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Abstract
The degrees of freedom afforded by nanocomposite materials and additive manufacturing allow for the precise control over the chromatic properties of gradient index (GRIN) optics. The ability to engineer nanocomposite optical materials using blends of three or more constituents makes it possible to independently specify the refractive index gradient and the dispersion of optical materials. The refractive index spectra of the primary nanocomposite feedstock are defined relative to one another using various concentrations of monomers and nanofillers. Inkjet deposition is then used to print-compose specific feedstock to form refractive index gradients with precise control over dispersion. Arrays of 4-mm-diameter spherical GRIN lenses were fabricated using different nanomaterial compositions. The ability to positively and negatively control dispersion and to obtain achromatic performance was demonstrated. Control over partial dispersion is also shown.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Additive manufacturing of optics has been a vibrant area of research [1–5]. It has recently been demonstrated that drop-on-demand inkjet print additive manufacturing can be used to fabricate gradient index (GRIN) optics [6–9]. A benefit of inkjet print fabrication is the versatility it provides for depositing different material compositions throughout the volume of an optical element. As the name implies, with drop-on-demand inkjet printing, a picolitre-scale drop of optical material is precisely deposited on a substrate according to a pre-determined print pattern. By varying the properties of the feedstock, as it is printed, or by print-composing heterogeneous materials by co-depositing, mixing, and inter-diffusing multiple feedstocks as the optic is composed, layer-by-layer, it is possible to create complex volumetric index gradients within the optical element [6,8].
GRIN offers potential degrees of freedom in the form of dimensionally varying refractive index gradients [10–15]. However, limited by available fabrication processes, GRIN geometries have historically been limited to simply radially symmetric index gradients [16–19]. In addition to enabling fabrication of radially symmetric spheric GRIN elements, inkjet print manufacturing is well suited for fabricating higher-order radially symmetric aspheric index gradients, as well as three-dimensional (3D) aspheric index gradients, wherein the gradient profiles may vary axially as function of their position on the optical axis, [6,8,20–23]. Inkjet print manufacturing is also particularly well suited for manufacturing freeform GRIN optical elements, in which there are no axes of symmetry in the gradient index profiles [24–30].
The properties of printable nanocomposite feedstock, which we refer to as “optical inks,” play a key role in inkjet print fabricated optics. To make GRIN optics, at minimum, a binary optical ink pair is required. A binary ink pair may consist of a “high index” ink, nhigh, and a “low index” ink, nlow. The difference in the index values of the two primary inks at a reference wavelength is the maximum refractive index contrast, Δn. Using print composition, intermediate refractive index values may then be created by locally depositing different drop concentrations of each primary optical ink and allowing them to mix on the substrate. The print composition process is conceptually similar to the halftoning techniques used in the graphics industry. Using halftoning, grayscale reproductions are created by varying the reflectivity of the substrate proportionally to the density of black drops. Similarly, when defining a refractive index gradient using a binary ink set, separate bitmaps are created for each printhead to control the deposition of each optical ink for each layer. The bitmaps determine the local droplet densities for the high-index ink and low-index ink droplets, such that when they mix on the substrate, the local composition takes on a complex refractive index spectrum that is the weighted average of the refractive index spectra of the constituents of both optical inks [6,8].
Most printers allow multiple inks to be printed; the variety of inks that can be simultaneously printed depends on the number of available printheads controlled by the printer. Consumer printers typically allow four inks to be simultaneously printed, but it is common for industrial printers to accommodate more inks [31,32]. The ability of a printer to simultaneously deposit and mix multiple optical inks adds degrees of freedom for optimizing gradient index optics. For example, the binary (i.e., high index and low index) ink set described above may be complemented with an ink formulated with an intermediate index value, nint, to allow for tri-level halftoning. Multi-level print composition allows for more precise control over the index gradient shapes and reduces the precision required for the deposition, mixing, interdiffusion, and polymerization processes necessary to form complex gradient profiles composed of a wide range of spatial frequencies.
The introduction of increased numbers of inks with different material compositions also adds dimensions to the design space for use in optimizing the chromatic properties of optics. In a binary blend of two materials, the refractive index and the change in index over wavelength (i.e., the dispersion) are dependent, so specifying one determines the other. The ability to print-compose gradients by co-depositing and mixing multiple primary optical inks, with specific refractive index spectra, provides the ability to break this dependency, so that index and dispersion may be controlled independently [30,33–35].
