Recovery of wavefront from multi-shear interferograms with different tilts

Yunfeng Guo; Jianpei Xia; Jianping Ding

doi:10.1364/OE.22.011407

1. Introduction

Lateral shearing interferometry is an important technique for evaluating optical wavefronts and is a useful tool in many applications [1–7]. Without the use of a reference wave, the shearing interferometry is highly resistant to vibration and more suitable in the absence of an available reference beam. The main drawback of lateral shearing interferometry is that it only measures the difference of the wavefront sheared with itself and an inverse problem thus has to be solved to extract the original phase. A variety of methods have been proposed to retrieve the two-dimensional (2D) wavefront from two phase difference maps in two orthogonal directions [8–14].

Among these approaches, the Fourier modal expansion method has been adopted extensively because of its fast computation, simplicity and accuracy. However, this method has the disadvantage that the signal’s Fourier spectrum at frequency periods corresponding to multiples of N/s (where N denotes the number of sampling points and s the shear amount) are lost. Therefore, Elster and Weingärtner first proposed a two-shear algorithm [8, 9], and subsequently, we proposed a multi-shear algorithm [14] to avoid spectral leaking. However, the effect of the tilt error in these algorithms is ignored. When changing shears to test a wavefront with lateral shearing interferograms, tilt errors of the wavefront could vary because of environment vibration, mechanical movement, misalignment of the setup, phase shift or phase unwrapping; the phase difference extraction from shearing interferograms may produce a constant bias, which is equivalent to a tilt in the wavefront, whether or not the original wavefront is tilted. Evaluation errors are introduced if multiple shearing interferograms are combined directly when the wavefront tilt differs between the shearing interferograms. Lateral shearing interferometry measures only the difference in the wavefront sheared with itself and wavefront tilt therefore does not show up as fringes but only as a bias to the fringe position. That is to say, the absolute tilt is difficult to determine directly from the shearing interferograms. Therefore the mismatch among the phase differences, due to tilt errors, can severely impair the reconstruction accuracy, particularly in multiple shearing interferometry. Schireiber [15] proposed a method to measure the wavefront tilt; however this method is not suitable to lateral shearing interferometers when the wavefront itself (rather than the tilt) is the main measurement goal.

In the case of one-dimensional (1D) wavefront recovery, methods have been established to eliminate tilt errors [16, 17]. To our best knowledge, no approach to the 2D multiple shearing interferometry that is robust against tilt variation has yet been reported. In this paper, after analyzing the effect of tilt errors, we propose a new method for the reconstruction of 2D wavefront with tilt. This improved algorithm can eliminate tilt errors adaptively and recover any wavefront with tilt exactly up to an arbitrary constant. The results of computer simulation and optical test confirm the effectiveness and accuracy of the proposed algorithm.

2. Wavefront reconstruction from multi phase differences without tilt error

The basic algorithm of the wavefront reconstruction from multiple phase differences has been given in the previous paper [14]. Here we concisely describe the algorithm. A series of phase differences from multiple shearing interferograms with different shears in the x and y directions are denoted by $D_{j}^{x} (m, n)$ and $D_{j}^{y} (m, n)$ respectively, with j denoting the j-th shearing. K groups of phase differences are expressed as

D_{j}^{x} (m, n) = {\begin{matrix} φ (m, n) - φ (m - s_{j}, n) & m \in [s_{j}, N - 1]; n \in [0, N - 1] \\ - \sum_{l = 1}^{N / s_{j} - 1} D_{j}^{x} (m + l s_{j}, n) & m \in [0, s_{j} - 1]; n \in [0, N - 1] \end{matrix}

D_{j}^{y} (m, n) = {\begin{matrix} φ (m, n) - φ (m, n - s_{j}) & n \in [s_{j}, N - 1]; m \in [0, N - 1] \\ - \sum_{l = 1}^{N / s_{j} - 1} D_{j}^{y} (m, n + l s_{j}) & n \in [0, s_{j} - 1]; m \in [0, N - 1] \end{matrix}

where j = 1, 2, …, K,

φ (m, n)

denotes the original phase discretized over a grid of dimensions

N \times N

, and s_j represents the j-th shear amount in the x and y directions. Here the “natural extension” is used to fill up the undefined values of phase differences [18]. The measured phase differences in the x and y directions are denoted

Δ_{j}^{x} (m, n)

and

Δ_{j}^{y} (m, n)

, respectively. We can try to estimate the wavefront by least-squares fitting these measured data to the theoretical phase differences and by minimizing the error

ε^{2} = \sum_{m, n} [\sum_{j = 1}^{K} {| Δ_{j}^{x} (m, n) - D_{j}^{x} (m, n) |}^{2} + \sum_{j = 1}^{K} {| Δ_{j}^{x} (m, n) - D_{j}^{y} (m, n) |}^{2}] .

