## Abstract

It is a key issue to measure the point-diffraction wavefront error, which determines the achievable accuracy of point-diffraction interferometer (PDI). A high-precision method based on shearing interferometry is proposed to measure submicron-aperture fiber point-diffraction wavefront with high numerical aperture (NA). To obtain the true shearing point-diffraction wavefront, a double-step calibration method based on three-dimensional coordinate reconstruction and symmetric lateral displacement compensation is proposed to calibrate the geometric aberration in the case of high NA and large lateral wavefront displacement. The calibration can be carried out without any prior knowledge about the system configuration parameters. With the true shearing wavefront, the differential Zernike polynomials fitting method is applied to reconstruct the point-diffraction wavefront. Numerical simulation and experiments have been carried out to demonstrate the accuracy and feasibility of the proposed measurement method, and a good measurement accuracy is achieved.

© 2016 Optical Society of America

## 1. Introduction

With the development of optical manufacturing, the interferometry has been widely applied in the optical testing. The traditional interferometer, such as Fizeau interferometer and Twyman-Green interferometer, generally employs a standard lens to generate the reference wavefront, and its achievable measurement accuracy is limited by the precision of standard lens. The point-diffraction interferometer (PDI) [1–5 ] utilizes point-diffraction wavefront as ideal spherical reference wavefront, the corresponding measurement precision can reach the order of subnanometer. The PDI overcomes the accuracy limitation due to reference elements, various setups have been developed to realize the high-precision testing of optical surfaces [6], three-dimensional coordinate measurement [7–10 ] and flow field detection [11]. Both the pinhole and single-mode fiber can be applied to generate the ideal spherical wavefront. However, the diffraction light power from pinhole is poor (transmittance <0.1‰) and the numerical aperture (NA) of diffraction wavefront with single-mode fiber is quite low (<0.20), and both approaches limit the measurement range of existing PDI [12]. A submicron-aperture (SMA) fiber with cone-shaped exit end has been proposed to obtain both the high diffraction light power and high-NA spherical wavefront, and it is considered as a feasible way to extend the measurement range of the system [13, 14 ].

Due to the fact that the achievable testing precision of PDI is mainly determined by the sphericity of diffraction wavefront, the analysis of diffraction-wavefront error has become a fundamental way to evaluate the performance of PDI. The numerical simulation based on vector diffraction theory, such as the finite difference time domain (FDTD) method [12, 15 ] and rigorous coupled wave (RCW) method [1], provides an easy and efficient way to estimate the point-diffraction wavefront error. However, both the computational accuracy and complex practical factors (such as environmental disturbance and performances of various optical parts) can introduce significant deviation from ideal case. Due to the accuracy limitation of standard optics, the traditional interferometers fail to measure the point-diffraction wavefront error, which is expected to be in the order of subnanometer or even smaller. Various experimental testing methods have been proposed to measure the point-diffraction wavefront error, the majority of which are based on null test [16] and hybrid method [17]. Typically, the hybrid method requires several measurements with the rotation and displacement of the optics under test, it is sensitive to environmental disturbance and cannot completely separate the systematic error. The null test is a self-reference method, and it is widely applied to reconstruct the point-diffraction wavefront and calibrate the PDI. However, it requires the foreknowledge about the system configuration to remove the high-order aberrations, which is introduced by the point-source separation and cannot be negligible in the case of high-NA wavefront.

In this paper, a high-precision measurement method, which is based on shearing interferometry, is presented to analyze the point-diffraction wavefront from SMA fiber. A point-diffraction projector consisting of four SMA fibers is applied to obtain the shearing wavefront in *x* and *y* directions. Based on the rigorous model for the SMA fiber projector configuration analysis, a modified double-step calibration method is proposed to remove the systematic error introduced by geometric aberration. Section 2 presents the principle of the proposed method for SMA point-diffraction wavefront measurement, including the basic theory of point-diffraction wavefront retrieval based on differential Zernike polynomials fitting method and the rigorous model for systematic error calibration. In Section 3 and 4, the results of computer simulation and laboratory experiments are given to demonstrate the feasibility of the proposed method, respectively. Some concluding remarks are drawn in Section 5.

