The shearing test is based on the analysis of differences in measurement results that occur when rotating or translating the test surface. The test results yield a collection of error maps. Each error map describes the sum of apparent reference errors and test surface errors for a particular position and orientation of the test surface. If the test part is rotated to *N* equally spaced positions about the optical axis and the resulting, we can get the averaged wavefront

$$ {T}_{ave}\left(\rho, \theta \right)=\frac{1}{N}{\displaystyle \sum_{i=0}^{N-1}{T}_i\left(\rho, \theta \right)}=\frac{1}{N}{\displaystyle \sum_{i=0}^{N-1}\left[R\left(\rho, \theta \right)+S\left(\rho, \theta \right)\right]} $$

(1)

where *R*(*ρ*, *θ*) is the systematic error including the reference surface, *S*(*ρ*, *θ*) is the surface error of the test part.

The wavefront of circular cross section can be expanded by polar coordinate polynomials in the following form

$$ W\left(\rho, \theta \right)={\displaystyle \sum_{k,l}{R}_l^k\left(\rho \right)\left({\alpha}_l^k \cos k\theta +{\alpha}_l^{-k} \sin k\theta \right)} $$

(2)

where \( {R}_l^k\left(\rho \right) \) are the radial terms of Zernike polynomials and coefficients \( {\alpha}_l^{\pm k} \) specify the magnitude of each term while the angular terms specify the angular part of the polynomial representation. *ρ* and *θ* are the normalized radial and angular coordinates.

From Eq. (2), if the wavefront is rotated to *N* equally spaced positions about the optical axis (*φ* = 2*π*/ *N*), the averaged resulting wavefront can be written as

$$ {W}_{ave}\left(\rho, \theta \right)=\frac{1}{N}{\displaystyle \sum_{j=0}^{N-1}W\left(\rho, \theta +j\frac{2\pi }{N}\right)}=\frac{1}{N}{\displaystyle \sum_{j,k,l}^{N-1}{R}_l^k\left(\rho \right)\left( \cos k\theta {\displaystyle \sum_{j=0}^{N-1}{\alpha}_l^{j\varphi, k}}+ \sin k\theta {\displaystyle \sum_{j=0}^{N-1}{\alpha}_l^{j\varphi, -k}}\right)} $$

(3)

where

$$ \left[\begin{array}{c}\hfill {\alpha}_l^{\varphi, k}\hfill \\ {}\hfill {\alpha}_l^{\varphi, -k}\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill \cos k\varphi \hfill & \hfill \sin k\varphi \hfill \\ {}\hfill \hbox{-} \sin k\varphi \hfill & \hfill \cos k\varphi \hfill \end{array}\right]\left[\begin{array}{c}\hfill {\alpha}_l^k\hfill \\ {}\hfill {\alpha}_l^{-k}\hfill \end{array}\right] $$

(4)

For *k* = 0 (i.e., for rotationally symmetric terms), it is the intuitively obvious result that the procedure has no influence on rotationally symmetric terms. For *k* ≠ 0, the series sum to zero for all cos*kφ* except *k* = *cN*(*i* = 1,2,3….) and for all sin*k φ*. It is easy to see that rotating a wavefront to *N* equally spaced positions and averaging removes nonrotationally symmetric terms of all angular orders except *kNθ*. The term *W*
_{
kNθ
}(*ρ*, *θ*) is the *N*th rotationally symmetric component (angular orders *kNθ*), which can be written as

$$ {W}_{kN\theta}\left(\rho, \theta \right)={\displaystyle \sum_{k,l}{\left(-1\right)}^{k\left(N+1\right)}{R}_l^{kN}\left(\rho \right)\left({\alpha}_l^{kN} \cos kN\theta +{\alpha}_l^{-kN} \sin kN\theta \right)} $$

(5)

So the averaged test wavefront can be rewritten as

$$ {T}_{ave}\left(\rho, \theta \right)=R\left(\rho, \theta \right)+{S}_{sym}\left(\rho, \theta \right)+{W}_{kN\theta}\left(\rho, \theta \right) $$

(6)

where *S*
_{
sym
}(*ρ*, *θ*) is the rotational symmetry surface deviation of the test part *S*(*ρ*, θ).

