1. Introduction

Polymer optics is widely used in applications where light weight, compact size, and low cost are the main priorities [1, 2]. The injection-molding process is a well-established technique for fabrication of plastic optics [3]. However, in addition to birefringence limitations that arise from the molding process, it requires a costly and time-consuming mold fabrication process by single-point diamond turning. One possible route to rapid and cost-efficient prototyping and small-series production of polymer optics is 3D printing. Indeed, there have been promising demonstrations of 3D printed optics using two-photon lithography [4, 5], micro-stereo lithography [6], a hanging droplet technique [7], and single-drop ink-jet printing [8]. Nevertheless, 3D printing of macroscopic (centimeter-scale) imaging-quality optics has remained a challenge due to the high requirements on surface smoothness and shape.

In our recent work [9, 10] we have applied a fast multi-printhead ink-jet 3D printing process, known as Luxexcel-Printoptical Technology [11], to realize freeform optics for non-imaging applications. Here we introduce an iterative fabrication scheme, which leads to an improvement of nearly two orders of magnitude in surface-profile accuracy of 3D printed optics fabricated by this technology. The advances reported here allow the fabrication of imaging-quality optics in macroscopic dimensions by 3D printing. In particular, we demonstrate an imaging lens with centimeter-scale diameter, which features optical performance comparable to that of a commercial glass lens with similar optical dimensions. Though we use a spherical singlet lens as a reference element for convenience, the 3D printing process described here is fully applicable to fabrication of aspheric (freeform) optics.

The paper is organized as follows. The manufacturing process is described in Sect. 2, with emphasis on the new developments that facilitate 3D printing of imaging-quality plastic optics. In Sect. 3 we demonstrate the high surface quality in terms of both roughness (in 1 nm scale) and shape (error in $\pm 500$ nm range), and in Sect. 4 we present measurements of the optical performance of the 3D printed lens. Finally, in Sect. 5, conclusions are drawn and some future directions of the work are outlined.

2. Manufacturing process

The basic 3D printing process, described in detail by Assefa et al. [9, 10], is a layer-by-layer ink-jet printing technique, where three printheads (each with 1000 parallel nozzles, all three printheads slightly misaligned to triple the resolution) deposit $17\mu$m diameter droplets of liquid polymer on the substrate. Adjacent droplets merge after deposition before curing, forming a layer with a thickness of around $4.1\mu$m (depending on how many droplets overlap when letting them to spread and merge before curing) and shape defined by the (sliced) printing data. The deposited layer is solidified by UV exposure, and subsequent layers are grown using the same procedure one by one until a staircase approximation of the desired surface is formed. The outer printed surface can be smoothed by controlling the fluid dynamics during UV-curing throughout the lens build-up process [9]. As a result, the 3D printed optical element is complete without any post-processing, like polishing or coating to reduce the surface roughness. Such finalization steps are typically performed in other 3D printing processes [12] using robot-based polishing methods or acetone treatment, at the cost of surface deformation and loss of rapidity, simplicity, and cost-effectiveness of the manufacturing process.

The polymer material that we use in the printing process is LUX-Opticlear. Its refractive index (n) at λ = 588 nm is 1.53 and its Abbe number is 45. To ensure proper attachment of the first droplet layer on the glass substrate, its surface is silanized prior to the deposition process. The silanization process also involves a 15-minute treatment in a mixture of HFE-7100 (by 3M) and trimethylhydroxysilane in nitrogen atmosphere, followed by rinsing in HFE-7100 for another period of 15 minutes [13].

Typically, the basic printing process of centimeter-scale optical elements such as lenses leads to surface-profile deviations of tens of micrometers from the desired shape [9, 10]. The improvements reported here result, on one hand, from process improvements gained by better understanding and control of the intricate fluid chemistry. On the other hand, we introduce an iterative manufacturing technique to correct surface shape errors by testing the 3D printed element against a reference element with a reasonably similar optical function. The testing setup (described in the following section) provides information on the output-wavefront error at subwavelength accuracy. Based on this information, the printing data or the sliced layer images are then modified to account for these errors and a new element is printed. This process is repeated until the measured wavefront error is at an acceptable level (typically 4–5 times). The iterative process is feasible because of the high speed of our 3D printing process: the writing time of one $4.1\mu$m thick polymer layer with an area of $6\times 7{\text{ cm}}^2$ is only 4 seconds, which implies 26 minutes for the lens with the height of 1.6 mm to be demonstrated in this paper. The complete ramp-up phase of the lens manufacturing, taking into account the 4-5 iterations with lens measurements, takes less than half a working day. After that we can produce 8 lenses (diameter of 2.54 cm and thickness of 1.6 mm) per hour, which can be easily peeled off from the glass substrate. The nominal resolution of the printer is 1080 dots per inch or the sides of the square pixels are 23.52 μm.

