# Purely geometric spin-orbit Laguerre-Gauss waveplates from 3D laser-nanostructured silica glass

Mufeeduzaman Chavilkkadan, Sergei Shevtsov, Peter Kazansky, Etienne

Brasselet

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Mufeeduzaman Chavilkkadan, Sergei Shevtsov, Peter Kazansky, Etienne Brasselet. Purely geometric spin-orbit Laguerre-Gauss waveplates from 3D laser-nanostructured silica glass. APL Photonics, 2025, 10 (4), 10.1063/5.0255787. [hal-05337359](https://hal.science/hal-05337359v1)

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Submitted on 29 Oct 2025

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# Purely geometric spin-orbit Laguerre-Gauss waveplates from 3D laser-nanostructured silica glass

Mufeeduzaman Chavilkkadan,<fnref n="1" />
Sergei Shevtsov,<fnref n="2" />
Peter Kazansky,<fnref n="2" /> and Etienne Brasselet<fnref n="1" />,ª)

## AFFILIATIONS

**Note:** This paper is part of the Special Topic on Angular Momentum of Light.

ª) Author to whom correspondence should be addressed: etienne.brasselet@u-bordeaux.fr

## ABSTRACT

We present the experimental demonstration of laser nanostructured silica modal waveplates that allow the generation of pure Laguerre-Gauss modes whose azimuthal order sign *l* is controlled by the spin state of an incident circularly polarized Gaussian beam of fixed radial order *p*. This overcomes the previous limitation of laser nanostructuring technology associated with the loss of modality for one of the two spin states due to unavoidable dynamic phase modulation from spatially varying retardance. It allows the spin-orbit nature of the fabricated optical element to be fully exploited. This is made possible by extending the usual 2D structuring strategy for laser-induced anisotropic optical properties to a 3D strategy. In addition, although the resulting modal waveplates are designed to operate for predetermined values of incident beam size and wavelength, we determine the extent to which the modal behavior remains of good quality when operating conditions deviate from the nominal specifications.

## I. INTRODUCTION

Beam shaping, whether related to the amplitude, phase, polarization state, or spectrum of light, is a fundamental requirement of optical technologies. When dealing with optical information and communications technologies, it is relevant to define the bits of information as optical modes forming an orthogonal basis, which implies to solve the light's propagation equation. Within the paraxial approximation, exact analytical solutions are obtained by solving the paraxial Helmholtz equation and one can find an infinite number of infinite-dimensional bases constituted of shape-invariant modal beams. This has been well-illustrated by Bandres and Guitiérrez-Vega who introduced the so-called Ince-Gauss beams.<fnref n="1" /> In fact, by using an elliptical coordinate system, they have obtained a continuum of bases that allows a continuous transition between two long-established families of modes, those of Hermite-Gauss and Laguerre-Gauss,<fnref n="2" /> corresponding to the Cartesian and cylindrical coordinate systems, respectively. With lasers, the creation of a modal

beam usually involves shaping a fundamental Gaussian beam with an extracavity device that modifies its structure in both amplitude and phase: spatial phase modulators. Nowadays, the most agile of these are electrically reconfigurable, pixel-based amplitude, and/or phase reflective masks that allow point-by-point digital control of optical field changes specific to the required mode. This digital holographic approach uses the first order of diffraction as the modal drop-port, with the other orders of diffraction distributing the part of the field that has not been used to shape the prescribed mode.³ It is also possible to use zeroth-order optical devices for modal beam shaping, using the polarization state of the light as a drop-port as proposed theoretically in 2017,⁴ which has the practical advantage of preserving an in-line optical configuration. This implies the use of doubly inhomogeneous anisotropic optical elements, for which both the slow axis orientation angle (ψ) and the birefringent phase retardation (Δ) are spatially varying. Several experimental demonstrations have been reported since then, using the Laguerre-Gauss basis as a prototypical example. Two types of technological routes

<footnote n="1">¹ Université de Bordeaux, CNRS, Laboratoire Ondes et Matière d'Aquitaine, F-33400 Talence, France</footnote>

<footnote n="2">²Optoelectronics Research Centre, University of Southampton, Southampton SO17 1BJ, United Kingdom</footnote>
were explored, based either on metasurfaces,⁵⁻⁷ or on laser nanos-structured silica glasses.⁸⁻¹⁰ It turns out that modal silica devices are the most successful to date, with clear apertures of macroscopic dimensions (several mm), robustness to high photon flux, and polychromatic performances enabling the modal shaping of supercontinuum lasers.¹⁰

However, a major limitation of silica modal beam shapers remains. Indeed, the optical spin-orbit process inherent in these devices has not yet been fully exploited, with a mode being obtained for only one of the two incident circular polarization states. In turn, this prevents the generation of a Laguerre-Gauss beam—uniquely defined by its azimuthal order $l \in \mathbb{Z}$ and radial order $p \in \mathbb{N}$—with spin-controlled sign of $l$ that is associated with the amount of orbital angular momentum carried by photon.¹¹ The technological bottleneck lies in the fact that the spatial modulation of the birefringent phase delay is accompanied by a spatial modulation of the dynamic phase.⁹ This leads to an aspheric shaping of the wavefront that can only be corrected by design for one of the two incident spin states.¹⁰ Building on the work of Pancharatnam in the 1950s,¹² here, we use a 3D design approach rather than the usual 2D strategy, from which we design, fabricate, and characterize spin-orbit Laguerre-Gauss beam shapers having a flat dynamic phase spatial profile over the entire clear aperture of the device. In this way, the spatial modulation of the amplitude and the wavefront of the incident fundamental Gaussian beam are decoupled by design, which gives access to spin-controlled azimuthal order.

