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Abstract
CCD arrays encode color information via uniformly distributed red, green and blue pixels. Therefore, even a perfectly achromatic system projecting an image onto a CCD plane cannot possibly associate a single object point with the 3 or more discrete pixels encoding color content. Here, we propose and demonstrate a micro-lens array (MLA) design that simultaneously corrects chromatic aberrations and separates color channels to spatially distinct pixels. Starting from a commercially available aspheric condenser lens, methods to design and assess the performance of a few microns deep MLA etched on the convex optical surface are detailed. Actual fabrication is carried out by fluid jet polishing, with an optical form deviation of 0.24 µm rms. Finally, the MLA is assessed with a narrowly collimated beam containing two wavelengths, which produces distinct spots of diameter 10-15 µm as expected.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In optical systems, an ideal lens should focus one point from the object plane to one point on the image plane exactly. However, in reality all lenses suffer from so-called geometric and chromatic aberrations. Because of the latter, light of different wavelengths generates focal spots at different spatial locations [1]. It is considered the most crucial problem for polychromatic optical applications. Up-to-now, various techniques for creating chromatic aberration free lenses have been investigated [2–4].
Achromatic doublets or triplets are the most common approach to approximately eliminate chromatic aberration [5]. However, this technique is cumbersome, because the number of different materials must equal the number of wavelengths selected for focus compensation. This significantly increases system complexity, size and weight [6]. Besides, a fraction of light intensity will be lost at each interface. Chromatic aberration can also be corrected through the use of diffractive optical elements (DOE) [7,8]. Zhan et al. proposed a broadband diffractive Pancharatnam-Berry phase lens to realize practical chromatic aberration correction in virtual reality displays [9]. Compared with conventional doublet lenses, the DOE can be fabricated on a thin plate. Besides, because of their relatively flat structure, diffractive optics reduce distortions and therefore have much more uniform point spread function (PSF) distribution across the image plane. However, the main drawback of DOE is its tendency to produce severe image distortions in broadband application. What is more, diffractive patterns are generally intricate, costly and time consuming to manufacture with high precision [10].
Recently, achromatic metalens have been used for broadband achromatic focusing [11,12], by accurately controlling the phase using many sub-wavelength spaced structures with thicknesses at the wavelength scale or below. Many compact optical devices based on meta-surfaces have been demonstrated, including flat lenses [13] and polarimeters. An achromatic metalens can be viewed as a diffractive focusing lens with only one diffractive order, that inherits all the advantages of a conventional diffractive element over their bulky, refractive counterparts. However, metalens are very small and cannot easily be applied to larger aperture application. Besides, a number of expensive fabrication devices and complex procedures are needed to fabricate highly accurate units.
With respect to chromatic aberrations correction, these methods have a common characteristic that the wavelengths targeted for compensation all come to focus at a single image point in the focal plane. This means that one object point corresponds exactly to one image point. However, physical CCD sensors are in fact discretized into red, blue and green channels when capturing images. For instance, an achromatic doublet is illustrated in Fig. 1(a). The red, green and blue channels on the CCD sensor are seen to correspond to three different object points. While this would not be an issue if the image was reproduced with the exact same arrangement of color channels, in most cases the arrangement of color channels in color cameras and color displays are quite dissimilar. Furthermore, fully accurate reproduction of the original object in print (whereby the separate color channels are superimposed into a single image point) is clearly impossible. In another example of an achromatic condenser shown in Fig. 1(b), all on-axis parallel rays will focus to a single point in the image plane. If the central pixel of the CCD array corresponds to the green channel, only green light can be captured with a well aligned optical system. Recently, some related research has focused on these problems. Cu-Nguyen et al. designed a hyperchromatic lens for hyperspectral imaging [14]. An object including red/green/blue parts is imaged to three different spots captured by spectrometer. Instead of capturing the focused and defocused images in one frame, they are collected separately by focusing at different narrow wavelength bands axially. Chen et al. designed a lens for chromatic aberration compensation by dividing the lens surface into several discrete areas [15]. A common method to counter this problem consists of slightly offsetting the CCD relative to the image plane, which allows capturing all colors, although this approach deteriorates the sharpness of each spot.
To address the above gap relating to the separation from a single object point (or ray direction) to spatially distinct pixels for each color channel, our work systematically considers a lens design and fabrication method for chromatically driven multi-focusing of light onto color CCD arrays. The design procedure is automated by means of geometric optical ray tracing and numerical optimization of a base lens surface. Details of the novel approach will be discussed below.
