A novel multiple directional shearing interferometry system with synchronous polarization phase shifting

Abstract

This paper explores a multi-directional (multiple directional) shearing synchronous polarization phase-shifting interferometer that utilizes a birefringent crystal displacer. This design effectively mitigates nonlinear issues and environmental influences commonly encountered in synchronous phase-shifting interferometry. Additionally, it enables the acquisition of shear wavefront information from multiple directions. By solving for the coefficients of the measured wavefront using multi-directional differential wavefront data, the system performs coefficient fitting to reconstruct the wavefront. The characteristics of crystal birefringence beam modulation, phase grating diffraction, and the polarization phase-shifting array for synchronous phase-shifting interferometry are investigated, based on the specific implementation methods of multi-directional shearing and synchronous polarization phase-shifting. The preprocessing method for the multi-directional shearing phase-shifting interferogram is then presented, followed by a detailed explanation of the spatial position registration technique for the shearing interferogram and the calculation method for the shear amount. The interferometer’s construction system was experimentally tested using a spherical optical component, and the results were compared with those obtained from ZYGO to verify its accuracy and testing reliability.

Introduction

The demand for high-precision optical components in modern optical systems is constantly increasing^1,2,3, which in turn imposes higher requirements on the accuracy of surface testing technology to effectively reflect the surface quality of optical components. The lateral shearing interferometry system, with its completely common-optical path, effectively mitigates the impact of vibration, making it ideal for on-line testing in complex environments. Additionally, it requires low coherence of the light source. The surface of optical components can be directly measured by adjusting the shear amount. Lateral shearing interferometry is versatile in measuring aspherical surfaces and particularly well-suited for testing surfaces^4,5. Hence, a comprehensive examination of lateral shearing interferometry, along with the advancement of measurement devices, will provide an efficient approach for measuring depth gradient optical components. Shearing interferometry is a technique that splits the wavefront under examination into two coherent wavefronts, which are displaced by a shearing device. Subsequently, the phase deviation of the wavefront is reconstructed using the interferometric fringe pattern by the displaced coherent wavefronts. By adjusting the sensitivity of the interferometry system, surface testing of optical components with large deviations can also be performed. However, the interferometric pattern in lateral shearing interferometry reflects the difference in the surface of the measured wavefront, and requiring a series of phase reconstruction calculations to complete the deviation reconstruction of the measured wavefront. It is not as simple and intuitive as traditional Fizeau-type or Twyman-Green-type interferometry patterns in reflecting the deviation of the measured wavefront. Synchronous polarization phase-shifting interferometry, as an effective surface testing method^6,7, offers high measurement accuracy and enables instantaneous testing of the surfaces of optical components⁸. To enhance the anti-interference performance of synchronous polarization phase-shifting interferometry in practical applications, combining it with lateral shearing interferometry technology^9,10 can significantly improve the robustness of the optical system^11,12 and ensure the accuracy of surface testing. Yong Bum SEO et al.¹³ proposed a free-form surface analyzer based on spatial phase-shifting lateral shearing interferometry for single exposure. The system employs birefringent materials to generate two laterally displaced beams and utilizes spatial phase-shifting technology with pixelated polarization cameras to analyze the interferometric fringes. By using only a few simple optical elements to generate lateral shearing waves, the surface profile of a free-form surface can be obtained in real time from the interferometric fringe patterns. Building on contributions in the literature, simultaneous two-step phase-shifting lateral shearing interferometry can be achieved using two birefringent crystals¹⁴. The measurement of an aspherical surface using this method can be accomplished with a micro-polarizer that has transmission polarization directions of 0° and 90°, ensuring a precise amount of phase shift. By utilizing polarization interference, noise and irrelevant information related to the measured wavefront can be effectively suppressed¹⁵. Additionally, a novel quadria-wave lateral shearing interferometry technique based on DBCs-BD¹⁶ is introduced. When the replica waves are incident on the analyzer and the transmission axis is oriented at 45° or 135°, an orthogonal lateral polarization interferogram is obtained. The feasibility of this interferometer has been demonstrated through theoretical derivation and empirical validation. This system alleviates the stringent quality requirements for optical devices in the interference optical path, effectively minimizing random and additional systematic errors. The testing accuracy is improved by introducing phase shift technology, a novel low-cost snapshot phase-shifting lateral shearing interferometer (SPLSI) with full-range continuously adjustable lateral shear ratio was proposed^17,18, a tunable shear plate, consisting of a polarizing beam splitter (PBS) and a reflective mirror, separates the orthogonally polarized states of the test wavefront into the original and sheared wavefronts. The lateral shear ratio can be adjusted by changing the spacing between the PBS and the mirror. A pixelated polarization camera simultaneously captures four π/2 phase-shifted interferograms for phase-shifting interferometric measurement. However, this interferometer does not use a polarizing beam splitter (PBS) to perform multi-directional shear or wavefront reconstruction via multi-directional shearing interferometry.

The density of interference field patterns in a lateral shearing interferometry system is not only determined by the wavefront being measured, but also by the lateral shift between the wavefront. Thus, by adjusting the lateral shift of each wavefront on the detector’s imaging plane, the issue of fringe pattern densification caused by non-null aspherical normal aberration can be effectively mitigated. This mechanism is used to achieve non-null aspherical surface testing in lateral shearing interferometry systems. A testing technique combining shearing interferometry with partial compensation was proposed for measuring deep aspherical surfaces¹⁹. In experiments using phase-shifting lateral shearing interferometry to measure aspherical surfaces, an RMS relative error of 0.0548% was achieved, despite variations in the shearing error amounts²⁰. In summary, shearing interferometry has become as a focal point of research in aspherical wavefront testing, particularly lateral shearing interferometry, due to its ability to acquire shearing interferograms in two orthogonal directions for reconstructing wavefront surface information. To enhance the spatial wavefront information conveyed by shearing interferograms, this paper presents a multi-directional lateral shearing synchronous polarization phase-shifting interferometry technique for the precise measurement of optical surfaces. The lateral shearing interferometry system combines the polarized birefringent crystal technique with synchronous polarization phase-shifting, enabling the acquisition of shear wavefront surface information through multi-directional lateral shearing. The lateral shearing device and the multi-directional lateral shearing synchronous polarization phase-shifting interferometry with a common optical path have been designed for the wavefront testing system.

In this paper, we propose the theory and implementation method for multi-directional lateral shearing polarization phase-shifting interferometry. “Methodology” section presents the methodology of multi-directional lateral shearing polarization phase-shifting interferometry. “Experimental system” section presents the experimental system, including a detailed explanation of the proposed method, the theoretical foundations of multi-directional shearing displacer and synchronous polarization phase-shifting, and experimental setup. “Results” section presents experimental results, demonstrating the feasibility and high precision of the proposed method for surface deviation measurement. “Discussion” section discusses the calibration of the transmission axis angle of polarizer array, preprocessing of phase-shifting interference patterns, and the calculation of the shear amount. “Conclusions” section concludes the paper.

Methodology

This paper proposes a multi-directional lateral shearing synchronous polarization phase-shifting interferometry to achieve wavefront interferometric measurement in a dynamic environment. The method compensates for missing data in other dimensions, effectively increases the number of data points, and completes phase reconstruction by utilizing multiple shear wavefront information from different shear directions. This approach effectively eliminates the influence of random system errors on the final reconstruction, enabling the acquisition of differential wavefront information along various shear directions on the tested surface. Consequently, this approach enables the accurate calculation of the final Zernike coefficient, thereby enhancing the precision of surface reconstruction. As shown in Fig. 1, this paper behind the principle of multi-directional lateral shearing synchronous polarization phase-shifting interferometry. It introduces the fundamentals of polarization phase-shifting interferometry and holographic diffraction technology, effectively leveraging the distinct phase shifts of left-handed and right-handed circularly polarized light when subjected to polarizers oriented in different directions. This allows for the realization of lateral shearing interferometry and dynamic phase-shifting.

