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Abstract
Point spread function engineering uses specially designed phase plates placed at the exit pupil of an imaging system to reduce defocusing sensitivity. A custom phase plate is typically required for each system to enable extended depth of field imaging, so methods enabling variable extended depth of field imaging are of particular interest. In this paper, we discuss the fabrication of previously designed fixed cubic phase plates and variable phase plate pairs with quartic surface profiles and present a novel application of a point source microscope for performance characterization. Experimental measurements of through-focus point spread functions are compared with predictions to demonstrate and characterize the extended depth of field for both fixed and variable freeform phase plates.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Depth of field (DoF), which is defined as the region in object space over which imaging systems can create a sharp image, decreases as the numerical aperture (NA) of the system increases [1,2]. Therefore, methods that enable Extended Depth of Field (EDoF) for improved image quality are of particular interest. Researchers have demonstrated variable focus systems utilizing, for example, deformable mirrors [3], liquid lenses [4], and thermal lenses [5] to enable extended depth of field. Point Spread Function (PSF) engineering is another technique that uses a special optical component at the exit pupil to enable EDoF in imaging systems, such as in microscopes, by modifying the PSF of the system to reduce its sensitivity to defocus at the expense of image quality. A high-quality image can then be retrieved using computational methods [6–8].
A broad range of phase plates with varying surface profiles have been used to engineer the PSF to enable EDoF imaging. Logarithmic aspheres [9,10], axicons [11,12], and binary and annular phase modulated components [13–16] are some examples with rotationally symmetric profiles, and the cubic [7,17–19] and logarithmic [20–23] surfaces as examples of asymmetric (freeform) EDoF phase plates. These phase plates are designed specifically for each optical system. As a result, techniques enabling tunable EDoF are desirable, potentially reducing the number of component and cost. Researchers have previously reported phase plate pairs that enable tunable EDoF by adjusting the relative location of the phase components. Phase plate pairs with polynomial, sinusoidal, or Gaussian surfaces are examples that benefit from phase element lateral translation or rotation to produce a variable EDoF effect [24–29].
We have previously reported on the design of freeform phase plates with variable EDoF for lenses with varying NA values, using a system geometry analogous to the Alvarez lens [24]. The design incorporated Cubic Phase Plates (CPP) as a baseline [7,30]. A Quartic Phase Plate Pair (QPPP) was then designed using an analytical approach [31] to enable EDoF for the range of lenses via relative translation of the phase plates along the x-axis. The freeform surface profiles of the of the CPP and QPPP are given by Eqs. (1) and (2), respectively:
where α and β are surface coefficients, and β is derived from the α coefficient for the highest NA lens so that the QPPPs shifted by variable distance d to create a net effect corresponding to the CPP for a given NA lens [24]. The phase plate pair was designed and manufactured to have no net optical functionality at d = 0. Figure 1 illustrates the concept around which the variable EDoF phase plate pair is based. Multiple fixed phase plates can be replaced with a single pair of shifted phase plates to enable EDoF for lenses with different NA values.
Fig. 1. Two fixed Phase Plates (PP1 and PP2) are replaced with one freeform phase plate pair to enable EDoF through the relative translation of the phase plates along the x-axis [24].
In this paper we report on fabrication and experimental characterization methods and results for CPP and QPPP designs from [24]. The surface parameters for the CPPs and QPPP are summarized in Tables 1 and 2.
Table 1. Cubic Phase Plate Optimization Results
Table 2. Quartic Phase Plate Pair Design Results
Since these freeform EDoF phase plates lack rotational symmetry, manufacturing processes require more degrees of freedom than those used in conventional optics. Advances in ultra-precision machining have enabled fabrication of freeform surfaces by allowing different machine geometries [32]. We discuss the design and fabrication of custom optomechanical fixturing, as well as the freeform EDoF phase plates.
