Control over circularly polarized light and orbital angular momentum (OAM) beams is of great importance for applications ranging from super-resolution microscopy1 and optical trapping2 to chiral biospectroscopy3 and spatial multiplexing in optical communication.4 Currently, most applications rely on free-space or flat optics with metasurfaces offering the most advanced control over the phase, amplitude, and polarization of the beam.5^,6 However, it was found that spin and OAM degrees of freedom can also be manipulated within an optical fiber by imparting an axial twist onto the waveguide. Crucially, the resulting structure becomes chiral, therefore breaking the symmetry between modes of left- and right-handed circular polarization (LCP/RCP) and modes with positive or negative OAM—states that are inevitably degenerate in an untwisted waveguide.

Clever use of this symmetry breaking led to numerous applications that are based on the circular birefringence and circular dichroism (CD) caused by the twist. Examples include circular polarization filtering,7^,8 twist and tension sensing,9^–11 or the protection of circular or linear polarization states against external perturbations (vibrations, mechanical stress, temperature changes), as required for fiber-optic sensors for detecting electric currents and magnetic fields.12^–14 More advanced applications additionally use OAM birefringence or other topological effects in twisted waveguides, enabling light guidance in the absence of a core,15^,16 conversion of OAM states,17^–19 suppression of higher-order fiber modes,20 and nonreciprocal isolation of OAM modes.21 Furthermore, several nonlinear optical applications have been demonstrated such as circularly polarized supercontinuum generation,22 generation of ultranarrow spectral dips in stimulated Brillouin scattering,23 or Raman lasing with tunable polarization states.24 Notably, all of these effects are purely of geometrical origin, independent of any torsional stress in the material, making them inherently robust.

Despite these advances, the widespread use of twisted waveguides has so far been limited by the lack of methods for on-chip integration, as planar lithography is incapable of realizing such intricate three-dimensional structures. Currently, virtually all twisted waveguides come in the form of fibers and are fabricated by spinning the preform during the drawing process of the fiber or in a thermal post-processing step.25 However, the twist rates achievable by these approaches have been limited, particularly for complex waveguide designs. For example, the highest reported twist rates to date are $2.9\text{turns}/\mathrm{mm}$ (twist period: $341\mu \mathrm{m}$) for solid-core photonic crystal fibers (PCFs)26 and $0.08\text{turns}/\mathrm{mm}$ (twist period: 11.9 mm) for hollow-core fibers.8 Despite the low achieved twist rates, it is important to note that hollow-core fibers represent a growing research direction within fiber optics and are highly attractive for applications due to the strong light–matter interactions within the gas- or liquid-filled core. These unique properties have led to numerous applications, e.g., in photochemistry,27^,28 quantum technologies,29 bioanalytics,30 or telecommunications.31

Here, we address these challenges by introducing the concept of twisted light cages, the first twisted hollow-core waveguides implemented directly on silicon substrates via two-photon polymerization-based 3D nanoprinting. Unlike glass-based fabrication, this method does not involve any mechanical rotation during the fabrication, thus enabling in principle arbitrarily high twist rates, and hence compact device footprints. In addition, the waveguides feature the recently reported light cage design, where the cladding is formed by an array of freely suspended polymer strands.32^,33 Such a cage structure cannot be realized in a fiber drawing process and enables direct lateral access to the light-guiding hollow core, which has been shown to strongly reduce exchange times of gaseous and liquid analytes.34^–36

As the theoretical foundation of this concept, we investigate the origin of the two main properties of twisted light cages—strong CD and circular birefringence. Through numerical analysis and theoretical modeling in the helicoidal coordinate frame, we explain that CD arises from twist-induced resonances between the fundamental and higher-order core modes. This novel analytical framework allows us to predict twist-induced resonances in any on-axis twisted waveguide based on the properties of its untwisted version. Furthermore, the presence of circular birefringence is discussed based on the analysis of the angular momentum flux of the fundamental mode.37^,38 Experimentally, we implement light cages with four different twist rates and measure their corresponding CD spectra.

Although two earlier studies have demonstrated the feasibility of implementing twisted waveguides using 3D nanoprinting—in the form of an off-axis twisted waveguide39 and an on-axis twisted solid-core PCF40—they lacked a full theoretical and experimental analysis of their optical properties. By contrast, this study offers the first in-depth analysis of 3D-nanoprinted twisted hollow-core waveguides. With these advancements, the concept of twisted light cages opens an avenue for translating the vast amount of research on twisted fibers to on-chip devices, with particular relevance to nonlinear and quantum optics, chiral spectroscopy, and further applications requiring strong interaction of matter with circularly polarized light or OAM beams.

Twisted waveguides are categorized based on whether their light-guiding core lies on or off the twist axis, resulting in markedly different optical properties (selected examples shown in Fig. S1 in the Supplementary Material). In off-axis twisted waveguides, light travels along a helical trajectory, which is associated with a strong topology-induced circular and OAM birefringence17 and can lead to bending loss at high twist rates.41 The twisted light cage concept, by contrast, features an on-axis twist where light still travels along a straight line, whereas the intensity distribution of the mode follows the spatial rotation of the waveguide. In this case, chiral effects are caused by a different mechanism, which is explained in Sec. 3.

The design of twisted light cages builds upon their untwisted version, which has been studied in several works.32^–36^,42^–46 In general, light cages consist of a hexagonal arrangement of 12 polymer strands, which are held together laterally by support rings placed at regular intervals along the waveguide axis. Such an open design enables any medium to diffuse passively into the hollow core and allows for intense light–matter interaction. This contrasts with traditional “tube-like” hollow-core fibers, where media can only be introduced from the end faces, thus making the light cage concept particularly interesting for diffusion-related applications. At the same time, light remains confined in the core by the antiresonance effect, resulting from a phase mismatch between the central core mode and the modes of the polymer cladding.32^,47 The waveguide itself is spatially separated from the substrate by solid blocks, which compensate for any tilt of the substrate during fabrication.

The key enabler of the concept is the freedom of 3D nanoprinting, which allows the realization of light cages with an axial twist directly on silicon substrates [Fig. 1(a)]. This implementation scheme eliminates the need for any post-processing since the twist is already incorporated in the design step, i.e., in the computer-aided design file that is processed by the printer (all steps are described in Sec. 10.1). Using this method, 5-mm-long light cages with four different twist rates from 0 to $\text{11.4\hspace{0.17em}\hspace{0.17em}}{\text{mm}}^{-1}$ have been realized. Note that we define the twist rate $1/P$ as the number of turns per unit length, where $P$ is the twist period. Scanning electron microscope (SEM) and photographic images of the waveguides are shown in Figs. 1(b) and 1(c). The polymer strands were implemented with a diameter of $2r_c=3.8\mu \mathrm{m}$ and spacing of $\mathrm{\Lambda }=7\mu \mathrm{m}$, as shown in Fig. 2(a).

Conceptually, a typical spectral feature of antiresonant waveguides is low-loss transmission bands limited by resonance dips at which the core and the lossy cladding modes phase-match, leading to modal hybridization and an anti-crossing of the dispersion of the involved modes. In light cages, these resonances occur at the cut-off wavelengths of the linearly polarized (LP) modes of the isolated polymer strands32 [example of an attenuation spectrum is shown in Fig. 2(c)].