We then obtain an optimal solution of the combined Fourier coefficients [11],

Φ (p, q)

, as below,

Φ (p, q) = \frac{\sum_{j = 1}^{K} (1 - e^{i 2 π p s_{j} / N}) Ζ_{j}^{x} (p, q) + \sum_{j = 1}^{K} (1 - e^{i 2 π q s_{j} / N}) Z_{j}^{y} (p, q)}{\sum_{j = 1}^{K} [4 \sin^{2} (\frac{π p s_{j}}{N}) + 4 \sin^{2} (\frac{π q s_{j}}{N})]},

where

Z_{j}^{x} (p, q)

and

Z_{j}^{y} (p, q)

represent the 2D Fourier transforms of measured phase differences,

Δ_{j}^{x} (m, n)

and

Δ_{j}^{y} (m, n)

, respectively. The wavefront phase

φ (m, n)

can be reconstructed over all frequencies by performing the inverse Fourier transform for the Fourier coefficients

Φ (p, q)

.

3. Improved algorithm with tilt from multiple phase differences

Assume a two-dimensional wavefront $φ (x, y) = φ_{0} (x, y) + a x + b y + c$ , where a and b are respectively tilt factors in the x and y directions and c is a piston term. Provided that we have K sets of shearing interferograms, we can then obtain K groups of phase differences of wavefront (with tilt) in the x and y directions, respectively, expressed as

Δ_{j}^{x} (m, n) = D_{j}^{x} (m, n) + a_{j} s_{j},

Δ_{j}^{y} (m, n) = D_{j}^{y} (m, n) + b_{j} s_{j}

Here a_j or b_j, which might differ in K shearing interferograms, is assumed to be unknown since they are difficult to test. To recover the wavefront

φ (x, y)

, we can use K shearing interferograms. However, we cannot combine these K shearing interferograms directly when a_j or b_j differs among interferograms. There will be error when we combine K phase differences directly, which is called tilt error. The effect of the tilt error should be analyzed and eliminated by a specially designed algorithm of wavefront recovery.

The phase difference extracted from the j-th shearing interferogram can be performed with Fourier transform of N points, and divided by the shearing transfer function respectively to obtain the 2D Fourier coefficient estimate of the wavefront in the x and y directions, respectively, expressed as follows,

Z_{j}^{x} (p, q) = \frac{\sum_{m, n = 0}^{N - 1} Δ_{j}^{x} (m, n) \exp [- \frac{i 2 π}{N} (m p + n q)]}{1 - \exp (- \frac{i 2 π s_{j} p}{N})},

Z_{j}^{y} (m, q) = \frac{\sum_{m, n = 0}^{N - 1} Δ_{j}^{y} (m, n) \exp [- \frac{i 2 π}{N} (m p + n q)]}{1 - \exp (- \frac{i 2 π s_{j} q}{N})} .

Substituting Eq. (4) into Eq. (5) and using Eq. (1), we obtain

\begin{matrix} Z_{j}^{x} (p, q) = [\sum_{n = 0}^{N - 1} \exp (- \frac{i 2 π n q}{N}) {\sum_{m = s_{j}}^{N - 1} [D_{j}^{x} (m, n) + a_{j} s_{j}] \exp (- \frac{i 2 π m p}{N}) \\ - \sum_{m = 0}^{s_{j} - 1} [\sum_{l = 1}^{N / s_{j} - 1} [D_{j}^{x} (m + l s_{j}, n) + a_{j} s_{j}]] \exp (- \frac{i 2 π m p}{N})}] / [1 - \exp (- \frac{i 2 π s_{j} p}{N})], \end{matrix}

\begin{matrix} Z_{j}^{y} (p, q) = [\sum_{m = 0}^{N - 1} \exp (- \frac{i 2 π m p}{N}) {\sum_{n = s_{j}}^{N - 1} [D_{j}^{y} (m, n) + b_{j} s_{j}] \exp (- \frac{i 2 π n q}{N}) \\ - \sum_{n = 0}^{s_{j} - 1} [\sum_{l = 1}^{N / s_{j} - 1} [D_{j}^{y} (m_{j}, n + l s) + b_{j} s_{j}]] \exp (- \frac{i 2 π n q}{N})}] / [1 - \exp (- \frac{i 2 π s_{j} q}{N})] . \end{matrix}

