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Abstract
We analyze the feasibility of using refractive free-form phase plates at the aperture stop of microscope objectives as an alternative to active alignment to compensate for assembly tolerances. The method involves the determination of misalignment-induced aberrations at the exit pupil, and transferring them to the aperture stop while taking pupil aberrations into consideration. We demonstrate that despite being able to correct only for field-independent aberrations, this method can restore near-diffraction-limited imaging performance of passively aligned systems with practical tolerances, given that the as-designed system is highly corrected. We confirm the results via numerical simulations for two different commercial objective designs.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
High-end optical systems such as microscope objectives (MOs) require micrometer-scale positioning precision in their assembly to maintain the as-designed diffraction limited optical performance. The state-of-the-art method to maintain this quality relies on cell-mounted assemblies [1–3]. In this method, each lens of the optical system is mounted separately within a mechanical cell often called a sub-assembly. For each sub-assembly, the lens can be centered involving different active alignment methods with different adjustment and/or compensation steps that vary in complexity [4,5]. Subsequently, all sub-assemblies are combined together to build the entire optical system, where further active alignment methods can be applied. These often use the same degrees-of-freedom (DOF) as for the sub-assembly, where the alignment is performed while optimizing a measurable figure of merit, such as the wavefront’s root-mean-square (RMS) error at a given field-of-view (FOV) point. However, the use of the principle DOFs for compensation is not always advantageous. Foremost, active alignment routines introduce additional fabrication costs. Furthermore, controlling one type of aberration by changing one any DOF can introduce others, since they are not linearly independent with respect to each other.
Rapid development of advanced manufacturing techniques for free-form optics has opened access to new design spaces, and has enabled new passive compensation methods in lens design [6]. As an example, aspherical compensation plates have been used for a long time in Schmidt-Cassegrain telescopes to compensate for spherical aberrations [7,8]. Fuerschbach et al. introduced and exploited free-form optics at off-pupil locations to optimize the design of refractive non-symmetrical optical systems [9,10]. Reimers et al. showed that by using free-form surfaces the compactness of optical systems can be significantly increased. [11]. However, the focus for all of these examples and so far in the literature was on the use of free-form optics for the system design and not on the facilitation of their assembly.
In this work, we propose that an assembly-specific free-form refractive phase plate located at the objective’s aperture stop can compensate for a significant portion of optical aberrations induced by passive alignment steps. By this approach, misalignment-related aberrations can be individually addressed to restore the as-designed performance during the assembly. The main advantage of the proposed method is to replace the active alignment procedures in the assembly of microscope objectives, which can reduce the assembly complexity and increase the fabrication yield. We present in detail the compensation method, and discuss how to incorporate pupil aberrations. With this background, we then present two case studies, in which we model the assembly scenarios for two high-end commercial microscope objectives, and demonstrate the advantages and limits of correcting misalignment-induced aberrations by a free-form phase plate at the aperture stop.
2. Refractive free-form phase plates for residual aberration correction
Figure 1 summarizes how a refractive free-from surface can conceptually be included in objective assembly process without active alignment. First, each sub-assembly are assembled with passive alignment only, with resulting system suffering from residual aberrations induced by the non-zero alignment tolerances. These aberrations are characterized at the exit pupil (EP) by a wavefront error measurement from an artificial guide star placed at the center of the FOV. This measurement of field-independent aberrations can be done with an interferometer or by using a Shack-Hartmann wavefront sensor (SHWS). Due to pupil aberrations (e.g. aberrations in the imaging of the aperture stop (AS) onto the exit pupil (EP)), however, the measured aberrations should be remapped on the aperture stop, where the phase plate will be located. Finally, with this wavefront data, a unit-specific compensation phase plate is fabricated and placed at the AP. An additional wavefront error measurement can be performed at this stage for validation. For each of these steps, however, there are important considerations and sub-steps, which are of significant importance that are thoroughly analyzed next.
Fig. 1. Schematic description of residual aberration correction using a refractive phase plate. During the assembly process the lenses are not perfectly aligned introducing imaging aberrations. These aberrations are characterized by the measurement of the wavefront at a specific FOV point and a phase plate is fabricated according that data and incorporating pupil aberrations. Finally, the phase plate is located within the imaging path of the lens system.
