A Model-Based Approach for Measuring Wavefront Aberrations Using Random Ball Residual Compensation

Abstract

The projection objective lens holds a pivotal role in lithography, directly influencing imaging system quality and, consequently, the lithography machine’s feature dimensions. Optical inspection methods for this lens require advancements in calibrating systematic error and enhancing alignment precision of auxiliary devices, given their impact on calibration accuracy. In the random averaging method, random ball can give rise to additional wavefront aberrations due to misalignment and numerical aperture mismatch. To mitigate these aberrations and enhance the accuracy of systematic error calibration, this paper introduces a random ball residual compensation (RBRC) model. Additionally, when combined with the random averaging technique, it elevates the calibration accuracy of the measured lens’s wavefront aberrations. The experimental results underscore the method’s effectiveness, accurately determining optical component eccentricities and numerical aperture errors. After eliminating these errors, more accurate values of lens wavefront aberrations are achieved. This research significantly contributes to enhancing error calibration of lithography objective lens systems.

1. Introduction

The projection lens is a core component of lithography machines, and the high-precision measurement of wavefront aberrations in the projection lens is a crucial step affecting the imaging quality [1]. In the modern microelectronics industry and lithography technology, optical wavefront sensing plays a critical role in projection optical systems. This is especially true in the era of low-k1 lithography, where the calibration of wavefront aberrations in projection optical systems becomes particularly important. As the numerical aperture (NA) of the projection optical system approaches 1.0 or greater, the wavefront aberrations need to be precisely controlled within a smaller range to ensure imaging quality and accuracy of feature sizes [2]. Therefore, developing a high-precision and efficient wavefront sensing system to measure wavefront aberrations in projection optical systems has become an urgent task.

Currently, commonly used methods for high-end photolithography machines include point diffraction interferometers, Hartmann sensors, and lateral shearing interferometers [3,4,5]. For optical systems with small and moderate numerical apertures, these methods have been successfully validated. However, for optical systems with large numerical apertures, the overall systematic error within the optical path significantly impacts wavefront aberration detection. Issues such as the presence of numerous components and multiple diffraction orders can contribute to this effect [6,7,8]. Ray et al. [9,10] primarily focused on diffraction interference of the 0th and ±1st orders. However, this limitation only provides a restricted amount of interference information, which hampers its applicability in high-precision measurements. These studies failed to encompass higher-order diffraction interference, which can often lead to limitations in measurement accuracy, particularly within complex optical environments. As a result, these investigations are unable to meet the demand for high-precision measurements in real-world scenarios, emphasizing the pressing need for more comprehensive and refined interference models to achieve a higher level of precision.

The Shack–Hartmann wavefront sensors are widely used for wavefront detection. They offer flexibility by allowing the measurement of aberrations using either plane waves or spherical waves [11]. For example, in optical environments such as astronomical imaging, combining algorithms with complex Shack–Hartmann wavefront sensors can produce clear images [12]. Additionally, the Shack–Hartmann wavefront sensor, when integrated with adaptive optics technology, enhances image information for biological samples in optical microscopes [13]. In the context of compensating for ocular aberrations, Shack–Hartmann wavefront sensors play a crucial role. They assist in measuring and compensating for high-order aberrations and scattered light in the human eye [14,15]. However, it is worth noting that Shack–Hartmann sensors have limited fields of view. They can only detect wavefront for targets near zero-degree viewing angles [16]. Furthermore, data processing for lateral shearing interferometers is complex, computationally intensive, and requires high-end hardware [17]. In summary, achieving a simple, flexible, and precise removal of inherent systematic error in wavefront measurement is of paramount importance to the entire optical measurement industry. The achievement of a simple, flexible, and accurate removal of inherent systematic error from wavefront measurement holds immense significance for the entire optical measurement industry.

The presence of eccentricity error can impact the imaging quality of optical systems. For optical lenses, due to their inherent rotational asymmetry, this can result in various aberrations such as astigmatism, coma, and distortion [18]. In lithography systems, the projection lens is susceptible to external environmental factors during the exposure process, including variations in temperature and air pressure, as well as vibrations from the surroundings. These external factors can lead to deviations in the optical elements, causing them to deviate from their ideal positions. Such deviations can have a detrimental effect on the performance of the optical system, subsequently affecting imaging quality and precision [19,20,21].

