## Abstract

Multi-transmitter aperture synthesis increases the effective aperture in coherent imaging by shifting the backscattered speckle field across a physical aperture or set of apertures. Through proper arrangement of the transmitter locations, it is possible to obtain speckle fields with overlapping regions, which allows fast computation of optical aberrations from wavefront differences. In this paper, we present a method where Zernike polynomials are used to model the aberrations and high-order aberrations are estimated without the need to do phase unwrapping of the difference fronts.

© 2012 OSA

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### Equations (6)

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(1)
$${U}_{i}\left(x,y\right)=P\left(x,y\right)\text{exp}\left(j2\pi {W}_{e}\left(x,y\right)\right){U}_{b}\left(x-{x}_{i},y-{y}_{i}\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,2,$$
(2)
$${W}_{i}\left(x,y\right)={W}_{e}\left(x,y\right)+{W}_{b}\left(x-{x}_{i},y-{y}_{i}\right),\hspace{0.17em}\hspace{0.17em}i=1,2.$$
(3)
$$\begin{array}{lll}\mathrm{\Delta}W\left(x,y\right)\hfill & =\hfill & {W}_{1}\left(x+{x}_{1},y+{y}_{1}\right)-{W}_{2}\left(x+{x}_{2},y+{y}_{2}\right)\hfill \\ \hfill & =\hfill & \left({W}_{e}\left(x+{x}_{1},y+{y}_{1}\right)+{W}_{b}\left(x,y\right)\right)-\left({W}_{e}\left(x+{x}_{2},y+{y}_{2}\right)+{W}_{b}\left(x,y\right)\right)\hfill \\ \hfill & =\hfill & {W}_{e}\left(x+{x}_{1},y+{y}_{1}\right)-{W}_{e}\left(x+{x}_{2},y+{y}_{2}\right).\hfill \end{array}$$
(4)
$$\begin{array}{lll}\mathrm{\Delta}W\left(x,y\right)\hfill & =\hfill & \sum _{k}{a}_{k}{Z}_{k}\left(x+{x}_{1},y+{y}_{1}\right)-\sum _{k}{a}_{k}{Z}_{k}\left(x+{x}_{2},y+{y}_{2}\right)\hfill \\ \hfill & =\hfill & \sum _{k}{a}_{k}\left({Z}_{k}\left(x+{x}_{1},y+{y}_{1}\right)-{Z}_{k}\left(x+{x}_{2},y+{y}_{2}\right)\right)\hfill \\ \hfill & =\hfill & \sum _{k}{a}_{k}\mathrm{\Delta}{Z}_{k}\left(x,y\right),\hfill \end{array}$$
(5)
$$\mathrm{\Delta}{W}^{\left(m\right)}\left({x}_{i}^{\left(m\right)},{y}_{i}^{\left(m\right)}\right)={\alpha}^{\left(m\right)}+\sum _{k}{a}_{k}\mathrm{\Delta}{Z}_{k}^{\left(m\right)}\left({x}_{i}^{\left(m\right)},{y}_{i}^{\left(m\right)}\right),$$
(6)
$$\left[\begin{array}{c}\mathrm{\Delta}{W}^{\left(1\right)}\left({x}_{1}^{\left(1\right)},{y}_{1}^{\left(1\right)}\right)\\ \vdots \\ \mathrm{\Delta}{W}^{\left(1\right)}\left({x}_{{N}_{1}}^{\left(1\right)},{y}_{{N}_{1}}^{\left(1\right)}\right)\\ \mathrm{\Delta}{W}^{\left(2\right)}\left({x}_{1}^{\left(2\right)},{y}_{1}^{\left(2\right)}\right)\\ \vdots \\ \mathrm{\Delta}{W}^{\left(2\right)}\left({x}_{{N}_{2}}^{\left(2\right)},{y}_{{N}_{2}}^{\left(2\right)}\right)\\ \vdots \\ \vdots \\ \vdots \\ \mathrm{\Delta}{W}^{\left(M\right)}\left({x}_{1}^{\left(M\right)},{y}_{1}^{\left(M\right)}\right)\\ \vdots \\ \mathrm{\Delta}{W}^{\left(M\right)}\left({x}_{{N}_{M}}^{\left(M\right)},{y}_{{N}_{M}}^{\left(M\right)}\right)\end{array}\right]=\left[\begin{array}{ccccccc}1& 0& \cdots & 0& \mathrm{\Delta}{Z}_{1}^{\left(1\right)}\left({x}_{1}^{\left(1\right)},{y}_{1}^{\left(1\right)}\right)& \cdots & \mathrm{\Delta}{Z}_{K}^{\left(1\right)}\left({x}_{1}^{\left(1\right)},{y}_{1}^{\left(1\right)}\right)\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1& 0& \cdots & 0& \mathrm{\Delta}{Z}_{1}^{\left(1\right)}\left({x}_{{N}_{1}}^{\left(1\right)},{y}_{{N}_{1}}^{\left(1\right)}\right)& \cdots & \mathrm{\Delta}{Z}_{K}^{\left(1\right)}\left({x}_{{N}_{1}}^{\left(1\right)},{y}_{{N}_{1}}^{\left(1\right)}\right)\\ 0& 1& \cdots & 0& \mathrm{\Delta}{Z}_{1}^{\left(2\right)}\left({x}_{1}^{\left(2\right)},{y}_{1}^{\left(2\right)}\right)& \cdots & \mathrm{\Delta}{Z}_{K}^{\left(2\right)}\left({x}_{1}^{\left(2\right)},{y}_{1}^{\left(2\right)}\right)\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0& 1& \cdots & 0& \mathrm{\Delta}{Z}_{1}^{\left(2\right)}\left({x}_{{N}_{2}}^{\left(2\right)},{y}_{{N}_{2}}^{\left(2\right)}\right)& \cdots & \mathrm{\Delta}{Z}_{K}^{\left(2\right)}\left({x}_{{N}_{2}}^{\left(2\right)},{y}_{{N}_{2}}^{\left(2\right)}\right)\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0& 0& \cdots & 1& \mathrm{\Delta}{Z}_{1}^{\left(M\right)}\left({x}_{1}^{\left(M\right)},{y}_{1}^{\left(M\right)}\right)& \cdots & \mathrm{\Delta}{Z}_{K}^{\left(M\right)}\left({x}_{1}^{\left(M\right)},{y}_{1}^{\left(M\right)}\right)\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0& 0& \cdots & 1& \mathrm{\Delta}{Z}_{1}^{\left(M\right)}\left({x}_{{N}_{M}}^{\left(M\right)},{y}_{{N}_{M}}^{\left(M\right)}\right)& \cdots & \mathrm{\Delta}{Z}_{K}^{\left(M\right)}\left({x}_{{N}_{M}}^{\left(M\right)},{y}_{{N}_{M}}^{\left(M\right)}\right)\end{array}\right]\left[\begin{array}{c}{\alpha}^{\left(1\right)}\\ {\alpha}^{\left(2\right)}\\ \vdots \\ {\alpha}^{\left(M\right)}\\ {a}_{1}\\ \vdots \\ {a}_{K}\end{array}\right]$$