The proposed approach is presented in Sec. II. The experimental demonstration is first reported using a set of distinct waveplates and then extended to a monolithic design in Sec. III. In Sec. IV, we go beyond the generation of single Laguerre-Gaussian modes and show that one can obtain polarization textures hosting Stokes skyrmions exhibiting additional features compared to that reported using non-monotonous radial birefringent profile.¹³ Finally, we gather our concluding remarks in Sec. V.

## II. FLAT DYNAMIC PHASE APPROACH

### A. Principle

From the work by Pancharatnam,¹² we learn that an effective waveplate with arbitrary birefringent phase retardation ($\Delta_{eff}$) ranging from 0 to $2\pi$ can be obtained by placing half-wave plate in between two quarter-wave plates whose slow axis are parallel to each other. In fact, by rotating the half-wave plate around its normal, one obtains an effective waveplate having its slow axis orientation ($\psi_{eff}$) at 45° from that of the quarter-wave plates and a retardance that continuously varies with the angle between the slow axis of the half-wave ($\psi_H$) and quarter-wave plates ($\psi_Q$), as depicted in Fig. 1, namely,

$$ \psi_{\text{eff}} = \psi_{\text{Q}} + \frac{\pi}{4}, \qquad (1) $$

$$ \Delta_{\text{eff}} = 2\pi + 4(\psi_{\text{H}} - \psi_{\text{Q}}). \qquad (2) $$

A few years ago, it was theoretically proposed¹⁴ to use this concept to design doubly inhomogeneous effective waveplates endowed with arbitrary spatial distribution of their slow axis orientation and birefringent phase retardation. The idea is to simply use a

<figure><img src="image_2.png" /><figcaption>FIG. 1. Illustration of how an effective waveplate with arbitrary birefringent phase retardation $\Delta_{eff} = 2\pi + 4\delta$ can be obtained by cascading three waveplates, the first and the third being quarter-wave plates (QWPs) and the second being half-wave plate (HWP) following the work by Pancharatnam.¹²</figcaption></figure>

space-variant half-wave plate sandwiched between two identical quarter-wave plates according to Eqs. (1) and (2). This allows one to consider the realization of waveplates with shaped optical anisotropic properties that would otherwise be difficult to obtain. In particular, if we note that each of the waveplates has uniform retardance, the overall dynamic phase turns out to be uniform as well.

In the present context of Laguerre-Gauss modal beam shaping from a circularly polarized incident fundamental Gaussian beam, and restricting our demonstration to modes with fundamental radial order $p = 0$, the effective design requirement is obtained from the derivation detailed in Ref. 10. This gives

$$ \psi_{\text{eff}}(\phi) = \frac{n}{2}\phi + \psi_0 \qquad (3) $$

and

$$ \sin\left[\frac{\Delta_{\text{eff}}(r)}{2}\right] = \left(\frac{r\sqrt{2}}{w_0}\right)^{|n|} \exp\left[|n|\left(\frac{1}{2} - \frac{r^2}{w_0^2}\right)\right], \qquad (4) $$

where $(r, \phi)$ refers to the polar coordinate in the transverse plane, $n$ is an integer, $\psi_0$ is a constant, and $w_0$ defines the incident beam waist radius to use in order to maximize the modal output power fraction. Denoting $\sigma = \pm 1$ as the helicity of the incident circularly polarized Gaussian beam, the output contracircularly polarized component is a Laguerre-Gaussian mode of order $(l, p) = (\sigma n, 0)$.

The design for the quarter-wave plates is obtained from Eqs. (1) and (3), which gives

$$ \psi_{\text{Q}}(\phi) = \frac{n}{2}\phi - \frac{\pi}{4} + \psi_0, \qquad (5) $$

while the design of the half-wave plate is given by the combination of Eqs. (2) and (4). We note that the arcsin function in the expression of $\Delta_{eff}(r)$ obtained from Eq. (4) makes it possible to choose between a monotonic or non-monotonic radial dependence for $\psi_H$. Here, in contrast to previous work,⁹,¹⁰ we opt for a monotonic design, which allows avoiding a kink at $r_0 = w_0/\sqrt{2}$, thus favoring a more accurate fabrication. We will also see in Sec. IV that such a choice also brings added value when considering the generation of skyrmionic polarization textures. This gives

$$ \psi_H(r < r_0, \phi) = \frac{n}{2}\phi + \frac{1}{2}\arcsin[f_n(r)] + \frac{\pi}{4} + \psi_0 \qquad (6) $$
and

$$ \psi_H(r > r_0, \phi) = \frac{n}{2}\phi - \frac{1}{2}\arcsin[f_n(r)] + \frac{3\pi}{4} + \psi_0, \quad (7) $$

with

$$ f_n(r) = \left( \frac{r\sqrt{2}}{w_0} \right)^{|n|} \exp \left[ |n| \left( \frac{1}{2} - \frac{r^2}{w_0^2} \right) \right]. \quad (8) $$

