## Abstract

Wavefront estimation from the slope-based sensing metrologies zis important in modern optical testing. A numerical orthogonal transformation method is proposed for deriving the numerical orthogonal gradient polynomials as numerical orthogonal basis functions for directly fitting the measured slope data and then converting to the wavefront in a straightforward way in the modal approach. The presented method can be employed in the wavefront estimation from its slopes over the general shaped aperture. Moreover, the numerical orthogonal transformation method could be applied to the wavefront estimation from its slope measurements over the dynamic varying aperture. The performance of the numerical orthogonal transformation method is discussed, demonstrated and verified by the examples. They indicate that the presented method is valid, accurate and easily implemented for wavefront estimation from its slopes.

© 2015 Optical Society of America

## 1. Introduction

Wavefront estimation from its slope measurements is a typical and classic problem in modern optical testing. It has been discussed widely in the literatures. There are some representative metrologies relating to measuring the wavefront slopes or local tilts which are extensively applied in the wavefront estimation, such as the Shack-Hartmann sensing [1–4], phase measuring deflectometry [5–7] or shearing interferometry [8–10]. Mathematical manipulation for reconstructing the wavefront sag information by integrating the slope data gained by those slope sensors is a critical process. Different kinds of approaches which are classified as zonal approaches and modal approaches have been presented to accomplish this issue. In the zonal approach, the wavefront is reconstructed directly from a set of slope measurements, for example the finite difference method. Three different kinds of grid sampling configurations [11–13] in the slope measurements were discussed in the zonal estimation. In order to improve the wavefront reconstruction accuracy by finite difference method, Li *et al*. [14] applied the higher order truncation errors in the Southwell geometry and Huang *et al*. [15] used an iterative compensation method. In the modal approach, the wavefront is decomposed as a linear combination of basis functions [16,17] and determined by the corresponding weighting coefficients from slope measurements. The discrete Fourier transform (DFT) [18–21] and discrete cosine transform (DCT)-based methods [22] were employed to compute the wavefront in the frequency domains. Besides, integration method with radial basis function from gradient data was presented by Ettl *et al*. [23] and further addressed by Huang *et al*. [24]. Recently, Mochi *et al*. [25] presented a method which was based on the projection of the measured gradients onto an orthonormal basis of polynomials derived from the gradient of a specific polynomial set using a modified Gram-Schmidt procedure in the application of shearing interferometry.

Sometimes wavefront under sensing with non-circular aperture could be encountered. Zhao *et al*. [26] presented a set of orthonormal vector polynomials for directly fitting slope data or mapping distortion over a unit circle. Applying this to reconstruct wavefront with non-circular aperture may be not proper in theory. With respect to wavefront estimation from its slope data across the non-circular aperture, iterative DFT and iterative DCT [22] methods were applied to the wavefront estimation over the non-circular aperture respectively, where they were employing the Gerchberg iteration [27]. Zou *et al*. [28] presented an iterative zonal wavefront estimation algorithm which was also the Gerchberg iteration to extrapolate the missing slope data outside the valid region.

In this paper, our work is mainly concentrated on the modal approach for wavefront estimation from its slopes over the general shaped aperture. Numerical orthogonal polynomials were regarded as base polynomials for wavefront reconstruction over the general shaped aperture [29,30]. The numerical orthogonal polynomial set is held on the discrete data points. In practice, the slope sensors record the slope or gradient data also in a pixelated format. Extending the numerical orthogonal transformation method to derive the numerical orthogonal gradient polynomials in *x* and *y* directions respectively is proposed. They can be directly used to fit the slope data recorded by slope sensors. At the same time, converting the fitted slope map to the wavefront is also straightforward. Further, the characters of presented method indicate that it can be applied to wavefront estimation from its slope measurements over the dynamic varying aperture.

The rest of paper is organized as follows. Section 2 illustrates the numerical orthogonal transformation method to obtain the numerical orthogonal gradient polynomials which are served as numerical orthogonal basis functions for directly fitting the measured slopes. In Section 3, modal wavefront estimation from the measured slopes is obtained straightforward. The performance of the presented method is discussed, demonstrated and verified by examples in Section 4. Section 5 concludes the work.

