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Abstract
Wavefront aberration is one important parameter for objective lenses. When the NA (Numerical Aperture) of the objective lens becomes larger than 0.8, wavefront aberration measurement with high accuracy and low cost is difficult to realize because of the lack of a reference sphere. In this paper, a new method is proposed to measure the wavefront aberration of a high NA objective lens. A cat-eye retroreflector with a plane mirror is used to reflect the wavefront. The plane mirror is tilted in at least three different directions by certain tilt angles to collect sufficient information of the wavefront aberration under test. Specific grid-combined Zernike polynomial is built for each set of tilt angles and directions to fit the corresponding returned wavefronts. The wavefront aberration can be reconstructed from the fitting results of the returned wavefronts. The measurement accuracy is influenced by the tilt angle, tilt angle error, NA, defocus amount of the plane mirror, detector’s resolution, and other random noise. The tilt angle error is the main source of the measurement error. The relative measurement error is within 5% and 1% when the relative tilt angle error is below 0.5% and 0.1% respectively. The feasibility of the proposed method is verified experimentally by measuring the wavefront aberrations of 0.14 NA, 0.65 NA, and 0.9 NA objective lenses. Wavefront aberration measurement for a high NA objective lens with high accuracy and low cost is achievable through this method.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Wavefront aberration is a key factor to evaluate the imaging quality of the optical system. Zernike polynomial is chosen as the representation of the wavefront aberration in most occasions due to its orthogonality on a uniformed circular pupil and fine correspondence with typical aberrations [1]. For some special cases where the pupil is rectangular, Chebyshev and Legendre polynomials are frequently used for wavefront approximation [2]. Different interferometers are used to generate interfering patterns where the phase information of the wavefront aberration is obtained by certain algorithms. Typical interferometers include Fizeau interferometer and Twyman-Green interferometer, or Mach-Zehnder interferometer and Ronchi interferometer which use lateral shearing interferometry methods [3, 4]. Some sophisticated systems like ILIAS [5], MISTI [6], and Ipot [7] are designed for photolithography objective lens (NA can reach 1.35) measurement.
The star test is one of the earliest and easiest methods to measure the quality of the objective lens. This method analyses the image of a point source after the tested lens to give the aberration information. Nowadays, the deep-learning algorithm is used to extract aberration information more accurately with the help of computer technology. However, the improvement of test accuracy is still limited, because the deep-learning method is too sensitive to the training data and the input data [8].
As shown in Fig. 1, the most popular method to test an objective lens is placing a spherical mirror at the confocal position of the exit pupil. The input wavefront is reflected by the spherical mirror and returns the same way it comes. The returned wavefront contains two times the wavefront aberration of the objective lens, which can be detected by the Fizeau interferometer or Twyman-Green interferometer [3]. The spherical mirror can be convex or concave. But considering the short working distance of the high NA objective lens, the concave spherical mirror is usually the only option. The minimum F-number of the commercial spherical mirrors is 0.65, corresponding to NA 0.77, which limits the available testing range of the NA. Since the lithography objective lens system or the bright field microscope inspection system possesses an NA over 0.9 [5, 9], and the wavefront aberration scaling in nm RMS, using the traditional method will require a spherical mirror with the surface accuracy better than λ/40 (λ=633nm, PV) and NA over 0.9, making the test not only expensive but also limited by the accuracy of the mirror surface. In Juskaitis’ work [10], an experimental setup is proposed to measure the high NA immersion objective lens without using the spherical mirror to avoid the physical limitation.
If replacing the spherical mirror with a plane mirror directly, the plane mirror must be placed on the focal plane of the objective lens as Fig. 2(a), which means that the reflected wavefront will be rotated by 180°. The returned wavefront in this configuration contains no odd aberration component, but only two times the even aberration components of the objective lens wavefront aberration. The advantage of this method is its ability to test high NA objective lens’ even aberration at a lower cost. This configuration with a mirror and a focus system is also known as a cat-eye retroreflector system [11–15], which reflects the wavefront to its source with few scattering. This physical property inspires us to design a new method to solve the high NA objective lens test problem.