We stress that, since the iterative process is based on wavefront-error analysis instead of direct surface-profile measurement, it automatically accounts for and corrects any errors caused by possible refractive-index variations inside the 3D printed element. Such variations can result from UV-curing dosage differences across the printing area. The refractive-index inhomogeneity of LUX Opticlear material is measured (for 0.5 mm thick plate) to be less than $2\times {10}^{-5}$, which is the threshold required for precision imaging applications [14].

We chose to demonstrate the accuracy of the 3D printing process by fabricating polymer lenses that are similar in geometry with a commercial plano-convex glass lens with 100 mm focal length, 51.5 mm radius of curvature, 2.54 cm aperture diameter, and 3.6 mm axial thickness (Thorlabs LA1509 made of N-BK7, n = 1.517 at λ = 588 nm). Since such a lens is essentially diffraction-limited only over an aperture diameter of $\sim 10$ mm, we stopped both the commercial lens and the 3D printed lens down to this aperture in the optical characterization experiments. Using an interferometric setup designed specifically for the present purposes, we measured and compensated a lens with an aperture of $\sim 12$ mm.

3. Surface roughness and shape

The surface roughness of the 3D printed lenses was characterized with an optical profilometer (Wyko NT9300) and typical results are presented in Fig. 1. Using the standard surface roughness definitions [15], the measured RMS roughness is $\text{Rq}=0.9\pm 0.3$ nm and the mean roughness is $\text{Ra}=0.7\pm 0.25$ nm (σ, $N=25$). These results, obtained without any post-processing of the 3D printed surface, represent an order-of magnitude improvement over our previously reported results [9].

 

Fig. 1 Surface roughness of 3D printed lens. (a) A 2D map of the surface over an small area (60 μm×50 μm). (b) Cross-sectional surface profile at the center in nanometer scale.

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Figure 2 illustrates the interferometric setup used to measure the wavefront errors produced by the 3D printed lens. A collimated leaser beam (λ = 633 nm) is split into two parts, one passing through the reference lens (Thorlabs LA1509) and the other through the 3D printed lens before recombination. Finally an imaging objective is used to observe the interference pattern of the two refracted wavefronts formed by the lenses. The lens positions are first adjusted along the optical axis such that the mean curvatures of the two output waves are equal, and then the reference lens is shifted slightly off-axis to introduce a fringe pattern with a convenient spatial frequency. The 3D printed lens produces the same wave front as the reference lens if the fringes are straight. We note that the focal lengths of the two lenses need not be precisely the same for this test to work, which arise from slight difference in refractive index between the lenses. Further, since the lenses are placed with the planar surface facing the collimated beam, the reference wavefront has spherical aberration. However, this does not matter in the present null-test setup.

 

Fig. 2 Null-test Mach-Zehnder interferometer.

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The wavefront error is retrieved from the interference pattern using the technique introduced by Takeda et al. [16]. To remove the effect of un-even illumination we capture three images: the first with only the reference arm open (intensity profile I₁), the second with only the object arm open (profile I₂), and the third with both arms open (profile I). The normalized fringe pattern to be analyzed further is then calculated using the formula

The result of this procedure is illustrated in Fig. 3(a). We then Fourier transform I_norm and crop the result to retain only one of the two Fourier peaks as shown in Fig. 3(b). Note that Eq. (1) normalizes the interferogram to an interval $\left[-1,+1\right]$, which removes the dc peak from the Fourier transform. We then shift the maximum value to the center of the Fourier coordinates and apply the inverse Fourier transform to reveal the wrapped phase difference between the reference and object wave fronts; see Fig. 3(c). An unwrapping procedure, which is based on adding the integer multiples of $2\pi$ into the wrapped phase gradient, yields the final result illustrated in Fig. 3(d).

 

Fig. 3 Phase-difference retrieval procedure. (a) Normalized interferogram I_norm, (b) Fourier transform of I_norm and the cropping area (the rectangle; the circle contains the retained Fourier peak). (c) Wrapped phase (in radians) after inverse Fourier transform. (d) Unwrapped phase.

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The measured surface profile deviation is then iteratively compensated on the design part, thus the printing data is modified for example as shown in Fig. 4. The black pixels are printed and the white ones are not. The uncompensated layer would be just a solid black disk.

 

Fig. 4 An example of a sliced layer image pixels after the error correction. The white pixels are left unprinted to compensate for the shape error. The diameter of the compensated area is ∼12 mm.