## 2. Wavefront retrieval from SMA-fiber point-diffraction Projector

Figure 1
shows the schematic diagram of SMA fiber projector used to evaluate the sphericity of point-diffraction wavefront. The SMA fiber can be applied to obtain both high diffraction light power and high-NA spherical wavefront. According to the structure of SMA fiber projector shown in Fig. 1(a), the SMA fiber taper surface is coated with Cr film and the exit aperture is formed from the polished tip, with the aperture size about 0.5 μm. Four parallel SMA single-mode fibers S_{1}, S_{2}, S_{3} and S_{4} with coplanar exit ends are integrated in a metal tube. To obtain the identical wavefront parameters, the four SMA fibers with same aperture size and cone angle are carefully chosen in the experiment to minimize the measurement error. The coherent beams are coupled into the SMA fibers, and the point-diffraction waves from the fibers interfere on a detection plane, as is shown in Fig. 1(b). By alternatively switching on the waves from the fiber pairs G_{1} (S_{1} and S_{2}) and G_{2} (S_{3} and S_{4}), the shearing interferograms in *x* and *y* directions can be obtained and the corresponding shearing wavefronts can be measured with phase-shifting method, respectively.

#### 2.1 Point-diffraction wavefront retrieval method

With the shearing wavefronts obtained from the projector shown in Fig. 1, the differential Zernike polynomials fitting method [18–20
] can be applied to retrieve the SMA fiber point-diffraction wavefront. Denoting the point-diffraction wavefront under test as $W(x,y)$, we have the shearing wavefronts $\Delta {W}_{x}(x,y)$ and $\Delta {W}_{y}(x,y)$ in *x* and *y* directions,

According to Eqs. (1) and (2) , the shearing wavefronts $\Delta {W}_{x}(x,y)$ and $\Delta {W}_{y}(x,y)$ can be expressed as

Denoting the shearing wavefronts, differential Zernike polynomials and the coefficients in Eq. (3) as $\Delta W={(\Delta {W}_{x},\Delta {W}_{y})}^{T}$, $\Delta Z={(\Delta {Z}_{i,x},\Delta {Z}_{i,y})}^{T}$ and $a$, respectively, Eq. (3) can be transformed into a matrix form,

Thus, the coefficients $\left\{{a}_{i}\right\}$ of Zernike polynomials in Eq. (2) can be obtained from the least-squares solution of Eq. (5),According to Eq. (6), the retrieval of point-diffraction wavefront $W(x,y)$ depends on the measurement precision of shearing wavefronts $\Delta {W}_{x}(x,y)$ and $\Delta {W}_{y}(x,y)$. Traditionally, the systematic error introduced by lateral displacement between SMA fibers can be calibrated by removing Zernike tilt and power terms. However, the residual high-order aberrations can significantly influence the measurement precision, especially in the case of high NA and large lateral displacement. Thus, a general and rigorous method for geometric error removal is required to realize the high-precision measurement of point-diffraction wavefront.

#### 2.2 High-precision method for systematic error calibration

In the null test of pinhole diffraction wavefront and single-mode fiber diffraction wavefront, the high wavefront NA in pinhole PDI and large lateral displacement between fibers in fiber PDI could introduce some high-order geometric aberrations, respectively. Different from traditional pinhole PDI and single-mode fiber PDI, the null test of SMA fiber diffraction wavefront involves both high NA and large lateral displacement, placing much higher requirement on the calibration of the systematic error introduced by lateral displacement between SMA fibers. A double-step calibration method based on three-dimensional coordinate reconstruction and symmetric lateral displacement compensation can be applied to completely remove systematic error. It should be noted that the possible longitudinal displacement between SMA fibers may also introduce certain systematic error, however it can be well minimized with the fine adjusting mechanism. Besides, the error introduced by the longitudinal displacement is low-order aberration, and it can be well calibrated with traditional misalignment calibration method by subtracting the Zernike piston, tilt and power terms.

### 2.2.1 First-step calibration based on three-dimensional coordinate reconstruction

Without loss of generality, we take the displacement in *x* direction between SMA fibers as the geometric error calibration model to be analyzed, as shown in Fig. 2
.

