## Abstract

Helical structures exhibit novel optical and mechanical properties and are commonly used in different fields such as metamaterials and microfluidics. A few methods exist for fabricating helical microstructures, but none of them has the throughput or flexibility required for patterning a large surface area with tunable pitch. In this paper, we report a method for fabricating helical structures with adjustable forms over large areas based on multiphoton polymerization (MPP) using single-exposure, three dimensionally structured, self-accelerating, axially tunable light fields. The light fields are generated as a superposition of high-order Bessel modes and have a closed-form expression relating the design of the phase mask to the rotation rate of the beam. The method is used to fabricate helices with different pitches and handedness in the material SU-8. Compared to point-by-point scanning, the method reported here can be used to reduce fabrication time by two orders of magnitude, paving the way for adopting MPP in many industrial applications.

© 2022 Chinese Laser Press

## 1. INTRODUCTION

In the last two decades, chiral metamaterials with helical structures have attracted significant attention because they exhibit novel optical properties, such as circular dichroism [1], optical activity [2], and negative refractive index [3,4]. Helical structures have many applications including broadband circular polarizers [2,5], circular polarization conversion [6], wave plates [7], and broadband polarization-insensitive absorbers [8]. Helical structures can also be found in microfluidic mixers [9,10], heat diffusors [11,12], and scaffolds for growing cells [13].

Producing helical structures requires nonplanar fabrication methods such as multiphoton polymerization (MPP). MPP is a direct laser writing technique that is increasingly used [1,5,6,14–18] because it provides a means for creating almost arbitrarily shaped, three-dimensional (3D) structures with high resolution. However, MPP is time consuming due to the point-by-point writing strategy. The slow fabrication speed of conventional MPP is the key factor that limits its wider adoption for industrial manufacturing over large surface areas. One approach for increasing fabrication speed is to reshape the laser beam so that one entire layer or even the complete structure could be fabricated with a single exposure. Layer-by-layer fabrication of various extruded tube-like 3D structures has been demonstrated by focusing shaped Bessel beams [19–23]. The fabrication is achieved by scanning the focused beam along the optical axis, but the increase in throughput is limited because axial scanning is still needed. Volumetric fabrication with a single exposure has also been demonstrated, including the fabrication of high-aspect-ratio structures using Bessel [24,25] and axilens beams [26]. However, the shape of the structures is limited by the needle-like focal shape and thus lacks variety. Vortex beams have been used to fabricate 3D helical structures with limited aspect ratios [27–29]. For good optical and mechanical functionality, helical structures are often required to have multiple pitches and high aspect ratios, such as those used as microfluidic mixers [9,10]. Metamaterials consisting of high-aspect-ratio helices with multiple pitches exhibit superior performance [2,5,14,30,31]. However, methods reported in Refs. [27–29] achieved only low-aspect-ratio helical structures with less than one pitch. So far, no method has the throughput required for fabricating over large areas with tall helices that have variable pitch.

This work reports a fabrication approach based on MPP with a single exposure using a new class of self-accelerating beams (SABs) with rotating intensity distribution [32–35]. The proposed SABs are based on the superposition of two high-order Bessel beams. Compared to the existing radial SABs with constant rotation rates, the proposed beams feature tunable angular acceleration (variable pitch). Moreover, the rotation rate (and hence pitch) can be tailored to follow a nearly arbitrary profile within the limit of optical arrangement, while other SABs with angular acceleration reported in Refs. [36–38] follow a specific equation with a single tuning parameter. We have developed an analytical model to describe the generation and propagation of the SABs. A closed-form solution for the intensity distribution and rotation rate is derived, which gives insight into the relationship between the computer-generated hologram (CGH) and the beam profile it generates. The analytic model makes it easier to synthesize 3D beams and contrasts with iterative algorithms used by others, which can produce useful solutions, but yield little intuition into how the beams are generated and are time consuming. Micro-helices with constant and varying pitches are fabricated with single exposures in the forms of individual helices and matrices of helices. With optimization, this method can be used to increase throughput by more than two orders of magnitude over conventional MPP and paves the way for mass microfabrication of helical structures for industrial applications, including metamaterials, microfluidics, and biomaterials.

