Adaptive wavefront interferometry for unknown free-form surfaces

Shuai Xue; Shanyong Chen; Zhanbin Fan; Dede Zhai

doi:10.1364/OE.26.021910

1. Introduction

The precondition of optical fabrication is surface figure error test technique. Wavefront measuring interferometers are widely used to obtain high accuracy surface figure test results. For testing aspheric and freeform surfaces [1–3], a null corrector is additionally required [4]. However, the null corrector is static and designed uniquely for the ideal test surface. When the test surface has large surface figure error (e.g., in-process and during the optics manufacturing process) surface, too dense fringes or invisible ones appear. Such surface figure error cannot be measured for beyond the dynamic range of conventional interferometers. Dynamic range means testable departure range [5]. It is limited by the pixel density of CCD. Theoretically, surface figure error with slope larger than a quarter wave per pixel [6] is beyond dynamic range. This kind of surface figure error can be considered to be severe. In fact, severe surface figure error often occurs at the initial stages of polishing. Take the process of computer controlled optical surfacing (CCOS) as an example, two kinds of large edge error shapes, i.e., the rolled edge and the turned-down edge often appear [7]. Testing severe surface figure error has become a bottleneck in advanced optical fabrication.

Resorting to detector arrays with the increased number of pixels [8] may extend the dynamic range. However, the enhancement is limited. The use of longer wavelength [9] or two-wavelength techniques [10] can decrease the fringe density, but with loss of accuracy and sensitivity. Sub-Nyquist interferometer [6] can improve the measurement range. However, it needs assumptions like, test surfaces have continuous derivatives. Tilt-Wave-Interferometry (TWI) can also improve the dynamic range [11]. However, it is currently complicated and difficult to calibrate.

Severe surface figure error is in nature unknown free-form surface varying during fabrication. Utilizing a variable null (VN) capable of generating variable aberrations seems a good idea. There are mainly two kinds of VN. The first kind is a combination of phase plates, optical wedge or other static optical elements. Hilbert et al. [12], Shafer [13], Chen et al. [14], QED Technologies [15], and Zhang et al. [16] have investigated this kind of VN. The second kind is AO elements such as deformable mirrors (DMs) and spatial light modulators (SLMs). It can generate more flexible aberration. Pruss and Tiziani [17], Fuerschbach et al. [18], He et al. [19], and Zhang et al. [20] have investigated using DMs as VNs. Cao et al. [21], Kacperski et al. [22], Cashmore et al. [23], and Ares et al. [24] have investigated using SLM as a VN. However, all these methods only compensate intrinsic aberration of test surfaces. Intrinsic aberration means the departure of the nominal test surface from standard reference wavefront. It can be easily calculated from the theoretical model. However, severe surface figure error of surfaces is unknown. That means compensation target is unknown. Hence the above VN methods are not applicable for testing severe surface figure error.

For solving this problem, pioneering contributions have been made by Huang et al. [25] through utilizing a DM as a full aperture dynamic null. The unknown compensation target is obtained by optimization algorithms. To overcome the limited accuracy of the DM, a deflectometry system (DS) [26,27] is additionally utilized. However, the DS requires delicate calibration thus complicating the test system. DMs only have reflection-type ones. This also restricts the application of the method. Moreover, the optimization algorithm is not analyzed in detail.

The method of this paper is to introduce wavefront sensor-less AO technique to traditional wavefront interferometry. This new test principle is called adaptive wavefront interferometry (AWI) by us. Based on the principle, a simple structured SLM-based AWI test system is established. The system is based on a commercial interferometer. No other elements are required besides the SLM. In this system, the calibrated high-accuracy SLM works as an adaptive correction element. For turning invisible fringes into resolvable ones, the SLM is close-loop controlled by a two steps Zernike mode stochastic parallel gradient descent (ZSPGD) algorithm [28]. Amplitudes of Zernike modes generated by the SLM are selected as optimization variables. Two image metrics of interferograms are analyzed by us. Compensation effects represented by these two metrics are real-time monitored to guarantee convergence of the iteration. The phase conjugation algorithm is additionally utilized for turning resolvable fringes into null ones. Finally, local severe surface figure error is extracted from the SLM phase and the null test result by reverse optimization based on ray trace model. For troubleshooting the limited aperture of the SLM, stitching technique is incorporated into the SLM-based AWI. Simulation and experiments on the SLM-based AWI are conducted. At the end of this paper, discussions and prospects of the AWI are provided to develop the proposed method.

2. Principle

Conventional interferometry test for surfaces utilizing a static null is shown as Fig. 1(a). Because unknown free-form severe surface figure error is beyond the dynamic range of the interferometer, invisible fringes region appears such as the upper one in Fig. 1(b). As a result, the corresponding surface figure error data in this region is not available as shown in Fig. 1(c). To test the surface figure error within this region, a transmissive SLM is inserted in the test scheme as adaptive null optics as shown in Fig. 1(d).

