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 [1,5,6,1418] 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 [1923]. 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 [2729]. 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. [2729] 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 [3235]. 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. [3638] 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 λ0=1030nm, 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, 1920×1150pixels, 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-f system composed of a plano–convex lens (f=750mm) and an objective lens (10×, NA=0.3, focal length=18mm) and then spatially filtered by an iris placed at the Fourier plane of the 4-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 30×30 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.

 figure: Fig. 1.

Fig. 1. High-throughput microfabrication of helical structures using self-accelerating beams (SABs). (a) Experimental setup: BBO, beta barium borate crystal; PBS, polarizing beam splitter; BE, beam expander; BS, non-polarizing beam splitter; SLM, spatial light modulator; M, mirror; L, lens; ID, iris diaphragm; Obj, objective lens; CMOS, complementary metal–oxide-semiconductor. (b) Calculation of a computer-generated hologram (CGH). Phase order l=10, m=9. (c) Illustration of exposure in the SU-8 sample. Green arrow shows the beam propagation direction. (d) Fourier spectrum of an SAB at the Fourier plane of the 4-f system. The white dashed circle represents the iris diaphragm acting as a low-pass filter to isolate the zeroth-order beam. (e) Simulation of iso-intensity and x-y intensity distribution of an SAB in SU-8. (f) SEM images of fabricated helices, including a 45° view of a 30×30 matrix (individual helix highlighted in yellow, scale bar corresponds to 30 μm.). The white-boxed inset shows a high-magnification view of a single helix.

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3. THEORY AND CGH DESIGN

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

I(r,θ,z)=I[r,θ+W(z)z],
where I(r,θ,z) is the intensity expressed in a cylindrical coordinate, and W(z)z is the phase term that results in rotation. The rotation rate ω(z) is
ω(z)=W(z)dW(z)dzz.

Various types of beams have been generated with constant rotation rates. Here we present a general method that produces SABs with axially tunable ω(z).

The foundation of our SABs is the superposition of two high-order Bessel beams and the use of radial-to-axial (“r-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 kr and their individual azimuthal phase components lθ and mθ [4043]:

BBPl=kr+lθ,BBPm=kr+mθ.

Superposing two Bessel beams having BBPl and BBPm generates a superposed Bessel beam [44,45] that is still “diffraction free,” and its transversal intensity profile at any z has |lm| petal-like local maxima that are |lm|-fold rotationally symmetric with respect to the optical axis. Inspired by the concept of geometrical r-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

RPMl=RPMm=lm2v(r)r.

The superposition of these two components gives a transmittance function T(r,θ)=exp[i(BBPl+RPMl)]+exp[i(BBPm+RPMm)] that is implemented using the SLM. As T(r,θ) 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

φ(r,θ)=(BBPl+RPMl)×BAMl+(BBPm+RPMm)×BAMm.

The encoded phase φ(r,θ) 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,θ) with demagnification M is obtained at the image plane of the 4-f system with a low-pass filter applied at the Fourier plane. The resulting transmittance function can be written as T(Mr,θπ).

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,θ,z) after the image plane (see Appendix A). The solution has a form similar to Eq. (1) except for one term, Iz(z)=zexp[2k2z2/(k2w02)], where k represents the wave vector, k=Mk, w0=w0/M, and w0 represents the beam waist of the Gaussian beam. Iz(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 ω(z) can be determined as

ω(z)=kkM2v(M2kkz)zkkM2v(M2kkz).

The terms on the right-hand side of Eq. (6) differ by vz and v, where v represents the first derivative of v with respect to z. Equation (6) establishes a “forward” relationship from v(r) to ω(z). When v is a constant, v=0 and ω=kM2v/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 ω(z), in other words, the “reverse” relationship between v(r) and ω(z). To find v(r) that corresponds to an arbitrary ω(z), one can rewrite Eq. (6) as an ordinary linear differential equation, which has a general solution of

v(r)=1rM2kkω(M2kkr)dr,
where r=M2kz/k. Equation (7) provides a reverse relationship between ω(z) and v(r) and is the key to the generation of SABs in this work. The function v(r) that is needed in the CGH can be calculated from Eq. (7) either analytically or numerically. This makes it possible to tailor the helical shape on demand without time-consuming phase-retrieval algorithms [51]. We also notice that the rotation of our SABs is not associated with orders l and m. This means that there are some degrees of freedom for tuning the transverse intensity profile by changing l and m while maintaining ω(z) (Fig. 7 in Appendix B).

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× objective lens shown in Fig. 1(a) is mounted on a motorized stage to image a series of transverse (x-y) intensity distributions at positions z along the optical axis [Figs. 2(c) and 2(g)].

 figure: Fig. 2.

Fig. 2. Comparison of simulated and measured helical beams with (a)–(d) constant and (e)–(h) decelerated rotation rate. Phase order l=10 and m=9. (a), (e) Iso-intensity contours of simulated SABs in air. The threshold intensity (normalized) is set to 0.5. (b), (f) Simulated and (c), (g) measured x-y intensity distribution. Scale bar represents 5 μm. (d), (h) Rotation rate ω along optical axis. Red curves, designed profiles; orange points, measured data; blue dots, simulation results.

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The first example is an SAB with a constant rotation rate ω=1.26×105radm1 in air, phase orders (l,m)=(10,9), and k=6.6×104m1. The minus sign of ω 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. 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 α from x-y intensity profiles over a fixed axial distance Δz and calculating the rotation rate as ω(z)=Δα/Δz. As shown in Fig. 2(d), the measured ω(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 ω(z) that varies with respect to z (“chirping”). Figures 2(e)–2(h) show an example with ω(z)=25.13z1radm1 in air, phase order (l,m)=(10,9), and k=6.6×104m1. This SAB rotates with a rate that decelerates as it propagates. The white sectors in Fig. 2(f) show that Δα(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 ω(z) exceeds 2×105radm1. 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-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 (150μm) 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-y plane does not change after the interface. Finally, the expression of rotation rate ω(z) can also be obtained by using Eq. (2). We observed that ω(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 λ0=515nm). The CGHs have phase order (l,m)=(10,9) and k=6.6×104m1. 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.

 figure: Fig. 3.

