## Abstract

We propose a radial shearing interferometric approach to measure spherical wavefronts as both of the reflective and transmissive optical configurations. The modified cyclic radial shearing interferometer uses a single lens in the optical layout, which can conveniently adjust the radial shearing ratio between two shearing spherical wavefronts, and the use of a polarization camera enables to reconstruct the wavefront by a single image. The wavefront mapped onto the camera plane can be identified and quantified throughout an optimized wavefront reconstruction algorithm. In the experiments, plano-convex lenses and concave mirrors were used to generate spherical wavefronts, and the proposed system was able to reconstruct the surface figures after system characterization and calibration. Further investigations were performed to evaluate the system measurement accuracy by the radius of curvature comparison with design value and a commercial Shack-Hartmann wavefront sensor.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The measurement of spherical wavefronts has been crucial for industrial fields such as optical surface profile detections [1–4] and laser beam quality diagnoses [5–7]. In addition to the Shack-Hartmann wavefront sensors [8] and pyramid techniques [9], which have the limitation of lateral resolution, several interferometric techniques have been used to reconstruct the wavefront by analyzing the interferograms based on their own measurement principles [10–12]. A point diffraction interferometry (PDI) was proposed to obtain spherical surfaces based on the reference wave generation by a pinhole. Although it is capable to precisely measure the wavefronts, however, the reference and measurement wave are separate in PDI, and the measurement results are sensitive to environmental factors such as vibrations [10]. In order to reduce these vibration noises, an optical shearing interferometer has been adopted. A lateral shearing interferometer (LSI) can obtain the spherical wavefront by the acquisition of two shearing interferograms, and the slopes of wavefronts are integrated for wavefront reconstruction. However, LSI has the limitation to obtain two shearing interferograms.

On the other hand, a radial shearing interferometer (RSI) has the benefit to reconstruct the wavefront with a single radial shearing interferogram [13]. Especially, a cyclic radial shearing interferometer (CRSI) has the advantages of compact optical configuration, common paths between two shearing beams, the immunity of vibrations and high system stability [14]. Typically, a telescope system with two lenses is adopted in CRSI, and two plane waves with different sizes are overlapped with each other at the imaging plane to obtain the radial shearing interferogram. However, the issues on spherical wavefront testing by utilizing CRSI have not been fully discussed yet because it was originally designed for measuring plane wavefronts and their slight deviations.

In order to achieve spherical wavefront measurements, an additional optical system should be placed in front of CRSI to adjust the original wavefront into plane wavefronts. Recently, a simple CRSI was proposed with a single lens inside of the cyclic configuration instead of using a telescope system for non-planar wavefront sensing. However, it focused on the optical system establishment and assessed the system feasibility compared to convention CRSI [15]. Therefore, a growing body of evidence suggests that it is highly desirable to design a spherical wavefront measurement system based on CRSI with maximizing its performance advantages and potential, and make it as a mature treatment for inspecting versatile spherical wavefront. Our proposed method outweighed the deficiencies of complex alignment, loss of spatial resolution, severe optical aberrations in convention structure discussed above, while maintaining the advantages of unsusceptible vibration operation, highly efficient data dynamic acquisition.

In this investigation, we describe spherical wavefront measurements by a modified CRSI. A single lens is used in the optical path of the cyclic scheme, and a polarization camera captures four phase-shifted interferograms with a single image. In section 2, we illustrate the configuration of spherical wavefront diagnosis system with the wavefront reconstruction algorithm, and analyze the spherical wavefront measurement difficulty in conventional CRSI. In section 3, the experiment results are discussed with the two kinds of spherical wavefront measurement approaches as a reflective and a transmissive type, respectively. Section 4 and 5 conclude and discuss some considerations.

