crystal along its optical axis (Fig. 1A) (42, 43), the light component traveling obliquely at an angle θ with respect to the optical axis experiences birefringence due to the anisotropic refractive indices, n_o and n_e(θ), as shown in Fig. 1B. As a result, the polarization state of the light oscillates between right and left circular polarizations depending on θ with its axisymmetric spatial distribution around the optical axis.

Fig. 1. Multiscale DS created by SOI of light in uniaxial solids.

(A) Tight-focus configuration of circularly polarized light in a uniaxial crystal. (B) Index ellipsoid of negative uniaxial crystal. The refractive indices on the extraordinary and ordinary axes are denoted by n_e and n_o, respectively. Wave vector components inclined by an angle θ relative to the optical axis experience axisymmetric birefringence with refractive indices of n_e(θ) and n_o. (C) Multiscale DS operation G_I on spatial distribution of the polarization state of the driving light field and GaSe crystal in a plane parallel to the x-y plane inside the crystal. The fundamental beam is assumed to have right-circular polarization (s₁ = 1). The seven ellipsoids with arrows represent the polarization and phase of the laser electric field at each spatial point. Successive operations of the time translation $\widehat{\mathrm{\tau }}$ , the microscopic operation $\widehat{r}$ , and the macroscopic rotation $\widehat{R}$ make the system consisting of light and solid identical to the original. (D) Photon diagram for angular momentum conservation in high harmonics. Spin angular momentum (s_m) or OAM (l_m) is conserved when micro- or macroscale symmetry is present. SOI of light entangles the angular momenta, leaving only the total angular momentum J_m conserved. Dashed lines represent virtual states, and the solid line represents the ground state in the electronic transitions. (E) Experimental configuration for the imaging of spin angular momentum–resolved high harmonics from GaSe crystal. The bottom left inset is the beam profile of the fundamental beam measured before it is focused with an aspherical lens. Normalized intensity is shown on a linear color scale. AQWP, achromatic quarter-wave plate; IR, infrared.

We show that multiscale DS predicts a total angular momentum conservation rule for both perturbative and nonperturbative HHG when a strong laser illuminates a uniaxial crystal under the tight focus conditions. When $\widehat{G}$ is a DS operator of an electron system interacting with an external light field, the electric fields of the high harmonics emitted by the system remain identical under the same DS operation $\widehat{G}$ (19). In situations where the crystal structure of the solid has n-fold rotational symmetry within the plane perpendicular to the optical axis and the laser beam has a spin angular momentum and OAM state of (s₁, l₁), the system in Fig. 1A is expected to have the following two DS operators:

$${\widehat{G}}_{\mathrm{I}}={\widehat{R}}_{n,1}{\widehat{r}}_{n,1}{\widehat{\mathrm{\tau }}}_{n,-s_1-l_1}$$ (2)

and

$${\widehat{G}}_{\text{II}}={\widehat{R}}_{2,1}{\widehat{\mathrm{\tau }}}_{2,-l_1}$$ (3)

The operator ${\widehat{G}}_{\mathrm{I}}$ is composed of three operations: a temporal translation ${\widehat{\mathrm{\tau }}}_{n,-s_1-l_1}$ of −(s₁ + l₁)/n times the period of the light field, a microscopic spatial rotation ${\widehat{r}}_{n,1}$ of 2π/n, associated with the crystal symmetry and polarization of light, and a macroscopic spatial rotation ${\widehat{R}}_{n,1}$ of 2π/n. Here, we assume that the electric field is periodic in time and has a negligible z component. Figure 1C shows an example of the spatial distribution of polarizations in a plane parallel to the x-y plane when a circularly polarized Gaussian beam is applied to a uniaxial crystal. In the previous research on solids, only microscopic DS operations (e.g., $\widehat{r}$ and $\widehat{\mathrm{\tau }}$ ) were considered in an effort to understand nonlinear responses because a driving field with a spatially uniform circular polarization was applied (16, 47). However, the distribution of polarizations becomes nonuniform in a system with SOI, which breaks local microscopic DS. Even under such a condition, by subsequently applying the macroscopic operation of $\widehat{R}$ , the system becomes identical to the original. The multiscale operation ${\widehat{G}}_{\mathrm{I}}$ predicts that the total angular momentum is conserved as

$$J_m=m{J}_1+\mathit{nQ}$$ (4)