For typical strand diameters achievable by 3D nanoprinting, the strands support multiple modes at visible wavelengths, resulting in several core-strand resonances in the transmission spectra. We refer to such light cages as multimode strand light cages (strand diameter: $2r_c=3.6\mu \mathrm{m}$). On the other hand, most simulations in the theoretical part of this work are performed for single-mode strand light cages, where the strand diameter is reduced to $2r_c=0.4\mu \mathrm{m}$, such that the strands only support the fundamental ${\mathrm{LP}}_{01}$ mode at the investigated wavelength of $\lambda =770\mathrm{nm}$. Because the ${\mathrm{LP}}_{01}$ strand mode does not have a cut-off, resonances between core and strand modes are absent, simplifying the following study of the additional resonances caused by twisting [see Fig. 2(c)]. For this analysis, it is important to note that both single-mode strand and multimode-strand light cages support multiple modes in the hollow core. To achieve comparable propagation loss for both waveguides, the number of strands was increased from 12 to 108 in the single-mode strand variant, such that the fraction of open space between the strands $1-2r_c/\mathrm{\Lambda }$ remains constant [see Figs. 2(a) and 2(b)]. The cross-section of the strands is chosen to be circular in the $xy$ plane at all twist rates rather than adopting a more complex strand type41 as explained in Sec. 10.1. Note that all simulations consider left-handed waveguides, except when comparing the results to the experimental data for right-handed light cages presented in Fig. 5.

To analyze the optical properties of twisted light cages, we map the twisted waveguide to a straight waveguide using helicoidal coordinates, which are implemented in the commercial finite element method (FEM) solver JCMwave. This allows us to calculate the modes of an infinitely extended twisted waveguide using its 2D cross-section, as described in more detail in Sec. 10.2.

We start by analyzing single-mode strand light cages, which do not feature any resonances between the core and the strand modes. Yet, when twisting the structure and simulating the optical properties at a fixed wavelength, resonances appear at certain twist rates [Figs. 3(a) and 3(b)]. There are two types of resonances. Some resonances are achiral, i.e., they affect the RCP and LCP fundamental core modes similarly. Others are chiral resonances, where only one of the two modes shows an increased loss, whereas the mode of opposite handedness is unaffected, resulting in CD. The origin of these twist-induced resonances is explained in the following, whereas the difference between chiral and achiral resonances is discussed at the end of this section.

By analyzing the crossings and anticrossings in the effective index of the involved modes, we find that the resonances are caused by a coupling of the low-loss fundamental core mode to a lossy higher-order core mode and occur whenever

1. •the effective refractive indices $n_{\mathrm{eff}}^{\text{helical}}$ of the two modes match;
2. •the total angular momentum $j$ of the modes differs by an integer multiple of $n$, where $n$ is the order of the rotational symmetry of the waveguide cross section $C_{nz}$ ($n=6$ for light cages).

Next, we simplify the first condition by noting that the strong splitting in $n_{\mathrm{eff}}^{\text{helical}}$ between modes of different total angular momenta is mostly not a waveguide-related effect but is primarily caused by the rotation of the helicoidal frame relative to the laboratory frame.38 As explained in Sec. 10.3, this allows us to approximate the effective indices in the helicoidal frame based on that of the untwisted waveguide: $$n_{\mathrm{eff}}^{\text{helical}}\approx n_{\mathrm{eff}}^{l,m}{|}_{\alpha =0}-(s+l)\frac{\alpha \lambda }{2\pi },$$(1)where $|\alpha |=2\pi /P$ denotes the angular twist rate and $n_{\mathrm{eff}}^{l,m}{|}_{\alpha =0}$ is the effective index of a core mode with OAM order $l\in \mathbb{Z}$, radial order $m\in \mathbb{N}$, and spin $s=\pm 1$ in the untwisted waveguide. Equation (1) is shown as the gray-dashed line in Fig. 3(a) and predicts the twist rates (or wavelengths) at which the indices of the two modes intersect. Therefore, resonances between the fundamental modes ($l=0$, $s=\pm 1$, $m=1$, $j=l+s$) and higher-order core modes $(\tilde{l},\tilde{s},\tilde{m},\tilde{j})$ can be calculated by the following two conditions that must be fulfilled simultaneously: $$\alpha \mathrm{\Delta }j=k_0(n_{\mathrm{eff}}^{\mathrm{0,1}}-n_{\mathrm{eff}}^{\tilde{l},\tilde{m}})(\text{grating condition}),$$(2)$$\mathrm{\Delta }j=6q\text{for}q\in \mathbb{Z}(\text{angular momentum selection rule}),$$(3)where $\mathrm{\Delta }j=j-\tilde{j}$ is the difference in total angular momentum between the modes. Note that the right-hand side of Eq. (2) is always positive, which imposes a condition on the sign of $\mathrm{\Delta }j$ depending on the handedness of the waveguide. For left-handed waveguides, $\alpha$ and, thus, $\mathrm{\Delta }j$, are positive, such that resonances can only be caused by higher-order modes with $\tilde{j}<0$.

Equation (2) effectively describes a diffraction grating as used in the context of fiber gratings.48 This is expected since twisting introduces a periodic modulation along the propagation direction, thus acting as a longitudinal grating.49 The left side of the equation is the grating wavevector for a period length $P/6$ and diffraction order $q$, and the right side describes the wavevector mismatch between the modes (see Fig. S3 in the Supplementary Material). This transfer of linear momentum can occur both in twisted waveguides and untwisted waveguides with a periodic index modulation.

Equation (4), on the other hand, describes a transfer of angular momentum mediated by the twist, which does not occur in untwisted waveguides. The origin of the angular momentum transfer $\mathrm{\Delta }j=6q$ [Eq. (4)] can be understood based on the symmetry of the modes of the untwisted waveguide.38 For circularly symmetric systems (e.g., step-index fibers with cylindrical symmetry), modes are eigenstates of the angular momentum operator with the eigenvalue $j$ being an integer. However, the rotational invariance is broken in light cages due to their hexagonal cross-section. This lower $n$-fold rotational symmetry implies that modes need to be constructed as a superposition of eigenstates of the angular momentum operator with integer eigenvalues $j_0+nq$$\forall q\in \mathbb{Z}$.38 These angular momentum harmonics therefore allow a mode with a certain dominant angular momentum $j_0$ (in previous equations simply referred to as $j$) to couple to all modes with a dominant angular momentum of $j_0+nq$. Without these harmonics, such a coupling would not be possible because the eigenstates of the angular momentum operator are mutually orthogonal.

With this model in place, we can now reveal why some of the twist-induced resonances are achiral and others are chiral. The required condition $\mathrm{\Delta }j=\mathrm{\Delta }s+\mathrm{\Delta }l=6q$ can be achieved in two ways (listed in Table 1). For $\mathrm{\Delta }s=0,\mathrm{\Delta }l=6q$, both the LCP and RCP fundamental modes couple to the corresponding higher-order modes with $\tilde{l}=-6q$ at the same twist rate, thus resulting in an achiral resonance. When a fundamental mode couples to a higher-order mode of opposite spin ($\mathrm{\Delta }s=-2,\mathrm{\Delta }l=6q+2$ or $\mathrm{\Delta }s=+2,\mathrm{\Delta }l=6q-2$), the resonances of the LCP and RCP fundamental modes occur at different twist rates resulting in chiral resonances [see Fig. 3(a)]. Because the higher-order modes of light cages feature nearly two orders of magnitude higher propagation loss than the fundamental modes, such a twist-induced chiral coupling leads to a resonance (i.e., transmission dip), which can lead to strong CD.

