Method and system for the centrifugal fabrication of low cost, polymeric, parabolic lenses

McCrae Wattinger; Paul Gordon; Masih Ghorayshi; Gerard Coté

doi:10.1364/OE.27.021405

1. Introduction

While many modern engineers have access to rapid prototyping using in- house equipment such as 3D or circuit board printers, the optical engineer is typically relegated to much slower options for either prototypical or end-use applications. Additive techniques for rapidly generating lenses have been proposed in the literature, but typically require extensive polishing processes similar to traditional lens manufacturing techniques. This polishing sub-process can be prohibitively tedious and expensive for a rapid prototyping technique [1]. Rapid micro-lens production has been proposed by holding and curing a volume of liquid polymer in the desired shape using material surface tension and pressure to create lenses with maximum diameters typically of several millimeters [2,3]. These systems range from single droplet extrusion [4], to leveraging the interface of two immiscible fluids [5], to the simultaneous production of full lens arrays which are generally much smaller in scale (order of microns in aperture diameter) [6,7]. While these micro-lens techniques avoid the need for any post-polishing processes, their reliance on forces of surface tension and sample pressures limit the theoretically maximum diameters possible, restricting applicability for many systems.

In order for a rapid lens fabrication system to be useful, it must be capable of producing lenses over a wide range of diameters and focal lengths, with controllable selection of the desired surface profile. It should produce lenses without a lengthy post-polishing step, in less than a day, with minimal costs, and should be useful with a variety of materials to allow for customization of the lens index of refraction. In this work, one such system is proposed, prototyped, and tested for potential utility as a lens rapid prototyping method.

The proposed system builds off of the operating principle of liquid mirror telescopes (LMT’s). LMT’s were first reported in 1909 by astrophysicist Dr. Robert Wood, who proposed and demonstrated a technique for shaping liquid metal into a highly uniform telescopic mirror [8]. LMT’s are built around a rigid, cylindrical container filled with a reflective fluid that is rotated about its axis of symmetry at a steady velocity. The resulting surface is a very smooth paraboloid, which can be used as a collection mirror in telescopic systems with a large range of aperture diameters [9]. The surface roughness of these mirrors are in the same neighborhood as spin-coated objects, which avoid the need for any polishing steps [10]. The production of liquid mirrors for aerospace and astrophysics applications has now been in development for over one-hundred years [11–14].

This same general principle can be used to produce lenses of three distinct geometric forms, each with at least one parabolic surface whose curvature is regulated by angular velocity. To accomplish this, a rotating chamber is filled with uncured, optically clear polymer, which is then shaped by centrifugal force and cured while at a steady-state rotational velocity. The range of possible diameters is dependent only on the cavity diameter of the rotating chamber, which makes the aperture diameter arbitrary to the fabrication engineer. In some ways this process is similar to spin-casting from a mold, which is often used to produce meniscus contact lenses [15]. A model for lens production using this technique is presented along with corresponding experimentally manufactured lenses. A number of evaluation processes are used to evaluate the merit of the technique, and a final discussion addresses the limitations of the method.

2. Investigation of the fabrication principle

The criterion for a lens to be classified as an asphere is dependent on the mathematical form of its surface height as a function of radial position, r. In particular, the surface height z(r) must take the form

(1)$$z(r) = \frac{{{r^2}}}{{{R_{curv}}\left( {1 + \sqrt {1 - (1 + \kappa )\frac{{{r^2}}}{{{R_{curv}}^2}}} } \right)}} + \sum\limits_i^{} {{\alpha _i}{r^i}}$$

where z(r) is an even polynomial with coefficients αi, when the i^th term has a degree greater than two. R_curv is defined as the vertex radius of curvature, and κ is the conic constant. The conic constant classifies z(r), for various values of κ [16].