## B. Femtosecond laser processing of silica glass

The space-variant anisotropic optical elements were fabricated by femtosecond laser processing of synthetic silica glass substrates, in which birefringent structures of type X with low scattering losses and polarization controlled slow axis orientation can be written.¹⁵ Two types of optical elements have been used in this work. First, in Sec. II C, we report experimentally on the proposal presented in Ref. 14 by using a set of three 2D structured waveplates corresponding to a commercial solution (Workshop of Photonics). Then, in Sec. III, we propose and implement a 3D structuring approach that we developed to obtain monolithic optical elements, which provides a more robust and user-friendly solution.

## C. Proof-of-principle demonstration: Cascading scheme

Following the principle shown in Fig. 1, we have combined three structured waveplates in a series, each of which satisfies Eqs. (5)–(7), respectively, with $w_0 = 1$ mm and a clear aperture of 6 mm diameter, for a nominal wavelength $\lambda_0 = 532$ nm. To minimize multiple reflections from the air-glass interfaces of the stacked assembly, a refractive index matching liquid is added between the waveplates; see Fig. 2(a). In addition, the demonstration is performed for $n = 2$, which corresponds to axisymmetric structured waveplates, thus limiting the mutual centering of the three individual optical elements to a 2D translation alignment problem. The experimental results are summarized in Fig. 2, where the structural assessment of the individual waveplates is shown in Figs. 2(b) and 2(c), where the maps of $\psi_Q$ and $\psi_H$ are determined by standard Stokes polarimetry. It consists of imaging the plates illuminated by a circularly polarized probe beam with $\lambda_{\text{probe}}$ wavelength and helicity $\sigma_{\text{probe}}$. Then, the reduced Stokes parameters $s_1$ and $s_2$ are evaluated four intensity measurements made by placing a linear polarizer oriented along directions oriented at (0°, ±45°, 90°) from the x axis after the sample.¹⁶ This allows determining the slow axis orientation retrieved from $\psi = (1/2) \arctan(s_2/s_1) - \text{sign}(\sin \Delta_{\text{probe}})\sigma_{\text{probe}}\pi/4$, here choosing $\lambda_{\text{probe}} = 633$ nm and $\sigma_{\text{probe}} = +1$. Note that all measured reduced Stokes parameters are renormalized according to $s_i \rightarrow s_i / (\sum_{i=1,2,3} s_i^2)^{1/2}$ in order to ensure that the experimental $s = (s_1, s_2, s_3)$ is a unit vector.

The effective behavior of the combination of waveplates was then tested by observation between crossed linear polarizers (XPOL) and crossed circular polarizers (XCPOL); see Figs. 2(d) and 2(e). The observed untwisted XPOL intensity pattern with fourfold rotational symmetry supports an effective left/right structural symmetry, while the doughnut-shaped XCPOL intensity profile qualitatively agrees with the expected intensity profile of a Laguerre-Gauss beam with $l = \pm 2$ and $p = 0$. Nevertheless, the imperfections clearly visible in

<figure><img src="image_3.png" /><figcaption>FIG. 2. (a) Sketch of three stacked 2D structured waveplates behaving as a modal waveplate. Panels (b) and (c): experimental distribution of the slow axis orientation (hue color) of the disk-shaped quarter-wave plates and square-shaped half-wave plate with $n = 2$ and $\psi_0 = \pi/4$. The brightness refers to retardance, so the black area refers to unstructured silica. Insets: simulations. Panels (d) and (e): experimental XPOL and XCPOL images of the effective waveplate observed under spatially extended illumination from a halogen lamp spectrally filtered at 532 wavelength. The white arrows refer to the configuration used for the polarizer and the analyzer. Note that the orientation of the square-shaped half-wave plate is unimportant since $n = 2$ implies axisymmetry. Scale bars: 2 mm.</figcaption></figure>

the central part of the device, as well as the fact that the extension to $n \neq 2$ is accompanied by additional constraints related to the required rotational alignment, have motivated the development of a more robust solution. To this end, the idea is to switch from stacking individual 2D-structured waveplates to a single 3D-structured monolithic optical element, as discussed in Sec. III.

## III. MONOLITHIC SPIN-ORBIT MODAL WAVEPLATES

### A. Design and fabrication

Laser-induced nanoporous birefringent structures of type X were imprinted using a mode-locked Yb:KGW ultrafast laser system (Pharos, Light Conversion Ltd.), operating at 1030 nm with a variable repetition rate from 1 kHz to 1 MHz and a pulse duration ranging from 190 fs to 10 ps. The beam was focused by an aspheric lens with a numerical aperture of 0.16 into the sample, which was mounted on a 3D translation stage (Aerotech Ltd.) to control its position and scanning speed. A raster scanning geometry was used to ensure smooth modifications. The slow axis of the type X structures, which is perpendicular to the polarization of the incident writing light, was controlled by adjusting the polarization of the incident light using a combination of a linear polarizer, an electro-optic modulator, and a quarter-wave plate.