Fig. 1

The primary diagram of the multi-directional lateral shearing polarization phase-shifting interferometry.

To ensure multi-directional shearing, the displacer structure is carefully designed to maintain the stability of the system’s polarization state during rotation. Additionally, a quarter-wave plate is incorporated to convert the ordinary light (o-light) and extraordinary light (e-light) split by the crystal into left-handed and right-handed circular polarization, respectively, thereby, facilitating synchronous phase-shifting interferometry. Lateral shearing is achieved through a single-axis birefringent crystal, with a polarizer positioned in front to control the polarization state of the incident light. By adjusting the transmission axis of the polarizer, polarization phase-shifting can be effectively achieved.

Measurement principle and optical path configuration

This paper proposes a multi-directional lateral shearing synchronous phase-shifting interferometry measurement system, as shown in Fig. 2, designed for the precise measurement of optical element surfaces.

Fig. 2

The optical path diagram of the multi-directional lateral shearing synchronous polarization phase-shifting interferometry system.

The measurement principle of the system is as follows: The light emitted from the He-Ne linearly polarized laser, which is linearly polarized, is first attenuated to adjust its intensity. It is then modulated into right-handed circularly polarized light by a quarter-wave plate before passing through a beam expansion and collimation system. Finally, the light is vertically incident on the beam splitter. After passing through the standard lens (a converging lens that defines the aperture area of the tested component based on its focal length), the light reaches the surface of the component under test. The test wavefront reflected from the surface then passes through the collimator, which consists of a polarizer, a parallel polarization beam displacer, and a quarter wave. After passing through a parallel polarization beam displacer, the test wavefront is split into two linearly polarized light beams with mutually orthogonal vibration directions, thereby achieving lateral polarization shearing of the two beams. The parallel beam displacer separates the two beams of light, which will then modulated into left- and right- handed circularly polarized light as they pass through a quarter-wave plate with a fast axis aligned at a 45° angle to the polarization direction. Upon reaching the grating, the beams undergo diffraction, resulting in the separation of the diffracted light into multiple diffraction orders. The first convex lens will focus the diffracted light, after which a small aperture stop can be placed at the back focal plane of the first convex lens and the front focal plane of the second convex lens. This setup will filter out the ± 1st order diffracted light, enabling further splitting of the test wavefront. The beam generated by this shearing displacer system will consist of four channels, each carrying identical wavefront information. However, since orthogonal polarization states do not produce interference patterns, the beam must pass through a directional polarizer array to enable interference. The directional polarizer array consists of a 2 × 2 grid of four polarizers made from the same material. The transmission axis of adjacent polarizers is oriented at a 45° angle, and light beams interfere at these four directions. As a result, four interference fringes with significant phase shifts can be observed on the polarizer array, and the interferometric image is captured using a CMOS camera. To extract surface information, the interferogram undergoes preprocessing, followed by phase unwrapping and wavefront fitting.

Based on the lateral shearing principle of birefringent crystals, it can be inferred that the two wavefronts of the displaced beam emerging from the crystal have orthogonal polarization states. Consequently, we can exploit the polarization phenomenon of light to achieve phase-shifting. By incorporating polarizing devices into the optical path, we can modify the polarization state of the beam, thus converting the polarization state modulation of the interferometric beam into phase modulation, enabling phase-shifting. The principle of lateral shearing polarization phase-shifting interferometry is illustrated in Fig. 3.

Fig. 3

The principle of lateral shearing polarization phase-shifting interferometry.

Taking the lateral shearing generated by a birefringent crystal as an example, when the surface under test is illuminated through a polarizer, the light is split into o-light and e-light, whose Jones vectors are respectively represented as follows:

$${E_o}=\left[ {\begin{array}{*{20}{c}} 0 \\ 1 \end{array}} \right],{E_e}=\left[ {\begin{array}{*{20}{c}} 1 \\ 0 \end{array}} \right]$$

(1)

By positioning a quarter-wave plate with its fast axis aligned at a 45° angle to the x-axis behind a shearing displacer, the o-light and e-light passing through the crystal are converted into left- and right- circularly polarized light, respectively, as represented by the Jones vector.

$$\begin{gathered} {E_{o1}}=\left[ {\begin{array}{*{20}{c}} 1&i \\ i&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 0 \\ 1 \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} 1 \\ { - i} \end{array}} \right] \hfill \\ {E_{e1}}=\left[ {\begin{array}{*{20}{c}} 1&i \\ i&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1 \\ 0 \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} 1 \\ i \end{array}} \right] \hfill \\ \end{gathered}$$

(2)

Assuming that the angle between the transmission axis of the analyzer and the x-axis is α, the light beam passing through the analyzer will undergo a phase shift and form interference. The Jones vector for the two shear wavefronts involved in interference is represented as:

$$\begin{gathered} {E_{o2}}=\left[ {\begin{array}{*{20}{c}} {{{\cos }^2}\alpha }&{\sin \alpha \cos \alpha } \\ {\sin \alpha \cos \alpha }&{{{\sin }^2}\alpha } \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} 1 \\ { - i} \end{array}} \right]=\exp \left[ {i(\frac{\pi }{2} - \alpha )} \right]\left[ {\begin{array}{*{20}{c}} {\cos \alpha } \\ {\sin \alpha } \end{array}} \right] \hfill \\ {E_{e2}}=\left[ {\begin{array}{*{20}{c}} {{{\cos }^2}\alpha }&{\sin \alpha \cos \alpha } \\ {\sin \alpha \cos \alpha }&{{{\sin }^2}\alpha } \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} 1 \\ i \end{array}} \right]=\exp (i\alpha )\left[ {\begin{array}{*{20}{c}} {\cos \alpha } \\ {\sin \alpha } \end{array}} \right] \hfill \\ \end{gathered}$$

(3)

The final interference intensity distribution is given by:

$$I=({E_{o2}}+{E_{e2}}) \cdot {({E_{o2}}+{E_{e2}})^ * }=E(2+\cos 2\alpha )+C$$

(4)

From Eq. (4), the interference fringes undergo a full period shift when the analyzer rotates by an angle of π. By varying the angle α of the analyzer’s transmission axis, phase-shifting interference can be effectively achieved. C represents constant background intensity, which doesn’t depend on the angle of polarization. By adjusting the angle of the analyzer’s transmission axis, a phase shift twice the magnitude of the variable can be introduced, enabling phase-shifting interference. By setting the angle to 0°, 45°, 90°, and 135°, phase shifts of 0, π/2, π, and 3π/2 are generated, thus achieving a four-step phase shift.