Finally, one might ask why modes of different spin angular momentum $s$ are allowed to couple as these states are mutually orthogonal in free space. This can be resolved when noting that each eigenstate of the angular momentum operator with eigenvalue $j$ does not necessarily contain just one spin state. In fact, even in round step-index fibers, the fundamental ${\mathrm{HE}}_{\mathrm{1,1}}$ mode with angular momentum $j=1$ is a superposition of three states: a dominant contribution with $s=1$ and OAM $l=0$, and two minor contributions with $(s=-1,l=2)$ and $(s=0,l=1)$, the latter one arising from the longitudinal field component.50^,51 The letters $s$ and $l$ used throughout the manuscript refer to the dominant contribution, whereas the weaker contributions enable the coupling between modes of opposite-handedness. As a result, hybrid modes containing both spin states form near chiral resonances [lower row of Fig. 3(c)].

To support this theoretical argument we perform an OAM modal decomposition of the fundamental mode, i.e., we analyze the contributions of the different OAM eigenstates of an untwisted light cage. To this end, the modes are decomposed into Bessel beams of spin $s$, radial order $p$, and azimuthal order $l$, where $l$ corresponds to the (integer) OAM order of the Bessel beam as explained in Sec. 10.4. The result of this analysis is a probability distribution $\sum _p|a_{l,p}{|}^2$ for finding a photon with angular momentum $l$ in the analyzed mode of the light cage. For the RCP mode ($j_0=-1$), this distribution is expected to contain OAM harmonics of the form $l=6q$ with $s=-1$, and $l=6q-2$ with an opposite spin $s=+1$. The result of the OAM decomposition for the RCP mode matches this pattern, as shown in Fig. 4(e) (decomposition of the LCP mode is available in Fig. S10 in the Supplementary Material). A further result of the decomposition is that contributions of the opposite spin with $l=6q-2$ (and $l=6q+2$ for the LCP mode) feature relatively weak amplitudes explaining why the coupling strength in chiral resonances is lower than in achiral resonances.

The angular momentum selection rule [Eq. (4)] provides a direct link between the rotational symmetry of the waveguide and the number of allowed resonances. To emphasize this point, additional simulations were performed for a geometry where the 108 strands of the single-mode light cage are arranged in a circle instead of a hexagon, resulting in a $C_{108z}$ rotational symmetry. Indeed, the OAM decomposition of the fundamental LCP and RCP modes shown in Fig. S11(b) in the Supplementary Material indicates that the first OAM harmonics occur only for $|l|=108$ and $|l|=108\pm 2$ (for the respective contributions of opposite spin). This larger spacing between the OAM harmonics directly translates into a reduction in the number of twist-induced resonances within a given range of twist rates. In fact, resonances are completely absent for the round structure in the range of investigated twist rates of 0 to $\text{3.5\hspace{0.17em}\hspace{0.17em}}{\text{m}\mathrm{m}}^{-1}$ [see Fig. S12(c) in the Supplementary Material]. On the other hand, it should be noted that the number of twist-induced resonances in a given twist rate (or wavelength) interval also depends on the effective index difference between the fundamental modes and the OAM modes in the untwisted structure [Eq. (2)]. For example, a triangular arrangement of the strands with a smaller mode area (i.e., large spacing in effective index) can have a lower density of resonances than a hexagonal arrangement with a larger mode area (Fig. S21 in the Supplementary Material).

To reveal the impact of the twist, we additionally analyzed the OAM distribution of the twisted waveguides (so far only untwisted waveguides were analyzed). For left-handed twisted waveguides, the amplitudes of the negative OAM orders increase for both LCP and RCP modes, in both the hexagonal and round light cages (see Figs. S10 and S11 in the Supplementary Material). Looking at the example of the RCP mode in the hexagonal light cage, the resulting average of the OAM distribution therefore decreases from $\overline{l}=-2.4\times {10}^{-4}$ in the untwisted waveguide to $\overline{l}=-2.3\times {10}^{-2}$ at a twist rate of $\text{3.5\hspace{0.17em}\hspace{0.17em}}{\text{m}\mathrm{m}}^{-1}$ [data taken from Fig. 4(e)]. Thus, a left-handed twist causes the fundamental modes to acquire a small overall negative OAM. A possible explanation for this effect might lie in modal hybridization. As previously discussed and shown in Fig. 3(a), only modes with negative OAM can couple to the fundamental mode for a left-handed twist, whereas the index difference to the modes with positive OAM increases. Therefore, even away from resonances, the fundamental mode will always be—to a small extent—hybridized with modes carrying negative OAM, thus explaining the shift of the OAM distribution.

Another key parameter that needs to be investigated is circular birefringence, which is a well-known feature of twisted waveguides. To reveal the experimentally measurable circular birefringence, the real part of the effective index is transformed back to the laboratory frame by Eq. (8) using the dominant values of $s$ and $l$. As the amplitudes of the OAM harmonics are several orders of magnitude smaller, they can be neglected in this transformation. The circular birefringence $B_C$ is then calculated as the difference in effective index $n_{\mathrm{eff}}^{\mathrm{lab}}$ between the LCP and RCP modes with $l=0$. $B_C$ increases from 0 in the untwisted waveguide to $7.5\times {10}^{-6}$ at a twist rate of $\text{3.5\hspace{0.17em}\hspace{0.17em}}{\text{m}\mathrm{m}}^{-1}$ [Fig. 4(b)]. This value is similar to the circular birefringence in commercially available spun optical fibers52 and is therefore sufficient to ensure robust propagation of circularly polarized light. In practical terms, the polarization direction of linearly polarized light would be rotated by an angle $\theta =B_C\pi z/\lambda \approx 9°$ for a waveguide length of $z=5$ mm at this twist rate.

The physical origin of circular birefringence in on-axis twisted waveguides is again related to the angular momentum of the modes. It turns out that even in an untwisted waveguide the total angular momentum distribution is not symmetric if the rotational invariance is broken, i.e., the amplitudes of contributions with $j_0+6q$ are different from those with $j_0-6q$.38 This asymmetry results in the mode having a total angular momentum flux that slightly deviates from $j_0$, which has been shown to be the cause of circular birefringence in on-axis twisted waveguides:37^,38$$B_C=\alpha (⟨j⟩-j_0)\frac{\lambda }{\pi },$$(4)where $⟨j⟩$ denotes the angular momentum flux of the RCP mode in the untwisted waveguide and is calculated as the sum of spin angular momentum flux $⟨s⟩$ and OAM flux $⟨l⟩$, as defined in Sec. 10.5. For the RCP mode in the hexagonal light cage, $⟨s⟩=-0.99947$, $⟨l⟩=3.8\times {10}^{-4}$, and $⟨j⟩=-0.99909=j_0+9.1\times {10}^{-4}$. We note that the value of $⟨l⟩$ differs from the average $\overline{l}$ obtained in the OAM decomposition. This discrepancy likely arises because in the OAM decomposition, only the transverse electric field components were analyzed while the calculation of $⟨l⟩$ involves all transverse electric and magnetic field components. Nevertheless, both $\overline{l}$ and $⟨l⟩$ decrease with increasing twist rate, confirming the earlier result.

The outcome of Eq. (4) is shown as a light purple line in Fig. 4(b), matching well with the simulated values in the absence of resonances. Furthermore, this theory correctly predicts that the circular birefringence is lower if the strands of the light cage are arranged in a circle instead of a hexagon since the $C_{108z}$ symmetry of the circular light cage is closer to complete rotational invariance where $B_C$ would be 0 (Fig. S12 in the Supplementary Material).