Researchers have previously evaluated EDoF phase plate performance by imaging a point source, spoke target, or other objects at different axial locations (defocus) [27,33–38]. In this paper, we demonstrate a novel use of a Point-Source Microscope (PSM) to characterize the performance of the designed phase plates [39]. The PSM can be thought of as a modern autostigmatic microscope [40], and can be used for a variety of purposes, including optical alignment and centration [41,42], wavefront quality measurement [43], and radius of curvature measurement [44]. For our work, the PSM is used in confocal mode to capture the through-focus spots for imaging lenses with and without EDoF phase plates. The measured spots are then analyzed to quantify the through-focus spot variation and compared to theoretical predictions.
2. Optical and optomechanical fabrication
2.1 Optical system architecture
As discussed above, freeform optical systems with variable functionality typically rely on relative translational motion between the freeform surfaces from flexures or precision translational mounts. To expedite manufacturing and allow for more stable alignment and testing, we chose instead to fabricate the freeform surfaces for the QPPP with pre-set shifts on the surfaces corresponding to the desired lens NA under test, as shown in Fig. 2.
Fig. 2. Simplified image of (a) freeform surface and (b) the QPPP surface, shifted in the x-direction by distance d.
The resulting optical system architecture (Fig. 3) consists of two optical component mounts, one for the aspheric lens and one for the CPP (or QPPP), two steel alignment pins, six magnets (three for each mount) and the optical components.
2.2 Manufacturing of optomechanical fixturing
Optical testing of the freeform phase plates requires a mounting system that encourages modularity. For this optomechanical design, housings were made for each of the aspheric lenses, CPPs, and QPPPs. The housings were all machined from the same piece of turned aluminum bar stock with 63.5 mm outer diameter to facilitate tolerance and alignment preservation. The requirement of specialty manufactured optomechanical housings came from the tight tolerances within this optical system design. Among all the optical systems (CPPs and QPPP), the smallest spacing tolerance was +/- 50 µm. Each mount was then machined on the HAAS Computer Numerically Controlled (CNC) Toolroom mill (Fig. 4).
For the asphere mounts, each lens with different numerical aperture (NA) was held in place by a slip ring. The airgap between the asphere and phase plate was required to be no greater than 1 mm. This tightly controlled distance was split between the asphere mount back face and the phase plate front face. Hardened steel machine pins with an outer diameter of 3.175 mm and neodymium magnets were used to clock and hold the housings together for testing. Magnets were used instead of fasteners (such as threaded rods or bolts) to avoid over-constraining the system and to enable modularity in the testing set-ups. For the phase plate mounts, the CPP tests required only the one optic, so all the CPP mounts were thinner than the QPPP mounts.
2.3 Optical fabrication
Two main limiting factors of freeform optical manufacturing are requirements for precision control and final surface finish. Freeform surface manufacturing requires at least three axes of motion [45]. For the freeforms in our system, this geometry is enabled through a precision manufacturing diamond turning lathe and coordinated multi-axis machining, aka Slow Tool Servo (STS). STS is a lathe-based machining method that leverages three separate axes (X, Z, C). As the diamond tool feeds across the rotating surface in the X direction, the machine synchronizes the angular location of the part (C axis) with the distance of the tool in the Z direction, relative to the part’s translational and angular position. Coordinated axis diamond turning typically leaves 2–15 nm RMS surface roughness, depending on the material, while enabling complex freeform surface manufacture [46,47].
For toolpath surface generation, NanoCAM4 (a precision manufacturing software package) was leveraged. NanoCAM4 allows for the direct import of the freeform surface equations to form a manufacturing toolpath, As shown above in Fig. 2 and Table 2, the optical datum within NanoCAM4 was offset by the required distance along the x-axis for each QPPP and the machining toolpath was processed and exported for usage.
Manufacture of the freeform phase plates were completed in two phases: (1) Rough shaping of optical blanks, and (2) Ultraprecision coordinated-axis diamond turning of the freeform optical surfaces. Rough machining methods are useful for quickly removing large amounts of material with moderate precision, while diamond turning is used for precision shaping capabilities and the fact that optical material can be turned to a specular finish with virtually no tool wear [48].