Fig. 2. Wavefront aberration measurement of the objective lens using a plane mirror. (a)Without tilt; (b) with tilt angle Δθ.
In this paper, a new method for a high NA objective lens wavefront aberration test is proposed using the cat-eye retroreflector system and Zernike polynomial fitting. The plane mirror with different tilt angles is used to obtain multiple returned wavefronts, which are then used to reconstruct the wavefront aberration of the tested objective lens by Zernike polynomial fitting. The new method has a simple measurement configuration and lower cost. Also, the testable range of NA is not limited by the components. Measurement result with a relative error below 10% is expectable via this method.
This paper is organized as follows. In Section 2, we present the principle and formulas to explain how this method works. In Section 3, simulations are performed to show the results in the idealized condition. In Section 4, the main error sources are analyzed. In Section 5, the experimental results using the proposed method and traditional method are compared.
2. Test method
The measurement configuration is shown in Fig. 2. The plane mirror is placed with different tilt angles and directions during the test to gather corresponding returned wavefronts. The returned wavefronts are collected by the Fizeau interferometer in this work. The requirement of the interferometer is discussed in Section 4.3. The grid-combined Zernike polynomials are built according to the propagation to fit the returned wavefronts. Finally, the wavefront aberration of the objective lens is reconstructed by the fitted Zernike coefficients. The ray-tracing illustrations when the mirrors are tilted in a random direction and X direction are given in Fig. 3 (a) and (b), respectively.
2.1 Grid-combined Zernike fitting method
According to the propagation shown in Fig. 3, the under-test wavefront aberration is flipped, shifted, stretched, and summed with the original wavefront aberration, then it is collected by the interferometer as the returned wavefront. By expanding the wavefront aberration to several Zernike terms, a series of linear equations can be established according to the relationship between the returned wavefront and the wavefront aberration, and the Zernike coefficients of the wavefront aberration can be solves from the equations.
If the under-test wavefront aberration Wlens can be represented by the first N terms of the Zernike polynomials, Wlens can be expressed as
where i represents polynomial term number, Ci is the i-th Zernike coefficient.In the proposed configuration, the returned wavefront W can be derived from the Wlens according to the ray-tracing illustrated in Fig. 3. It should be noted that Fig. 3 shows an equivalent geometrical relationship for the objective lens, and the actual ray-tracing could be more complex.
According to the ray-tracing given in Fig. 3, the returned ray through the point (xOUT, yOUT) carries the summed phase at (xOUT, yOUT) and (xIN, yIN). Thus the returned wavefront can be written as
2.2 Coordinate calculation
The key issue in the Zernike fitting method introduced above is deriving (xIN, yIN) from the known (xOUT, yOUT), which will be discussed in this section. The essence of this section is the use of ray analysis to obtain the relationship between the returned wavefront and the wavefront aberration of the tested objective lens.
In Fig. 3 (a), the focal point of the objective lens is K, and KC is perpendicular to the reflecting surface. The XMY plane is defined to help calculate the transformation between the plane and spherical wavefronts. Fig. 3 (b) shows a simpler situation where the mirror is only tilted in X direction and the definition of point M is clearer. The focal length of the objective lens is f. A(xA, yA) and B(xB, yB) are the intersections where the incident ray and reflected ray hit the X-Y plane. Fig. 3(b) shows the reflected ray in the AMK plane, which gives a better view of the geometrical relationship between POUT (xOUT, yOUT) and A(xA, yA). With the known point POUT (xOUT, yOUT), A(xA, yA) can be expressed as
2.3 Discussion
Eq. (6) shows how to build an equation with multiple returned wavefronts and solve the coefficients of the under-test wavefront aberration. At least three returned wavefronts corresponding to different tilt directions are required to ensure that sufficient information is collected. In this work, the mirror is not tilted or tilted in the X/Y direction for convenience purposes. Especially, when the plane mirror is not tilted, the returned wavefront can be expressed as
The proposed method is similar to the method using lateral shearing interferometry. The lateral shearing interferometry creates a copy of the test wavefront and then misaligns these two wavefronts by a certain shear amount to obtain the interfering pattern. By extracting the phase information from the interfering patterns, the test wavefront can be reconstructed with different reconstruction algorithms [16–20]. In the proposed method, the returned wavefront can be considered as the sum of the flipped, shifted, stretched wavefront aberration and the original wavefront aberration.