After some trivial deductions, we get the following relation

Z_{j}^{x} (p, q) = \frac{\sum_{m, n = 0}^{N - 1} D_{j}^{x} (m, n) \exp (- \frac{i 2 π m p}{N})}{1 - \exp (- \frac{i 2 π s_{j} p}{N})} + \frac{a_{j}}{s_{j}} \sum_{m = 0}^{N - 1} m \exp (- \frac{i 2 π m p}{N}),

Z_{j}^{y} (p, q) = \frac{\sum_{m, n = 0}^{N - 1} D_{j}^{y} (m, n) \exp (- \frac{i 2 π n q}{N})}{1 - \exp (- \frac{i 2 π s_{j} q}{N})} + \frac{b_{j}}{s_{j}} \sum_{n = 0}^{N - 1} n \exp (- \frac{i 2 π n q}{N}) .

In the above expression, the left side and the first term in the right side represent the 2D Fourier coefficient of the wavefront with and without tilt, respectively, in the corresponding direction. The difference between Fourier coefficient of the wavefront with and without tilt relates to the second term that is associated with the tilt factor a_j or b_j, multiplying by the 1D Fourier transform of a continuous natural number sequence whose length is N. For brevity's sake, we use

W_{j}^{x}

and

W_{j}^{y}

to denote the Fourier coefficients of the wavefront without tilt, namely the first term in the right side of Eq. (7), and assume

Y (p) = \sum_{m = 0}^{N - 1} m \exp (- i 2 π m p / N)

. Then Eq. (7) can be rewritten by

Z_{j}^{x} (p, q) = W_{j}^{x} (p, q) + \frac{a_{j}}{s_{j}} Y (p),

Z_{j}^{y} (p, q) = W_{j}^{y} (p, q) + \frac{b_{j}}{s_{j}} Y (q) .

From Eq. (8) we know that the uncertainty of tilt factor would introduce the error into the Fourier coefficient estimate of wavefront and thus impair the accuracy of wavefront reconstruction. Our approach aims to eliminate the influence of tilt and propose an improved wavefront recovery algorithm that is robust against tilts.

Inasmuch as the Fourier coefficient of a real wavefront without tilt must keep unchanged for different shears (i.e., $W_{j}^{x}$ and $W_{j}^{y}$ are independent to j), by subtracting $Z_{1}^{x} (Z_{1}^{y})$ from $Z_{j}^{x} (Z_{j}^{y})$ expressed in (8), we get the following expression

\frac{Z_{j}^{x} (p, q) - Z_{1}^{x} (p, q)}{Y (p)} = α_{j},

\frac{Z_{j}^{y} (p, q) - Z_{1}^{y} (p, q)}{Y (p)} = β_{j},

where

α_{j} = (a_{j} / s_{j} - a_{1} / s_{1})

and

β_{j} = (b_{j} / s_{j} - b_{1} / s_{1})

represent the compensation factors that can be used to eliminate the influence of tilt mismatch. These compensation factors can be exactly calculated according to Eq. (9) because the quantities in the left side can be determined during the measurement. It should be pointed out that our aim is to remove the tilt mismatch among different shearing amounts, rather than to determine the absolute slanted bias of wavefront. Hence, we can assume

a_{1} = 0

and

b_{1} = 0

(i. e.

α_{j} = a_{j} / s_{j}

and

β_{j} = b_{j} / s_{j}

) and rewrite Eq. (8) as

Z_{j}^{x} (p, q) = W_{j}^{x} (p, q) + α_{j} Y (p),

Z_{j}^{y} (p, q) = W_{j}^{y} (p, q) + β_{j} Y (q) .

The second term in the Eq. (10) represents the unbiased estimators–removing the tilt mismatch between multi-shear interferograms–for Fourier coefficients of the phase differences. Substituting Eq. (10) into Eq. (3), we obtain the modified Fourier coefficients

Φ (p, q) = \frac{\sum_{j = 1}^{K} (1 - e^{\frac{i 2 π p s_{j}}{N}}) [W_{j}^{x} (p, q) + α_{j} Y (p)] + \sum_{j = 1}^{K} (1 - e^{\frac{i 2 π q s_{j}}{N}}) [W_{j}^{y} (p, q) + β_{j} Y (q)]}{\sum_{j = 1}^{K} [4 \sin^{2} (\frac{π p s_{j}}{N}) + 4 \sin^{2} (\frac{π q s_{j}}{N})]} .

