Radial shearing interferometers (RSIs) have been widely exploited in the optical shop metrology field, primarily attributed to their advantages of reference-free measurement and common-path configuration [1–4]. An additional distinctive attribute of RSIs is their capability to perform single-shearing step measurements on rotationally symmetric optics. It contrasts with lateral shearing interferometers, in which typical two-shearing steps or sophisticated dual optical paths are required [5–7]. The RSI can be implemented to generate two wavefronts named magnification and demagnification, as depicted in Fig. 1(a), and the shearing amount between them is controlled to reduce the wavefront slope, which effectively avoids the Nyquist fringe problem [8]. A widely adopted configuration is by combining a Sagnac loop with a lens pair having different focal lengths, as shown in Fig. 1(b) to shear the original wavefront undertest into magnified and demagnified wavefront by the clockwise and counterclockwise, respectively [9, 10]. Four interferograms are captured simultaneously by dynamic interferometer technology which employs a polarized camera to optimize the phase-shifting data collection, effectively avoiding strong environmental vibrations [11–14]. However, this interferogram wavefront does not directly represent the original wavefront under test, which is a significant demerit of the RSI. Hence, an efficient wavefront reconstruction method is a core issue in achieving a high-accuracy measurement [15, 16].

Figure 1. Radial shearing interferometers (RSIs). (a) Basic principle of the radial shearing interferometer. (b) Magnified and demagnified wavefront generated by Sagnag loop. BS, beam splitter; L₁ and L₂, lens 1 and lens 2 (with different focal lengths); M₁ and M₂, mirror 1 and mirror 2.

The zonal wavefront analysis-based methods have been widely proposed and verified. It focuses on describing the relationship between height and slope data based on an iterative or least-square method [17–19]. Another approach is the modal method using Zernike polynomials to describe the reconstruction of the original wavefront through a transformation matrix [20–24]. Zernike coefficients representing the target wavefront are directly calculated by simple multiplication between the transformation matrix and the coefficients of an RSI interferogram wavefront [25].

In this study, we present a method for improving the accuracy of the modal wavefront reconstruction in the RSIs. We present our efforts to enhance the accuracy of the RSI wavefront reconstruction method using high-order Zernike polynomials. A mathematical relationship between the target and the interferogram wavefront is derived and extracted as the unlimited-length transformation matrix. We generated the complex RSI interferogram wavefront to evaluate the performance of an algorithm and show the importance of including high-order Zernike polynomials for accurate wavefront reconstruction [26–28]. It is demonstrated through cases where the wavefront is reconstructed with insufficient and sufficient orders of Zernike polynomials. Subsequently, the peak-to-valley (PV) and root mean square (RMS) of the reconstructed errors are quantified and evaluated.

The two electrical fields in the Sagnac loop of the RSIs, as well as the resulting measurement field, are denoted as E₁, E₂, and E_M, respectively, using the Eq. (1):

where E₁₀, E₂₀, and E_M0 represent the amplitudes of the target, magnified, and measurement fields, respectively. W₁, W₂, and W_M correspond to the wavefronts of each field, and k represents the wavenumber. The wavefronts can be represented using Zernike polynomials, as follows:

where Z_j (ρ, φ) is the Zernike polynomials, µ_j, v_j, u_j are Zernike coefficients for the single index j by Noll’s notation. The measurement field results from the difference between the target and magnified fields, as follows:

The radially magnified wavefront W₂ can be expressed by using the target wavefront W₁ with the shearing amount a, which target wavefront W₁ with the shearing amount a The radially the is determined by the square of the lens focal length ratio as follows [24]:

where R_j(ρ) and P_j(φ) represent radial and angular Zernike polynomials, respectively. The radially magnified wavefront is a function of reduced radial polynomials, denoted as R_j(aρ), where the radial variable ρ is replaced by aρ. Then the coefficients of both the radial polynomials µ_j and reduced radial polynomials v_j can be described by the matrix formation as follows [23]:

To obtain the elements M_i,j in high-order, we solved the following matrix equation.