Suppose that the distances between exit apertures of fiber pairs G_{1} (S_{1} and S_{2}) and an arbitrary point $P(x,y,z)$ on the detection plane are ${R}_{1}$ and ${R}_{2}$, and we have the corresponding optical path difference OPD,

_{1}and S

_{2}, $(x,y,z)$ is that of the arbitrary known point $P$. To simplify the analysis, the origin of the coordinate system is located at S

_{1}, the distance between the SMA fiber projector and CCD detector is $D$. Thus, the OPD in Eq. (7) can be simplified as

According to the one-to-one correspondence of the OPD distribution on the detection plane and the coordinate of fiber apertures, the double-iterative method based on Levenbery-Marquardt (L-M) algorithm [10] can be applied to reconstruct the three-dimensional coordinates in Eq. (7), that is the global minimum ${\Phi}^{*}$ of the residual function $F(\Phi )$,

*k*indicates the point number on the detection plane, ${\stackrel{\u2322}{\xi}}_{k}$ is the measured OPD and ${\text{OPD}}_{k}$ is the OPD reconstructed from coordinates $\Phi $ according to Eq. (8). With the reconstructed coordinates ${\Phi}^{*}$, the systematic error can be preliminarily calibrated,

However, the reconstruction accuracy of fiber coordinates can only reach the order of submicron in the practical case, resulting in obvious residual error in the calibration result. The root mean square (RMS) value of the residual error in the preliminary calibration is $0.0077\lambda $ corresponding to the 0.5 μm coordinate reconstruction error for 0.60 NA fibers and 250 μm lateral displacement between two fibers. To further remove the residual systematic error, a second-step calibration, which is based on symmetric lateral displacement compensation, needs to be carried out.

### 2.2.2 Second-step calibration based on symmetric lateral displacement compensation

To simplify the analysis, the expression for OPD in Eq. (7) can be transformed to the cylindrical coordinate system. Let $x=r\mathrm{cos}\theta $ and $y=r\mathrm{sin}\theta $, where $r$ and $\theta $ are the polar radius and polar angle. OPD in Eq. (8) can be written as

Denoting the radius of maximum inscribed circle of the shearing interferogram as ${R}_{m}$ and the half aperture angle of point-diffraction wavefront as ${\phi}_{m}$, we have the normalized radius $\rho \text{=}r/{R}_{m}$, numerical aperture $\text{NA}=\mathrm{sin}{\phi}_{m}$ and $t={R}_{m}/D$. Based on Taylor expansion and Zernike polynomials fitting, a high-order approximation can be applied to Eq. (11) and OPD can be simplified as

where ${Z}_{2}$ refers to*x*tilt terms, ${Z}_{9}$, ${Z}_{19}$, and ${Z}_{33}$ are Zernike primary, secondary and tertiary

*x*coma terms, respectively, ${a}_{2}$, ${a}_{9}$, ${a}_{19}$ and ${a}_{33}$ are the corresponding coefficients,

According to Eq. (12), the major systematic error introduced by the lateral displacement includes tilt and coma terms, they cannot be completely removed with traditional misalignment calibration method [21] by subtracting the Zernike piston, tilt and power terms. From Eq. (12), the residual coma aberrations due to lateral displacement depend on the lateral displacement *s*, NA and distance $D$, the corresponding Zernike coefficients are odd functions about *s*. Thus, the superposition of geometric errors ${\text{OPD}}^{(s)}$ and ${\text{OPD}}^{(-s)}$ corresponding to opposite shear directions can be expressed as

According to the analysis above, the geometric error can be further reduced by superposing two measurements with opposite shear directions, corresponding to lateral displacement $s$ and $-s$, respectively. Denoting the preliminarily calibrated wavefront data corresponding to lateral displacement $s$ and $-s$ as ${W}_{m1}^{(s)}$ and ${W}_{m1}^{(-s)}$, and the true shearing wavefront under test $\Delta W$, we have

