## Abstract

Freeform optical components enable advanced manipulation of light that is not possible with traditional optical systems. However, their fabrication relies on machining processes that are complex, time-consuming, and require significant infrastructure. Here we present the ability to shape liquid volumes and solidify them into desired freeform components, enabling rapid prototyping of freeform components with high surface quality. The method is based on controlling the minimum energy state of the interface between a curable optical liquid and an immersion liquid, by dictating a geometrical boundary constraint. We provide an analytical solution for the resulting topography given a predefined boundary and demonstrate the fabrication of freeform components with sub-nanometer surface roughness within minutes. Such a fabrication capability, that allows for rapid prototyping of high-quality components, has the potential to answer an unmet need in the optical design industry—allowing researchers and engineers to rapidly test freeform design concepts. It can be further envisioned to be expanded to an industrial scale, allowing for mold-less fabrication of freeform optics.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Freeform optics is a broad term that refers to any optical component, reflective or refractive, in which one or more of its optical surfaces perform complex phase manipulation on the incoming wavefront, beyond that achievable using traditional optical components. A single freeform optical component can replace the functionality of multiple traditional lenses (spherical or aspherical) within an optical system, allowing significant reduction in dimensions and assembly complexity. Moreover, freeform surfaces can provide new functionality that is not attainable using standard optics [1–6].

In recent years, the advent in computer-based optical design enabled the simulation of complex components and led to tremendous growth in the use of freeform optics for new applications [7]. These include multifocal corrective eyewear [8,9], telescopes [10–12], beam shaping [13], ultra-short projection lenses [14,15], panoramic imaging [16], solar energy concentration [17], and photolithography [18].

Small freeform components can be fabricated by using micro-structuring techniques such as micromachining, lithography, and etching [19,20]. Yet these methods are limited to characteristic scales of up to tens of micrometers in depth, and up to several ${\rm{mm}}^2$ in area [20]. The fabrication of larger components relies on machining approaches such as grinding, milling, and turning, followed by polishing or finishing [7,21–24]—processes that remain complex, expensive, and time-consuming. Additive manufacturing is a natural candidate when seeking to construct arbitrary three-dimensional configurations. However, existing 3D printing technologies are primarily based on layer-by-layer fabrication and cannot yet provide the required surface quality for optical applications. Better surface quality can be obtained using post-printing processes such as reflow or coatings [25–28], but at the cost of additional complexity, fabrication time, and reduced control over the precise shape of the surface. Furthermore, since printing time is proportional to the volume of the object (in contrast to mechanical processing that is roughly proportional to the surface area), practical considerations limit the fabrication to small lenses [25–28]. An additive manufacturing method that enables us to achieve smooth optical surfaces was demonstrated by two-photon polymerization, yet this method is also limited to the scale of micrometers [29,30].

Recently, work by Frumkin and Bercovici suggested a new method of fabricating optical components based on shaping of liquid volumes [31]. Their method is based on injection of a curable optical fluid into a rigid bounding frame contained within an immiscible immersion liquid environment. The authors showed that using a simple cylindrical frame, the balance between gravitational forces, hydrostatic forces, and surface tension forces, enables the creation of spherical and aspherical lenses. The method was shown to be scale-invariant, allowing the production of lenses at any size while maintaining a surface roughness of 1 nm, without the need for any further mechanical processes. Furthermore, the production time only weakly depends on the size of the produced lens.

In this work, we introduce an extension of this scale-invariant fluidic shaping method to non-axisymmetric boundary conditions, allowing for rapid fabrication of freeform optical components. We study the case of a cylindrical bounding frame whose height varies azimuthally, and based on free-energy minimization, derive an analytical model that relates the shape of this bounding frame to that of the enclosed liquid interface forming the freeform surface. These boundary conditions, which can be represented as a sum of azimuthal waves, together with the injected volume and effective buoyancy, provide an infinite number of degrees of freedom which translate into a family of freeform surfaces. These can be naturally represented by a sum of polar–Bessel functions in a similar manner to the commonly used Zernike polynomials [32]. Using bounding frames created with a standard 3D printer, we demonstrate the ease of fabrication of several freeform components. We use frames defined by a single wavenumber to demonstrate common freeform surfaces such as saddles, tilts (bifocal), and quatrefoils, as well as frames defined by superposition of waves yielding arbitrary desired surfaces. All those components enjoy a surface quality of the order of 1 nm, characteristic of the fluidic shaping method, without the need for any subsequent polishing steps.