Polarization states of the interferometry system

This paper describes the relationship between polarization state and phase through the application of the Jones matrix. The Jones vector ${E_i}={\left[ {\begin{array}{*{20}{c}} 1&0 \end{array}} \right]^T}$ represents the linear polarization output of a He-Ne laser, where T denotes the transposed matrix. The quarter wave plate’s fast axis is aligned at a 45°-degree angle to the polarization direction of the laser output, and its Jones matrix is ${G_{\lambda /4}}$, where λ represents the working wavelength of the laser.

$${G_{\lambda /4}}=\frac{1}{{\sqrt 2 }}\left[ {\begin{array}{*{20}{c}} 1&{ - i} \\ { - i}&1 \end{array}} \right]$$

(5)

The light emitted by the laser is modulated into circularly polarized light and then incident vertically on the beam splitter prism. After passing through the standard lens, it reaches the surface of the measured component. The wavefront surface information of the measured surface is represented by: $E=A\exp \left[ {iW\left( {x,y} \right)} \right]$, where A presents the amplitude, and $W\left( {x,y} \right)$ presents the phase information. The Jones vector represents the test wavefront reflected by the test optical surface under test.

$${E_0}={E_i} \cdot {G_{\lambda /4}} \cdot E=\left[ {\begin{array}{*{20}{c}} 1 \\ 0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1&{ - i} \\ { - i}&1 \end{array}} \right] \cdot A\exp \left[ {iW\left( {x,y} \right)} \right]=A\exp \left[ {iW\left( {x,y} \right)} \right]\left[ {\begin{array}{*{20}{c}} 1 \\ { - i} \end{array}} \right]$$

(6)

The test wavefront passes through a shearing displacer, which consists of a polarizer with a polarization direction of θ, a birefringent crystal, and a quarter wave plate, resulting in lateral shearing. The Jones matrix is given as follows:

$$P=\left[ {\begin{array}{*{20}{c}} {{{\cos }^2}\theta }&{\sin \theta \cos \theta } \\ {\sin \theta \cos \theta }&{si{n^2}\theta } \end{array}} \right]$$

(7)

The transmission axis of the polarizer is aligned with the optical axis of the crystal, forming a 45° angle with the x-axis. The Jones vector of the test wavefront after passing through the polarizer, is denoted as:

$${E_1}=P \cdot E=\left[ {\begin{array}{*{20}{c}} 1&1 \\ 1&1 \end{array}} \right]A\exp \left[ {iW\left( {x,y} \right)} \right]\left( {1 - i} \right)\left[ {\begin{array}{*{20}{c}} 1 \\ { - i} \end{array}} \right]=A\exp \left[ {iW\left( {x,y} \right)} \right]\left( {1 - i} \right)\left[ {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right]$$

(8)

After passing through the birefringent crystal, lateral shearing occurs. During this process, the propagation direction of the ordinary (o) -light remains unaltered, representing the original light wave. Meanwhile, the extraordinary (e)-light experiences a lateral displacement of s-distance, producing a replicated light wave of the original one. The Jones vector for the sheared light wave emerging from the crystal is denoted as:

$$\begin{gathered} {E_o}=\left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&0 \end{array}} \right]A\exp \left[ {iW\left( {x,y} \right)} \right]\left( {1 - i} \right)\left[ {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right]=A\exp \left[ {iW\left( {x,y} \right)} \right]\left[ {\begin{array}{*{20}{c}} 1 \\ 0 \end{array}} \right] \hfill \\ {E_e}=\left[ {\begin{array}{*{20}{c}} 0&0 \\ 0&1 \end{array}} \right]A\exp \left[ {iW\left( {x+s,y} \right)} \right]\left( {1 - i} \right)\left[ {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right]=A\exp \left[ {iW\left( {x+s,y} \right)} \right]\left[ {\begin{array}{*{20}{c}} 0 \\ 1 \end{array}} \right] \hfill \\ \end{gathered}$$

(9)

After passing through a quarter-wave plate with its fast axis aligned at a 45^° angle to the x-axis, the Jones vector of the transmitted light beam is given by:

$$\begin{gathered} {E_r}={G_{\lambda /4}} \cdot {E_o}=\frac{1}{{\sqrt 2 }}\left[ {\begin{array}{*{20}{c}} 1&{ - i} \\ { - i}&1 \end{array}} \right]A\exp \left[ {iW\left( {x,y} \right)} \right]\left[ {\begin{array}{*{20}{c}} 1 \\ 0 \end{array}} \right]=\frac{A}{{\sqrt 2 }}\exp \left[ {iW\left( {x,y} \right)} \right]\left[ {\begin{array}{*{20}{c}} 1 \\ i \end{array}} \right] \hfill \\ {E_t}={G_{\lambda /4}} \cdot {E_e}=\frac{1}{{\sqrt 2 }}\left[ {\begin{array}{*{20}{c}} 1&{ - i} \\ { - i}&1 \end{array}} \right]A\exp \left[ {iW\left( {x+s,y} \right)} \right]\left[ {\begin{array}{*{20}{c}} 0 \\ 1 \end{array}} \right]=\frac{A}{{\sqrt 2 }}\exp \left[ {iW\left( {x+s,y} \right)} \right]\left[ {\begin{array}{*{20}{c}} { - 1} \\ i \end{array}} \right] \hfill \\ \end{gathered}$$

(10)

According to the Jones matrix formalism, the polarization states of the two beams, which carry phase information from the measured surface, are orthogonal to each other. Therefore, these two polarized lights waves are in orthogonal polarization states.

$${E_r}^{T} \cdot {E_t}^{*}=0$$

(11)

When orthogonal polarized light passes through a polarizer array, it acquires Jones matrices ${E_r}^{\prime }$ and ${E_t}^{\prime }$ corresponding to the direction of the electric field. The Jones matrix of a polarizer can be expressed as:

$$P=\left[ {\begin{array}{*{20}{c}} {{{\cos }^2}\alpha }&{\sin \alpha \cos \alpha } \\ {\sin \alpha \cos \alpha }&{{{\sin }^2}\alpha } \end{array}} \right]$$

(12)

After passing through the polarizer array, the Jones vectors of ${E_r}^{\prime }$ and ${E_t}^{\prime }$ are:

$$\begin{aligned} & E_{r} ^{\prime } = P \cdot E_{r} = \left[ {\begin{array}{*{20}c} {\cos ^{2} \alpha } & {\sin \alpha \cos \alpha } \\ {\sin \alpha \cos \alpha } & {\sin ^{2} \alpha } \\ \end{array} } \right]\frac{A}{{\sqrt 2 }}\exp \left[ {iW(x,y)} \right]\left[ {\begin{array}{*{20}c} 1 \\ i \\ \end{array} } \right] \\ & \quad \quad \quad \quad \quad = \frac{A}{{\sqrt 2 }}\exp \left[ {iW(x,y) + \alpha } \right]\left[ {\begin{array}{*{20}c} {\cos \alpha } \\ {\sin \alpha } \\ \end{array} } \right] \\ & E_{t} ^{\prime } = P \cdot E_{t} = \left[ {\begin{array}{*{20}c} {\cos ^{2} \alpha } & {\sin \alpha \cos \alpha } \\ {\sin \alpha \cos \alpha } & {\sin ^{2} \alpha } \\ \end{array} } \right]\frac{A}{{\sqrt 2 }}\exp \left[ {iW(x + s,y)} \right]\left[ {\begin{array}{*{20}c} { - 1} \\ i \\ \end{array} } \right] \\ & \quad \quad \quad \quad \quad = - \frac{A}{{\sqrt 2 }}\exp \left[ {iW(x + s,y) - \alpha } \right]\left[ {\begin{array}{*{20}c} {\cos \alpha } \\ {\sin \alpha } \\ \end{array} } \right] \\ \end{aligned}$$

(13)

From this, the distribution of interference fringes on the detector imaging plane can be obtained, and the intensity of the interference fringes can be expressed as follows:

$$I=({E_r}^{\prime }+{E_t}^{\prime }){({E_r}^{\prime }+{E_t}^{\prime })^*}=\frac{A}{2}\left\{ {1+\cos \left[ {W(x,y) - W(x+s,y)+2\alpha } \right]} \right\}+C$$

(14)