To get the measurement data with the lateral displacement $-s$, one may rotate the SMA fiber projector by 180°. The systematic error introduced by the lateral displacement changes oppositely with respect to the original position. Due to the approximately axial symmetric structure of fiber tips, the shearing wavefront map corresponding to the point-diffraction wavefronts under measurement is supposed to remain unchanged after the 180° rotation. Figure 3
shows the general procedure for the high-precision retrieval of SMA fiber point-diffraction wavefront. The shearing interferograms in *x* and *y* directions are acquired by switching the waves from the fiber pair G_{1} (S_{1} and S_{2}) to the fiber pair G_{2} (S_{3} and S_{4}), with the corresponding demodulated original wavefronts ${W}_{m0,x}^{(s)}$ and ${W}_{m0,y}^{(s)}$, respectively. Subsequently, the projector is rotated by 180° and the waves from the fiber pairs G_{1} and G_{2} are switched to get another original wavefronts ${W}_{m0,x}^{(-s)}$ and ${W}_{m0,y}^{(-s)}$, corresponding to the lateral displacement $-s$. The original wavefronts (${W}_{m0,x}^{(s)}$, ${W}_{m0,x}^{(-s)}$) and (${W}_{m0,y}^{(s)}$, ${W}_{m0,y}^{(-s)}$) in *x* and *y* directions are preliminarily calibrated with the method based on three-dimensional coordinate reconstruction, with the acquired wavefronts denoted (${W}_{m1,x}^{(s)}$, ${W}_{m1,x}^{(-s)}$) and (${W}_{m1,y}^{(s)}$, ${W}_{m1,y}^{(-s)}$). Zernike piston, tilt and power terms are removed from the pre-calibrated wavefronts (${W}_{m1,x}^{(s)}$, ${W}_{m1,x}^{(-s)}$) and (${W}_{m1,y}^{(s)}$, ${W}_{m1,y}^{(-s)}$). Then the symmetric lateral displacement compensation is carried out to further calibrate the residual error, the pre-calibrated wavefronts (${W}_{m1,x}^{(s)}$, ${W}_{m1,x}^{(-s)}$) and (${W}_{m1,y}^{(s)}$, ${W}_{m1,y}^{(-s)}$) are superposed to remove the high-order aberrations according to Eq. (18). Finally, the true shearing wavefronts $\Delta {W}_{x}$ and $\Delta {W}_{y}$ can be obtained, the differential Zernike polynomials fitting method is employed to retrieve the point-diffraction wavefront ${W}_{0}$ under test.

## 3. Numerical simulation results

According to the ray tracing method, a point-diffraction projector (the wavelength *λ* is 532 nm), as shown in Fig. 1 is modeled to demonstrate the feasibility of the proposed high-precision method for SMA fiber point-diffraction wavefront measurement, both the systematic error calibration and point-diffraction wavefront retrieval methods are analyzed. The detection plane is set at 100 mm away from the projector. Figure 4
shows the simulation results of the systematic error calibration for the case of 250 μm lateral displacement of two ideal point sources with 0.60 NA. Figure 4(a) shows the residual error obtained with the traditional misalignment calibration method, in which the Zernike piston, tilt and power terms are removed from original wavefront. The residual errors obtained from the first-step calibration (with lateral displacement reconstruction error 0.5 μm) and the second-step calibration using the proposed method introduced in Subsection 2.2 are shown in Figs. 4(b) and 4(c). According to Fig. 4(a), a significant residual error, with the PV value $\text{19}.\text{7395}\lambda $ and RMS $3.8406\lambda $, can be seen in the calibration systematic error with traditional calibration method. It can be seen from Fig. 4(b) that the first-calibration based on three-dimensional coordinate reconstruction can significantly reduce the systematic error compared to the traditional method, with the corresponding PV and RMS value being $0.0395\lambda $ and $0.0077\lambda $, respectively. However, further calibration is required to minimize the residual systematic error. With the second-step calibration based on the symmetric lateral displacement compensation, the PV and RMS of residual error can be further reduced to $3.4666\times {10}^{-4}\lambda $ and $0.8253\times {10}^{-4}\lambda $, respectively. Thus, it is applicable for the high-precision measurement of point-diffraction wavefront with the accuracy better than $1\times {10}^{-4}\lambda $.