## 2. EXPERIMENTAL SETUP

Figure 1(a) shows the fabrication system. The light-source is a femtosecond laser (Pharos, Light Conversion) that generates pulses having a temporal full-width at half-maximum (FWHM) of 170 fs, center wavelength of ${\lambda}_{0}=1030\text{\hspace{0.17em}}\mathrm{nm}$, and repetition rate of 100 kHz. The output beam is frequency-doubled by a beta barium borate (BBO) crystal, expanded and collimated, and sent to a spatial light modulator (SLM), which is a reflecting, phase-only, liquid-crystal-on-silicon device (Meadowlark Optics, $1920\times 1150\text{\hspace{0.17em}}\mathrm{pixels}$, 256 gray levels, 9.2 μm pixel-pitch). CGHs generated with the proposed method are displayed on the SLM. The CGHs are the key to generating the proposed SABs (see Section 3). The frequency-doubled beam is reflected off the SLM and directed into a $4\text{-}f$ system composed of a plano–convex lens ($f=750\text{\hspace{0.17em}}\mathrm{mm}$) and an objective lens ($10\times $, $\mathrm{NA}=0.3$, focal $\mathrm{length}=18\text{\hspace{0.17em}}\mathrm{mm}$) and then spatially filtered by an iris placed at the Fourier plane of the $4\text{-}f$ system [Fig. 1(d)]. The beam near the focal plane of the objective lens is a demagnified replica of that formed on the SLM. The demagnified beam is directed into a layer of photoresist (SU-8 2075, Kayaku Advanced Materials) that was spin-coated on a glass substrate [Fig. 1(c)] and baked prior to exposure to remove solvent. After exposure, the sample is baked to activate cross-linking and developed in propylene glycol methyl ether acetate (PGMEA) to remove unexposed material. Figures 1(e) and 1(f) show a comparison between the simulated beam shape and fabricated micro-helices. The Fresnel diffraction theory is used to simulate the SABs in the photoresist. An example of an iso-intensity contour of the simulated SABs is plotted in Fig. 1(e) together with cross sections of the intensity distribution. Figure 1(f) shows a $30\times 30$ matrix of helices with a single helix highlighted in yellow. Each helix is formed volumetrically with a single exposure of five laser pulses. The matrix is fabricated by translating the sample after each helix is exposed.

## 3. THEORY AND CGH DESIGN

A light field with rotating intensity distribution can be generally described as

where $I(r,\theta ,z)$ is the intensity expressed in a cylindrical coordinate, and $W(z)z$ is the phase term that results in rotation. The rotation rate $\omega (z)$ isVarious types of beams have been generated with constant rotation rates. Here we present a general method that produces SABs with axially tunable $\omega (z)$.

The foundation of our SABs is the superposition of two high-order Bessel beams and the use of radial-to-axial (“$r\text{-}z$”) mapping [39]. The Bessel-beam phase [BBP, Fig. 1(b)] of two Bessel beams with orders $l$ and $m$ can be expressed in terms of a common radial phase component ${k}_{\perp}r$ and their individual azimuthal phase components $l\theta $ and $m\theta $ [40–43]:

Superposing two Bessel beams having ${\mathrm{BBP}}_{l}$ and ${\mathrm{BBP}}_{m}$ generates a superposed Bessel beam [44,45] that is still “diffraction free,” and its transversal intensity profile at any $z$ has $|l-m|$ petal-like local maxima that are $|l-m|$-fold rotationally symmetric with respect to the optical axis. Inspired by the concept of geometrical $r\text{-}z$ mapping [39], we introduce a rotation-tuning function $v(r)$ to modulate the phase along the radial direction. This extra phase term, which we call “radial phase modulation” (RPM), is written as

The superposition of these two components gives a transmittance function $T(r,\theta )=\mathrm{exp}[-i({\mathrm{BBP}}_{l}+{\mathrm{RPM}}_{l})]+\mathrm{exp}[-i({\mathrm{BBP}}_{m}+{\mathrm{RPM}}_{m})]$ that is implemented using the SLM. As $T(r,\theta )$ has both phase and amplitude information, an encoding method is needed for a phase-only SLM. The method we use is based on a pair of complementary “binary amplitude masks” [BAMs, Fig. 1(b)] with checkerboard patterns [46]. After encoding, the phase becomes