Monolithic modal waveplates were fabricated by structuring the orientation of the laser-induced slow axis in a silica substrate along the three spatial directions using a 12-layer design, as shown in Fig. 3. This choice results from the fact that birefringent modifications of type X (nanopores) transform into another type of modification (nanogratings) exhibiting substantial light scattering above an energy threshold. In turn, this limits the maximum
<figure><img src="image_4.png" /><figcaption>**FIG. 3.** Left panel: illustration of the monolithic design (top) and image of the substrate hosting three monolithic 3D structured waveplates with $n = (-1, -2, -3)$ and $\psi_0 = 0$. Illumination comes from the sample side, making the structures easily visible by light scattering, while the bright-field transmission image through the main faces of the sample shows barely perceptible scattering (not shown). Right panel: XPOL (top) and XCPOL (bottom) images of the three optical elements.</figcaption></figure>

retardance that can be imprinted using type X modification to ~50 nm.¹⁷ In addition, from a fabrication efficiency perspective, minimizing the number of layers is preferable, resulting in a total of 12 layers with 44.3 nm retardance per layer, which corresponds to a birefringent phase retardation of $\pi/6$ for the nominal wavelength $\lambda_0 = 532$ nm. The first three and last three layers have a slow axis distribution given by Eq. (5), while the slow axis of the six central layers follows a distribution given by Eqs. (6) and (7).

In practice, the spacing between the closest points of two consecutive birefringent layers is empirically chosen to be 50 µm in order to prevent any impact on the previously written birefringent layer while writing the next one above it. Indeed, the transformation of the structures of type X (nanopores associated with low scattering losses) into structures of type II (nanogratings associated with high scattering losses)¹⁷ must be avoided. If the layers are too close, this risk increases as we are operating the laser processing close to the condition of maximum retardance for the structures of type X. This gives a periodic layered arrangement along the z axis with a period of $\Lambda \approx 94$ µm; see Fig. 3. In addition, we checked that the effects of spherical aberrations on birefringence and structure thickness, and consequently, on retardance, are negligible when the writing depth is less than 1.3 mm, which is the maximum layer depth of the fabricated optical elements.

Three spin-orbit modal waveplates with $R = 1$ mm radius were produced, using the parameters $w_0 = 500$ µm and $n = (-1, -2, -3)$; see the bright field image of the produced optical elements in Fig. 3. In addition, the XPOL and XCPOL pictures highlight the expected left/right symmetry and axisymmetry of the effective structure, respectively.

## B. Structural characterization

The quantitative assessment of the structural features of the fabricated modal waveplates was done experimentally by polarimetric analysis, here taking care of using $\lambda_{\text{probe}} = \lambda_0$ since the effective behavior of the 3D structures waveplates depends on

<figure><img src="image_5.png" /><figcaption>**FIG. 4.** Experimental (black curves) and theoretical (red curves) profiles of the azimuthal dependence of the slow axis orientation angle $\psi_{\text{eff}}(\phi)$ evaluated at $r = R/4$ (a) and of the radial dependence of the azimuth-averaged birefringent phase retardation $\Delta_{\text{eff}}(r)$ (b). The data for increasing charges $n = (-1, -2, -3)$ are shown.</figcaption></figure>

wavelength, as discussed in Sec. III D. The results are summarized in Fig. 4, which shows the experimental (black curves) and simulated (red curves) azimuthal profiles of $\psi_{\text{eff}}$ [panel (a)] and radial profile of $\Delta_{\text{eff}}$ [panel (b)]. On the one hand, $\psi_{\text{eff}}(\phi)$ is evaluated according to the method mentioned in Sec. II C. On the other hand, $\Delta_{\text{eff}}(r)$ is extracted from the additional measurement of the third reduced Stokes parameter $s_3$, which is evaluated from the intensity measurements of the cocircularly and contracircularly polarized components,¹⁶ using $\text{mod}(\Delta_{\text{eff}}, \pi) = \arccos(\sigma_{\text{probe}} s_3)$. This was done by the azimuth average of $s_3$ in order to evaluate $\Delta_{\text{eff}}(r < r_0) = \arccos[\sigma_{\text{probe}}(s_3)\phi(r)]$ and $\Delta_{\text{eff}}(r > r_0) = 2\pi - \arccos[\sigma_{\text{probe}}(s_3)\phi(r)]$, recalling that $\Delta_{\text{eff}}(r_0) = \pi$ by design. The results collected in Fig. 4 allow us to conclude that the monolithic fabrication process is of satisfactory quality. Next, we report on the modal beam shaping characteristics obtained by studying the propagation behavior of the beam generated under the XCPOL configuration.