Fig. 3. Comparison between helical beams simulated in SU-8 and fabrication results obtained with five SABs having different types of rotation. The contrast of each SEM image was adjusted individually to improve visibility and aid comparison. Scale bar corresponds to 20 μm.

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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 ω=5.61×104radm1 and 7.85×104radm1, 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 ω=4.12×108(z1.12×104)radm1 (“accelerating”) and ω=25.13z1radm1 (“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.

 figure: Fig. 4.

Fig. 4. Characterization of a fabricated helix. (a) SEM image of the helix with one edge highlighted. (b) One edge (red solid curve) of the helix is extracted from the optical microscopic image, and the distance d from the optical axis (white dotted-dashed line) is measured. Scale bar corresponds to 10 μm. (c) Comparison of d between measurement (orange triangle) and theoretical prediction (blue curve). Error bars represent ± one standard deviation.

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By performing a series of single exposures with lateral translation, helical matrices can be fabricated. Figure 5 shows several matrices, each consisting of 30×30 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.

 figure: Fig. 5.

Fig. 5. Matrices of various helical structures fabricated by a combination of exposure and linear translation.

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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-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

I(r,θ,z)=I[r,θ+W(z)z],
where I(r,θ,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 ω(z). The rotation rate is given by
ω(z)=W(z)dW(z)dzz.

A special group of SABs with constant ω has been discussed previously [32,33]. In the present work, SABs are defined in terms of an axially tuned rotation rate ω(z) using a phase-only SLM. We start from an amplitude transmittance function:

t(r,θ)=eikreilθ,
where (r,θ) are the polar coordinates in the plane z=0, k is the radial component of the wave vector, and l is an integer that represents the order of a Bessel beam [4043]. 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 |lm| petal-like local maxima with |lm|-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 ω(z). Inspired by the geometrical radial-to-axial (“r-z”) mapping between phase maps and generated Bessel beams [39], we introduce a rotation-tuning function υ(r) to modulate the phase along the radial direction. The transmittance functions can then be expressed as
tl(r,θ)=eiφ1=exp{i[klm2υ(r)]r}exp(ilθ),
tm(r,θ)=eiφ2=exp{i[k+lm2υ(r)]r}exp(imθ).

The superposition tlm(r,θ)=tl(r,θ)+tm(r,θ) is the desired transmittance function with amplitude and phase information. The task at hand is to find the relationship between υ(r) and ω(z). To this end, we derive the light field intensity resulting from tlm(r,θ) and extract such a relationship. Since tlm(r,θ) 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) M1 and M2 are used:

M1,2=12α=β=Λ1,2(α,β)ei2παxpei2πβyp,
where
Λ1,2(α,β)cos[π(α±β)2]sinc(α2)sinc(β2).

In Eqs. (A6) and (A7), sinc(x)=sin(πx)/πx, and p is the period of the binary gratings. The transmittance functions tl and tm are spatially sampled by M1 and M2, respectively, as

eiφ(x,y)=M1(x,y)eφ1(x,y)+M2(x,y)eφ2(x,y),
where (x,y) are the Cartesian coordinates. φ(x,y) is the phase function that will be encoded on the SLM. As M1 and M2 are complementary, the following relation holds true:
φ(x,y)=M1(x,y)φ1(x,y)+M2(x,y)φ2(x,y).

A CGH described by Eq. (A9) is displayed on the SLM, which is illuminated by a Gaussian beam with a 1/e2 radius w0 and clipped by an aperture with a radius R. We use a 4-f system with demagnification 1/M=f2/f1 to obtain the Fourier spectrum H(u,ν) of eiφ(x,y) at the Fourier plane (Fig. 6):

H(u,ν)=U0(x1,y1)ei2π(x1xf+y1yf)f1λdx1dy1=circ(x12+y12R)ex12+y12w02eiφ(x1,y1)×ei2π(x1xf+y1yf)f1λdx1dy1.
 figure: Fig. 6.

Fig. 6. Propagation model used in the derivation.

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In Eq. (A10), (x1,y1) and (xf,yf) are the coordinates at the SLM and Fourier plane of the 4-f, respectively; U0(x1,y1) is the complex field at the input plane of the 4-f system. The circ function is unity for r1 and zero elsewhere; λ is the wavelength in vacuum; (u,ν) are spatial frequencies defined as u=xf/f1λ and ν=yf/f1λ. Combining Eqs. (A6)–(A8) and applying the convolution theorem, we can obtain

H(u,ν)=[12α=β=Λ1(α,β)Ψ(uαp,νβp)+12α=β=Λ2(α,β)Ω(uαp,νβp)]Γ(u,ν),
where Ψ(u,ν)=F{exp[iφ1(x1,y1)]}, Ω(u,ν)=F{exp[iφ2(x1,y1)]} and Γ(u,ν)=F{circ(x12+y12/R)·exp[(x12+y12)/w02]} are the Fourier transforms of the corresponding complex fields; represents the convolution operation. Since the size of the SLM aperture and the Gaussian beam size w0 are both 10mm in the experiment, Γ(u,ν) is close to a delta function. Transmittance function tl,m(r,θ) is assumed to be band limited at spatial frequency, so the spatial frequency separation 1/p can be set large enough that the adjacent diffraction orders in H(u,ν) will not overlap with each other. It is easy to see that if we use a spatial filter P(u,ν) to transmit only the zeroth order, the Fourier spectrum becomes
H(u,ν)P(u,ν)=12F{tlm(r,θ)}Γ(u,ν),
where
F{tlm(r,θ)}=Ψ(u,ν)+Ω(u,ν).