where J_m is the total angular momentum of the mth-order harmonics defined by J_m = l_m + s_m, l_m and s_m = ±1 represent the indices of OAM and spin angular momentum of the mth harmonics, respectively, and Q is an integer. In addition, ${\widehat{G}}_{\text{II}}$ predicts the following restriction:

$$l_m=m{l}_1+2Q\prime$$ (5)

where Q′ is an integer. This equation restricts l_m to even integers when l₁ takes an even integer value (see the Supplementary Materials for a detailed derivation). If the laser beam interacts with gas media with spherical symmetry, then the nQ term on the right-hand side of Eq. 4 becomes zero (11, 23, 48) for conserving the angular momenta between the incident and emitted photons. In crystalline solids, on the other hand, an nQ degree of freedom arises from their discrete rotational symmetry (47, 49, 50). Previous research has shown that in the absence of the SOI, the spin angular momentum and OAM are independently conserved in solids (Fig. 1D) (47, 51). On the other hand, in the presence of the SOI, it is possible to control both spin angular momentum and OAM with the combined symmetry of light and crystal.

To investigate the above processes in a real material, we spectrally and spatially measured the higher harmonics generated by focusing a laser beam tightly on a uniaxial crystal (Fig. 1E). The driver laser was a right-circularly polarized (RCP; s₁ = 1) infrared pulse with a photon energy of 0.51 eV and had a Gaussian-like beam profile with l₁ = 0 (inset of Fig. 1E). The sample was GaSe with a bandgap energy of 2.2 eV and in-plane threefold rotational symmetry (n = 3 in Eq. 4). The GaSe crystal is a standard platform for investigating HHG (36–38) and has ideal properties for studying the SOI of light because of its large uniaxial anisotropy (n_e = 2.41, n_o = 2.74 at 0.51 eV) (52). Here, we strategically used a crystal with a thickness of 2 mm to induce a strong SOI. Thick crystals are not usually chosen for HHG research because of the difficulty in handling the complicated cascade processes and phase matching (39). In all of the experiments reported below, we optimized the focus point so that the above-bandgap harmonics from the back surface of the crystal were maximized.

Effect of SOI on harmonic spectra

The circular polarization–resolved harmonic spectra are shown in Fig. 2A. The fundamental beam was tightly focused with an external Gaussian divergence angle of 402 mrad. Both polarization components appeared in the harmonics up to sixth order. Both even- and odd-order harmonics appeared because the GaSe crystal lacks inversion symmetry. Since the bandgap energy is around the photon energy of the fourth-order harmonics, we expected the conversion process to be substantially different for each order: i.e., we expected that the third-order and lower harmonics would mainly be generated as the light propagated in the crystal, while the fourth-order and higher harmonics would mainly be generated from the back surface due to reabsorption in the crystal. To confirm the effect of SOI, we compared the measured spectra with those of a loose focus condition with a 12.7-mrad external divergence angle, as shown in the inset of Fig. 2A. In contrast to the tight focus condition, only the RCP fourth-order harmonics and left-circularly polarized (LCP) fifth-order harmonics appeared, while the other harmonics were largely suppressed in accordance with the spin angular momentum conservation rules (47). Thus, our observations clearly demonstrate the effect of the tight focus in mixing different polarization components. As calculated in Fig. 2B, the divergence angle of 402 mrad is large enough to generate almost equal amounts of counter-rotating circular polarization components for the fundamental beam. To confirm the nonlinearity of the observed HHG process, we measured the dependence of the harmonic intensity on the incident pulse energy (Fig. 2C). The focused intensity of the infrared driving pulse was estimated to be 0.15 TW cm⁻² at the pulse energy of 0.1 μJ, which is enough to reach the extreme nonlinear regime (36–38). Actually, the intensity of the mth (m = 3, 4, 5, and 6)–order harmonics, I_m, especially the higher-order ones, saturated with increasing incident pulse energy, P, and deviated from the power law predicted by perturbation theory (I_m ∝ P^m). This saturation is characteristic behavior that indicates the nonperturbative nature of HHG (2). All experiments reported below were performed in the extreme nonlinear regime (0.1 μJ; gray line in Fig. 2C).

Fig. 2. High harmonics in extreme nonlinear regime generated by tightly focused paraxial Gaussian beam.