The encoded phase $\phi (r,\theta )$ is displayed on the SLM as it is illuminated with a Gaussian beam that is expanded to almost fill the aperture of the device. A replica of $T(r,\theta )$ with demagnification $M$ is obtained at the image plane of the $4\text{-}f$ system with a low-pass filter applied at the Fourier plane. The resulting transmittance function can be written as $T(Mr,\theta -\pi )$.

Using the Fresnel diffraction theory and the stationary phase method [40,47,48], we can derive a closed-form solution for the intensity distribution at point $(r,\theta ,z)$ after the image plane (see Appendix A). The solution has a form similar to Eq. (1) except for one term, ${I}_{z}(z)=z\text{\hspace{0.17em}}\mathrm{exp}[-2{k}_{\perp}^{\prime}2{z}^{2}/({k}^{2}{w}_{0}^{\prime}2)]$, where $k$ represents the wave vector, ${k}_{\perp}^{\prime}=M{k}_{\perp}$, ${w}_{0}^{\prime}={w}_{0}/M$, and ${w}_{0}$ represents the beam waist of the Gaussian beam. ${I}_{z}(z)$ describes the intensity variation along the $z$ axis. The depth of field is finite due to Gaussian illumination, which is similar to the property of a Gaussian–Bessel beam generated by an axicon lens or axicon phase [42,49,50]. To study the rotation of the intensity distribution, this term can be neglected for now. Using Eq. (2), the rotation rate $\omega (z)$ can be determined as

The terms on the right-hand side of Eq. (6) differ by ${v}^{\prime}z$ and $v$, where ${v}^{\prime}$ represents the first derivative of $v$ with respect to $z$. Equation (6) establishes a “forward” relationship from $v(r)$ to $\omega (z)$. When $v$ is a constant, ${v}^{\prime}=0$ and $\omega =-{k}_{\perp}{M}^{2}v/k$, and Eq. (6) then reduces to the conclusion in Ref. [33] under the paraxial approximation.

We are more interested in generating an SAB that has a pre-defined $\omega (z)$, in other words, the “reverse” relationship between $v(r)$ and $\omega (z)$. To find $v(r)$ that corresponds to an arbitrary $\omega (z)$, one can rewrite Eq. (6) as an ordinary linear differential equation, which has a general solution of

## 4. CHARACTERIZATION OF SELF-ACCELERATING BEAMS IN FREE SPACE

The SABs generated by the proposed method are measured in free space to validate the theory. A CMOS camera with a $20\times $ objective lens shown in Fig. 1(a) is mounted on a motorized stage to image a series of transverse ($x\text{-}y$) intensity distributions at positions $z$ along the optical axis [Figs. 2(c) and 2(g)].

The first example is an SAB with a constant rotation rate $\omega =-1.26\times {10}^{5}\text{\hspace{0.17em}}\mathrm{rad}\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$ in air, phase orders $(l,m)=(-10,-9)$, and ${k}_{\perp}=6.6\times {10}^{4}\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$. The minus sign of $\omega $ means the intensity profile rotates clockwise when viewed opposite from the propagation direction. The experimentally measured transverse intensity profile [Fig. 2(b)] agrees well with the simulation results [Fig. 2(c)]. The SAB has a “C”-shaped transverse profile with a radius around 5 μm. While the beam propagates along the optical axis, the C shape rotates clockwise. Note that the radius can be controlled independently by the choice of phase orders $l$ and $m$ (see Appendix B), or by changing ${k}_{\perp}$. The SAB shows a long depth of focus of over 300 μm, which is limited by the SLM aperture and Gaussian beam width. In 3D, the intensity distribution of SABs has a helical shape that resembles a twisted strap. We quantify the rotation by extracting the directional angle $\alpha $ from $x\text{-}y$ intensity profiles over a fixed axial distance $\mathrm{\Delta}z$ and calculating the rotation rate as $\omega (z)=\mathrm{\Delta}\alpha /\mathrm{\Delta}z$. As shown in Fig. 2(d), the measured $\omega (z)$ is close to the designed value. The fluctuation observed in Fig. 2(d) is due to the distortions in the beam profiles, which affects the accuracy of extracted angles.