Additionally, the optical inks, themselves, may be formulated by blending multiple constituent materials. Using different concentrations of an ensemble of monomers, nanoparticles, ligands, and surfactants, each contributing, on a weighted average, its own refractive index spectrum, makes it possible to precisely tailor the refractive index spectra, nλ, of primary optical inks relative to one another.
Print composing multiple heterogeneous primary optical inks with complementary refractive index spectra, provides significant degrees of freedom for optimizing the chromatic properties of GRIN optic designs. For example, the refractive index spectra of primary optical inks may be composed such that the print-composed index gradient and dispersion are independent of one another. Print composition of multiple heterogenous primary optical inks also makes it possible to design optical materials with a large variety of anomalous partial dispersion values that are not available in standard glass or plastic materials [36–38].
In this context, the formulations of nanocomposite materials for GRIN optics creates a multi-dimensional design space. A simple three-dimensional design space may include the orthogonal dimensions of refractive index, dispersion, and partial dispersion. In this case, the specific properties of miscible high-index and low-index primary optical inks are reflected on the refractive index axis as well as on the dispersion and partial dispersion axis dimensions. The volume of the design space is then bounded by the three-dimensional coordinates of the primary inks.
Within the bounds imposed by the properties of the primary optical inks, print composition of different concentration ratios of two, or more, high and low index primary ink droplets creates intermediate index gradients, which define application-specific lines, curves, planes, or contoured surface solutions within the three-dimensional design space. These solutions may compensate for the dispersion originating from the shaped surfaces of the lens, or from prior lenses.
In this paper, implementing several of these degrees of freedom, we show the ability to independently control the index and dispersion of GRIN elements, and we demonstrate an achromatic singlet lens. We show the ability to formulate nanocomposite primary optical inks with specific relative refractive index spectra, such that when the optical inks are co-deposited on the substrate, index gradients with positive, negative, or neutral (i.e., achromatic) dispersion are realized. For each of four primary ink pairs, 45-mm x 35-mm plano-plano optical elements were printed, which each consisted of a nine by seven element array of 4-mm-diameter radially-symmetric spheric GRIN lenses centered on a 5-mm square pitch. These lenslet arrays may be classified as structured freeform refractive optics, as there are no axes of symmetry in the gradient profiles that comprise the optical elements. The ability to independently control the dispersion of index gradients is demonstrated by measuring the focal lengths of the optics at three wavelengths. Control over partial dispersion is also shown.
2. Dispersion controlled optical inks
The refractive index of any transparent material is a function of the wavelength. Therefore, a lens made in one single material shows different positions of focus at each wavelength. The difference in position of these focal points is known as the longitudinal primary chromatic aberration. To correct this aberration, achromatic lenses are usually manufactured using two lenses of different material having different Abbe numbers combined to form an achromatic doublet [39]. However, unless the material pair is carefully chosen, secondary color will remain [40]. Triplets can then be employed to correct some higher orders of chromatic aberrations, such as secondary color [41], at the expense of increased system size, weight, and cost.
The advances in the fabrication of GRIN optical elements have prompted interest in their ability to correct chromatic aberrations [22,34–42,42,43]. Both the properties of the constituent materials and the fabrication process used to create a GRIN profile contribute to the chromatic properties of the final element. However, fundamental material properties and process constraints have previously limited the ability to independently control optical properties such as index and dispersion [13,44–46].
The degrees of freedom afforded by multi-material composition of optical inks breaks this dependency, allowing for the index and dispersion properties of gradients to be controlled independently. Nanocomposite optical inks may be formulated to have specific index, dispersion, and secondary color properties. Nanocomposites are made by embedding various concentrations of one or more organic or inorganic nanoparticles in blends of low-viscosity photocuring monomers [6,8]. Each nanoparticle is smaller than 10-nm, about 1/30th the shortest wavelength of light passing through the optic, and is chemically coated to eliminate agglomeration, such that Rayleigh and Mie scattering are insignificant [6]. Additionally, the inks are formulated with the rheological properties necessary for precise printing [47–49].