In Eq. (14), $W(x,y) - W(x+s,y)$ represents differential phase, denoted by $\Delta W$, which is the phase measured by lateral shearing interferometry. Similarly, by rotating the polarized lateral shearing beam displacer along the vertical direction of the x-axis, lateral shearing along the y-axis can be achieved. Here, C accounts for the constant background intensity (for instance, environmental light or stray light), and the term $\frac{A}{2}\left\{ {1+\cos \left[ {W(x,y) - W(x+s,y)+2\alpha } \right]} \right\}$describes the varying part of the light intensity based on polarization. To acquire the phase data of the test surface, it is essential to use differential phase measurement to compute the original phase. Due to the 45° angle difference between the transmission axes of the four regions of the polarizer array, the four beams of light passing through the shearing displacer system will exhibit phase-shifted interference when they pass through the polarizer array. From Eq. (14), the light intensity of the four phase-shifted interference fringe patterns can be derived.

$$\left\{ \begin{aligned} & I_{{0^{^\circ } }} = \frac{A}{2}\left[ {1 + \cos \left( {\Delta W} \right)} \right] + C \\ & I_{{45^{^\circ } }} = \frac{A}{2}\left[ {1 + \cos \left( {\Delta W + \frac{\pi }{2}} \right)} \right] + C \\ & I_{{90^{^\circ } }} = \frac{A}{2}\left[ {1 + \cos \left( {\Delta W + \pi } \right)} \right] + C \\ & I_{{135^{^\circ } }} = \frac{A}{2}\left[ {1 + \cos \left( {\Delta W + \frac{{3\pi }}{2}} \right)} \right] + C \\ \end{aligned} \right.$$

(15)

The differential phase $\Delta W$can be obtained using the four-step phase-shifting algorithm.

$$\Delta W=\arctan \left[ {\frac{{{I_{{{135}^ \circ }}} - {I_{{{45}^ \circ }}}}}{{{I_{{0^ \circ }}} - {I_{{{90}^ \circ }}}}}} \right]$$

(16)

Experimental system

Based on the optical path of the multi-directional lateral shearing synchronous phase-shifting interferometry, the structural design of the multi-directional shearing displacer component was developed. This includes the spatial splitting technology and the design of the polarizer array design to achieve polarization synchronous phase-shifting.

Modularization of the multi-directional shearing displacer

The multi-directional shearing device utilizes a uniaxial birefringent crystal to shear the wavefront. It consists of a polarizer, a birefringent crystal, and a quarter wave plate. By adopting a modular design, the polarization state of the system remains unchanged throughout the multi-directional shearing process. This minimizes the impact on the polarization state and effectively prevents system errors caused by the rotation of individual components.

Designing the lateral shear amount in birefringent crystals

The selection of the shear amount is a critical parameter in shearing interferometry. The magnitude of the shear determines the number of sampling points required for phase recovery, which significantly affects the final surface reconstruction. The shear amount is primarily determined by the thickness of the birefringent crystal, with the crystal’s properties being established based on crystal optics theory and the practical requirements of the system.

The displacement principle of a uniaxial birefringent crystal is illustrated in Fig. 4. When a beam of monochromatic light is incident on the crystal surface at an angle of $\omega$ between the optic axis and the normal to the incident surface at ${\theta _i}$, it produces two parallel beams with mutually orthogonal directions, known as the o-light and e-light, respectively. ${k_e}$ presents the wave normal of e-light, ${\theta ^{\prime\prime}_t}$ presents the refraction angle of o-light, ${\theta ^{\prime}_t}$ presents the refraction angle of e-light, ${\theta _t}$ presents the discrete angle of waves normal for both o-light and e-light. The d presents the thickness of the crystal PBD, the principal refractive index of o-light denoted as ${n_o}$, and the principal refractive index of e-light denoted as ${n_e}$. In case, the S represents the displacement distance between o-light and e-light, it signifies the shift amount of lateral sheared of the crystal PBD along the x-direction or y-direction. When light traveled inside a crystal, the refractive index of o-light remains unchanged as ${n_o}$. However, the refractive index of e-light varied with the discrete angle ${\theta _t}$ between waves normal of o-light and e-light. Therefore, there is ${n^{\prime}_e}\left( {{\theta _t}} \right)={{{n_o}{n_e}} \mathord{\left/ {\vphantom {{{n_o}{n_e}} {\sqrt {n_{o}^{2}{{\sin }^2}\left( {{\theta _t}} \right)+n_{e}^{2}{{\cos }^2}\left( {{\theta _t}} \right)} }}} \right. \kern-0pt} {\sqrt {n_{o}^{2}{{\sin }^2}\left( {{\theta _t}} \right)+n_{e}^{2}{{\cos }^2}\left( {{\theta _t}} \right)} }}$.

Fig. 4

The displacement principle of the crystal PBD. (Reprinted with permission from Ref.¹⁶, Fig. 9. © 2024 Optical Society of America), (a) general, (b) normal incidence while maintaining a 45° angle with the crystal optical axis.

When the wavefronts normal of the o-light and e-light are aligned, and the thickness of the crystal is sufficient, the two light beams emanating from the crystal’s lower surface will experience a lateral displacement. As shown in Fig. 4b, the Huygen’s principle is applied. The wave normal direction of both o-light and e-light within the crystal are identical, meaning the incident light is perpendicular to the crystal surface at an angle of ${\theta _i}={0^{{\circ }}}$degree. In the case of o-light, its wave normal is aligned with the same direction, thereby making it perpendicular to the crystal surface. However, for the e-light, there is a certain angle, ${\theta _t}$, between the direction of the e-light and its normal. In the main section of the crystal, the transmitted e-light will experience a lateral displacement relative to the incident light. The included angle between the e-light and the e-light wave normal can be calculated by Eq. (17) (Reprinted with permission from Ref¹⁶. , Eq. (25). © 2024 Optical Society of America).

$$\tan {\theta _t}=\left( {1 - \frac{{n_{o}^{2}}}{{n_{e}^{2}}}} \right) \cdot \frac{{\tan \left( {{{\theta ^{\prime}}_t}+\omega } \right)}}{{\left( {1+\frac{{n_{o}^{2}}}{{n_{e}^{2}}}} \right){{\tan }^2}\left( {{{\theta ^{\prime}}_t}+\omega } \right)}}$$

(17)

Since the wave normal of the e-light wave is the same as that of the o-light, ${\theta _t}$ presents the separation angle between the o-light and e-light. Therefore, the crystal shear displacement can be calculated using Eq. (18).

$$s=d \cdot \tan {\theta _t}$$

(18)

According to the birefringence theory of uniaxial crystals, ${\theta _t}$presents the deflection angle, ${n_e}(\omega )$presents the refractive index, S presents shear amount, and $\Delta \phi$presents the phase difference of the e-light. These parameters satisfy the following conditions. The relationship between the incident angle and the shear displacement is simulated using Eq. (19), as shown in Fig. 6. When the crystal’s optical axis is aligned at $\omega$ =  45°, the crystal attains maximum shear displacement. After careful calculation, it can be concluded that a birefringent crystal with a refractive index ${n_o}$ = 1.65578 and${n_e}$  = 1.48520, and a thickness of d = 9.28 mm, is required. When the optical axis $\omega$ is oriented at a 45° angle to the crystal surface, the shear amount S reaches a maximum of 1.005 mm.

$$\left\{ \begin{aligned} & S = d \cdot tan\theta _{t} = d \cdot \left( {1 - \frac{{n_{o}^{2} }}{{n_{e}^{2} }}} \right) \cdot \frac{{tan(\theta _{t}^{\prime } + \omega )}}{{\left( {1 + \frac{{n_{o}^{2} }}{{n_{e}^{2} }}} \right)tan^{2} (\theta _{t}^{\prime } + \omega )}} \\ & n_{e} (\omega ) = \frac{{n_{o} n_{e} }}{{\sqrt {n_{o}^{2} \sin ^{2} \omega + n_{e}^{2} \cos ^{2} \omega } }} \\ & \Delta \phi = \frac{{2\pi }}{\lambda }\frac{d}{{\cos \frac{{\theta _{t} }}{2}}}\left[ {n_{e} (\omega ) - n_{o} } \right] \\ \end{aligned} \right.$$

(19)

Fig. 5

The relationship between incident angle and shear displacement.