2. Materials and methods
2.1 Optical design method
We consider an aspheric condenser lens operating at three different wavelengths (λ=450nm,520nm and 637nm), with each color channel focusing to a separate pixel on the CCD array. A schematic view of basic principle for the new approach is shown in Fig. 2. The aspherical base surface is divided into many hexagonal aspherical micro-lenses arrayed concentrically from center to edge. The ommatidia located on the same circumference of a circle are assumed to be located on the same layer, i.e. number 0,1 in Fig. 2(a) means layer0, layer1 and so on. These hexagonal ommatidia all have the same size L and are uniformly arranged. But the aspheric profile, spatial orientation, and color filter are adjusted for each ommatidium individually, such that rays passing through it may focus to a particular group of pixels. On each layer, the ommatidia alternate between the color filters corresponding to the different wavelengths. These sub-areas of the lens surface can therefore be labelled as red, green and blue groups, as shown with different colors in Fig. 2(a). Different groups only allow the corresponding wavelength band to pass through. Overlaid color filters are adjusted in accordance with these bands. The parameters of each ommatidium will be optimized separately.
The overall coordinate system (O-XYZ) is defined at the base of the initial aspherical condenser lens. Local coordinate systems (O-X'Y'Z’) are located at the center of each hexagonal ommatidium. Both base lens and hexagonal micro-lenses, in their respective coordinate systems, can be expressed as:
For optimal design of the aspherical micro-lenses, particle swarm optimization (PSO) algorithm is adopted [16]. The central tenet in PSO rests upon the concepts of natural selection and survival of the fittest. According to a fitness function, the PSO process begins by randomly selecting an initial population and allowing reproduction, crossover, and mutation to proceed. In this model, chromatic multi-focusing mostly depends on selecting proper scale and orientation for the micro-lens array (MLA) elements. Five parameters are considered for each MLA element: the curvature C, conic constant k, tilt angle ($\theta _x^{\prime},\; \theta _y^{\prime}$) around the $X^{\prime}$ and $Y^{\prime}$ axis, and relative position along $\Delta \textrm{H}$ the Z axis, as shown in Fig. 2(b). The PSO algorithm searches for the best arrangement of parameters. The size and location of the focal spot predicted through geometric optical ray tracing is the main criterion for achieving a chromatically multi-focal MLA. The target function of the ${i^{th}}$ element is used to calculate an object value:
Because of the rather low removal rate in processing by FJP, the maximum removal height was set to less than 10 $\mathrm{\mu}\textrm{m}$, which brings some constraints on the parameters as shown in Table 1. To fulfill the PSO algorithm, the solution space is firstly initialized at random in the range of constraint conditions in Table 1. Assume that the number of particles is N, and the position ${X_n}$ and ${V_n}$ of the ${n^{th}}$ particle can be expressed as:
Table 1. Constraint conditions of each parameters
The best position of each particle can be expressed as:
For all particles in the solution space, the best position is written as: Each particle updates its velocity and position according to the following formulae:2.2 Optical performance of ideal MLA
In order to validate the proposed optical design, studies of the optical spot diagrams were carried out. A base aspherical lens with diameter of 12.5 mm and thickness of 4 mm was optimized and simulated. The length L of each hexagonal region was set as 3 mm. The display order of the designed ommatidia is shown in Fig. 3(a): there are two layers of hexagons. The center is layer 0, which belongs to blue group. The 1st layer has 6 hexagonal regions divided into 3 groups of different color. The optimized parameters of green and red ommatidia are illustrated in Table 2 (the blue group does not require optimization by PSO). The three-dimension optimized lens profile is shown in Fig. 3(b).
Fig. 3. (a) Top and (b) perspective view of MLA. Simulated spots for (c) different wavelengths passing through the corresponding ommatidia, and (d) blue light passing through the green ommatidia.
Table 2. The optimized parameters for each ommatidium
For on-axis performance, spots on the image plane were simulated for different wavelengths passing through the corresponding ommatidia as shown in Fig. 3(c). The different spot colors correspond to the different groups at their respective offset locations in the image plane (-25, 0 and +25 µm). According to this simulation, each spot size is around 1∼2 µm, a value well below the size of a single pixel on the target CCD ($5.2 \times 5.2$ µm). When the length L of each ommatidium changes, the overall spot size remains almost the same. It should be noted that different lens groups should only allow the corresponding wavelength band to pass through. Figure 3(d) shows simulated spots when blue light passes through the green ommatidia. The rays passing through ${G_1}$ and ${G_2}$ cannot focus on one point. Besides, the spot size is approximately 15 $\mathrm{\mu}\textrm{m}$ and the spot position deviates from the design position. Therefore, color filters must be overlaid in front of the lens to insure that the light wavelength bands and lens groups are matched up.