(A) Circular polarization–resolved harmonic spectra from GaSe crystal up to the sixth order. External divergence angle of incidence was chosen to be 402 mrad using an aspherical lens with a focal length of 6 mm. RCP and LCP components are shown in red and blue, respectively. The dashed line represents the bandgap energy E_g of the GaSe crystal; an incident pulse energy of 0.1 μJ was used. Inset: Spectra corresponding to loosely focused driving beam with an external beam divergence angle of 12.7 mrad. A lens with a focal length of 200 mm and an incident pulse energy of 3.7 μJ was used for the loose focus. (B) Calculated power transfer between RCP and LCP components of the fundamental wave in the GaSe crystal due to SOI of light. The red and blue lines are the powers of the RCP and LCP components, respectively, in a slice at a depth of z from the front surface of the GaSe crystal. These powers are normalized by the total power of the fundamental wave. Solid and dashed lines represent calculated results for Gaussian external divergence angle of 402 and 12.7 mrad. (C) Dependence of the harmonic intensity on the incident pulse energy P. The gray line represents the incident power used in the experiments. The input pulse energy of 0.1 μJ corresponds to the intensity of 0.15 TW cm⁻² at the focal plane in vacuum. Each colored dashed line is a guide for the eye showing the power law, where the mth harmonic intensity is proportional to P^m. arb. units, arbitrary units.

Spin-dependent structured high harmonics linked to the crystal symmetry

The spatial profiles of high harmonics are crucial information for characterizing the mixing of spin states through the SOI of light. Figure 3A displays the RCP and LCP components of the spatial profiles of the second-, third-, fourth-, and fifth-order harmonics obtained under the tight focus condition. In contrast to the circularly symmetrical incident beam, a variety of structured light appeared. In the LCP components of the second- and fifth-order harmonics and RCP component of the fourth-order harmonics, bright spots appeared at the center of the beam, signifying the presence of the l = 0 mode. These polarization components are present even in the absence of SOI (47). Although the other components in Fig. 3A should be forbidden in the absence of SOI, donut-shaped patterns were clearly observed in our experiments. These observations imply that nonzero OAM modes were generated. We also observed characteristic structures with six nodes in the azimuthal direction. These observations are much different from those acquired under the loose focus condition, where only circular profiles were observed (see fig. S1). Mathematically, the observed sixfold symmetry implies that the OAM spectrum contains only mode separations that are multiples of six. Reflecting this principle, the observed structures are successfully reproduced as shown in Fig. 3B by fittings consisting of sums of Laguerre-Gaussian modes containing similar proportions of OAM modes with these separations. We confirmed the inversion of the spiral structures by inverting the polarization of the fundamental beam (see fig. S3). This symmetric behavior corresponds to reversing the sign of the relative phase difference between the radial modes.

Fig. 3. Spatial imaging of vectorially structured high harmonics.

(A) Spatial profiles of right and left circularly polarized components of high harmonics. (B) Reconstructed spatial profiles made from fittings with two OAM and three radial modes of the Laguerre-Gaussian series.

The observed symmetric light structures show an unprecedented link to the crystal’s structure. When we rotated the GaSe crystal around the optical axis, the spatial profiles of the harmonics rotated accordingly, as represented by fourth-order RCP harmonic in Fig. 4. This rotation corresponds to a shift in the relative phases between the multiplexed OAM modes. Note that this link is unique to phenomena in the combination of nonlinear optics and SOI. In linear optics, GaSe crystals have a uniform refractive index in the in-plane directions due to the highly symmetric structure of threefold rotational symmetry. Thus, the linear optical response of the GaSe crystal does not affect the light structures when the crystal is rotated. In nonlinear optics, however, the polarization of light becomes sensitive to even finer details of this highly symmetric structure on a microscopic scale and, in turn, sensitive to the rotation of the crystal. Furthermore, through SOI, the polarization correlates with macroscopic spatial light structures. Therefore, the combination of nonlinear optics and SOI enables us to create the link between the macroscopic light structure and the microscopic crystal’s structure.

Fig. 4. Macro- and microscopic structures of light and crystal linked through SOI.

Spatial profiles of RCP component of the fourth harmonic at different crystal orientations. Orientation of Ga─Se bonding in the crystal is represented by the sides of white triangles. White dashed lines are guides for the eye that represent the azimuthal phase of spiral structures.