Although the circular birefringence was analyzed in the lab frame, it is important to note that any intersections in the effective indices of the fundamental modes and the OAM modes are absent after performing the transformation to the lab frame using Eq. (8) [see Fig. S8(a) in the Supplementary Material]. This is to be expected because we only transformed the index of the dominant angular momentum contribution, whereas the angular momentum harmonics—which are responsible for the coupling—would be assigned different indices in the lab frame because the transformation depends on the angular momentum (see Sec. 10.3). Intersections with the effective index of an OAM mode would therefore only occur if each angular momentum harmonic of the fundamental mode is transformed individually. To avoid such a complex analysis, it is generally best to describe on-axis twisted waveguides in the helicoidal frame where all angular momentum harmonics feature the same effective index.

The presented models for the position of the twist-induced resonances and the circular birefringence rely solely on the properties of the untwisted waveguide. Yet, Eqs. (2) and (4) require knowledge of the relevant higher-order modes in the untwisted waveguide. To reduce computational time, we show in Sec. S12 in the Supplementary Material that the effective indices of these higher-order modes can alternatively be obtained from those of the fundamental mode by applying a recently reported model for antiresonant waveguides, which approximates the waveguide cladding as a tube.47 In this case, the effective index of the untwisted waveguide modes can be estimated as $$n_{\mathrm{eff}}^{l,m}\approx 1-A{u}_{l,m}^2,$$(5)where $A(\lambda )$ contains the core-cladding resonances but does not depend on the order of the mode and can be obtained from a fit of the dispersion of the fundamental mode of the untwisted waveguide (see Sec. S12 in the Supplementary Material). $u_{l,m}$ is the $m$’th root of the $l$’th order Bessel function of the first kind, and $l=\dots ,-1,0,1,\dots$ and $m=1,2,\dots$ refer to the azimuthal and radial order of the modes, respectively, akin to the definition of LP modes.

Plugging this relation into Eq. (2) then allows us to determine the twist rates at which resonances occur. The results are shown as vertical lines in Fig. 4(c), matching well with the simulated resonances at low twist rates. At higher twist rates, the model projects that more and more resonances occur but the prediction of the exact twist rates becomes less reliable.

Having studied the underlying physics of twisted light cages through the simpler example of the single-mode strand configuration, we now turn to the characterization of the fabricated waveguides, which consist of 12 multimode strands of a larger diameter for reasons of mechanical stability. These multimode strand light cages feature additional resonances between core and strand modes leading to more complex transmission spectra. However, accompanying simulations show that the amplitude and spectral location of the core-strand resonances are only negligibly affected by twisting (see Fig. S16 in the Supplementary Material). Therefore, the analytical framework developed for single-mode strand light cages applies identically to multimode-strand light cages. In practice, we investigated samples with four different twist rates ranging from 0 to $\text{11.4\hspace{0.17em}\hspace{0.17em}}{\text{m}\mathrm{m}}^{-1}$, all with a right-handed twist direction [SEM images shown in Fig. 5(b)]. The implemented strand diameter was determined to be $2r_c=3.814\mu \mathrm{m}$, based on the measured spectral position of the core-strand resonances of the untwisted waveguide (procedure described in Ref. 33).

As a first verification of the theoretical modeling, the LCP fundamental mode was excited in one of the twisted waveguides and mode images were recorded at different distances from the end face of the waveguide. Because the mode is invariant in the helicoidal frame, its intensity distribution is supposed to follow the right-handed twist of the waveguide in the lab frame. This rotation was confirmed in the measurement as shown in Fig. 5(a), with the full measurement being available as a Supplementary Video (Video 1, MP4, 853 kB).

Next, the CD was determined using the white light transmission setup described in Sec. 10.6. CD is here defined as the absolute value of the difference in loss between the LCP and RCP modes. In Figs. 5(c) and 5(d), the results are compared with numerical simulations of the modal attenuation of waveguides with identical properties. Three different regimes can be distinguished. (1) In the untwisted waveguide only core-strand resonances are present and the transmission is identical for LCP and RCP light. (2) At intermediate twist rates (0 to $\text{1.3\hspace{0.17em}\hspace{0.17em}}{\text{m}\mathrm{m}}^{-1}$), simulations indicate the formation of the first twist-induced core-core resonances (Fig. S16 in the Supplementary Material). However, for the experimentally realized twist rate of $\text{0.95\hspace{0.17em}\hspace{0.17em}}{\text{m}\mathrm{m}}^{-1}$, these resonances fall outside of the investigated wavelength range. The core-strand resonances can still be clearly distinguished and remain nearly unaffected by twisting, except for small spin-dependent shifts (evaluated based on the simulated data in Fig. S17 in the Supplementary Material). These shifts give rise to weak CD but were not investigated further. (3) At high twist rates (1.5 to $\text{10\hspace{0.17em}\hspace{0.17em}}{\text{m}\mathrm{m}}^{-1}$), more and more twist-induced resonances appear, which result in strong CD and overall higher loss (more details in Fig. S18 in the Supplementary Material). Due to the large number of core-core resonances, the individual core-strand resonances cannot be distinguished anymore in the loss spectra. Toward the highest investigated twist rate, a slight reduction of the CD is observed, likely caused by the spectral overlap of RCP and LCP chiral resonances. Overall, the experimental results clearly confirm the presence of CD in twisted light cages, reaching values of up to 0.8 dB/mm at a twist rate of $\text{5.7\hspace{0.17em}\hspace{0.17em}}{\text{m}\mathrm{m}}^{-1}$.

Compared with previous works on twisted waveguides, our fabricated samples with a twist period of $90\mu \mathrm{m}$ surpass the twist rate of different classes of glass-based twisted waveguides, such as commercially available spun-optical fibers, off-axis twisted waveguides, and fibers with complex microstructures such as PCFs, as visualized in Fig. 6 and listed in detail in Table S1 in the Supplementary Material.

Of these works, only two have so far demonstrated twisted waveguides with hollow cores, in the form of glass-based single-ring hollow-core PCFs,8 with a twist period that is more than two orders of magnitude larger. In terms of the two previous realizations of 3D-nanoprinted twisted waveguides,39^,40 we overcome the twist period by a factor of more than two.

The measured CD in twisted light cages is about two orders of magnitude larger than in the previously reported twisted hollow-core fiber by Roth et al.8 One explanation for this large difference lies in the fact that CD in light cages is caused by the coupling of the fundamental core mode to another (lossy) core mode, which have a large spatial overlap. By contrast, Roth et al. used a cladding mode as the loss channel for modal discrimination, which consequently limits the overlap and coupling strength with the core mode. Nevertheless, we note that the overall loss at which we achieve this CD is comparably high in the current experimental realization. On the one hand, to reach a 10-dB discrimination between LCP and RCP light, the two modes would be attenuated by 58 and 68 dB, respectively (Table 2). On the other hand, the associated simulations indicate that the optical properties of the investigated twisted light cages can be considerably improved. Specifically, a CD of 5.4 dB/mm can potentially be reached, yielding losses of 2.9 and 12.9 dB for the two polarizations after a propagation distance of 1.9 mm [see Fig. 5(d) and Table 2]. Instead, in the glass-based realization of Ref. 8, a length of 1.2 m is required to achieve the same discrimination, which precludes its use in compact devices.