Furthermore, the asymmetric component of the test surface can be derived as

$$ {S}_{asy}\left(\rho, \theta \right)={T}_i\left(\rho, \theta \right)-{T}_{ave}\left(\rho, \theta \right)+{W}_{kN\theta}\left(\rho, \theta \right) $$

(7)

The errors of angular variation *kNθ* can be represented based on Zernike polynomials and additional shear rotation measurement [9]. And it may be always neglected in the multi-angle averaging method, when *N* is large enough.

Additional measurements provide redundancies to improve and characterize measurement uncertainties. However, the rotation of the test part also introduces uncertainties related to azimuthal errors of the rotational angle and lateral displacement of the part with respect to the optical axis of the interferometer. The effect of uncertainties will arise from uncertainties in the rotational angle. Moreover, there are challenges to rotate the test surface accurately to the desired positions, especially for large optics, and keep the environment and metrology system stable during the multi-measurements.

So the estimation algorithm should be presentd to eliminate azimuthal errors caused by rotation inaccuracy. And the unknown relative alignment of the measurements also can be estimated through the differences in measurement results at overlapping areas.

The difference *W* between the shear rotation measurements can be written as

$$ \begin{array}{c}W=R\left(\rho, \theta \right)+{S}_i\left(\rho, \theta \right)-R\left(\rho, \theta \right)-{S}_j\left(\rho, \theta +\varphi \right)\\ {}={S}_i\left(\rho, \theta \right)-{S}_j\left(\rho, \theta +\varphi \right)\\ {}={\displaystyle \sum_{k,l}{R}_l^k\left(\rho \right)\left(\Delta {\alpha}_l^k \cos k\theta +\Delta {\alpha}_l^{-k} \sin k\theta \right)}\end{array} $$

(8)

where \( \Delta {\alpha}_l^{\pm k} \) is the differences of the coefficients between two measurements.

It is trivially obvious to find \( {\alpha}_l^{\pm k} \) in terms of \( \Delta {\alpha}_l^{\pm k} \) from the difference of two measurements from Eqs. (4) and (8)

$$ {\alpha}_l^{\pm k}=-\frac{1}{2}\left[\Delta {\alpha}_l^{\pm k}\pm \frac{\Delta {\alpha}_l^{\mp k} \sin k\varphi }{\left(1- \cos k\varphi \right)}\right] $$

(9)

This shows that the azimuthal terms of the wavefront can be determined from the azimuthal terms of the difference between the original wavefront and itself after rotation by *φ*. So the wavefront can be represented based on Zernike polynomials. Futermore, the *kNθ* variations of surface deviation *W*
_{
kNθ
}(*ρ*, *θ*) neglected in the multi-angle averaging method can also be obtained by additional rotation testing with a suitable selection of rotation angles *θ*
_{0} with *k = cN* and *kθ*
_{0} ≠ 2*mπ* (*m* is an integer).

The differences of the coefficients between two measurements can be written as

$$ \Delta {\alpha}_l^{\pm k}={\alpha}_l^{\pm k}\left( \cos k{\varphi}_i-1\right)\pm {\alpha}_l^{\mp k} \sin k{\varphi}_i $$

(10)

For azimuthal position error correction, the angle *φ*
_{
i
} can be treated as additional unknowns together with the coefficients \( {\alpha}_l^{\pm k} \). Then their actual values can be determined from the measured difference wavefront by least-squares method. Then the estimation algorithm adopts least-squares technique to eliminate azimuthal errors caused by rotation inaccuracy.