To achieve higher precision in wavefront aberration measurements, it is necessary to subtract the system’s inherent errors from the measurement results. Parks et al. demonstrated the feasibility of the random ball method for systematic error calibration [22,23,24]. Ping argued that this method is easy to implement and does not require additional optical components. Averaging multiple measurement values effectively reduces random noise in experiments and improves measurement accuracy [25]. Griesmann and colleagues [26] developed a robotic arm for conducting random ball experiments, verifying the randomness of the random ball, and conducting experiments on the influence of the number of random iterations on the results. Hou et al. proposed a theoretical analysis model based on dynamic random ball testing. Using the dynamic random ball testing method, it is possible to achieve higher precision measurements of optical elements with low precision reference surfaces [27]. Because the Random Ball Test offers high measurement precision, Soons compared it to shear testing and obtained satisfactory results [28]. Zhou et al. mentioned that the random ball test can also be used to calibrate slope-dependent errors in profilometers such as the Scanning White Light Interferometer (SWLI) [29].

In this study, the random ball method with the utilization of the random ball residual compensation (RBRC) model is employed for high-precision calibration of lens wavefront aberrations. The specific methodology is as follows: Firstly, in a single-channel experiment, a lens with a specific curvature radius is used, and initial systematic errors are measured using the random averaging technique. Subsequently, the lens is replaced by the lens under test to measure its wavefront aberration. Next, while keeping the lens position the same, the wavefront sensor at the rear of the single-channel optical path is replaced with a high-precision reflector. The wavefront aberration of the lens is measured using a dual-channel measurement approach. The difference between the two measurements is used as the dataset for the least-squares model optimization objective. This allows the determination of the wavefront error caused by eccentricity and numerical aperture mismatch during measurements, achieving error subtraction for higher-precision wavefront aberration calibration. This model significantly enhances the flexibility in designing the random ball, features a straightforward data processing approach, and mitigates eccentricity errors even in low-precision alignment conditions.

2. Theoretical Analysis

Because the system error in single-channel measurements is large, the model proposed in this paper aims to reduce the aberrations of the objective lens in single-channel measurement mode. The measurement of the random ball in single-channel mode and the measurement of the objective lens are shown in Figure 1. The optical path for the random ball setup is shown in Figure 1a. Inside the 4D interferometer, there is a built-in light source with a wavelength of 632.8 nm. The random ball device contains a small ball lens (R = 20.00 mm) for testing purposes, and at the bottom of this ball lens, there are three additional small ball lenses that allow for random rotation of the small balls. The light from the 4D interferometer passes through the random ball and enters the SID4 wavefront sensor. After image reconstruction, wavefront aberration data can be obtained. After completing the random ball measurement, the random ball is replaced with the objective lens for another measurement. The specific optical path is shown in Figure 1b. The model data is obtained from the residual single-channel data and dual-channel data.

In this method, the numerical aperture (NA) of the random ball should be ensured to match that of the tested lens, which can minimize the error caused by NA mismatch. However, in the model presented in this paper, this error can be accurately removed. The principle of this method is shown in Figure 2, where all the images are represented in the form of phase maps. Data obtained from both single-channel and dual-channel measurements are processed to calculate the residual, which serves as the model data and is used as the optimization target for the Zernike Fringe coefficient. This process yields the calibrated wavefront error. Finally, this error is subtracted from the wavefront of the test mirror, resulting in the calibrated wavefront of the objective lens.

A.