2.1 Physics of the fabrication technique

The system operates upon two physical mechanisms. The first is known as Bernoulli’s rotation of fluid in a static cylinder, which is solely responsible for the aspheric nature and parabolic profiles of the lenses [17]. A derivation of the profile starts with the Naiver-Stokes equation (the fluid analog of Newton’s second law), which describes the pressure field (P) of a body of fluid in terms of fluid density (ρ), its motion subjected to gravity (g), and acceleration due to rotation (a_rot),

(2)$$\mathop \nabla \limits^ \to P(r,z) = \rho ({g\hat{z} - {a_{rot}}\hat{r}} ).$$

At a time prior to rotation, the fluid is static, as depicted in Fig. 1(a). Rotation of the chamber results in a centripetal force directed inward at the axis of rotation. The cartoon in Fig. 1(b), displays the fluid’s opposing reaction, pushing the fluid toward the wall of the crucible. The resulting void is filled with atmosphere and causes a pressure gradient with the change in depth, resulting in the following equations of motion [17],

(3)$$\frac{{\partial \rho }}{{\partial r}} = \rho {\omega ^2}r$$

(4)$$\frac{{\partial \rho }}{{\partial z}} = - \rho g.$$

Solving the second equation for ρ, and moving all the partial differentials to one side yields a single differential equation in z, and integration of this equation produces a single profile function z(r), characteristic of the fluid height in the cylinder under steady state rotation [17],

(5)$$z(r,\omega ) = {h_0} - \frac{{{D^2}}}{{16{R_{curv}}(\omega )}} + \frac{{{r^2}}}{{2{R_{curv}}(\omega )}}.$$

Fig. 1. Physical sketch of static layout (a), and sketch of bodies undergoing rotation (b).

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In this equation, D defines the crucible diameter. R_curv takes on a physical relationship with the motor’s angular velocity of rotation ω in units of radians per second and the gravitational constant [16],

(6)$${R_{curv}}(\omega ) \equiv \frac{g}{{{\omega ^2}}}$$

(7)$$\Delta {R_{curv}} = 2{R_{curv}}\left( {\frac{{\Delta \omega }}{\omega }} \right).$$

The uncertainty in R_curv, as it varies with speed is also defined in Eq. (7). Variances in the speed of rotation yield changes in the parabolic surface. Figure 2 later depicts several different surface profiles that can be achieved for different rates of rotation. The gravitational constant takes on the value 9800 mm-s^-2, adjusted from m-s^-2 for the appropriate scale of the optics.

Fig. 2. Mathematically derived lens properties: surface profiles for a several motor velocities at a volume of 3 mL (a), focal length and R_curv vs motor velocity (b), NA vs velocity (c), diffraction limit vs velocity (d), and R_curv uncertainty vs motor velocity (e).

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It should be emphasized that this work’s focus is on the centrifugal lens production technique as a novel manufacturing method and thus uses the radius of curvature as one of the primary metrics of interest for evaluating the lenses. Other characteristics of lens quality such as wavefront error and aberration coefficients [18] are obviously more insightful parameters for evaluating lens quality, however they are left for future analysis as the goal of this paper is to prove the use of the described physical concepts toward lens-making. The R_curv is used because it highlights connections to the surface geometry of the lens, the velocity of the fabrication, and is used to build into other optical properties like focal length, etc.

This form of asphere is parabolic in nature. To determine the type of asphere this surface represents, simply substituting -1 for κ into Eq. (1) yields

(8)$$z(r) = \frac{{{r^2}}}{{2{R_{curv}}}} + const.$$

Comparing the result to Eq. (5) verifies that the surface generated by this particular application is indeed an asphere of parabolic form. Assuming that higher order terms from vibrations and off-axis rotations are minimized, the profile should theoretically be axially symmetric. The possible effects of perturbations on the lenses will be revisited in the error analysis portion of the discussion in this paper.

The second mechanism to be considered is surface wetting that occurs at the boundary between the chamber, air, and liquid polymer. Surface tension continues to act on the fluid interface, but the strength of its interaction is dependent upon the speed of rotation. Dr. Vlado A. Lubarda treats this speed dependence rigorously in [19]. It is necessary to define a ratio between the standard capillary length (due to gravity) and an effective capillary length due to rotation of the fluid. This ratio is defined below and altered to represent the problem in terms of R_curv,

(9)$${\left( {\frac{{{l_o}}}{{{l_\omega }}}} \right)^2} = \frac{{D{\omega ^2}}}{{2g}} = \frac{D}{{2{R_{Curv}}}}.$$

where l_o represents the standard capillary length due to gravity and l_ω represents the capillary length due to rotation.