## 2. THEORY

We consider a configuration similar to that described in Fig. 1, where an optical liquid of density $\rho$ and volume $V$ is injected into a cylindrical bounding surface of radius ${R_0}$ and height $h({R_0},\theta) = f(\theta)$. The liquid is suspended in an immersion liquid of density ${\rho _{\textit{im}}}$, resulting in an effective density of $\Delta \rho = \rho - {\rho _{\textit{im}}}$.

We assume that the liquid wets the inner walls of the frame and forms an interface $h(r,\,\theta)$ with the immersion liquid. The shape of that interface is determined by a balance between surface tension, hydrostatic forces, and gravitational forces. The relative importance of the body force to surface forces can be expressed by the dimensionless Bond number, $Bo = \frac{{R_0^2}}{{\ell _c^2}} = \frac{{| {\Delta \rho} |gR_0^2}}{\gamma},$ where ${{\ell }_{c}}=\sqrt{\frac{\gamma }{| \vartriangle \rho |g}}$ is the capillary length, $\gamma$ is the interfacial energy between the two liquids, and $g$ is Earth's gravity directed in the negative $\hat z$ direction.

At the steady state, the fluidic interface $h(r,\theta)$ will take a shape that minimizes the free energy of the system under a fixed volume constraint. The free energy functional, $\Pi$, is given by

At equilibrium, the first variation of the energy functional vanishes, i.e., $\delta \Pi = 0$, yielding the standard Euler–Lagrange equation [33],

We define the following dimensionless variables:

For most optical elements the characteristic deformation length ${d_c}$ is significantly smaller than the component’s radius, thus $\varepsilon = {({\frac{{{d_c}}}{{{R_0}}}})^2} \ll 1$. Therefore, at the leading order in $\varepsilon$, and by defining $x = R\sqrt {Bo}$ and $\tilde H = H - P$, Eq. (4) reduces to the Helmholtz equation, $\tilde H{x^2} + {x^2}{\tilde H_{\textit{xx}}} + x{\tilde H_x} + {\tilde H_{\Theta \Theta}} = 0$, the general solution to which is given by [34]

Expressing the bounding surface height as a Fourier expansion, $f(\Theta) = {a_0} + \sum_{n = 1}^\infty {{a_n}\cos n\Theta} + \sum_{n = 1}^\infty {{b_n}\sin n\Theta} ,$ the mean average of the bounding height is ${h_0} = {a_0},$ and the constants ${A_n}$ and ${B_n}$ can be obtained by requiring Eq. (5) to satisfy the boundary condition $\tilde H(\sqrt {Bo} ,\Theta) = \frac{{f(\Theta) - {a_0}}}{{{d_c}}} - P$, yielding

In dimensional form, the solution is given by

Figure 2(a) presents solutions of Eq. (7) for several Bond numbers and several values of $n$, representing bounding frame heights with a single azimuthal frequency. We note that in contrast to $n$, which are discrete, the Bond number is continuous, and the values presented here are a subset chosen to illustrate its qualitative effect on the resulting surface. These basic modes can be conveniently arranged in a pyramid structure in accordance with the wavenumber of the periodic boundary condition, with the outermost layer of the pyramid corresponding to neutral buoyancy conditions, i.e., $Bo = 0$, and an increasing Bond number toward the center. Each layer in Fig. 2(a), corresponding to a fixed $Bo$ number, represents a group of orthogonal surfaces that can be superposed to form a freeform surface, $h(r,\theta) = \sum_{n = 0}^\infty {{h^{(n)}}}$, where

We note that ${h^{(0)}}$ serves as a baseline surface on which the other modes are constructed and is also the only component in the solution that is a function of the injected volume. A particular case is that of ${P^*} = 0$ (i.e., $V = \pi {a_0}R_0^2$), in which the base term reduces to a constant, meaning that the base term does not contribute to optical power. For any other volume, the baseline solution represents a Bessel surface, which for the particular case of $Bo = 0$ is a spherical lens. In Fig. 2(a) the baseline surface appears at the center column of the pyramid, representing the case of a homogenous boundary condition, corresponding to a cylindrical frame with a uniform height. For the case of $Bo = 0$, the solution can be further simplified and represented conveniently using the power series (see Supplement 1 Section S2):

This representation is also equivalent to representation by Zernike polynomials of the order $Z_n^{|m| = n}$ [32]. All other surfaces in the pyramid are not equivalent to Zernike polynomials, yet as Fig. 2(a) shows, have similar optical functionality.

Figure 2(b) shows that the extent to which the boundary shape affects the inner regions of the surface is a strong function of the wavenumber, $n$. As $n$ increases, the influence of the boundary is increasingly more localized, as can also be seen by the ${r^n}$ dependence in Eq. (9). This behavior holds also for other values of $Bo$. Clearly, the method cannot be used to create high-resolution features at the center of the component, but at the same time the solutions are insensitive to high-frequency deviations on the bounding frame (e.g., its surface roughness).