Random averaging method

In the actual processing of the ball lens, there must be an error of uneven surface shape. By randomly rotating the spherical lens N times, N wavefront maps can be obtained, and the average value can be expressed as:

$$\overline{W}=W_{\mathrm{system}}+W_{SPH}+W_{i,S}+\mathsf{\Delta }W$$

(1)

where $W_{\mathrm{system}}$ is the systematic error containing a light source and sensor information, $W_{\mathrm{S}\mathrm{P}\mathrm{H}}$ is the inherent spherical aberration of the globe lens, $W_{i,S}$ is the errors due to surface non-uniformity are obtained, where i represents the number of random rotations, and as i approaches infinity, the random error tends to zero, $\mathsf{\Delta }W$ represents the other errors of the random ball. $W_{SPH}$ and $W_{i,S}$ can be subtracted by the random average method. Finally, $\overline{W}$ can be expressed as:

$$\overline{W}=W_{\mathrm{system}}+\Delta W$$

(2)

As shown in Figure 3a,b, the ball lens is replaced with the tested plano-convex lens. The optical path error of the tested mirror $W_{\mathrm{lens}}^{\prime }$ in this single-channel measurement can be expressed as:

$$W_{\mathrm{lens}}^{\prime }=W_{\mathrm{system}}+W_{\mathrm{lens}}$$

(3)

Based on the systematic error calibration results mentioned above, the objective lens result is ultimately represented as:

$$W_{\mathrm{lens}}=(W_{\mathrm{lens}}^{\prime }-\overline{W})+\Delta W$$

(4)

B.

High-precision measurement

To validate the accuracy of the single-channel mode measurements, the wavefront aberration of the plano-convex lens is additionally measured using a high-precision 4D interferometer in dual-channel mode. The 4D interferometer used in the experiments described in the paper is the PhaseCam^® 6100, which is a highly compact and dynamic laser interferometer designed for the measurement of optical components and optical systems. It utilizes a high-speed camera and laser source ($\mathsf{\lambda }=632.8\mathrm{nm}$) for wavefront measurements, providing precise information about optical elements, including wavefront shape and aberrations. Typically, 4D interferometers have rapid data acquisition capabilities, making them well-suited for applications with real-time requirements.

We consider that the position and angle of the tested objective lens stay the same, and only the SID4 wavefront sensor is replaced with a high-precision reflecto. $W_{\mathrm{lens-double}}^{\prime }$ is the wavefront maps in dual-channel mode.

$$W_{\mathrm{lens-double}}^{\prime }=W_{\mathrm{system}}^{\prime }+W_{\mathrm{lens}}$$

(5)

where $W_{\mathrm{system}}^{\prime }$ is the systematic error of the light source and the wavefront sensor, $W_{\mathrm{lens}}$ is the errors of the tested objective.

Because the dual-channel measurement mode uses a high-precision reflector where systematic errors are considered to be approximately zero, i.e., $W_{\mathrm{system}}^{\prime }\approx 0$, the dual-channel results can be approximated as:

$$W_{\mathrm{lens-double}}^{\prime }=W_{\mathrm{lens}}$$

(6)

C.

Model analysis

We believe that in both single-channel and dual-channel wavefront measurements, the position and angle of the objective lens stay the same. Therefore, the residuals of the two channels’ measurements should be zero. The residual $\mathsf{\Delta }W$ can be expressed as:

$$\mathsf{\Delta }W=W_{\mathrm{lens-double}}^{\prime }-(W_{lens}^{\prime }-W_{system})$$

(7)

If the value of $\mathsf{\Delta }W$ is non-zero, it considers that some of the systematic errors in the single-channel measurement have not been fully removed. The remaining systematic error is caused by the eccentricity and NA mismatch between the random ball and the objective lens. These two slight variations will cause a large number of changes in the wavefront measurement results, such as low-order astigmatism and spherical aberration [30].

We think that the value of $\mathsf{\Delta }W$ is actually a combination of Zernike Fringe coefficients, reflecting the residual tilt, coma, spherical aberration, and other aberrations in the measurement results. It can be expressed as:

$$\mathsf{\Delta }W=\{Z_1,Z_2,\cdots ,Z_{37}\}=\mathsf{\Delta }{W}_{NA}+\mathsf{\Delta }{W}_{\mathrm{Center}}$$

(8)

where $\mathsf{\Delta }{W}_{NA}$ represents wavefront aberrations caused by numerical aperture mismatch, and $\mathsf{\Delta }{W}_{\mathrm{Center}}$ corresponds to wavefront aberrations caused by eccentricity.