The Zernike fitting algorithm described in Section 2.1 can be considered as a modified version of DZF method [1]. Similarly, the orthogonality of the basis function does not influence the result’s accuracy when the N terms of Zernike polynomials used for fitting are sufficient to represent the under-test wavefront aberration [20]. The only difference is that the under-test wavefront is sheared differently in those two methods, and therefore the new Zernike basis (difference Zernike basis in the DZF method, and grid-combined Zernike basis in the proposed method) is constructed differently from the original circular Zernike basis.
By combining the cat-eye retroreflector and the modified Zernike fitting method, the proposed method can test the high NA objective lens with simple optical components. Similar to the DZF method, the proposed method should be applied in the application where the under-test wavefront aberration can be represented by finite Zernike polynomials. If this condition is not satisfied, the result could be inaccurate. Considering that the lateral shearing interfering method requires a sophisticated control system or high accuracy grating which needs to be custom design to get a proper shear amount, the proposed method is cheaper and easier to realize.
3. Simulation
Let the wavefront aberration of the objective lens in the simulation contains the first 100 Zernike wavefronts, and the coefficients are randomly assigned, then the wavefront aberration is generated as Fig. 4 shows. Assuming that the NA and focal length is 0.9 and 3mm respectively, the three returned wavefronts corresponding to the mirror orientations of no-tilt and 10° tilted in X/Y direction are given in Fig. 5. Only part of the reflected wavefront can pass through the objective lens again when the mirror is tilted, therefore the corresponding returned wavefronts are spindle-shaped (wide in the center and narrow on the ends). All the wavefronts and coefficients are scaled in wavelength in this paper.
Fig. 4. The wavefront aberration of the objective lens and its 1∼100th Zernike coefficients for the simulation.
Fig. 5. The different returned wavefronts when NA = 0.9. (a) No-tilt; (b) 10° tilted in X direction; (c) 10° tilted in Y direction.
Fig. 6 shows the relationship between the tilt angle and the interfering area’s width:
After building up the grid-combined Zernike polynomials and put them in Eq. (6) with the returned wavefronts in Fig. 5, the fitting result is calculated and shown in Fig. 7. The RMS error of the fitted wavefront is only 1.10×10−6 λ, which is negligible compared with the test wavefront shown in Fig. 4.
Fig. 7. NA=0.9, fitting result in the simulation. (a) Fitted wavefront; (b) Fitted wavefront error; (c) Fitted coefficient; (d) Fitted coefficient error.
Simulation for 0.1 NA objective lens is also performed, where the pupil size remains the same, but the tilt angle among X/Y direction is set to 1°. The simulation result is shown in Fig. 8. Similar to the result of the 0.9 NA simulation, the RMS error is only 1.15×10−6 λ in this case.
To quantitative evaluate the wavefront fitting error, the relative RMS error is defined as
where WF is the fitted wavefront generated by the coefficient solved from Eq. (6). Because Eqs. (2)∼(6) are linear operations, theoretically, the relative RMS error will not be influenced by the amplitude of the Wlens, which means that the absolute error is less for a smaller Wlens and larger for a bigger Wlens. Therefore the relative RMS error could be an appropriate parameter to evaluate the measurement error.According to Eq. (24), the relative RMS errors are 0.00015% and 0.00016% in the simulation when the NA is 0.9 and 0.1 respectively. Besides, another two groups of random wavefronts consist of 1∼16 Zernike terms and 1∼100 Zernike terms are simulated. Each group contains 100 wavefronts with random Zernike coefficients. The RMS values of the random wavefronts vary between 0.05∼0.5 λ, and the relative RMS errors have the same magnitude as the results in Fig. 7 and Fig. 8, which proves the feasibility of this new method under idealized conditions. If not specially explained, the “fitting error” mentioned below corresponds to the relative RMS error defined in Eq. (24).