Another class of SAB has rotation rate $\omega (z)$ that varies with respect to $z$ (“chirping”). Figures 2(e)–2(h) show an example with $\omega (z)=-25.13{z}^{-1}\text{\hspace{0.17em}}\mathrm{rad}\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$ in air, phase order $(l,m)=(-10,-9)$, and ${k}_{\perp}=6.6\times {10}^{4}\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$. This SAB rotates with a rate that decelerates as it propagates. The white sectors in Fig. 2(f) show that $\mathrm{\Delta}\alpha (z)$ reduced between two 10 μm intervals. We also observe a deviation from the usual C shape in both experiment and simulation at the start of the SABs when $\omega (z)$ exceeds $2\times {10}^{5}\text{\hspace{0.17em}}\mathrm{rad}\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$. Here the beam appears as a “double helix” [left side of Fig. 2(e)] and this indicates the existence of an optimal axial range within which good helical shapes can be obtained with this method. In theory, the highest achievable rotation rate is determined by the SLM pixel size and the demagnification ratio of the $4\text{-}f$ system. However, there is a limit of the highest rotation rate, beyond which the beam starts to get distorted as shown in Fig. 2(e) (between $z=0.1$ and 0.15 mm). The SABs reported here could be used in applications beyond fabrication, such as particle manipulation [52] and microfluidic pumping [53,54].

## 5. PROPAGATION THROUGH INTERFACES

The discussion so far is in free space. In actual fabrication, the SABs pass through multiple interfaces (air–glass, glass–SU-8). Therefore, it is important to understand how SABs propagate through a planar interface of two media with different refractive indices $n$. Because the glass substrate is thin ($\sim 150\text{\hspace{0.17em}}\mathrm{\mu m}$) and its index is close to that of the SU-8, only the air–SU-8 transition is considered. By applying the angular spectrum method [55] and paraxial approximation, the intensity distribution is derived for the SAB after propagating into the photoresist (see Appendix C), from which a few observations can be made. First, the depth of focus elongates by a factor of $n$ compared to that in free space. Second, the intensity pattern in the $x\text{-}y$ plane does not change after the interface. Finally, the expression of rotation rate $\omega (z)$ can also be obtained by using Eq. (2). We observed that $\omega (z)$ becomes smaller when the beams propagate into a medium with higher $n$. As seen in Appendix C, good agreement is found between the theoretical and simulated profiles (Fig. 9).

## 6. FABRICATION OF MICRO-HELICES

Equation (7) was used to design CGHs that generate SABs for fabricating helices in SU-8 ($n=1.6$ at ${\lambda}_{0}=515\text{\hspace{0.17em}}\mathrm{nm}$). The CGHs have phase order $(l,m)=(-10,-9)$ and ${k}_{\perp}=6.6\times {10}^{4}\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$. Each helix was patterned with static exposure (no scanning) and an exposure time of less than 0.15 ms. The resulting structures were imaged by scanning electron microscopy (SEM). The results are shown in Fig. 3. The length of helical structures is limited by the thickness of the spin-coated SU-8 layer, and variations are due to “edge beads” [56]. The lengths all exceed 150 μm, resulting in aspect ratios over 15. Because of the high aspect ratios, these helices have weak mechanical strength and do not withstand capillary forces or fluid-flow during development. They drift with the solvent and are found lying on the glass substrate after development.

