Wavefront Reconstruction for Fractional Lateral Shear Measurements using Weighted Integer Shear Averages

TL;DR

The paper tackles the challenge of reconstructing wavefronts from lateral shearing interferometry when the applied shear is fractional rather than an integer multiple of the sampling interval. It introduces a weighted integer shear averaging method, showing analytically how using multiple nearby integer shears and appropriate weights cancels the leading error terms: two-shear averaging eliminates first-order errors and three-shear averaging removes second-order terms. Numerical simulations with a test wavefront confirm substantial RMS-error reductions compared with conventional single-shear reconstructions, validating both the theory and the method's practicality. The approach is simple, computationally efficient, extension-friendly to two dimensions, and holds significant potential for robust wavefront measurements in systems where fractional shears are unavoidable.

Abstract

Wavefront reconstruction in lateral shearing interferometry typically assumes that the shear amount is an integer multiple of the sampling interval. When the shear is fractional, approximating it with the nearest integer value leads to noticeable reconstruction errors. To address this, we propose a weighted integer shear averaging method. The approach combines reconstructions from nearby integer shears with carefully chosen weights designed to cancel the dominant error terms. Analytical error analysis shows that two-shear averaging removes first-order errors, while three-shear averaging removes second-order errors. Numerical simulations with a test wavefront confirm that the method achieves significantly lower RMS error than conventional single-shear reconstruction. The technique is simple, computationally efficient, and can be readily extended to two-dimensional interferometry. This makes weighted integer shear averaging a practical and accurate tool for wavefront reconstruction when fractional shear is unavoidable.

Figures (5)

• Figure 1: The simulation data as true wavefront $\phi^{\rm True}(x)$ and differential phase $f^{\rm M}(x,s)$ for $s = 1.5$.
• Figure 2: Comparisons of errors between the wavefront without and with scaling: (a) Reconstructed wavefronts by the basic method without scaling, $\phi_{s,S}(x)$ and their errors, $\Delta\phi_{s;S}(x)\equiv\phi_{s,S}-\phi^{\rm True}(x)$. (b) Scaled wavefronts, $\phi_{s;S}^\dagger(x)=\frac{S}{s}\phi_{s;S}(x)$ and their errors, $\Delta\phi_{s;S}^\dagger(x)$. Note: The errors range in the lower panels differ between subfigures (a) and (b).
• Figure 3: Errors by the weighted averaging method. The superscripts 1, 2, and 3 of ${\boldsymbol{S}}^{}$ express the number of the scaled wavefronts employed in the weighted averaging method. Note: The vertical scales differ between the panels.
• Figure 4: Shear dependence of relative reconstruction errors. (a) Reconstructed wavefront using the basic method. (b)-(d) Reconstructed wavefronts evaluated by the proposed method using single, two and three integer shear values, respectively. Open and solid symbols denote integer and fractional shears, respectively. The dashed lines in subfigures represent the theoretical error obtained by least-squares fitting for solid points with linear, quadratic, or cubic functions. The scales of vertical axes differ between subfigures. The interval between evaluations for $s$ is $\delta_{x}/10$.
• Figure 5: Shear dependence of relative errors. The open symbols at the local minima and solid symbols represent the case where $s$ are integer shears and fractional shears, respectively.