## 2. Numerical orthogonal transformation method for deriving the numerical orthogonal gradient polynomials

The measured slope data obtained by slope sensor is always in a pixelated format, which are discrete data points. What we expect in the concept of modal wavefront estimation is that the acquisition slope data can be fitted by a set of orthogonal basis functions and then directly transformed to the corresponding wavefront. In this section, we first recall the numerical orthogonal polynomials which are orthogonal over the discrete data points and then obtain the derivatives of numerical orthogonal polynomials in *x* and *y* directions by partial differential. Using the derivatives of numerical orthogonal polynomials as basis functions, the numerical orthogonal gradient polynomials can also be obtained by the numerical orthogonal transform method. They can be further employed in the wavefront estimation from its gradients over the general shaped aperture.

As we know, Zernike circle polynomials form a complete and orthogonal set, numerical orthogonal polynomials $\left\{{P}_{l}({x}_{n},{y}_{n}),\text{\hspace{0.17em}}n=1,2,\mathrm{...},N\right\}$ can be expressed as a linear combination of Zernike circle polynomials [31] in Eq. (1) and are orthonormal over the discrete data points. In theory, numerical orthogonal polynomials here can be represented as other complete set of orthogonal polynomials. We choose Zernike circle polynomials which are widely applied in optical testing.

where ${C}_{lj}$ is the transform coefficient and $J$ is the number of Zernike circle polynomials denoted by $Z$. Equation (1) can be written in a matrix form aswhere $C$ is the $J\times J$ transform matrix which is a lower triangle matrix and ${C}^{T}$ is the transpose of matrix $C$. $P$ and $Z$ are two $N\times J$ matrices that cover the $N$ data points respectively.With respect to the Eq. (1), we don’t concern the subscript index $n$ temporarily that indicates the numbering of the data point across the valid region. The partial derivatives ${P}_{l}^{x}$ and ${P}_{l}^{y}$ of each term of numerical orthogonal polynomials in *x* and *y* directions are

*x*and

*y*directions respectively. Here the derivatives in the

*x*and

*y*directions are at the same sag point following the Southwell grid sampling geometry.

Next, ${P}_{l}^{x}$ and ${P}_{l}^{y}\text{\hspace{0.05em}}(l=2,3,\cdots ,J)$ will be used as numerical basis functions in *x* and *y* directions except the first piston term, as illustrated in Eq. (5) and Eq. (6). Because ${P}_{1}^{x}=0$ and ${P}_{1}^{y}=0$, they are not used to construct the numerical orthogonal gradient polynomials.

*x*and

*y*directions respectively. The subscript index $i$ of numerical orthogonal gradient polynomials $G$ is ordering from 2 to $J$ except the piston term where it is in accordance with numerical orthogonal polynomials or Zernike circle polynomials.

Further, Eq. (3) and Eq. (4) can be combined together and written into a matrix,

Therefore, the numerical orthogonal gradient polynomials are derived simply by substituting Eq. (9) to Eq. (10), as illustrated as Eq. (11).

We can observe that the numerical orthogonal gradient polynomials are related to the gradients of Zernike circle polynomials. At the same time, the gradients of Zernike circle polynomials are the linear combination of certain terms of Zernike circle polynomials, as shown by Zhao [26].The following important step is to derive the transposed transformation matrix ${\stackrel{\u2322}{C}}^{T}$ and ${D}^{T}$ respectively. As we know, matrix $P$ is orthonormal over the discrete data points, thus we can obtain ${P}^{T}P=NI$, where $I$ is a $J\times J$ unit matrix. Substituting Eq. (2) into it, we can get

The matrix ${P}^{T}Z{C}^{T}$ can be further changed in Eq. (13) based on the matrix basic principle.Thus, we can obtain from Eq. (13)Here, we let $C={({Q}^{T})}^{-1}$ [32], Eq. (14) can be converted asThe matrix ${Z}^{T}Z$ is symmetric and positive definite, Eq. (15) can be solved uniquely by Cholesky decomposition method for obtaining matrix $Q$ and then the transform matrix $C$ is gained by $C={({Q}^{T})}^{-1}$. Thus, the numerical orthogonal polynomial can be obtained by Eq. (2). The gradients $\tilde{P}$ of numerical orthogonal polynomials for constructing the numerical orthogonal gradient polynomials can be derived by Eq. (9).The transform matrix $D$ can also obtained by applying the same process as that of getting the transform matrix $C$. Let $D={({R}^{T})}^{-1}$, then we can obtain

The matrix ${\tilde{P}}^{T}\tilde{P}$ is also symmetric and positive definite, therefore, the intermediate matrix $R$can also be solved and then the transform matrix $D$ is derived by matrix equation$D={({R}^{T})}^{-1}$. Thereby the numerical orthonormal gradient polynomials $\tilde{G}$ can be achieved by Eq. (11) easily.## 3. Modal wavefront estimation from the measured slopes