The left column of Fig. 3 shows the beams used to produce the helical structures. The first two helices [Figs. 3(a) and 3(b)] have constant rotation rate $\omega =-5.61\times {10}^{4}\text{\hspace{0.17em}}\mathrm{rad}\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$ and $-7.85\times {10}^{4}\text{\hspace{0.17em}}\mathrm{rad}\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$, and the corresponding pitch is 112 μm and 80 μm, respectively. We can clearly see the difference in pitch from the SEM images. The third helix [Fig. 3(c)] has the opposite rotation rate (handedness) to that in Fig. 3(b). The change of handedness is achieved simply by reversing the sign in Eq. (7). The last two helices [Figs. 3(d) and 3(e)] have axially variable rotation rates $\omega =4.12\times {10}^{8}(z-1.12\times {10}^{-4})\text{\hspace{0.17em}}\mathrm{rad}\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$ (“accelerating”) and $\omega =-25.13{z}^{-1}\text{\hspace{0.17em}}\mathrm{rad}\text{\hspace{0.17em}}{\mathrm{m}}^{-1}$ (“decelerating”), respectively. Careful examination (see below) shows that the pitches of these helices match the beam shapes. Note that the threshold used in the iso-intensity contours is set arbitrarily for illustration purposes, and it might differ from the actual threshold in fabrication. Therefore, deviation in shape between SABs and fabricated helices is expected. There is slight bending and deformation in the fabricated helices due to the high aspect ratios. With positive-tone photoresists [57], this method could be used to fabricate helical microchannels for microfluidic mixers.

To examine the shape of fabricated helices quantitatively, we use images of single helices and trace their edges. An example is shown in Fig. 4 for a decelerating helix. We have tested various edge-detection algorithms (see Appendix D) and found that their accuracy varies with the specific SEM images given. Therefore, we have decided to extract the edge manually. Since part of the edge is unseen from the SEM image [Fig. 4(a)], a transmission optical microscopic image is used to perform measurements [Fig. 4(b)]. Then the distance $d$ to the optical axis is measured at different $z$ locations, as shown in Fig. 4(b). The measurement is compared with theoretical prediction calculated by Eq. (6), and a good agreement is found.

By performing a series of single exposures with lateral translation, helical matrices can be fabricated. Figure 5 shows several matrices, each consisting of $30\times 30$ helices with a certain pitch and handedness. The fabrication time for each matrix is approximately 15 min, and most of the time is spent on translating the sample. In terms of fabrication volume per unit time, our method is more than 100 times higher than point-by-point scanning for similar structures reported in Ref. [1] (detailed discussion in Appendix E). The fabricaiton time could be further reduced with a galvanometer scanner. The helical structures shown in Fig. 5 can be used as metamaterials at terahertz frequencies [58,59]. Further reduction in pitch can be achieved by tighter focusing (which may require the use of vectorial diffraction theories) and will enable application at the optical frequencies.

## 7. CONCLUSION

We have demonstrated a method for rapid microfabrication of helical structures with tunable axial shapes by developing a new class of SABs and adapting them to MPP. The SABs are based on the superposition of high-order Bessel modes and form non-diffracting irradiance profiles that rotate along the optical axis. An algorithm was developed to design CGHs that form the SABs. A closed-form expression was derived that can be used to directly synthesize targeted CGH and SAB without resorting to iterative algorithms that yield non-intuitive results. SABs can be generated with independently adjustable transverse width and rotational pitch. Using the concept of $r\text{-}z$ mapping, the pitch of the helix can be made to vary with propagation distance $z$. Single and matrices of helices with various pitch and handedness were fabricated in SU-8 with good agreement with the beam shapes. Such helical structures could have applications in metamaterials [1,2,5], microfluidics [9,10], and biomaterials [13]. A reduction in fabrication time exceeding two orders of magnitude over conventional point-by-point scanning MPP is estimated for a fully optimized system. As MPP is maturing towards industrial applications, our method addresses the issue of throughput, and is a step forward for mass microfabrication over large surface areas.