As the optimization objective of the model, $\mathsf{\Delta }W$ is minimized using Zernike coefficients as targets through a least squares optimization process. This yields the parameters for the eccentricity and NA mismatch of the globe lens, allowing the calculation of the wavefront aberration caused by eccentricity and NA mismatch during the measurement. The final expression for the tested objective can be represented as:

$$W_{\mathrm{lens-}final}=W_{\mathrm{lens}}^{\prime }-W_{system}+\Delta W$$

(9)

The error caused by the mismatch between the numerical aperture (NA) of the spherical lens and the objective lens $\mathsf{\Delta }{W}_{NA}$ can be expressed as

$$\mathsf{\Delta }{W}_{NA}=W_{NA-20}-W_{NA-17}$$

(10)

where $W_{NA-20}$ represents the wavefront data of the numerical aperture error of experimental spherical lens, and $W_{NA-17}$ represents the wavefront data of the numerical aperture error of ideal spherical lens.

As shown in Figure 4a, the black dashed part represents the ideal alignment measurement position in the XOY plane, while the red dashed part represents the eccentric position that occurs in the experiment due to incomplete alignment. In Figure 4b, the gray area represents the aligned positions of the random ball in the YOZ plane, while the blue area indicates positions with eccentricity. Since the eccentric displacement is small, the wavefront residual generated by the experimental eccentricity in simulation can be equivalent to the wavefront residuals due to X, Y, and Z-axis displacements from the central position. This is denoted as $\mathsf{\Delta }{W}_{\mathrm{Center}}$. It can be expressed as:

$$\mathsf{\Delta }{W}_{\mathrm{Center}}=W_{\mathrm{Zernike}}^C-W_{\mathrm{Zernike}}^I$$

(11)

where $W_{\mathrm{Zernike}}^C$ represents the wavefront data of the simulation beam after experiencing displacement, $W_{\mathrm{Zernike}}^I$ represents the wavefront data of the simulation beam at the central position, and $\mathsf{\Delta }{W}_{\mathrm{Center}}$ represents the error caused by eccentricity, the data of which can be obtained from the experiment. The wavefront of the ideal sphere without eccentricity $W_{\mathrm{Zernike}}^I$ is represented as:

$$W_{\mathrm{Zernike}}^I=[Z_5^I,Z_6^I,Z_7^I,Z_8^I,Z_9^I]$$

(12)

The residual $\mathsf{\Delta }{W}_{\mathrm{Center}}$ is represented as:

$$\mathsf{\Delta }{W}_{\mathrm{Center}}=[\mathsf{\Delta }{Z}_5,\mathsf{\Delta }{Z}_6,\mathsf{\Delta }{Z}_7,\mathsf{\Delta }{Z}_8,\mathsf{\Delta }{Z}_9]$$

(13)

Since the random ball test method tends to converge the wavefront of the entire globe lens to that of an ideal sphere after multiple random rotations, we can simulate the wavefront of the ideal sphere in the RBT method. The residual results of the experiment, denoted as $\mathsf{\Delta }{W}_{\mathrm{Center}}$, represent the difference in Zernike coefficients between the eccentric sphere and the non-eccentric sphere. The difference observed in this experiment is consistent with the difference observed in simulations. Similarly, given the premise of knowing the Zernike coefficients of the ideal sphere without eccentricity in simulations, adding this difference allows us to obtain the Zernike coefficients of the eccentric sphere lens in simulations. Finally, by inversely solving these coefficients, we can determine the eccentric position.

In the simulation experiment, the ideal sphere without eccentricity is set with eccentricities x = 0, y = 0, and z = 0. Because the error is mainly concentrated in Z5–Z9, the optimization targets only need to be set to these.

According to Equation (14), by adding the residual $\mathsf{\Delta }{W}_{\mathrm{Center}}$ to the ideal model $W_{\mathrm{Zernike}}^I$, with $W_{\mathrm{Zernike}}^C$ as the optimization target using the least squares method, we can obtain the parameters of eccentricity and the Zernike coefficients at that eccentricity.

$$W_{\mathrm{Zernike}}^C=\mathsf{\Delta }{W}_{\mathrm{Center}}+W_{\mathrm{Zernike}}^I$$

(14)

For single-channel wavefront measurement $W_{\mathrm{lens}}^{\prime }$, and dual-channel wavefront measurement $W_{\mathrm{lens-double}}^{\prime }$, because the random ball in the random ball test method is positioned eccentrically, the eccentricity error $\mathsf{\Delta }{W}_{\mathrm{Center}}$ of the random ball is not subtracted when removing the systematic error and the numerical aperture error.

