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Abstract
Recently, the development of motion-free 3D microscopy utilizing focus tunable lenses (FTL) has been rapid. However, the downgrade of optical performance due to FTL and its gravity effect are rarely discussed in detail. Also, color dispersion is usually maintained purely depending on the FTL material without further correction. In this manuscript, we provide a quantitative evaluation of the impact of FTL on the optical performance of the microscope. The evaluation is based on both optical ray tracing simulations and lab experiments. In addition, we derive the first order conditions to correct axial color aberration of FTL through its entire power tuning range. Secondary spectrum correction is also possible and an apochromatic motion-free 3D microscope with 2 additional doublets is demonstrated. This study will serve a guidance in utilizing FTL as a motion-free element for 3D microscopy.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
For the last 2 decades, focus tunable lenses (FTL) are being rapidly developed, from concepts and prototypes on paper to commercially available products. The development of FTL went down three different paths. The first type of tunable lens is liquid crystal (LC) tunable lens, which is based on large optical birefringence property of the nematic liquid crystal (NLC). Since the orientation of liquid crystal is different along the optical path when light propagate through the LC panel, the lens effect is achieved. By changing the applied electric field, the effective focal length can be tuned. The concept of focus tunable LC lens was introduced in the late 1970s and is still being extensively studied today [1–4]. The major drawback of the LC tunable lens is its sensitivity to light polarization, slow in response time and low in optical damage threshold [5]. Another way of achieving FTL is by the liquid-liquid deformation using electrowetting principle [6]. Two non-miscible liquids with different refractive index form a natural liquid-liquid interface, the curvature of the interface is tunable when applying voltage, thus changing the focal length. However, these two liquids are required to have same density. The lens will show thermal effects due to differences in thermal expansion coefficients of the liquids [7]. To minimize the effect, this type of FTL suffers from strong aperture size limitation, usually less than 5 mm. The newest approach to the FTLs, leading by a Swiss company named Optotune AG, is shape-changing polymer lens. The concept of shape-changing polymer lens is similar to the liquid-liquid lens based on electrowetting principle. However, the liquids are replaced with polymer, and the liquid-liquid interface is now built by a thin membrane. The curvature of the membrane changes by applying pressure either mechanically or electrically. This type of FTL has less thermal issue thus supports larger aperture. Also, surface deformation due to gravity has less impact to the polymer lens, compared to liquid lens, due to relatively strong retaining force of the membrane.
A wide range of industrial and scientific applications have been developed utilizing the FTLs [8–15], especially for microscopy. FTLs can be used for different purposes in microscopy, includes tunable illumination, electrically controlled zoom and autofocus. In this manuscript, we are focusing on 3D microscopy using FTL. For a nominal infinite conjugate microscope setup, the illumination light is focused on the specimen placed at the front focal plane of the microscope objective as shown in Fig. 1(a). In order to image the specimen at different depths, either the objective or the specimen traditionally has to be moved mechanically along Z-axis, significantly reducing the imaging speed. In recent years, as shown in Fig. 1(b), an FTL setup with a tuning range from negative to positive focal length has been developed to focus and image the specimen at different depths. Compare to the mechanical scanning, motion-free scanning with FTL reduces vibration to the specimen during the movement, and the scanning speed could be faster. One major drawback of this setup is that the new objective loses its telecentricity, which is standard for most of the high-end refractive infinite conjugate objectives. However, since now the objective scans through multiple focal planes, the impact caused by the non-telecentric nature of the system is not significant. An alternate solution is to relay the system with a 4F system and put the FTL at its conjugate pupil. Nevertheless, this solution is much less practical than putting the FTL directly above the objective.
Fig. 1. Light propagation through infinite conjugate microscope objectives. a) A conventional infinite conjugate objective with a fixed focus plane. b) A conventional infinite conjugate objective pairs with an FTL with a tuning range from negative to positive focal length to image the specimen at different depth.
Recently, there are a number of studies on motion-less 3D microscopy using FTLs [16–18]. Nevertheless, most papers seem to focus on the capability of 3D scanning rather than the optical performance. Since high-end microscope objectives are mostly diffraction limited, and the FTLs are designed without the consideration of aberration control, attaching FTL on the objective will certainly impact the optical performance of the objective. Also, chromatic aberration correction was rarely discussed. It seems the manufacturers of FTLs are purely relying on the low dispersion material for maintaining low chromatic aberration without any further color correction. Furthermore, when discussing the applications of FTLs in general, the impact caused by gravity when positioning FTLs in the horizontal direction is often overlooked by designers [19]. In this manuscript, we address these issues by providing a comprehensive evaluation on optical performance of 3D microscopy using FTL. We perform the evaluation both in software simulation and experiment. And we purpose a novel method for correcting chromatic aberration of FTL at the end of the manuscript.