Optical structure of the multi-directional shearing displacer

A shearing displacer, composed of a polarizer, a birefringent crystal, and a quarter wave plate, is a core device for achieving multi-directional shearing. The polarizer is used to modify the polarization state of light before it enters the crystal. Based on the principle of two-beam interference, high-contrast interference fringes can be obtained when the light intensities of both beams are comparable. The angle between the transmission axis of the polarizer and the optical axis of the crystal is set to 45°. To ensure equal intensity for both the o-light and e-light, the two beams with orthogonal polarization states emitted from the crystal must pass through a quarter-wave plate, with its fast axis aligned to the optical axis of the crystal. This setup converts the light into left-handed and right-handed circularly polarized light. Polarization phase-shifting can then be achieved by adding polarizing plates at various angles. The positional and angular relationships of each component in the shearing displacer are illustrated in Fig. 6. The shearing displacer has the ability to modulate the incident beam into two light waves with lateral offset. As the displacer rotates, it effectively achieves wavefront shearing in different directions.

Fig. 6

Schematic diagram of the relative positions and angle relationships between the components of the shearing displacer.

The transmission characteristics of a multi-directional shearing beam displacer are as follows: The wavefront returned by the test optical surface is modulated into linearly polarized light through a polarizer, which can be represented as a Jones vector.

$${E_i}=\left[ {\begin{array}{*{20}{c}} {{E_x}} \\ {{E_y}} \end{array}} \right]={E_0}\exp \left[ {iW(x,y)} \right]\left[ {\begin{array}{*{20}{c}} {\cos \theta } \\ {\sin \theta } \end{array}} \right]$$

(20)

After passing through a birefringent crystal, the linearly polarized light is split into o-light and e-light, with orthogonal vibration directions, denoted as ${E_o}$ and ${E_e}$, respectively. Their Jones vectors are

$${E_o}=\left[ {\begin{array}{*{20}{c}} {\cos \theta } \\ 0 \end{array}} \right]\exp \left[ {iW(x,y)} \right]$$

(21)

$${E_e}=\left[ {\begin{array}{*{20}{c}} 0 \\ {\sin \theta } \end{array}} \right]\exp \left[ {iW(x+s,y)} \right]$$

(22)

The o-light and e-light pass through a quarter-wave plate with its fast axis aligned at a 45° angle to the x- axis, which can be represented using a Jones matrix.

$$\begin{aligned} E^{\prime } _{o} = G_{{\frac{\lambda }{4}}} \cdot E_{o} & = \frac{1}{{\sqrt 2 }}\left[ {\begin{array}{*{20}c} 1 & i \\ i & 1 \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}c} {\cos \theta } \\ 0 \\ \end{array} } \right]\exp \left[ {iW(x,y)} \right] \\ & = \frac{{\cos \theta \cdot \exp \left[ {iW(x,y)} \right]}}{{\sqrt 2 }}\left[ {\begin{array}{*{20}c} 1 \\ i \\ \end{array} } \right] \\ \end{aligned}$$

(23)

$$\begin{aligned} E_{e}^{\prime } = G_{{\frac{\lambda }{4}}} \cdot E_{e} & = \frac{1}{{\sqrt 2 }}\left[ {\begin{array}{*{20}c} 1 & {amp;i} \\ i & {amp;1} \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}c} 0 \\ {\sin \theta } \\ \end{array} } \right]\exp \left[ {iW(x + s,y)} \right] \\ & = \frac{{\sin \theta \cdot \exp \left[ {iW(x + s,y)} \right]}}{{\sqrt 2 }}\left[ {\begin{array}{*{20}c} { - 1} \\ i \\ \end{array} } \right] \\ \end{aligned}$$

(24)

According to Eqs. (23) and (24), the shearing displacer modulates the incident light wave into left-handed and right-handed circularly polarized light. Phase-shifting interferometry can be achieved by introducing a phase shift through a polarizer placed after the shearing device. By demodulating the interferogram information and applying a reconstruction algorithm, the differential phase information of the optical elements under test can be obtained. The optical axis of a birefringent crystal lies along the x-axis, so the polarization direction of the polarizer should be aligned at a 45° angle to the crystal’s optical axis. This alignment ensures that the o-light and e-light emitted from the crystal have equal components, which guarantees excellent contrast in the fringes during subsequent interference. The fast axis of the quarter-wave plate is aligned at a 45° angle to the x-axis, enabling the modulation of the ${E_o}$ and ${E_e}$ emitted by the crystal into left- and right- handed circularly polarized light. The multi-directional shearing displacer features a cylindrical structure design. In the experiment, a polarizer with an extinction ratio of 800:1 and a diameter of 25.4 mm, along with a zero-order quarter-wave plate (also with a 25.4 mm diameter, @ THORLABS Inc.), were used. The dimensions of the single-axis crystal are 15 mm×15 mm ± 0.2 mm, with a thickness of 9.28 mm ± 0.2 mm.

Design of spatial beam splitting and synchronous polarization phase-shifting

Light splitter system

The structure of the light splitter system, as shown in Fig. 7, consists of a two-dimensional grating, a converging lens, and a small aperture. When a plane wave is incident on a two-dimensional phase checkerboard grating, zero order diffraction light and four orders of diffraction light can be obtained, all with identical intensity distribution, for measurement purposes.

Fig. 7

Schematic diagram of the light splitter structure using a two-dimensional grating and an aperture diameter.

In the light splitter structure, the diffraction properties of two-dimensional gratings are primarily used to generate four identical beams of light. To optimize light energy utilization, a two-dimensional chessboard grating is employed as the splitter device. The splitting effect is tested by designing and implementing a two-dimensional phase grating based on the optical path structure shown in Fig. 8a, resulting in four interferograms with a π/2 phase shift in a single frame. The phase-shifted interferometry images, obtained through the spatial splitter phase-shifting scheme, are shown in Fig. 8b.

Fig. 8

The structure and phase-shifted interferograms of synchronous phase-shifting interferometry scheme. (a) Schematic diagram of synchronous polarization phase-shifting scheme structure, (b) phase shifted interferograms obtained through synchronous polarization phase-shifting interferometry.

Polarizer array

The phase shift scheme of the multi-directional shearing polarization phase-shifting interferometry measurement system utilizes a polarizer array composed of four polarizers spliced together. The schematic diagram of its structure is shown in Fig. 9a, while the physical image is shown in Fig. 9b. By using this array to modify the phase shifts of the four light beams after they pass through the filtering system, with each polarizer in the array oriented at a 45° difference, a phase shift of π/2 be achieved.

Fig. 9

The structure of polarizer array for synchronous polarization phase-shifting interferometry. (a) Schematic diagram of polarization array, (b) physical map of polarizer array.

The customized polarizer array used in the experiment has dimensions of 20 × 20 mm and a thickness ranging from 3 mm to 4 mm. As shown in Fig. 9b, it consists of four linear polarizers, each with an optical axis size of 10 × 10 mm, arranged and spliced together. The optical axis angles are 0°, 45°, 90°, and 135°, with a splicing gap of 0.2 mm and an optical axis accuracy of ± 2°.

