## Abstract

We propose a highly stable and single-shot wavefront sensor using vectorial shearing interferometry. We design it using a pair of cascaded Sagnac interferometers that allow spatial frequency multiplexing in the interferogram in order to detect the vector gradients simultaneously and independently. The scheme avoids wavelength dispersion in sheared copies as compared to the vectorial shearing interferometer schemes based on diffractive optical elements. Capability to tune the amount of shear merely through the axial displacement of the imaging sensor, and with the added advantage of adjustable resolution controlled by the spatial frequency introduced, the near common-path geometry opens up its applications in optical testing even in a vibration sensitive environment.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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### Equations (9)

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(1)
$${u}_{A}({x}^{\mathrm{\prime}},{y}^{\mathrm{\prime}})={F}_{({f}_{1})}^{-1}\left\{{F}_{({f}_{1})}[{u}_{o}({x}^{\mathrm{\prime}},{y}^{\mathrm{\prime}})]\times exp\left(-i\frac{2\pi {z}_{1}}{\lambda}\sqrt{1-\frac{{\hat{x}}^{2}}{{f}_{1}^{2}}-\frac{{\hat{y}}^{2}}{{f}_{1}^{2}}}\right)\right\}$$
(2)
$${U}_{F}\left(\frac{\hat{x}}{\lambda {f}_{1}},\frac{\hat{y}}{\lambda {f}_{1}}\right)={F}_{({f}_{1})}[{u}_{O}({x}^{\mathrm{\prime}},{y}^{\mathrm{\prime}})]\times exp\left(-i\frac{2\pi {z}_{1}}{\lambda}\sqrt{1-\frac{{\hat{x}}^{2}}{{f}_{1}^{2}}-\frac{{\hat{y}}^{2}}{{f}_{1}^{2}}}\right)$$
(3)
$$\begin{array}{rl}{U}_{F}\left(\frac{\hat{x}-{\hat{x}}_{s,n}}{\lambda {f}_{1}},\frac{\hat{y}-{\hat{y}}_{s,n}}{\lambda {f}_{1}}\right)& ={U}_{O}\left(\frac{\hat{x}-{\hat{x}}_{s,n}}{\lambda {f}_{1}},\frac{\hat{y}-{\hat{y}}_{s,n}}{\lambda {f}_{1}}\right)\times \\ & exp\left(-i\frac{2\pi {z}_{1}}{\lambda}\sqrt{1-\frac{{(\hat{x}-{\hat{x}}_{s,n})}^{2}}{{f}_{1}^{2}}-\frac{{(\hat{y}-{\hat{y}}_{s,n})}^{2}}{{f}_{1}^{2}}}\right)\end{array}$$
(4)
$${u}_{B,n}(x,y)={F}_{({f}_{2})}\left\{{U}_{F}\left(\frac{\hat{x}-{\hat{x}}_{s,n}}{\lambda {f}_{1}},\frac{\hat{y}-{\hat{y}}_{s,n}}{\lambda {f}_{1}}\right)\right\}$$
(5)
$${u}_{I,n}(x,y)={F}_{({f}_{2})}^{-1}\left\{{F}_{({f}_{2})}[{u}_{B,n}(x,y)]\times \mathrm{exp}\left(i\frac{2\pi {z}_{2}}{\lambda}\sqrt{1-\frac{{\hat{x}}^{2}}{{f}_{2}^{2}}-\frac{{\hat{y}}^{2}}{{f}_{2}^{2}}}\right)\right\}$$
(6)
$${u}_{I,n}(x,y)={F}_{({f}_{2})}^{-1}\left[{F}_{({f}_{2})}\left[{F}_{({f}_{2})}\left\{{U}_{F}\left(\frac{\hat{x}-{\hat{x}}_{s,n}}{\lambda {f}_{1}},\frac{\hat{y}-{\hat{y}}_{s,n}}{\lambda {f}_{1}}\right)\right\}\right]\times \mathrm{exp}\left(i\frac{2\pi {z}_{2}}{\lambda}\sqrt{1-\frac{{\hat{x}}^{2}}{{f}_{2}^{2}}-\frac{{\hat{y}}^{2}}{{f}_{2}^{2}}}\right)\right]$$
(7)
$${u}_{I,n}(x,y)={F}_{({f}_{2})}^{-1}\left[\begin{array}{l}{U}_{O}\left(\frac{-(\hat{x}-{\hat{x}}_{s,n})}{\lambda {f}_{1}},\frac{-(\hat{y}-{\hat{y}}_{s,n})}{\lambda {f}_{1}}\right)\times \mathrm{exp}\left(-i\frac{2\pi {z}_{1}}{\lambda}\sqrt{1-\frac{{(\hat{x}-{\hat{x}}_{s,n})}^{2}}{{f}_{1}^{2}}-\frac{{(\hat{y}-{\hat{y}}_{s,n})}^{2}}{{f}_{1}^{2}}}\right)\\ \times \mathrm{exp}\left(i\frac{2\pi {z}_{2}}{\lambda}\sqrt{1-\frac{{\hat{x}}^{2}}{{f}_{2}^{2}}-\frac{{\hat{y}}^{2}}{{f}_{2}^{2}}}\right)\end{array}\right]$$
(8)
$$\begin{array}{rl}{u}_{I,n}(x,y)& =exp\left(i\frac{2\pi {z}_{2}}{\lambda}\left(1-\frac{{f}_{1}^{2}}{{{f}_{2}}^{2}}+\frac{{\hat{x}}_{s,n}^{2}+{\hat{y}}_{s,n}^{2}}{2{{f}_{2}}^{2}}\right)\right)exp\left(-i\frac{2\pi}{\lambda {f}_{2}}(x{\hat{x}}_{s,n}+y{\hat{y}}_{s,n})\right)\\ & \times \phantom{\rule{thinmathspace}{0ex}}{u}_{O}\left(\frac{-1}{m}\left(x-\frac{{\hat{x}}_{s,n}{z}_{2}}{{f}_{2}}\right),\frac{-1}{m}\left(y-\frac{{\hat{y}}_{s,n}{z}_{2}}{{f}_{2}}\right)\right)\end{array}$$
(9)
$${I}_{I}(x,y)=|{u}_{I,1}(x,y)+{u}_{I,2}(x,y)+{u}_{I,3}(x,y){|}^{2}$$