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

U(r0,θ0)=T(Mr0,θπ)circ(Mr0R)eM2r02w02,
where (r0,θ0) are the polar coordinates in the image plane. In what follows, we continue using polar coordinates because they enable convenient calculation of the Fresnel diffraction integral. Given that U(r0,θ0) is the superposition of two complex fields, an individual field of order m can be written as
Um(r0,θ0)=eimπei[k+lm2v(Mr0)]Mr0eimθ0circ(Mr0R)eM2r02w02.

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

Um(r,θ,z)=1iλzeimπeikzeikr22z0RMdr0r0eikr022zeM2r02w02ei[k+lm2v(Mr0)]Mr002πdθ0eimθ0eikrr0zcos(θθ0),
where k=2π/λ is the wavenumber. For convenience, we define
RRM,
kMk,
w0w0M,
V(r0)Mv(Mr0).

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

Um(r,θ,z)=1iλzeimπeikzeikr22z0Rdr0r0eikr022zer02w02ei[k+lm2v(r0)]r002πdθ0eimθ0eikrr0zcos(θθ0).

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

02πdθ0eimθ0eikrr0zcos(θθ0)=2π(i)meimθ0Jm(krr0z),
where Jm(x) is an mth-order Bessel function of the first kind. Inserting Eq. (A22) into (A21) gives
Um(r,θ,z)=1iλzeimπeikzeikr22z2π(i)meimθ0×0Rdr0f(r0)eikμ(r0),
where functions f(r0) and μ(r0) are defined as
f(r0)=r0er02w02Jm(krr0z)eilm2v(r0)r0,
and
μ(r0)=r022zkkr0.

Because the phase term μ(r0) in Eq. (A23) varies rapidly compared to f(r0) 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 μ(r0) is zero. There is one critical point rc=zk/k in the interval (0,R) when 0<z<Rk/k. The leading contribution of the integral in Eq. (A23) is

0Rdr0f(r0)eikμ(r0)f(rc)eikμ(rc)2πikμ(rc),
where μ(rc) represents the second derivative of μ(r0) at point r0=rc. The approximation is valid only when the variation of f(r0) in the regions of stationary phase is small, i.e., r02zλ/4 [42]. By combining the results of Eqs. (A23)–(A26), we obtain an expression for the complex field at distance z after the image plane as
Um(r,θ,z)=(2π)32(i)12(i)m1λkk32z12ek2k2w02z2×ei(kz+kr22zk22kz)eilm2v(kkz)kkzeimθJm(kr).

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

I(r,θ,z)l+m=|Um(r,φ,z)+Ul(r,φ,z)|2=ze2k2k2w02z2{|Cl|2Jl2(kr)+|Cm|2Jm2(kr)+2|ClCm|Jl(kr)Jm(kr)×cos{(lm)[θ+Mv(Mkkz)kkz]+θlm}},
where Cl and Cm are
Cl=(2π)32(i)12(i)l1λkk32,
Cm=(2π)32(i)12(i)m1λkk32.

It is obvious that Cl/Cm=|Cl|/|Cm|eiθlm. Note that the term zexp[2k2z2/(k2w02)] 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(kw0)/(2k). To study the rotation of the intensity profile on a transverse plane, we temporarily drop this term, and Eq. (A28) then becomes

I(r,θ,z)l+m=|Cl|2Jl2(kr)+|Cm|2Jm2(kr)+2|ClCm|Jl(kr)Jm(kr)×cos{(lm)[θ+Mv(Mkkz)kkz]+θlm}.

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:

θ+Mv(kkz)kkz=const.

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

θ+M2v(M2kkz)kkz=const.

When the function v is a constant, the rotation rate ω=M2vk/k, which is also a constant. For more general cases, the axially dependent rotation rate ω(z) can be determined as

ω(z)=dθdz=kkM2v(M2kkz)zkkM2v(M2kkz),
where v represents the first derivative of v with respect to z. Equation (A34) establishes a “forward” relationship between the rotation rate ω(z) and function v(r), with the “r-z” mapping relationship r=M2kkz. As we are interested in a beam with predefined ω(z), a substitution of r=M2kkz can be made:
ω(M2kkr)=kkM2v(r)rkkM2v(r).

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

v(r)=1rM2kkω(M2kkr)dr.

Equation (A36) provides a “reverse” relationship between ω(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.

 figure: Fig. 7.

Fig. 7. Simulation results of two beams with (a) large and (b) small helical diameters generated by changing the phase order (l,m). The transverse intensity profiles shown on the left are obtained at z=0.2mm. Scale bar, 5 μm.

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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 (120μm) 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-f system discussed in the previous section is located in air and at a distance z0 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

A(fx,fy;z=0)=U(x0,y0)ei2π(x0fx+y0fy)dx0dy0,
where (x0,y0) are the Cartesian coordinates of the image plane. After propagating over distance of z0 to the air–photoresist interface, the angular spectrum [55] of A can be written as
A(fx,fy;z0)=A(fx,fy;0)ei2πλ1(λfx)2(λfy)2z0.
 figure: Fig. 8.

Fig. 8. Schematic for the propagation of helical beams into photoresist. L2 is the second lens of the 4-f system. Variable n is the refractive index of the photoresist, and z is the coordinate along the optical axis.

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When |fx|1 and |fy|1, the paraxial approximation can be applied, and Eq. (C1) becomes

A(fx,fy;z0)=A(fx,fy;0)ei2πλz0eiπλ[(λfx)2(λfy)2]z0.

Equation (C3) can also be obtained by Fourier transforming the Fresnel diffraction impulse response. After propagation distance of z1 in the photoresist, the angular spectrum becomes

A(fx,fy;z1+z0)=T(fx,fy)A(fx,fy;0)×ei2πλz0eiπλ[(λfx)2(λfy)2]z0ein2πλz1ei1nπλ[(λfx)2(λfy)2]z1=T(fx,fy)A(fx,fy;0)ei2πλz0(1n2)×eikn(z1+nz0)eikn2n2[(λfx)2(λfy)2](z1+nz0).