4. Error analysis
During practical applications, the grid-combined Zernike polynomial is derived by the tilt angle measured from phase shape or read from rotation stage. Therefore the tilt angle accuracy will directly influence the results solved from Eq. (6). Besides, the deviation between the plane mirror and the focal point, or the system’s stability will also cause errors in the test results. The tilt angle error and the defocused mirror are systematical errors, which can be analyzed by numerical simulations.
4.1 Tilt angle error
First, the tilt angle ratio Rθ and relative tilt angle error ΔθR are defined as
where θmax is the maximum tilt angle, Δθ is the tilt angle error, and θ is the tilt angle of the mirror.For the objective lens under test with different NA values, the shapes of the returned wavefronts are approximately the same once the tilt angle ratio Rθ and relative tilt angle error $\Delta {\theta _R}$ are confirmed. Therefore the conclusion based on Rθ and ΔθR can be applied to different NA values.
According to Fig. 6, the theoretical range of Rθ is from 0 to 1, but the interfering area could be too small when Rθ is close to 1, which will reduce the information contained in the returned wavefront. Besides, the short working distance (1 mm level or less) of a high NA objective lens makes it infeasible to place the plane mirror if Rθ is close to 1, which could damage the test lens and reduce the accuracy. For a high NA objective lens, the practical feasible Rθ is usually below 0.25.
In this section, the lateral resolution of the interfering area is set to 256${\times} $256 pixels. Additionally, if the tilt angle error is below zero, the selected interfering area will be larger than the actual interfering area. In this case, invalid data values outside the actual interfering area are involved during the fitting procedure. The negative relative tilt angle error can be avoided and is no longer considered in this section.
In the following simulation, the typical Zernike wavefronts’ error response to the tilt angle error is analyzed by assuming that the test objective lens’s aberration only consists of the chosen Zernike term whose coefficient is set to 1λ. To find the relationship between test error and tilt angle error more clearly, only three different tilting statuses are used to collect different returned wavefronts: no tilt, tilt in X (horizontal) direction, and tilt in Y (vertical) direction. The tilt angle error is introduced only when the mirror is tilted in X/Y directions, and the relative tilt angle error is between 0.1∼0.5%. The propagation of the wavefront follows the ray-tracing in section 2. Under such simulation conditions, several typical Zernike wavefronts’ fitting errors are simulated and shown in Figs. 9–12. From the simulation results, the following conclusions can be summarized:
- 1. According to Fig. 9, the fitting error is proportional to the relative tilt angle error ΔθR. Fitting error is larger when ΔθR increases;
- 2. According to Fig. 10, the Zernike wavefront with a higher radial polynomial order also has a higher fitting error under the same relative tilt angle error ΔθR. As shown in Fig. 10, the fitting errors of the first 16 Zernike wavefronts are below 2%, while the fitting error of the 100th Zernike wavefront is 5∼6%. Fig. 11 compares the fitting errors of those Zernike wavefronts which correspond to the spherical aberrations and shows the same pattern. The reason is that the spatial frequency of the Zernike wavefront increases as the polynomial order increases, and makes the high-order Zernike wavefront more sensitive to the tilt angle error.
- 3. The ideal range of Rθ is 0.15∼0.3 when the test wavefront aberration mainly comprises typical Zernike wavefronts Z5∼Z16. As shown in Fig. 12, the tendencies of different Zernike wavefronts fitting errors are different: The fitting errors of astigmatism Z5 and Z6 are always below 0.5% and 0.02%, respectively; Z7/10/14’s fitting errors decrease as Rθ increases, while Z9/13/16 show opposite patterns. The fitting errors of Z5∼Z16 are all relatively small and steady when the Rθ is between 0.15∼0.3, where the relative tilt angle error ΔθR is the main cause of the fitting error.
Fig. 10. Typical Zernike wavefronts’ fitting error under various tilt angle ratios when the relative tilt angle error is 0.15%.
Fig. 11. The fitting error of Zernike wavefronts corresponding to spherical aberrations when the relative tilt angle error is 0.1%.
Fig. 12. Typical Zernike wavefronts’ fitting error under different tilt angle ratio and relative tilt angle error.