The original wave-front

φ (m, n)

can be reconstructed exactly by performing the inverse Fourier transform of the Fourier coefficients

Φ (p, q) .

The influence of the possible tilt errors have be removed adaptively by using the improved algorithm.

4. Computer simulation

To evaluate the reliability of the proposed algorithm, we first use the phase function

φ (x, y) = 2 π [0.15 (x^{2} + y^{2} - 2) (x^{2} + y^{2} + 1) - 0.6 (x^{2} - y^{2}) (x^{2} + y^{2} - 1.25)],

where x and y are within the range of [–1, 1] and there are 420 × 420 grids. The phase distribution is shown in Fig. 1(a).The phase differences with a shear of 4 pixels in two orthogonal directions are shown in Fig. 1(b) and Fig. 1(c). The multi-shear s_j are chosen to be 1, 2, 3, 4, and 5 pixels in this case. The array of phase differences is added by a uniform constant that is used to imitate the effect of tilt. Multiple group phase differences are added using normal random noise (mean of zero and standard deviation of 0.0001), which is assumed to be tilts for different phase differences. The reconstructed phase is shown in Fig. 1(d) using the old multi-shear reconstruction algorithm. The phase distribution reconstructed using the improved algorithm is plotted in Fig. 1(e). The deviation of the reconstructed phase from the original phase is computed, using the root mean square (RMS) as the error measure. The RMS values are 0.0058 and 7.8 × 10⁻⁹ for the old and proposed algorithms, respectively. To visualize the performance of the two algorithms from the plot, we calculate the difference between the original and recovered phases in the 200-th row. The deviation is plotted in Fig. 1(f). The peak-to-valley value of deviation in the 200-th row turns out to be 0.64 for the old and 2 × 10⁻⁷ for the improved algorithm, respectively. We find the old algorithm give rise to larger errors, particularly in the border region of the wavefront. It is known that the dimension of the phase difference is inherently smaller than that of the original phase; hence, the natural extension of the difference data is required to remove this dimensional mismatch. During the data extrapolation process, the phase differences that are undefined in the border region are obtained by summing the available phase differences [18]; accordingly, the bias in the phase differences accumulates within the border region. If the bias mismatch takes place in the multiple phase differences – this is so with the old algorithm, a bulge of deviation appears at the border region. Because the improved algorithm can adaptively eliminates the tilt errors of wavefront (i.e. bias mismatch in the phase differences), it produce a high-precision reconstruction of phase, as shown in Figs. 1(e) and 1(f).

Fig. 1 Simulation results for a shear combination s = 1, 2, 3, 4, 5 and N = 420 × 420. (a) Original phase; (b) x-directional phase difference (4-pixel shear); (c) y-directional phase difference (4-pixel shear); (d) phase reconstructed with the old algorithm; (e) phase reconstructed with the new improved algorithm; (f) difference between the original and recovered phases in the 200-th row using the two algorithms.

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Second, we choose a discontinuous phase function for simulation. Owing to its step-like phase map, we choose the letter E for the reconstruction simulations with 420 × 420 grid points. We choose the same shear amounts and tilts as in the above example. The original phase is plotted in Fig. 2(a).In the simulation, the obtained RMS error is 0.0058 for the old algorithm and zero for the improved algorithm. The improved algorithm demonstrates superior performance in recovering the phase distribution when there is tilt error. To visualize the performance of the two algorithms from the plot, we use the deviation between the original phase and reconstructed phase. The deviation obtained using the previous algorithm is shown in Fig. 2(b), and the deviation obtained using the improved algorithm is shown in Fig. 2(c). An illusion of two surfaces in Fig. 2(b) is due to fact that large positive and negative deviations at the border region cast a perspective over the three-dimensional plot. In order to better see the difference in performance between two algorithms, we plot in Fig. 2(d) the deviation of the reconstructed to the original phase in the 200-th row. In this case, the peak-to-valley value of deviation in the 200-th row is 0.66 for the old and 0 for the improved algorithm, respectively. Again, the high precision of the improved algorithm demonstrates that the algorithm can automatically eliminate the effects of the tilt errors.