There are three possible explanations for the higher loss in the fabricated waveguides. (1) As previously investigated, the surface roughness of the strands leads to additional scattering loss explaining why the off-resonance loss in the untwisted waveguide is about one order of magnitude larger than in simulations.32^,33 (2) The cross-section of the twisted strands varies with the axial position in the waveguide, which results in a broadening of the core-strand resonances leading to higher losses. This would explain the absence of clear core-strand resonances in the sample with the intermediate twist rate of $0.9\text{5\hspace{0.17em}\hspace{0.17em}}{\text{m}\mathrm{m}}^{-1}$ [see Figs. 5(c) and 5(d)]. (3) Support rings and support blocks are excluded from the simulations because their inclusion would break the translational invariance of the waveguide, thus requiring a shift to computationally impractical three-dimensional simulations. However, previous experimental and theoretical investigations on untwisted 3D-nanoprinted antiresonant waveguides have demonstrated that the additional losses induced by these structural elements are negligible in comparison to the other loss mechanisms.34^,53 This conclusion is further supported by the fact that the optical power confined within the hexagonal core is close to unity, with minimal power present in the region of the support structures, for both untwisted and twisted light cages in single-mode and multimode configurations (see Fig. S20 in the Supplementary Material).

Future work will therefore focus on solving these fabrication-related challenges, e.g., by changing the fabrication direction from horizontal to vertical (i.e., perpendicular to the substrate). With this adjustment, the shape of the voxel within the cross-sectional plane of the waveguide changes from elliptical to circular, thus enhancing the accuracy of the fabrication. Furthermore, strategies to reduce the overall propagation loss could be explored, e.g., by reducing the spacing between strands, adding multiple layers of strands (untwisted dual-ring light cages with reduced losses have been demonstrated in Ref. 42), performing multi-dimensional parameter optimizations (e.g., by incorporating multiple different strand diameters using a quasi-analytical model for efficient computation),54^,55 or applying techniques for reducing the surface roughness of the polymer (e.g., Ref. 56). In addition, emerging methods offer solutions for enhancing scalability and cost-effectiveness of the fabrication process. These include the use of multi-focal arrays for parallelized fabrication57^,58 and the emerging two-step absorption-based 3D nanoprinting where the expensive femtosecond laser is replaced by a cost-effective continuous wave diode while maintaining high spatial resolution.59

Regarding the theoretical analysis of twist-induced resonances, we observed a discrepancy between our findings and those presented in earlier works on twisted PCFs,25^,26 an issue previously addressed in Ref. 60. We want to extend this discussion to the theoretical analysis in Ref. 8, which states that only modes of the same total angular momentum are allowed to couple. Yet, a visual inspection of Fig. 6 of their work indicates the coupling of a core mode with angular momenta $s=+1,l=0,j=+1$ with a cladding mode with $s=-1,l=+12,j=+11$, seemingly contradicting their claim. Using the angular momentum selection rule Eq. (4) for their five-fold rotationally symmetric fiber, however, would explain this coupling correctly as a chiral resonance.

In this context, we note that the conditions for twist-induced mode coupling, Eqs. (2) and (4), have first been derived for $q=1$ in the context of chiral fiber gratings (i.e., on-axis twisted solid-core fibers) using first-order perturbation theory.18^,49 Although our derivation can successfully predict the spectral locations of twist-induced resonances for arbitrary values of $q$, a perturbative approach would give access to additional details, such as the hybridization of modes and the formation of anti-crossings in the real part of the effective index. To our knowledge, such a comprehensive analysis is still lacking for on-axis twisted waveguides, but a conceptual outline is available for off-axis twisted waveguides.17 Finally, we note that Eqs. (2) and (4) have previously been validated for resonances occurring in on-axis twisted PCFs that are caused by a coupling between core and cladding modes.60 Our work additionally demonstrates the applicability to resonances caused by coupling between two core modes.

We also want to address whether the high twist rates achievable with 3D nanoprinting offer an advantage over the lower twist rates reached through fiber drawing or thermal post-processing techniques. As evident from Eq. (2), higher twist rates are generally beneficial as they enable coupling of the fundamental mode to modes of very high OAM order, which feature higher losses and can therefore result in stronger CD. In the current design, the coupling strength to such modes is limited, as their amplitudes in the OAM decomposition of the fundamental mode are comparably small (cf. Fig. S11 in the Supplementary Material). Increasing the amplitudes of these higher-order OAM contributions using non-polygonal arrangements of the strands (e.g., star-shaped) therefore provides a path to achieve even stronger CD and circular birefringence. On the contrary, higher twist rates also increase the number of resonances per wavelength interval (Fig. S18 in the Supplementary Material), which can lead to a spectral overlap of RCP and LCP chiral resonances, thus reducing the CD. Therefore, there is an optimal twist rate at which the CD is maximized, which can be tuned by adjusting the core size. For example, smaller core sizes increase the spacing in the effective index between the fundamental and higher-order modes in the untwisted waveguide, reducing the number of resonances per wavelength interval and thereby mitigating the resonance overlap.

Further improvements in CD could in principle be reached by adopting more advanced waveguide designs, such as negative curvature fibers 61^,62 or hollow-core nested antiresonant nodeless fibers,63 where the difference in propagation loss between the fundamental and higher-order modes is larger. Although fabricating the thin-walled hollow cylinders required for these designs at visible wavelengths is still challenging, realizations by 3D nanoprinting at near-infrared wavelengths are conceivable.

Possible applications of twisted light cages are demonstrated in Fig. 7 based on the example of the single-mode strand light cage, as the effects of twist-induced resonances are more clearly observed. Although the small diameter of single-mode strands is difficult to realize at visible wavelengths, such a structure might be feasible in the mid-infrared spectral range. Regardless, all of the mentioned applications are also feasible with the multimode strand light cages, which are less complex to fabricate. As a first application, the CD of chiral resonances shown in Fig. 7(a) allows twisted light cages to act as circular polarization filters.

These filters can be placed in line with existing on-chip waveguides in devices requiring a pure circular polarization state, e.g., in areas such as optical communication, chiral sensing, or chiral quantum optics.64 For example, circular polarization states can be used in free-space optical communication to encode information via polarization shift keying.65 In this context, twisted light cages could be used as analyzers in the receiving device, complementing on-chip analyzers for linear polarization states.66 As another example, the measurement of the weak CD of biomolecules or chiral nanoparticles demands a highly pure circular polarization state, as even a slight ellipticity in the polarization can interact with the potentially stronger linear dichroism of the analyte, causing false-positive CD measurements.67 The unique design of light cages with its open space between the strands offers two key advantages over glass-based fabrication techniques in this regard. First, this openness allows for quick passive sample exchange without relying on external pumps, which are required to transport analytes through tube-like hollow-core fibers.34^,35 Specifically, such an open design has been shown to result in nearly unrestricted diffusion of gases into the core and achieves a six-fold reduction in analyte exchange time in aqueous environments compared with similarly sized hollow-core fibers.34^,53 Second, the top strands of the light cage can be completely removed in a short section of the light cage, such that objects interacting with the light in the core can be imaged through this opening.45 As only light scattered by the analyte is detected, this corresponds to dark-field detection and allows for the characterization of chiral nanoparticles or viruses.68

Furthermore, light cages are 3D chiral structures (as opposed to planar structures with 2D chirality),69^,70 meaning that they suppress polarization of a specific handedness both for forward- and backward-propagating light. Improved designs with lower propagation loss could therefore be of relevance for the realization of single-handedness chiral cavities, which benefit from spin-dependent loss.71^,72 As before, the use of light cages in this application would be particularly beneficial as atoms or molecules can be introduced into the cavity via the open space between the strands, enabling a novel platform for chiral sensing.