Because the Zernike Fringe coefficients in the simulation results are not actual measured outcomes, only the deviation between the eccentric wavefront and the ideal wavefront remains constant. Therefore, the subtraction should be done between the two wavefronts’ differences, not the Zernike coefficients of the wavefront.

The calculation of the difference $\mathsf{\Delta }{W}_{\mathrm{Center}}$ has been mentioned earlier and can be obtained from the simulation model:

$$\mathsf{\Delta }{W}_{\mathrm{Center}}=W_{\mathrm{Zernike}}^C-W_{Zernike}^I=W_{\mathrm{lens-double}}^{\prime }-(W_{lens}^{\prime }-W_{system})-\mathsf{\Delta }{W}_{NA}$$

(15)

Based on the optimization of $\mathsf{\Delta }{W}_{\mathrm{Center}}$, a new well-behaved $\mathsf{\Delta }{W}_{\mathrm{Center}}^{\prime }$ can be obtained, and after subtracting it, a new and more accurate wavefront aberration of the objective lens can be achieved. The final calibration result of the objective lens should be expressed as:

$$W_{\mathrm{lens-final}}=W_{lens}^{\prime }-W_{system}+\mathsf{\Delta }{W}_{NA}+\mathsf{\Delta }{W}_{\mathrm{Center}}$$

(16)

D.

Uncertainty analysis

In the verification process, we take the average of 100 wavefront maps as the reference or ideal value, and each individual wavefront map as a sample. We record the difference in Zernike coefficients between each sample and the averaged wavefront map as $\mathsf{\Delta }{W}_i$

$$\mathsf{\Delta }{W}_i=[\mathsf{\Delta }{z}_5,\mathsf{\Delta }{z}_6,\mathsf{\Delta }{z}_7,\mathsf{\Delta }{z}_8,\mathsf{\Delta }{z}_9]$$

(17)

As shown in Figure 5, it demonstrates the x, y, and z eccentricities of the random ball during its motion. All dashed circles in different colors in the figure represent potential eccentric positions of the random sphere. The red solid circle indicates the ideal position of the random sphere. By substituting the Zernike coefficients of each wavefront map into the RBRC model, specific parameters for the three coordinates can be obtained. The positional parameters for each individual wavefront map are represented as:

$$[\mathsf{\Delta }{x}_i,\mathsf{\Delta }{y}_i,\mathsf{\Delta }{z}_i]=ZO [\mathsf{\Delta }{Z}_5^i,\mathsf{\Delta }{Z}_6^i,\mathsf{\Delta }{Z}_7^i,\mathsf{\Delta }{Z}_8^i,\mathsf{\Delta }{Z}_9^i]$$

(18)

In the 100 randomly selected maps, 26 maps were used to calculate the corresponding $\mathsf{\Delta }x$, $\mathsf{\Delta }y$, and $\mathsf{\Delta }z$, calculating the magnitude of changes for each of the three positions separately.

To objectively evaluate the proposed eccentricity error compensation method in this paper, Root Mean Square Error (RMSE) is employed to assess the dispersion of the eccentricity positions obtained through back-solving the Zernike coefficients of the random ball wavefront. Additionally, Type A uncertainty calculation is used to evaluate the effectiveness of the model’s computations.

The specific formula for calculating RMSE is as follows:

$$RMSE=\sqrt{\frac{1}{n}{{\sum }_{i=1}^n\left(x_i-\overline{x}\right)}^2 }$$

(19)

In the Equation (19), $x_i$ represents the actual value of the i-th data point, is the mean of the data, and n is the number of samples. The unit of RMSE is the same as that of the original data. In statistics, a smaller RMSE value indicates better data performance [31].