The interpretation of this ratio (via Lubarda’s results), implies that surface tension will be a contributing factor to the surface shape when the ratio is much larger than 1. In other words, small diameter chambers at high speeds, or large diameter chambers at slower speeds will introduce variations in the surface profile via the effects of surface tension. As such, this study shall focus on the cases in which the rotational forces dominate over surface tension, for the sake of simplicity. This also means the minimum of the parabola should be at least 1-2 mm above the chamber bottom, for the same reason. One could certainly attempt to control the combined surface effects from surface tension and fluid rotation to produce variations from the ideal parabolic surface described by Eq. (8). Accounting for the added complexity requires altering Eq. (5), an example of this process is demonstrated in [20]. To reiterate, when Eq. (9) is much greater than 1, the surface variation from Eq. (5) due to surface tension will be non-negligible. This would reduce an effective aperture of the fabricated lens well below the industry standard of 90 percent [21].

2.2 Theoretical lens performance via optical specifications

The high-end rotational velocity limit for lens production was selected based on the limiting amount of liquid in the crucible, derived in [17]. This upper bound on velocity can be thought of as a spilling limit as any motor velocities higher than this would result in fluid climbing up the walls and out of the container. For a given volume or initial fluid height, this maximum fluid volume can be modeled as

(10)$${\nu _{\max }} = \frac{{2\sqrt {{h_0}g} }}{{\pi D}} = \frac{{4\sqrt {gV} }}{{{{({\pi D} )}^2}}}.$$

Parabolic profiles described by Eq. (8) can form either concave or convex faces, depending on the manufacturing technique employed. For plano-convex cases, additional insight into capabilities of the manufacturing technique were gained through studying the lens R_curv, numerical aperture, and diffraction limits over the range of possible combinations of fabrication variables. Varying both the R_curv and focal length of this lens with motor velocity is described in Fig. 2(b). These variances can be modeled using the lens-maker’s equation, which can be simplified by assuming that the second sides of the concave and convex lenses are planar, such that the second radius of curvature is infinite, thereby reducing the second radius term to zero. Thus, knowing that the R_curv varies with the inverse square of angular velocity, the focal length (f) of the lens can be related in much the same way using

(11)$$\frac{1}{f} = \frac{{({{n_{mat}} - 1} )}}{{{R_{curv}}(\omega )}} = ({{n_{mat}} - 1} )\frac{{{\omega ^2}}}{g}.$$

The diameter of the lens-spinning crucible and thus the diameter of the produced lenses may be freely chosen, establishing another degree of freedom within the lens selection process. Effects of variances in the lens diameter are analyzed using multiple curves to describe both the theoretically maximal NA and corresponding diffraction limit of each lens (each curve corresponding to a unique diameter) in Fig. 2(c) and 2(d). The NA of each lens is described as

(12)$$NA = {n_{air}}\sin \left( {\arctan \left( {\frac{{{\omega^2}({{n_{mat}} - 1} )D}}{{2g}}} \right)} \right).$$

Examining Fig. 2(c) for changes in NA with motor velocity, it should be noted that different volumes of liquid polymer in the chamber can shift each curve to the left or right. A confirmation for the accuracy of each combination of volume and motor speed is seen in the fact that each speed lies within the range of possible combinations dictated by the spill velocity in Eq. (10). Going one step further, it is predicted that the spill velocity represents the fastest lens per chamber diameter and liquid volume pair. Figure 2(d) summarizes the theoretical single lens diffraction limit capabilities of this manufacturing system. Using Rayleigh’s criterion for defining the diffraction limit of this asphere requires two physical dimensions of the lens – aperture diameter and focal length [22]. By expressing the focal length in terms of rotational velocity, the diffraction limit is described as,

(13)$${\ell _{\lim }} = \frac{{1.22f\lambda }}{D} = \frac{{1.22g\lambda }}{{{\omega ^2}({{n_{mat}} - 1} )D}}$$

where λ is the wavelength of light used for analysis.

3. Methods

3.1 System design

The main requirements for physical system design were maintenance of constant motor velocity with small uncertainty, data collection, and minimizing axial disturbances during rotation. The system is composed of a LabVIEW graphical user interface (GUI), the electronics required to control the motor and collect data, a crucible which may be detached from a vertically rotating drive shaft, drive train connecting the drive shaft to an electric motor, and crucible heating component (a heat gun, in this case). The crucible consists of an aluminum cylindrical cavity with several small ejection ports in the base and a polished disc-shaped insert that is deposited into the bottom of the crucible prior to lens manufacture. The polished disc simultaneously provides a polished surface to mold lenses against and the ability to eject lenses from the crucible once they are cured. The arrangement of the system is depicted in Fig. 3.