The Bond number represents the deviation from neutral buoyancy, and as shown in Fig. 2(c), tends to accentuate the effect of the bounding frame geometry. As the figure shows, increasing the Bond number can also invert the local curvature and create new local extrema. For a fixed Bond number, the injected volume can also have a significant effect on the shape of the resulting component, as illustrated by Fig. 2(d). By increasing the volume, the surface is transformed completely—from having a central valley with two side peaks to having a central peak with two side valleys. Combined, these three effects—the frame shape, the Bond number, and the volume, which are captured by Eq. (7)—provide significant degrees of freedom in the design of desired surfaces.

## 3. EXPERIMENTAL

Figure 3 illustrates the fabrication process of freeform surfaces using the fluidic shaping method. We use a 3D printer to print a rigid cylindrical bounding frame with a desired height variation along the azimuthal direction. We seal the bottom of the frame with a flat glass substrate and position it at the bottom of a larger container. We inject a desired volume of optical liquid into the frame and fill the container with immersion liquid until the frame is entirely submerged. Finally, we illuminate the container with UV light for several minutes to solidify the optical liquid. The optical component is then ready and can be removed from the immersion liquid. The bounding frame can be detached from the component and reused, or it can remain attached to it and serve as a mechanical interface.

Figure 4 presents the design and fabrication of a freeform optical component using the fluidic shaping method, and a comparison of its surface topography with the theoretical prediction. Figure 4(a) presents the frame providing the boundary conditions for fabricating the component, printed on a consumer-grade 3D printer. Figure 4(b) presents the expected resulting surface as obtained from Eq. (7) for a bounding frame with four sinusoidal periods, i.e., $h({R_0},\theta) = {b_4}\sin ({4\theta})$, where ${b_4} = 0.55\;{\rm mm},\,{R_0} = 17.5\;{\rm mm} $, a Bond number of $Bo = 3$, and a volume of $V = 3\,{\rm mL}$. Figure 4(c) presents the resulting component after solidification. The entire fabrication process required 40 min to complete, of which 30 min were spent on 3D printing of the frame, and 10 min on manual injection of the liquids, curing, removal from the container, and drying. Using a digital holographic microscope (DHM R-1000 LynceeTec, Switzerland) we measured the topography of the resulting freeform surface along three radial lines (0°, 30°, 60°). A detailed description of the fabrication process and the measurement technique is available in Supplement 1 Sections S3 and S4.

Since our process is currently entirely manual, we expect inaccuracies in the injected volume and in the density of the immersion liquid. In addition, the shape of the boundary is affected by the ${\pm}25\,\,{\unicode{x00B5}{\rm m}}$ accuracy of the 3D printer. Hence, we first turn to extract the global parameters—the Bond number, the injected volume, and the amplitude of the frame-height variation, from the resulting component. Using a least means squares fit between the three measured lines and the theoretical model, we obtain $Bo = 2.91,\,\, V = 3.21\,{\rm ml},\,\,{b_4} = 0.56\,\,{\rm mm}$, which are well within the expected error of our manual process.

Figure 4(d) presents the difference between the measured surface height along the three radial lines, and the theoretical values along these lines using the extracted parameters. The results show that for radii smaller than 8 mm the differences between theory and measurements are capped at approximately 1 µm and grow to 4 µm toward the edges of the frame. The RMS of the error, over all three lines, is 690 nm. Figure 4(e) presents the measurement of the surface roughness of the resulting component over an area of $4.4 \times 4.4\;\unicode{x00B5}{\rm m}$, using atomic force microscopy (AFM), showing an RMS value of 0.98 nm, as expected due to the smoothness of the liquid–liquid interface.

## 4. DISCUSSION

We introduced a new method for the design and fabrication of freeform components based on shaping of fluidic interfaces. We showed that the steady-state topography of the liquid interface is dictated by the shape of a bounding frame, the volume of the optical liquid, and the dimensionless Bond number. The fabrication time, dominated by the 3D printing of the frame, is of the order of tens of minutes and is independent of the complexity of the frame shape. Owing to the natural smoothness of liquid–liquid interfaces, the resulting surface roughness is of the order of 1 nm without the need for any post-polishing steps. In addition, while the results presented in this work were obtained using a specific polymer, in principle, any curable liquid can be used to form a freeform component, provided that an appropriate immersion liquid can be identified. We have successfully implemented the method using silicone oils and a variety of optical adhesives having different optical and mechanical properties.