Figures 5(a) and 5(b) show the residual systematic errors corresponding to various lateral displacement amounts with different NAs with traditional method and the proposed double-step calibration method. According to Fig. 5(a), the traditional method is not valid when the lateral displacement is over 50 μm because the residual error RMS is larger than $2.0\times {10}^{-3}\lambda $ with the 0.10 NA fiber. The residual error also increases significantly with the lateral displacement, especially for the case with high NA fiber. In Fig. 5(b) the residual error is less than $1.0\times {10}^{-4}\lambda $ RMS within a 300 μm lateral displacement even for the 0.60 NA fiber, it confirms the feasibility of the proposed double-step calibration in Subsection 2.2.

The simulation results for the point-diffraction wavefront retrieval algorithm based on the differential Zernike polynomials fitting method is shown in Fig. 6 . Figure 6(a) is the original input wavefront for shearing interference; the lateral displacement $s$ and NA in the simulation are 250 μm and 0.60, respectively. The reconstructed wavefront is shown in Fig. 6(b) and the residual error, which is the difference of the original wavefront map and reconstructed wavefront map, is shown in Fig. 6(c). According to Fig. 6, the high-precision retrieval of shearing wavefront is realized with the differential Zernike polynomials fitting method, the corresponding peak-to-valley (PV) and RMS value of residual wavefront retrieval error are $\text{1}\text{.7257}\times {10}^{\text{-}14}\lambda $ and $\text{4}\text{.2928}\times {10}^{\text{-}15}\lambda $, respectively.

## 4. Experimental results

As shown in Fig. 7
, an experimental SMA fiber point-diffraction interferometer has been set up to verify the validity of the proposed method for point-diffraction wavefront measurement. The aperture size of the four SMA fiber point-diffraction sources is about 0.5 μm and the measured full aperture angle of diffraction wavefront obtained is 131°. After passing through the polarized beam splitter PBS, beam splitter BS1 and BS2, the circularly polarized beams at 532 nm are coupled into the SMA fibers S1, S2, S3 and S4. The exit ends of four SMA fibers are integrated in a projector, and the corresponding end displacements in the fiber pair G_{1} (S_{1} and S_{2}) and fiber pair G_{2} (S_{3} and S_{4}) are 44.9 μm and 48.3 μm, respectively. The difference of fiber tips due to fabrication error can result in different point-diffraction wavefont errors. To minimize the effect of difference among SMA fibers, the fiber tips with nearly the same aperture size and cone angle are used in the measurement system [22]. A 14-bit CCD camera with a minimum S/N ratio 60dB and the active area 10.56 mm (H) × 5.94 mm (V) (pixel size 5.5 μm (H) × 5.5 μm (V)) is used to detect the interferogram directly without imaging lens, minimizing the effect from optical components. By alternatively switching on the fiber pairs (G_{1} and G_{2}) and translating the PZT scanner, the original shearing wavefronts in *x* and *y* directions can be obtained with phase shifting method.

To obtain point-diffraction spherical wavefront, the projector should be placed at the far-field zone [23], that is the distance $D>>2{(2d)}^{2}/\lambda $, where *d* is the aperture size. The projector is placed at the position about 25 mm away from the CCD camera, and the corresponding detectable NA of point-diffraction wavefront is 0.12. With the unwrapped phase distribution, the systematic error calibration is carried out to obtain the shearing wavefront with the high-precision double-step calibration method introduced in Subsection 2.2. The unwrapped original wavefront before and after 180-degree rotation in *x* and *y* directions are shown in Figs. 8(a)-8(d)
, and the corresponding pre-calibrated shearing wavefronts after the first-step calibration are shown in Figs. 8(f)-8(i), respectively. Figures 8(e) and 8(j) are the true shearing wavefronts in *x* and *y* directions obtained with the double-step calibration method. The measurement results of the shearing wavefront are summarized in Table 1
.

According to Fig. 8 and Table 1, significant residual errors can be observed in the obtained wavefronts after first-step calibration based on three-dimensional coordinate reconstruction, the RMS value of pre-calibrated shearing wavefront in *x* direction is $0.0254\lambda $ and $0.0194\lambda $ in *y* direction. After the second-step calibration based on symmetric lateral displacement compensation, the residual error is further removed from the measured shearing wavefronts, the RMS value of true shearing wavefront in *x* direction is $1.38\times {10}^{-4}\lambda $ and $1.36\times {10}^{-4}$ in *y* direction. In addition, no significant odd error is observed in the finally retrieved wavefronts from Figs. 8(e) and 8(j). Thus, the high-precision calibration of systematic error introduced by the lateral displacement of SMA fibers is achieved with the proposed method.