Regarding off-axis performance, Fig. 4 shows the simulated spots for different incidence angles. In most cases, the designed lens can focus reasonably well the different groups to their respective offset locations in the image plane. Besides, the offset along Y direction for the different groups is uniform. The spot size increases from 1 µm to 6 µm, as the incidence angle goes from ${0.1^\circ }$ to ${2^\circ }$. That is below the size of a single pixel on the target CCD and indicates a well-focused spot in the image plane. For larger incidence angles, the lens groups cannot focus to one unique CCD pixel any longer, as shown in Fig. 4(d). Since this affects even the blue group, for which we did not perform optimization, it would seem that the base aspherical lens chosen for this study has rather limited off-axis performance.
Fig. 4. Simulated spots for off-axis performance (a)${\; }{0.1^\circ }$ (b)${\; }{1^\circ }$ (c)${\; }{2^\circ }$ (d)${\; }{4^\circ }$
Modulation transfer function (MTF) is one of the most important parameters by which image quality is measured. Figure 5 shows the simulated MTF for the different lens groups. The curve shows lens contrast over a frequency range from 0 $\textrm{lp}/\textrm{mm}$ to 150 $\textrm{lp}/\textrm{mm}$ (for the Sony IXC625 sensor, the resolution limit is 145 $\textrm{lp}/\textrm{mm}$). The blue lens group, which did not require optimization, shows the best MTF performance of around 80% at the maximum 150 lp/mm. Limited by the constraints put on optimization parameters, the MTF curves for red and green groups have lower spatial frequency performance, dropping to 80% at around 30 lp/mm and 40% at around 60 lp/mm. These limitations should be taken into account when using the MLA in actual application, but it appears nevertheless that practical usage is possible.
2.3 Optical performance of simulated MLA
For fabrication of the MLA on top of a base asphere, fluid jet polishing (FJP) method is adopted. FJP is a computer-controlled optical surfacing (CCOS) technology. Micro-sized abrasive particles are mixed with water and impinge on a small area of the workpiece surface, allowing removal of material at the nanoscale within a Gaussian like footprint area [17,18]. This method is simple and low-cost, though there are some machining limitations such as the impossibility of producing sharp features by overlap of the Gaussian footprint. Influence of the fabrication process is considered in our simulation model, in order to assess the spot diagram of the fabricated optic. From the MLA surface described in the previous section, a desired FJP material removal map is obtained by subtracting the base asphere as shown in Fig. 6(a). The principle of FJP removal process is demonstrated in Fig. 6(b). A tool influence function (TIF) is obtained by polishing at a static location on a spherical lens of similar curvature to the base aspheric lens, for a given amount of time. The TIF therefore represents the spatial distribution of material removal per unit of time. When polishing an actual aspheric lens, the TIF trajectory across the workpiece surface is discretized as a n×m matrix. The distance between points can be divided by the tool velocity to get equivalent dwell times (DT) at each point of the matrix.
Fig. 6. (a) Target removal profile by FJP. (b) Schematic diagram of the material removal process in FJP. (c) Simulated residual error of polished MLA. (d) Simulated spot diagram of polished MLA.