First, we explicitly expanded the reduced radial Zernike polynomial R_j(aρ) as a function of a and ρ by utilizing the radial Zernike polynomial R_j(ρ). Subsequently, by substituting the explicit functions of the reduced radial polynomial and radial polynomials into Eq. (6), we determined the elements of the matrix M_ij that nullify the equation. We used MATLAB^® to solve the Eq. (6), while designating the variables a, ρ, and the elements M_ij as the symbolic variables. The derived reduced radial polynomials and matrix M are represented in Table 1 and Table 2, respectively. During our expansion of the polynomials and matrix M up to the 400^th order, the expansion to an infinite order is achievable through the code accessible on the website specified in the data avaiablity section.

TABLE 1. Mathematical expression for the radial and reduced radial polynomials with the corresponding Noll’s notation.

Noll’s Notation (j)	Radial Polynomials, Rj(ρ)	Reduced Radial Polynomials, Rj(aρ)
1	R1(ρ) = 1	R1(aρ) = 1 = R1(ρ)
2	R2(ρ) = 2ρ	R2(aρ) = 2aρ = aR2(ρ)
3	R3(ρ) = 2ρ	R3(aρ) = 2aρ = aR3(ρ)
4	R4(ρ) = 3(2ρ2 − 1)	R4(aρ) = 3(2a2ρ2 − 1) = a2R4(ρ) + 3(a2 − 1)R1(ρ)
5	R5(ρ) = 6ρ2	R5(aρ) = 6a2ρ2 = a2R5(ρ)
6	R6(ρ) = 6ρ2	R6(aρ) = 6a2ρ2 = a2R6(ρ)
7	R7(ρ) = 8(3ρ3 − 2ρ)	R7(aρ) = 8(3a3ρ3 − 2aρ) = a3R7(ρ) − 8a(a2 − 1)R2(ρ)
8	R8(ρ) = 8(3ρ3 − 2ρ)	R8(aρ) = 8(3a3ρ3 − 2aρ) = a3R8(ρ) − 8a(a2 − 1)R3(ρ)
9	R9(ρ) = 8ρ3	R9(aρ) = 8a3ρ3 = a3R9(ρ)
10	R11(ρ) = 5(6ρ4 − 6ρ2 + 1)	R10(aρ) = 8a3ρ3 = a3R10(ρ)
100	R100(ρ) = 28(78ρ13 − 132ρ11 + 55ρ9)	R100(aρ) = 28(78a13ρ13 − 132a11ρ11 + 55a9ρ9) = 35a9(13a4 − 24a2 + 11)R54(ρ) + 242a11(a2 − 1)R76(ρ) + a13R100(ρ)
300	R300(ρ) = 48ρ23	R300(aρ) = 48a23ρ23 = a23R300(ρ)
400	R400(ρ) = 56(2,925ρ27 − 7,800ρ25 + 6,900ρ23 − 2,400ρ21)	R400(aρ) = 56(2,925a27ρ27 − 7,800a25ρ25 + 6,900a23ρ23 − 2,400a21ρ21)= 2154a21(117a6 − 325a4 + 300a2 − 92)R252(ρ) + 242a23(27a4 − 52a2 + 25)R298(ρ) + 2182a25(a2 − 1)R346(ρ) + a27R400(ρ)

TABLE 2. Mathematical expression of transformation matrix by shear amount.