2.1 Location of the phase plate
In principle, there are infinitely many locations where a compensation plate can be placed within the optical train. For the present method, however, we propose the AS as the location of choice due to two main reasons: First, the AS is a real, physical and mostly accessible plane of a microscope objective corresponding to the image of the pupil. Second, at this plane light from all fields overlap completely, and the wavefront of the entire FOV can be modulated simultaneously without inducing additional field-dependent aberrations [9,12]. Therefore, our method can be classified as a version of refractive pupil adaptive optics (AO) as in [13,14] with a static rather than dynamic correction device placed directly at the AS.
A correction plate at the aperture stop introduces an additional optical path difference (OPD) to all points in the FOV, which is given by
where $n_1$ is the refractive index of the compensation plate, and $n_2$ is the refractive index of air. The surface profile of the compensation plate $z(\vec {\rho })$ is represented as a Zernike decomposition, where $c_{mn}$ corresponds to contribution of Zernike mode $Z_n^m(\vec {\rho })$ given in polar coordinates normalized to the AS diameter. The Zernike modes are a set of orthogonal polynomials with radial and azimuthal dependency commonly used in aberration theory. Any arbitrary wavefront over a circular aperture can be represented as a linear combination of these modes. In this work, we use the Zernike Standard numbering scheme according to Noll, which is defined as [15]Due to the similarity of our method to classical pupil-AO, it inherits its limitations as well. In pupil-AO, it is well known that the correctable FOV range is restricted due the isoplanatic patch, which results from the fact that only field-independent contributions of the wavefront aberrations can be corrected at the pupil [16]. Figure 2 illustrates this by plotting representative misalignment-induced Zernike coefficients as a function of field position for an optical system with primary aberrations. Different Zernike coefficients with different field-dependencies up to the first-order are depicted in (a). In part (b) and (c) the coefficients of the present Zernike modes were compensated for arbitrarily chosen FOV points 1 and 2, respectively. It can be seen that the compensation for one FOV point reduces the Zernike coefficients for that particular point, but on the other hand increases them for other FOV points without changing the slopes of the curves. For a strongly aberrated system, this means that only a limited portion of the FOV centered around the correction point would experience improvement, while the image quality might even be degraded for rest of the FOV. If, on the other hand, alignment tolerances are sufficiently high, the isoplanatic patch can cover the entire FOV.
Fig. 2. (a) Misalignment-induced aberrations in an optical system have both field-dependent and field-independent contributions. The former is manifested as "DC offsets" in aberration amplitude as a function on field position, while the latter vary with linear or higher order dependence. (b)-(c) With a compensation plate placed at the AS, perfect correction can only be obtained at a single point, since this method only compensates for the field-dependent aberrations and remove the "DC component". The region around the perfect correction point within which the contribution of aberrations is negligible is called the isoplanatic patch.
2.2 Incorporation of pupil aberrations
In our method, the wavefront error is determined at the EP, but its compensation is performed at the AS. Due to this difference, pupil aberrations must be taken into account. Pupil aberrations are in general mapping errors between points in coordinate systems located at positions conjugated to each other, and are the result of imaging non-linearities [17,18]. Therefore, if a measured wavefront error from one plane is transferred onto another without considering this remapping, additional higher-order aberrations can be induced, and the correction quality might be degraded. For incorporating the pupil aberrations, the coordinate distortion between AS and EP is determined by ray-tracing through the ideal, as-designed model of the optical system. Considering that alignment errors are small, this approach assumes that their impact on pupil aberrations are of second order, and can be neglected. We demonstrate the validity of this assumption through Monte Carlo simulations in the following section. We trace a set of rays emanating from equidistant points on the AS to the EP, and record their location of intersection with the output plane. Subsequently, each location in each coordinate system is normalized to its maximum radial intersection distance and the mapping distortion $d(\vec {\rho })$ is calculated by
where $\vec {\rho }$ is the normalized radial coordinate at the AS and EP. This procedure is depicted in Fig. 3 using a ray bundle originated from equidistant points on the AS 3(a), arriving at at the EP on a non-equidistant grid 3(b). The resulting distortion is shown in 3(c).Fig. 3. Schematic determination of mapping error between the equidistantly sampled coordinate system of the AS in a) and its image the EP in b). Part c) show their relative coordinate distortion, which is also called pupil aberration.