The distribution of material removal is the two-dimensional convolution between the tool influence function R(x,y) matrix and the equivalent dwell time D(x,y) matrix, expressed as:
Therefore, residual error on the surface after FJP polishing can be expressed as: where H0 (x,y) is the target material removal distribution. According to these equations, the equivalent dwell time D(x,y) at each trajectory point can be computed by deconvolution method. Convergence of the problem depends on the complexity of the deconvolution method. Several approaches exist, such as linear matrix equation [19], sparse matrix [20], or even Zernike coefficients [21]. Dwell-time algorithms are generally based on linear equation model or discrete convolution, which are time consuming and usually have poor convergence rate in actual fabrication. In the employed optimization method, the analytical relationship between target removal depth, path parameters, volumetric removal rate of the TIF, and feedrate is used to directly and efficiently schedule feedrate without first calculating a dwell time map. The proposed analytical model can improve computational performance by 2 orders of magnitude compared with dwell methods, and is used to seed into a numerical optimization routine that refines the feed profile through an objective function that distributes material removal as function of feed. More details on this approach can be found in the literature [22]. The predicted residual error after polishing is shown in Fig. 6(c). The surface accuracy is better at the center of ommatidia than at the edges, as the convolution of a Gaussian signal cannot reproduce exactly the sharp slope transitions between MLA elements. Finally, the spot diagram of the FJP polished MLA was simulated as shown in Fig. 6(d). The spread increases to around 5∼10 µm, which is still acceptable for the target CCD in this work. In case narrower spread is required, it is possible to improve surface accuracy by reducing the TIF size (for example a smaller fluid jet nozzle or ion beam).3. Experimental results
3.1 Fabrication of MLA
To demonstrate feasibility of the proposed chromatically multi-focal MLA, a commercially available precision aspheric condenser lens was polished with an ultra-precision fluid jet polishing (FJP) machine, as shown in Fig. 7. The Zeeko IRP 200 machine consists of three linear axes (X, Y and Z axis), three rotational axes (A, B and C axis), and one spindle axis (H axis). A nozzle with a laser drilled sapphire insert of outlet diameter 0.25mm was mounted on the H axis. Processing parameters are listed in Table 3. The 3D surface profile of the machined MLA was evaluated with an ultra-precision coordinate measurement machine (Panasonic UA3P).
Table 3. Experimental parameters
A photograph of the fabricated MLA is shown in Fig. 8(a). Measurement of the material removal distribution was obtained by measuring the surface and subtracting the base aspheric shape, as shown in Fig. 8(b). After surface matching [23], Fig. 8(c) shows the residual error between the actual and target removal distribution. The Peak-to-Valley (PV) and root-mean-square (RMS) values of the residual error are 0.96 µm and 0.24 µm respectively, which indicates overall optical surface accuracy close to 1λ (628 nm). The simulated spot diagram for the actually polished MLA is shown in Fig. 8(d). While the spread is slightly larger, there is reasonably good agreement with the predicted spots of the simulated fabrication in Fig. 6(d). In order to analyze the relationship between focus spot and lens accuracy quantitatively, the diameter of spot was defined as the diameter of minimum envelope circle which contains 90% of rays [24]. The relationship between focus spot and lens accuracy, i.e. RMS or PV value of the form error, are shown below. The spot diameter increases with the increase in surface accuracy. When the PV value of form error increases to 1.44 $\mu m$ or RMS value reaches 0.36 $\mu m$, the spot diameter will increase to 25 $\mu m$ approximately, which is equal to the target distance between spots.
Fig. 8. (a) The aspherical lens after polishing, (b) actually polished material, (c) residual error, and (d) spot diagram of actually polished MLA. (e) The relationship between focus spot diameter and lens accuracy
3.2 Optical performance of fabricated MLA
The optical setup shown in Fig. 9(a) was purpose-built to assess the performance of the fabricated chromatically multi-focal MLA condenser lens. In this system, an on-axis collimated and expanded laser beam is incident to the optical axis of the fabricated lens. The switchable laser beam is generated by connecting the emission from two fiber-coupled laser sources (λ = 520nm and 637nm) to a RGB combiner (RGB46HA, Thorlabs), such that a single fiber output is collimated using an adjustable achromatic aspheric fiber port collimator for the wavelength range 350–700 nm (PAF2-A4A, Thorlabs). A blue laser source can be easily added to this setup (the more expensive blue source was not available for this experiment due to budget restrictions). For optimum measurement of spots on the CCD, an iris is placed before the lens in order to match with the aperture of the MLA region of the aspheric lens. To visualize spots in the focal plane, a CCD image sensor (DCC1545M, Thorlabs) with a resolution of 5.2um per pixel is used. Because the CCD camera has a damage threshold, an absorptive filter is used to attenuate the transmitted beam to 0.01% of the incident intensity. The CCD camera and aspherical lens are mounted on a 5-axis stage. The criterion for locating the focal plane consists of minimizing the size of the spots generated with the two laser sources, by varying the position and orientation of the MLA optic. Generally speaking, in the case of commercial products a physical film could be deposited over the entire surface of the lens. Different color filters must be matched with the ommatidia wavelength bands. For this specific case study, the optical performance of the polished lens was measured by installing through-holes before the lens that match the size and position of lens groups, to allow light from the lasers to only pass through selected ommatidia.