Mi,j	Value	Mi,j	Value
M1,1	1	M24,12	35a4(a2 − 1)
M2,2	a	M24,24	a6
M3,3	a	M25,15	35a4(a2 − 1)
M4,1	3(a2 − 1)	M25,25	a6
M4,4	a2	M26,14	35a4(a2 − 1)
M5,5	a2	M26,26	a6
M7,3	8a(a2 − 1)	M28,28	a6
M7,7	a3	M29,3	4a(7a6 − 15a4 + 10a2 − 2)
M8,2	8a(a2 − 1)	M29,7	8a3(7a4 − 15a2 + 5)
M8,8	a3	M29,17	48a5(a2 − 1)
M10,10	a3	M30,2	4a(7a6 − 15a4 + 10a2 − 2)
M11,1	5(a4 − 3a2 + 1)	M30,8	8a3(7a4 − 15a2 + 5)
M11,4	15a2(a2 − 1)	M30,16	48a5(a2 − 1)
M11,11	a4	M30,30	a7
M12,6	15a2(a2 − 1)	M31,9	8a3(7a4 − 15a2 + 5)
M12,12	a4	M31,19	48a5(a2 − 1)
M13,5	15a2(a2 − 1)	M31,31	a7
M13,13	a4	M32,10	8a3(7a4 − 15a2 + 5)
M14,14	a4	M31,18	48a5(a2 − 1)
M15,15	a4	M32,32	a7
M16,2	3a(5a4 − 8a2 + 3)	M33,21	48a5(a2 − 1)
M16,8	24a3(a2 − 1)	M33,33	a7
M16,16	a5	M34,20	48a5(a2 − 1)
M17,3	3a(5a4 − 8a2 + 3)	M34,34	a7
M17,7	24a3(a2 − 1)	M35,35	a7
M17,17	a5	M36,36	a7
M18,10	24a3(a2 − 1)	M37,1	42a8 − 105a6 + 90a4 − 30a2 + 3
M18,18	a5	M37,4	3a2(28a6 − 63a4 + 45a2 − 10)
M19,9	24a3(a2 − 1)	M37,11	45a4(4a4 − 7a2 + 3)
M19,19	a5	M37,22	63a6(a2 − 1)
M21,21	a5	M37,22	63a6(a2 − 1)
M22,1	7(5a6 − 10a4 + 6a2 − 1)	M37,37	a8
M22,4	21a2(3a4 − 5a2 + 2)	...	...
M22,11	35a4(a2 − 1)	M100,54	35a9(13a4 − 24a2 + 11)
M22,22	a6	...	...
M23,5	21a2(3a4 − 5a2 + 2)	M200,76	15a11(969a8 − 3,264a6 + 4,080a4 − 2,240a2 + 455)
M23,13	35a4(a2 − 1)
M23,23	a6	...	...
M24,6	21a2(3a4 − 5a2 + 2)	M400,400	a27

The coefficients µ_j of the target wavefront can be calculated using the derived matrix M and the coefficients of the measured wavefronts u_j obtained through the RSI, as follows [24]:

We verified the improved accuracy of wavefront reconstruction in the RSIs by using the high-order radial Zernike polynomials. An arbitrary target wavefront was prepared by summing the first 50^th orders of Zernike polynomials with randomly selected coefficients across a 512 × 512 lateral resolution, as shown in Fig. 2(a). In this target wavefront preparation, the piston, tip, and tilt coefficients were not applied. The coefficients of the Zernike polynomials for the target aberration were set to have values of 8.136λ PV and 1.354λ RMS, as shown in Fig. 2(b). Subsequently, the magnified wavefront was prepared by expanding the target wavefront with the magnification factor. In this study, we set the shearing amount a = 0.8. The cropping and interpolation were performed on the expanded wavefront to ensure an image size and resolution identical to the target wavefront, as shown in Fig. 2(c). The measured wavefront was acquired by subtracting the magnified wavefront from the target wavefronts, as shown in Fig. 2(d).

Figure 2. The generation of the target, magnified, and measured wavefronts. (a) Target wavefront with 8.136λ PV and 1.354λ RMS, (b) coefficients of 50^th order of the radial Zernike polynomials to generate the target wavefront. The coefficients of the piston, tip and tilt were not applied. (c) The magnified wavefront, (d) the measured wavefront by subtraction of the wavefront (a) and (c).