Fig. 1 Illustration of the SLM-based AWI for freeform surfaces with severe surface figure error. (a) The conventional test of a freeform surface utilizing a static null. (b) The full aperture interferogram of which the upper region cannot be resolved by the interferometer. (c) The surface figure error map of which the upper region is not available. (d) The SLM-based AWI for testing unknown severe local surface figure error. (e) The initial interferogram of the local region. (f) The final interferogram of the local region nulled by the SLM. (g) The surface figure error of the local region. (h) Full aperture surface figure error map stitching result.

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Firstly, inherit wavefront distortion (WFD) W_d of the SLM is in situ calibrated by inserting a high-accuracy flat between the SLM and the static null. + 1 order diffractive wavefront is preferred. To null WFD of SLM, phase Ф_d = 2πW_d/λ (λ is the wavelength of the interferometer) is loaded to the SLM by encoding the phase in interference type CGH [29]. The principle of the interference type CGH is based on optical hologram as shown in Fig. 2. Object wavefront is flat wavefront with the desired aberration. The complex amplitude of the object wavefront, at the recording plane is defined as U_o(X,Y). Reference wavefront is standard flat wavefront. The complex amplitude of the reference wavefront, at the recording plane is defined as U_r(X,Y). Their interferences, i.e., H = |U_r + U_o|² = |U_r |² + | U_o|² + U_r*U_o + U_rU_o*, are recorded in the recording plane. It includes three orders: the 0-order is composed of terms |U_r |² + | U_o|², the + 1 order is the term U_r*U_o and the −1 order is the term U_rU_o* [29]. When the SLM is illuminated by the reference wavefront, emergent wavefront is the wavefront with the desired aberration. To realize separation of diffraction orders, tilt carrier is added to the desired phase. The + 1 order can be spatially separated from −1 order and 0-order in the focal plane with an appropriate spatial filter [23]. The primary advantage of this encoding method is that nonlinearity phase response and spatially varying phase response of SLM need not to be calibrated. It is fringes pattern of the hologram image that determines the accuracy of the generated wavefront, hence the phase control accuracy of SLM is relatively high.

Fig. 2 Principle of encoding aberration based on optical hologram.

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Secondly, lateral movement is conducted to make the work region (a circular or rectangular region) of the SLM enclose one of the unknown severe figure error local regions. Initial interferogram within the work region of SLM is shown as Fig. 1(e). To make the departure between the test and measurement wavefronts in this region within the dynamic range of the interferometer, non-null test condition is firstly achieved. This is done by utilizing the SLM to compensate dominant low- to mid-spatial frequency of the unknown large local surface figure error. Note the unknown large local surface figure error is speculated intelligently by a two steps ZSPGD algorithm. Furthermore, null or near-null test condition is achieved by utilizing the same SLM controlled by phase conjugation algorithm. After compensated by the SLM, null or near-null interferogram as shown in Fig. 1(f) is obtained. Then surface figure error of the local region as shown in Fig. 1(g) can be obtained by reverse optimization based on ray trace model. The adaptive null algorithm (includes the ZSPGD and phase conjugation algorithms) and phase extraction method are two key points of our method. These will be analyzed in detail after the text of whole test procedures in this part.

Finally, stitching algorithm is applied to stitch results of Figs. 1(c) and 1(g) to obtain the stitched full aperture surface figure error as shown in Fig. 1(h). The stitching algorithm we utilized is very similar with the sub-aperture stitching algorithms [30–33]. Two key points of sub-aperture stitching algorithms are to determine misalignment aberrations among subapertures and stitching subapertures utilizing the overlap regions. In subaperture stitching, every subaperture has six-degree-of-freedom misalignments due to relative movement between the test surface with the interferometer. However, in our method, the test surface and the interferometer have no relative movement. It is the movement of SLM that introduces misalignment aberrations. The misalignment aberrations caused by the SLM is much smaller since the phase contribution of the SLM is much smaller than that of static null and test surface. We can simply modify the stitching algorithm proposed by QED [30] and apply it to our stitching process. Suppose the two sets of measured optical path difference (OPD) data of local region i and full aperture without local region k are denoted by w_i = {(u_j,i, v_j,i, φ_j,i), j = 1,2,…,N_i} and w_k = {(u_j,k, v_j,k, φ_j,k), j = 1,2,…,N_k}, respectively. The variable φ_j,i is the measured OPD on the pixel (u_j,i, v_j,i). N_i and N_k are the total number of measurement points of local region i and full aperture without local region k, respectively. The two sets of data can be represented in the same coordinate frame with certain components compensated as illustrated in Eq. (1)