## 2. Principle of spherical wavefront measurements

#### 2.1 Modified cyclic radial shearing interferometer

Figure 1 shows the optical configuration based on the cyclic radial shearing interferometer which we modified in this investigation. The plane wave from the collimated beam of the optical source is converted into spherical wavefront by using a reflective or a transmissive configuration, denoted as the blue and the green boxes in Fig. 1, respectively. Then, the beam is incident to the cyclic scheme to generate two radially sheared beams. In this case, the incident beam is divided by a polarizing beam splitter (PBS), and each beam travels in opposite direction with each other. Opposed to the traditional CRSI with the telescope system, of which focal length ratio can determine the radial shearing ratio, a single lens is put on the optical path inside of the cyclic interferometer, and the radial shearing ratio can be conveniently adjusted. After combined by the PBS, they go through a quarter wave plate (QWP), and are detected by a polarization camera (PCMOS). Because of polarizer array with four different transmission axes of 0°, 45°, 90° and 135° in the PCMOS, four phase-shifted interferograms can be obtained at once, and the phase is extracted by the phase shifting algorithm.

In this modified CRSI, the use of a single lens can lead to the compact system, convenient alignment of the optical components and the efficient measurements of spherical wavefronts despite of its slight change compared to the conventional ones. In addition, our approach provides the versatility of being able to perform optimization in both reflected and transmitted spherical wavefront measurement. For example, the transmitted spherical wavefront generated by a plano-convex lens (can be seen in the green box), and back aperture of the sample can be imaged by the PCMOS. For the comparison of measurement results, it is also imaged onto the Shack-Hartmann wavefront sensor (SHWFS) by using a 4f system. On the other hand, a concave mirror is used to obtain the reflected spherical wavefront as shown in blue box in Fig. 1, and the modified CRSI can directly measure the wavefront. In this case, the PCMOS is also placed in the conjugated plane with the surface of the concave mirror. Under this circumstance, the measurement result is consistent with the surface profile of concave mirror, which means the proposed system can be used for measuring the surface figure of the sample. This measurement approach is really different from that of the conventional CRSI, which uses an additional lens to convert the spherical wavefront into plane one.

#### 2.2 Simulation of radial shearing ratio adjustment

In conventional CRSI, it is very difficult to measure the spherical wavefronts unless a sophisticated zooming lens system is involved, which increase the system cost and complexity. It’s because the incident wavefront to the cyclic scheme is not a plane wave any more, which leads to the generation of two non-planar sheared wavefronts, without any specific precaution. Then, the radial shearing ratio becomes unknown, which prevents the subsequent wavefront reconstruction. Therefore, the conventional CRSI has two main limitations in measuring spherical wavefronts: one is unknown radial shearing ratio, and the other is incomputable of the measurement areas where two wavefronts are overlapped.

However, in our modified CRSI, spherical wavefront can be measured without these limitations because both the radial shearing ratio and the measurement areas are calculated using a simple thin lens equation. We set the optical path length between lens and object with appropriate specific value, which regarded as object distance *p*. With the help of the known quantity, single-lens focal length *f*, the image distance *q* can be calculated. Besides, we expressed the single-lens positions with *ΔL*, which defined as the lens deviation distance from midpoint of geometric cyclic optical beam path. Therefore, two beams with orthogonal polarization states split from PBS have different image distance, *q*_{1} and *q*_{2}, respectively. Note that the *q* can be either positive or negative. Here, we choose a convention where *q *> 0 for a real image, and *q *< 0 for a virtual image. If the lens equation yields a negative image distance, then the image means a virtual image on the same side of object. In this case, iterative calculation is needed substituting the absolute value of image distance as an object distance until image distance be a positive to make the real image.

After obtaining the image distance, then deducing the linear magnification relationship to calculate the size of two illumination beams (*d*_{1}, *d*_{2}), and the system ratio can be predicted. The ratio relationship can be described as:

*l*is the distance between single lens and CCD plane,

*f*is the focal length of single lens. Based on the mathematical modeling of the system, we took the simulations to determine the radial shearing ratios under three circumstances of single-lens locations (

*ΔL*). The Fig. 2 demonstrated the relationship between the system ratio and CCD position adjustment.

From the Fig. 2, we can see that firstly, the shearing ratio can be easily adjusted close to 1, and this feature is beneficial to measure the spherical wavefront because low ratio (close to 1) can generate a raw interferogram with high visibility. Second, in the whole cases of the focal length, the shearing ratio has quite smooth variation in a narrow range (from 1 to 1.5), the ratio will be larger with the increment of the lens deviation position. More importantly, in each focal length case, the ratio is barely susceptible to the adjustment of camera position, which are quite convenient to adopt in the reconstruction algorithm with less measurement error, and this perfectly meets our expectations.