To validate the effectiveness of the high-order polynomials, we conducted the reconstruction using 30^th order polynomials. This order is insufficient compared to the target wavefront’s coefficient order. As shown in Figs. 3(a) and 3(b), the reconstructed wavefront exhibits a notable discrepancy when compared to the target wavefront in Fig. 2(a). This discrepancy is attributed to the insufficient polynomial order to reconstruct the complex target wavefront. The error between the reconstructed and target wavefronts was significantly high, with values reaching up to 6.139λ in PV and 1.123λ in RMS, as shown in Figs. 3(c) and 3(d).

Figure 3. Wavefront reconstruction by 30^th order Zernike polynomials. (a) The reconstructed wavefront by the 30^th order Zernike polynomials, (b) the radial Zernike coefficients comparison between the target wavefront and the reconstructed wavefront, (c) the reconstructed error between the target wavefront and the reconstructed wavefront, and (d) the coefficient error of the target wavefront and the reconstructed wavefront.

The reconstructed error with an insufficient set of Zernike coefficients can be overcome when increasing them to higher orders. Figures 4(a) and 4(b) illustrate the successfully reconstructed wavefront by using the radial Zernike polynomial up to 50^th order, which matches the number of orders used to generate the target wavefront. It showed a good agreement with the target wavefront as shown in Fig. 2(a), with the expected PV and RMS of 8.131λ and 1.354λ, respectively. Specifically, in this effort, the reconstruction process when incorporating high-order Zernike polynomials has been demonstrated to show the negligible reconstructed error with PV and RMS of 0.011λ and 0.001λ, respectively in Fig. 4(c). The coefficient error is shown on a 10⁻⁴ unit in Fig. 4(d), which significantly improves the accuracy by a thousandfold compared to the previous test when applied with the same target wavefront.

Figure 4. Wavefront reconstruction by 50^th order Zernike polynomials. (a) The reconstructed wavefront by the 50^th order Zernike polynomials, (b) the radial Zernike coefficients comparison between the target wavefront and the reconstructed wavefront, (c) the reconstructed error between the target wavefront and the reconstructed wavefront, and (d) the coefficient error of the target wavefront and the reconstructed wavefront.

We further performed an extensive simulation using hundreds of orders of Zernike polynomials. Multiple transformation matrices were calculated, spanning orders from the 10^th to the 500^th. Subsequently, we iterated the identical reconstruction processes and error estimation for each reconstructed wavefront. The results of the PV and RMS errors with respect to the applied order of the radial Zernike polynomials are shown in Fig. 5. The data points were plotted at intervals of 10 up to the 100^th order, and subsequently at intervals of 100 for orders beyond the 100^th order. Both PV and RMS errors were dramatically decreased as the order of the radial Zernike polynomials reached 50, which is coincident with the orders used for the initial target wavefront. Following the 50^th-order reconstruction, the PV and RMS errors exhibited a continuous decline, as shown in the inset diagram of Fig. 5. Nonetheless, considering that the PV and RMS errors were already obtained at magnitudes of 10⁻² and 10⁻³, respectively, the consequent decrease does not bear substantial significance. The convergence of the PV and RMS towards the infinitesimal values emphasizes the critical importance of employing sufficiently high-order polynomials to achieve accurate reconstruction of the target wavefront. Notably, an unignorable reconstructed error is obtained at several wavelength levels, when the polynomial order is insufficient to reconstruct a wavefront with high spatial frequency aberration. Analyzing reconstruction error concerning the polynomial orders can contribute to determining the optimized value of the order for successful reconstruction, while achieving PV errors at the nanometer level and RMS errors below a sub-nanometer level.

Figure 5. Peak-to-valley (PV) and root mean square (RMS) values of error of the reconstructed wavefront with respect to the order of the radial Zernike polynomials in use. The wavefront error is dramatically reduced when using radial Zernike polynomials of order 50^th or higher.

