1. Introduction

Fusion energy is a potential, inexhaustible and clear energy source. Research shows that it is hopeful to achieve commercial power generation in the 21st century [1], ICF (Inertial Confinement Fusion) was first proposed in the early 1960s after the laser had come out, it is as one of main ways to achieve fusion energy, and so far it has made significant progress in the defense, environment, and economy. In order to realize the goal of the ICF, it requires the laser driver in ICF system not only has a high energy output, but also the more important is that the output of the laser driver ought to meet an excellent beams quality. Due to its characteristics of high power, short time pulse, high repetition, and large wave front distortion in Near-Infrared region, the laser presents unique challenges to conventional wave front diagnosing methods. In the existing methods, the phase shift interferometer is mainly applied to measure the surface shape of optical element, and Hartman wavefront sensor can also be used to serve this task but it is limited by its spatial resolution. A lateral shearing interferometer must utilize two lateral shearing interferograms in perpendicular directions. The system will be exceedingly difficult to be complemented when real time and high repetition measurements are required [2]. While on the other hand, a cyclical radial shearing interferometer (CRSI) can diagnose the transient pulse laser with high speed and good accuracy at one time due to its common path, no additional reference beam and simple configuration. Besides, it can suppress the noise to a great extent, and overcome the deficiencies of traditional wavefront aberration sensing and controlling system in the existing ICF laser driver. So it receives a considerable amount of attention [3–5 ].

The wavefront under test need to be reconstructed from the interference fringe pattern that taken by radial shearing interferometer. The universal algorithms for this purpose can be mainly divided into iterative method and polynomial reconstruction method. In 2008, Li et al proposed a mathematic formula that can be applied to work out the amplitude distribution of a tested laser beam from its radial shearing interferogram. And an improved algorithm was proposed so that it could treat the RSI interferogram with the presence of lateral displacement between the test beam and the reference beam in 2010 [6, 7 ]. Through the analysis of the relationship between the radial and reduced radial Zernike polynomials, Teong et al used matrix formalism to calculate the Zernike coefficients of a wavefront under test and shew the validity of reconstructing an arbitrary wavefront aberration from a CRS interferogram [8].

A similar approach described in [9] is based on standard Zernike polynomials and its matrix formalism to reconstruct the wavefront under test in a radial shearing interferometer, which has a certain amount of lateral shear in two orthogonal directions. Dai and V. N. Mahajan developed an approach to determine orthonormal polynomials over any integrable domain, which is better than the classical Gram–Schmidt orthogonalization process. Because it is nonrecursvie and can be performed rapidly with matrix transformations. And they have successfully extended this method to hexagonal, elliptical, rectangular pupil, and determined the polynomials that are orthonormal over them and represent balanced aberrations for such pupils [10, 11 ]. F. Z. Dai extended this nonrecursvie algorithm to discrete domain, and proposed a numerical orthogonal transformation method for reconstructing a wavefront by use of Zernike polynomials in lateral shearing interferometry, and analyzed the impact of the neglected high-order terms on the outcomes of the lower-order terms by using of numerical simulation and theoretical analysis [12].

Some of the optical components have square or rectangular apertures, such as NIF (Nation Ignition Facility), which is designed for ICF to achieve ignition, have more than 7,000 large optical components, most of which have rectangular apertures [13].And the wavefront under test is not always rotationally symmetrical due to the components illuminated by the pulse of light from the flash lamps, and effect of the gravity et al [14]. For systems with square aperture pupils, the Zernike circle polynomials are neither orthogonal nor do they represent balanced aberrations. On the other hand, however, the products of the $x$ and $y$ Legendre polynomials are suitable for expanding an aberration function and represent balanced aberrations for a rectangular pupil. So it was widely used in wavefront fitting [15–17 ].

This paper is organized as follows. In Section 2 the theory of Cyclical Radial Shearing Interferometer is described as well as the principles of wavefront reconstruction from difference fronts based on Legendre polynomials. This followed in Section 3 by a numerical simulation prove the correctness of the algorithm. The experimental results is presented in section 4, both the Li’s method referred in [6] and the algorithm proposed in this paper are used to obtain the original wavefront. The error impact matrix, error propagation coefficients and mode aliasing matrix T are studied in section 5. The relationship between ratio shear, sampling points, terms of polynomials and noise propagation coefficients, and the relationship between ratio shear, sampling points and norms of T matrix are both analyzed in section 6. This leads to Section 7 summarized the full text.