Fig. 9. Experimental characterization of MLA: (a) Schematic of optical setup. (b) Spot for red laser. (e) Spot for green laser. (c)(f) Horizontal intensity distribution. (d)(g) Vertical intensity distribution.
The pitch of original data captured by the CCD is 5.2 $\mathrm{\mu}\textrm{m}$. In order to obtain an accurate Gaussian fitting result, the original data was processed with the GetData image processing software. Figures 9(b)–(g) shows the intensity distribution of the focal spots after processing along horizontal and vertical direction, with the on-axis red and green laser beams. The spot size for both wavelength is in the range 10∼15 $\mu m$, indicating excellent focusing and a spread in agreement with the simulated spots in Fig. 8(d). The slightly large spread of the actual lens reflects additional sources of uncertainty in manufacturing, such as volumetric errors of the machine tool and stochastic fluctuations of the TIF over time. Then the distance between spots was obtained from the horizontal intensity distribution. After Gaussian fitting, the calculated peaks for red and green in X direction are 14.31 $\mathrm{\mu}\textrm{m}$ and 62.24 $\mathrm{\mu}\textrm{m}$. Therefore, the separation distance is 47.93 $\mathrm{\mu}\textrm{m}$, which is in fair agreement with the expected 50 $\mu m$ distance. We can therefore conclude that the proposed optical design can correctly separate focal spots as a function of wavelength.
4. Discussion
From an optical design point-of-view, the conventional approach in broadband optical design has up until now mostly consisted of chromatic aberrations reduction through the application of doublet, triplet or diffractive optics. While these methods are largely successful in redirecting most of the light spectrum to a well-focused spot in the image plane, they do not attempt to differentiate the way in which different parts of the spectrum are treated. This realization is in stark contrast with the physical characteristics of light capture devices such as CCDs, which spatially discretize the collection of light from different parts of the spectrum. Our proposed scheme in which the lens area is sub-divided into ommatidia functionally associated with different parts of the spectrum is a simple yet novel approach. The number of functional groups can be as low as 2 wavelength bands and as high as dozens of wavelength bands, since the small size of each sub-area (2∼3 mm) allows filling a medium-sized optical surface (20 mm aperture) with hundreds of ommatidia. While this paper focused on the design of a condenser lens capable of focusing parts of the spectrum from a collimated beam to spatially distinct groups of CCD pixels, the MLA optimization method presented here is applicable to a multitude of broadband optical systems including ultra-compact spectroscopy, replacing Fabry-Perot cavities for multiplexed communication signals, and of course consumer camera systems. In the latter case, ability to split the RGB contributions from a single object point into distinct RGB pixel on the image plane of the CCD opens the door to a factor of 3 improvement in resolution when reproducing the image by color-overlay printing.
From a fabrication point-of-view the presented approach, in which a base aspheric lens is subsequently transformed into an MLA by polishing out a few microns of material, offers tantalizing possibilities in prototyping and reconfiguring of optical systems. The numerical method for dwell time deconvolution from a target removal map and TIF is applicable to several kinds of polishing systems including fluid jet, ion beam, and laser etching. The common characteristics of these processes are the time dependent nature of removal, and tunability of the TIF width and depth (allowing a range of removal depths on the physical workpiece as low as few nanometers and as large as few microns). When considering the number of design parameters in MLAs (which can be 5 or more for each individual ommatidium), the ability to test out various configurations by modifying serially produced aspheres is attractive, with subsequent finalization of the practical design and manufacturing a new set of molds for serial production by plastic injection of glass-press molding.
5. Summary
In this paper, a novel class of chromatically multi-focal optics based on micro-lens array design was proposed, that can adjust the focal point of wavelengths bands according to their position on a CCD sensor array. The included automated method to derive an MLA design from a base aspheric lens represents a fresh advance for the state of the art in chromatic aberration correction. The fabrication method based on FJP polishing showed that high form accuracy and excellent focusing of the discrete wavelength spots can be achieved, with the location of the spots experimentally captured on CCD showing reasonable agreement with simulated spot diagrams.
Funding
Japan Society for the Promotion of Science (17K14571).
Acknowledgments
This research was supported by Zhejiang University while the first author visited Kyoto University. The authors acknowledge financial support through donation from DMG Mori, and support from Zeeko Ltd. in loaning the fluid jet polishing system and measuring equipment.
Disclosures
The authors declare no conflicts of interest.
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