Fig. 3. Prototype fabrication system: CAD model of the fabrication system emphasizing (from left to right) the detachable aluminum crucible with sliding bottom flat, heat gun, IR temperature sensor, chain drive, electric motor and control circuitry (a). Experimental motor velocity data gathered over time using 8-bit speed encoder (b).

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For motor velocity control, a proportional, integrative, and derivative (PID) feedback control scheme was implemented [23]. After locating the proper PID coefficients, a sweep of viable motor speeds was performed to verify that the RMS speed was constant over the range of 200 to 300-rpms as depicted in Fig. 3(b). Constant motor velocities were achieved between 80-1000-rpm. Over the range of velocities from 70 to 300-rpms, the PID tuning achieved a minimal variation in RMS velocity of ± 1 rpm, corresponding to the velocity profiles captured in Fig. 2(e).

3.2 Fabrication process

Sylgard 184 polydimethylsiloxane (PDMS) was chosen to demonstrate the system capabilities although the system has applicability for a variety of materials. The thermo-set polymer was prepared externally from the system by combining the standard of 10 to 1 base to curing agent in volumes that correspond to a desired lens thickness. Once prepared, a thin layer of PDMS may be deposited and cured onto the still base surface of the polished disc to provide a smoother planar surface to mold lenses against. Uncured PDMS for the lens is then deposited into the aluminum cylindrical chamber, which initiates the fabrication process. The crucible is accelerated by the motor until the desired angular velocity is reached. Once steady state velocity is achieved, a uniform parabolic surface manifests across the liquid polymer, as a direct consequence of the centrifugal pressure field generated at the PDMS-air interface.

The heat gun is then turned on, and the crucible reaches a constant temperature of approximately 150°C after four minutes. The duration of this heating varies depending on the volume/thickness of lenses desired. After heating is over, the heat gun is turned off, and the crucible is removed from the motor shaft and allowed to cool on a large metal counter top for 10-12 minutes to room temperature (approximately 21°C). The removable aluminum platform and cured lens are elevated up and out of the chamber. The lens is then carefully detached from the face of the removable aluminum platform, and is thereafter only handled by the sides of the cured elastomer, as shown in Fig. 4(c). This process yields a single planar-convex aspheric lens, over the span of a half hour in total (not including material prep), most of which is spent heating and cooling the chamber. Each step of the process is illustrated in Fig. 4.

Fig. 4. Cross section of lens fabrication process: concave lens prep (a), concave lens cure step (b), and concave lens ejection (c). Alternative process: convex lens cure (d), convex lens ejection and removal from mold (e). Alternative process: meniscus lens cure (f), meniscus lens ejection and removal from mold (g).

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The described procedure yields a plano-concave lens. Procedures for generating plano-convex and meniscus geometries use the planar-concave lens as a casting mold. After the mold is prepared, uncured PDMS for the second lens is injected into the chamber-mold ensemble. To produce plano-convex lenses, the chamber is kept static (not rotating), and a homogeneous heating condition is applied to initiate lens curing. Heterogeneous heating during this step was found to produce surface irregularities in the planar face due to thermal expansion. It was also found that lower temperatures and longer curing times reduced incidence of such abnormalities when casting PDMS lenses. To create a meniscus lens, the entire crucible containing cured mold and uncured PDMS is instead rotated and cured using the heat gun as previously described for the plano-concave case. After the secondary lenses are cured, they can be carefully peeled off of the concave molds, generating a perfect casting of the concave surface on a now convex surface. Vapor-deposited mold release may be used to improve the ease of lens separation, but it was found to compromise surface roughness and was not necessary to separate lenses from each other.

At this point it may be considered what effects of viscosity and spatial thermal gradients within the liquid polymer play into the thermoset curing process. Because the rotating fluid is ideally a hydrostatic problem, as shown in Eq. (2), viscosity does not explicitly appear in Eq. (5). Viscosity’s role arises when any mechanical vibrations or eccentricity (chamber axis tilt) occurs, causing fluid to flow. In this case higher viscosity is favorable, since it damps small amplitude vibrations, preventing them from appearing on the lens surface. Slow, homogenous heating throughout the PDMS is preferable to prevent surface irregularities which may arise from uneven curing. A Seek thermal imaging camera was used to observe the heat propagation within a curing lens and to reposition the heat gun to ensure homogenous sample heating.