## C. Spin-controlled modal beam generation

The modal property of the generated contracircularly polarized light field was tested by recording its intensity profile during its propagation along the z axis. As shown in Fig. 5, the intensity profile is propagation invariant up to a stretching factor and independent of the incident spin state $\sigma = \pm 1$, as expected from the theory of Laguerre-Gauss modes. In addition, the spin-controlled azimuthal index $l = \sigma n$ is verified by interfering the generated beam with a coherent Gaussian reference beam. This is illustrated in Fig. 6, where the $|n|$-fork intensity pattern is reversed as the handedness of the incident polarization state is flipped.

## D. Robustness to nominal specifications

Pure modal shaping for $p = 0$ is not altered when the incident beam waist radius $w_{\text{in}}$ departs from the nominal value $w_0$ as long as the nominal wavelength is used but is associated with another waist value given by $w_{\text{out}} = w_0/\sqrt{|l| + w_0^2/w_{\text{in}}^2}$ as is the case for 2D structured waveplates with a non-monotonous retardance
<figure><img src="image_6.png" /><figcaption>FIG. 5. Experimental and simulated normalized azimuth-average intensity distribution in the (z, r) plane for right-handed ($\sigma = -1$) and left-handed ($\sigma = +1$) circularly polarized incident Gaussian beam with beam waist radius $w_0$ and wavelength $\lambda_0$. The data were recorded by moving a fixed system made of lens, a circular polarizer, and a camera, along the z axis (dashed line) by steps of 1 cm over a distance of 1.3 m. Vertical scale bar: 1 mm. Horizontal scale bar: 30 cm.</figcaption></figure>

profile.¹⁰ However, here, in contrast to previous work, the structuring of the anisotropic optical properties of the material is no longer purely transverse but also varies longitudinally. This makes the optical element departing from its modal beam shaping behavior for $\lambda \neq \lambda_0$. This is qualitatively illustrated in Fig. 7, which shows white light XPOL imaging of the monolithic devices. The twisted characteristics observed, which depend on the facet used as the incident facet, are in striking contrast to the untwisted XPOL intensity patterns at $\lambda = \lambda_0$ shown in Fig. 3. This is due to the piecewise twisted nature of the structured medium along the z axis for $\lambda \neq \lambda_0$, whereas the effective structured medium is untwisted for $\lambda = \lambda_0$; see Eq. (3).

<figure><img src="image_7.png" /><figcaption>FIG. 6. Experimental forked interference intensity patterns between the modal field at a distance $z = 0.5$ m from the modal waveplate for right-handed ($\sigma = -1$) and left-handed ($\sigma = +1$) circularly polarized incident Gaussian beam with beam waist radius $w_0$ and wavelength $\lambda_0$ (see Fig. 6), and an obliquely Gaussian reference beam.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>FIG. 7. XPOL images of the three monolithic waveplates for white light natural illumination from the halogen lamp of the used optical microscope and for both orientation of the waveplates with respect to the incident light. Insets: simulated XPOL patterns according to Eq. (12), where the row and column vectors ($1, i\sigma$) are replaced by (0, 1) and (1, 0), respectively, using $\lambda = 633, 532$, and 450 nm for the red, green, and blue RGB channels, respectively.</figcaption></figure>

Therefore, a given incident circularly polarized component experiences a structural twist reversal when the sample is turned upside down.

The loss of modality is numerically evaluated at $z = 0$ by recalling the Laguerre-Gauss modal decomposition of an arbitrary paraxial beam,

$$ \tilde{E}_{\text{out}}(r, \phi) = \sum_{l \in \mathbb{Z}, p \in \mathbb{N}} c_{l,p} \tilde{E}_{l,p}(r, \phi; w_{\text{out}}), \quad (9) $$

where the tilde symbol refers to normalized fields that satisfy $\iint |\tilde{E}|^2 rdrd\phi = 1$ and $w_{\text{out}}$ is the output beam waist radius, which is a free parameter. However, in the search of maximal modal purity, the modal decomposition must be made using the value of $w_{\text{out}}$ that maximizes the modal power fraction $|c_{l,0}|^2$ carried by the targeted mode with indices $(l, 0)$, namely,

$$ |c_{l,0}|^2 = \max_{w_{\text{out}}} \left[ \left| \iint \tilde{E}_{\text{out}}(r, \phi) \tilde{E}_{l,0}^*(r, \phi; w_{\text{out}}) rdrd\phi \right|^2 \right], \quad (10) $$

where

$$ \tilde{E}_{l,0}(r, \phi; w_{\text{out}}) = \frac{1}{w_{\text{out}}} \sqrt{\frac{2^{|l|+1}}{\pi^{|l|!}}} \left(\frac{r}{w_{\text{out}}}\right)^{|l|} \exp\left(-\frac{r^2}{w_{\text{out}}^2} + il\phi\right) \quad (11) $$

and

$$ \tilde{E}_{\text{out}}(r, \phi) = \frac{1}{w_{\text{in}}} \sqrt{\frac{1}{2\pi} \left(\frac{1}{i\sigma}\right)^{\mathrm{T}}} \left[ \hat{J}_{\mathrm{Q}} \hat{J}_{\mathrm{H}} \hat{J}_{\mathrm{Q}} \left(\frac{1}{i\sigma}\right) \right] \exp\left(-\frac{r^2}{w_{\text{in}}^2}\right), \quad (12) $$