2. Simulation and experiment setup
2.1 Simulation setup
Figure 2 demonstrate the setup for the simulation study. The FTL that we used is the Optotune EL-10-30-TC. The theoretical power tuning range of this lens is + 8.3 dpt to + 20 dpt (+120 mm to + 50 mm in focal length). However, the lens shape loses stability at the extreme of its tuning range, so the power tuning range is limited between + 9 dpt and + 16 dpt. Note that power tuning range of this lens is entirely positive, so an offset lens (OL) is paired with the FTL to shift the tuning range to include both positive and negative power. A −75mm focal length plano-concave lens is used as the OL, which brings the power tuning range of the combined setup to from −3.65 dpt to 4 dpt (−274 mm to -∞ mm and + 250 mm to +∞ mm in focal length). It will be ideal to evaluate the performance with real lens data of microscope objectives, however almost all of the microscope objective vendors do not disclose the lens data for their infinite conjugate objectives. With our best effort, we are able to acquire 4 “black box” Zemax files from Thorlabs, including a 2X (TL2X-SAP, NA = 0.1) and a 4X (TL4X-SAP, NA = 0.2) refractive apochromatic objectives, a 15X (LMM-15X, NA = 0.3) and a 40X (LMM-40X, NA = 0.5) reflective objectives for our simulation. Visible spectrum (F, d, C) are used as the wavelengths in the simulation. For each set of simulation, on-axis MTF, chromatic aberration (longitudinal focal shift) and focal plane scanning range are evaluated. The MTF and chromatic aberration are evaluated at 5 different FTL power settings at 9 dpt, 11 dpt, 12.43 dpt, 14 dpt and 16 dpt. 12.43 dpt is chosen because at this power, the combination of FTL and OL is afocal.
Fig. 2. Simulation setup in Zemax. “Black box” models provided by Thorlabs were used as the objective lenses. An OL is paired with FTL to expand its focus tuning range.
2.2 Experiment setup
2.2.1 Spherical aberration and chromatic focal shift
Figure 3(a) demonstrates the setup to evaluate the spherical aberration and chromatic focal shift introduced by the FTL. A Twyman-Green type interferometer is constructed and positioned vertically to minimize the gravity effect caused to the FTL. The polarization camera essentially provides phase shift and capture the interferogram at four different phases [20,21]. A 5X Mitutoyo plan apochromatic objective with NA = 0.14 is used in this experiment. A collimated laser beam with enough beam diameter to cover the entire objective NA is used as light source. Perfect collimation is checked with shear plate for each wavelength. All the data measured in this experiment went through reproducibility test, where 5 sets of data were measured and averaged for each data point, with the objective and spherical mirror being re-aligned for every measurement. The measured wavefront is estimated from the final image and fit with Zernike polynomials in Matlab. Same as in the simulation, the spherical aberration and chromatic focal shift are evaluated at 5 FTL power settings at 9 dpt, 11 dpt, 12.43 dpt, 14 dpt and 16 dpt. Also, we evaluate the objective without FTL to serve as a reference.
Fig. 3. Experiment setups. a) A Twyman-Green interferometer to measure the spherical aberration and chromatic focal shift of the 3D microscopy setup in vertical direction. b) Measure the image contrast with a USAF 1951 target to evaluate system resolution. Illumination system shares the common path with imaging system to provide same NA. c) Measure aberration and contrast of the 3D microscopy setup in horizontal direction to exam gravity effect.
While spherical aberration is straight forward and can be extract from the ninth Zernike coefficient ${Z_8}$, the focus shift between different wavelengths is more complicated to measure. First, 3 Lasers with different wavelengths across the visible spectrum (488nm, 543nm and 633nm) are used as the light source. Zernike coefficients are measured for each wavelength without changing the distance between the spherical mirror and objective. Then the field-independent wavefront aberration coefficient for focus is calculated by
where $\rho $ is radial distance, ${Z_3}$, ${Z_4}$, ${Z_5}$ and ${Z_8}$ are the fourth, fifth, sixth and ninth Zernike coefficient, and the sign is chosen to minimize the magnitude of the coefficient [22]. To accurately get the linear relationship between the field-independent wavefront aberration coefficient and the focus in mm, the system is calibrated for each wavelength independently by intentionally defocus the spherical mirror and record its new coefficient. Then the focus shift between different wavelengths can be calculated.2.2.2 Resolution
Figure 3(b) is the setup to test the impact caused to system resolution by the FTL setup. This is done by evaluating the image contrast with an USAF1951 target up to 9th group. A blue LED (peak at 455 nm) is used as the light source. A condenser lens is placed after the LED to collimate the divergence beam from the LED to fill the pupil of the objective. The illuminated target is imaged by the objective and a 200 mm tube lens to the camera. The image contrast is evaluated at 5 FTL power settings at 9 dpt, 11 dpt, 12.43 dpt, 14 dpt and 16 dpt, and is also evaluated without FTL as a reference. The experiment result and analysis will be discussed in Section 3.2.