The range of surfaces that can be produced using the method can be described by summation of any number of Fourier–Bessel modes that share a single Bond number. Additional degrees of freedom can be gained by utilizing frames that vary azimuthally on both ends. The optical liquid injected into the frame would thus have two contact surfaces with the immersion liquid, creating components with freeform surfaces on both sides. Moreover, enclosing each end of the frame within a different immersion liquid would result in surfaces that are based on different Bond numbers, allowing to further increase the degrees of freedom of the component.

Finally, while our work here focused on fabrication of solidified components, the optical fluid may purposely remain in its fluidic state, allowing the implementation of dynamically controlled optical components. Simple real-time changes may be possible by injecting or aspirating optical liquid, or by adjusting the immersion liquid density. A higher level of control could perhaps be achieved by implementing a deformable bounding frame whose shape can be modified in time.

## Funding

European Research Council (678734).

## Acknowledgment

This project has received funding from the European Research Council under the European Union’s Horizon 2020 Research and Innovation Programme, grant agreement 678734 (MetamorphChip). We thank Baruch Rofman for performing the AFM surface quality measurements, and Rishabh Das for critically reviewing an earlier version of the derivation.

## Disclosures

The authors declare no conflicts of interest.

## Data Availability

Data underlying the results presented in this paper are available in Dataset 1, Ref. [35].

## Supplemental document

See Supplement 1 for supporting content.

## REFERENCES

**1. **J. P. Rolland, M. A. Davies, T. J. Suleski, C. Evans, A. Bauer, J. C. Lambropoulos, and K. Falaggis, “Freeform optics for
imaging,” Optica **8**,
161–176 (2021). [CrossRef]

**2. **F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement
of freeform optics,” CIRP Ann. **62**, 823–846
(2013). [CrossRef]

**3. **J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling
increased compactness,” Light Sci.
Appl. **6**, e17026
(2017). [CrossRef]

**4. **K. Takahashi, “Development of ultrawide-angle
compact camera using free-form optics,” Opt.
Rev. **18**, 55–59
(2011). [CrossRef]

**5. **E. R. Dowski and W. T. Cathey, “Extended depth of field
through wave-front coding,” Appl.
Opt. **34**,
1859–1866 (1995). [CrossRef]

**6. **Y. Wang, Z. Li, X. Liu, F. Fang, and X. Zhang, “Freeform-objective Chernin
multipass cell: application of a freeform surface on assembly
simplification,” Appl. Opt. **56**, 8541–8546
(2017). [CrossRef]

**7. **J. Ye, L. Chen, X. Li, Q. Yuan, and Z. Gao, “Review of optical freeform
surface representation technique and its application,”
Opt. Eng. **56**, 110901
(2017). [CrossRef]

**8. **A. Cao, J. Wang, H. Pang, M. Zhang, L. Shi, Q. Deng, and S. Hu, “Design and fabrication of a
multifocal bionic compound eye for imaging,”
Bioinspir. Biomim. **13**,
026012
(2018). [CrossRef]

**9. **W. Jia, B. Zhang, and S. Li, “Progressive multifocal liquid
lenses based on asymmetric freeform surface structure of nonuniform
thickness elastic membranes with different
constraints,” Int. J. Opt. **2019**, 1–12
(2019). [CrossRef]

**10. **B. V. Rao, K. V. Sriram, and C. S. Narayanamurthy, “Design of two-mirror telescope
systems with freeform surfaces: modified configurations and
analysis,” Proc. SPIE **7**, 014002 (2021). [CrossRef]

**11. **N. Bregenzer, M. Bawart, and S. Bernet, “Zoom system by rotation of
toroidal lenses,” Opt. Express **28**, 3258–3269
(2020). [CrossRef]

**12. **S. Bernet, “Zoomable telescope by rotation
of toroidal lenses,” Appl. Opt. **57**, 8087–8095
(2018). [CrossRef]

**13. **Z. Feng, L. Huang, M. Gong, and G. Jin, “Beam shaping system design
using double freeform,” Opt. Express **21**, 14728–14735
(2013). [CrossRef]

**14. **B. Yang, K. Lu, W. Zhang, and F. Dai, “Design of a free-form lens
system for short distance projection,” Proc.
SPIE **8128**, 81280E
(2011). [CrossRef]

**15. **Z. Zhuang, Y. Chen, F. Yu, and X. Sun, “Field curvature correction
method for ultrashort throw ratio projection optics design using an
odd polynomial mirror surface,” Appl.
Opt. **53**,
E69–E76 (2014). [CrossRef]

**16. **T. Ma, J. Yu, P. Liang, and C. Wang, “Design of a freeform varifocal
panoramic optical system with specified annular center of field of
view,” Opt. Express **19**, 3843–3853
(2011). [CrossRef]

**17. **P. Zamora, A. Cvetkovic, M. Buljan, M. Hernández, P. Benítez, J. C. Miñano, O. Dross, R. Alvarez, and A. Santamaría, “Advanced PV
concentrators,” in *34th IEEE Photovoltaic
Specialists Conference (PVSC)*
(2009).