In Eq. (C4), kn=nk is the wavenumber in the photoresist, and T(fx,fy) is the transmittance function considering reflection loss at the interface. If we ignore the constant phase shift, which is inconsequential, and if T(fx,fy) does not vary significantly for different frequencies (fx,fy), 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 z1+nz0. 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 kn in the photoresist, and the propagation distance becomes z+nz0. Note that the quantity z+nz0 is not the usual optical path length, which should read z0+nz. Then we can write the intensity distribution as

I(r,φ,z=z1+z0)l+m=(z1+nz0)e2k2kn2w02(z1+nz0)2×{|Cl|2Jl2(kr)+|Cm|2Jm2(kr)+2|ClCm|Jl(kr)Jm(kr)×cos{(lm){θ+Mv[Mkkn(z1+nz0)]kkn(z1+nz0)}+θlm}}.

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 (z1+nz0)exp[2k2(z1+nz0)2/(kn2w02)]. Second, the intensity pattern in the x-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=z1+z0 as

ω(z=z1+z0)=kknM2v[M2kkn(z1+nz0)](z1+nz0)kknM2v[M2kkn(z1+nz0)],
where v is with respect to z1. Equations (C5) and (C6) have been verified with simulations using Fresnel integrals [Figs. 9(a) and 9(b)]. It is found that propagation into a medium with a higher refractive index slows down the rotation rate [decreases the magnitude of ω, Fig. 9(c)]. The aforementioned procedure can be applied successively to account for transmission through multiple interfaces.
 figure: Fig. 9.

Fig. 9. (a) Theory and (b) simulation of intensity distribution on the x-z plane of an SAB with a decelerating rotation rate propagating from air to the photoresist. White dashed lines indicate the interface between the two media. (c) Rotation rate ω at different z locations. Solid line is from Eq. (C6). Open circles are from the simulation. Discrepancies at small z are due to limitations of the algorithm used to extract ω for low intensities.

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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).

 figure: Fig. 10.

Fig. 10. Characterization of a fabricated helix from an SEM image. (a) Original image. (b) The edges are detected from the image by edge-detection algorithms. (c) Noise is suppressed and only one edge is displayed. (d) Locations along the edge are plotted.

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 figure: Fig. 11.

Fig. 11. Characterization of a fabricated helix from a transmissive optical microscopic image. Red curve overlayed on the image is used to illustrate the edge that is extracted.

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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 100mms1. More recently, XY writing speeds of 400mms1 [62] and 8000mms1 [63] have been realized using resonant galvo-mirrors. Given that the galvo-mirror scans the beam only in the x-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 100μms1 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 30mms1 [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 30mms1. 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 30mms1 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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References

  • View by:

  1. M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19, 207–210 (2007).
    [Crossref]
  2. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
    [Crossref]
  3. J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
    [Crossref]
  4. E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79, 035407 (2009).
    [Crossref]
  5. J. K. Gansel, M. Latzel, A. Frölich, J. Kaschke, M. Thiel, and M. Wegener, “Tapered gold-helix metamaterials as improved circular polarizers,” Appl. Phys. Lett. 100, 101109 (2012).
    [Crossref]
  6. J. Kaschke, L. Blume, L. Wu, M. Thiel, K. Bade, Z. Yang, and M. Wegener, “A helical metamaterial for broadband circular polarization conversion,” Adv. Opt. Mater. 3, 1411–1417 (2015).
    [Crossref]
  7. C. Wu, H. Li, X. Yu, F. Li, H. Chen, and C. T. Chan, “Metallic helix array as a broadband wave plate,” Phys. Rev. Lett. 107, 177401 (2011).
    [Crossref]
  8. Z. Lu, M. Zhao, Z. Yang, L. Wu, P. Zhang, Y. Zheng, and J. Duan, “Broadband polarization-insensitive absorbers in 0.3–25 μm using helical metamaterials,” J. Opt. Soc. Am. B 30, 1368–1372 (2013).
    [Crossref]
  9. M. K. S. Verma, S. R. Ganneboyina, Rakshith, and A. Ghatak, “Three-dimensional multihelical microfluidic mixers for rapid mixing of liquids,” Langmuir 24, 2248–2251 (2008).
    [Crossref]
  10. C. Shan, F. Chen, Q. Yang, Z. Jiang, and X. Hou, “3D multi-microchannel helical mixer fabricated by femtosecond laser inside fused silica,” Micromachines 9, 29 (2018).
    [Crossref]
  11. M. I. Hussain, G. H. Lee, and B. Engineering, “Numerical thermal analysis of helical-shaped heat exchanger to improve thermal stratification inside solar,” in International Conference on Agricultural and Environmental Engineering (2014), pp. 6–10.
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2021 (1)

H. Cheng, C. Xia, S. M. Kuebler, P. Golvari, M. Sun, M. Zhang, and X. Yu, “Generation of Bessel-beam arrays for parallel fabrication in two-photon polymerization,” J. Laser Appl. 33, 012040 (2021).
[Crossref]

2020 (2)

H. Cheng, C. Xia, S. M. Kuebler, and X. Yu, “Aberration correction for SLM-generated Bessel beams propagating through tilted interfaces,” Opt. Commun. 475, 126213 (2020).
[Crossref]

D. Pan, B. Xu, S. Liu, J. Li, Y. Hu, D. Wu, and J. Chu, “Amplitude-phase optimized long depth of focus femtosecond axilens beam for single-exposure fabrication of high-aspect-ratio microstructures,” Opt. Lett. 45, 2584–2587 (2020).
[Crossref]

2019 (2)

H. Cheng, C. Xia, M. Zhang, S. M. Kuebler, and X. Yu, “Fabrication of high-aspect-ratio structures using Bessel-beam-activated photopolymerization,” Appl. Opt. 58, D91–D97 (2019).
[Crossref]