## APPENDIX A: GENERATION AND PROPAGATION OF SABS IN FREE SPACE

An SAB with rotating intensity distribution can be generally described as

where $I(r,\theta ,z)$ is the intensity expressed in cylindrical coordinates. The intensity distribution of such a beam remains unchanged along $z$ when observing in a coordinate frame that rotates at the same rate $\omega (z)$. The rotation rate is given byA special group of SABs with constant $\omega $ has been discussed previously [32,33]. In the present work, SABs are defined in terms of an axially tuned rotation rate $\omega (z)$ using a phase-only SLM. We start from an amplitude transmittance function:

where $(r,\theta )$ are the polar coordinates in the plane $z=0$, ${k}_{\perp}$ is the radial component of the wave vector, and $l$ is an integer that represents the order of a Bessel beam [40–43]. Superposition of two such transmittance functions generates superposed Bessel beams with orders of $l$ and $m$ [44,45]. Such beams are still “diffraction-free,” and their transverse intensity profiles at any given $z$ location have $|l-m|$ petal-like local maxima with $|l-m|$-fold rotational symmetry with respect to the optical axis. We will build the SAB based on such transversal intensity profiles. First, we need to find a method to control the rotation rate $\omega (z)$. Inspired by the geometrical radial-to-axial (“$r\text{-}z$”) mapping between phase maps and generated Bessel beams [39], we introduce a rotation-tuning function $\upsilon (r)$ to modulate the phase along the radial direction. The transmittance functions can then be expressed asThe superposition ${t}_{\mathit{lm}}(r,\theta )={t}_{l}(r,\theta )+{t}_{m}(r,\theta )$ is the desired transmittance function with amplitude and phase information. The task at hand is to find the relationship between $\upsilon (r)$ and $\omega (z)$. To this end, we derive the light field intensity resulting from ${t}_{\mathit{lm}}(r,\theta )$ and extract such a relationship. Since ${t}_{\mathit{lm}}(r,\theta )$ contains amplitude information that cannot be generated directly by a phase-only SLM, we use the double-phase-hologram technique reported in Ref. [46]. Two complementary binary amplitude gratings (checkerboard patterns) ${M}_{1}$ and ${M}_{2}$ are used:

In Eqs. (A6) and (A7), $\mathrm{sinc}(x)=\mathrm{sin}(\pi x)/\pi x$, and $p$ is the period of the binary gratings. The transmittance functions ${t}_{l}$ and ${t}_{m}$ are spatially sampled by ${M}_{1}$ and ${M}_{2}$, respectively, as

where $(x,y)$ are the Cartesian coordinates. $\phi (x,y)$ is the phase function that will be encoded on the SLM. As ${M}_{1}$ and ${M}_{2}$ are complementary, the following relation holds true:A CGH described by Eq. (A9) is displayed on the SLM, which is illuminated by a Gaussian beam with a $1/{e}^{2}$ radius ${w}_{0}$ and clipped by an aperture with a radius $R$. We use a $4\text{-}f$ system with demagnification $1/M={f}_{2}/{f}_{1}$ to obtain the Fourier spectrum $H(u,\nu )$ of ${e}^{i\phi (x,y)}$ at the Fourier plane (Fig. 6):

In Eq. (A10), $({x}_{1},{y}_{1})$ and $({x}_{f},{y}_{f})$ are the coordinates at the SLM and Fourier plane of the $4\text{-}f$, respectively; ${U}_{0}({x}_{1},{y}_{1})$ is the complex field at the input plane of the $4\text{-}f$ system. The circ function is unity for $r\le 1$ and zero elsewhere; $\lambda $ is the wavelength in vacuum; $(u,\nu )$ are spatial frequencies defined as $u={x}_{f}/{f}_{1}\lambda $ and $\nu ={y}_{f}/{f}_{1}\lambda $. Combining Eqs. (A6)–(A8) and applying the convolution theorem, we can obtain

Equations (A12) and (A13) show that we get exact retrieval of the Fourier spectrum of the original transmittance function ${t}_{l,m}(r,\theta )$ after spatial filtering. At the image plane of the $4\text{-}f$ system, without including the amplitude, we obtain a demagnified and spatially reversed complex field:

Using the Fresnel diffraction integral in cylindrical coordinates, we can write the complex field at point $(r,\theta ,z)$ after the image plane as

Substituting Eqs. (A17)–(A20) into Eq. (A16) yields

Using an integral representation of Bessel functions [60], we can calculate the second integral in Eq. (A21) as