The set of numerical orthogonal gradient polynomials can be served as numerical orthogonal basis functions to fit the measured slope data directly recorded by slope sensor. Here the measured slope data ${W}^{x}+{\mu}^{x}$ and ${W}^{y}+{\mu}^{y}$ in *x* and *y* directions can be represented as a matrix in Eq. (17), where ${W}^{x}$ and ${W}^{y}$ are the ideal slopes and ${\mu}^{x}$ and ${\mu}^{y}$ are the inevitable noise errors in two directions in the measurement. The measured slopes can be directly decomposed as a linear combination of numerical orthogonal gradient polynomials in Eq. (18).

When the measured slope data is recorded over a general shaped aperture, the estimated wavefront can be decomposed as a linear combination of numerical orthogonal polynomials except the piston term and written as a matrix form in Eq. (20).

where $\stackrel{\u2322}{P}$ is an $N\times (J-1)$ matrix denoting each of $J-1$ numerical orthogonal polynomial terms over $N$ sampling data points. $\beta $is a $(J-1)\times 1$ vector which can be related to the estimator $\widehat{\alpha}$ in Eq. (21).Substituting Eq. (21) to Eq. (20), the estimated wavefront $W$ can be obtained straightforward over a general shaped aperture from its measured slopes.When the measured slope data is recorded over a circular aperture, which is a common aperture in optical testing, the estimated wavefront over the circular aperture can be written as a linear combination of Zernike circle polynomials in Eq. (22).

Substituting $\stackrel{\u2322}{P}=Z{\stackrel{\u2322}{C}}^{T}$ where $Z$ is an $N\times J$ matrix into Eq. (20), and comparing that with Eq. (22), we can obtainwhere $\gamma $ is a $J\times 1$ vector. The magnitude of the first coefficient of the piston term approximates zero. In other words, the piston term is also not concerned. Thereby, substituting Eq. (23) to Eq. (22), the estimated wavefront over a circular aperture is directly mapping from its gradients.From the proceeding analyses, modal wavefront estimation from its slope measurements could be realized by numerical orthogonal transformation method over general shaped aperture. The key steps are shown in Fig. 1 for fitting the measured slope data by numerical orthogonal gradient polynomials directly and converting the fitted slope map to wavefront map which can be represented in terms of numerical orthogonal polynomials over the general shaped aperture in Eq. (20) or in terms of Zernike circle polynomials over the circular aperture in Eq. (22). We also note that recently Mochi *et al*. [25] developed a method which is different from our work. The first step of Mochi’s method is to choose a suitable set of 2D orthogonal polynomials that are expected to accurately describe the measured wavefront over the measurement domain. Generally speaking, obtaining a set of 2D orthogonal polynomials relating to the wavefront over the general shape aperture especially for the non-circular apertures or irregular apertures is not very convenient. In our work, numerical orthogonal polynomials could be served as numerical base functions for wavefront estimation over general shaped aperture in modal approach [30]. When the wavefront under testing with the complex aperture, using numerical orthogonal polynomials could be direct and do not need to choose or derive a particular 2D polynomial set which is orthogonal over this complex aperture. In this paper, the intent of the presented method for deriving the numerical orthogonal gradient polynomials is a well extension for wavefront estimation from its slopes over the general shaped aperture. For other methods which belong to the zonal approaches for addressing the wavefront estimation from its slopes, we will further make suitable comparisons between our work and the zonal approaches in the future work.

## 4. Examples and discussions

To evaluate and verify the capabilities of numerical orthogonal transformation method to derive the numerical orthogonal gradient polynomials for fitting the measured slope or gradient data and then transforming to the reconstructed wavefront directly over the general shaped aperture, a few examples are demonstrated clearly. Root mean square (*RMS*) and peak-to-valley (*PV*) values are served as the quality criteria to quantify the differences between reconstructed and original wavefronts.

#### 4.1 Wavefront reconstruction from its slopes over a circular aperture

A random wavefront over the most common circular aperture without noise impact firstly is generated by the first 22 Zernike circle polynomials. The corresponding coefficient of each Zernike circle polynomial term varies randomly over the interval $[-0.5,0.5]$ (unit: wave). In theory, the number of Zernike circle polynomial terms can be reached infinitely and can be chosen arbitrary. Here, in our simulation analysis, we demonstrate that condition as an example, as shown in Fig. 2. Meanwhile the corresponding slope maps are also displayed in Fig. 2(c) and Fig. 2(d). There are $256\times 256$ sampling pixels which are uniformly distributed over the full unit square domain. The testing digitized wavefront is filtered by a unit circle.