With this known distortion $d(\vec {\rho })$, we can remap the measured wavefront error from EP plane to the AS, assuming it is misalignment-invariant. For this, we only need to sample the measured wavefront error at the EP with the warped grid, since the location at these points correspond to equidistant sampling in the AS. These sampled points are then fitted to Zernike coefficients up to 7th radial-order, and Eq. (1) is used to transfer these coefficients from OPD to sag height $z$ of the desired compensation plate at the AS.
3. Case studies: application to commercial high NA objectives
We validated the proposed compensation method for two different microscope objectives with different properties and pupil aberrations in a ray-tracing simulation environment (OpticStudio from Zemax, LLC). Both models were high-NA, bright-field imaging objectives, with similar, diffraction-limited performance within the design FOV in different imaging media (immersion oil vs. air). In their initial state, both were rotationally symmetric, and were modeled as infinite-conjugate optical systems.
The first investigated system was an Olympus plan apochromat objective with NA of 0.95, 100x magnification, and chromatic correction range from F to C line [19]. With an 8-element optical train, it was designed to image in an air without the use of a cover glass. The second system was a plan apochromat from Nikon, designed to image trough a cover glass with a thickness of 170 µm and an oil-immersion with a refractive index $n_d =$ 1.515 at the d-line wavelength $\lambda _d$ of 587.5 nm. It also consists 8 elements, with NA of 1.4, a 60x magnification, and a chromatic correction range same as the Olympus objective [20].
The application of the phase plate compensation method required an accessible AS. In both objectives, however, this plane was not readily available for a phase plate since the AS was located inside the glass of one of its lens elements. To make the AS plane accessible, both models were slightly modified in two steps. First, the AS was shifted to a location between those two lenses with the closest distance to the aperture’s initial position. Then, the lens distance at its new position was fixed to 1 mm, and the system’s lens radii and distances between them were re-optimized to maintain the original object-sided telecentricity, working distance, spherochromatism, magnification, NA, and imaging performance. The criterion for the last item was that the wavefront errors remained below the Marechal criterion (RMS $\lambda /14$) over the entire as-designed FOV, with a diameter of 120 µm.
Figure 4(a) shows a cross section of the objectives from Olympus and Nikon before and after the radii optimization with an compensation plate located at the AS. The longitudinal chromatic aberrations, and their polychromatic, image-sided modulation transfer functions (MTF) for the F, d, and C line are depicted in Fig. 4(b)&c, respectively. All MTF plots contain tangential and sagittal MTFs for equally distributed points in the FOV. Figure 4(d) plots the extracted EP coordinate distortion, which is needed for the transfer of the wavefront error from EP to AS. From the longitudinal aberration and MTF curves, it can be seen that the modified models again have diffraction-limited performance for all FOV points in their as-designed state. Notably, the objective from Olympus shows strong pupil aberrations relative to the negligible ones from Nikon.
Fig. 4. Performance comparison of the microscope objectives used as case studies before and after re-optimization. (a) The cross-section view of the objectives from Olympus [19] and Nikon [20]. (b) Longitudinal chromatic aberration of the objectives are well control in cases. (c) Modulation transfer function (MTF) curves averaged over the FOV shows that re-optimized designs have a slightly better performance. d) The coordinate distortion between the AS and EP is much larger for the Olympus objective. Thus, pupil aberrations are a significant factor for this device when transferring the measured wavefront error from EX to AS.
In a next step, the compensation method was evaluated with 300 Monte Carlo (MC) simulations for both objectives, where we modeled the assembly tolerances to be typical for modern cell-mounted assembly techniques. Each sub-assembly’s borehole can have radial and axial run-out, as well as the change of their inner diameter, which allows each lens to experience in total 5 DOF: the decentering along $x-$, $y-$, and $z$-axes and tilt about $x$ and $y$-axes. These DOFs are depicted in Fig. 5, with their associated max. and min. tolerance values used for the MC simulations. There, each range corresponded to a normal tolerance distribution with a confidence interval of 95 % around the nominal value. The vendors of both systems do not provide any qualitative or quantitative information about the used assembly technology. For that purpose, we have oriented along sparsely available literature treating the mechanical tolerance of MOs and assumed along with [3] their ranges. In that publication, Frolov et al. treat the assemblies of the front parts of high NA MOs and their tolerances. We averaged those values and distributed them equally over all lenses within the optical train. Such equal distribution is rather unusual, however, to our best knowledge, we think that our assigned tolerance ranges averagely represent a good average model for the cell-mounted assembly procedure.