The constituents of the nanocomposite optical inks define their complex refractive index spectra. A simple linear two-material composition model [50–53] allows the index to be approximated at each wavelength as a function of two constituent materials, n0(λ) and n1(λ) as
The dispersion properties of the nanocomposite optical materials are defined by the variation of the refractive index as a function of wavelength, i.e., n = f(λ). Most optical materials have positive (normal) dispersion, which means that the refractive index decreases at longer wavelengths. A simple measure of the chromatic dispersion of an optical material is the coefficient of mean dispersion, known as its Abbe number, V, which is obtained by measuring the index at several key wavelengths, i.e., V = f(n(λ)). When the wavelengths aren’t explicitly modeled, the Abbe number can be defined for a particular waveband according to
The standard solution to correct axial color aberrations is to replace a homogeneous singlet with a doublet composed of two different materials with different Abbe numbers. Generally, in order to minimize the optical power required of each individual element when correcting color, the ratio of the two Abbe numbers should be as large as possible. In doing so, the differing dispersions of the two materials are able to balance one another in order to bring two wavelengths to the same focus.
In order to correct secondary axial color, three wavelengths need be brought to the same focus. This requires definition of an additional property known as the partial dispersion (P) of a material. The partial dispersion for a portion ${n_{{\lambda _{i\; }}}} - {n_{{\lambda _{j\; }}}}$ of the wavelength range ${n_{{\lambda _{i\; }}}} - {n_{{\lambda _{j\; }}}}$ may defined as
For ordinary optical glasses, the relationship between the partial dispersion ratios and the Abbe number is roughly linear. Secondary color chromatic aberrations in lens doublets or triplets may be controlled using optical glasses with anomalous relative partial dispersion, P, which departs from the normal straight-line relationship with the Abbe number on a P = f (V) diagram.
For GRIN optics, wherein optical power is obtained by index gradients, a metric similar to the Abbe number can be used to characterize the primary differences in optical power over a wavelength range:
where Δnshort, $\Delta $ nmid and $\Delta $ nlong is the change of the index of refraction at three relative wavelengths. Similarly, a partial dispersion can be defined for a GRIN material [11]:The standard dispersion definitions from Eq. (2) and Eq. (3) still apply for the optical power generated by surface curvature, where the index is evaluated at the surface vertex. Therefore, a single gradient index element may have two contributions to optical power, each with different dispersion properties.
The degrees of freedom available with nanocomposites allow for the refractive index spectra of optical inks to be precisely defined. When optimizing the optical inks for use in gradient index optics, it is possible to vary the constituents of the high or low index inks to precisely define the refractive index spectra of their difference, such that a wide range of precisely defined positive and negative ${V_{GRIN}}$ values, small and large, are available to optimize dispersion. For example, a straightforward way of realizing achromatic GRIN elements is to compose the high and low index optical inks with nearly the same index slope ($\mathrm{\Delta }{n_{short}} - \mathrm{\Delta }{n_{long}}\; \to 0$) such that large absolute values of VGRIN are achieved and chromatic aberrations are reduced. The index slopes of the two primary optical materials may also be composed relative to one another, by introduction of constituents with complementary refractive index spectra, to achieve large dispersive values. For example, when Δnshort < Δnlong, the small negative VGRIN value indicates negative dispersion and when Δnshort > Δnlong, the small positive VGRIN value indicates positive dispersion.
Similarly, heterogenous primary optical inks may be composed with refractive index spectra that, in relationship to one another, precisely determine the partial dispersion, PGRIN, of the optical element. As it is the difference in the refractive index spectra of the two optical inks that defines Δnλ at each of the three wavelengths defined in Eq. (4) and Eq. (5), by introducing additional constituents to one or both of the optical inks, it is possible to relax the dependence of the PGRIN values relative to the values, making possible GRIN materials with a wide range of anomalous partial dispersions that are not available in homogeneous optical materials. Apochromatic ink sets are constructed similarly.
These degrees of freedom afford the possibility of realizing monolithic GRIN optical elements that replace the optical functionality of multiple surface-figured homogeneous-index lenses [22,27,28,30,33,42].
3. Nanocomposite Optical Ink Formulations
NanoVox has developed a series of inkjet printable optical inks which are composed of different types and concentrations of ceramic or organic nanoparticle types mixed with one or more types of monomers, ligands, and surfactants. For this effort, an ink set consisting of three heterogenous optical inks, NanoVox models VZAXX250, VZXXX000, and VZBXX250, was selected. The feedstock optical inks, which for simplicity are assigned reference numbers (1, 3, and 6), have the optical properties shown in Table 1, where λ short = 486.1 nm, λ mid = 587.56 nm, and λ long = 656.3 nm.
Table 1. Properties of feedstock optical inks.