2. Theoretical basis of Cyclical Radial Shearing wavefront reconstruction using Legendre polynomials

The layout of the cyclical radial shearing interferometer (RSI) is illustrated in Fig. 1 . A beam to be tested with a square aperture enters through the beam splitter and is contracted by the Keplerian telescope to serve as an object beam (solid line). The portion of the incident beam that reflects off the splitter is expanded by the telescope and travels through the path in the opposite direction to serve as a reference beam (dashed line). The object beam and reference beam are reflected by the$M_2$, $M_1$and propagate in the opposite direction. After that, the expanded and contracted wavefront meet and interfere on the interface of the beam splitter and form radial shear as shown in Fig. 2 .

 

Fig. 1 Principle diagram of cyclic radial shearing interferometry with square aperture.

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Fig. 2 Schematic of the RS with lateral shearing and the coordinate system in square aperture.

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In Fig. 2, the small square denotes the contracted wavefront and the phase difference data of the test wave fronts will be obtained in this area. A Cartesian coordinate system is established and the origin $O$ of it is located at the center of the contracted wavefront. The larger square denotes the expanded beam. The center $O^{\prime }$ of it is shifted laterally and the sheared amount is $-x_0$ in the $x$ direction and $-y_0$ in the $y$ direction. The focal lengthsof $L_1$ and $L_2$ are $f_1$ and $f_2$ respectively, and $s=f_2/f_1<1$ is the ratio shear. The reciprocal of square of $s$ is the magnification of the RSI system. When a laser output enters into this RSI, the tested wavefront $A(x,y)\exp \{ik[W(x,y)]\}$will be contracted to$A(x/s,y/s)\exp \{ik[W(x/s,y/s)]\}$and the reflected wavefront will be expanded to$A(xs,ys)\exp \{ik[W(xs,ys)]\}$ [6]. Only are there coordinate scaling transformation between the contracted wavefront and wavefront under test, and there no information lost in $A(x/s,y/s)\exp \{ik[W(x/s,y/s)]\}$. So we can consider the contracted wavefront to be as wavefront under test. If we use $W(x/s,y/s)$ and $W(xs-x_0,ys-y_0)$ to denote the contracted and expanded wavefront phases and $\Delta W(x/s,y/s)$to denote the phase difference of the two wavefront in their common area, it can be expressed as

If we rewrite the contracted and expanded wavefront phases as $W_1=W_1\left(x,y\right)=W\left(x/s,y/s\right)$ and$W_2=W_2\left(x,y\right)=W\left(xs-x_0,ys-y_0\right)$, so Eq. (1) becomes

Here, we use Legendre polynomial to represent W₁ and W₂ and reconstruct the coefficients of W₁ based on the phase difference $\Delta W\left(x,y\right)$ in Eq. (2), which is measured by RSI in advance. Due to their orthogonality of Legendre polynomial over square area. Their coefficients yield properties as described below: (1) The piston coefficient represents the mean value of the aberration function. (2) The sum of the squares of the coefficients (excluding the piston coefficient) yields the variance of the aberration function. (3) The value of a coefficient does not depend on the number of polynomials used in the expansion [18].

The one dimensional Legendre polynomials $P_n\left(x\right)$ are orthogonal over the interval [-1, 1] according to

V. N. Mahajan used the Legendre polynomials $P_n\left(x\right)$ to generate$L_n\left(x\right)$, which is given by$L_n\left(x\right)=\sqrt{2n+1}{P}_n\left(x\right)$. They define the products of Legendre polynomials in $x$ and $y$ variables as two dimensional orthonormal polynomials over square pupil [14]

Table 1. The first 15 orthonormal Legendre polynomials for a system with square pupil.