\begin{array}{l} z_{j, i} = φ_{j, i} + a_{i} + b_{i} \cdot u_{j, i} + c_{i} \cdot v_{j, i} + s_{i} \frac{\partial φ_{j, i}}{\partial u_{j, i}} + t_{i} \frac{\partial φ_{j, i}}{\partial v_{j, i}} + δ_{i} [u_{j, i} \frac{\partial φ_{j, i}}{\partial v_{j, i}} - v_{j, i} \frac{\partial φ_{j, i}}{\partial u_{j, i}}], \\ z_{j, k} = φ_{j, k} + a_{k} + b_{k} \cdot u_{j, k} + c_{i} \cdot v_{j, k} + s_{k} \frac{\partial φ_{j, k}}{\partial u_{j, k}} + t_{k} \frac{\partial φ_{j, k}}{\partial v_{j, k}} + δ_{k} [u_{j, k} \frac{\partial φ_{j, k}}{\partial v_{j, k}} - v_{j, k} \frac{\partial φ_{j, k}}{\partial u_{j, k}}], \end{array}

where, a_i, a_k are coefficients of piston, b_i, b_k are coefficients of x-tilt, c_i, c_k are coefficients of y-tilt, s_i, s_k are coefficients of x-shift positioning errors of the data, t_i, t_k are coefficients of y-shift, δ_i, δ_k are coefficients of clock.

The inconsistency of z_j,i and z_j,k in the overlapping region is minimized in the least-square (LS) sense, yielding the following linear LS problem

\min F = \sum_{j o = 1}^{N o} {(z_{j o, i} - z_{j o, k})}^{2},

where j_o is the index in the overlapped region, and N_o is the overlapped data number. By solving the LS problem, coefficients of compensator terms can be obtained. Then the stitching process is completed. Note this stitching algorithm usually demands the microns level movement accuracy.

The whole test procedures of the SLM-based AWI for testing surfaces with unknown severe surface figure error are shown in Fig. 3.

Fig. 3 Flowchart of the SLM-based AWI.

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One key point of the above test procedures is the adaptive null algorithm. It includes three major steps as shown in Fig. 3. The first step is fringes restoration based on ZSPGD. Fringes restoration means to turn the invisible fringes state into resolvable fringes state. The second step is dense fringes relaxation based on ZSPGD, Fringes relaxation means to turn the dense fringes state into sparse fringes state. The last step is null fringes based on phase conjugate algorithm.

In the first step, if the work region of SLM is chosen as circular, then the loaded phase to SLM can be a linear combination of Zernike terms. Resolvable fringes can be obtained if the dominant low- to mid-spatial frequency error is compensated by the SLM. Utilizing ZSPGD, fringes of the interferogram within the work region of SLM, which is denoted by A⁽⁰⁾ hereafter, is restored. In ZSPGD algorithm, coefficients of Zernike terms are optimization variables, and the objective is to maximize J which is the sum of squared gray level differences between any two pixels of the interferogram. ZSPGD is defined as

Z^{(k+1)} = Z^{(k)} + γ δ J δ Z^{(k)},

δ J = J (Z^{(k)} + δ Z^{(k)}) - J (Z^{(k)}),

J = \sum_{all (i, j)} {(g_{i} - g_{j})}^{2} / 2.

where Z = {z₁, z₂, …, z_n} is the coefficients of Zernike terms, the combination of which constitutes the phase of SLM; γ is the gain coefficient; δZ^(k) = {δz₁, δz₂, …, δz_n }^(k) are small random perturbations having identical amplitudes α and a Bernoulli probability distribution; i and j are two pixel index numbers of the interferogram; g_i and g_j are the grayscale values of the two pixels.

The initial interferogram of A⁽⁰⁾ before fringes restoration step is shown as Fig. 4(a). During the optimization, the interferogram is divided into two sub-regions, i.e., invisible fringes sub-region A_n⁽⁰⁾ and resolvable fringes sub-region A_r⁽⁰⁾. Define the pixel number of the invisible fringes sub-region as M_n⁽⁰⁾, the pixel number of the resolvable fringes sub-region as M_r⁽⁰⁾, and the total pixel number of the interferogram as M⁽⁰⁾. Since ZSPGD restores fringes by optimizing the objective function, the objective function can be regarded as a function of M_r⁽⁰⁾. This function should be uni-modal because monotonicity of the objective function determines the probability of getting stuck in local minima. Moreover, the slope amplitude determines how quickly the objective function changes as M_r⁽⁰⁾. Hence it is important to analyze J and obtain its application range and dynamic performance.

Fig. 4 Interferograms of the unknown large local surface figure error region within the work region of the SLM at the three steps of adaptive null algorithm. (a) The initial interferogram before fringes restoration step. (b) The interferogram after fringes restoration step. (c) The interferogram after dense fringes relaxation step. (d) The interferogram after phase conjugation algorithm.

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J can be regarded as a sum of three parts as:

J = J_{1} + J_{2} + J_{3} = \sum_{i \in A_{r}^{(0)}, j \in A_{r}^{(0)}} {(g_{i} - g_{j})}^{2} / 2 + \sum_{i \in A_{n}^{(0)}, j \in A_{n}^{(0)}} {(g_{i} - g_{j})}^{2} / 2 + \sum_{i \in A_{r}^{(0)}, j \in A_{n}^{(0)}} {(g_{i} - g_{j})}^{2} .