To demonstrate gradient index lenses with different primary and partial dispersive characteristics, a set of primary optical ink compositions was considered based on the three feedstock optical inks. These primary optical inks were optimized for deposition using inkjet printheads. Table 2 shows the compositions of the five primary optical inks, which are defined by the volume percentage of optical inks shown in Table 1. The index, Abbe number, and Partial Dispersion are listed for each primary optical ink.
Table 2. Composition and properties of primary optical inks.
Figure 1 (left) shows the index values of the primary inks plotted against their Abbe numbers. Figure 1 (right) shows a smaller portion of the primary ink design space, where the index is plotted versus the index slope, nλshort - nλlong. Plotting the index versus the index slope is a convenient way to characterize the dispersive properties of inks for use in optimizing GRIN lens designs [22,34–43]. In addition to the properties of the individual primary optical inks, the plots in Fig. 1 show the properties of compositional blends of the five ink pairs (4-3), (4-7), (4-8), (4-15), and (4-16). These ink pairs share a common high index ink, Ink 4, making it convenient for demonstrating index gradients with independent control over dispersion through formulation of ensembled refractive index spectra. With a common high index ink, the specific refractive index spectra of the low index inks may be tailored by composition, to determine the dispersion properties of the pair. The plots show the print composed intermediate index values whereby the concentration of Ink 4 is varied 100% to 0% relative to the other primary paired ink.
Fig. 1. Left: plot of index (λmid = 588-nm) versus Abbe number, with solid lines the print-composed inks used in this study. Right: a portion of the chart on left, with the slope (nλ=486 - nλ=656-nm) used as the dispersion metric. The lines connecting the points represent different blends of the inks.
As can be seen in Fig. 1 (right), the compositional slope of ink pairs (4-3), (4-7), and (4-8) have a positive dispersion slope, wherein the highest index value (100% Ink 4) has more dispersion than the other compositional mixes of the pairs. For ink pair (4-15) the compositional dispersion slope is negative, whereby the 100% Ink 4, the highest index value, has a lower dispersion than the other compositional mixes. For ink pair (4-16), the intermediate valued blends all have the same dispersion value such that the refractive index slope (nshort - nlong) is constant for all compositional mixes, allowing for the possibility of achromatic gradient index elements.
Figure 2 shows the Primary Dispersion and Partial Dispersion values of the various ink pairs plotted against the index values of the blended ink pairs. The plot shows that it is possible to formulate primary optical inks which, when print-composed in different ratios, form index gradients with different primary and partial dispersions values.
Fig. 2. Plot of Primary Dispersion defined by the index slope (nλshort - nλlong) and Partial Dispersion [(nλshort - nλmid)/ [(nλshort - nλlong)] plotted as a function of index for the various ink pairs (dashed lines are Primary Dispersion; solid lines are Partial Dispersion). The lines represent the compositional blends; the endpoints represent 100% concentration of one ink.
Figure 3 shows Δnλ of the ink pairs plotted over the 450 nm to 850 nm wavelength range. As can see seen, the (4-3) and (4-7) ink pairs are characterized by Δnshort > Δnlong. When these ink pairs are configured in a positive GRIN lens, the optical power experienced by the short wavelength light is larger than the optical power experienced by the long wavelength light, resulting in focal lengths that are shorter at the shorter wavelengths. However, when the same gradient index profile is fabricated from ink pair (4-15), wherein Δnshort < Δnlong, the shorter wavelength light is focused a longer distance than the longer wavelength light. Ink pair (4-16) has Δnshort = Δnlong, which means that the focal lengths of the short and long wavelength light are the same, such that the optic is achromatic.
The GRIN properties for the five ink pairs are shown in Table 3.
Table 3. Optical properties of GRIN Ink pairs (λmid = 587.56-nm, λshort = 486.13-nm, λlong = 656.27-nm)
4. Printed Gradient Index Lenses
Nine by seven element arrays of 4-mm-diameter radially symmetric GRIN lenses, on a 5-mm square pitch, were constructed using ink pairs (4-3), (4-7), (4-8), and (4-9). The 45-mm x 35-mm plano-plano lens arrays were printed 0.170-mm thick.