3. Experiment

3.1. Random Ball Test

In order to compute the intrinsic spherical aberration of the spherical lens, various parameters of the lens are constrained and simulated, resulting in an inherent spherical aberration of 0.58204051λ. The experimental results of single-channel random rotation with 100 repetitions of the random ball and averaging are shown in Figure 6a. After subtracting the inherent spherical aberration, the remaining wavefront represents the calibrated systematic error, as shown in Figure 6b.

As shown in Figure 7a–f, from the 100 wavefront data, we randomly selected 1, 10, 20, 30, 40, and 50 sets of data and averaged them separately. We then subtracted each of these averages from the average of 100 wavefronts to obtain residuals.

After applying a logarithmic transformation to the residuals, we performed a linear fit, and the results are shown in Figure 8. The fitted slope of the straight line is −0.49, indicating that the outcomes from the random spheres are reasonable.

3.2. Single-Channel Measurement of the Lens

After completing the random ball experiment, we replaced the ball lens with an objective lens and measured this wavefront aberration. The wavefront aberration results of the plano-convex lens are shown in Figure 9a–c. After subtracting the systematic error calibrated by the random ball test, the actual wavefront aberration results of the lens are shown in Figure 9d–f.

3.3. Dual-Channel Measurement of the Lens

The purpose of the dual-channel measurement of the lens aberration is to validate the results obtained from the single-channel measurement and retrieve the model’s data. Both experiments used the light source provided by the same 4D interferometer. The process of switching from single-channel mode measurement to dual-channel mode measurement is shown in Figure 10a. The objective’s position and angle remain unchanged, but the SID4 wavefront sensor is replaced with a spherical mirror. The results of measuring the lens are shown in Figure 10b,d.

The subtraction of the single-channel measurement results from the dual-channel measurement results is shown in Figure 11a. After removing the calibrated systematic error from the single-channel measurement results and then subtracting them from the dual-channel measurement results, the specific results are shown in Figure 11b.

Theoretically, after removing the systematic error of calibration, the residual result should be 0 nm because the objective lens and the light source do not change. However, the presence of 47.373 nm indicates the remaining systematic error. This discrepancy may be due to the random ball maintaining its eccentric position during the random rotation process, causing errors in the calibrated wavefront aberration. The residual result represents the difference between the actual result and the ideal result, which serves as the dataset for our model optimization. By adding this residual to the results from the ideal model, the actual eccentricity differences in the x, y, and z directions can be obtained, denoted as $\mathsf{\Delta }{W}_{\mathrm{Center}}$.

4. Experimental Results and Discussion

4.1. NA Error Model

Because the ball lens purchased in practice has a different curvature radius than the one designed, the wavefront error in the experimental data processing will include the wavefront error caused by this inconsistent NA.

Table 1. NA parameters of different ball lens.

	Distance	Radius	Material	RAID
Ideal	50 mm	17.77	N-BK7	5.626°
Actual	50 mm	20.00	N-BK7	4.856°

displays the specification differences between the designed globe lens and the actually purchased globe lens, both placed at a distance of 50mm from the light source on the front surface of the ball lens. RAID represents the entrance pupil angle of a random ball. The material of this component is N-BK7. As mentioned in Chapter 2, the NA error of the ideal ball lens is denoted as $W_{NA-17}$, and the NA error of the actual ball lens is denoted as $W_{NA-20}$, both of which can be obtained from simulation.

The data for the model can be obtained from simulation, and the simulation 3D model is shown in Figure 12a,b. The result after subtracting this error is shown in Figure 12c.

4.2. Eccentricity Model

In Section 4.1, after removing the NA error from the residual, the remaining errors are all due to eccentricity, which is the inherent error caused by the eccentricity of the random ball.

The optimized parameters and results of the model are shown in

Table 2. Model Parameters and Result.

	X/mm	Y/mm	Z/mm	Z5	Z6	Z7	Z8	Z9
Ideal	0	0	0	0	0	0	0	0.5820
Target	-	-	-	−0.0510	−0.0250	−0.0830	−0.1510	0.7490
Result	0.053	0.114	5.242	−0.0049	0.0058	−0.0727	−0.1573	0.7499
△W	-	-	-	−0.0049	0.0058	−0.0727	−0.1573	0.1679

.