3.3 Lens samples

The independent variables controlled for this technique were the liquid polymer volume, rotational velocity, heating profile, and lens geometry. The curing temperature of the heat gun was held constant at 150 °C, cavity diameter held constant at 25 mm, and liquid PDMS volume held constant at three milliliters. Five lenses made for each planar-concave and resulting planar-convex case according to the parameters laid out in Table 1. It is important to note that each concave mold was produced for every repetition of a planar-convex lens in order to see how repeatable the system was in generating desired optical characteristics. Samples of both concave and convex lenses for each speed are shown in Fig. 5.

Fig. 5. Showcase of fabricated lenses. The planar-convex lenses are situated in the front row, each cured under static conditions, with the adjacent planar-convex mold/lens responsible for its production in the back row to its right.

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Table 1. Lens fabrication test cases and relevant metrics.

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3.4 Profilometry measurements

To assess lens R_curv, a Zygo white light interferometer was used. Several caveats for the interferometric method must be addressed. Firstly, surfaces of parabolic form have steep surface gradients, causing fringe density to increase very rapidly with radial distance from the vertex until it exceeds the resolution of the imaging system. The result is that the scannable area of these lenses is confined to a maximum of 1.5 mm² across the apex of the lens, limiting the amount of information that can be gathered for the overall lens profile. However, it does provide high-confidence validation of R_curv and thus confirmation of the theoretical model and overall method of lens manufacture. To gather information for the shape of the entire lens surface, a Mitutoyo PH-A14 profile projector was also used to project a focused shadow of the lens onto an electronic coordinate measuring system. The system’s lateral resolution was 2.54 microns, and surface profile height was collected every 2.54 µm across. Two orthogonal profiles were collected for each lens. This method provides bulk profiles spanning the full diameter of the lens, and while having less sampling precision than an interferometer, the data can be accurately fit to an aspheric equation used to solve for R_curv and the conic constant of the lens.

3.5 Surface roughness measurements

A Bruker Dimension Icon atomic force microscope (AFM) was used to characterize the root-mean-squared (RMS) surface roughness of the lenses. The point of this measurement is to identify the typical surface roughness produced at each interface in aggregate over several lenses with the same fabrication conditions. Two unique surface types are of interest for surface roughness – statically cured and dynamically cured surfaces. It was shown that PDMS surfaces molded against another take on the same surface roughness as the original surface. If intrinsically low surface roughness can be demonstrated for these lenses, undesirable post-polishing steps may be avoided. The RMS roughness of each interface is then compared to industry standards. The equation for surface roughness is defined as,

(14)$${R_q} = \sqrt {\frac{1}{{NM}}\sum\limits_{}^{} {_n\sum\limits_{}^{} {_m{{({z({n,m} )- \overline z } )}^2}} } } .$$

Where z-bar is the average surface height of a grid of discrete heights z(n,m), such that n is the row index, and m is the column index. N and M make up the total number of rows and total number of columns in the grid respectively.

3.6 USAF target imaging – resolution and modulation contrast

The purpose of this test is to assess the lens quality and performance by collecting images that will show changes in magnification and resolution capabilities for each lens that result from the varying focal distances and radii of curvature. Images of a negative contrast 1951 US Air Force Target (USAFT) were collected, using each fabricated convex lens as a single-unit objective. Polymeric lenses from the study were fitted into lens tubes and attached as replacement objectives for an inverted brightfield microscope, and images of the AFT were collected using a 16-bit monochromatic sCMOS camera.

4. Results

4.1 R_curv and conic constant

Measurements for R_curv were taken for each of the convex lenses across all three rotational velocities in order test the correlation between angular velocity and the respective optical parameters as theorized. For each of the three speeds, five lenses were tested for repeatability. Figure 6 contains information for R_curv as well as a sample of the bulk profilometry results.

Fig. 6. Lens vertex profile and R_curv measurements: 200-rpm lens apex profile found using interferometer (a), average of measured R_curv against theoretically predicted values (b), aspheric fit line and theoretical profile for 300 rpm projected profile, with conic constant -0.99960 and R_curv of 10.5310 (c), and the error (absolute difference) between the aspheric fit line and the ideal profile (d).