Usually, the adjusting accuracy of the tilt angle can reach $5 \times {10^{ - 6}}$ rad and therefore the main source of the tilt angle error is introduced when setting the initial tilt angle to zero. Judging from the interfering area’s shape is a simple way to distinguish whether the tilt angle is zero or not. In this way, the tilt angle measuring accuracy is determined by the angle of the pixels at the edge of the interfering area. According to Fig. 13, the field angle changes as the resolution changes. The angle of the edge pixel under different NA values and resolutions is illustrated in Fig. 14(a), which shows that the angle with higher NA is larger. When the resolution is 256${\times} $256 pixels and NA=0.9, the angle of the edge pixel is about 0.9°, while the angle of 0.1 NA is about 0.05°. Since the changing amount of the ray direction is two times the changing amount of the tilt angle, the maximum tilt angle error should be half of the angle of the edge pixel, whose corresponding ΔθR is given in Fig. 14(b). Therefore the higher NA value causes a higher relative tilt angle error ΔθR. For example, the ΔθR when NA=0.9 is 1.75 times the ΔθR when NA=0.1. Correspondingly, the fitting error when NA=0.9 should be 1.75 times the error when NA=0.1 according to Fig. 9.
Fig. 14. The field angle of the pixel at the edge and the relative tilt angle of the half field angle under different NA values and resolutions. (a) The field angle of the pixel at the edge; (b) relative tilt angle of the half field angle.
Diffraction effect is a common issue existed in optical test and has been discussed in existing work [21]. In the proposed method, the diffraction effect can be reduced by adjusting the interferometer’s imaging plane. The proposed method uses the Zernike polynomial to fit the aberration, which is not sensitive to the diffraction fringe as shown in section 5.
The results in Fig. 15 are obtained under the condition where the lateral resolution is 256${\times} $256 pixels, Rθ=0.23, and the tilt angle is half of the edge pixel’s field angle. The fitting error of the Zernike wavefront increases as NA increases, and the fitting error of 0.9 NA is about 1.3∼1.7 times the error of 0.1 NA. The fitting errors of typical Zernike wavefronts are all below 8%. The ratio of the fitting error when NA=0.9 to the error when NA=0.1 is 1.3∼1.6, which is consistent with the previous discussion. The tilt angle error can be reduced by using a high-resolution detector, choosing high accuracy angle adjusting device, or setting the tilt angle to a proper range.
Fig. 15. The fitting errors of the typical Zernike wavefronts when the tilt angle error is half field angle of the pixel at the edge and Rθ=0.23.
To analyze the influence of the tilt angle error when the under-test wavefront consists of multiple Zernike terms, a test group of 100 wavefronts is generated by assigning random Z1∼16 coefficients. The coefficients are given in Fig. 16, where each column represents the Z1∼16 coefficients of a corresponding wavefront. The relative errors when ΔθR is 0.1% and 0.5% are shown in Fig. 17, where the lines represent average values, and the blocks represent the distribution of the values. The NA is set to 0.9 during the simulation. The relative RMS errors are around 1% and 7% when the relative tilt angle errors are 0.1% and 0.5%, respectively. The minimum relative RMS error appears when Rθ=0.3. The result is consistent with the previous discussion.
Fig. 16. Coefficients of 100 randomly generated wavefront. Each column represents the Z1∼16 coefficients of a corresponding wavefront.
Fig. 17. Relative RMS error of the random wavefronts under different tilt angle ratio and relative tilt angle error when NA=0.9. Line is the average value. Block is the distribution area of the values.
4.2 Defocus error
If the deviation amount between the mirror and the focal point is ΔL, the ray path changes as Fig. 18 shows. If the mirror is not tilted, the ray path will be like the one given in Fig. 18(a). When the mirror is tilted in the X direction, the ray path is changed to the one in Fig. 18(b). No matter the mirror is tilted or not, the convergence point of the reflected wavefront is always shifted along the normal by distance KK’, which means that the defocused mirror will introduce power aberration (Z4) to the reflected wavefront. If the tilt angle is θ, the coefficient of the introduced Z4 term will be [22]
Fig. 18. The ray path change caused by mirror defocusing. (a) Observing along the X-axis when the mirror is not tilted; (b) observing along the Y-axis when the mirror is tilted in the Y direction.