Using the numerical orthogonal gradient polynomials $\tilde{G}$ as numerical orthogonal basis functions and the algorithm discussed in Sec. 2 and Sec. 3, the slopes are fitted by Eq. (19) at first hand. Then the reconstructed wavefront can be obtained easily. Because the circular aperture also belongs to the category of general shaped apertures, the estimation of $\gamma $ and $\beta $ are similar to each other theoretically. In fact, the result in our analysis is in line with the theoretical analysis. The transform matrix $C$ from Zernike circle polynomials to numerical orthogonal polynomials is very close to a unit matrix when the aperture is a circle, as shown in Fig. 3(a). The differences of the reconstructed coefficients $\gamma $ (Difference 1) or $\beta $ (Difference 2) with the original randomly generated coefficients except that of piston term are displayed in Fig. 3(b). Difference 2 is a bit larger than that of Difference 1 but the maximum deviation of $\beta $ is still lower than 0.001. This is mainly because the transform matrix $C$ is not a real unit matrix in reality. Nevertheless, the reconstruction performance is similar to each other by Zernike circle polynomials or numerical orthogonal polynomials over the circular aperture. The *RMS* and *PV* errors are the floating point round-off errors. Residual error maps are shown in Fig. 3(c) and Fig. 3(d) respectively where the piston and tilts are removed from the wavefront.

Besides, we also detect the matrix ${({\stackrel{\u2322}{C}}^{T}{D}^{T})}^{T}$ in Eq. (11) to obtain the numerical orthogonal gradient polynomials from the gradients of Zernike circle polynomials in the circle aperture, as shown in Fig. 4. The magnitude of each element in the matrix ${({\stackrel{\u2322}{C}}^{T}{D}^{T})}^{T}$ has a constant scale factor to that in the Table 3 of Zhao [26]. That is to say, in our analysis, the result of presented method is in accordance with that of Zhao over the circular aperture.

#### 4.2 Wavefront reconstruction from its slopes over a general shaped aperture

For other non-circular apertures, such as square, annulus, ellipse and hexagon and other regular shaped apertures, we just filter out the valid data positions in those apertures and set them as input data as illustrated in Fig. 1, the wavefront estimation over such apertures from its slope data can be carried out by the numerical orthogonal transformation method. In this section, a complex wavefront over a normalized area in Eq. (24) filtered by a general shaped aperture is taken as an example, as shown in Fig. 5, which is characterized by exponential functions adding to the base cylindrical wavefront.

The general shaped aperture is with different shaped ‘holes’ which are considered as the missing or incomplete data encountered during the wavefront sensing. The complexity of the testing wavefront in Fig. 5(b) is more complicate than that of the random generated wavefront in Fig. 2(b). Reconstruction abilities of the presented method using the numerical orthogonal gradient polynomials for slope data fitting over the general shaped aperture is displayed in Fig. 6(a). In all residual error maps, the piston and tilts are removed from the complex wavefronts. Due to the complexity of the testing wavefront in this section and the characters of base functions (the gradient of Zernike circle polynomials) for transforming to the numerical orthogonal gradient polynomials, the reconstruction accuracy here is limited. Nevertheless, in order to increasing the reconstruction accuracy for the complex surface as illustrated in Eq. (24), we have three handling ways in our mind for complex wavefront estimation from its slope data: (1) choosing the proper base functions and the characters of their gradients are relating to the testing wavefront; (2) using the iterative compensation modal method; and (3) adding more polynomial terms into the estimation. Here the number of numerical orthogonal gradient polynomials is increased up to the 37th term. Its reconstruction performance is improved, as shown in Fig. 6(b). More works with way (1) and (2) as mentioned above will be further realized in the future.

#### 4.3 Impact of noise error

Inevitably, during the wavefront sensing by slope sensors, the measured slope data could be contaminated by the random measurement noise. In all the preceding analyses, the slope data was assumed as noise-free. This section demonstrates the noise effects in different levels. The random noise with zero mean and normal distribution is added to the ideal slope data. The signal (slope data)-to-noise-ratio (*SNR*) regarded as the noise level is defined as a ratio of the mean of slope data to the standard deviation of the random noise. In practical sense, the mean of slope data could not be zero. The condition of random generated wavefront is the same as that of Section 4.1 but it is over the general shaped aperture. A normalized reconstruction error $\epsilon $ is considered as the performance criterion, as illustrated in Eq. (25), which is defined as the *RMS* of the difference between the reconstructed wavefront ${W}_{R}$ and original wavefront ${W}_{o}$ to the *RMS* of original wavefront.