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Figure 6(a) shows an interferometrically measured lens apex depicting excellent surface roughness and uniformity across the region. Each fabricated lens was measured and the Zygo software computed R_curv for each profile. Figure 6(b) contains the average R_curv values over a set of five lenses of the same motor velocity compared to the corresponding theoretically expected values. The error bars in the expected cases are the theoretical uncertainty in R_curv displayed in Fig. 2(e). The error bars associated with the averages are the standard deviation in radius of curvature of lenses fabricated from a single speed. The deviation across different speeds decreased according to the trend predicted in Fig. 2(e). In all cases the measured R_curv were systematically smaller than what the model predicted, with the differences between experimentally measured and the theoretically predicted R_curv being 0.8604 mm, 1.4905 mm and 1.1216 mm for the 200, 250 and 300 rpm cases respectively. The larger-than-expected deviations and smaller radii of curvature suggest the presence of minor variables in the fabrication prototype not accounted for by the theoretically perfect physical model of the system.

Each bulk profile slice gathered using the profile projector was processed in MATLAB to acquire a well-conditioned aspheric fit. The conic constant, radius of curvature, and the first three higher order even polynomial coefficients were obtained using MATLAB’s built in least squares regression curve fitting employing the Levenberg-Marquardt algorithm. Pre-processing the data (prior to fitting), consisted of translating the coordinate system to align the origin to the parabolic apex and truncating the last 2-3 points on either side of the profile to ensure symmetry for the fitting algorithm. The aspheric parameters for each profile are listed in Table 2.

Table 2. Aspheric fitting parameters of 2D lens profiles (shadow measurements).

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Each of the 2D profile slices produced a conic constant of negative one, indicating that the slices are indeed parabolic, as predicted by Eq. (5). This result was only achieved once the fitting algorithm incorporated the additional even polynomials, which increased the fitting constrains of the problem. The measured radius of curvature values varied in the opposite trend from Fig. 6(b); in this case they were almost always larger than the expected value, although the error between the expected and measured values did decrease with the increase in speed (as expected). Also, between profiles from the same lens (0 to 90 degrees), the radius of curvature varied, suggesting either systematic error in the measurement technique or lens astigmatism not shown by the interferometric measurements. It is hypothesized that minor axial tilt (even on the micron scale) affects the computed radius of curvature values. These data validate the fabrication technique’s ability to produce the desired parabolic profiles across the lens surfaces.

4.2 AFM – RMS surface roughness

RMS surface is an important parameter for overall lens quality and was assessed using atomic force microscopy for both static and dynamic surfaces. Examples of the surface profiles per interface are described in Fig. 7 below. Each profile samples a square with a side length of 10 µm. RMS surface roughness values for each interface type are listed in comparison with industry standard lens values in Table 3.

Fig. 7. Diagram of surface interface varieties (a), Surface roughness map for dynamic air-PDMS interface with area of 10 × 10 µm (b), Surface roughness map for static air-PDMS interface with area of 10 × 10 µm (c).

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Table 3. RMS Roughness measurements and industry standards [24].

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4.3 US Air Force target imaging

Convex lenses molded inside the spun concave lenses were used to image the USAFT in order to demonstrate their utility towards single-lens imaging and to validate their performance. ImageJ was used to select the smallest resolvable group element on the target. Figure 8 contains images from typical lenses for 200 rpm, 250 rpm and 300 rpm lenses, in which the resolving ability and magnification clearly increase as the rotational speed increases. This validates the theory described earlier which indicates that R_curv and focal length will decrease with increasing speed, leading to higher NA lenses and, thus, better resolution.

Fig. 8. USAFT images for 200-rpm (a), 250-rpm (b), and 300-rpm (c), which resolve elements 5-5, 6-2, and 6-5, respectively. Uncropped images presented, with magnified subsets below each.

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5. Discussion

5.1 Lens quality assessment

The lenses produced appear to exhibit excellent clarity across their entire width. Obviously, the overall clarity and refractive properties are largely dependent on the material used to fabricate the lenses. Although only PDMS was used to demonstrate the system capabilities, other compounds including other thermoset polymers as well as photo-cure or chemically cured materials could just as easily be used. Use of photo-curable polymers offer promise because they can avoid possibly detrimental effects of material thermal expansion during curing. Each PDMS lens costs an estimated 40 cents/lens to produce, with the possible cost of any lens produced tied directly to its bulk material cost. The polymeric lenses exhibit flexibility when compared to glasses, but mechanical properties could be easily manipulated to suit the appropriate needs by varying the material composition. Indeed, the deformability of lenses may have intrinsic value for adaptive optics applications. It is also worthy of note that care must be taken to prevent dust from adhering to the lens surfaces due to static charge accumulation.