The positive GRIN lenses were fabricated with a simple spherical index profile,
where n0 is the highest index of the ink pair, Δn is the difference between the high index and the low index inks, and r is the radius referenced from the optical axis through the center of the lens. To maintain optimal control and simplify analysis, the radial coefficient Cr2 = 0.25 was maintained constant for the four lens designs, and as part of the experiment, the gradient index profiles were deliberately not modified to achieve the same focal length for each ink pair. Rather the positive GRIN elements were designed with optical inks implemented over their full composition range (i.e., the center of the lens was 100% Ink 4 and the edge of the lens was 100% of the complementary ink). The index profiles of each of the four lenses is shown in Fig. 4.Fig. 4. Index profile of the different lenses fabricated. The color map – not to scale- represents transition from high index to low index; red is high index (100% Ink 4 is at 0-mm) and green at the edge is low index (0% Ink 4).
Before printing, we measured the refractive index values for the inks using an Atago Abbe refractometer. The ink compositions were verified using a TA Instruments TGA-2950 Thermo Gravimetric Analyzers with a TA Instruments DSC-2920 Differential Scanning Calorimeters (DSC Q2000.
The rheological properties of the inks were also characterized. To enable droplet ejection and formation, it is important for the inks to be composed with the proper rheological properties so that they may be properly ejected from the printhead. Upon an electric signal, droplets with characteristic tail formations are ejected from the printhead nozzle. Relative to the nozzle geometrics, the electric voltage magnitude and pulse shape influence the droplet formations [54]. The droplet formation is further influenced by the velocity and size of the droplet and by properties of the ink’s fluid mixture—viscosity, surface tension, and density [52].
The printability of an ink can be calculated by a combination of dimensionless numbers, which depend on various physical–chemical properties of the printable fluid and dimensions of the printing orifice [55–60]. The Reynolds number, Re, and the Weber number, We, specify the relative magnitude of the fluid’s interfacial, viscous and inertial forces [56]:
The limitations of drop ejection with respect to the interfacial, viscous and inertial properties of the fluid is given by Z, the inverse of the Ohnesorge number, Oh, and is defined as the ratio of the Reynolds number and the square root of the Weber number:
The inks are formulated for properties that are optimized for the range of Reynolds and Weber numbers, which can be summarized with the Ohnesorge number. The viscosity was measured using a RheoSense HVROC-S microVISV viscometer. A Model 190 Rame-Hart goniomter / tensiometer was used for testing surface tension. The density was directly measured using a Precision Electric Microbalance Model #AUW120D. The “printability” plots of the Weber number versus the Reynolds numbers for the printheads used is shown in Fig. 5.
Fig. 5. Printability plot showing parametric bounds of the printheads. The Weber number is plotted against the Reynolds for the inks used.
The lens arrays were printed on a customized one-meter-format commercial graphics inkjet printer, using only two printheads to construct the gradient profiles: one printhead for each ink of the pair. For simplicity, the gradient index profiles were fabricated using binary (i.e., two index levels: “high index” and “low index”) print composition to create the gradients intermediate refractive index values. The printhead nozzles were configured for 21-pL ink drops, which in the multi-pass printer achieves 600 dpi print resolution. To prepare for inkjet print fabrication, the planar radial GRIN lens designs were reduced, using “halftoning,” to a set of bitmaps that defined the placement of each ink’s droplets. The bi-level patterns used to construct the radially symmetric spheric index gradients were optimized to promote mixing and inter-diffusion of the two inks so that, after polymerization, sub-wavelength accurate, smooth gradient index patterns were produced.
As the design was radially symmetric with no axial variation in the gradient profiles, the same bitmaps were used to print each layer. As noted above, the same spheric gradient was used for each ink pair; this allowed the same bitmap to also be used to fabricate each optical element.
During each print pass, the bitmaps are communicated to the printheads to control the firing sequence of each nozzle so that the spatial locations of each ink’s droplets are precisely defined. After deposition, neighboring droplets are inter-diffused. By locally defining the density of droplets for each ink type, control over their inter-diffusion makes it is possible to precisely control the local refractive index spectra based on the weighted percent volume concentration of the constituents of the co-deposited inks. After the inks inter-diffuse for a fixed period of time, the ink patterns are partially locked into place using partial ultraviolet (UV) photonic curing. In this way, two-dimensional spatial patterns of varying material compositions are created that precisely define refractive index gradients with specific chromatic properties. Successive layer-by-layer printing of spatially patterned materials over the underlying partially cured lower layers, after vitrification and solidification, creates smooth 3D index gradients throughout the volume of the optical element.