Identification of OAMs for comparison with conservation rule

To identify the OAM forming the structured harmonics, we disentangled it with a spatial light modulator (SLM), as illustrated in fig. S5. By applying additional phase factors exp(iΔlϕ) to the collimated harmonics depending on the azimuthal phase ϕ, the spin-orbit states transformed as (s_m, l_m) → (s_m, l_m + Δl). Subsequently, the spatial image of the reflected high harmonics was Fourier-transformed by a lens, and the resulting image was detected by a camera. The OAM contained in the harmonics were identified by observing the central spot that became bright when Δl equalled −l_m (45).

Figure 5A displays measured images of harmonics of all orders by applying Δl values from 2 to −10 to both circular polarization components. Here, the spatial profiles varied in accordance with the phase variations of the SLM. The images enclosed in the red squares are spatial patterns whose intensity distribution concentrates at the center. These images directly correspond with the OAM components l_m = −Δl forming the harmonics. For example, they indicate that the fifth-order RCP component is a superposition of three OAM states, l = −2, 4, and 10. Notably, the positions of the red squares are limited to even-number values of l_m and show clear patterns with respect to OAM l_m and the harmonic order m, dependent on the polarization. This result indicates the presence of angular momentum selection rules.

Fig. 5. Spin-orbit tomography for high harmonics and selection rule.

(A) Imaging of the phase-modulated harmonics reflected from SLM through Fourier transformation by a lens. The red squares represent experimental data that show bright spots at the beam center. Bottom axis shows OAM corresponding to the phase patterns displayed on the SLM, i.e., l_m = −Δl. (B) Azimuthal angle–averaged plots for (A) as a function of radial positions for the fourth-order harmonics (top, RCP component; bottom, LCP component). To determine the original points of the polar coordinates, the center positions of the respective images are chosen for the respective harmonic order and polarization components. Intensity normalized for each polarization is shown on a log color scale. White dashed lines represent OAM components allowed by both the total angular momentum conservation rule derived from the multiscale DS in Eq. 4 and the conditions in Eq. 5. (C) Table of spin angular momentum and OAM of light in high harmonics. The red square experimental data points are calculated by integrating the harmonic signal in 10 pixels by 10 pixels on the center spots of the original 740-pixel by 740-pixel images in (A) (top, RCP; bottom, LCP). Intensity normalized for each harmonic order and polarization component is shown on a log color scale. Black squares represent the allowed states by the total angular momentum conservation rule derived from multiscale DS. The gray area indicates the region where components allowed by the cascaded process of harmonic generation and SOI of light are present.

Further clues to the interpretation of the results were obtained by examining the radial dependence of the angle-averaged harmonic intensity. In Fig. 5B, the fourth-order RCP components depict clear increases in the radius of the intensity distribution as Δl deviates from 0 and 6. This observation accords with the fact that OAM modes with larger l_m exhibit a ring-shaped intensity distribution with a larger radius in the focal plane. This finding supports the description of the obtained harmonics as a sum of OAM components with l = 0 and 6. Similar results support the presence of l = 2 and 8 for the fourth-order LCP components (fig. S6 shows results for other orders).

Figure 5C summarizes the observed angular momentum states obtained by integrating the harmonic intensity around the center of the images in Fig. 5A. All of the experimental data points show clear peaks that lie at the conditions predicted by the total angular momentum conservation rule (Eq. 4) with n = 3 and the conditions restricted by the Eq. 5. Note that this remarkable agreement requires experimental assurance of the purity of both spin and orbital momentum states of the incident beams (see the Supplementary Materials).