**18. **R. Wu, H. Li, Z. Zheng, and X. Liu, “Freeform lens arrays for
off-axis illumination in an optical lithography
system,” Appl. Opt. **50**, 725–732
(2011). [CrossRef]

**19. **F. Fang, N. Zhang, and X. Zhang, “Precision injection molding of
freeform optics,” Adv. Opt. Technol. **5**, 305–309
(2016). [CrossRef]

**20. **M. Roeder, T. Guenther, and A. Zimmermann, “Review on fabrication
technologies for optical mold inserts,”
Micromachines **10**, 233
(2019). [CrossRef]

**21. **L. Zhu, Z. Li, F. Fang, S. Huang, and X. Zhang, “Review on fast tool servo
machining of optical freeform surfaces,” Int.
J. Adv. Manuf. Technol. **95**,
2071–2092 (2018). [CrossRef]

**22. **Z. Xia, F. Fang, E. Ahearne, and M. Tao, “Advances in polishing of
optical freeform surfaces: a review,” J.
Mater. Process. Technol. **286**,
116828 (2020). [CrossRef]

**23. **V. Mishra, D. Sabui, D. R. Burada, V. Karar, S. Jha, and G. S. Khan, “Experimental investigations on
slow tool servo process parameters for freeform optics
machining,” Mater. Manuf. Process. **35**, 797–810
(2020). [CrossRef]

**24. **H. Feng, R. Xia, Y. Li, J. Chen, Y. Yuan, D. Zhu, S. Chen, and H. Chen, “Fabrication of freeform
progressive addition lenses using a self-developed long stroke fast
tool servo,” Int. J. Adv. Manuf.
Technol. **91**,
3799–3806 (2017). [CrossRef]

**25. **G. Shao, R. Hai, and C. Sun, “3D printing customized optical
lens in minutes,” Adv. Opt. Mater. **8**, 1901646
(2020). [CrossRef]

**26. **R. Kirchner and H. Schift, “Thermal reflow of polymers for
innovative and smart 3D structures: a review,”
Mater. Sci. Semicond. Process. **92**,
58–72 (2019). [CrossRef]

**27. **Z. Hong and R. Liang, “IR-laser assisted additive
freeform optics manufacturing,” Sci.
Rep. **7**, 7145
(2017). [CrossRef]

**28. **B. G. Assefa, T. Saastamoinen, J. Biskop, M. Kuittinen, J. Turunen, and J. Saarinen, “3D printed plano-freeform
optics for non-coherent discontinuous beam shaping,”
Opt. Rev. **25**,
456–462
(2018). [CrossRef]

**29. **F. Kotz, A. S. Quick, P. Risch, T. Martin, T. Hoose, M. Thiel, D. Helmer, and B. E. Rapp, “Two-photon polymerization of
nanocomposites for the fabrication of transparent fused silica glass
microstructures,” Adv. Mater. **33**, 2006341
(2021). [CrossRef]

**30. **T. Gissibl, S. Thiele, A. Herkommer, and H. Giessen, “Two-photon direct laser
writing of ultracompact multi-lens objectives,”
Nat. Photonics **10**,
554–560 (2016). [CrossRef]

**31. **V. Frumkin and M. Bercovici, “Fluidic shaping of optical
components,” Flow **1**,
E2 (2021). [CrossRef]

**32. **V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a
guide,” J. Mod. Opt. **58**, 545–561
(2011). [CrossRef]

**33. **M. Gelfand and S. V. Fomin, *Calculus of Variations*
(Prentice-Hall,
1963).

**34. **A. D. Polyanin and V. E. Nazaikinskii, *Handbook of Linear Partial
Differential Equations* (Taylor and
Francis, 2016).

**35. **M. Elgarisi, V. Frumkin, O. Luria, and M. Bercovici, “Measurements of the freeform
surface (R_0=17.5 mm, b_4=0.56 mm, V=3.21 mL, Bo=2.91,
phi_0=21.45 deg), and its difference from the theoretical
surface,” figshare (2021)
https://doi.org/10.6084/m9.figshare.16713010.