Experimental setup

The experimental setup was constructed based on the optical path of the multi-directional lateral shearing synchronous polarization phase-shifting interferometry system. The completed specific optical path is shown in Fig. 10.

Fig. 10

Experimental setup of the testing system for multi-directional lateral shearing synchronous phase-shifting interferometry.

The experimental system was constructed based on the design of key components, with the parameters and selection of these components outlined as follows. In the experimental setup, the light emitted by the laser must be processed. Microscopic objectives and pinhole filters are used for fine adjustments. The laser beam is focused by the microscopic objective and then diffracted through a micrometer-sized aperture. By utilizing diffracted 0th-order light, the light source becomes more uniform, effectively minimizing noise during the interference process. In the optical setup measurement system, a He-Ne laser (model: HRS015B, wavelength: 632.992 nm, output power: 1.2–2.7mW, power stability: > 1.2 mW, @Thorlabs Inc.), an attenuator (model: NDC-50 S-1, wavelength range: 240–1200 nm, Attenuation patch size: Ø50 mm, optical density range: 0.04–1.0, @Thorlabs Inc.), a quarter-wave plate (Zero-level quarter wave plate, Ø25.4 mm, @Thorlabs Inc.), a microscopic objective (model: GCO-2131, magnification: 10×, numerical aperture: 0.25, work distance: 15.12, focal distance: 15.381. Aperture model: GCO-P05A, aperture size: 5 μm, @Thorlabs Inc.), a pinhole filter (diameter: Ø25.4 mm, aperture size: 5 μm, @Thorlabs Inc.), a beam splitter. Then, one side of the interferometer includes a collimating lens, an aperture, a standard lens (convergent lens, aperture: 50.8 mm, focal length: 200 mm. @ Thorlabs Inc.), and the optical component under test. Another side of the interferometer includes a collimating lens (focal distance: 200 mm, Ø50.8 mm, @Thorlabs Inc.), a plane mirror, a multi-directional shearing beam displacer, it contains a linear polarizer, a polarization shearing beam displacer (model: CN2103091401, design size: 15 mm × 15 mm ± 0.2 mm, thickness: 9.28 mm ± 0.2 mm, @Union Optic Inc.), and a quarter-wave plate (for details, see Fig. 6). Splitting beam component of phase grating, it contains a chessboard two-dimensional phase grating (refractive index: 1.457, duty ratio: 1:1, period: 20 μm, etching depth: 692 nm, diffraction efficiency of four first-order: > 60%), a converging lens (LB1596-A, Ø = 25.4, f = 100 mm, @ Thorlabs Inc.) and an aperture-diameter. After this, it presents a same converging lens and a polarizer array (design size: 20 mm × 20 mm, polarization angles: 0°, 45°, 90° and 135°, accuracy of optical axis: < 2°), an imaging lens (LB1596-A, Ø = 25.4, f = 150 mm, @ Thorlabs Inc.), and a CCD camera (2/3 inch Sony CMOS Pregius sensor (IMX250), resolution ratio: 2,448 × 2,048 (5 MP), 75fps. @ Imaging Source, Inc.). Finally, by precisely adjusting the optical path, the various optical elements are aligned along the same optical axis and positioned accurately relative to each other, resulting in clear interference fringe patterns.

Results

To verify the accuracy of the multi-directional lateral shearing polarization synchronous phase-shifting interferometry setup, measurements were conducted on a spherical mirror with a diameter of 50 mm and a curvature radius of 400 mm, using both the constructed setup and the ZYGO interferometer. The multi-directional polarization synchronized phase-shifting shearing interference patterns obtained from the experimental measurements are shown in Fig. 11.

Fig. 11

Phase-shifting interferograms with different shear directions obtained from experiments. (a) 0°, (b) 45°, (c) 90°, (d) 135°, (e) 180°, (f) 225°, (g) 270°, (h) 315°.

After preprocessing the shearing interferogram, the shear amount was calculated to be 36 pixels, and the actual interference area was determined. Phase extraction and unwrapping were performed on the interferogram. Finally, the differential Zernike algorithm was applied to fit the differential wavefront in two directions, effectively eliminating adjustment errors and defocusing. Figure 12 illustrates the reconstruction results for various shear directions. Additionally, when the testing light tilts and incident on the polarization shearing beam displacer (PBD) within the range of − 0.4° ~ 0.4°, the shear distance be corrected for wavefront reconstruction, yielding high-precision wavefront testing result. This part is not discussed in detail, for further information, please refer to the latest Ref.²¹.

Fig. 12

Reconstruction results for various shear directions. (a) 0° and 45°, (b) 90° and 135°, (c) 180° and 225°, (d) 270° and 315°.

The Zernike coefficients, obtained by solving the differential information in various directions, are averaged. By processing phase-shifted fringe interferograms in different shear directions and applying the four-step phase-shifting algorithm to extract phase information, differential phases in multiple shear directions can be obtained. Using the obtained differential phases, the differential information in any two directions is employed to fit the differential Zernike polynomial, which reconstructs the original wavefront and yields the 36 Zernike coefficients for surface fitting^22,23,24. These coefficients, referred to as fusion coefficients, are then used to generate the final reconstructed result, as shown in Fig. 13a. Specifically, the PV value is 0.296 λ and the RMS value is 0.039 λ. Meanwhile, the measurement result obtained from the ZYGO interferometer on identical spherical mirror as shown in Fig. 13b, revealed a PV value of 0.248 λ and an RMS value of 0.039 λ.

Fig. 13

Measurement results of (a) multi-directional lateral shearing interferometry and (b) ZYGO interferometer.

To verify the accuracy of the constructed measurement setup, the section line results at the central position of the measurement data are extracted and shown in Fig. 14a. To further optimize the number of directional derivatives for wavefront reconstruction, we identify the set of directional derivatives that minimizes the normalized error. Specifically, interference patterns are obtained from different shearing directions, and the error for each direction is calculated. The directional with the smallest error is then selected for wavefront reconstruction. By choosing the direction with the least derivative error, the accuracy of the wavefront reconstruction is improved. Finally, the wavefront is reconstructed using the differential Zernike polynomial fitting algorithm, as shown in Fig. 14b. The core implementation code is as follows:

Fig. 14

Section line results at central position of (a) directional derivatives with constant normalized error, (b) after determining the optimal number of directional derivatives, (c) four directional derivatives, (d) sixteen directional derivatives.

By utilizing shear wavefront information from various orthogonal shear directions, we compute the multiple directional derivatives from the multiple orthogonal shear directions, and comparison between the result of using multiple derivatives, as shown in Fig. 14c,d. After performing the derivatives and comparing the reconstruction results for the same spherical object, the corresponding fitting data are presented in Table 1. Based on the steepness of the surface of the tested component, the PV and RMS values of the reconstructed wavefront remain almost unchanged after sixteen or more directional derivatives. The best direction index corresponding to the minimum measurement error for this tested component is sixteen directional derivatives. It is evident that there is minimal relative deviation between the two testing results, with the ΔRMS surpassing 0.01λ.

Table 1 Reconstructing data using multiple orthogonal shearing derivatives and the ZYGO interferometer (λ = 632.8 nm).

From the measurement results show that the outcomes of multi-directional lateral shearing interferometry closely align with those obtained from the ZYGO interferometer. However, due to the inherent presence of systematic errors in the actual measurement system, some deviations between the final measurement results and those from the ZYGO remain. Specifically, after the calculation, the deviation in the ΔPV value between the constructed measurement setup and ZYGO interferometer is 0.0448λ, while the deviation of the ΔRMS value is 0.002λ. This indicates that the multi-directional lateral shearing polarization synchronous phase-shifting interferometry system developed in our study is accurate. When compared to the ZYGO interferometer, our system achieves a PV error of λ/20 and an RMS error of λ/100.