To calculate J₁, define random variables X and Y as grayscale value of a pixel in sub-region A_r⁽⁰⁾. So g_i and g_j are sample values of X and Y, respectively. Define a new random variable Z = X −Y. Hence in statistical sense, J₁ can be calculated as

J_{1} = C_{M_{r}^{(0)}}^{2} E (Z^{2}),

where E(Z²) means the expectation value of Z², and

C_{M_{r}^{(0)}}^{2} = M_{r}^{(0)} (M_{r}^{(0)} - 1) / 2

. For simplicity, X and Y both satisfy a uniform distribution, i.e., X~U(0,1) and Y~U(0,1). According to the distribution of a function of two random variables, E(Z²) is calculated as 1/6. Substituting this value into Eq. (7), we can obtain J₁ = M_r⁽⁰⁾ (M_r⁽⁰⁾ −1)/12.

To calculate J₂, since g_i and g_j are grayscale values of two pixels in sub-region A_n, they are equal. Then J₂ = 0.

Since g_j = 0, J₃ can be calculated statistically as:

J_{3} = C_{M_{r}^{(0)}}^{1} C_{M_{n}^{(0)}}^{1} \cdot E (X^{2}),

where E(X²) means the expectation value of X²,

C_{M_{r}^{(0)}}^{1} = M_{r}^{(0)}

, and

C_{M_{n}^{(0)}}^{1} = M_{n}^{(0)}

. According to the expectation of a uniform distribution, E(X²) is calculated as 1/3. Substituting this value into Eq. (8), we can obtain J₃ = M_r⁽⁰⁾ (M⁽⁰⁾ − M_r⁽⁰⁾)/3.

Substituting the values of J₁, J₂, and J₃ into Eq. (6), we can obtain:

J \approx - M_{r}^{(0)} (3 M_{r}^{(0)} - 4 M^{(0)}) / 12 .

According to Eq. (9), following conclusions can be drawn:

1) In statistical sense, J increases as M_r increases from 0 to 2M⁽⁰⁾/3, and J decreases as M_r⁽⁰⁾ increases from 2M⁽⁰⁾/3 to M⁽⁰⁾, thus the applicable range of J is 0 ≤ M_r ⁽⁰⁾≤ 2M⁽⁰⁾/3.
2) In statistical sense, the increase rate of J decreases as M_r⁽⁰⁾ increases from 0 to 2M⁽⁰⁾/3, and the theoretical increase rate equals zero when M_r⁽⁰⁾ = 2M⁽⁰⁾/3.

Hence, the ZSPGD algorithm must be modified to enhance convergence and improve the convergence rate. Firstly, the calculation area of J is A⁽⁰⁾, and the iteration of Eqs. (3)-(5) are conducted until M_r⁽⁰⁾ = ξ∙M⁽⁰⁾, where ξ is a fraction smaller than 2/3. Then invisible fringes area A⁽¹⁾ within A⁽⁰⁾ is identified as new calculation area of J, and iteration of Eqs. (3)-(5) are also conducted until M_r⁽¹⁾ = ξ∙M⁽¹⁾, where M_r⁽¹⁾ is pixel number of the resolvable fringes sub-region within A⁽¹⁾, and M⁽¹⁾ is total pixel number within A⁽¹⁾. This J value calculation area iteration is conducted until the pixel number of invisible fringes area is smaller than a threshold. During every single ZSPGD iteration, the search is not stopped until the pixel number of invisible fringes area is smaller than a threshold.

Based on Eq. (9), we can further obtain that J increases from about [M^(q)]²/60 to [M^(q)]²/9 as M_r^(q) increases from M^(q)/20 to 2M^(q)/3, where q denotes the q-th J value calculation area iteration. That means J does not have significant changes in order of magnitude. This is helpful to choose a fixed value of γ^(q).

After this step, fringes within A⁽⁰⁾ are restored completely as shown in Fig. 4(b). However, the fringe density is still too high. Influenced by vibrations and noise, complete surface figure error map within A⁽⁰⁾ still cannot be acquired by the interferometer. During the second step, the similar ZSPGD algorithm and SLM phase updating is applied. The judgment criterion J is simply replaced by energy gradient function, which is an assessment criterion of image definition as

J = \sum_{i \in A^{(0)}} {(\frac{\partial g_{i}}{\partial x})}^{2} + {(\frac{\partial g_{i}}{\partial y})}^{2} .

where g_i and g_j are the grayscale values of the two pixels.

The loaded phase to the SLM after the second step is denoted as Ф₂ (exclusive of the inherit wavefront distortion phase Ф_d). After the second step, the dense fringes within A⁽⁰⁾ are relaxed as shown in Fig. 4(c). The corresponding wavefront error map W_r2 can be easily obtained by the interferometer. Since non-null test will introduce retrace errors [34] to the test results, a further null or near-null step is additionally required. Fitting the test result W_r2 by Zernike modes and denote the fitting result as W_r2f, then the phase Ф_r2f = 2πW_r2f/λ can be loaded to the SLM. Finally, null or near-null condition is met as shown in Fig. 4(d). The corresponding wavefront error map can be acquired by the interferometer and denoted as W_r3.