Another frequently explored application of on- and off-axis twisted waveguides is OAM generation.17^–19 However, in the case of twisted PCFs, the OAM is carried by a lossy cladding mode in most works,8^,25^,26^,73 which limits their use as mode converters. An exception is Ref. 74, where OAM beams were generated in the core of the twisted fiber but used an additional fiber Bragg grating for mode coupling. Twisted light cages, on the contrary, offer two advantages in this regard: (1) OAM modes can be generated directly in the light-guiding core and (2) 3D nanoprinting provides a straightforward path to implement adiabatic mode conversion by enabling the fabrication of structures with spatially varying twist rates. Figure 7(b) shows an example where an adiabatically increasing twist rate would result in the conversion of a mode with $l=0$ to a mode with $l=-6$. Such adiabatic coupling would yield a broad operating bandwidth, which is an advantage over resonant coupling at a fixed twist rate. Preferably, adiabatic mode coupling should be implemented for an achiral resonance as its larger coupling strength allows for shorter device lengths.75^,76 Furthermore, changing the core size and twist rate along the propagation direction could be used to tune the core-core mode coupling to an exceptional point with applications, e.g., in sensing or for realizing on-chip devices with asymmetric transmission.77

In terms of sensing applications, twisted light cages are sensitive to any phenomenon that affects the helical pitch $P$, such as torsion and tension. According to Eq. (2), the wavelength ${\lambda }_r$ at which resonances occur is equal to ${\lambda }_r=P \mathrm{\Delta }n/\mathrm{\Delta }j$, where $\mathrm{\Delta }n(\lambda )$ and $\mathrm{\Delta }j$ are the mismatch in effective index and angular momentum of the modes in the untwisted waveguide. The wavelength dependence of $\mathrm{\Delta }n$ can be described by the tube waveguide model discussed in Sec. S12 in the Supplementary Material, which predicts a shift of the resonances to longer wavelengths as the twist rate increases, matching well with the simulated values in Fig. 7(c). As $\mathrm{\Delta }n$ grows approximately quadratically in $l$ while $\mathrm{\Delta }j$ grows linearly in $l$, higher-order resonances generally feature a higher sensitivity to changes in $P$. For the first achiral resonance ($\mathrm{\Delta }j=6$), we find a torsion sensitivity $\mathrm{\Delta }{\lambda }_r/\mathrm{\Delta }\alpha$ of 0.11 nm/(rad/m). In other words, the resonance wavelength increases by 1 nm if $P$ decreases by $1.5\mu \mathrm{m}$. This value lies within the range of sensitivities between 0.03 and 0.5 nm/(rad/m) reported for fiber-based measurements.9^–11^,78

We note that some of the above applications have already been realized using chiral fiber gratings, which generally achieve high-performance metrics. Examples include a CD of $3\mathrm{dB}/\mathrm{mm}$ over a bandwidth of more than 80 nm7 (induced by coupling of the core mode to lossy cladding modes), a torsion sensitivity of 0.47 nm/(rad/m) for a 24-mm-long grating with a resonance contrast of 32 dB,11 OAM generation with high coupling efficiency,79 and polarizers based on adiabatically twisted fibers.80 Due to the simple geometry of these fibers (a single solid core embedded in a homogeneous cladding), very high twist rates can be achieved corresponding to twist periods down to $24\mu \mathrm{m}$.81

However, in contrast to such twisted solid-core fibers, hollow-core light cages allow for a strong interaction of gases or liquids with the light in the core. Such an interaction is for example used in nonlinear frequency conversion and multidimensional soliton generation with gas-filled hollow-core waveguides.82^,83 This highlights the need for future studies to thoroughly analyze the phase matching and validate the above conditions in the case where the twisted light cages are filled with highly nonlinear materials, especially when considering phase-matching sensitive effects such as degenerate four-wave mixing or84 ultrafast soliton fission seeded dispersive wave generation.85 The benefit of using twisted waveguides in these applications is mostly related to their circular birefringence, which allows circularly polarized supercontinuum generation ($B_C=1.1\times {10}^{-6}$)22 or light sources with pressure-tunable polarization states based on Raman scattering ($B_C=3\times {10}^{-8}$).24 For twisted light cages, simulations indicate that $B_C$ is on the order of ${10}^{-6}-{10}^{-5}$ and could be further increased using larger twist rates [cf. Eq. (4)]. As a result, twisted light cages offer an opportunity for the chip integration of the aforementioned works while also increasing the robustness of the polarization state against environmental fluctuations.

Furthermore, only a low fraction of the optical power below ${10}^{-3}$ is guided inside the potentially absorbing strand material of the light cages.32 This enables high-power applications and access to wavelength ranges where a material platform is considered too lossy for solid-core guidance, e.g., in the technically relevant mid-infrared86 or extreme ultraviolet range.87

Finally, we want to emphasize that 3D nanoprinting provides a clear path for interfacing light cages with other on-chip waveguides via photonic wire bonding or88 free-form tapers.89 This avoids additional processing steps required for interfacing twisted fibers with photonic chips (fiber stripping, cleaving, and mechanical alignment) and addresses the large mode diameter mismatch that precludes simple edge coupling of twisted hollow-core fibers. Moreover, integration of light cages with fibers has been successfully demonstrated using V-grooves on silicon chips36^,45 or fabrication directly onto fiber-end faces.46 As a consequence, twisted light cages present a promising platform for transitioning research on twisted fibers to integrated optical devices, whereas their unique open cage structure allows accessing novel applications by introducing gases or liquids into the hollow core of this chiral waveguide.

In summary, this work introduced the concept of twisted light cages as a new platform for integrated chiral photonics and constitutes the first experimental demonstration of CD in an on-chip hollow-core waveguide. Building on previous works, the origin of CD17^,18^,49^,60 and circular birefringence37^,38 in these waveguides have been explained based on the presence of higher-order OAM states in the fundamental mode of the untwisted waveguide.

The CD was found to be caused by twist-induced chiral resonances, which result from the coupling of higher-order core modes with the fundamental mode of opposite spin. This method provides high spatial overlap between the coupled modes in contrast to previous works using resonances between core and cladding modes. Furthermore, a mode-coupling selection rule was verified, which shows that resonances only occur if the total angular momentum of the involved modes differs by multiples of the order of the rotational symmetry $n$ of the waveguide ($n=6$ for light cages). In this context, we presented a derivation for the mode coupling condition based on the properties of the helicoidal coordinate frame, which is valid for both, core-core and core-cladding mode coupling in on-axis twisted waveguides. The occurrence of such resonances was shown to be determined by the properties of the OAM modes of the untwisted waveguide and can be predicted analytically by approximating the geometry of light cages as a tube.47

Experimentally, we measured a large CD of $0.8\mathrm{dB}/\mathrm{mm}$, which is accompanied by overall high modal attenuation, which will be reduced in future experimental studies. Combined with the unique cage structure, enabling the introduction of liquids or gases into the hollow core, 3D-nanoprinted twisted light cages open up exciting prospects for translating years of research on glass-based twisted fibers into complex on-chip devices with unprecedented properties. Applications of twisted light cages include waveguide-integrated and broadband generation of circularly polarized and OAM beams, nonlinear frequency conversion with circularly polarized light,22^,24 twist and strain sensing, and chiral spectroscopy.

The twisted light cages were fabricated on diced silicon wafer substrates using a commercial two-photon-absorption direct laser writing system (Photonic Professional GT, Nanoscribe GmbH). The system was operated in the dip-in configuration using a liquid negative-tone photoresist (IP-Dip, Nanoscribe GmbH). Twisted waveguide segments of length $178\mu \mathrm{m}$ were defined in an STL file and translated into printer movements using the Describe software package (Nanoscribe GmbH) with the settings shown in Table 3. Waveguides of 5 mm length are realized by stitching together individual segments with an overlap of $2\mu \mathrm{m}$ using the onboard mechanical stage. Further information on the process can be obtained from Ref. 32. High repeatability of the fabrication method with intra-chip variations in the realized dimensions of 2 nm was demonstrated in Ref. 33.