Initial blanking began with 31.75 mm round bar stock PMMA rough-cut on a horizontal band saw into approximately 5 mm thick disks. Both sides of these disks were then ‘faced’ (cut smooth) with a diamond tool with 1.008 mm nose radius on the Moore Nanotech 350FG using a 38.1 mm diameter aluminum vacuum chuck. The Moore Nanotech 350FG (Fig. 5), is a 5-axis precision diamond machining center with 3 linear axes (X, Y, Z) with 0.034 nm resolution and two rotary axes (B, C) with 1.75 nanoradian resolution. Total manufacturing volume on this machine is 350 mm by 150 mm by 300 mm (X, Y, Z). The machine is enclosed in a temperature-controlled room at 20 C +/- 0.1 degrees C at 50% relative humidity to minimize thermal variations during manufacturing.
Once the rough-cut blanks were faced on the front and back sides, the parts were milled on a HAAS Toolroom Mill (Fig. 4) to cut clocking flats for angular alignment of the parts. A clocking flat was made using the HAAS mill and two steel pins pressed into two drilled and reamed holes in a machined aluminum block. The Outer Diameters (OD) of the steel machine pins aligned with a circumscribed circle which matched with the optic’s OD. This design allowed for an endmill with a known diameter to cut off a flat on the optic blank’s outer diameter. The optical blanks were next taken back to the Moore Nanotech 350FG for final machining. Both faces of the optical blanks were machined to the desired thickness with parallelism of +/- 1 µm. To generalize, these optical surfaces are comprised of a raised outer ring, ranging from 5 to 100 µm raised above the freeform clear aperture, as shown in Fig. 6. The main function of the outer ring is to help set the air gap between the QPPP components.
The final diamond machining of the freeform CPP and QPPP optical surfaces specified in Eqs. (1) and (2) and Tables 1 and 2 were completed on the Moore 350FG using the NanoCAM4 generated toolpaths. Both CPPs and QPPPs required up to four rough STS cutting passes at a Depth of Cut (DOC) of 20 µm with a stepover of 15 µm, and an angular density of 1 degree. These rough cutting passes were followed with single precision finishing pass with 15 µm DOC 5 µm stepover, and an angular density of 0.4 degrees. The optics were mounted for final surface cutting used a 10 PSI vacuum and a layout fluid. Layout fluid is a thin liquid, useful for mounting small parts on a vacuum chuck. When it dries, the layout fluid can act as a thin adhesive with no adverse effects on PMMA, unlike some other glues.
We were unable to measure the form of the resulting freeform optics at this time, but from prior experience we expect the form accuracy on the Moore Nanotech 350FG to be better than 0.25 µm Peak-to-Valley (PV) [49]. Surface finish was measured using the Zygo Zegage Plus 3D optical surface profiler (Fig. 7). The optical surfaces were measured using 20x and 50x objectives with 3 averages and a Gaussian bandpass filter of 2.5 to 80 µm, following ISO 10110-8 [50]. These filters were chosen to isolate surface roughness from form or waviness. The average root mean squared (RMS) surface roughness across the phase plates is 11 nm.
Fig. 7. (a) Image of a finished QPP, and interferometric measurement of a finished QPPP using the Zygo Zegage Plus 3D optical surface profiler and the average surface roughness measurements for the (b) 20x and (c) 50x objectives.
3. Performance characterization method and results
3.1 Autostigmatic and point source microscopes
PSM can be thought of as a modern realization of an Autostigmatic Microscope (ASM) [40]. In an ASM, a point source of light is imaged perfectly by a microscope objective after reflection off a beam splitter. The focused light is then re-imaged at the eyepiece, forming a perfect image. This configuration is illustrated in Fig. 8, modeled after Ref. [40].
ASM can be used in a variety of applications, including measurement and alignment. For example, the radius of curvature of a convex mirror can be found by first positioning the focus of ASM objective at the center of curvature of the spherical surface (confocal reflection), and then moving the ASM to focus on the surface of the sphere (cat’s eye reflection). The distance between the two focus spots is the radius of the spherical surface, as shown in Fig. 9 [40].