Overall, the surface qualities produced by this method exhibit excellent RMS surface roughness. Results of the AFM measurements show that the dynamic air-PDMS interface has a higher RMS roughness than the static air-PDMS interface, which seems logical, as the dynamic case adds opportunities for mechanical vibrations or axial asymmetries of rotation to perturb the surface as it is curing. The typical roughness value for microscope precision optics is on the same order of magnitude as the typical roughness induced from the prototype fabrication system, without the need for any precision polishing, which is a significant advantage for lenses generated using this method.

5.2 Sources of error

Lens R_curv displayed good agreement with parabolic profiles as predicted by the theoretical models. It was shown that the precision of R_curv decreased as the fabrication velocity increased, as predicted by the uncertainty defined in Eq. (7). However, discrepancies exist between predicted R_curv values and experimental measurements from the interferometer. All measured values of R_curv are consistently less than the model-predicted values for the corresponding speeds, which suggests that either the model is overly simplistic or there is a systematic error occurrence during fabrication.

Possible sources of error identified within the system are velocity control of the motor, misalignment between crucible and drive shaft axes of rotation, control of volume of uncured polymer deposited in the crucible, and possible thermal gradients present within the material. As previously discussed in the system design section, the uncertainty of the RMS motor velocity for all 3 speeds was measured at 1 rpm. Using propagation of error, it was found that any misalignment between the chamber and the rotating shaft can introduce precession for the container. This effect has already been modeled in [25], and its effects can be captured by analyzing influence on the rotational velocity. The effect of the eccentricity on the rotation speeds can be summarized by Eqs. (15-17), upon the condition that the angular displacements occurring during precession are no larger than 3-5 degrees (so the small angle approximation can be employed).

(15)$${\omega _z} \cong \omega$$

(16)$${\omega _r} \cong \varphi \omega$$

(17)$$\frac{{\Delta {R_{curv}}}}{{{R_{curv}}}} = 2\sqrt {{{\left( {\frac{{\Delta \omega }}{\omega }} \right)}^2} + {{({\Delta \varphi } )}^2}}$$

The tilt in φ of the radial axis can affect the magnitude of the uncertainty. At a maximum angular displacement of 5 degrees, the second term dominates, and produces an uncertainty 4-5 times R_curv for a particular speed. The actual angular displacements witnessed during the precession of this particular chamber were roughly 1-3 degrees off-axis, which could account for R_curv discrepancies between theoretical and measured values.

5.3 Surface discontinuity (“ring” phenomenon)

Inspection of cured lenses revealed an occasional anomaly in the fabrication results. Several lenses display a rotationally symmetric discontinuity that varies in diameter and uniformity across samples. Examples of lenses both with and without this “ring” can be viewed in Fig. 9. The leading hypotheses for ring appearance are based on suspected inhomogeneous heating of uncured polymer during fabrication. This could affect lens quality by inducing the phenomena known as vorticity rings, which can occur as thermal gradients cause convection-flow that drives fluid up and radially outward towards the walls of the crucible, giving rise to a toroidal vortex within the polymer [26]. Another possibility is that the uneven heating of fluid causes subtle abnormalities in the lenses due to the thermal expansion of the PDMS. Further tests to control internal thermal gradients are needed to test this hypothesis.

Fig. 9. Top view of three convex lenses all fabricated at 300 rpms demonstrating the presence of rings. Leftmost lens is an example without a ring. The middle and rightmost lenses contain the ring feature, indicated by the arrows.

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6. Conclusion

In this study, a method and system for the production of polymeric aspheric lenses in concave, convex, and meniscus forms with a range of customizable parabolic profiles was demonstrated. The physical phenomenon was modeled mathematically and implications on lens R_curv, focal length, diffraction limit, and numerical aperture were described. Lenses were produced for a number of test cases, with results that generally validate the constructed models. The surface quality of the lenses was assessed via RMS roughness using an atomic force microscope, with statically and dynamically cured surfaces showing surface roughness values comparable to those reported for precision tier optics in industry. Lens profiles showed consistently parabolic conic constants with some variability in radii of curvature. In some lenses, a surface anomaly was observed, and a source was hypothesized to be thermally-driven curing abnormalities. Overall, the proposed lens manufacturing technique has potential to be able to rapidly produce aspheric lenses with comparable quality to current manufacturing technologies, but more sophisticated mechanical and thermal control are merited to improve the precision and consistency of lenses produced.