Another key point of the above whole test procedures is extraction of local surface figure error from the phase of SLM and null test result. The total phase (exclusive of the inherit wavefront distortion phase Ф_d) applied to SLM is denoted as Ф, where Ф = Ф_r2f + Ф_2. To extract the unknown surface figure error W_u within A⁽⁰⁾, the straightforward calculation W_u = W_r3 + λФ/2π is not correct. This calculation does not take the propagation effects into consideration. The relation of the SLM phase with the surface figure error of the test surface is not always equal. The generated phase on SLM varies considerably, and thus the paths through the system change. This will be truer when in addition to the SLM a static null is used.

Extraction of the local severe surface figure error W_u from the SLM phase and the null test result is by reverse optimization based on ray trace model. This method has been proposed for retrace error correction in non-null aspheric interferometry [35,36]. The basic idea is applied to reconstruct the local severe surface figure error. When the phase conjugation step is done, the test result W_r3 is firstly fitted by 37 Zernike polynomials. Then W_r3 can be expressed as W_{r3 =} W_r3f + W_r3r, where W_r3f denotes the fitting results, and W_r3r denotes the residual of W_r3 after Zernike fitting. Then W_r3f can be expressed by an implicit function as

W_{r3f} = f (W_{uf}, λ Ф / 2 π),

where W_uf denotes the local severe surface figure error fitting by 37 Zernike polynomials, and Ф denotes the total phase (exclusive of the inherit wavefront distortion phase Ф_d) applied to SLM.

According to the system nominal parameters, the testing system can be modeled in ray-tracing software. In the model, Eq. (11) would be

{\bar{W}}_{r 3 f} = f ({\bar{W}}_{u f}, λ \bar{Ф} / 2 π),

where

{\bar{W}}_{r 3 f}

,

{\bar{W}}_{u f}

and

\bar{Ф}

are the simulated counterparts in the model corresponding to Eq. (11), respectively.

Note that $\bar{Ф}$ = Ф. An optimization function is set with Zernike polynomial coefficients of the fitted local severe surface figure error ( ${\bar{W}}_{u f}$ ) as variables and those of the detected experimental wavefront (W_r3f) as optimization targets. Through the reverse optimization process with iterative ray tracing, ${\bar{W}}_{u f}$ approaches W_uf along with the simulated wavefront ( ${\bar{W}}_{r 3 f}$ ) approaches to the actual one (W_r3f) in the experiment. The optimal solutions can be extracted, and the fitted local severe surface figure error is reconstructed with a simple fitting procedure. Since W_r3r denotes the residual of W_r3 after Zernike fitting process, this can be simply regarded as the residual of W_u after Zernike fitting. Finally, the local severe surface figure error can be extracted as

W_{u} = {\bar{W}}_{u f} + W_{r 3 r} .

One main error of the ray trace method is modeling error. It includes inaccurate element structure parameters and element position inconsistencies with the actual system. Element structure parameters can be measured by different commercial measurement instruments. Alignment and monitoring of element position and posture are a little complicated. However, many publications have talked about this. The whole test system without the SLM is a typical non-null test system, and the alignment of the static null and test surface has been discussed in detail in [35]. Alignment of the SLM can be completed before the phase is loaded to SLM. That is, it would be calibrated in advance. This process can follow the method Zhang et al. use [20]. The nominal thickness of the SLM and the distance between the SLM and the test surface is measured by a distance measuring set LenScan LS600 [37]. LenScan LS600 can measure center thicknesses of optical elements and air gaps along the optical axis. Its measurement range is 600mm with absolute accuracy of ± 1μm. Simulation shows the modeling misalignment aberration due to the distance measurement error ( ± 1μm) is negligible.

3. Simulations

To verify the performance of the ZSPGD algorithm, the fringes restoration and fringes relaxation processes based on ZSPGD algorithm are simulated utilizing Zemax and Matlab. The severe surface figure error is represented by a Zernike standard sag surface with Z₄ = 11.5λ, Z₁₁ = −3.5λ (λ = 632.8nm). The initial test wavefront error map is shown in Fig. 5(a), with 41.107λ (peak-to-valley, PV). The corresponding interferogram generated by Zemax is shown as Fig. 5(b). Note that the sample grid is 128 × 128, and moire fringe appears due to too dense fringes. However, this is not the practical interferogram obtained by the interferometer. The practical interferogram is subjected to maximum allowable fringe frequency, which is equal to Nyquist frequency of the CCD. That means wavefront with slope higher than a half wave per pixel is beyond the dynamic range of the CCD. On this basis, the wavefront slope is calculated and pixels that have slope higher than a half wave per pixel are identified. Gray values of the corresponding pixels on the interferogram are modified to be equal to zeros. The modified interferogram as shown in Fig. 5(c) is more similar with the practical interferogram obtained by an interferometer. SLM is simulated by a flat lens with Zernike standard phase surface. Communication between Matlab and Zemax is realized by Dynamic Data Exchange (DDE). Wavefront map data and interferogram data generated by Zemax are transferred to Matlab, and phase of the SLM calculated by Matlab is transferred to Zemax. The amplitude α of small random perturbations equals 0.1λ, γ^(q) = 1/ ([M^(q)]² /9/20), the J value calculation area varying parameter ξ equals 0.5.