In practical experiments, the results of measured wavefront include noise originating from various sources, including light sources, detectors, and environmental factors. To demonstrate the robustness of the proposed reconstruction method under noisy conditions, we added random noise with the signal-to-noise ratio (SNR) of 10, which is more severe condition compared to the typical noise level in laser-based interferometers, as shown in Fig. 6(a). The magnified wavefront and measured wavefront are shown in Figs. 6(b) and 6(c), respectively are generated similarly to those discussed in section 3; It is not repeated in this section. We will conduct reconstruction with noised measured wavefront in Fig. 6(c) to perform the method’s potential. Figures 7(a) and 7(b) illustrate the successfully reconstructed wavefront by using the radial Zernike polynomial up to the 50^th order, which is a sufficient order of target wavefront. It also shows a good agreement with the target wavefront as shown in Fig. 2(a), with the expected PV and RMS of 8.114λ and 1.355λ, respectively. Specifically, in this effort, the wavefront reconstruction including noise showed a reasonable reconstructed error with PV and RMS of 0.097λ and 0.013λ, respectively in Fig. 7(c); They are all under the sub-wavelength scale. The notable agreement of the coefficient shows not only the amplitude but also the sign convention as shown in Fig. 7(b) and the coefficient error is shown on a 10⁻³ unit in Fig. 7(d). The fact that the algorithm performs well and provides a reasonable reconstruction result. In addition, we also discuss the dependence of the reconstruction error with the SNR as presented in Fig. 8. PV and RMS of the error become saturated by 10⁻² λ and 0.001λ when SNR is greater than 20. otherwise, for the investigated SNR value range from 1 to 60, the error always shows up with wavelength and sub-wavelength scale, which shows the proposal method is consistent with actual measurements.

Figure 6. The preparation of the wavefronts under noisy condition. (a) Target wavefront with random noise with signal-to-noise ratio of 10, (b) the magnified wavefront generated from the target wavefront with noise, and (c) the measured wavefront by subtraction of the noised wavefront (a) and (b).

Figure 7. Wavefront reconstruction with noised target wavefront by 50^th order Zernike polynomials. (a) The reconstructed wavefront by the 50^th order Zernike polynomials, (b) the radial Zernike coefficients comparison between the target wavefront and the reconstructed wavefront, (c) the reconstructed error between the target wavefront and the reconstructed wavefront, and (d) the coefficient error of the target wavefront and the reconstructed wavefront.

Figure 8. The dependence of the reconstruction error with the signal to noise ratio.

In this study, we formulated high-order reduced radial Zernike polynomials to improve the accuracy of modal wavefront reconstruction in radial shearing interferometers. We expanded the reduced polynomials up to infinite order with symbolic variables of the radial and shearing amount, as well as the elements of the transformation matrix. For the simulation of the modal wavefront reconstruction, we generated a target wavefront. Afterward, magnified, measured wavefronts were generated. To validate the effectiveness of the high-order Zernike polynomials, we applied both low-order and high-order polynomials to the wavefront reconstruction process. Consequently, the PV and RMS errors notably decreased with values of 0.011λ and 0.001λ, respectively, as the order of the radial Zernike polynomial increased. Such high-order Zernike polynomials are expected to enhance the accuracy of wavefront reconstruction with high spatial frequencies in conventional RSIs. In addition, we also show the practical applicability of the algorithm in the experiment by modeling the target wavefront with random noise and they show a notable reconstructed error with PV and RMS of 0.097λ and 0.013λ, respectively. Consequently, this advanced mathematical tool can potentially improve the measurement accuracy for rotational symmetric aspheric and freeform optics exhibiting high slope characteristics.

National Research Foundation of Korea (Grant no. NRF-2021R1A4A1031660), Ministry of Trade, Industry and Energy (Grant no. MOTIE-20014784, 20018441), KOITA grant funded by MSIT (Grant no. 1711199141), and Hanoi University of Science and Technology (Grant no. HUST, T2022-PC-021).

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The datasets generated and analyzed during the current study are available in https://shorturl.at/xG279.