By substituting the parameters into Equations (15) and (16) and incorporating them into the RBRC model, the eccentricity error $\mathsf{\Delta }{W}_{\mathrm{Center}}$ can be derived. The final residual result of the objective lens, after subtracting this error term along with NA error, is shown in Figure 13a. In conclusion, the calibration result for the objective lens is shown in Figure 13b, with an RMS value of 222.018 nm.

4.3. Uncertainty Analysis

Due to alignment and fixation issues, the random sphere may experience eccentricity during each rotation caused by vibration and sliding. The proposed model accurately determines the parameters of the eccentric lens. When applying this method in the experimental random ball test process, after multiple averaging iterations, the x, y, and z positions of all individual wavefront maps should gradually converge to the averaged position.

According to Equation (18), calculate and record the motion of $\mathsf{\Delta }x$, $\mathsf{\Delta }y$, and $\mathsf{\Delta }z$ as obtained. Figure 14 displays the variations of x, y, and z eccentricities during the motion of the random sphere, as depicted. The theoretical mean positions should approach 0, 0, and 50, and the calculated mean values are quite close to the theoretical values and remain stable. Therefore, the obtained $\mathsf{\Delta }x$, $\mathsf{\Delta }y$, and $\mathsf{\Delta }z$ from this method are reasonable.

Based on Equation (19), the calculated results for the Root Mean Square Error (RMSE) in the x, y, and z directions are shown in Figure 15. The calculated eccentricity data remains relatively stable across different measurement iterations, indicating that the eccentricity positions obtained through the reverse Zernike calculation are reasonable.

4.4. Discussion

This paper presents a wavefront error compensation method based on a random ball residual compensation model. Experimental results demonstrate that the combination of this model and the random ball method can enhance the accuracy of systematic error calibration and consequently improve the detection precision of the projection lens.

Yang [32] pointed out the limitations of using random balls for interferometer error calibration, as the mismatch between the curvature radius of the random ball and the test optics can lead to measurement inaccuracies. As pointed out by Evans et al. [21], during the alignment of the random ball with the interferometer, misalignments and errors such as tilt and eccentricity can lead to aberrations like coma and spherical aberration. These aberrations should be removed when processing random ball data.

The model proposed in this paper effectively addresses these issues. Additionally, when combined with their approach, it allows for a more flexible selection of random balls with different curvatures, reducing costs and enhancing the calibration accuracy of the objective lens.

However, in the experiments, the surface smoothness of the random balls was not particularly high, which may have resulted in some uncorrected spherical aberrations. Moreover, the data acquisition process for the model was somewhat cumbersome. In the test, it was achieved through cross-validation by switching between single-channel and dual-channel modes, which could be challenging to implement in practical engineering scenarios. Additionally, the manual rotation of the random balls in our experiments was labor-intensive and time-consuming. To simplify the data acquisition process for the model, it is possible to incorporate precise data obtained from the random balls, such as surface smoothness, curvature radius, refractive index of the medium, and others, into the model parameters. This would result in a more accurate random ball model and eliminate the need for one of the detection modes, either single-channel or dual-channel. Consequently, the measured results can be directly subtracted from the Zernike coefficients of the precise random ball model, yielding a more comprehensive dataset. As for the frequency of rotations, predictive algorithms can be employed to reduce the number of rotations required.

In the field of optical wavefront measurements, this research provides valuable insights into the calibration of certain computationally compiled optical element errors. It also introduces a novel approach for cross-validation experiments involving error calibration.

5. Conclusions

This paper proposes a method for error separation and compensation in wavefront measurement based on a random ball residual compensation model. To compensate for wavefront errors caused by improper equipment selection and limited experimental conditions, NA compensation and eccentricity compensation methods based on random ball residual compensation models are proposed. The wavefront errors are represented in the Zernike form. Additionally, in high-precision machining and measurement, using this method with computer assistance can improve alignment accuracy and expand the selection range of devices. For instance, when measuring with lenses of inconsistent curvature radii, using this method with computer assistance can compensate for the errors between the actual lens and the design lens. However, this method is currently limited to the use of random ball tests or high precision.