Fig. 2 Simulation results for the shear combination s = 1, 2, 3, 4, 5 and N = 420 × 420. (a) Original phase map; (b) deviation between the original phase and phase reconstructed with the old algorithm; (c) deviation between the original phase and phase reconstructed with the new improved algorithm; (d) difference between the original and recovered phases in the 200-th row using the two algorithms.

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The numerical simulation also shows that the proposed multi-shear algorithm performs even better in recovering the discontinuous phase than the continuous phase. As explained in [11], this effect is due to the fact that a discontinuous function has a broader Fourier frequency range than does a smooth function. A step-like phase, which has more high-frequency components in the Fourier domain, is more likely susceptible to frequency-leaking damage, which occurs when the single-shear algorithm is used but can be avoided completely by using the multi-shear algorithm.

5. Optical experiments

We carry out optical experiments using a three-wave lateral shearing interferometer based on a spatial light modulator (SLM) [19], as schematically shown in Fig. 3.A cosine grating is displayed on the SLM and controlled by a computer. The flexibility of the SLM can be fully used in the sense that the phase shift, as well as the direction and amount of shear, can be dynamically controlled. Therefore, the interferometer can avoid mechanical movement and attain better alignment precision. The interferograms are captured by a charge-coupled device (CCD) camera. The phase differences are extracted by an eight-step phase-shifting algorithm [19] and then unwrapped. The phase differences in the perpendicular direction are obtained by rotating the grating through 90 degrees. Multiple phase differences are acquired for different shear amounts, and are then used to reconstruct the desired phase. In the optical test, the misalignments of the setup and the vibration of the environment are difficult to avoid completely by hardware calibration, which introduces tilt errors of the wavefront under test. Besides, phase shifting and phase unwrapping can only extract phase differences with uncertain biases which are equivalent to tilt errors in the wavefront. Hence, the improved multi-shear algorithm can be helpful to improve the precision of the reconstructed wavefront.

Fig. 3 Three-wave lateral shearing interferometer based on an SLM with four lenses (L1, L2, L3 and L4) and a rotating ground glass that lowers the spatial coherence of light to reduce speckle noise.

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The object under test is a binary optical element that is produced by photoetching quartz glass. The object is placed in the object plane. The shearing interferograms are captured by the CCD camera and sampled within a 420 × 420 pixel domain. Shear amounts of 1, 2, 3, 4 and 5 pixels are chosen for the multiple-shearing interference test, and the reconstructed phase is shown in Fig. 4(a).The phase reconstructed using the new improved algorithm is shown in Fig. 4(b). For comparison, the relief profile along the 210-th row of the measured surface map is presented in Fig. 4(c). Here, again, the old algorithm incurs large errors in the border region, wherein abnormal variations appear. By contrast, the improved multi-shear reconstruction method can greatly reduce the misinterpretation of data. From the measured topography shown in Fig. 4, we can infer the surface information of the sample. Height d between the top and bottom of the binary relief is related to the measured phase by d = λφ/(2πΔn), where Δn is the difference between the refractive indexes of the glass and air. The refractive index of the glass is 1.457 at the working wavelength λ ( = 632.8 nm). We can read the corresponding relief height of the etched pattern from the relief profile; the corresponding relief height of the etched pattern is around 500 nm.

Fig. 4 Optic surface testing with the old and improved algorithms: (a) 2D phase map reconstructed using the old algorithm; (b) 2D phase map reconstructed using the new improved algorithm; (c) 1D relief profile: dotted line obtained using the old algorithm, dashed line obtained using the new algorithm.

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7. Conclusion

Tilt errors resulting from phase difference measurement are inevitable, which can seriously impair the precision of multiple shearing interferometry. An improved algorithm based on the multi-shear interferometry has been proposed to reconstruct a two-dimensional wavefront. The effects of tilt errors on the wavefront reconstruction are analyzed and a compensation method is developed. The proposed algorithm automatically eliminates the error and reconstructs any wavefront with tilt exactly up to an arbitrary constant. Numerical and optical tests have confirmed the excellent wavefront recovery capability of the proposed method.

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China under Grant Nos. 11074116 and 10874078. We thank Dr. Xin Chen (Department of Instrument Science and Engineering, Shanghai Jiao Tong University) for helpful discussions.

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Recovery of wavefront from multi-shear interferograms with different tilts

Abstract

1. Introduction

2. Wavefront reconstruction from multi phase differences without tilt error

3. Improved algorithm with tilt from multiple phase differences

4. Computer simulation

5. Optical experiments

7. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (20)

Optics Express