2.2.3 Gravity effect
Figure 3(c) shows two setups to evaluate the gravity effect with the FTL setup positioned in horizontal direction. Both interferogram and image contrast are evaluated with procedure similar to the previous evaluations. When the FTL is positioned in the horizontal direction, significant coma will appear due to the gravity pulling the polymer lens towards the ground. For interferometric measurements, only 632.8 nm laser is used since chromatic aberration has no significant impact on monochromatic aberrations. Spherical aberration (${Z_8}$) and coma in both directions (${Z_6}$ and ${Z_7}$) are calculated from the interferograms. Both the aberration coefficients and the image contrast are compared with the data measured in the previous experiments where FTL is positioned in the vertical direction.
3. Results
3.1 Simulation findings
Figure 4 shows the scanning range and the on-axis MTF plots of the 4 chosen microscope objectives at 5 FTL power settings: 9 dpt, 11 dpt, 12.43 dpt, 14 dpt and 16 dpt. On-axis MTFs for the objective themselves without attaching the FTL are also provided as references. All of the 4 objectives are able to produce diffraction-limited performance before attaching the FTL. After the FTL is attached, on-axis MTF performance downgrades as expected. The decrease in contrast is more significant when paring FTL with low magnification objectives. Contrast suffers less but still is noticeable when higher magnification objective lenses are used. At the same time, scanning range decreases when the objective magnification is increased. Between different power settings of the FTL, it seems for all the objectives, the optimal contrast is achieved when the power of FTL is 12.43 dpt, where the total power of the FTL and OL is 0 dpt (afocal).
Fig. 4. On-axis MTF of the FTL and OL paired with a) Thorlabs 2X refractive objective, b) Thorlabs 4X refractive objective, c) Thorlabs 15X reflective objective and d) Thorlabs 40X reflective objective at different FTL power setting. Performance of the systems without FTL are provided as reference.
Figure 5(a) shows the chromatic focal shift within the visible spectrum of the 4 chosen microscope objectives at the same 5 power settings. Focus shifts between the F-line and d-line (486nm and 588nm) and between the F-line and C-line (486nm and 656nm) are recorded at each FTL power setting. Significant chromatic shift is observed. As the power of FTL increases, the effective focal length of the entire objective setup decreases, and the chromatic focal shift reduces with the effective focal length. Another way to assess the chromatic aberration correction is to read the plot of chromatic focal shift vs. focal length. Figure 5(b) compares the chromatic focal shifts between the objectives without FTL and with FTL set to 12.43 dpt. The evaluation spectrum is extended to 365 nm – 750 nm to further evaluate the chromatic focal shift. Both refractive objectives show clear sign of apochromatic correction while both reflective objectives have no chromatic focal shift due to their reflective nature. However, after attaching the FTL, the relationships between chromatic focal shift and wavelength are almost linear for all the objectives. This means, despite the low dispersion of the material used by the FTL (Abbe Number ∼ 105), the objective setup with the FTL shows no achromatic correction at all.
Fig. 5. Chromatic aberration evaluation from the simulation. a) Chromatic focal shifts between F-line and C-line (blue bar) and between F-line and d-line (orange bar) for 4 chosen objectives with FTL and OL. b) Chromatic focal shifts for the 2 refractive objectives without and 4 objectives with FTL. Note the scale of refractive objectives after attaching FTL is 10X of the original objectives without FTL. The reflective objectives without FTL has no chromatic aberration due to their reflective natures, thus the curves are not shown.
3.2 Experiment results
Figure 6(a) shows the spherical aberration at each FTL power setting by measuring the 9th Zernike coefficient Z8 using out Twyman-Green interferometer. The FTL setup is placed vertically so the gravity effect is minimum, and the surface shape of the FTL remains almost spherical within the clear aperture. Significant spherical aberration is observed in the experiment. Since the microscope objective without FTL setup is diffraction limited, spherical aberration observed in the experiment is contributed almost entirely by the FTL and OL. As we increase the power of the FTL, the curvature of the curved surface of FTL is also increasing, resulting a rapid increase of spherical aberration. In addition, the system shows noticeable spherochromatism, which is the color variation of the spherical aberration, at each FTL power setting.