We generate 500 random wavefronts over the general shaped aperture in each noise level *SNR* to statistically estimate the normalized error $\epsilon $, as show in Fig. 7. The *SNR* is from 1 to 30. When the *SNR* is larger than 4, the normalized error $\epsilon $ is lower than 0.001. When the *SNR* is increased, the noise impact is decreased and converged quickly.

#### 4.4 Impact of data sampling size

We also demonstrate the reconstruction performance of the presented method under different data sampling sizes over the square domain which is filtered by the corresponding circular aperture. At the same time, the noise impact analysis is also included. In Fig. 8, we show the *RMS* of the normalized error $\epsilon $ under different *SNR* levels ($SNR=1,4,16,64,256$ respectively) and different data sampling sizes. The vertical coordinate is taken as the base 10 logarithm of the *RMS* normalized error $\epsilon $. In each *SNR* level, 100 random wavefronts are produced to obtain the estimation of $\epsilon $. In the identical *SNR* level, when the data sampling size is larger, the performance is better. This is to say, during slope sensing for wavefront reconstruction, if the unavoidable noise is comparatively large, we can use a dense data sampling in order to retrieve the wavefront more accurately. In this condition, we could cost more computation memories obviously. On the other hand, when the *SNR* is 256 under the data sampling size $32\times 32$, the performance is no better than that of larger data sampling size under smaller *SNR* (e.g.: $SNR=64$when data sampling size is $128\times 128$ or $SNR=16$when data sampling size is $512\times 512$). Therefore, in the slope-based testing, we should try our best to reduce the random noise impact to compromise the computation storage and speed with proper data sampling size for a more accurate wavefront estimation.

#### 4.5 Applying the presented method to estimate the wavefront over the dynamic varying aperture

As discussed above, the numerical orthogonal transformation method is valid and accurate for deriving the numerical orthogonal gradient polynomial for fitting the measured slopes directly in modal approach and afterward transferring to the wavefront easily. We note that the positions of valid data points over the general shaped aperture are important to the presented method. Occasionally, we could encounter that some optical slope tests perform over the dynamic varying apertures. The positions of valid data points in this testing are changed in-time. Our work can be applied to reconstruct the wavefront from its slopes over the dynamic varying aperture. In the simulation, the random generated wavefront in Fig. 2(b) is filtered by the dynamic varying aperture. Figure 9(a)-9(d) show a single frame from the simulated video (Visualization 1) to demonstrate the nice ability of the presented method for wavefront estimation from its slopes over dynamic varying aperture. Four frames of all the residual reconstruction error maps as examples are displayed from Fig. 9(e) to Fig. 9(h) over the dynamic varying aperture.

## 5. Conclusion

The slope-based sensing metrologies are widely used in the wavefront measurement in the field of adaptive optics, ophthalmology and optical testing. The measured slope data gathered by slope sensor can be directly fitted by numerical orthogonal gradient polynomials as numerical orthogonal basis functions which are derived by numerical orthogonal transformation method. Meanwhile, converting to the reconstructed wavefront in modal approach is also very straightforward in terms of numerical orthogonal polynomials over the general shaped aperture or in terms of Zernike circle polynomials across the most common circular aperture. The numerical orthogonal transformation method is a non-iterative method. The capabilities of the presented method are demonstrated and determined by examples and discussions. The impacts of random noise and sampling size during the testing by the presented method are also discussed by Monte Carlo simulation. The results in our work show that the numerical orthogonal transformation method is elegantly effective, accurate, and conveniently implemented in the wavefront estimation from its slopes over the general shaped aperture. Further, it can be employed in the slope-based wavefront sensing over the dynamic varying aperture.

## Acknowledgments

The authors would like to thank the National Natural Science Foundation of China (NSFC, No: 60977008, 61377015), the Doctoral Foundation of Ministry of Education (No: 20103219110014) and Open Foundation of Chinese Academy of Science (CAS) Key Laboratory (No: 2008DP173445) for financial support. Jingfei Ye is grateful for the financial support of the China Scholarship Council (CSC: 201406840022). Authors also want to thank Dr. Lei Huang in Brookhaven National Laboratory for helpful discussions. Authors also want to thank the reviewers for their valuable comments and suggestions.

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