The measured OAM demonstrates that the cascade harmonic generation preserving the DS provides fundamental insights into the dynamics of the angular momentum and frequency conversion processes. In the presence of SOI in temporally periodic extreme nonlinear processes, all spin-orbit states satisfying Eqs. 4 and 5 are generally allowed. However, the dominant processes among them can be attributed to cascade processes involving linear SOI and harmonic generation. Through linear SOI, transformation of spin components between +1 and −1 occurs, accompanied by the transfer of the remaining angular momentum to the orbital component (42–44). In the harmonic generation process, spin angular momentum and OAM states undergo a transformation to follow the conservation rules of respective angular momentum, as shown in left-hand side of Fig. 1D (47, 51). Here, we denote the state of the spin angular momentum and OAM in the mth-order harmonics as (m; s, l) to illustrate energy and angular momentum conservations. For example, when light propagates in the crystal, part of the fundamental light (1; 1,0) is converted into the (1; −1,2) component due to the linear SOI. The observed harmonics are expected to come primarily from these two photons. This explains why most of OAM components in Fig. 5 have positive signs. For example, the presence of the (5; 1,10) component can be attributed to the fifth multiple of the (1; −1,2) component. However, generating harmonic components with negative OAM is more complicated. For example, generating (3; −1, −2) requires three steps, involving second-harmonic generation of (2; −1,0) from (1; 1,0), generation of (2; 1, −2) from (2; −1,0) via the SOI, and sum frequency generation of (3; −1, −2) from the (2; 1, −2) and (1; 1,0). Thus, our observation of negative OAM components reveals that the cascade process makes a crucial contribution to HHG in bulk crystals even for above-bandgap nonperturbative harmonics. The gray-shaded area in Fig. 5C shows the OAM components that can be generated by the abovementioned cascade processes. Most of the observed components lie around the central part of the shaded area since there are more cascade conversion paths to create OAM components around the central part than those around edge of the shaded area. Note that Eqs. 2 and 3, i.e., Eqs. 4 and 5, are derived by simply considering the symmetry of the crystal and the fundamental frequency component of the propagating beam, i.e., without considering complex modulations of the light field inside the crystal. If symmetry is maintained at a given z section in the crystal, then the nonlinear polarization at that section will have the same symmetry. Consequently, at a section z + Δz, the total electric field can have different spatiotemporal profiles from those at z, while still maintaining the same DS. This property ensures the robustness of DS even in the presence of complex cascading processes.

DISCUSSION

In summary, we observed high harmonics with a spatial structure linked to the crystal symmetry of solids by incorporating the SOI of light. We showed that the modified total angular momentum conservation rule, reflecting discrete crystal symmetry, provides essential insights into the spin-dependent OAM control in general nonlinear processes, including cascade and extreme nonlinear phenomena. Moreover, we demonstrated that multiscale DS effectively works as the combined symmetry of solid and light in these phenomena. Our results pave the way for solid-based engineering of structured light pulses and exploration of their topological properties in the extreme ultraviolet region (5, 53). In addition, the dynamic modulation of Floquet states by an intense pulsed laser field may offer a method for ultrafast temporal shaping of vectorial structures (31, 32, 54). Furthermore, synchronizing the symmetry of crystals with metasurfaces, photonic crystals, and optoelectronic devices may present ways of expanding the functionalities of solids in nonlinear photonics (19, 28, 33, 34, 55).

MATERIALS AND METHODS

Experimental setup

The infrared HHG driver was generated by a β-BaB₂O₄–based optical parametric amplifier (TOPAS-PRIME, LIGHT CONVERSION) pumped by a Ti:sapphire amplifier (center wavelength, 800 nm; pulse energy, 3 mJ; pulse duration, 20 fs; repetition rate, 3 kHz). The center wavelength of the driver source was around 2.4 μm, and the pulse duration was 80 fs as estimated from the Fourier transform limit at full width of half maximum. To eliminate the undesired beam from the optical parametric amplifier, the spectra shorter than a wavelength of 1650 nm was blocked by a long-pass filter. The nearly collimated infrared driver source was spatially filtered with a 2.5-mm aperture iris to shape the beam profile as close to a Gaussian profile as possible. The power of the fundamental beam was controlled by wire-grid polarizers, and the beam was converted to a circularly polarized one with an achromatic quarter-wave plate (SAQWP05M-1700, Thorlabs). To achieve an external divergence angle of 402 mrad, which satisfies the conditions for substantial oblique incidence to induce SOI of light in a 2-mm-thick GaSe crystal (EKSMA Optics), the fundamental beam with a diameter of 3 mm (full width of half maximum) was focused using an aspherical lens with a focal length of 6 mm. The resultant high harmonics from the GaSe were refocused by an objective lens (×20, Nikon) onto a spectrometer (QE Pro, Ocean Insight) or a color complementary metal-oxide semiconductor (CMOS) camera (CS165CU/M, Thorlabs) to obtain harmonic spectra and spatial profiles. The focal position of the beam relative to the crystal was optimized to maximize the intensity of the sixth harmonic. The polarization of the high harmonics was analyzed by an achromatic quarter-wave plate (SAQWP05M-700, Thorlabs) and a wire-grid polarizer (WP25M-UB, Thorlabs). To acquire the spatial images of the high harmonics, pairs of short-pass and long-pass filters were inserted in front of the CMOS camera to pick up each order of harmonics. Since the peak wavelength of the second harmonics lies on the edge of the detectable spectral region of Si sensors, the short wavelength tail of the second harmonics around 1000 nm was detected by the camera. The focused beam spot size was estimated to be 4 μm from the spot size of the LCP component of the second-harmonic generation. The estimation assumes that the intensity of the second harmonics is proportional to the square of the intensity of the fundamental beam and equivalent amounts of RCP and LCP components are present at the focal plane. The intensity at the focus point was estimated using this beam spot size. In the loose-focus setup designed for comparatively showing the effect of SOI, an incident beam with a Gaussian divergence angle of 12.7 mrad was selected using a lens with a focal length of 200 mm. This condition results in a relatively large beam waist in the crystal and lower beam intensity. We therefore used a higher incident pulse energy of 3.7 μJ to generate the high harmonics. For the collimation of the high harmonics in this setup, a plano-convex lens with a focal length of 30 mm was used instead of the objective lens.