The PSM operates in the same way as an ASM, except that it has two light paths. In the PSM, a point source (fiber coupled laser diode at 635 nm) is accompanied with a Kohler light source (LEDs) to create a powerful reflected light imaging microscope. The addition of the Kohler light source enables imaging of opaque surfaces and uniform illumination of the sample under test. Figure 10 shows the schematic of a PSM [40].
3.2 Use of PSM to characterize through-focus performance
We consider the use of PSM in the configuration shown in Fig. 11(a) to characterize the through-focus performance of the EDoF phase plates. To the best of our knowledge, the PSM has not been previously used to measure through-focus spots to investigate the effects of EDoF phase plates on PSF variation. The PSM enables the creation of a point-like source and, when displaced along the optic axis, enables measurement of the PSF through focus. The microscope objective and the imaging system under test have their foci aligned in a confocal configuration. The imaging system (lens and phase plates) is displaced along the optic axis using a translation stage with micrometer and the spot variation is measured to perform a focus scan and characterize the imaging system's performance.
Fig. 11. (a) Schematic of experimental setup (top view) for characterization of variable EDoF imaging, and (b) experimental setup (side view) for phase plate characterization.
It is anticipated that the addition of EDoF phase plates (CPPs and QPPPs) will result in larger through-focus spots with less variation than would be observed in the absence of phase plates. Figure 11 (b) shows the experimental setup. Translation and rotation stages are used in the experimental setup for both through-focus scanning and alignment. The NA of the objective and imaging system under test are matched for the measurement.
3.3 Performance characterization
3.3.1 Through-focus spot measurements
The PSM was used to explore through-focus performance of the three different lens system configurations (0.25, 0.33, and 0.5 NA) both with and without freeform EDoF phase plates. We note that the 0.5 NA system was extremely sensitive to alignment, and we were unable to collect meaningful test data. Results from the 0.25 NA and 0.33 NA systems were qualitatively similar. With this in mind, we now consider in detail the through-focus spot measurements for the 0.33 NA configuration with and without EDoF phase plates.
The through-focus range of interest for the 0.33 NA lens is from -35 µm to +35 µm (-6Δz to +6Δz), where Δz is calculated from [2]:
where n is the refractive index of the surrounding medium (n = 1 for air), λ=635 nm (design wavelength), and NA = 0.33 (lens numerical aperture). Due to the micrometer resolution of 1 µm, the experimental defocus scan range is from -36 µm to +36 µm, in 6 µm steps. Figure 12 shows the PSM measurements of through-focus spot for the 0.33 NA lens with and without the corresponding CPP and QPPP elements (Tables 1 and 2). As discussed previously, the QPPP elements are manufactured with 0.7 mm shift in opposite directions to enable EDoF, which eliminates misalignment errors caused by additional degree of freedom required for phase plate pair translation. As expected, qualitative examination of Fig. 12 shows observable through-focus spot variation for the aspheric lens with no phase plate. With the addition of CPP and QPPP, larger spots are created with less variation as the lens is displaced along the optic axis and moved away from the optimum best focus. We present quantitative analyses of these observations in the next section.Fig. 12. Through-focus spot measurements from -36 µm to +36 µm (-6Δz to +6Δz), for the 0.33 NA lens with and without EDoF phase plates.
3.3.2 Root-mean-square (RMS) deviation of spots
For a qualitative performance comparison of the EDoF phase plates, we introduce the Root-Mean-Square (RMS) of the normalized spots through focus [24]. To calculate this parameter, the through-focus spot intensity measurement matrices (I) are first normalized in MATLAB, using the “rescale” function, which scales a data array to the interval [a b]. Each intensity matrix is rescaled using the equation below:
where a and b are the lower and upper boundaries that the array is to be normalized to (0 and 1 in our calculations), I is the data array to be normalized, and Imin and Imax are the minimum and maximum values of the input array I [51]. After the intensity measurements for all through-focus and optimum focus locations are normalized, the RMS variation of the normalized intensities are calculated, using Eq. (4):Fig. 13. Experimental RMS deviation of normalized spots from -36 µm to +36 µm (-6Δz to +6Δz), for the 0.33 NA lens with and without EDoF phase plates.