Regarding the specific geometry of the polymer strands, we chose their cross-section to be circular in the $xy$ plane at all twist rates [Figs. 2(a) and 2(b)]. This choice ensures the highest robustness against fabrication inaccuracies as the printer operates on a Cartesian grid (i.e., the variation of the cross-section between individual strands is minimal). Another option would be to use strands with a circular cross-section in the plane perpendicular to the helical trajectory of the strands (i.e., the plane in which the wavefronts of the strand modes lie). However, this would require the cross-sections to be elliptical in the $xy$ plane, be tilted with respect to each other, and feature a twist-rate-dependent ellipticity. A detailed study on the differences between these two strand geometries (termed helicoidal waveguide and Frenet-Serret waveguide) can be found in Ref. 41. Keeping in mind that the shape of the 3D-nanoprinted voxel is also elliptical, it is more challenging to ensure that all strands have identical properties with the latter geometry. Furthermore, this more advanced geometry was not required in this work because (1) in the case of single-mode strand light cages, core and strand modes do not interact, and (2) in the case of multimode strand light cages, the difference between the optical properties of the helicoidal and Frenet-Serret strand geometry is negligible due to the large area of the cross-section.41

A commercial FEM solver (PropagatingMode module of JCMwave) with native support for calculations in helicoidal coordinates was used to simulate the optical properties of all waveguides. The strand material is assumed to be lossless with the dispersion defined by a Sellmeier equation in Sec. S6 in the Supplementary Material. The waveguide is surrounded by air, followed by a perfectly matched layer (PML) to absorb the outgoing power of the leaky modes. The solver features an appropriate definition of a PML for helicoidal coordinates. Convergence of the results in terms of mesh size in the strands and the core, and the distance between the waveguide and PML has been checked (Sec. S7 in the Supplementary Material). It is important to note that the mesh size in the core needs to be reduced as the twist rate increases because the mode develops more and more fine spatial features [cf. Fig. 4(d) and Fig. S7 in the Supplementary Material].

The used helicoidal coordinates $({\xi }_1,{\xi }_2,{\xi }_3)$ are related to Cartesian coordinates $(x,y,z)$ via25$$(x,y,z)=({\xi }_1\cos (\alpha {\xi }_3)+{\xi }_2\sin (\alpha {\xi }_3),-{\xi }_1\sin (\alpha {\xi }_3)+{\xi }_2\cos (\alpha {\xi }_3),{\xi }_3).$$(6)

Application of this coordinate transformation maps a twisted waveguide to a straight waveguide, which is invariant along the ${\xi }_3$ coordinate. In this case, the effect of twisting is encoded in an anisotropic permittivity and permeability tensor, as described in more detail in Sec. S8 in the Supplementary Material.

The results returned by the mode solver are the fields in the $xy$ plane at $z=0$ of the lab frame and the effective index $n_{\mathrm{eff}}^{\text{helical}}$ such that the electric (or magnetic) field $\tilde{\mathbf{F}}$ in the helicoidal frame satisfies38$$\tilde{\mathbf{F}}({\xi }_1,{\xi }_2,{\xi }_3)={\mathrm{e}}^{\mathrm{i}k{\xi }_3{n}_{\mathrm{eff}}^{\text{helical}}}\mathbf{F}({\xi }_1,{\xi }_2).$$(7)

Modes of a twisted waveguide can in general not be described by an effective index $n_{\mathrm{eff}}^{\mathrm{lab}}$ in the lab frame (for a detailed discussion see the Supplementary Material of Ref. 41). In brief, because the spatial phase and intensity profile of the mode follow the rotation of the waveguide, the phase at a certain point $xy$ in the lab frame might not increase linearly in $z$. In particular, if you choose a point $xy$ close to the corners of the hexagon, this point might lie outside of the waveguide core if you change $z$. Therefore, only coordinate systems in which the waveguide is invariant along one coordinate, as it is in the helicoidal frame, can accurately describe modes in twisted waveguides.

Nonetheless, it is possible to dissect a mode in a twisted waveguide into its different angular momentum states and define an individual effective index in the lab frame for each of these states, given that they are rotationally invariant (i.e., a mode of a twisted waveguide has multiple effective indices in the lab frame). For an angular momentum state with OAM $l\in \mathbb{Z}$ and spin $s=\pm 1$, its effective index in the lab frame $n_{\mathrm{eff}}^{\mathrm{lab}}$ is related to the effective index of the mode in the helicoidal frame $n_{\mathrm{eff}}^{\text{helical}}$ (see full derivation in Ref. 41): $$n_{\mathrm{eff}}^{\text{lab}}(s,l)=n_{\mathrm{eff}}^{\text{helical}}+(s+l)\frac{\alpha \lambda }{2\pi }.$$(8)

Note that the imaginary part of the effective index is identical in the two coordinate frames. In the case of light cages, the results of the OAM decomposition in Sec. 4 show that there is only a single dominant OAM state with the amplitude of the remaining OAM harmonics being at least three orders of magnitude smaller. Therefore, it is justified to neglect these higher-order states and define a single effective index $n_{\mathrm{eff}}^{\text{lab}}$ using only the dominant orders $s$ and $l$ in Eq. (8). This fact is used in Sec. 5 to evaluate the measurable circular birefringence in the lab frame, which is about three orders of magnitude smaller than the splitting in $n_{\mathrm{eff}}^{\text{helical}}$. The twist rate dependence of $n_{\mathrm{eff}}^{\text{helical}}$ is therefore dominated by the term $(s+l)\frac{\alpha \lambda }{2\pi }$. By neglecting the twist rate dependence of $n_{\mathrm{eff}}^{\text{lab}}$, Eq. (8) can therefore be used in Sec. 3 to approximate the effective index in the helicoidal frame based on the effective index of the untwisted waveguide.

To evaluate the angular momentum components of a waveguide mode in Sec. 4, an orthonormal basis set with OAM-carrying basis states is required. Here, we use Bessel beams, because they are closely related to the Fourier basis used in Cartesian coordinates.90 Because the decomposition is carried out for simulation results defined in a finite area, boundary conditions need to be imposed on the basis states. Here, we set the basis states ${\mathrm{\Psi }}_{lp}$ to zero at a certain radius $R_0$ from the origin, yielding90$${\mathrm{\Psi }}_{lp}(\rho ,\varphi )=\frac{1}{\sqrt{N_{lp}}}{J}_l\left(\frac{\rho }{R_0}{u}_{l,p}\right){\mathrm{e}}^{\mathrm{i}l\varphi },$$(9)where $J_l(x)$ is the $l$’th order Bessel function of the first kind, $u_{l,p}$ is the $p$’th root of $J_l(x)$, and $N_{lp}=\pi R_0^2{J}_{l+1}^2(u_{l,p})$ is a normalization constant. We then dissect the transverse electric field components of the waveguide mode into its spin components $E_{s=+1}$ and $E_{s=-1}$ by projecting onto the circular basis $|s⟩=1/\sqrt{2}(1,s\mathrm{i})$. These components $E_s(\rho ,\varphi )$ are then individually expanded as $$E_s(\rho ,\varphi )=\sum _{l=-\infty }^{\infty }\sum _{p=1}^{\infty }{a}_{l,p}{\mathrm{\Psi }}_{lp}(\rho ,\varphi ),$$(10)with the complex amplitudes $$a_{l,p}={\int }_0^{R_0}\mathrm{d}\rho {\int }_0^{2\pi }\mathrm{d}\varphi E_s(\rho ,\varphi ){\mathrm{\Psi }}_{lp}^{*}(\rho ,\varphi )\rho .$$(11)