#### 2.3 Phase extraction and spatial resolution improvement

The PCMOS, of which image sensor has an array of four linear polarizers, is used in the modified CRSI, and four phase-shifted interferograms with the consecutive π/2 are obtained at once. In this case, the interferograms obtained by the PCMOS can be mathematically described with the radial wavefront difference (*ΔW*) as

*a*(

*x, y*) is the background intensity,

*b*(

*x, y*) is the fringe visibility, and

*n*= 0,1,2,3 represents the number of phase-shifted frames separated from four different pixel sets of PCMOS. Then,

*ΔW*can be calculated by four-step phase shifting algorithm [13] as

Based on the spatial phase shifting technique, we can obtain *ΔW* with a single image, which avoids measurement errors caused by mechanical motions for lateral shearing of wavefronts in LSI or temporal phase shifting in RSI. However, the spatial resolution of the system should decrease because of the fundamental structure of the PCMOS, which has (2 × 2) unit cell. For instance, the pixel density of the PCMOS is (1600 x1600), but the wavefront difference map has (800 × 800) pixels while the image size is the same.

In previous phase calculation algorithm, we set four adjacent super pixels as a (2 × 2) unit cell and results in a phase map with 1/4 of the raw data. To improve the spatial resolution of the system, we can combine nine super pixels as a (3 × 3) convolution matrix. For the center matrix element, notice that one (3 × 3) convolution matrix includes four (2 × 2) unit cells, namely, there are more than one polarizer array pixels in the same orientations in this convolution matrix and the summation operation is necessary. It is worth noting that for the edge elements in the first and last row/column of matrix, the summation region is different which is contained within two (2 × 2) unit cells; for the four corner elements of the matrix, the summation region is only within (2 × 2) unit cell, and it can be used to determine the phase with the conventional four-step phase-shifting algorithm. By calculating the sum of corresponding magnitudes in the same polarization elements, the derivation equation of phase information is as follow:

This is the (2 × 2) convolution kernel method, which Neal J. Brock, etc. [16] have shown that this is the best condition for improving spatial resolution, we can successfully get the wavefront difference results which have the same size with original selected interferogram image data, thereby improving the spatial resolution. Additionally, errors caused by vibration and turbulence was averaged together successfully. After that, we can demodulate the wrapped phase distribution and obtain the actual radial wavefront difference pattern.

#### 2.4 Optimized modal wavefront reconstruction algorithm

In the conventional CRSI, two radial shearing beams are contracted and expanded with the shearing ratio determined by the focal length ratio of two lenses, respectively. In this case, the output wavefront is nearly planar if the incident wavefront is a plane. Then, the interference region restricts to the contracted beam area, and the wavefront difference can be described with the original wavefront (*W _{0}*) and the radial shearing ratio ($s = {f_1}/{f_2}$) as:

To describe two radial shearing beams in our system, we set initial beam diameter as *d*_{0}, and the diameters of two shearing beams are set as *d _{c}* and

*d*, respectively. Then, the two radially sheared wavefronts (

_{e}*W*and

_{c}*W*) can be expressed as:

_{e}In the overlapping area to generate the interference, the wavefront difference can be written as:

*s*is the ratio between the relatively contracted beam and the original beam while

_{c}*s*is the ratio between the relatively expanded beam and the original beam. When each wavefront is expressed with the Zernike polynomials, the unit circle of the Zernike polynomials needs to be enlarged or reduced according to the beam size considering the radial shearing ratios,

_{e}*s*and

_{c}*s*as

_{e}*α*is the weighting coefficient of Zernike polynomials. By substituting Eq. (9) and Eq. (10) for Eq. (8), the wavefront difference (Δ

_{i}*W*) can also be fitted by the combined Zernike polynomials with different unit circle sizes, denoted as the wavefront difference radial polynomials, and Eq. (8) can be re-written with a matrix expression as where

**U**is the wavefront difference radial polynomials matrix, and

**α**is the weighting coefficient matrix. By the matrix operation,

**α**can be calculated with known

**U**as

Finally, **W _{0}** is obtained by the multiplication of

**α**and

**Z**as

It is noted that **ΔW** contains the additional phase difference between two converging or diverging beams caused by using a single lens. Therefore, this phase difference denoted as quadrature phase should be considered to be compensated during the wavefront recovery calculation process [15].