Figure 14 compares the normalized PSF and spot measurement at best focus and optimum focus for the 0.33 NA lens with and without CPP and QPPP. We note that the simulation shows diffraction limited performance and PSF for the case of the 0.33 NA lens with no phase plate, while the experiment shows a larger spot. This difference could be attributed to the differences in simulation and test system layouts, shown in Fig. 1 and Fig. 10(a), respectively. In simulations, the wavefront passes through the lens once, whereas in the experimental setup, the wavefront passes through the same lens twice. This difference results in additional spherical aberration and a larger spot.
Fig. 14. Simulated PSF at best focus and measured spot at optimum focus for the 0.33 NA lens with and without EDoF phase plates.
We observe a rotation in the measured spot compared to the simulated PSF for the 0.33 NA lens with the CPP, which is caused by the CPP's orientation along the optic axis with respect to the aspheric lens (about 45 degrees clocking), as well as a change in the shape of the spot. The shape of the measured spot is similar to that of a family of beams with cubic wavefronts and Seidel coma aberration [52]. We observe a similar rotation of the spot for the 0.33 NA lens when using the QPPP, as well as a difference in the shape of the measured spot. Further investigation of the manufactured form and quantification of misalignment errors may aid in gaining a better understanding of the shapes of the measured spots.
Figure 15 compares the RMS deviation of normalized spots to the previously simulated PSFs [24]. This comparison demonstrates reasonable agreement between the simulated and experimental results and shows smaller through-focus variation for the CPP and shifted QPPP than the lens alone, with QPPP resulting in larger RMS deviation compared to CPP. However, the RMS deviation of the experimental spot measurements have larger values compared to the simulation data. This difference could be due to the difference in simulation layout and experimental set up, as well as potential misalignments present in the system.
Fig. 15. Comparison between the RMS deviation of normalized PSF (simulation) and spots (experimental) from -36 µm to +36 µm (-6Δz to +6Δz), for (a) 0.33 NA lens, (b) 0.33 NA lens with CPP and (c) 0.33 NA lens with QPPP.
4. Discussion and conclusion
We have reported on the manufacturing and characterization of an optical system based on previously designed EDoF freeform phase plates [24]. Custom optomechanical fixturing was designed and fabricated. The freeform surface profiles of the optical phase plates were manufactured using ultraprecision coordinated axis diamond turning, with measured average RMS surface roughness of 11 nm.
We also demonstrated a novel use of a Point Source Microscope to experimentally measure the through-focus spots of the optical components under test. Experimental measurements confirm more consistent but larger through-focus spots with the addition of the EDoF phase plates. This can be seen qualitatively through visual comparison of the through-focus spots, and quantitatively by comparing the RMS deviation of the normalized spots. We note that the measured spots at optimal best focus do not perfectly match predictions from software, and that the spots for the 0.33 NA lens with CPP exhibit characteristics of cubic wavefronts (Airy beams) and Seidel comatic beams [52]. Measuring the form error of the manufactured CPPs and quantifying the misalignments in the test setup may provide a better understanding of the spot shape degradation [53]. We note that the configuration in which the PSM was used to determine the through-focus spot variation is slightly different from the layout used for optical design of the freeform phase plates. However, the results still show that the PSM can provide useful information about through-focus performance.
Funding
National Science Foundation IUCRC Center for Freeform Optics (IIP-1822026, IIP-1822049).
Acknowledgments
The authors would like to acknowledge valuable discussions with Dr. Christoph Menke from Carl Zeiss AG, Dr. Sébastien Héron from THALES, and Mike Dhooghe from the University of North Carolina at Charlotte.
Disclosures
The authors declare no conflicts of interest.
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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