After printing the arrays, the lenslets were visually inspected. Figure 6 shows a picture of lenslets from a portion of one of the nine by seven element arrays. The 0.170-mm thickness of the optics was confirmed using a profilometer. A Zygo ZeScope was used to measure the surface. As shown in Fig. 7(a), the surface irregularity of the “as-printed” part, without polishing, was measured to be 0.08 micron (P-V) across the entire optical area; this contributed to 0.07-waves (λ/14) RMS error. The horizontal line profile through the center of the optic shows only 0.03 microns (P-V) of surface irregularity.
Fig. 6. Pictures of 7 × 11 element lens arrays at different magnification. Each element is a 4-mm-diagonal radial GRIN lens printed on a 5-mm square pitch.
Fig. 7. (a) Zygo ZeScope measurement of “as-printed” plano-plano optical element showing 0.07 RMS wave error from surface; (b) optical path difference (OPD) measurement of pupil plane showing gradient profiles; (c) OPD deviation (irregularity) from intended GRIN design; (d) vertical and horizontal plots of OPD deviation from fitted GRIN curve ($\lambda$ = 633 nm).
To compare the optical function of the printed GRIN parts to that of the intended spheric index design, wavefront analysis was performed using a custom-built digital homographic microscope (DHM). The map of the measured optical path difference (OPD) in Fig. 7(b) shows the accumulated phase delays attributable to the deposited concentrations of nancomposite materials. The spheric refractive index gradient profiles are clearly seen in the OPD measurements. The wavefront error (WFE) at the pupil is plotted in Fig. 7(c). Wavefront error is defined as the difference between the reference spherical wavefront phase and the detected wavefront phase. The total RMS wavefront error of the unpolished parts, including the 0.07 waves of error from the surface, is below 0.15 waves (i.e., λ/6.67), which indicates good overall print accuracy. For reference, this part was fabricated with the printheads moving horizontally and the platen moving vertically relative to the orientation on the page. The horizontal and vertical cross-sections of the pupil plane OPD measurements are compared to the fitted GRIN curve in Fig. 7(d). The curves show directional bias in the WFE contributions, which may be attributable to the unfinished surface or to uncompensated printer biases and diffusion biases resulting from the multi-pass print process. The plots also show WFE contributions from the border.
Figure 8 shows Zernike polynomial contributions to the WFE. The Zernike polynomials are a set of functions that are orthogonal over the unit circle. The individual Zernike basis functions (i.e., modes) correspond to classical optical aberrations, such as defocus, astigmatism, coma, and spherical aberration. Consequently, a Zernike expansion provides a convenient accounting scheme in which the total root mean squared (RMS) wavefront error is equal to the square root of the sum of the squares of the individual coefficients in the Zernike spectrum of a wavefront aberration map [61–63]. It is important to note that while Strehl ratio can be calculated from RMS wavefront error, it cannot be directly linked to a surface measurement without an understanding of the exact nature of the error.
The Zernike decomposition of the WFE shown in Fig. 8 indicates that, of the first fifteen Zernike modes, which are generally attributable to low frequency figure (LSF) error, the WFE contributions, all lower than about +/- 0.025 RMS waves, is dominated by spherical (Z8), x-coma (Z6), y-coma (Z7), horizontal astigmatism (Z4), and diagonal astigmatism (Z5). This is not surprising, as to simplify analysis a simple spherical gradient index profile was implemented. The degrees of freedom afforded by a three-dimensional (3D) aspheric gradient profiles, including both radial and axial components, would reduce spherical and coma aberrations, similarly to how aspheric-surfaced homogeneous-index element may be used to reduce aberrations [21].
Furthermore, only binary-halftoning was used and no optimization of the bitmaps was performed. The bitmaps were generated directly from the gradient index profiles, with no compensation for known concentration-dependent diffusion effects or for printer specific biases. Moreover, all lenslet array test and characterization was performed on “as printed” parts, with no post fabrication polishing.
The astigmatism aberrations, which were also evident in the WFE line profiles shown in Fig. 7(d), indeed, include the contributions from the unpolished surfaces quantified above. However, supported by other studies, we find that astigmatism aberrations may also result from uncompensated printer biases, such as printhead nozzle alignment and offsets in the mechanism used to move the platen between passes. While outside the scope of this effort, these errors may be reduced by calibrating the printer, by optimizing the process parameters, and by compensating the print maps.
Figure 9 (top) shows the tangential and sagittal modulation transfer function (MTF) curves obtained from the DHM measurements of the point spread function (PSF), measured at the focal plane, compared to diffraction limited performance. The 4-mm lenses have an average focal length of about 216 mm (f/54), which results in a diffraction limited spot size of 77.1 microns and a limiting resolution of 31.55 lp/mm.