## 3. Experimental results

In order to evaluate the performance of spherical wavefront measurements, we constructed transmissive and reflective types of the modified CRSI configuration. The light source was a femtosecond laser (Light Conversion, Lithuania) with 1030** **nm center wavelength, 8.2** **nm bandwidth and 223 fs pulse duration. The output laser pulse went through a BBO nonlinear crystal (1030** **nm SHG@Edmund Optics) for frequency doubling, and the center wavelength was converted into 515** **nm. The beam was collimated and the spot diameter was enlarged to 8** **mm by employing a beam expander (GBE05-A@Thorlabs).

#### 3.1 Transmitted spherical wavefront measurement

Figure 3 shows the photograph of transmissive type of the modified CRSI constructed in this investigation. A commercial Shack-Hartmann wavefront sensor (WFS40-14AR@Thorlabs, SHWFS) was used to compare the measurement results.

To generate a spherical wavefront, a plano-convex lens (L_{1}) was put in front of the source part, and various lenses were used as L_{1} for adjusting the radius of curvature (ROC) of the spherical wavefront. By using a non-polarizing beam splitter (BS), the beam was divided into two beams. The transmitted one reached to wavefront sensor eventually by going through a 4f system, which consisted of two lenses (L_{3} and L_{4}) with 200** **mm focal lengths, while the reflected one was incident to the cyclic configuration with a single lens (L_{2}), where it was separated into two orthogonal polarization states with opposite direction with each other for two radial shearing beams. With the adjustment of quarter-wave plate (WPQ10M-514@Thorlabs), the polarization camera (BFS-U3-51S5P-C@FLIR) was able to capture the four phase-shifted radial shearing interferograms.

Applying the phase extraction and wavefront reconstruction explained in the Section 2, the spherical wavefronts were reconstructed as shown in Fig. 4(a) as 3D wavefront profiles and their projections. As shown in Fig. 4(a), the measurement results were successfully reconstructed as spherical wavefronts, and the ROC of the wavefront becomes smaller as the focal length of L_{1} is shorter, as expected. The measurement area was (8 **× **8) mm^{2}, which is consistent with original beam diameter we controlled with a level-actuated iris. In the meantime, the wavefront sensor instrument can measure the wavefront in the area of (11.26 **× **11.26) mm^{2} and we selected (8 **× **8) mm^{2} as the adequate measured area of our system. For the quantitative comparison between the results of our system and the SHWFS, Fig. 4(b) compared the cross-sectional curve of reconstructed wavefront profile under both methods. The results demonstrated the good consistency.

In the further, we calculated the ROC of wavefronts with the aid of the sphere fitting model. Table 1 summarizes the experiment results of various spherical wavefronts as PV and ROC values, and the measurement results are nearly identical between our method and referenced wavefront sensor.

To estimate the stability of the measurement results, we implemented ten times consecutive measurements with the 60s sampling interval. The experimental setup was on the active vibration isolation optical table to prevent the floor vibrations. Figure 5 shows the ROC fluctuations of our method and SHWFS measurement results. The grey region on the plot represents the standard deviations (STD) of ten measurements. As the focal length of L_{1} increases, the STD becomes higher in both measurement results, which reveals that the system stability is lower as the ROC of wavefront to be tested is larger. In our proposed method, the STD is no more than 0.08** **mm as shown in Fig. 5(a) while it reaches to 2.47** **mm in the SHWFS. As the result, the proposed method has the better stability than SHWFS.