where $(\cdot)^{\mathrm{T}}$ is the transpose operator, and $\hat{J}_{\mathrm{X}=\mathrm{Q}, \mathrm{H}}$ are the Jones matrices associated with the structured quarter-wave and half-wave
<figure><img src="image_9.png" /><figcaption>FIG. 8. (a) Modal purity as a function of the normalized incident beam waist for the nominal wavelength $\lambda_0 = 532$ nm and a selection of wavelengths, for $l = \pm 1$. (b) Corresponding normalized optimal output beam waist. The dashed line refers to the analytical expression at nominal wavelength (see the text). (c) Modal purity spectrum for the nominal beam waist $w_0 = 500$ µm for $|l| = (1, 2, 3)$. The dotted lines refer to the modal purity of standard q-plates, given by the expression $\frac{2^{|l|} |l|^2}{1+|l|} \left( \frac{1+|l|}{2+|l|} \right)^{2+|l|} \frac{|l|^2 (|l|/2)}{\Gamma(1+|l|)}$, where $\Gamma(\cdot)$ is the gamma function.¹⁸ (d) Corresponding normalized optimal output beam waist. The dotted line refers to the nominal wavelength.</figcaption></figure>

plates for nominal wavelength $\lambda_0$, respectively, but used at arbitrary wavelength $\lambda$, namely,

$$ \hat{f}_X = \begin{pmatrix} \cos \frac{\Delta_X}{2} + i \sin \frac{\Delta_X}{2} \cos 2\psi_X & i \sin \frac{\Delta_X}{2} \sin 2\psi_X \\ i \sin \frac{\Delta_X}{2} \sin 2\psi_X & \cos \frac{\Delta_X}{2} - i \sin \frac{\Delta_X}{2} \cos 2\psi_X \end{pmatrix}, \quad (13) $$

where on neglecting the dispersion of the birefringence, the dependence of the retardance on the wavelength is accounted according to

$$ \Delta_Q(\lambda) = \frac{\pi \lambda_0}{2 \lambda}, \quad (14) $$

$$ \Delta_H(\lambda) = \pi \frac{\lambda_0}{\lambda} \quad (15) $$

and where $\psi_Q(\phi)$ is given by Eq. (5) and $\psi_H(r, \phi)$ is given by Eqs. (6)-(8).

The loss of modality as a function of beam waist is shown in Figs. 8(a) and 8(b) for a set of wavelengths spanning the entire visible range, noting that the modal purity does not depend on the handedness of the incident circular polarization. This indicates that deviation from the nominal incident beam waist is substantially detrimental to a broadband modal behavior, typically beyond the range $\lambda_0 \pm 100$ nm. However, for $w_{in} = w_0$, one obtains a modal behavior over the entire visible domain whose purity exceeds that obtained from standard 2D structured waveplates with a slow axis orientation given by $\psi = \pm(|l|/2)\phi + \psi_0$ and uniform retardance (i.e., q-plates¹⁹), as shown in Figs. 8(c) and 8(d).

Experimentally, polychromatic beam shaping is investigated by using a circularly polarized supercontinuum incident laser beam prepared with a waist radius $w_0$ at $\lambda_0$ (keeping in mind that there is a few percent relative variation of the beam waist across the visible domain), and the waveplates are placed between achromatic crossed circular polarizers. The experimental intensity profiles at the output of the waveplates and in the far field are collected in Fig. 9.

<figure><img src="image_10.png" /><figcaption>FIG. 9. Supercontinuum modal beam shaping. The experimental intensity profiles of the generated beams are shown at the output of the waveplates ($z = 0$, data obtained by imaging the exit facet of the sample) and in the far field ($z = \infty$, data obtained by optical Fourier transform using a lens), for $\sigma = -1$ (upper row) and for $\sigma = +1$ (bottom row). Leftmost inset: incident beam intensity profile. Other insets: simulations; see the text for details. Scale bars: 0.5 mm.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>FIG. 10. Supercontinuum modal beam shaping at intermediate propagation distances. (a) The experimental intensity profiles of the generated beams are shown after a 50 cm propagation distance for $\sigma = -1$ (upper row) and for $\sigma = +1$ (bottom row). Scale bar: 1 mm. (b) Calculated radial dependence of the phase of the output field given by $\Phi_{\text{out}}(r) = \arg[\tilde{E}_{\text{out}}(r, 0)] - \arg[\tilde{E}_{\text{out}}(0, 0)]$ based on Eq. (12) for the three selected RBG channels. Insets: illustration of the spin-dependent converging/diverging effect when $\lambda \neq \lambda_0$, for $l = \pm 1$. (c) Calculated spin-dependent propagation dependence of the beam waist radius for the three selected RGB channels for $l = \pm 1$.</figcaption></figure>

The observed dependence on $\sigma$ is only marginal both for $z = 0$ and $z = \infty$, as supported by the simulations for $l = \pm 3$ (see the insets) performed by constructing polychromatic RGB images obtained using $\lambda = 633, 532, \text{ and } 450$ nm for the red, green, and blue channels, respectively. The simulations at $z = 0$ directly refer to Eq. (12) while the far field is calculated by Fourier transform of the field obtained at $z = 0$. The color dispersion in the far field along the radial coordinate basically comes from the Gaussian divergence of each constituting monochromatic beam that scales as $\lambda$, hence giving rise to a reddish (bluish) beam on the outside (inside) in the far field.