B. W. Pearre, C. Michas, J. M. Tsang, T. J. Gardner, and T. M. Otchy, “Fast micron-scale 3D printing with a resonant-scanning two-photon microscope,” Addit. Manuf. 30, 100887 (2019).
[Crossref]

2018 (5)

2017 (9)

E. S. Shukri, “Numerical comparison of temperature distribution in an annular diffuser fitted with helical screw-tape hub and pimpled hub,” Energy Proc. 141, 625–629 (2017).
[Crossref]

S. Ji, L. Yang, Y. Hu, J. Ni, W. Du, J. Li, G. Zhao, D. Wu, and J. Chu, “Dimension-controllable microtube arrays by dynamic holographic processing as 3D yeast culture scaffolds for asymmetrical growth regulation,” Small 13, 1701190 (2017).
[Crossref]

L. Yang, D. Qian, C. Xin, Z. Hu, S. Ji, D. Wu, Y. Hu, J. Li, W. Huang, and J. Chu, “Two-photon polymerization of microstructures by a non-diffraction multifoci pattern generated from a superposed Bessel beam,” Opt. Lett. 42, 743–746 (2017).
[Crossref]

L. Yang, D. Qian, C. Xin, Z. Hu, S. Ji, D. Wu, Y. Hu, J. Li, W. Huang, and J. Chu, “Direct laser writing of complex microtubes using femtosecond vortex beams,” Appl. Phys. Lett. 110, 221103 (2017).
[Crossref]

J. Ni, C. Wang, C. Zhang, Y. Hu, L. Yang, Z. Lao, B. Xu, J. Li, D. Wu, and J. Chu, “Three-dimensional chiral microstructures fabricated by structured optical vortices in isotropic material,” Light Sci. Appl. 6, e17011 (2017).
[Crossref]

Y. Li, L. Liu, D. Yang, Q. Zhang, H. Yang, and Q. Gong, “Femtosecond laser nano/microfabrication via three-dimensional focal field engineering,” Proc. SPIE 10092, 100920B (2017).
[Crossref]

L. Yang, S. Ji, K. Xie, W. Du, B. Liu, Y. Hu, J. Li, G. Zhao, D. Wu, W. Huang, S. Liu, H. Jiang, and J. Chu, “High efficiency fabrication of complex microtube arrays by scanning focused femtosecond laser Bessel beam for trapping/releasing biological cells,” Opt. Express 25, 8144–8157 (2017).
[Crossref]

C. Vetter, A. Dudley, A. Szameit, and A. Forbes, “Real and virtual propagation dynamics of angular accelerating white light beams,” Opt. Express 25, 20530–20540 (2017).
[Crossref]

J. Webster, C. Rosales-Guzmán, and A. Forbes, “Radially dependent angular acceleration of twisted light,” Opt. Lett. 42, 675–678 (2017).
[Crossref]

2016 (2)

X. Yu, C. A. Trallero-Herrero, and S. Lei, “Materials processing with superposed Bessel beams,” Appl. Surf. Sci. 360, 833–839 (2016).
[Crossref]

M. A. Skylar-Scott, M. C. Liu, Y. Wu, A. Dixit, and M. F. Yanik, “Guided homing of cells in multi-photon microfabricated bioscaffolds,” Adv. Healthc. Mater. 5, 1233–1243 (2016).
[Crossref]

2015 (5)

C. Schulze, F. S. Roux, A. Dudley, R. Rop, M. Duparré, and A. Forbes, “Accelerated rotation with orbital angular momentum modes,” Phys. Rev. A 91, 043821 (2015).
[Crossref]

C. Vetter, T. Eichelkraut, M. Ornigotti, and A. Szameit, “Optimization and control of two-component radially self-accelerating beams,” Appl. Phys. Lett. 107, 211104 (2015).
[Crossref]

A. V. Do, B. Khorsand, S. M. Geary, and A. K. Salem, “3D printing of scaffolds for tissue regeneration applications,” Adv. Healthcare Mater. 4, 1742–1762 (2015).
[Crossref]

J. Kaschke and M. Wegener, “Gold triple-helix mid-infrared metamaterial by STED-inspired laser lithography,” Opt. Lett. 40, 3986–3989 (2015).
[Crossref]

J. Kaschke, L. Blume, L. Wu, M. Thiel, K. Bade, Z. Yang, and M. Wegener, “A helical metamaterial for broadband circular polarization conversion,” Adv. Opt. Mater. 3, 1411–1417 (2015).
[Crossref]

2014 (3)

C. Vetter, T. Eichelkraut, M. Ornigotti, and A. Szameit, “Generalized radially self-accelerating helicon beams,” Phys. Rev. Lett. 113, 183901 (2014).
[Crossref]

S.-J. Zhang, Y. Li, Z.-P. Liu, J.-L. Ren, Y.-F. Xiao, H. Yang, and Q. Gong, “Two-photon polymerization of a three dimensional structure using beams with orbital angular momentum,” Appl. Phys. Lett. 105, 061101 (2014).
[Crossref]

O. Mendoza-Yero, G. Mínguez-Vega, and J. Lancis, “Encoding complex fields by using a phase-only optical element,” Opt. Lett. 39, 1740–1743 (2014).
[Crossref]

2013 (1)

2012 (3)

J. K. Gansel, M. Latzel, A. Frölich, J. Kaschke, M. Thiel, and M. Wegener, “Tapered gold-helix metamaterials as improved circular polarizers,” Appl. Phys. Lett. 100, 101109 (2012).
[Crossref]

R. Rop, A. Dudley, C. López-Mariscal, and A. Forbes, “Measuring the rotation rates of superpositions of higher-order Bessel beams,” J. Mod. Opt. 59, 259–267 (2012).
[Crossref]