Fig. 6. Optical performance evaluation of 3D microscopy setup in vertical direction. a) Spherical aberration by evaluate Zernike coefficient Z8 at different FTL power setting. b) Chromatic focal shift between 488 nm and 633 nm (blue bar), and between 488 nm and 543 nm (orange bar). c) Image of USAF1951 resolution target at different FTL settings. d) Contrast of group 7 element 1 of the resolution chart image.
To further investigate the chromatic aberration, color variation of the focal plane, also known as the focal shift, is evaluated by measuring the defocus of each wavelength using the interferometer. The results are showed in Fig. 6(b). Compared to the simulation result showed in Fig. 5(a), similar trend is observed. Chromatic focal shift with up to 58 times larger is observed when the power of FTL is set at 9 dpt, compared to the original microscope objective without FTL attached. As the power of FTL increases, system focal length decreases, resulting in a reduction of chromatic focal shift. However, even at the largest FTL power setting of 16 dpt, the chromatic focal shift between the two extreme wavelengths (488 nm and 633 nm) is still more than 5 time larger than that of the original microscope objective.
Figure 6(c) shows the contrast evaluation of the original microscope objective and the objective with FTL and OL attached at 5 FTL power settings. The images of USAF target are cropped to focus on groups 6, 7, 8 and 9. Figure 6(d) shows the quantitative evaluation on the average contrast (sagittal and tangential) of the line pairs of group 7 element 1. A 50% to 80% contrast reduction, depending on the power setting of FTL, is observed after FTL and OL are attached. The highest contrast is observed when the FTL has a power of 12.43, which agrees with the finding from the simulation.
The evaluation of the FTL setup with gravity effect is shown in Fig. 7. The wavefront map calculated from the interferogram shows that coma becomes the dominate aberration when FTL setup is positioned horizontally. The Zernike coefficient of 3rd order spherical aberration Z8, and 3rd order coma Z6 and Z7 are presented in Fig. 7(a). The orientation of the FTL is set up so that Z7 indicates the coma along the direction of gravity, where Z6 is the coma orthogonal to the direction of gravity. Spherical aberration has small change compared to the vertical FTL setup. Coma coefficient Z7 reduces slightly with the increase of FTL power, but the difference is not significant. Note the presence of coma orthogonal to the direction of gravity, this may be caused by the imperfection of the FTL itself. This effect can also be observed from the wavefront map in Fig. 7(b), where the direction of coma has a 6-degree departure from the direction of gravity. The presence of coma caused by gravity has a large impact on the image contrast. Figure 7(c) shows the images of 1951 USAF resolution target with 5 FTL settings, the image quality is degraded significantly. Figure 7(d) plots the corresponding average contrasts, showing the contrast decreases rapidly as the power of FTL increases. This experiment indicates that this type of FTL cannot be used horizonatally.
Fig. 7. Optical performance evaluation of 3D microscopy setup in horizontal direction. a) Spherical aberration and coma by evaluate Zernike coefficient Z6, Z7 and Z8 at different FTL power setting. b) Wavefront map from interferometer shows strong coma. Coma direction departures from gravity direction by about 6 degrees c) Image of USAF1951 resolution target at different FTL settings. d) Contrast of group 7 element 1 of the resolution chart image.
4. Correction of chromatic aberration
From both simulation and experiment results, we come to conclusion that despite the low dispersion of FTL material, the induced chromatic aberration is too severe to be ignored and must be corrected. There are two different types of chromatic aberrations to be corrected: chromatic change of magnification (lateral color) and chromatic change of focus (axial color). The chromatic change of magnification is the color variation of the image magnification, and it can be partially corrected through the post image processing. Since changing the focal length will result a magnification change of image during the scanning, post processing to adjust magnification at each FTL setting is necessary. Chromatic change of focus, on the other hand, is hard to eliminate during the post processing, thus requires to be corrected before the image is taken.
Correcting chromatic change of focus for FTLs are more complicated than conventional lenses. Axial color for conventional lenses can be easily corrected by an achromatic doublet that is designed with specific combination of lens materials and lens powers. However, for FTLs, lens power changes with different current settings, but the lens material, specifically the Abbe number remains constant. That means if we design an achromatic doublet to correct axial color at one FTL power setting, the correction will be not effective once the FTL power is changed. Currently, the correction of chromatic aberration of FTL has rarely been discussed due to this difficulty. Philipp Waibel proposed an achromatic FTL system with multi-chamber membrane lenses [23]. However, this method requires designing the FTL with optimal refractive index and Abbe number, which is not entirely practical. In this section, we propose a method to correct axial color of FTLs with conventional lens design technique.