For each MC run, the pupil aberrations were first measured at the center of the FOV to be used in the remapping of the measured wavefront error from EP to AS. The field-independent system aberrations were then measured at the EP for the same on-axis point, and represented as a Zernike decomposition using up to the 7th radial-order at $\lambda _d$ of 587.5 nm. The Zernike coefficients were then translated using Eq. (1) to a sag profile of the compensation plate at the AS in a conjugated manner, while considering the pupil aberrations, as shown in the 3rd step of Fig. 1. The optical properties of the phase plate were chosen to be same as those of a 860 µm thin, transparent phase modulator recently developed by our group, since we plan to use this device for experimental demonstration of our method [21]. Therefore, the chosen refractive index was set to $n_d = 1.434$ with an Abbe-number of $\upsilon _d = 55.1$ at the same wavelength.
The effect of the tolerance compensation was evaluated by comparing the imaging performances of the objectives before and after the insertion of the compensating plate. The performance was quantified by image-sided, polychromatic MTF curves and the polychromatic wavefront RMS error. The latter was obtained by first calculating the RMS wavefront error for the F,d and C line at 20 by 20 equidistantly sampled points in the FOV, and then determining the mean of their RMS. To investigate the impact on the isoplanatic patch of the performed correction, we used two different sizes for the FOV. The first size had a diameter of 20 µm and the second as in the as-designed case of 120 µm.
The results of the evaluation are shown in Fig. 6 and 7. Figure 6 depicts overlaid sagittal and tangential MTF curves for both objectives for the two FOV ranges. The relative opacity of the shading in these figures can be interpreted as a probability density for the expected the MTF of an assembled system for a given FOV size. Figure 7 shows the distribution of the RMS wavefront error for the same evaluated FOV sizes and conditions.
Fig. 6. Overlaid sagittal and tangential polychromatic MTF plots of all of the 300 misaligned and compensated objectives from Olympus and Nikon. The middle and right column correspond to the FOV size indicated in their legend. The color depth of the curves can be interpreted as an indicator, where to expect the system’s performance after the assembly in their misaligned and compensated state.
Fig. 7. Histograms of the polychromatic RMS wavefront error averaged over 20 by 20 equidistantly sampled points in the indicated FOVs. The shown "Mean" and "Std." within the plot account for the corresponding histogram and FOV size, where ${\lambda } = {\lambda }_d$.
Figure 6 and 7 clearly indicate that without any compensation, the alignment errors have drastic impact on the imaging performance. In Fig. 6, the most probable systems end up with a cut-off frequency only $\leq$ 5 % of that of the as-designed case. Figure 7 presents the same behavior for both objectives, only this time in terms of a statistical distribution of the polychromatic RMS wavefront error over the FOV. There, in both cases the mean RMS error is $\approx$ 7x larger than the Marechal criterion which corresponds to a Strehl ratio of 0 over the entire FOV. These results indicate that the fabrication yield of both objectives without any compensation is expected to be close to zero, since the claimed design performance criteria are not fulfilled.
On the other hand, with phase plate compensation, the fabrication yield can be increased significantly. This can be observed in Fig. 6 from the MTF curves and in Fig. 7 from the RMS wavefront error for both FOV sizes. For all MC runs, the probability of acceptable performance drastically increases, since the MTF curves approach the diffraction-limit, with a major improvement in cut-off frequency. Also, the mean RMS wavefront error over the FOV and its standard deviation is also decreased. For the objective from Olympus, the mean RMS error changed from 0.541 ${\lambda }$ to 0.083 ${\lambda }$ and 0.090 ${\lambda }$ for a FOV size of 20 µm and 120 µm, respectively. For the Nikon objective, it changed from 0.590 ${\lambda }$ to 0.102 ${\lambda }$ and 0.127 ${\lambda }$ for equivalent FOV sizes. Moreover, the standard deviation for the both investigated objectives reduced. For Olympus from 0.239 ${\lambda }$ to 0.021 ${\lambda }$ and 0.021 ${\lambda }$, and for Nikon from 0.239 ${\lambda }$ to 0.044 ${\lambda }$ and 0.046 ${\lambda }$.