J. Kaschke, J. K. Gansel, and M. Wegener, “On metamaterial circular polarizers based on metal N-helices,” Opt. Express 20, 26012–26020 (2012).
[Crossref]

2011 (5)

C. Wu, H. Li, X. Yu, F. Li, H. Chen, and C. T. Chan, “Metallic helix array as a broadband wave plate,” Phys. Rev. Lett. 107, 177401 (2011).
[Crossref]

A. Ovsianikov, A. Deiwick, S. Van Vlierberghe, P. Dubruel, L. Möller, G. Drager, and B. Chichkov, “Laser fabrication of three-dimensional CAD scaffolds from photosensitive gelatin for applications in tissue engineering,” Biomacromolecules 12, 851–858 (2011).
[Crossref]

H. Lee, K. Lee, B. Ahn, J. Xu, L. Xu, and K. W. Oh, “A new fabrication process for uniform SU-8 thick photoresist structures by simultaneously removing edge bead and air bubbles,” J. Micromech. Microeng. 21, 125006 (2011).
[Crossref]

S. Li, Z. Yang, J. Wang, and M. Zhao, “Broadband terahertz circular polarizers with single- and double-helical array metamaterials,” J. Opt. Soc. Am. A 28, 19–23 (2011).
[Crossref]

Y. Yu, Z. Yang, M. Zhao, and P. Lu, “Broadband optical circular polarizers in the terahertz region using helical metamaterials,” J. Opt. 13, 055104 (2011).
[Crossref]

2010 (2)

G. Kumi, C. O. Yanez, K. D. Belfield, and J. T. Fourkas, “High-speed multiphoton absorption polymerization: fabrication of microfluidic channels with arbitrary cross-sections and high aspect ratios,” Lab Chip 10, 1057–1060 (2010).
[Crossref]

J. K. Gansel, M. Wegener, S. Burger, and S. Linden, “Gold helix photonic metamaterials: a numerical parameter study,” Opt. Express 18, 1059–1069 (2010).
[Crossref]

2009 (4)

R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express 17, 23389–23395 (2009).
[Crossref]

M. Thiel, M. S. Rill, G. von Freymann, and M. Wegener, “Three-dimensional bi-chiral photonic crystals,” Adv. Mater. 21, 4680–4682 (2009).
[Crossref]

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79, 035407 (2009).
[Crossref]

2008 (1)

M. K. S. Verma, S. R. Ganneboyina, Rakshith, and A. Ghatak, “Three-dimensional multihelical microfluidic mixers for rapid mixing of liquids,” Langmuir 24, 2248–2251 (2008).
[Crossref]

2007 (1)

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19, 207–210 (2007).
[Crossref]

2006 (1)

2005 (1)

G. Whyte and J. Courtial, “Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg–Saxton algorithm,” New J. Phys. 7, 117 (2005).
[Crossref]

2004 (1)

J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
[Crossref]

2003 (1)

2002 (3)

A. Terray, “Microfluidic control using colloidal devices,” Science 296, 1841–1844 (2002).
[Crossref]

P. Galajda and P. Ormos, “Rotors produced and driven in laser tweezers with reversed direction of rotation,” Appl. Phys. Lett. 80, 4653–4655 (2002).
[Crossref]

W. Zhou, “An efficient two-photon-generated photoacid applied to positive-tone 3D microfabrication,” Science 296, 1106–1109 (2002).
[Crossref]

2001 (1)

S. Kawata, H. B. Sun, T. Tanaka, and K. Takada, “Finer features for functional microdevices,” Nature 412, 697–698 (2001).
[Crossref]

2000 (2)

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[Crossref]

V. Jarutis, R. Paškauskas, and A. Stabinis, “Focusing of Laguerre–Gaussian beams by axicon,” Opt. Commun. 184, 105–112 (2000).
[Crossref]

1996 (1)

C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
[Crossref]

1989 (1)

Admon, T.

Ahn, B.

H. Lee, K. Lee, B. Ahn, J. Xu, L. Xu, and K. W. Oh, “A new fabrication process for uniform SU-8 thick photoresist structures by simultaneously removing edge bead and air bubbles,” J. Micromech. Microeng. 21, 125006 (2011).
[Crossref]

Arlt, J.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[Crossref]

Bade, K.

J. Kaschke, L. Blume, L. Wu, M. Thiel, K. Bade, Z. Yang, and M. Wegener, “A helical metamaterial for broadband circular polarization conversion,” Adv. Opt. Mater. 3, 1411–1417 (2015).
[Crossref]

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

Bahabad, A.

Belfield, K. D.

G. Kumi, C. O. Yanez, K. D. Belfield, and J. T. Fourkas, “High-speed multiphoton absorption polymerization: fabrication of microfluidic channels with arbitrary cross-sections and high aspect ratios,” Lab Chip 10, 1057–1060 (2010).
[Crossref]

Blume, L.

J. Kaschke, L. Blume, L. Wu, M. Thiel, K. Bade, Z. Yang, and M. Wegener, “A helical metamaterial for broadband circular polarization conversion,” Adv. Opt. Mater. 3, 1411–1417 (2015).
[Crossref]

Burger, S.

Cai, Z.

Campbell, J.

Y. Liu, J. Campbell, O. Stein, L. Jiang, J. Hund, and Y. Lu, “Deformation behavior of foam laser targets fabricated by two-photon polymerization,” Nanomaterials 8, 498 (2018).
[Crossref]

Chan, C. T.

C. Wu, H. Li, X. Yu, F. Li, H. Chen, and C. T. Chan, “Metallic helix array as a broadband wave plate,” Phys. Rev. Lett. 107, 177401 (2011).
[Crossref]

Chattrapiban, N.

Chen, F.

C. Shan, F. Chen, Q. Yang, Z. Jiang, and X. Hou, “3D multi-microchannel helical mixer fabricated by femtosecond laser inside fused silica,” Micromachines 9, 29 (2018).
[Crossref]

Chen, H.