Figure 8(a) describes the problem we are trying to solve. Assume we are putting an achromatic doublet in front of the FTL to bring F-line and C-line into a common focus, this means the back focal distance (BFD) for F-line and C-line must be the same. To derive conditions to fulfil this requirement, we assume all elements in this figure are thin lenses, start with basic Gaussian reduction
where ${{\Phi }_1}$ and ${{\Phi }_2}$ are the powers of 2 lens elements of the doublet, ${{\Phi }_3}$ is the power of the FTL, ${{\Phi }_{total}}$ is the power of the system, t is the distance between the doublet and FTL, $d^{\prime}$ is the distance between FTL and rear principle plane, ${{\Phi }_{nC}}$, ${{\Phi }_{nd}}$ and ${{\Phi }_{nF}}$ are the powers of nth element regarding to the specific wavelengths, and ${V_n}$ is the Abbe number of the nth element. According to Eq. 2, we haveFig. 8. Chromatic aberration correction for FTL. a) Initial setup for equation derivation. b) Achromatic FTL system with 2 thin doublets c) Apochromatic 3D microscopy system. d) Chromatic focal shift for the chromatic FTL system. Maximum plot scale is 100 µm. e) Chromatic focal shift for the apochromatic 3D microscopy system. Maximum plot scale is 2 µm. f) On-axis MTF performance for the apochromatic 3d microscopy system
Then
andNext, we verify this condition in Zemax. We setup the system in Fig. 7(a) in Zemax with the same FTL we use for the previous simulation. The merit function is to have achromatic condition for visible spectrum throughout the entire tunable range of the FTL. For optimization, we set surface curvatures of the doublet and the distance between the doublet and FTL as variables. In addition, we set the glass materials of the doublet as substitution and use hammer optimization in Zemax to find various glass combinations. Then we evaluate each system to check how well it follows our derived conditions. The results are shown in Table 1.
Table 1. Comparison between Zemax solution and derived conditions. Condition 1 from Eq. 18 is highlighted in red and condition 2 from Eq. 21 is highlighted in green.
From Table 1, we can see that each solution found by Zemax meets the 2 conditions we derive within 3% error. This study validates that if the two conditions are met, the chromatic focal shift can be corrected and is independent of FTL power. However, we also notice that for all the solutions that Zemax finds, the combined focus tunable ranges are very small, and the FTL is positioned close to the image plane. To explain this, we start with calculation of the BFD for intermediate wavelength d-line:
Figure 8(b) shows an achromatic focus tunable system with one FTL and 2 doublets. For simulation purpose, the full tunable range of the FTL (8.33 dpt to 20 dpt) is sampled with 5 even increments and thin lenses are used for the doublets. The chromatic focal shift of this system is shown in Fig. 8(d), the system is able to maintain achromatic correction at all FTL power settings. And the tunable range for the combined system is reasonable from 8.33 dpt to 16.5 dpt.
Another perk of using 2 doublets is to achieve apochromatism. A previous paper discussed a method to easily achieve apochromatism with 2 doublets [24]. Figure 8(c) shows an apochromatic tunable microscope objective setup. A paraxial surface with 10 mm focal length is used to simulate a 20X diffraction limited objective, and the design spectrum expands from visible spectrum to 400 nm – 700 nm. Note that the system achieves apochromatism at each FTL power setting as shown in Fig. 8(e). Also, as shown in Fig. 8(f), the on-axis MTF is diffraction limited at most of the FTL power setting except the largest FTL power setting, where the spherical aberration starts to lose control and results in a downgrade in image quality.
5. Conclusion
In this paper, we provide a detailed evaluation of 3D microscopy with focus tunable lens. We evaluate both the image quality and chromatic aberration using optical simulation with Zemax. We are able to confirm our findings from the simulation via experiments with the custom Twyman-Green interferometer. We realize that the microscopy system with FTL will suffer from significant image qualify downgrade. Also, the FTL must be setup vertically since gravity will cause large coma when FTL is positioned in horizontal direction. In addition, we realize despite the low dispersion material used for the FTL, the chromatic aberration is severe and must be corrected. In this paper, we also propose a novel method to correct axial color throughout the entire focus tunable range with 2 achromatic doublets. The image quality would also be improved with the additional lens elements. We hope this paper can serve as a proper guidance for future study on 3D microscopy with FTL.
Funding
National Institutes of Health (1S10OD018061-01A1); National Institute of Dental and Craniofacial Research (R21DE028734-01A1); National Science Foundation (1918260).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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