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Their orthonormality over square pupil is expressed by

Here we suppose that the contracted wavefront $W_1$ can be described by t order Legendre polynomials. Since the terms N and the order t of Legendre polynomials have the mathematical relationship of$N=(2+t)\times (1+t)/2$, so $W_1$ can be expressed as

Here, $P_k$ is a portion of Legendre polynomials in expanded area. According to the research of R. Ragazzoni et al [19], the relationship between the polynomials over twoareas satisfy the following equation:

Substituting Eq. (6), Eq. (7) and Eq. (8) into Eq. (2), we obtain Eq. (9) as follows

Equation (9) can be rewritten in its matrix form and it is expressed as

Where $A={\left\{a_1,a_2\cdots a_N\right\}}^T$is the Legendre coefficients vector of $W_1$ and $B$ is a transform matrix, which is given by

$B$can be calculated by

Suppose that the difference phase $\Delta W$ can be decomposed into a linear combination of Legendre polynomials and the coefficient vector is$C$. From Eq. (10), we can get

So the coefficients vector $A$ can be solved by [9]

22. Numerical simulations

To validate the proposed method and the theory discussed above, an arbitrary wavefront under test is generated, which is the combination of 6 orders (28 terms) Legendre polynomials with the corresponding random coefficients. Assume that the shear ratio is S = 0.75 and the later shear in $x$ direction is $x_0=0.1$ and $y_0=0.3$ in $y$ direction. The 2D plots of the contracted wavefront $W_1$, the expanded wavefront $W_2$ and the phase difference $\Delta W$ are shown in Fig. 3 .

 

Fig. 3 Original data of radial shearing interference with s = 0.75, $x_0=$0.1and$y_0$ = 0.3. (a) Contracted wavefront phase. (b) Expanded wavefront phase. (c) The phase difference$\Delta W$ .

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After $\Delta W$ is decomposed into linear combination of 28 terms Legendre polynomials, the corresponding coefficients is$C$.

According to Eq. (12), the matrix $B$ can be calculated and shown in Fig. 4(a) . It shows that matrix $B$ is an upper triangular matrix, which we can predict from Eq. (11). According to Eq. (14), the coefficients vector $A$ is obtained. And then, the contracted wavefront $W_1$is reconstructed from Eq. (6) and it is plotted in Fig. 4.

 

Fig. 4 Reconstruction of radial shearing interference. (a) Coefficients transform matrix B. (b) Reconstructed wavefront. (c) Difference between original wavefront and reconstructed wavefront. (d) Relative error of Legendre coefficients between original wavefront and reconstructed wavefront.

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Figure 4(c) shows the residual error between reconstructed wavefront and the contracted wavefront. And its order of magnitude of pv and rms reaches ${10}^{-14}$nm. The value of percentage error between the original coefficients and the corresponding coefficients of the contracted wavefront is also plotted in Fig. 4 (d) and the order of magnitude is${10}^{-13}$.It can be neglected.

4. Experiment

In order to verify the algorithm proposed in this paper, a setup based on the CRSI with ratio shear s = 200mm/300mm is established according to Fig. 1. The FT(Fourier Transform) method [20],which is effective and usually adopted to treat fringe pattern,is used to get the test wavefront in this paper. Therefore, a carrier frequency fringe pattern nearly along the horizontal direction is generated by slightly tilting the BS in Fig. 1. Simultaneously, a crosshair is inserted into the light path and located at the input plane of the CRSI, then its image is recorded by a CCD, and the lateral shearing amount due to tilting BS can be read out from the image. It is about$y_0=15$pixels and $x_0=-1$pixels in the two directions in our setup. Experimentally, two interferograms are recorded by the CCD. One fringe pattern is generated by putting an ordinary window glass with 4mm thickness in the input plane and is shown Fig. 5(a) , and the other interferogram without this glass is recorded and is applied to cancel out the system error in our setup. After processing these two interferograms by Fourier transform, we can work out the phase difference $\Delta W\left(x/s,y/s\right)$ which is caused from the distorted wavefront of the ordinary window glass. The unwrapped $\Delta W\left(x/s,y/s\right)$ is shown in Fig. 5(b). After removing the piston and tilt in the $\Delta W\left(x/s,y/s\right)$ shown in Fig. 5(b), we can get the result as shown in Fig. 5(c). Without the piston and tilt, the reconstructed wavefront by the iterative algorithm proposed in [6] is shown in Fig. 5(d) and the reconstructed wavefront by the algorithm in this paper is shown in Fig. 5(e), respectively.

 

Fig. 5 Experiment results. (a) CRSI interferogram. (b) Phase difference. (c) Phase difference after remove piston and tilt. (d) Reconstructed phase by the iterative algorithm proposed in [6]. (e) Reconstructed phase using Legendre polynomials. (f) The difference between the two reconstructed wavefronts by using two different methods.