From Eq. (8), the wavefront difference can be further written as

$$ \begin{array}{c}{W}_i^k={\displaystyle \sum_{k,l}{R}_l^k\left(\rho \right)\Big\{\left( \cos k{\varphi}_i-1\right)\left({\alpha}_l^k \cos k\theta +{\alpha}_l^{-k} \sin k\theta \right)}+ \sin k{\varphi}_i\left({\alpha}_l^{-k} \cos k\theta +{\alpha}_l^k \sin k\theta \right)\Big\}\\ {}={\displaystyle \sum_{k,l}\Big\{{\gamma}_{0l}^k{\mathrm{Z}}_l^k\left(\rho, \theta \right)\left( \cos k{\varphi}_i-1\right)+{\tilde{\gamma}}_{0l}^k{\mathrm{Z}}_l^k\left(\rho, \theta \right) \sin k{\varphi}_i}\Big\}\\ {}={\displaystyle \sum_{k,l}{\xi}_{li}^k{\mathrm{Z}}_l^k\left(\rho, \theta \right)}\end{array} $$

(11)

The cost functions can be obtained by least squares method and be minimized to determine the true values of the unknowns of \( {\gamma}_{0l}^k \), \( {\tilde{\gamma}}_{0l}^k \) and *φ*
_{
i
}, as discussed in [12].

$$ \left[\begin{array}{cc}\hfill {\displaystyle \sum_{i=0}^{N-1}{\left[ \cos \left(k{\varphi}_i\right)-1\right]}^2}\hfill & \hfill {\displaystyle \sum_{i=0}^{N-1} \sin \left(k{\varphi}_i\right)\left[ \cos \left(k{\varphi}_i\right)-1\right]}\hfill \\ {}\hfill {\displaystyle \sum_{i=0}^{N-1} \sin \left(k{\varphi}_i\right)\left[ \cos \left(k{\varphi}_i\right)-1\right]}\hfill & \hfill {\displaystyle \sum_{i=0}^{N-1}{ \sin}^2\left(k{\varphi}_i\right)}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\gamma}_{0l}^k\hfill \\ {}\hfill {\tilde{\gamma}}_{0l}^k\hfill \end{array}\right]=\left[\begin{array}{c}\hfill {\displaystyle \sum_{i=0}^{N-1}{\widehat{X}}_{li}^k\left[ \cos \left(k{\varphi}_i\right)-1\right]}\hfill \\ {}\hfill {\displaystyle \sum_{i=0}^{N-1}{\widehat{X}}_{li}^k \sin \left(k{\varphi}_i\right)}\hfill \end{array}\right] $$

(12)

$$ \left[\begin{array}{cc}\hfill {\displaystyle \sum_l^{L(k)}{\left[{\gamma}_{0l}^k\right]}^2}\hfill & \hfill {\displaystyle \sum_l^{L(k)}{\gamma}_{0l}^k{\tilde{\gamma}}_{0l}^k}\hfill \\ {}\hfill {\displaystyle \sum_l^{L(k)}{\gamma}_{0l}^k{\tilde{\gamma}}_{0l}^k}\hfill & \hfill {\displaystyle \sum_l^{L(k)}{\left[{\tilde{\gamma}}_{0l}^k\right]}^2}\hfill \end{array}\right]\left[\begin{array}{c}\hfill \cos \left(k{\varphi}_i\right)\hfill \\ {}\hfill \sin \left(k{\varphi}_i\right)\hfill \end{array}\right]=\left[\begin{array}{c}\hfill {\displaystyle \sum_l^{L(k)}\left\{{\widehat{X}}_{li}^k{\gamma}_{0l}^k+{\left[{\gamma}_{0l}^k\right]}^2\right\}}\hfill \\ {}\hfill {\displaystyle \sum_l^{L(k)}\left\{{\widehat{X}}_{li}^k{\widehat{\gamma}}_{0l}^k+{\gamma}_{0l}^k{\widehat{\gamma}}_{0l}^k\right\}}\hfill \end{array}\right] $$

(13)

This generalized algorithm adopts least-squares technique to determine the true azimuthal positions of part rotation and consequently eliminates testing errors caused by rotation inaccuracy. The true values of the unknowns of \( {\gamma}_{0l}^k \), \( {\tilde{\gamma}}_{0l}^k \) and *φ*
_{
i
} can be obtained by the iterative procedure. The total computational time is influenced by the number of terms of Zernike polynomials in consideration (maximum *l* and *k*), the number of rotation *N*, and the precision of the initial guess of *φ*
_{
i
}. Finally, the testing errors caused by rotation inaccuracy can be compensated by the solutions of \( {\gamma}_{0l}^k \), \( {\tilde{\gamma}}_{0l}^k \) and *φ*
_{
i
}.