#### 3.2 Reflected spherical wavefront measurement

The spherical wavefront measurements were also implemented with the reflective type of the modified CRSI. We used various concave mirrors to generate spherical wavefronts, and the reflected beams were incident to the cyclic scheme as same as the transmissive type. In this case, the measurement results can represent the surface profile of the concave mirrors with the aid of imaging theory. Figure 6 shows the flow chart of the experiment process for reflected spherical wavefronts in this investigation. As the concave mirrors, the mirrors with focal lengths of 250** **mm (CM254-250@Thorlabs), 500** **mm (CM254-500@Thorlabs) and 750** **mm (CM254-750@Thorlabs) were used. First of all, we determine the location of L_{1} inside the system and fix it. Then, a plane mirror (BB1-E02@Thorlabs) was used to obtain the quadrature wavefront for the additional phase calibration. After the calibration, the concave mirrors were measured with adjusting the location of L_{1} along the beam propagation direction repeatedly. The aim of moving the lens is to confirm the validity of measurement results and appraise the system measurement accuracy.

For the several concave mirrors, the proposed system was able to reconstruct their 3D surface profiles successfully as shown in Fig. 7. Compared to the specification of the manufacturer, the deviations of the ROC were less than 1 mm. Under the three circumstances of lens position adjustment, the mirrors were repeatedly measured, and the ROCs were calculated as presented in Table 2. Regardless of the lens location, the proposed system was able to reconstruct the spherical wavefronts without any significant errors.

## 4. Discussion

As shown in Table 2, the root-mean-square errors (RMSE) of the ROC for three mirrors are less than 0.60 mm, which still needs the further improvement in measurement accuracy. There are several factors may introduce the measurement errors in the proposed system. First of all, the initial beam is not ideal plane wavefront, which may contain distortions after undergoes the BBO crystal and beam expander. As we explained in the Section 2.4, we made the calibration process during the wavefront reconstruction. Secondly, imperfection of some optical components such as the extinction ratio of the linear polarizer and retardance errors of the quarter induce the wavefront distortions [17].

The dominant error factor in our configuration is the determination of the radial shearing ratio. In the experiment, the radial shearing ratio was calculated by the number of PCMOS pixels considering the overlapping regions of two beams. However, due to the edge of two deformed beams mapped on the PCMOS array plane is ambiguous in some cases, the uncertainty of calculated radial shearing ratios usually within 5-pixel sizes. Because of the fractional error caused by inaccurate radial shearing ratio, the calculated ROCs were deviated from the nominal values. Based on the Eq. (1), the relative error of the shearing ratio in our system can be expressed as:

*r*, Δ

*d*

_{1}and Δ

*d*

_{2}are the errors of shearing ratio and two deformed beam diameters. Based on our experiment data in Section 3, we analyzed the relative errors quantitatively, the Fig. 8 illustrated the calculated radial shearing ratio relative error within five-pixel deviations both in transmitted and reflected wavefront. It can be seen that the relative inaccuracy is no more than 0.8% even though for the strong aberrated wavefront (small focal length).

In addition, slight misalignment of optical elements might affect to the cyclic structure, and it was the practical error source to generate the lateral shearing. The lateral shearing between two wavefronts is also practically inevitable, so the iterative reconstruction method [18,19] should be considered in the further investigation.

## 5. Summary

We described a modified radial shearing interferometer to measure spherical wavefronts as both of the reflective and transmissive optical configurations. The modified cyclic radial shearing interferometer uses a single lens in the optical layout, which can conveniently adjust the radial shearing ratio between two shearing spherical wavefronts, and the use of a polarization camera enables to reconstruct the wavefront by a single image. The wavefront mapped onto the camera plane can be identified and quantified throughout an optimized wavefront reconstruction algorithm. In the experiments, plano-convex lenses and concave mirrors were used to generate spherical wavefronts, and the proposed system was able to reconstruct the surface figures after system characterization and calibration. Further investigations were performed to evaluate the system measurement accuracy by the radius of curvature comparison with design value and a commercial Shack-Hartmann wavefront sensor.

## Funding

National Natural Science Foundation of China (52005147, 62005071); China Postdoctoral Science Foundation (2020M682258); National Key Research and Development Program of China (2019YFE0107400).