Fig. 5 Simulation conditions. (a) The simulated test wavefront reflected from the large figure error local region of the full aperture test surface and transmits through the static null. (b) The corresponding interferogram generated by Zemax. (c) The modified interferogram which simulates the practical interferogram acquired by the interferometer.

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After fringes restoration, the interferogram is shown as Fig. 6(d). Figures 6(a)-6(c) are interferograms during the fringes restoration process. To relax the dense fringes, fringes relaxation algorithm is conducted. The interferogram after this step is shown as Fig. 7(d). Figures 7(a)-7(c) are interferograms during the fringes relaxation process. A media is given in Visualization 1 showing interferograms during the full process. This simulation validates enhanced convergence and improved convergence rate of the modified ZSPGD algorithm.

Fig. 6 Interferograms during the fringe restoration optimization. (a)−(c) The fringes that have not been restored completely. (d) The fringes that have been restored completely.

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Fig. 7 Interferograms during the fringe relaxation optimization (see Visualization 1). (a)−(c) The fringes that have not been relaxed completely. (d) The final interferogram after fringe relaxation.

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4. Experiment

Experiments are conducted to validate the SLM-based AWI. The SLM Holoeye^TM LC 2012 that we used for this work is a transmissive type one, with 1024 × 768 pixels, and pixel pitch of 36μm. The interferometer is a 4” Zygo GPI interferometer.

The experiment is to utilize the SLM-based AWI system to test a Φ61mm flat aluminum mirror with unknown severe local surface figure error. The mirror is fabricated by diamond turning [38], and severe local surface figure error is deliberately fabricated within the central region. The test setup is shown in Fig. 8. The test setup is very simple. SLM is inserted between the Zygo interferometer and the test surface.

Fig. 8 SLM-based AWI experiment setup to test a Φ61mm flat mirror with unknown severe local surface figure error.

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Firstly, WFD self-compensating procedures are conducted by inserting a high-accuracy TF (material is glass) between the test mirror and the SLM. WFD of the SLM is measured by the interferometer. Figure 9(a) shows the interferogram of WFD. Figure 9(b) shows the WFD map of SLM within the regularly used circular aperture. To self-compensate the WFD, the WFD is encoded as an interference type CGH. Then the hologram image with tilt carrier as shown in Fig. 10 is loaded to SLM. The interferogram after self-compensating is shown as Fig. 11(a). Figure 11(b) shows the residual error with 0.090λ (PV). These results not only show the self-compensating accuracy of the SLM, but also validate the high control accuracy of SLM by the method of encoding phase in an interference type CGH. Note this has also verified the diffraction efficiency of SLM is sufficient for test optics with glass material.

Fig. 9 Measurement results of WFD of SLM. (a) Interferogram of WFD of SLM, where the green dot-dash circle stands for the usually used apertures, with size of Φ26mm. (b) WFD map of SLM within the circular aperture.

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Fig. 10 Interference type CGH image encoding WFD phase with tilt carrier.

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Fig. 11 Measurement results after self-compensating WFD of SLM. (a) Interferogram of full aperture after self-compensating. (b) Residual error within the circular aperture after self-compensating.

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Secondly, the full aperture of the flat mirror is tested by removing the SLM. The interferogram is shown as Fig. 12(a), note that the majority area within the central region is invisible fringes. The corresponding surface figure error within this region is not available as shown in Fig. 12(b). To test the surface figure error within the invisible fringes region, the calibrated SLM is inserted. Since the central Φ26mm circular aperture of the SLM can enclose all invisible fringes area, it is selected as the work region. Figure 13(a) is the interferogram at initial. Firstly, fringes restoration is conducted. The parameters of ZSPGD at this step are chosen as: amplitudes α = 0.4λ, γ^(q) = 1/ ([M^(q)]² /300), J value calculation area varying parameter ξ equals 0.5. The interferogram after fringes restoration is shown as Fig. 13(b). The fringes are still too dense. Then fringes relaxation is conducted. The parameters of ZSPGD at this step are chosen as: the amplitude α = 0.2λ, γ = 1/ (J₀ /20), where J₀ means the initial value of J at this step. The interferogram after fringes relaxation is shown as Fig. 13(c). Since + 1 order diffractive wavefront is utilized, the same phase map test result as Fig. 13(c) is encoded and added to the already existed phase map on SLM. Near-null condition is met as shown in Fig. 13(d). A media is given in Visualization 2 to show the interferograms during the full process.