Because the phase term $\mu ({r}_{0})$ in Eq. (A23) varies rapidly compared to $f({r}_{0})$ over the range of integration, the integral can be evaluated asymptotically using the stationary phase method [47,48]. The critical points in the integration are those at which the first derivative of $\mu ({r}_{0})$ is zero. There is one critical point ${r}_{c}=z{k}_{\perp}^{\prime}/k$ in the interval $(0,{R}^{\prime})$ when $0<z<{R}^{\prime}k/{k}_{\perp}^{\prime}$. The leading contribution of the integral in Eq. (A23) is

Now we can superpose Eq. (A27) with another complex field with an order $l$, and derive the intensity of the superposed fields as

It is obvious that ${C}_{l}/{C}_{m}=|{C}_{l}|/|{C}_{m}|{e}^{i{\theta}_{\mathit{lm}}}$. Note that the term $z\text{\hspace{0.17em}}\mathrm{exp}[-2{k}_{\perp}^{\prime 2}{z}^{2}/({k}^{2}{w}_{0}^{\prime 2})]$ in Eq. (A28) describes the intensity variation along the $z$ axis, and it means that the depth of focus is limited by size of Gaussian illumination and size of the SLM. This phenomenon is similar to the axial intensity profile observed for Gaussian–Bessel beams that are generated by focusing a Gaussian beam using an axicon lens [49,50]. The FWHM of the depth of focus is $1.13(k{w}_{0}^{\prime})/(\sqrt{2}{k}_{\perp}^{\prime})$. To study the rotation of the intensity profile on a transverse plane, we temporarily drop this term, and Eq. (A28) then becomes

Equation (A31) has the same form as Eq. (A1), so we can conclude that all iso-intensity points of the beam will fulfill the following condition:

Substituting Eq. (A18) into Eq. (A32) yields

When the function $v$ is a constant, the rotation rate $\omega =-{M}^{2}v{k}_{\perp}^{\prime}/k$, which is also a constant. For more general cases, the axially dependent rotation rate $\omega (z)$ can be determined as

Treating $v(r)$ as unknown, we can rewrite Eq. (A34) as an ordinary linear differential equation that has the following general solution:

Equation (A36) provides a “reverse” relationship between $\omega (z)$ and $v(r)$. This equation can be solved analytically or numerically to obtain a CGH that produces a beam with a specific rotation profile.

## APPENDIX B: TUNING TRANSVERSE BEAM PROFILE

Notice that Eqs. (A34) and (A36) do not contain the Bessel beam order $l$ or $m$, and this means that we can change the transverse profile by using different $l$ or $m$ without affecting the rotation rate. For example, the diameter of the helical shape can be independently selected, as shown in Fig. 7.

## APPENDIX C: PROPAGATION THROUGH AN INTERFACE

Now we consider the situation in which the beam propagates from air through the air–photoresist interface. Here we ignore the glass substrate because it is thin ($\sim 120\text{\hspace{0.17em}}\mathrm{\mu m}$) and has a refractive index similar to that of the photoresist SU-8 (the difference is 0.1 at 515 nm). We assume that (1) the image plane of the $4\text{-}f$ system discussed in the previous section is located in air and at a distance ${z}_{0}$ before the planar air–photoresist interface, and (2) the photoresist is homogeneous with a refractive index $n$ (Fig. 8). We already know the complex field at the image plane from Eq. (A14), and its Fourier spectrum is given by

When $|{f}_{x}|\ll 1$ and $|{f}_{y}|\ll 1$, the paraxial approximation can be applied, and Eq. (C1) becomes

Equation (C3) can also be obtained by Fourier transforming the Fresnel diffraction impulse response. After propagation distance of ${z}_{1}$ in the photoresist, the angular spectrum becomes

In Eq. (C4), ${k}_{n}=nk$ is the wavenumber in the photoresist, and $T({f}_{x},{f}_{y})$ is the transmittance function considering reflection loss at the interface. If we ignore the constant phase shift, which is inconsequential, and if $T({f}_{x},{f}_{y})$ does not vary significantly for different frequencies $({f}_{x},{f}_{y})$, Eq. (C4) has the same form as Eq. (C2), as if the angular spectrum has propagated in the photoresist from the start for a distance of ${z}_{1}+n{z}_{0}$. Now we perform the inverse Fourier transform on Eq. (C4), and we obtain a result similar to Eq. (A28), except that the wavenumber $k$ in air becomes ${k}_{n}$ in the photoresist, and the propagation distance becomes $z+n{z}_{0}$. Note that the quantity $z+n{z}_{0}$ is not the usual optical path length, which should read ${z}_{0}+nz$. Then we can write the intensity distribution as