C. Wu, H. Li, X. Yu, F. Li, H. Chen, and C. T. Chan, “Metallic helix array as a broadband wave plate,” Phys. Rev. Lett. 107, 177401 (2011).
[Crossref]

Cheng, H.

H. Cheng, C. Xia, S. M. Kuebler, P. Golvari, M. Sun, M. Zhang, and X. Yu, “Generation of Bessel-beam arrays for parallel fabrication in two-photon polymerization,” J. Laser Appl. 33, 012040 (2021).
[Crossref]

H. Cheng, C. Xia, S. M. Kuebler, and X. Yu, “Aberration correction for SLM-generated Bessel beams propagating through tilted interfaces,” Opt. Commun. 475, 126213 (2020).
[Crossref]

H. Cheng, C. Xia, M. Zhang, S. M. Kuebler, and X. Yu, “Fabrication of high-aspect-ratio structures using Bessel-beam-activated photopolymerization,” Appl. Opt. 58, D91–D97 (2019).
[Crossref]

Chichkov, B.

A. Ovsianikov, A. Deiwick, S. Van Vlierberghe, P. Dubruel, L. Möller, G. Drager, and B. Chichkov, “Laser fabrication of three-dimensional CAD scaffolds from photosensitive gelatin for applications in tissue engineering,” Biomacromolecules 12, 851–858 (2011).
[Crossref]

Chu, J.

D. Pan, B. Xu, S. Liu, J. Li, Y. Hu, D. Wu, and J. Chu, “Amplitude-phase optimized long depth of focus femtosecond axilens beam for single-exposure fabrication of high-aspect-ratio microstructures,” Opt. Lett. 45, 2584–2587 (2020).
[Crossref]

S. Ji, L. Yang, C. Zhang, Z. Cai, Y. Hu, J. Li, D. Wu, and J. Chu, “High-aspect-ratio microtubes with variable diameter and uniform wall thickness by compressing Bessel hologram phase depth,” Opt. Lett. 43, 3514–3517 (2018).
[Crossref]

L. Yang, S. Ji, K. Xie, W. Du, B. Liu, Y. Hu, J. Li, G. Zhao, D. Wu, W. Huang, S. Liu, H. Jiang, and J. Chu, “High efficiency fabrication of complex microtube arrays by scanning focused femtosecond laser Bessel beam for trapping/releasing biological cells,” Opt. Express 25, 8144–8157 (2017).
[Crossref]

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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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Figures (11)

Fig. 1.
Fig. 1. High-throughput microfabrication of helical structures using self-accelerating beams (SABs). (a) Experimental setup: BBO, beta barium borate crystal; PBS, polarizing beam splitter; BE, beam expander; BS, non-polarizing beam splitter; SLM, spatial light modulator; M, mirror; L, lens; ID, iris diaphragm; Obj, objective lens; CMOS, complementary metal–oxide-semiconductor. (b) Calculation of a computer-generated hologram (CGH). Phase order l=10, m=9. (c) Illustration of exposure in the SU-8 sample. Green arrow shows the beam propagation direction. (d) Fourier spectrum of an SAB at the Fourier plane of the 4-f system. The white dashed circle represents the iris diaphragm acting as a low-pass filter to isolate the zeroth-order beam. (e) Simulation of iso-intensity and x-y intensity distribution of an SAB in SU-8. (f) SEM images of fabricated helices, including a 45° view of a 30×30 matrix (individual helix highlighted in yellow, scale bar corresponds to 30 μm.). The white-boxed inset shows a high-magnification view of a single helix.
Fig. 2.
Fig. 2. Comparison of simulated and measured helical beams with (a)–(d) constant and (e)–(h) decelerated rotation rate. Phase order l=10 and m=9. (a), (e) Iso-intensity contours of simulated SABs in air. The threshold intensity (normalized) is set to 0.5. (b), (f) Simulated and (c), (g) measured x-y intensity distribution. Scale bar represents 5 μm. (d), (h) Rotation rate ω along optical axis. Red curves, designed profiles; orange points, measured data; blue dots, simulation results.
Fig. 3.
Fig. 3. Comparison between helical beams simulated in SU-8 and fabrication results obtained with five SABs having different types of rotation. The contrast of each SEM image was adjusted individually to improve visibility and aid comparison. Scale bar corresponds to 20 μm.
Fig. 4.
Fig. 4. Characterization of a fabricated helix. (a) SEM image of the helix with one edge highlighted. (b) One edge (red solid curve) of the helix is extracted from the optical microscopic image, and the distance d from the optical axis (white dotted-dashed line) is measured. Scale bar corresponds to 10 μm. (c) Comparison of d between measurement (orange triangle) and theoretical prediction (blue curve). Error bars represent ± one standard deviation.
Fig. 5.
Fig. 5. Matrices of various helical structures fabricated by a combination of exposure and linear translation.
Fig. 6.
Fig. 6. Propagation model used in the derivation.
Fig. 7.
Fig. 7. Simulation results of two beams with (a) large and (b) small helical diameters generated by changing the phase order (l,m). The transverse intensity profiles shown on the left are obtained at z=0.2mm. Scale bar, 5 μm.
Fig. 8.
Fig. 8. Schematic for the propagation of helical beams into photoresist. L2 is the second lens of the 4-f system. Variable n is the refractive index of the photoresist, and z is the coordinate along the optical axis.
Fig. 9.
Fig. 9. (a) Theory and (b) simulation of intensity distribution on the x-z plane of an SAB with a decelerating rotation rate propagating from air to the photoresist. White dashed lines indicate the interface between the two media. (c) Rotation rate ω at different z locations. Solid line is from Eq. (C6). Open circles are from the simulation. Discrepancies at small z are due to limitations of the algorithm used to extract ω for low intensities.
Fig. 10.
Fig. 10. Characterization of a fabricated helix from an SEM image. (a) Original image. (b) The edges are detected from the image by edge-detection algorithms. (c) Noise is suppressed and only one edge is displayed. (d) Locations along the edge are plotted.
Fig. 11.
Fig. 11. Characterization of a fabricated helix from a transmissive optical microscopic image. Red curve overlayed on the image is used to illustrate the edge that is extracted.