Fig. 12 Test results of measuring the test surface with interferometer directly. (a) Interferogram. (b) Corresponding surface figure error map.

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Fig. 13 Interferograms during SLM-based AWI test (see Visualization 2). (a) At initial. (b) After fringes restoration step. (c) After fringes relaxation step. (d) After phase conjugation step.

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Figure 14(a) shows the reconstructed surface figure error result within the central Φ26mm circular aperture. Finally, the stitching algorithm shown in the Principle part is utilized to obtain full aperture surface error map as shown in Fig. 14(b), with 33.331λ (PV) and 5.974λ (RMS).

Fig. 14 Test results by SLM-based AWI and on machine measurement. (a) The central Φ26mm circular local region surface figure error obtained by AWI. (b) Full aperture surface figure error obtained by AWI. (c) Full aperture surface figure error obtained by on machine measurement. (d) Difference map between the two test results.

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To add a degree of traceability to the proposed method, cross test is completed by on machine measurement method. The test setup is shown in Fig. 15. On machine test method is based on a four axis CNC ultra-precision lathe. The lathe we utilized is Nanotech 450UPL. Spindle motion accuracies along axial and radial direction are both 12.5nm. Straightness of X-axis is 0.3 microns over 350mm full travel. Straightness of Z-axis is 0.3 microns over 300mm full travel. Straightness of Y-axis is 0.2 microns over 100mm full travel. The cutting tool is replaced by a spectrum confocal displacement sensor- STIL Initial 0.4 CL2MG140. Max error of the sensor within the entire 0.4mm measuring range is 80nm. The sensor moves back and forth to follow the surface shape as the diamond tool does, but with lower rotation speed and larger pitch. The accuracy of the on machine test method is about 0.2 microns within 150mm aperture since we have tested a D150mm flat and compared the result using a traceable Zygo GPI wavefront interferometer with a standard transmissive flat. The full aperture test result by on machine measurement is shown in Fig. 14(c), with 33.293λ (PV) and 5.970λ (RMS). Figure 14(d) shows the difference between Fig. 14(b) and 14(c), with 0.280λ (PV) and 0.047λ (RMS). Note that the PV value of 0.280λ is nearly equivalent to the test accuracy of on machine test.

Fig. 15 On machine measurement based on an ultra-precision lathe.

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5. Discussion

This SLM-based AWI has shown the ability of utilizing wavefront sensor-less AO technique to extend the dynamic range and the measurable range of conventional interferometry. The principle is verified to be useful. Compared with the existing method such as two-wavelength interferometer and TWI, one main merit of our method is the structure is simple. It can be established based on commercial interferometers. Only an SLM is required to be inserted into the test setup. Moreover, the SLM can be inserted to null test setup with a static null to test aspheres and free-forms with severe surface figure error. However, there still exist several challenges to develop an AWI system that operates robustly and quickly for a wide selection of test surfaces with high test accuracy.

Follows are considerations about further extending the measurement range and shortening the measurement time.

(1) Unknown free-form shape severe surface figure error can be tested by the AWI method. Note that we mean the surface figure error (not the test surface) is freeform shape. The test surfaces can be flats, spheres, aspheres, and free-form surfaces. For aspheres and free-form surfaces, it is not the SLM that compensates the intrinsic aberration (theoretical departure from spheres or flats) of the test surfaces. The magnitude of the intrinsic aberration is usually too large for SLM to compensate. It is the additional static null such as an Offner null Dall null or CGH that compensates the intrinsic aberration. Our verification experiment is conducted on a flat with large local surface figure error. Experiments of applying our method to asphere or free-forms are a little more complex, since there exists an additional static null in the test system.
(2) The adaptive null optics – SLM can intelligently speculate and compensate the wavefront error (double passes) reflected from the test surface at the SLM plane with error about 40 microns (our SLM) within about 26mm circular aperture. If the test surface has surface figure error larger than that, the method is not applicable. The compensation ability is limited by the pixel array size of the SLM. Our SLM has a two-dimensional array of 1024 × 768 pixels. As far as we know, SLM with 10 megapixel (4K) resolution [39] is available at Holoeye^TM, and it can generate a larger amount of compensation.
(3) Combining the proposed AWI system with variable null will be very meaningful. The combined system is promising to test a series of surfaces with unknown free-form surface figure error. Variable null such as variable spherical aberration null and variable off-axis aberration null can generate a relatively wide range of variable spherical, coma, and astigmatic aberrations. Hence, they can compensate inherent aberration of different surfaces. However, the number of aberration modes that a certain type of variable null can generate is limited to one or two. Thus, variable null cannot be utilized to compensate the unknown free-form severe surface figure error. As for our proposed AWI system, unknown free-form severe surface figure error can be tested by adaptive null optics such as SLMs and DMs. However, AWI cannot test a wide range of surfaces which have different surface shape parameters, because the amount of aberrations that adaptive null optics can generate is very small compared with inherent aberrations of different surfaces. By combining AWI with variable null, the new test system will overcome the deficiency of both methods and combine the advantages of both methods simultaneously. This technique is under research by us, and we hope to report it soon.
(4) A more appropriate objective function is required to enhance the convergence rate and robustness of control algorithms. Image metrics based on interferogram are utilized by us as the objective function. The merit function used during fringes restoration is not a monotonous function of the resolvable fringes pixel number. Hence the optimization process is a little complex. Finding a merit function that is a monotonously increasing function of the resolvable fringes pixel number will simplify and accelerate the optimization process [40]. This is under researched by us. Moreover, merit function based on interferogram is very sensitive to noise and vibration. In contrast, the objective functions commonly utilized in AO telescopes are based on image metrics of far-field spot such as Strehl Ratio (SR), and mean radius of spot [41]. Considering the far-field spot is less sensitive to noise and vibration than interferogram, utilizing image metrics based on the far-field spot may be a better choice.
(5) Control Algorithms with shorter search time and easier implementation are vital to the practical application of AWI. SPGD algorithm can search for the optimum values of multi-dimension parameter in a parallel way, which accelerates the convergence of algorithms, and have some stochastic specialties, which can help the algorithm escape from local extremes to some extent. However, adjustment of parameters of SPGD is complicated, experience-dependent and time-consuming. Improper parameters will lead to very slow convergence rate or repeated oscillations. This is the main obstacle in applying model-free algorithms to the practical use of AWI. Some other model-free stochastic parallel algorithms, such as, Genetic Algorithm (GA), Simulated Annealing (SA) and Algorithm of Pattern Extraction (Alopex) can also be used as control algorithms in AO systems [42]. Comparison of these algorithms performance for our technique is worthy to find a suitable algorithm for different conditions. Compared with model-free algorithms, model-based algorithms may be a much better choice for the practical application. One model-based approach proposed by Booth is capable of correcting aberrations with a minimum of N + 1 photodetector measurements for N aberration modes [43]. What is more, fewer parameters required to be adjusted make model-based algorithms more attractive for practical application of AWI.