Fig. 9. Point spread functions (left), frequency analysis (middle), and wavefront error (right) measured over a 3-mm aperture using DHM. The top row shows measured data. The middle row shows data including contribution from the first 36 Zernike polynomials (only). The bottom row shows residual (higher order Zernike polynomial) contributions to data. The Strehl ratio is listed for each case.
A 0.15 wave RMS error was measured. A Strehl ratio (S) of 0.5 was measured over a 3-mm clear aperture of the lenslets. The Strehl ratio is the ratio of maximum focal spot irradiance of the actual optic from a point source to the ideal maximum irradiance from a theoretical diffraction-limited optic [64]. To analyze the WFE contributions, an aberration transfer function (ATF) curve was created from the high order Zernike polynomials.
In terms of frequency analysis, the frequency response of an optical system is reduced by phase distortion within the passband. The plot of Fig. 9 (middle) shows the MTF performance with contributions from only the first thirty-six Zernike polynomials. The data shows about 0.10 RMS total wavefront error attributable to these Zernike polynomials. A Strehl ratio of 0.802 was calculated when only the LSF figure errors were included. A Strehl ratio S ≥ 0.8 is generally considered to correspond to diffraction-limited performance.
Figure 9 (bottom) shows the MTF curve that includes contributions from only the higher order (i.e., larger than Z36) Zernike polynomials. A Strehl ratio of 0.614 was calculated when only contributions from mid-spatial-frequency (MSF) and high-spatial-frequency (HSF) Zernike polynomials were included in the measurement. The data shows that the higher frequency errors significantly contributed to the overall optical power of the WFE. The data shows about 0.12 wave RMS error from the higher frequency bands.
The high spatial frequency errors may be reduced in several ways. First, a finer resolution printer would allow for more process control. Whereas a 600-dpi printer was used in this effort, increasing the print resolution to 1384 dpi, which is available in current industrial printers, would decrease the drop size 57%, and after optimizing the inter-diffusion, would contribute to reducing granularity. Additionally, use of multi-level thresholding and optimization of the halftone print maps may further reduce the high frequency WFE.
To analyze dispersion, a Thorlabs BC106 Beam Profiler was used to measure the lenslet focal lengths, sampled across the area of the array, at three separate wavelengths. To measure the focal lengths, a white light source was configured with a 500-micron diameter pinhole, at the focus of an achromatic doublet (fL = 200 mm) model #32-917 [65]. A set of 10-nm bandwidth filters with center wavelengths of 486 nm, 589 nm, and 656 nm were used to define the wavelength.
The lenslet focal lengths can be predicted using the lens formula:
where k is the gradient constant, z is the thickness (i.e., 0.170 mm), and n0 is the index at the center of the array. The gradient constant, k, is defined byThe measured focal length data is shown in Table 4. The measured data match the focal lengths predicted by Eq. (10), when calculated using the properties of the ink pairs listed in Table 3, within the range of error. The measured data also match the focal lengths modeled using the Zemax ray trace engine, modified with a GRIN plug-in developed by NRL [42]. By design, the focal lengths of the four lenses differed because the radial coefficient Cr2 = 0.25, was constant for all designs, and the four ink sets each had a different Δn values. The focal length data normalized to the λ = 486 nm data, is listed in Table 4 and is plotted in Fig. 10.
Fig. 10. λ= 866 nm normalized average focal length (FL) data measured at three wavelengths for arrays fabricated for each ink pair.
Table 4. Measured average focal length data, including standard deviation. Data on the lower rows is normalized to the 486-nm data to compensate for the different Δn values of each ink pair. A metric of dispersion is shown in the last column.
To show the uniformity available with inkjet print fabrication, lenses selected from across the area of the 6 × 9 lenslet optical elements were tested. The focal lengths were measured from a common set of eleven lenslets which consisted of a centered vertical five-element stripe and a centered horizontal seven-element stripe of elements that each included the center lenslet of the arrays. The standard deviation of the focal length data is shown in Table 4. Across all wavelengths, the average standard deviation was about 1.4%, which is exceptional for such long focal length lenses and is within commercial tolerances. The control over the focal lengths of the elements of the arrays shows control over the printed gradient index profiles that is comparable, or better, than the control over the sag surface curvature achieved by commercial machining of homogeneous index lenses, especially those configured in an array.