In contrast, the color appearance of the shaped polychromatic beam is spin-dependent at intermediate distances, as illustrated in Fig. 10(a) for $z = 50$ cm. Interestingly, the color dispersion is almost completely compensated for one of the two spin states. The spin-dependent visual appearance of the beam is reversed by turning the sample upside-down (not shown here), which echoes the XPOL observations reported in Fig. 7. Such a chromatic behavior is due to the twisted structural nature of the anisotropic medium when $\lambda \neq \lambda_0$, as discussed in Sec. III D, which imparts the output field with an extra spin-dependent phase profile that depends on the wavelength. This is illustrated in Fig. 10(b) that shows the calculated radial profile of the phase of the output field given by Eq. (12) for the three selected RGB channels. The obtained aspheric dynamic phase profile with a spin-dependent sign of the wavefront curvature when $\lambda \neq \lambda_0$ explains the color appearance, which depends both on the incident spin state and the propagation distance. This is illustrated for $l = \pm 1$ in Fig. 10(c) that shows the dependence of the beam waist radius $w(z)$ vs $z$ for the three selected RGB channels. There, $w(z)$ is evaluated by fitting every calculated intensity profile obtained by using the Fourier propagation method with a Laguerre-Gauss profile.

It should be noted that such a spin-dependent chromatic effect, which depends on the propagation distance, is no longer relevant in the far field, where only the angular distribution of the wavevectors that constitute the field counts. In this case, modifying the wavefront by changing only the sign of the curvature has no effect on

the intensity distribution in the far field. Finally, we recall that the present analysis neglects the dispersion of the birefringence, which would have to be taken into account for a refined analysis of the propagation characteristics of shaped polychromatic beams.

Finally, we propose a geometric analysis of the spin/wavelength-dependent phase shaping phenomenon by looking at the trace of the output-reduced Stokes vector $\mathbf{s} = (s_1, s_2, s_3)$ defined as

$$ \mathbf{s} = \frac{(|E_x|^2 - |E_y|^2, 2\text{Re}(E_x^* E_y), 2\text{Im}(E_x^* E_y))}{|E_x|^2 + |E_y|^2} \quad (16) $$

<figure><img src="image_12.png" /><figcaption>FIG. 11. Trace of the output reduced Stokes vector on the Poincaré sphere of polarization along the half-line given by $(r, \phi = 0)$ for a modal plate with $|n| = 1$, for an incident circularly polarized beam with helicity $\sigma = +1$ (a) and $\sigma = -1$ (b), for $\lambda = (450, 502, 532, 562, 633)$ nm. The starting point at $r = 0$ corresponds to the north/south pole for $\sigma = \pm 1$, and the black arrows indicate the way trajectories start. The circle markers refers to $r = w_0$ and the square markers refer to $r = \infty$, noting that the asymptotic behavior is rapidly reached (no difference is visible already at $r = 4w_0$).</figcaption></figure>
on the Poincaré sphere as a function of the distance from the origin for the nominal wavelength and four detuned ones, for $\sigma = \pm 1$, as shown in Fig. 11. All traces end on the great circle defined by $s_2 = 0$, as expected from an effective full-wave plate at a nominal wavelength having its slow axis oriented at $-\pi/4$ from the x axis for modal plates with $\psi_0 = 0$; see Eqs. (5) and (7). Such geometrical description gives an alternative way to evaluate to the phase profiles reported in Fig. 10(b). Indeed, considering a given wavelength $\lambda$, $|\Phi_{\text{out}}(r)|$ equals half the solid angle subtended by the two geodesic arcs joining the north pole to the south pole of the Poincaré sphere (i.e., half great circles) that pass through the points defined by $(r, \lambda_0)$ and $(r, \lambda)$, respectively.

## IV. MODAL SUPERPOSITIONS AND SKYRMIONS

As discussed in previous studies, spin-orbit optical element with both inhomogeneous optical axis orientation and retardance provide a way to generate polarization-controlled Stokes optical skyrmions.13,20 Here, we show how laser nanostructured silica waveplates with monotonous design for the radial dependence of the retardance is an add-on for skyrmionic beam shaping. The experimental detection of the output Stokes textures is carried out at a wavelength of 532 nm. The chosen color coding associates a unique color to every point of the Poincaré sphere of polarization, as depicted in Fig. 12, which helps visualizing how the Stokes vector field covers it. As shown in Fig. 12, one obtains