A few important observations follow from Eq. (C5). First, we notice that the depth of focus is elongated by a factor of $n$ from the term $({z}_{1}+n{z}_{0})\mathrm{exp}[-2{k}_{\perp}^{\prime 2}{({z}_{1}+n{z}_{0})}^{2}/({k}_{n}^{2}{w}_{0}^{\prime 2})]$. Second, the intensity pattern in the $x\text{-}y$ plane does not change after propagation through the interface. Finally, following the same derivation used for propagation in air, we obtain the rotation rate at $z={z}_{1}+{z}_{0}$ as

## APPENDIX D: CHARACTERIZATION OF HELICAL STRUCTURES

To examine the shape of the fabricated helices quantitatively, we use images of single helices and trace their edges. An example is shown in Figs. 10 and 11 for a decelerating helix (the pitch increases with propagation). An edge-detection algorithm can be used to identify the profile, but only partial segments are obtained from SEM images because the backside of the helix is obscured. Complete profiles can be extracted from transmission optical microscopy images (Fig. 11), but the accuracy is poor because the image has low contrast and sharpness. Ultimately, profiles were extracted by manual inspection of transmission images. The distance $d$ of the edge to the optical axis at various locations was measured, and a representative plot is shown in Fig. 11(b).

## APPENDIX E: ESTIMATION OF FABRICATION TIME

Here we compare the time required to fabricate one helical structure using the high-throughput method reported in this work to that required for MPP by conventional point-by-point scanning. The scanning rate is the major bottleneck in the fabrication time for MPL, and it can be improved by moving the beam using galvo-mirrors, instead of translating the sample with a piezo-stage [61]. Galvo-mirrors routinely enable MPP with horizontal scanning speed up to $100\text{\hspace{0.17em}}\mathrm{mm}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$. More recently, *XY* writing speeds of $400\text{\hspace{0.17em}}\mathrm{mm}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$ [62] and $8000\text{\hspace{0.17em}}\mathrm{mm}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$ [63] have been realized using resonant galvo-mirrors. Given that the galvo-mirror scans the beam only in the $x\text{-}y$ plane, the fabrication speed is still limited by the speed with which the focus can be scanned in the $z$ direction. While $z$ axis scan rates of just a few $100\text{\hspace{0.17em}}\mathrm{\mu m}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$ are typical of commercial MPP systems and most commonly reported in the literature, a $z$-axis linear stage has been used to write super-micrometer features with speeds reportedly as high as $30\text{\hspace{0.17em}}\mathrm{mm}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$ [64]. To determine the highest possible throughput for fabricating helical structures by state-of-the-art MPP, we therefore assume that the helical structure is fabricated with $z$-axis scanning speed of $30\text{\hspace{0.17em}}\mathrm{mm}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$. Each helix in Fig. 5 has a length (along $z$ axis) of roughly 200 μm and is fabricated with five consecutive pulses. Given that the pulse repetition rate is 100 kHz, the corresponding fabrication time is 0.05 ms using the proposed high-throughput method, regardless of its length. In contrast, fabricating the same helix with conventional point-by-point MPP at a scan-speed of $30\text{\hspace{0.17em}}\mathrm{mm}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$ requires a minimum of 6.67 ms. Thus, the high-throughput method reported here increases throughput by two orders of magnitude. It should be pointed out that the fabrication length is limited by the thickness of the photoresist layer and can, with materials and optics optimization, approach the length of the helical beam (ca. 400 μm), for the same total fabrication time, and therefore further increase throughput.

## Funding

National Science Foundation (1711356, 1846671).

## Disclosures

The authors declare no conflicts of interest.

## Data Availability

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

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