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Comparing the PV and RMS values in the two figures, we can see the Legendre polynomials can be used to reconstruct the phase of a beam to be tested from its CRSI interferogram over a square area. The difference between the two reconstructed wavefronts by using the two different methods is shown in Fig. 5(f). It shows that the RMS and PV values of the difference are 0.03rad and 0.191rad. As we know, if the wavefront under test is complicated, the wavefront reconstruction error will relatively increase when we used the iterative algorithm proposed in [6], and this conclusion can be found in [21]. Additionally, even though we can enable the two focal point coincident with each other completely in our CRSI, it is hard to say the actual ratio shear is exactly 200mm/300mm. And the lateral shearing amount measurement error, the random additive noise contained in the fringe pattern, and the lack of terms of Legendre polynomials used to reconstruct the wavefront will cause the difference happens.

5. Error analyses

The simulation above are based on the assumption that there are no random noise in phase difference $\Delta W$ and the terms of Legendre polynomials of the simulated wavefront is equal to the terms of Legendre polynomials which we used to reconstruct. However, in general, this condition cannot be met in practical situation. Actually, it is inevitable that the wavefront under test contain random additive noise. And since the phase difference data cannot be fitted with infinite terms of Legendre polynomials, the estimations of lower-order coefficients are affected by the high-order coefficients.

5.1 Error propagation coefficients

Assuming that the phase difference $\Delta W$ contain random noise $\sigma$and the coefficients error is $\xi$ when $\Delta W$ is expanded by Legendre polynomials, then we have

Here, $\Delta W$ and $C$ are assumed to be the true value. And then, we obtain

Suppose the final coefficients error of $W_1$ that produced due to $\sigma$ is $\epsilon$,from Eq. (14), so we get

Then we have

So the resulting error $\delta$ in the contracted wavefront $W_1$ can be determined by

If we let $E=L{B}^{+}{L}^{+}$ and suppose there are $N^2$ sampling points, in addition, the phase errors in phase difference $\Delta W$ are independent and uncorrelated [17], the root mean square of phase error can be obtained by

Then the noise coefficients is defined by

${\Vert E\Vert }_F$ is the F norm of $E$.

5.2 Mode aliasing

In practice, we cannot known how many terms of Legendre polynomials are needed to fit the wavefront under test in advance. When the terms of Legendre polynomials used to reconstruct wavefront under test is less than the real one, this reconstruction wavefront will contain an error [12].

Assume that the terms of Legendre polynomials can be infinity, therefore, any practical difference phase $\Delta W$ can be represented by

Actually, phase difference cannot be fitted by infinity terms of Legendre polynomials. Here, we assume that it is fitted by first f terms of Legendre polynomials, so the $L$ and $C$ are divided into two parts. A formula can be obtained from Eq. (22) as

Where $L_f$ and $L_r$ are two parts containing first f and remaining columns of $L$, and $C_f$ and $C_r$ are corresponding coefficients. Then the estimation of $C_f$ can be calculated by

where

Here, $L_f{}^{+}$ is the generalized inverse of $L_f$ It can be written as $L_f{}^{+}={(L^T{}_f{L}_f)}^{-1}{L}_f{}^T$.

The matrix $B$ in Eq. (13) can also be divided into four parts as follows

Because $B$ is a upper triangular matrix, so $B_{r1}=0$.We have

The estimation of first f terms Legendre coefficients of $W_1$ can be obtained by Eq. (14), and the coefficients ${\widehat{a}}_f$is given by

Substituting Eq. (24) and Eq. (27) into Eq. (28), we obtain Eq. (29) as follows

Here, we define

Due to the orthogonality of Legendre polynomials over square aperture, we substitute $D=0$into Eq. (30), so it becomes

The Eq. (29) is changed into

The Eq. (32) can be expressed in matrix as

$\Delta a_i(i=1,2\cdots f)$ is the coefficients difference between the reconstructed wavefront and the original wavefront. As can be seen from the Eq. (33), $\Delta a_i$ is not only associated with T, but also related to the high order terms $a_{f+1},a_{f+2}\dots a_{f+r}$ of $W_1$. This error will not occur unless $f$ is greater or equal to the terms which the tested wavefront $W_1$ contains. In order to investigate the impact of shear ratio S and sampling points on the coefficients reconstruction error, we can calculate the F norm of T, namely$T_{norm}$, to evaluate the impact. The $T_{norm}$ can be described by

6. Analysis of relationships between sampling points, shear ratio, terms of polynomials and $e_p$,$T_{norm}$

As can be seen from expression of $E$,noise coefficients is associated with the Legendre polynomials values at discrete points and transform matrix $B$. Since $B$ is affected by the ratio shear, so it is necessary to analyze the relationship between noise coefficients, the number of sampling points and the terms of Legendre polynomials.