Discussion

Calibration of the transmission axis angle of the polarizer array

The azimuth error of a polarizer array is defined as the angular deviation between the direction of any polarizer and its ideal direction, denoted by ${\theta _i}$. Consequently, when an angle error is present in the polarizer, the intensity distribution of the interference field can be expressed as:

$${I_i}=\frac{1}{2}\left[ {{a^2}+{b^2}+2ab\sin \left( {\varphi +2{\theta _i}} \right)} \right]+C$$

(25)

In which, a and b represent the amplitudes of the two beams of light after shearing, ${\theta _i}$ represents the azimuth angle of the polarizer, and $\varphi$ represents the differential phase. By differentiating ${\theta _i}$ in Eq. (25), we can obtain:

$$\frac{{\partial {I_i}}}{{\partial {\theta _i}}}=2ab\cos (\varphi +2{\theta _i})$$

(26)

Thus, the phase difference is obtained as:

$$\Delta \varphi =\left[ {\frac{{\cos \varphi }}{{2ab}}\left( {\frac{{\partial {I_1}}}{{\partial {\theta _i}}} - \frac{{\partial {I_3}}}{{\partial {\theta _i}}}} \right)+\frac{{\sin \varphi }}{{2ab}}\left( {\frac{{\partial {I_4}}}{{\partial {\theta _i}}} - \frac{{\partial {I_2}}}{{\partial {\theta _i}}}} \right)} \right]\Delta {\theta _i}$$

(27)

Let ${\theta _2}={\theta _1}+{45^ \circ }$, ${\theta _3}={\theta _1}+{90^ \circ }$, ${\theta _4}={\theta _1}+{135^ \circ }$, calculate to obtain,

$$\Delta \varphi =\varphi \cos \left( {\varphi +2{\theta _1}} \right)\Delta {\theta _3}+\sin \varphi \sin \left( {\varphi +2{\theta _1}} \right)\left( {\Delta {\theta _4}+\Delta {\theta _2}} \right)$$

(28)

From the calculation results, it evident that, ${(\Delta \varphi )_\theta }_{{\hbox{max} }} \approx 3\Delta \theta$, and the azimuth error of the polarizer is comparable in magnitude to the measurement error, thereby significantly influence on the measurement results. Therefore, it is necessary to focus on controlling and reducing the azimuth error, and obtain the azimuth deviation via a calibration scheme.

The calibration principle of polarization array involves collecting instantaneous phase-shifting fringe data, acquiring grayscale data at the same position point from two adjacent images, and then fitting a light intensity curve using the least squares method. Based on this light intensity curve, a Lissajous figure is created, allowing for a qualitative determination of the deviation between the azimuth angle difference of adjacent polarizers and 90°. Specifically, by calculating the phase difference of the sine curve, we can obtain a quantitative value for the azimuth angle difference, which serves to verify of the phase shift accuracy in the polarization array. The principle of Lissajous curve fitting is as follows:

$$\begin{gathered} {I_1}={a_1}+{A_1}\cos ({\omega _1}x+{\varphi _1})+C \hfill \\ {I_2}={a_2}+{A_2}\cos ({\omega _2}x+{\varphi _2})+C \hfill \\ \end{gathered}$$

(29)

In which, ${a_1}$ and ${a_2}$ represent the background light intensity of the interference pattern, ${A_1}$ and ${A_2}$ are the amplitudes of the two interfering light waves, ${\omega _1}$ and ${\omega _2}$are the vibration frequencies of two light beams, ${\varphi _1}$ and ${\varphi _2}$represent the initial phase. C presents for any constant background intensity. The curve depicted in Fig. 15 was simulated based on the Lissajous principle.

Fig. 15

Lissajous simulation curve demonstrates that the phase difference between ${I_1}$ and ${I_2}$ is (a) 90°, (b) not 90°.

A perfect circle is achieved as the fitting result when the phase difference between ${I_1}$ and ${I_2}$ is 90°, On the other hand, an ellipse is the fitting result when the phase difference between ${I_1}$ and ${I_2}$ is not 90°. Therefore, the deviation between the two phases and 90^° can be ascertained by evaluating whether their fitting results form a perfect circle. The experiment utilizes the Lissajous figure scheme to calibrate the transmission axis angle of the polarizer array. The interference pattern obtained from the experiment is used to extract the light intensity, and the light intensity distribution is fitted with least squares method, yielding the following results:

$$\left[ {\begin{array}{*{20}{c}} N&{\sum {\cos ({\varphi _i})} }&{\sum {\sin ({\varphi _i})} } \\ {\sum {\cos ({\varphi _i})} }&{\sum {{{\cos }^2}({\varphi _i})} }&{\sum {\frac{{\sin (2{\varphi _i})}}{2}} } \\ {\sum {\sin ({\varphi _i})} }&{\sum {\frac{{\sin (2{\varphi _i})}}{2}} }&{\sum {\sin ({\varphi _i})} } \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} a \\ m \\ n \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {\sum {{I_i}} } \\ {\sum {{I_i}\cos ({\varphi _i})} } \\ {\sum {{I_i}\sin ({\varphi _i})} } \end{array}} \right]$$

(30)

In this case, there are$m=b\cos \delta$, $n= - b\sin \delta$, according to the Eq. (30), the specific phase shift amount between synchronous phase shift interferograms can be calibrated.

According to the above analysis, it can be inferred that when the azimuth angle difference between two polarizers in the polarizer array is 45°, the phase shift between each pair of synchronized phase-shifting interferograms should be π/2. The precision of the phase shift depends on the accuracy of the polarizer azimuth angle. The experiment was conducted using a differential interferometry measurement system for polarization synchronous phase-shifting. Figure 16a shows the synchronous phase-shifting interferometry images collected by the experimental system. By extracting the gray values from the four phase-shifting interference images within the same area, it was found that the intensity of the four beams of light is nearly identical, as shown in Fig. 16b. This result confirms that the angle adjustment of the quarter-wave plate was accurate during the calibration process in the experiment.

Fig. 16

Analysis of interferograms collected during experiments. (a) Four step phase-shifting interferogram, (b) Intensity curve corresponding to the four-step phase-shifting interferogram.

The gray values of each phase-shifted interferogram are extracted and fitted using the least squares method to obtain the intensity curve, as shown in Fig. 17. By fitting the actual light intensity curve to the Lissajous figure, the deviation between the phase shift from a π/2 can be determined. Figure 18 displays the Lissajous figure derived from the light intensity fitting curves of adjacent polarizers. After fitting, the resulting Lissajous figures closely resemble perfect circle, indicating that the phase shift produced by the polarizer array in the experimental system is approximately π/2.

Fig. 17

Light intensity curve fitted to the measured values. (a) I₁, (b) I₂, (c) I₄, (d) I₃.

Fig. 18

Fitting Lissajous figures to the curve of realistic light intensity. (a) I₁ and I₂, (b) I₂ and I₃, (c) I₂ and I₄, (d) I₄ and I₁.

By calculating the phase difference of the sine curve, the azimuth angle differences were obtained as follows: ${\delta _{12}}$ = 89.1°, ${\delta _{23}}$ = 90.8°, ${\delta _{34}}$= 91.2°, and ${\delta _{41}}$= 89.3°. MATLAB simulations indicated that the phase shift error did not exceed 0.02λ. This method can be used to calibrate the transmission axis angles of the polarizer array, effectively correcting phase shift errors during the reconstruction process.