4.3 Polarization Effect
The reflection on the mirror may change the phase and light intensity distribution of the wavefront according to Dubois’s work [23]. The phase change corresponding to different incident angles is ignorable even for the metal mirror [23]. Thus the main consideration is focused on the uneven intensity distribution caused by the different reflection index values for different polarization statuses. This effect could be removed with a polarizer (or polarizing beamsplitter) and a 45° rotated quarter-wave retarder.
For example, Fig. 19 shows a combination of a polarizing beamsplitter and a quarter-wave retarder used in classic Fizeau interferometer, where other components are omitted since they do not affect the polarization status of the wavefront. By using VirtualLabs Fusion to simulate the wavefront propagating between the interferometer and the test setup shown in Fig. 2(b), the polarization and electrical field amplitude of the wavefront are obtained and listed in Fig. 19(a)∼(e). The NA of the objective lens is 0.9, assuming no aberration existed. The plane mirror in the test setup is made of fused silica, which introduces no extra phase shift to the reflected wavefront except half-wave lost according to the Fresnel equation [24].
Fig. 19. Polarization status illustration using the proposed method and classic Fizeau interferometer. Electrical field distributions in X/Y directions are listed corresponding to wavefront 3∼5. The ellipses and short lines in the electrical field distributions denote different polarization statuses, and the black and white colors indicate different polarizing rotations. The values near the colorbar indicate relative amplitude intensity. (a)∼(e) are electrical field distributions in x/y polarization direction simulated by VirtualLabs.
The output wavefront from the interferometer is circularly polarized. After being reflected as shown in Fig. 2(b), the polarization and electrical field amplitude of the wavefront are changed. As shown in Fig. 19(a) and Fig. 20, the polarization status and the electrical field amplitude become uniformed after passing the polarizing beamsplitter again, which proves that the reflection in the proposed method will not cause phase error or interfering pattern error.
Fig. 20. Simulation results of the wavefront after passing through the beam splitter again in Fig. 17(a). (a) Light intensity; (b) phase.
According to the simulation above, the proposed method requires a interferometer with circular polarized output to avoid the error introduced during the reflection.
4.4 Random error
As shown in Fig. 5, the three returned wavefronts’ PV and RMS values are in the same order of magnitude as the test wavefront, because the returned wavefront in this method is generated by adding different pairs of locations on the test wavefront instead of taking wavefront derivation like the lateral shearing interferometry. Thus, the test repeatability of this method is approximately the same as the interferometer, which means that the random error in the returned wavefront will not be amplified in the final result. This is one significant difference between the proposed method and lateral shearing interferometry.
4.5 Summary
The possible error sources in this method are listed in Table 1. The main error source is the tilt angle error. If the relative tilt angle can be controlled within 0.5% (0.0056 rad when NA=0.9, 0.0005 rad when NA=0.1), the single Zernike wavefront’s fitting error will be less than 12%; If the relative tilt angle is below 0.1% (0.0011 rad when NA=0.9, 0.0001 rad when NA=0.1), the single Zernike wavefront’s fitting error will be less than 2%.
Table 1. Possible error sources and solutions
Mirror defocus will introduce defocus aberration in the returned wavefront. When the interfering pattern is close to zero-fringe status, this influence will not change the shape of the interfering area and can be eliminated by removing the defocus aberration in the returned wavefront.
The repeatability of the interferometer is a key factor of the test accuracy, which influences the test result randomly. This error will not be amplified in the test result and can be compensated by taking the average of multiple measurements.
The smoothness of the reflection surface could also affect the test result. If there is a scratch in the reflecting area, the interfering pattern will be significantly blurred. This effect is difficult to simulate because the reflecting direction is random when the light hits the scratch. Besides, this can be avoided by simply moving the scratched area away.