Funding

National Science Foundation (NSF) (1402846, 1648451, 2017240275); Precise Advanced Technologies and Health Systems for Underserved Populations (PATHS-UP) Engineering Research Center.

Acknowledgments

The authors would like to thank Dr. Tomasz Tkaczyk, Dr. Michal Pawlowski, and Gregory Berglund of Rice University for their support in collecting interferometric surface measurements, and Cody Lewis and Richard Horner of the Texas A&M Engineering Experiment Station for their support in creating the mechanical and electrical centrifuge systems. The authors would also like to thank Dr. Wayne Hung for his support in gathering the surface profile cross sections.

Disclosures

The authors have disclosed technologies in this publication in a provisional patent application.

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Motor velocity (rpms)	Radius of curvature (mm)	Convex lens thickness (mm)	Focal length (mm)
200	22.341	7.825	55.853
250	14.298	8.789	35.746
300	9.929	9.967	24.823

Speed (RPMS)	Lens Number	Orientation (Degrees)	R_CURV (mm)	κ	α₄ (r⁴)	α₆ (r⁶)	α₈ (r⁸)
200	1	0	25.2468	-0.99720	1.22E-05	-2.40E-07	2.18E-09
200	1	90	22.9102	-1.00300	-9.16E-05	1.42E-06	-6.26E-09
200	2	0	23.3990	-1.00420	-5.55E-05	7.82E-07	-2.48E-09
200	2	90	22.9208	-0.99670	-8.78E-05	1.23E-06	-4.24E-09
250	1	0	14.3235	-1.00060	-1.21E-04	1.51E-06	-5.99E-09
250	1	90	15.0187	-1.00190	-2.89E-05	1.70E-07	6.30E-11
250	2	0	15.8829	-0.99860	4.65E-05	-9.35E-07	5.13E-09
250	2	90	16.1768	-0.99980	3.53E-05	-4.94E-07	2.52E-09
300	1	0	9.9468	-0.99870	-1.66E-04	2.22E-06	-9.06E-09
300	1	90	10.5313	-0.99960	-3.70E-06	-2.72E-07	1.93E-09
300	2	0	10.4332	-0.99940	-2.92E-05	2.26E-07	-5.58E-10
300	2	90	10.1842	-0.99970	-9.04E-05	1.07E-06	-4.40E-01

Lens Surface	RMS Surface Roughness (nm)
Edmund Optic – Typical Lens	5.0
Edmund Optic – Precision Lens	2.0
Centrifugal PDMS Lens - Dynamic	2.3
Centrifugal PDMS Lens - Static	1.58

Motor velocity (rpms)	Radius of curvature (mm)	Convex lens thickness (mm)	Focal length (mm)
200	22.341	7.825	55.853
250	14.298	8.789	35.746
300	9.929	9.967	24.823

Speed (RPMS)	Lens Number	Orientation (Degrees)	R_CURV (mm)	κ	α₄ (r⁴)	α₆ (r⁶)	α₈ (r⁸)
200	1	0	25.2468	-0.99720	1.22E-05	-2.40E-07	2.18E-09
200	1	90	22.9102	-1.00300	-9.16E-05	1.42E-06	-6.26E-09
200	2	0	23.3990	-1.00420	-5.55E-05	7.82E-07	-2.48E-09
200	2	90	22.9208	-0.99670	-8.78E-05	1.23E-06	-4.24E-09
250	1	0	14.3235	-1.00060	-1.21E-04	1.51E-06	-5.99E-09
250	1	90	15.0187	-1.00190	-2.89E-05	1.70E-07	6.30E-11
250	2	0	15.8829	-0.99860	4.65E-05	-9.35E-07	5.13E-09
250	2	90	16.1768	-0.99980	3.53E-05	-4.94E-07	2.52E-09
300	1	0	9.9468	-0.99870	-1.66E-04	2.22E-06	-9.06E-09
300	1	90	10.5313	-0.99960	-3.70E-06	-2.72E-07	1.93E-09
300	2	0	10.4332	-0.99940	-2.92E-05	2.26E-07	-5.58E-10
300	2	90	10.1842	-0.99970	-9.04E-05	1.07E-06	-4.40E-01

Method and system for the centrifugal fabrication of low cost, polymeric, parabolic lenses

Abstract

1. Introduction