The full results of such a decomposition, including examples of some relevant Bessel basis functions, are shown in Sec. S10 in the Supplementary Material for the RCP fundamental mode of the untwisted light cage ($j_0=-1$). Convergence of the decomposition was checked by computing the sum of all probabilities, yielding $\sum _l\sum _p|a_{l,p}{|}^2=(1-3.5)\times {10}^{-5}$. The maximal order of $p$ was chosen such that any further rise in $p$ would increase the sum of probabilities by about the same amount as a further rise in the maximal value of $|l|$ [see Figs. S9(c) and S9(d) in the Supplementary Material]. A further choice that has to be made is the value of $R_0$, which is the radius of the circle, on which the Bessel functions are defined. Changing $R_0$ mostly changes the amplitude distribution among the different radial orders $p$ but has little impact on the OAM distribution $\sum _p|a_{l,p}{|}^2$. To minimize the impact of this ambiguous choice, all OAM decompositions were performed for 10 different values of $R_0$ ranging from 13 to $16\mu \mathrm{m}$. The resulting standard deviations are shown as error bars in Figs. S10 and S11 in the Supplementary Material and indicate an increasing error for larger values of $|l|$.

To evaluate the angular momentum flux $⟨j⟩$ of the waveguide mode in Eq. (4), we use the following conservation law relating the total angular momentum density $\mathbf{j}$ to the angular momentum flux density $\underset{\_}{\mathbf{M}}$: $$\frac{\partial j_i}{\partial t}+\sum _l\frac{\partial M_{l,i}}{\partial x_l}=0,$$(12)where $M_{l,i}$ describes the flux of the $i$ component of the angular momentum through a surface oriented perpendicular to the $l$ direction. The components of the tensor $\underset{\_}{\mathbf{M}}$ are defined in Ref. 91. The measurable total angular momentum in a waveguide is characterized by $M_{z,z}$, which can be decomposed into spin and OAM contributions. Integrated over the whole beam, the spin contribution ${\mathcal{M}}_{z,z}^s$ and OAM contribution ${\mathcal{M}}_{z,z}^l$ read91$${\mathcal{M}}_{z,z}^s=\frac{1}{2\omega }\mathfrak{I}\iint \mathrm{d}x\mathrm{d}y(E_x{H}_x^{*}+E_y{H}_y^{*}),$$(13a)$${\mathcal{M}}_{z,z}^l=\frac{1}{4\omega }\mathfrak{I}\iint \mathrm{d}x\mathrm{d}y(-H_x^{*}\frac{\partial E_y}{\partial \varphi }+E_y\frac{\partial H_x^{*}}{\partial \varphi }-E_x\frac{\partial H_y^{*}}{\partial \varphi }+H_y^{*}\frac{\partial E_x}{\partial \varphi }).$$(13b)

To calculate the angular momentum flux $⟨j⟩$, this result needs to be normalized by the total energy flux of the beam $P_z=\iint \mathrm{d}x\mathrm{d}y⟨{\mathbf{S}}_{\mathit{z}}⟩$, yielding $$⟨s⟩=\frac{{\mathcal{M}}_{z,z}^s\omega }{P_z},⟨l⟩=\frac{{\mathcal{M}}_{z,z}^l\omega }{P_z},⟨j⟩=⟨l⟩+⟨s⟩.$$(14)

To determine the optical properties of the fabricated samples, the transmission of white light through the waveguides was measured as shown in Fig. S19 in the Supplementary Material. The setups consist of a broadband supercontinuum laser source (SuperK Fianium, NKT Photonics, wavelength range: 390 to 2400 nm, repetition rate: 152 kHz to 80 MHz, output power: 5 to 15 mW in a 10 nm window), in- and out coupling objectives mounted on 3D translation stages [Olympus, 20×, numerical aperture $(\text{NA})=0.4$; Olympus, 10×, $\text{NA}=0.25$], a charge-coupled device (CCD) camera (Thorlabs DCU223C) for imaging the waveguide mode, and a spectrometer (Princeton Instruments Acton MicroSpec 2500i, grating period: $300\mathrm{g}/\mathrm{mm}$, blaze angle: 750 nm, spectral resolution: $\mathrm{\Delta }\lambda =0.13\mathrm{nm}$, detector: Princeton Instruments Acton Pixis 100) connected to a multimode-fiber (M15L05, core size: $105\mu \mathrm{m}$). Light is coupled to the fundamental mode of the waveguide, which is optimized by beam steering and shifting the objective on a 3D translation stage (Elliot Martock MDE122). The process is monitored by imaging the core mode onto the camera and optimizing for the highest pixel intensity while preserving the shape of the fundamental mode. In the second step, the power coupled to the fiber of the spectrometer is maximized. All recorded spectra are normalized to a reference spectrum taken without a sample and the objectives moved closer together to compensate for the missing length of the waveguide. Mode images at different wavelengths were recorded using the wavelength selector of the supercontinuum source (SuperK SELECT, smallest transmission bandwidth: 10 nm). To measure the transmission of circularly polarized light, two beams of opposite circular polarization are prepared using an interferometer arrangement with two linear polarizers (Thorlabs LPVIS100, 550 to 1500 nm) and a broadband quarter waveplate (Thorlabs AHWP05M-980, 690 to 1200 nm), as explained in Sec. S15 in the Supplementary Material. By blocking one arm of the interferometer, a certain circular polarization state (LCP or RCP) can be selected without making any mechanical movements on the optical components. This ensures that the incoupling conditions to the waveguide are identical for both polarizations.

Johannes Bürger received his PhD in physics from Ludwig-Maximilians-Universität Munich (Germany) in 2024, specializing in micro- and nanophotonics. His PhD research focused on 3D nanoprinting, hollow-core waveguides, metasurfaces for OAM holography, chiral BIC metasurfaces, and nanoparticle spectroscopy. Currently, he is a postdoctoral researcher at CNR Nanotec (Italy), investigating very strong coupling phenomena in two-dimensional material heterostructures.

Jisoo Kim completed his PhD in 2023 at Friedrich Schiller University of Jena (Germany) and the Leibniz Institute for Photonic Technologies (IPHT), focusing on 3D nanoprinting of hollow-core waveguides. His research explored applications of these waveguides in liquid sensing, fluorescence spectroscopy, and nanoparticle tracking. Since graduating, Jisoo has been working as an optical engineer at ASML in Veldhoven (Netherlands), specializing in predevelopment projects to drive advancements in optical system technologies.

Thomas Weiss is a professor of theoretical physics at the University of Graz (Austria). Before that, he was an assistant professor at the University of Stuttgart and a postdoctoral researcher at the Max-Planck Institute for the Science of Light in Erlangen (both in Germany). His research is centered around theoretical micro- and nanooptics, with a special focus on resonant phenomena in open systems, chiral light–matter interaction, and integrated photonics.

Stefan A. Maier graduated with his PhD in applied physics from Caltech in 2003. He currently is the head of the School of Physics and Astronomy at Monash University and the Lee-Lucas Chair in Experimental Physics at Imperial College London.

Markus A. Schmidt is a professor of fiber optics at the Friedrich Schiller University of Jena (Germany) and head of the Department of Fiber Photonics at the Leibniz Institute for Photonic Technologies (IPHT). His research interests lie in the field of photonics in combination with nano- and microstructured waveguides with applications in areas such as nano- and biophotonics and nonlinear optics, with a current focus on 3D-nanoprinted holograms and metasurfaces on optical fiber endfaces.