The OAM states of the harmonics were disentangled using an SLM (SLM-200, Santec). The optical setup for using the SLM is shown in fig. S5. The harmonics were collimated by the objective lens and were reflected by the SLM to apply an additional azimuthal phase to the harmonics. The reflected light beam was focused by a lens onto the CMOS camera to apply a Fourier transformation of the spatial patterns of the beam. To control the wavefront of the second, third, fourth, and fifth harmonics, corresponding phase patterns at wavelengths of 1000, 800, 600, and 480 nm were displayed on the SLM.

Analysis of color images

All color images of the harmonics are red-green-blue (RGB) color data. For visibility, the spatial images of the harmonics were normalized by their maximum values after subtraction of the background signal and then processed by a gamma correction with a Γ value of 2.2. The white balance of the RGB data was determined by multiplying the raw data obtained by the color CMOS camera, which had the same gain for each color sensor, by coefficients of 2.7, 1, and 3.15.

The fitting of the two-dimensional (2D) images in Fig. 3 was performed by considering Laguerre-Gaussian modes. The fitting functions were

$$S(\mathrm{\rho },\mathrm{\varphi })={\mid {\sum }_{p,l}{C}_{p,l}{U}_{p,l}(\mathrm{\rho })e^{\mathit{il}\mathrm{\varphi }}\mid }^2$$ (6)

with

$$U_{p,l}(\mathrm{\rho })=\sqrt{\frac{2p!}{\mathrm{\pi }(\mid l\mid +p)!}}\frac{1}{w_0}{\left(\frac{\mathrm{\rho }\sqrt{2}}{w_0}\right)}^{\mid l\mid }{L}_p^{\mid l\mid }\left(\frac{2{\mathrm{\rho }}^2}{w_0^2}\right)\text{exp}(-\frac{{\mathrm{\rho }}^2}{w_0^2})$$ (7)

where $L_p^l$ denotes the associated Laguerre polynomial. The indices p and l denote the radial mode and OAM mode, respectively. Polar coordinates around the center pixel in the images are defined by the radial position ρ and azimuthal angle φ. The fitting parameters were the complex coefficients C_p,l, beam waist w₀, and center pixel in the images. Two OAM modes and three radial modes were considered. The OAM modes were determined so as to match the two shown in Fig. 5. For the second-order LCP and third-order RCP components, only single OAM modes were considered for the fitting, as the interference structures between different OAM modes were absent in these observed images. The fitting was performed on 2D data obtained by averaging the three values for RGB colors. The relative intensities between the RGB colors were determined to be those that fit the experimental results. All parameters used to reconstruct Fig. 3 are shown in table S1.

Calculation for spin-orbit mixing of fundamental beam

Our estimation of the power exchange between RCP (l = 0) and LCP (l = −2) relies on the Ciattoni-Cincotti-Palma scheme (42) for a paraxial beam propagating along the optical axis of a uniaxial medium. In the related literature (42, 43), the mixing of spin angular momentum and OAM of light is assumed to happen at the beam waist of a Gaussian beam. On the contrary, the mixing starts from a plane off the beam waist in our experimental situation. We customized the derived formula to be applicable to our experimental situation as described below.