We also investigated the effect of the incorporation of pupil aberrations in the process, with the results depicted in Fig. 8. There the plot shows for each MC run the additional arising RMS wavefront error when transferring the wavefront from the EP to the AS, while incorporating pupil aberrations and not. For an objective design with considerable pupil aberrations, such as the Olympus one here, a remapping of measured aberrations from EP to AS, according to section 2.2 has a major influence on the resulting performance. For this system, the residual wavefront error is on average 0.142 ${\lambda }$ larger in RMS value than without an appropriate coordinate remapping. On contrary, this average is reduced below the Marechal criterion of 0.031 ${\lambda }$ with a remapping according to section 2.2. On the other hand, the improvement of the remapping is negligible for the Nikon objective, since this design has very small pupil aberrations.
Fig. 8. Influence of the incorporation of pupil aberrations into the compensation procedure as coordinate remapping for each MC run. The RMS error describes the additional error without and with incorporating coordinate remapping relative to the reference. The latter was determined at equidistantly sampled aperture stop for the on-axis FOV point.
Finally, we also investigated for both objectives the variation of the resulting compensating phase plate profiles. Figure 9 depicts the statistical distribution of different Zernike coefficients obtained by the decomposition of the optimum phase plate profiles for the 300 MC runs. As expected, all modes have a mean value of zero, which is due to the normal distribution of alignment tolerances over the simulation runs. For both objectives the 4th Zernike mode, which corresponds to defocus, has the largest variation, resulting from a sensitivity to axial positioning of the lenses. This type of sensitivity also induces error in the mode number 11 and 21, corresponding to first and second-order spherical aberration, respectively. However, they show advantageously a moderate sensitivity. The mode numbers 7,8 16,17,29, and 30 have the next largest contributions after defocus for both objectives, corresponding to first and higher-order linear coma.
Fig. 9. Statistical evaluation of the normalized Zernike sag coefficients of the compensation phase plates, where $m$ is the mean and $\sigma$ the standard deviation. The red error bars illustrate the occurred max. and min. values. Additionally, a sag profile of the phase plate is shown with its worst case peak-to-valley value $PV_{wc}$ which accounts only for the largest coefficients occurred during the Monte-Carlo simulation.
4. Discussion
The implementation of the method we discussed in this work can only be practical with a precise, rapid, and most importantly cost-effective fabrication technique of free-form phase plates. The shape fidelity and the roughness must be well controlled, since they are reflected one-to-one in additional induced aberrations. For example, according to Eq. (1) a RMS shape deviation of $\lambda$ of a certain Zernike mode, would induce a wavefront error of ${\lambda }\Delta n$ RMS.
There are a few promising precise fabrication techniques, which can be used for the feasibility tests of our method. The first one is an ion-exchange based method, where a structured metal mask is used to control the pattern and the diffusion depth of ions into a glass substrate, which locally changes the refractive index [22,23]. This technology is commercialized by SMOS-Smart Microoptical Solutions, Germany. The second one is a laser ablation method, where fused silica glass is ablated using a femtosecond laser scanned over the surface. To generate the desired surface profile, the laser power is modulated during the scan according to a given profile [24]. The technology is commercialized by PowerPhotonic Ltd., UK. Both methods are capable of manufacturing phase plates with the necessary range and precision, and therefore would suit for a proof-of-principle study.
Conventional objective assemblies do not in general provide an access aperture stop after assembly. Thus, in order to avoid any mechanical changes to the assembly before the insertion of the phase plate, the the objective barrel should be designed to allow a post-assembly insertion of the phase-plate.
In our analysis, we considered a FOV of 120 µm for both objectives. For larger FOV, the field-dependent aberrations will begin to dominate, limiting the attainable imaging quality significantly. To address these field-dependent aberrations, one could incorporate a second free-form phase plate at an off-pupil/off-aperture stop position to enable field-dependent aberration correction.
5. Conclusion
The need for active alignment is one of the main cost drivers for high-end microscope objectives. In this work, we demonstrated that a device-specific free-form phase plate located at the aperture stop can be an effective method for compensation of misalignment induced aberrations for such systems, potentially reducing their complexity while increasing their manufacturing yield.
Acknowledgments
The authors would like to thank Dr. Yuenqian Zhang, currently with Carl Zeiss AG, for sharing the ray-trace models of the two microscope objectives that he implemented for his dissertation.
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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