With the true shearing wavefronts in *x* and *y* directions after systematic error calibration, the point-diffraction wavefront can be reconstructed based on the differential Zernike polynomials fitting method introduced in Subsection 2.1. The measured wavefront error compared with an ideal sphere is shown in Fig. 9(a)
, PV and RMS are $\text{9}\text{.20}\times {\text{10}}^{\text{-4}}\lambda $ and $\text{1}\text{.54}\times {\text{10}}^{\text{-4}}\lambda $, respectively. To demonstrate the validity of the measured point-diffraction wavefront, the projector is rotated 45° about its axis and translated about 0.5 mm along -*z* direction (far away from the CCD camera) to show changes of wavefront deformation. If the measured wavefront is related to the beam passing through the SMA fibers, the wavefront would rotate corresponding to the rotation of projector, and it remains unchanged within certain NA for the projector translation. Figures 9(b) and 9(c) show the measured wavefronts after the rotation and translation of projector, respectively, and Table 2
illustrates the corresponding results.

It can be seen from Fig. 9 and Table 2 that the measured wavefront shape in the original position is similar to those obtained with projector rotation and translation, demonstrating the measurement repeatability and accuracy. The projector rotation is consistent with the wavefront shape, the absolute RMS difference with respect to that in the original position is smaller than $1.0\times {\text{10}}^{\text{-5}}\lambda $. Due to the reduction in the measurement NA of point-diffraction wavefront, the measured wavefront error after projector translation is a bit smaller than that in the original position, the absolute RMS difference is about $0.18\times {\text{10}}^{\text{-4}}\lambda $. Several factors can also result in the differences among the measurement results, including the environmental disturbance and the effect from the protective glass on CCD detector and CCD noise [24]. To minimize the effect of environmental disturbance and CCD noise, the experimental setup is placed on an active vibration isolation table and shielded in a heat-insulating box, and the measurement data is averaged from 64 measurements. The protective glass can introduce additional deformation on point-diffraction wavefront under test. By employing a CCD detector without protective glass, the measurement accuracy is expected to be further improved. It should be noted that the measured wavefront data should be taken as a macro-parameter due to the smoothing effect of each pixel, in which the relatively large pixel acts as a low-pass filter. To minimize this effect, the center of each pixel is taken as the point to be analyzed. It can also be further decreased by adopting the CCD with smaller pixel size and enlarging the distance $D$ between the SMA fiber projector and CCD detector.

## 5. Conclusion

A high-precision method with SMA fiber point-diffraction projector, which is based on shearing interferometry and double-step calibration of systematic error, is presented for the SMA fiber point-diffraction wavefront measurement. The double-step calibration method based on three-dimensional coordinate reconstruction and symmetric lateral displacement compensation is proposed to remove the systematic error from the shearing wavefront, the differential Zernike polynomials fitting method is applied to reconstruct the point-diffraction wavefront under measurement. The calibration of systematic error can be carried out without any prior knowledge about the system configuration parameters. Both the numerical simulation and experiments have been carried out to demonstrate the feasibility of the proposed measurement method, and a good measurement accuracy is achieved. The proposed method can obtain the high-precision measurement of point-diffraction wavefront, and also provide a feasible way to calibrate the geometric aberration in the interferometric testing system with case of no imaging lens, especially those with high NA and large wavefront displacement.

## Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (NSFC) (11404312, 51476154, 51404223 and 51375467), the program of China Scholarships Council (201408330449), Zhejiang Provincial Natural Science Foundation of China (LY13E060006, Q14E060016), Zhejiang Key Discipline of Instrument Science and Technology (JL150508, JL150502), Guangxi Key Laboratory of Automatic Detecting Technology and Instruments (YQ15204), Guangxi Colleges and Universities Key Laboratory of Optoelectronic Information Processing (KFJJ2014-03).

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