The incident electric field in the vacuum is assumed to be a Gaussian beam ${\mathbf{E}}^{\text{vac}}(r,\mathrm{\varphi },z,t)\mathfrak{=}\mathfrak{R}\left[{\mathbf{e}}_{+}{E}_{\text{Gb}}(r,\mathrm{\varphi },z)e^{-i\mathrm{\omega }t}\right],{\mathbf{e}}_{+}=(x+\mathit{iy})/\sqrt{2}$ propagating along the z axis, where (r, ϕ) is the radial coordinate for the x-y plane. The Gaussian envelope focused on z_f with waist w₀ is expressed as E_Gb(r, ϕ, z) = E₀[w₀/w(z − z_f)]e^{−r2/w2(z−z_f)}e^{ik₀(z−z_f)}e^{ik₀r2/2R(z−z_f)}e^{−iψ(z−z_f)}, where $w(z)=w_0\sqrt{1+{(z/z_{\mathrm{R}})}^2}$, $z_{\mathrm{R}}=\mathrm{\pi }{w}_0^2/\mathrm{\lambda },1/R(z)=z/(z^2+z_{\mathrm{R}}^2),\mathrm{\psi }(z)=\text{arctan}(z/z_{\mathrm{R}})$. We put the sample at 0 ≤ z < 2 mm. The complex field envelope is

$$E_{\text{Gb}}(r,\mathrm{\varphi },z=0)=\frac{{E\prime }_0{w}_0}{w(-z_{\mathrm{f}})}{e}^{-r^2/2s^2(-z_{\mathrm{f}})}$$

$$s^2(z)={[\frac{2}{w^2(z)}-\frac{i{k}_0}{R(z)}]}^{-1}=\frac{w_0^2}{2}(1+i\frac{z}{z_{\mathrm{R}}})$$ (8)

where the irrelevant phase not depending on r is absorbed in the phase of ${E\prime }_0$ . The complex waist squared function s²(z) becomes real-valued only at the focal point. According to equations 1 and 2 in (42), the field over the whole medium is determined by the Fourier transformation on the x-y plane at z = 0. The Fourier transformation of our Gaussian envelope is ${\tilde{E}}_{\text{Gb}}(k,z=0)=[{E\prime }_0{w}_0{s}^2(-z_{\mathrm{f}})/2\mathrm{\pi }w(-z_{\mathrm{f}})]e^{-k^2{s}^2(-z_{\mathrm{f}})/2}$. We obtain the absolute square as

$$\begin{array}{c}{\mid {\tilde{E}}_{\text{Gb}}(k,z=0)\mid }^2=\frac{{\mid {E\prime }_0\mid }^2{w}_0^2{\mid s^2(-z_{\mathrm{f}})\mid }^2}{{(2\mathrm{\pi })}^2{w}^2(-z_{\mathrm{f}})}{e}^{-k^2\mathfrak{R}[s^2(-z_{\mathrm{f}})]}\\ =\frac{{\mid {E\prime }_0\mid }^2{w}_0^2}{16{\mathrm{\pi }}^2}{e}^{-k^2{w}_0^2/2}\end{array}$$ (9)

One should note that the absolute square does not depend on z_f any more. The powers of RCP(+) and LCP(−) components, equations 22 and B4 in (42), is given as

$$\begin{array}{c}{W}_{\pm }(z)=\frac{1}{2}{W}_{\text{tot}}\pm 4{\mathrm{\pi }}^3{\displaystyle \underset{0}{\overset{\infty }{\int }}}\mathit{dk} k \text{cos}\left(\frac{z\mathrm{\Delta }}{2k_0{n}_o}{k}^2\right){\mid {\tilde{E}}_{\text{Gb}}(k,z=0)\mid }^2,\\ W_{\text{tot}}=8{\mathrm{\pi }}^3{\displaystyle \underset{0}{\overset{\infty }{\int }}}\mathrm{d}k k{\mid {\tilde{E}}_{\text{Gb}}(k,z=0)\mid }^2\end{array}$$ (10)

where $\mathrm{\Delta }=n_{\mathrm{e}}^2/n_{\mathrm{o}}^2-1$ . We obtain the powers of the RCP and LCP components as functions of z as

$$\frac{W_{\pm }(z)}{W_{\text{tot}}}=1\pm \frac{1}{1+{(z/L)}^2},W_{\text{tot}}=\frac{\mathrm{\pi }{\mid {E\prime }_0\mid }^2{w}_0^2}{2},L=\frac{k_0{n}_o{w}_0^2}{\mathrm{\Delta }}$$ (11)

Supplementary Materials

This PDF file includes:

Supplementary Text

Figs. S1 to S6

Table S1