<figure><img src="image_13.png" /><figcaption>FIG. 12. Experimental Stokes textures of the total field at the output of the three monolithic modal waveplate for right-handed ($\sigma = -1$) and left-handed ($\sigma = +1$) circularly polarized incident Gaussian beam with beam waist radius $w \sim 3w_0$ and wavelength $\lambda_0$. The dashed circles refer to the disk of radius $r^*$ that defines an $|n|$th-order skyrmion of charge $N^* = \sigma n$ (note that $r^* = r_0$ at $z = 0$). Insets: experimental data from previous work using a non-monotonous design for the radial profile of the retardance,13 but for $n = +1$.</figcaption></figure>

skyrmionium,21 which corresponds to a Stokes vector, making a full rotation over a great circle of the Poincaré sphere passing through the poles ($s_3 = \pm 1$) along any straight line running from the origin to the infinity. Its polarity is controlled by the incident helicity $\sigma$; its vorticity is given by $2\sigma n$; and its skyrm number given by

$$N = \frac{1}{4\pi} \iint \mathbf{s} \cdot \left( \frac{\partial \mathbf{s}}{\partial x} \times \frac{\partial \mathbf{s}}{\partial y} \right) dxdy \quad (17)$$

is zero, which is verified experimentally up to a typical precision of ~1% in all six situations that correspond to $\sigma = \pm 1$ and $n = (-1, -2, -3)$. This contrasts with the results obtained with non-monotonous designs,13 as shown in the insets of Fig. 12 for the first-order skyrmioniums, where $N = 0$ results from the Stokes vector making a $\pi$ rotation back and forth starting from one of the two poles of the Poincaré sphere. However, $|n|$th-order skyrmions with Skyrmme number $N^* = \sigma n$ are produced in the central part of the output beam defined as a disk of radius $r^*$ are produced in both cases, indicated by the dashed circles in Fig. 12, which is also ascertained experimentally up to ~1% by evaluating Eq. (17) over the disk of radius $r^*$.

## V. SUMMARY AND CONCLUDING REMARKS

We have experimentally demonstrated the possibility to overcome a major, previously unresolved limitation related to the mandatory use of a fixed incident polarization state in the realization of Laguerre-Gauss modal waveplates fabricated via femtosecond laser nanostructuring of silica glass substrates.10 This is achieved by structuring light-induced anisotropic optical properties in all three spatial dimensions, exploiting a long-standing trick originally proposed by Pancharatnam to realize effective waveplates with arbitrary values of anisotropic retardation.12 This makes it possible to obtain Laguerre-Gauss modes with the spin-controlled azimuthal index, which now allows taking full advantage of the inherent spin-orbit optical nature of the designed waveplates. The resulting optical elements have low absorption and scattering losses, are robust to high photon fluxes, offer large clear apertures, are easy to use, and exhibit quasi-modal broadband behavior that has been numerically studied over the visible domain. The present proof-of-principle demonstration between crossed circular polarizers can be further simplified by adding a quarter-wave plate on either side of the monolithic design, thus enabling use between simple crossed linear polarizers. In addition, we discussed how the proposed design allows considering the realization of unitary spin-orbit skyrmionic waveplates with improved functionalities with respect to recent attempts using the same material processing technology.13

Importantly, the generic nature of the proposed approach makes it directly applicable not only to arbitrary wavelength and arbitrary spatial phase modulation of the complex amplitude of an incident field but also to any other type of technology capable of structuring in 2D the orientation of the optical axis of a naturally or artificially uniaxial medium, whether obtained naturally or artificially. In particular, this paves the way for the use of multilayer liquid crystal photopatterning technologies, previously developed in the context of the achromatization of diffractive polarization gratings,22 for the design of arbitrary doubly inhomogeneous monolithic spin-orbit waveplates.
Finally, we note that the possible production of robust com-
plex amplitude optical elements to generate arbitrary superposition
of modes can be considered for various kinds of applications for
which compact setup are desirable. For instance, let us mention
the development of optical tweezers with enhanced trapping fea-
tures owing to the use of Laguerre-Gauss modes<sup>23</sup> of superpositions
of them,<sup>24</sup> and laser material processing using Bessel<sup>25</sup> and Airy<sup>26</sup>
modal beams.

ACKNOWLEDGMENTS

M.C. and E.B. acknowledge the support of IdEx University of
Bordeaux and its Grand Research Program GPR LIGHT, Agence
Nationale de la Recherche (Grant No. ANR-21-CE24-0014-01), and
thank Delphine Coursault for the discussions. P.K. acknowledges
support from the European Research Council (ENIGMA, Grant No.
789116).

AUTHOR DECLARATIONS

Conflict of Interests

The authors have no conflicts to disclose.

Author Contributions

Mufeeduzaman Chavilkkadan: Formal analysis (equal); Investiga-
tion (equal); Validation (equal); Writing – review & editing (equal).

**Sergei Shevtsov:** Investigation (supporting); Resources (equal); Writing – review & editing (equal). **Peter Kazansky:** Resources (equal); Supervision (supporting); Writing – review & editing (equal). **Etienne Brasselet:** Conceptualization (equal); Formal analysis (equal); Methodology (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available
from the corresponding author upon reasonable request.

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