Registration of phase-shifting interference patterns

The registration of interference images is a crucial step in the preprocessing of interferometric data. In a single frame on the CCD, four phase-shifted interferograms are captured based on the system’s optical path. However, to extract phase information, it is necessary to perform image segmentation and registration on the four interference images. This ensures that each point in the interferogram corresponds one-to-one. The images registration diagram of interferograms is shown in Fig. 19.

Fig. 19

Schematic diagram of interferogram position registration.

Four phase-shifting interferograms are obtained by segmenting a single frame into four phase-shifting interferograms. Specifically, using one interferogram as a reference location, the coordinates of the test points corresponding to the other interferograms will exhibit a lateral spatial position mismatch. To achieve spatial position registration of the phase-shifting interferograms, the phase correlation method²⁵ is employed. The central concept of phase correlation is that when an image undergoes translational motion, its spatial spectrum experiences a linear phase shift relative to the reference image. If the original image represents ${f_0}$ and the translational motioned image represents${f_1}$, then a phase shift exists between their spatial frequency.

$${f_0}=f(x,y)$$

(31)

$${f_1}=f(x - \Delta x,y - \Delta y)$$

(32)

In which, (∆x, ∆y) represents the relative translation amounts along the x-axis and y-axis, respectively. Fourier transforms are applied to ${f_0}$ and ${f_1}$, respectively. Assuming the image size is$M \times N$, we can obtain as follow:

$${F_0}(u,v)=F(u,v)$$

(33)

$${F_1}(u,v)=F(u,v)\exp \left[ { - j2\pi (\frac{{u\Delta x}}{M}+\frac{{v\Delta y}}{N})} \right]$$

(34)

From the above Eqs. (33) and (34), the normalized power spectrum of ${F_0}(u,v)$and ${F_1}(u,v)$ were calculated separately. Subsequently, an inverse Fourier transform was applied to the normalized power spectrum to obtain the phase correlation function.

$$C(x,y)=F{T^{ - 1}}\left( {\frac{{FT\{ {f_1}\} \cdot FT\{ {f_0}\} *}}{{|FT\{ {f_1}\} | \cdot |FT\{ {f_0}\} |}}} \right)=\delta (x - \Delta x,y - \Delta y)$$

(35)

In which, * represents the conjugate operation, and δ represents the Dirac function. From the following Eqs. (36) and (37), the translation amounts of ${f_1}$ relative to ${f_0}$ relative to the x-axis and y-axis can be obtained, which are (∆x, ∆y).

$$\left\{ {\begin{array}{*{20}{c}} {\Delta x= - (M - y - 1),}&{y>N/2} \\ {\Delta x=y - 1,}&{y \leqslant N/2} \end{array}} \right.$$

(36)

$$\left\{ {\begin{array}{*{20}{c}} {\Delta y= - (M - x - 1),}&{x>M/2} \\ {\Delta y=x - 1,}&{x \leqslant M/2} \end{array}} \right.$$

(37)

To find the position of the maximum value from Eqs. (36) and (37), calculate the spatial position mismatch between the two images relative to the reference position, and perform position registration between the interferograms.

As shown in Fig. 20a, four phase-shifting interferograms of a single frame captured by a CCD are simulated. A specific offset is then introduced to the segmented interferograms, and the algorithm’s feasibility is verified using the offset determined through the phase correlation method. Using ${I_1}$, as the reference, the spatial position mismatch between ${I_2}$ and ${I_1}$ is set to (30 pixels, 10 pixels), the spatial position mismatch between ${I_3}$ and ${I_1}$ is set to (−10 pixels, −20 pixels), and the spatial position mismatch between ${I_4}$ and ${I_1}$ is set to (5 pixels, 8 pixels).

Fig. 20

Simulated spot patterns: (a) Four spot patterns collected on a CCD through simulation, (b) spot patterns with positional mismatch error after segmentation.

By utilizing the phase correlation method to compute the speckle pattern with positional mismatch error, as shown in Fig. 20b, the peak coordinates of the pulse function, located at (891 pixels, 871 pixels), were acquired through a coordinate addressing operation, as illustrated in Fig. 21a. Then, using Eqs. (36) and (37), the positional mismatch between ${I_2}$ and ${I_1}$ was calculated to be (30 pixel, 10 pixel). Similarly, based on ${I_1}$, the spatial position mismatch between ${I_3}$ and ${I_1}$ was calculated as (−10 pixels, −20 pixels), as illustrated in Fig. 21b. While the spatial position mismatch between ${I_4}$ and ${I_1}$ was found to be (5 pixels, 8 pixels), corresponds to the preset mismatch value, as illustrated in Fig. 21c. After calibrating the offset, the results are shown in Fig. 22.

Fig. 21

The pulse function was obtained using the phase correlation method. (a) ${I_2}$ and ${I_1}$, (b) ${I_3}$and ${I_1}$, (c) ${I_4}$and ${I_1}$.

Fig. 22

The result following registration.

The experimental system was verified by using a CCD to capture four phase-shifted interferograms within a single frame for registration. The phase correlation method was applied to resolve positional mismatches by segmenting and registering the collected test spot patterns. The results of the interferogram registration using phase correlation method are shown in Fig. 23.

Fig. 23

To segment and register the test spot patterns, the following results were obtained: (a) registered spot patterns, (b) spot patterns after shearing, and (c) registered phase-shifted interferograms.

Calculation of the shear displacement amount

In lateral shearing interferometry, the phase reconstruction of the test wavefront depends on the shear amount. As a result, the shear amount plays a crucial role in determining the accuracy of wavefront reconstruction for the lateral shearing interferometry test wavefront. To determine the shear amount of the interferogram, various methods are typically used, including manual interpretation, edge detection, the Radon transform²⁶, and other techniques²⁷.

The interference pattern generated by lateral shearing has an effective area defined by the overlapping region of two shearing spots. Initially, the area of the two circular shearing spots can be determined, followed by the calculation of the overlapping area to accurately assess the shearing area and shear amount. The lateral shearing interferogram collected during spherical measurement is shown in Fig. 24a,b presents the binary processing result obtained through superposition, while Fig. 24c highlights the region of the acquired interferogram. The results of binary processing provide the leftmost and rightmost edge points of the interferogram. Using these edge points, the regions of the two shear apertures can be fitted (as shown in Fig. 24d,e). Ultimately, the effective interference region of lateral shearing interference is determined, as shown in Fig. 24f.

Fig. 24

Identification of interference regions in lateral shearing interferograms. (a) Original interferogram, (b) binary image, (c) identified area, (d) sheared spot area (left), (e) sheared spot area (right), (f) effective sheared interference area.

Conclusions

In conclusion, a multi-directional shearing synchronous polarization phase-shifting interferometer was studied. The methodology of multi-directional shearing synchronous polarization phase-shifting interferometry first described in detail, with a focus on the measurement optical path and polarization states of the interferometric system. Next, the experimental setup for wavefront testing using this technique was investigated. Key components, including the modular design of the multi-directional shearing displacer, spatial beam splitting scheme, and the synchronous polarization phase-shifting process, were presented. The experimental test was conducted on spherical surface mirrors, and the measurement results were compared with those obtained from a ZYGO interferometer. expressed as the RMS value, was found to be better than 0.01λ, demonstrating the effectiveness and precision of the proposed method for wavefront testing. Additionally, the azimuth error calibration of the polarizer array was discussed, along with an analysis of the numerical results regarding position mismatches and shear amount calculations. These findings further validate the feasibility and accuracy of multi-directional shearing synchronous polarization phase-shifting interferometry for wavefront testing, confirming its potential for higher theoretical precision in optical measurements.