The data show refractive index gradients with independent control over the dispersion. As predicted, the focal length data show that the lenses fabricated using ink pair (4-15) had opposite dispersion than the lenses fabricated using ink pairs (4-3) and (4-8). Significantly, the ink pair (4-7) shows reasonable achromatic performance; about 0.39% variation in focal length was measured over the three wavelengths. This demonstrates that using material pairs with matching index slopes (i.e., high VGRIN values) it is possible to reduce the chromatic aberration of GRIN lenses by bringing wavelength groups into a common focal plane.
As was discussed above, ink pairs with a composition closer to that of (4-16), a formulation introduced only for reference, would be expected to achieve even better focal length uniformity across the three wavebands. Nevertheless, the ability to compose and tailor the compositions of optical inks such that the index gradients and the dispersion could be precisely controlled is clearly demonstrated.
5. Conclusion
The versatility of using inkjet print additive manufacturing with precisely formulated optical inks for the fabrication gradient index optics was demonstrated. In conventional optical systems, to compensate for the dispersion, optical systems consist of multiple convex and concave lenses and are made from different glass types with varying dispersion levels. It is shown in this work, that the added degrees of freedom of printed gradient index optics, make it possible to replace multiple surface-shaped homogeneous lenses with a single GRIN lens.
Nanocomposites formulated using multiple constituents offer the ability to precisely control the refractive index, dispersion, and secondary color characteristics of optical inks. The added degrees of freedom afforded by print-composition of two or more nanocomposite optical inks with complementary refractive index spectra allow for application-specific chromatic optimization of monolithic GRIN lenses.
In this effort, we demonstrated several of these degrees of freedom. We formulated high-index and low-index-sets of heterogeneous optical inks with refractive index spectral properties optimized relative to one another such that using binary print composition, index gradients were created with specific dispersion characteristics. GRIN lenses with positive, negative, and achromatic dispersion were demonstrated.
Control over partial dispersion was also shown. Nanocomposite materials make possible optical ink pairs with relative refractive index spectra that offer anomalous partial dispersion properties. This level of independent control over secondary color has heretofore not been possible.
The primary effort of this effort was to demonstrate the degrees of freedom afforded by custom engineering of nanocomposite feedstock materials themselves. The effort did not attempt to fully optimize the performance of an achromatic GRIN lens. To allow for unambiguous demonstration and analysis of dispersion effects, simple plano-plano optical elements were designed with a simple spheric GRIN profile that was common for all ink pairs. While good optical performance, as indicated by the measured Strehl ratios and MTF data, were demonstrated, implementation of three-dimensional aspheric index gradient, with both radial and axial gradient functions, would further reduce aberrations and improve performance.
Design for manufacturing and process optimization were also outside the scope of this effort. From other efforts, we know that calibration of the printer, optimization of the process parameters, and compensation of the bitmap artwork will improve optical performance. Also, use of a higher resolution industrial printer will increase the degrees of manufacturing control and will enable WFE to be further reduced.
While in this effort the gradient index optics were demonstrated using binary print composition, the approach can be easily extended to fill the available number of printheads available in industrial ink jet printers (e.g., between six and twelve). This makes available multi-component print composition that provides additional degrees of freeform for application-specific optimization of primary and secondary color, including that introduced by the lens surface shapes. The ability to color-compensate aspheric-surfaced GRIN optics, embedding high-order 3D GRIN functions, with both radial and axial terms, offer significant possibilities for realizing diffraction limited achromatic singlet lenses. The approach further extends itself to freeform refractive optics, wherein chromatic control is necessary.
More generally, it was shown that a primary set of high-index and low-index heterogenous optical inks, with index spectra tailored relative to one another, creates a multi-dimensional design volume, in which print composition of multiple primary optical inks allows for application-specific solutions to be mapped in the form of gradient lines, curves, planes, or contoured shapes. This capability makes possible new types of spectrally engineered optical devices.
Ongoing research is being conducted to expand the design space of the optical inks, allowing for a broader range of index values, with fine control over a wide range of dispersion and secondary color properties. Combined with high resolution printers configured with more printheads that accommodate multiple primary optical print patterning, these will make possible sophisticated radially symmetric and freeform 3D GRIN optics that may replace a number of conventional homogeneous lenses in optical system designs.
Disclosures
George M. Williams: Allegro Microsystems Inc (E, I, P), NanoVox LLC (S, E, I, P)
John P. Harmon: NanoVox LLC (E, P), George Fox University (S)
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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