Follows are considerations about further improving the test accuracy.

(1) High accuracy phase control technique is needed. The remarkable advantages of the SLM control approach we adopt in the experiments are that the phase control accuracy does not rely on delicate nonlinearity response calibration. It is the fringes pattern of the hologram image that determines the generated wavefront. Hence the control accuracy is relatively high as shown in our experiments. However, the specific influence factors are still unknown when very large phase is loaded to SLM. Hence, further research is required.
(2) Simulation and experiments show SLM-based AWI works well for correcting low-order surface figure error. However, surface figure error is not confined to low-spatial frequency. Simultaneously adaptive correcting low- and high-order surface figure error will be very meaningful for practical application. However, this is hindered by the AO technique now. No AO elements can generate large amplitude high-order phase modes with high accuracy. Moreover, the convergence time and computation ability of the computer will be unbearable if more optimization variables are utilized [44]. At this point, we entirely agree with the sentiment of Booth’s that “Inspiration might be drawn from developments in astronomical ‘extreme’ AO where new techniques are being proposed for high-resolution, high-accuracy wavefront correction in applications such as exoplanet detection.” [45].
(3) Only a preliminary experiment is conducted to verify the proposed test method. When a static null exists in addition to the SLM, or not only one severe surface figure error local region exists, the test system will be more complicated. In that condition, careful alignment of elements position and posture, extraction of the local surface figure error by reverse optimization based on ray trace model and a more sophisticated stitching algorithm are important to guarantee the test accuracy. These demand further research.

Conclusion

By innovatively combining wavefront sensor-less AO technique and conventional interferometry technique, our proposed SLM-based AWI can test surfaces with severe surface figure error which are beyond the dynamic range of the conventional interferometer. In this system, a calibrated high-accuracy SLM works as adaptive null. Experiments show that severe surface figure error with amplitude larger than 20 microns within Φ26 mm circular aperture can be tested after only about 3 minutes iterative search process. This method is meaningful for guiding optics fabrication at the initial stages, at which the deterministic process is a little uncertain and severe surface figure error often occurs.

Funding

Hunan Provincial Natural Science Foundation of China (2016JJ1003); National Natural Science Foundation of China (51375488).

Acknowledgments

Our deepest gratitude goes to the anonymous reviewers for their careful work and thoughtful suggestions that have helped improve this paper substantially. We would like to thank Stefan Osten at Holoeye Corporation for he is always ready to help us with the use of SLM.

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Name	Description
Visualization 1	This video shows the full process in the simulation to turn invisible fringes to resolvable fringes utilizing AWI.
Visualization 2	This video shows the full process in the experiment to turn invisible fringes to resolvable fringes utilizing AWI.

Adaptive wavefront interferometry for unknown free-form surfaces

Abstract

1. Introduction

2. Principle

3. Simulations

4. Experiment

5. Discussion

Conclusion

Funding

Acknowledgments

References

Supplementary Material (2)

Cited By

Figures (15)

Equations (13)

Optics Express