5. Experiments
In this section, we test three different objective lenses listed in Table 2. Objective lenses A (0.14 NA) and B (0.65 NA) are tested using our proposed method and the traditional method using a spherical mirror. The tested results of the two methods are compared. Objective lens C (0.9 NA) is tested using the proposed method only. The experimental setup using the proposed method is shown in Fig. 21. The irregularities of the plane mirror and the spherical mirror’s reflective surfaces are both less than 1/20λ PV (λ=632.8nm), which are negligible for the experiments. To increase stability, the test results will be calculated using five returned wavefronts, which correspond to no-tilt, tilt left and right in horizontal (or X) direction, tilt up and down in vertical (or Y) direction. Besides, the Zygo Fizeau interferometer installed in our experiment uses the phase-shift method to extract phase data from the interfering patterns, which can obtain stable results under unevenly distributed light intensity [25].
Table 2. Parameters of tested objective lenses
5.1 Objective lens A (0.14 NA) measurement
The wavefront aberration of objective lens A tested by the spherical mirror method is shown in Fig. 22 (a). The corresponding fitted wavefront using 1∼100th Zernike wavefronts and fitted error are given in Fig. 22 (b)(c), respectively. After Zernike fitting, the high-frequency noise introduced by dust and diffraction at the edge is removed. The relative RMS error of the fitted phase compared with the raw phase data is about 8.3%.
Fig. 22. Spherical mirror test result of objective lens A. (a) Raw phase data obtained from interferometer; (b) fitted phase using 1∼100th Zernike wavefronts; (c) difference between (a) and (b).
By setting the plane mirror’s tilt angle to 1° and 2° in different directions, a series of returned wavefronts are obtained as shown in Fig. 23. There are 16 different combinations to pick four tilt returned wavefronts and the no tilt returned wavefront to calculate the test results, which are all listed in Fig. 24 (a). Also, the differences between the results in Fig. 24 (a) and Fig. 22 (b) are given in Fig. 24 (b), and the corresponding relative RMS errors are plotted in Fig. 32.
Fig. 23. Phases of returned wavefronts when testing objective lens A. XL/XR means the plane mirror is tilted left/right in X direction. YU/YD means the plane mirror is tilted up/down in Y direction. The corresponding tilt angle is read from the rotation stage and given in the bracket.
Fig. 24. Test results of objective lens A using different returned wavefronts. For example, the first result on the left top uses the following returned wavefront listed in Fig. 23: No Tilt, YU1, YD1, XL1, and XR1. (a) Test Result; (b) difference between (a) and Fig. 22 (b).
5.2 Objective lens B (0.65 NA) measurement
The procedures of testing the objective lens B are the same as the procedures described in section 5.1 except that the tilt angle is set to 5° and 10° in different directions. The returned wavefronts corresponding to each tilt angle are shown in Fig. 26. The test results obtained by using the spherical mirror and plane mirror are given in Fig. 25 and Fig. 27 (a), respectively. The relative fitted error of the spherical mirror test result is about 4.4%. The relative RMS error of the results in Fig. 27 (a) compared with the reference wavefront in Fig. 25 (b) is given in Fig. 32, and the wavefront difference is given in Fig. 27 (b).
Fig. 25. Spherical mirror test result of objective lens B. (a) Raw phase data obtained from interferometer; (b) fitted phase using 1∼100th Zernike wavefronts; (c) difference between (a) and (b).
Fig. 27. Test results of objective lens B using different returned wavefronts. (a) Test results; (b) differences between (a) and Fig. 25(b).
5.3 Objective lens C (0.9 NA) measurement
Since there is no such spherical mirror that corresponds to 0.9 NA, the reference wavefront for the objective lens C’s test results to compare with is generated using all returned wavefronts listed in Fig. 28. The calculated reference wavefront is shown in Fig. 29. Just like the procedures done in sections 5.1 and 5.2, there are 16 different combinations of returned wavefronts are used to calculate the wavefront aberration results of objective lens C, which are listed in Fig. 30 (a). The difference wavefronts corresponding to each test result compared with the reference wavefront are given in Fig. 30 (b). The relative RMS errors of the test results compared with the reference wavefront in Fig. 29 are given and discussed in the next section.
Fig. 28. Phases of returned wavefronts when testing objective lens C. Corresponding tilt angle is calculated from Eq. (23) and given in the bracket.
Fig. 29. Test result of objective lens C. All returned wavefronts are used. This wavefront is also used as a reference wavefront.