Figure 6 shows the noise coefficients change as a function of sampling points for ratio shear S = 0.75, 0.6, 0.45, 0.3, respectively, the terms of Legendre polynomials used here is 6 orders (28 terms).The noise coefficients decrease with the increase in $N$(the number of sampling points is $N^2$). And the noise coefficients grows with the increase of ratio shear s. The noise coefficients decreases rapidly when $N$ vary from 10 to 50. However, as $N$continuously increases and approaches 100 when a small shear ratio is adopted in RSI system, the noise coefficients becomes much smaller and does not change significantly.

 

Fig. 6 The relationship between the noise coefficients and sampling points using Legendre polynomials over square area. Curves A, B, C, D corresponding to the shear ratio s = 0.75,0.6, 0.45,0.3, respectively.

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Noise coefficients $e_p$ changing as a function of terms of polynomials that $\Delta W$ contains is shown in Fig. 7 . $e_p$ is calculated under the case of s = 0.75, 0.6, 0.45, 0.3, respectively, and the sampling points is $N^2=50\times 50$. The number of terms of polynomials varies from 2 to 8 orders, namely from 6 to 45 terms.

 

Fig. 7 The relationship between noise coefficients and terms of Legendre polynomials, shear ratio s = 0.75, 0.6, 0.45, 0.3,sampling points = 50*50.

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From these curves we can see that the noise coefficients $e_p$ grows with the increase of the ratio shear s and almost linearly increase with the terms of polynomials raising. And $e_p$ is more sensitive to the change in shear ratio when there is a bigger shear ratio value.

Suppose that 21 terms of Legendre polynomials are contained in $\Delta W$but only 15 terms are used to fit the phase difference. It will produce a coefficient error which can be calculated by use of Eq. (33). If $T_{norm}$ is large, it indicates that the remaining high-order terms will have a bigger impact on the coefficients of the lower-order terms. $T_{norm}$ as a function of $N$ for shear ratio S = 0.75, 0.6,0.45,0.3 separately is shown in Fig. 8 . Figure 8 shows that $T_{norm}$ decreases as $N$ increases and it trends to nearly keep a constant when $N$ increases from more than 30, but it relatively has a large amount of increase as the shear ratio increases.

 

Fig. 8 The relationship between F matrix norm of T and sampling points. Curves A,B,C,D correspond to the shear ratio s = 0.75,0.6,0.45,0.3,respectively.

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That is to say, to control this error, we need to adopt a RSI system with a small shear ratio.

7. Conclusions

This article proposes to use Legendre polynomials as basis function to retrieve the wavefront from the phase difference obtained by a radial shearing interferometer over square area. And the algorithm has been proved to be correct by using a simulation experiments that is a reconstruction of an arbitrary wavefront under test. The experimental results also show that he Legendre polynomials can be used to reconstruct the phase of a beam to be tested from its CRSI interferogram.

Next, the noise coefficients expression is deduced based on the assumption of the noise in phase difference which is independent and uncorrelated. Meanwhile the aliasing matrix T is evaluated. And we can use F norm of T as a reference to determine the effect of remaining high-order on the outcomes coefficients of low-order terms. The result of simulation shows that noise coefficients increase with the increase of the number of sampling points, and grows with the increase of ratio shear as well. Also we can see that the noise coefficients almost linearly increase with the increase of the terms of polynomials. Numerical analysis also shows that the norm of T is mainly influenced by the shear ratio, but less affected by the sampling points. And $T_{norm}$ increases as the shear ratio increases.

In practice, in order to reduce the coefficient reconstruction error caused by T, we need to increase the number of polynomial terms as far as possible. But it will increase the noise propagation coefficients, so the number of items need to be considered in a compromise way. Using a small shear ratio is an effective way to reduce the wavefront reconstruction error due to the noise propagation and the lack of use of Legendre polynomials terms.