## 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.

## References

**1. **M. V. R. K. Murty, “A Compact Radial Shearing Interferometer Based on the Law of Refraction,” Appl. Opt. **3**(7), 853–857 (1964). [CrossRef]

**2. **M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. **24**(24), 4439–4442 (1985). [CrossRef]

**3. **S. Sarkar, N. Ghosh, S. Chakraborty, and K. Bhattacharya, “Self-referenced rectangular path cyclic interferometer with polarization phase shifting,” Appl. Opt. **51**(1), 126–132 (2012). [CrossRef]

**4. **H. S. Chang and C. W. Liang, “High resolution full field wavefront measurement of a misaligned miniature lens,” Opt. Express **26**(24), 31209–31221 (2018). [CrossRef]

**5. **D. Liu, Y. Y. Yang, L. Wang, and Y. M. Zhuo, “Real time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt. **46**(34), 8305–8314 (2007). [CrossRef]

**6. **T. Ling, D. Liu, Y. Y. Yang, L. Sun, C. Tian, and Y. B. Shen, “Off-axis cyclic radial shearing interferometer for measurement of centrally blocked transient wavefront,” Opt. Lett. **38**(14), 2493–2495 (2013). [CrossRef]

**7. **X. Y. Chen, Y. Y. Yang, C. Wang, D. Liu, J. Bai, and Y. B. Shen, “Aberration calibration in high-NA spherical surfaces measurement on point diffraction interferometry,” Appl. Opt. **54**(13), 3877–3885 (2015). [CrossRef]

**8. **M. Aftab, H. Choi, R. Liang, and D. W. Kim, “Adaptive Shack-Hartmann wavefront sensor accommodating large wavefront variations,” Opt. Express **26**(26), 34428–34441 (2018). [CrossRef]

**9. **S. Berchtold, C. Böhm, and H. P. Kriegel, “The Pyramid-Technique: Towards Breaking the Curse of Dimensionality,” Proc. SIGMOD **27**(2), 142–153 (1998). [CrossRef]

**10. **D. Wang, Y. Yang, C. Chen, and Y. Zhuo, “Point diffraction interferometer with adjustable fringe contrast for testing spherical surfaces,” Appl. Opt. **50**(16), 2342–2348 (2011). [CrossRef]

**11. **Z. Y. Wang, S. Wang, P. Yang, and B. Xu, “Three-wave radial shearing interferometer,” Opt. Lett. **44**(8), 1996–1999 (2019). [CrossRef]

**12. **Y. B. Seo, H. B. Jeong, H. G. Rhee, Y. S. Ghim, and K. N. Joo, “Single-shot freeform surface profiler,” Opt. Express **28**(3), 3401–3409 (2020). [CrossRef]

**13. **P. Hariharan and D. Sen, “Radial shearing interferometer,” J. Sci. Instrum. **38**(11), 428–432 (1961). [CrossRef]

**14. **N. Gu, B. Yao, L. Huang, and C. Rao, “Compact single-shot radial shearing interferometer with random phase shift,” Opt. Lett. **42**(18), 3622–3625 (2017). [CrossRef]

**15. **D. Bian, D. H. Kim, B. G. Kim, L. D. Yu, K. N. Joo, and S. W. Kim, “Diverging cyclic radial shearing interferometry for single-shot wavefront sensing,” Appl. Opt. **59**(28), 9067–9074 (2020). [CrossRef]

**16. **N. Brock, B. T. Kimbrough, and J. E. Millerd, “A pixelated micropolarizer-based camera for instantaneous interferometric measurements,” Proc. SPIE **8160**, 81600W (2011). [CrossRef]

**17. **M. Novak, J. Millerd, N. Brock, M. N. Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. **44**(32), 6861–6868 (2005). [CrossRef]

**18. **C. Elster, “Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears,” Appl. Opt. **39**(29), 5353–5359 (2000). [CrossRef]

**19. **F. Z. Dai, F. Tang, X. Z. Wang, and O. Sasaki, “Generalized zonal wavefront reconstruction for high spatial resolution in lateral shearing interferometry,” J. Opt. Soc. Am. A **29**(9), 2038–2047 (2012). [CrossRef]