Fig. 30. Test results of objective lens C using different returned wavefronts. (a) Test results; (b) differences between (a) and Fig. 29.
The corresponding interfering patterns of some returned wavefronts are shown in Fig. 31. The size of the circular mask is decided by the pupil diameter given in Table 2. By adjusting the focus of the interferometer, the edge of the pattern is clear enough for tilt angle calculation. This also proves that the diffraction effect is not a major concern.
5.4 Summary
The relative RMS errors of the test results in Fig. 24 (b), Fig. 27 (b), and Fig. 30 (b) are shown in Fig. 32 and Table 3. The average relative RMS errors of the proposed method compared with the traditional method using a spherical mirror are 13.9% and 16.4% when testing objective lens A (0.14 NA) and B (0.65NA), respectively. The relative RMS error varying from 8.5% to 21.4% during the comparison proves that the proposed method is able to obtain a convincing result and has acceptable stability.
Fig. 32. Relative RMS error of different objective lenses test results. Result number indicates the position in Fig. 24 (b), Fig. 27 (b), and Fig. 31 (b). For example, 1 means the left top result, 2 means the result to the right of result 1, etc.
Table 3. Relative RMS error of different objective lenses test results
Due to the limitation of the experiment condition, the tilt angles cannot be exactly configured. Tilt angle error will cause deviation to the test result as discussed in section 4.1. Besides, the insufficient stability of the adjusting structure could cause misalignment in an unpredicted way, which could also lower the accuracy. The systematic error of the interferometer is another potential error source.
When the proposed method is used to test objective lens C (0.9 NA), whose NA is too high to find a suitable spherical mirror to get a reference result, the results using 5 returned wavefronts are compared with the result using all of the returned wavefronts to analyze the repeatability. The relative RMS error of objective lens C’s test results varies from 13.7% to 23.3%, and its average value is 19.4%.
Judging tilt angle from the shape of the tilt phase could also introduce error. Therefore it is harder to guarantee the tilt angle accuracy when testing 0.9 NA objective lens in the experiments.
It can be seen from Fig. 32 and Table 3 that the relative RMS error is larger when the NA of the under test objective lens is higher. This is mainly caused by the tilt angle error, which is increasing as the tilt angle setting value is increasing due to the experimental limitation. It can be expected to get a better result through the proposed method by increasing the accuracy of the tilt angle adjustment. For example, an array of reflective surfaces with different orientations could be pre-fabricated on a board with high accuracy, then different returned wavefronts could be collected by simply shifting the board to put different reflective surfaces at the focal point. Theoretically, this configuration could be used to test the immersion objective lens by covering the board with the immersion liquid.
During the experiments, we noticed that it is difficult to put the spherical mirror to the exact confocal position when testing the objective lenses B and C due to their high NA design. It takes a lot of effort to align the mirror and get the correct interfering pattern. However, it is easy to get the plane mirror intersected with the focal point in the proposed method, which proves the convenience of the proposed method.
6. Conclusion
In this paper, a new method to test the objective lens with cat-eye retroreflector and Zernike polynomial is proposed. By reflecting the converging wavefront with the plane mirror in more than three groups of different tilt angles and directions, the wavefront aberration of the test objective lens can be reconstructed by applying the least-square fitting to the returned wavefronts. Objective lenses with different NA values are tested using the proposed method, and the test results are compared with the result obtained using a spherical mirror when testing objective lenses A (0.14 NA) and B (0.65 NA). The comparison proves that the proposed method is able to obtain convincing and stable results. The test result of objective lens C (0.9 NA) shows the ability of the proposed method in the high NA objective lens test.
The relative RMS error is the appropriate parameter to evaluate the performance of this method. Tilt angle error is the key factor that influences the accuracy of the final test result. If the relative tilt angle error of the plane mirror is less than 0.5%, the single Zernike wavefront fitting error will be less than 5%; if the tilt angle error is below 0.1%, the single Zernike wavefront fitting error will be less than 1%. Besides, the accuracy can be further improved by using a detector with a higher lateral resolution or increasing the accuracy of the rotation stages.
Funding
National Science and Technology Major Project (2017ZX02101006).
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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