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Figure 5(a) shows a finished 3D printed lens, while 5(b) and 5(c) illustrate the surface quality of a 3D printed lens before and after the iterative process. Here the wavefront error data retrieved from the interferograms is transformed to surface-profile errors assuming that the lens material has a uniform refractive index. The error without iteration is more than 5 μm, but after five iteration steps it is reduced by nearly an order of magnitude to a range of approximately $\pm 500$ nm.

 

Fig. 5 (a) A 3D printed lens (F# = 8.4) with 25 mm clear aperture. (b) Surface profile error before iteration. (c) Surface deviation after five iterations.

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4. Optical performance

The 3D printed lens was first characterized, and compared with the commercial reference lens, by measuring its modulation transfer function (MTF) by taking pictures of a standard USAF 1951-IX MTF resolution target in reflection mode. The target was illuminated using essentially incoherent light with a bandpass filter centered at 550 nm and having a 100 nm passband. Figure 6 illustrates the results in comparison with the commercial lens. Considering the human-eye image resolution power, i.e., 10%

of modulation transfer function (MTF), the 2 elements of Group 7 of the USAF resolution target are resolved by the 3D-printed lens clearly, which corresponds to 143.7 lp mm${ }^{-1}$. However, qualitatively, the commercial lens has somewhat superior performance. The smallest details shown in (e) and (g) are visible but the contrast in the image of the 3D printed lens is somewhat lower. This could be due to the surface profile deviations or material uniformity inside the printed lens relative to the commercial glass lens.

 

Fig. 6 Images of a USAF 1951-1X MTF resolution target with (a) the 3D printed lens and (b) the commercial reference lens with 12 mm aperture diameter using a green band-pass filter. Magnified images of Groups 6 and 7 of the resolution target with (c) 3D-printed and (d) commercial lens. Magnified images of Group 7 in (e) and (g), and cross-sectional intensity profiles in (f), the red line marks the results with the printed lens and light blue with the reference lens.

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We have also checked the optical performance of the 3D-printed lens using Trioptics ImageMaster HR imaging quality testing device. Figure 7 shows the experimentally measured MTF values of the 3D-printed lens against the reference lens. The measurement result show that the MTF curve for 3D printed and reference lens are close to each other but below the diffraction limit. At the 10% MTF threshold, the imaging resolutions are recorded to be 125 lp mm${ }^{-1}$ and 135 lp mm${ }^{-1}$, respectively. However, the difference in MTF curves results could be attributed to practical polymer optics limitations such as scratch and ageing that could be minimized by using anti-reflection coating. In addition, the measured focal length of the printed lens is within ±3 % tolerance value from the target, relative to ± 1 % for the commercial lens.

 

Fig. 7 Ideal MTF resolution simulations and experimental results for the N-BK7 and LUX-Opticlear lenses using ∅ 10mm aperture at λ = 546 nm.

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5. Discussion and conclusions

We have shown that imaging-quality optics can be 3D printed by modified ink-jet technology: singlet lenses with wavefront errors in the range $\pm 1$ wavelengths in the visible region were demonstrated using an iterative error-correction process. Further tuning of the process may allow us to approach diffraction-limited performance in the near future.

Although we presented results on fabricating plano-convex lenses with a spherical top surface, the process can be readily applied to generation of free-form optics. As long as the deviation of the desired wavefront from a spherical reference wavefront does not exceed a few tens of wavelengths, the present Mach–Zehnder testing setup can be employed as such, using a pre-computed interference pattern as a target. Alternatively, one may employ a null test by adding an appropriately designed diffractive element (computer-generated hologram) in the reference arm of the interferometer (this type of null tests are described in Ref. [17]). The diffractive elements for this purpose could be fabricated rapidly and cost-effectively in binary-amplitude form using, e.g., the present 3D printing process, followed by metal-film deposition and lift-off.

At present only one polymer material (LUX-Opticlear with refractive index $n_{\mathrm{d}}\approx 1.53$ and Abbe number $V_{\mathrm{d}}\approx 45$) is available for the presently considered ink-jet process, which prohibits direct 3D printing of achromatic lens systems. However, manufacturing such systems is possible by combination of 3D printing with silicon molding and vacuum casting techniques [10, 18]. Polymers with Abbe numbers ranging from 30-60 can be employed in such a process. Another option is the use of hybrid refractive-diffractive optics. However, 3D printing of conventional modulo $2\pi$ diffractive surfaces is not currently possible due to the large ($4.1\mu$m) single-layer thickness in the Printoptical process; hence molds for diffractive surfaces would have to fabricated by more expensive techniques such as diamond turning or lithography.