Equations (49)

Equations on this page are rendered with MathJax. Learn more.

I(r,θ,z)=I[r,θ+W(z)z],
ω(z)=W(z)dW(z)dzz.
BBPl=kr+lθ,BBPm=kr+mθ.
RPMl=RPMm=lm2v(r)r.
φ(r,θ)=(BBPl+RPMl)×BAMl+(BBPm+RPMm)×BAMm.
ω(z)=kkM2v(M2kkz)zkkM2v(M2kkz).
v(r)=1rM2kkω(M2kkr)dr,
I(r,θ,z)=I[r,θ+W(z)z],
ω(z)=W(z)dW(z)dzz.
t(r,θ)=eikreilθ,
tl(r,θ)=eiφ1=exp{i[klm2υ(r)]r}exp(ilθ),
tm(r,θ)=eiφ2=exp{i[k+lm2υ(r)]r}exp(imθ).
M1,2=12α=β=Λ1,2(α,β)ei2παxpei2πβyp,
Λ1,2(α,β)cos[π(α±β)2]sinc(α2)sinc(β2).
eiφ(x,y)=M1(x,y)eφ1(x,y)+M2(x,y)eφ2(x,y),
φ(x,y)=M1(x,y)φ1(x,y)+M2(x,y)φ2(x,y).
H(u,ν)=U0(x1,y1)ei2π(x1xf+y1yf)f1λdx1dy1=circ(x12+y12R)ex12+y12w02eiφ(x1,y1)×ei2π(x1xf+y1yf)f1λdx1dy1.
H(u,ν)=[12α=β=Λ1(α,β)Ψ(uαp,νβp)+12α=β=Λ2(α,β)Ω(uαp,νβp)]Γ(u,ν),
H(u,ν)P(u,ν)=12F{tlm(r,θ)}Γ(u,ν),
F{tlm(r,θ)}=Ψ(u,ν)+Ω(u,ν).
U(r0,θ0)=T(Mr0,θπ)circ(Mr0R)eM2r02w02,
Um(r0,θ0)=eimπei[k+lm2v(Mr0)]Mr0eimθ0circ(Mr0R)eM2r02w02.
Um(r,θ,z)=1iλzeimπeikzeikr22z0RMdr0r0eikr022zeM2r02w02ei[k+lm2v(Mr0)]Mr002πdθ0eimθ0eikrr0zcos(θθ0),
RRM,
kMk,
w0w0M,
V(r0)Mv(Mr0).
Um(r,θ,z)=1iλzeimπeikzeikr22z0Rdr0r0eikr022zer02w02ei[k+lm2v(r0)]r002πdθ0eimθ0eikrr0zcos(θθ0).
02πdθ0eimθ0eikrr0zcos(θθ0)=2π(i)meimθ0Jm(krr0z),
Um(r,θ,z)=1iλzeimπeikzeikr22z2π(i)meimθ0×0Rdr0f(r0)eikμ(r0),
f(r0)=r0er02w02Jm(krr0z)eilm2v(r0)r0,
μ(r0)=r022zkkr0.
0Rdr0f(r0)eikμ(r0)f(rc)eikμ(rc)2πikμ(rc),
Um(r,θ,z)=(2π)32(i)12(i)m1λkk32z12ek2k2w02z2×ei(kz+kr22zk22kz)eilm2v(kkz)kkzeimθJm(kr).
I(r,θ,z)l+m=|Um(r,φ,z)+Ul(r,φ,z)|2=ze2k2k2w02z2{|Cl|2Jl2(kr)+|Cm|2Jm2(kr)+2|ClCm|Jl(kr)Jm(kr)×cos{(lm)[θ+Mv(Mkkz)kkz]+θlm}},
Cl=(2π)32(i)12(i)l1λkk32,
Cm=(2π)32(i)12(i)m1λkk32.
I(r,θ,z)l+m=|Cl|2Jl2(kr)+|Cm|2Jm2(kr)+2|ClCm|Jl(kr)Jm(kr)×cos{(lm)[θ+Mv(Mkkz)kkz]+θlm}.
θ+Mv(kkz)kkz=const.
θ+M2v(M2kkz)kkz=const.
ω(z)=dθdz=kkM2v(M2kkz)zkkM2v(M2kkz),
ω(M2kkr)=kkM2v(r)rkkM2v(r).
v(r)=1rM2kkω(M2kkr)dr.
A(fx,fy;z=0)=U(x0,y0)ei2π(x0fx+y0fy)dx0dy0,
A(fx,fy;z0)=A(fx,fy;0)ei2πλ1(λfx)2(λfy)2z0.
A(fx,fy;z0)=A(fx,fy;0)ei2πλz0eiπλ[(λfx)2(λfy)2]z0.
A(fx,fy;z1+z0)=T(fx,fy)A(fx,fy;0)×ei2πλz0eiπλ[(λfx)2(λfy)2]z0ein2πλz1ei1nπλ[(λfx)2(λfy)2]z1=T(fx,fy)A(fx,fy;0)ei2πλz0(1n2)×eikn(z1+nz0)eikn2n2[(λfx)2(λfy)2](z1+nz0).
I(r,φ,z=z1+z0)l+m=(z1+nz0)e2k2kn2w02(z1+nz0)2×{|Cl|2Jl2(kr)+|Cm|2Jm2(kr)+2|ClCm|Jl(kr)Jm(kr)×cos{(lm){θ+Mv[Mkkn(z1+nz0)]kkn(z1+nz0)}+θlm}}.
ω(z=z1+z0)=kknM2v[M2kkn(z1+nz0)](z1+nz0)kknM2v[M2kkn(z1+nz0)],