## Abstract

A technique based on superresolution by digital holographic microscopic imaging is presented. We used a two dimensional (2-D) vertical-cavity self-emitting laser (VCSEL) array as spherical-wave illumination sources. The method is defined in terms of an incoherent superposition of tilted wavefronts. The tilted spherical wave originating from the 2-D VCSEL elements illuminates the target in transmission mode to obtain a hologram in a Mach–Zehnder interferometer configuration. Superresolved images of the input object above the common lens diffraction limit are generated by sequential recording of the individual holograms and numerical reconstruction of the image with the extended spatial frequency range. We have experimentally tested the approach for a microscope objective with an exact 2-D reconstruction image of the input object. The proposed approach has implementation advantages for applications in biological imaging or the microelectronic industry in which structured targets are being inspected.

© 2006 Optical Society of America

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### Equations (6)

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(1)
$${U}_{m,n}^{\mathrm{CCD}}\text{\hspace{0.17em}}(x,y)=\mathbf{\left(}f(-\frac{x}{M},-\frac{y}{M})\mathrm{exp}\{j\text{\hspace{0.17em}}{{\frac{k}{2Md}\text{\hspace{0.17em}}[(x-{x}_{m})}^{2}+(y-{y}_{n})}^{2}]\}\mathbf{\right)}\otimes \text{disk}(\mathrm{\Delta \nu}\mathrm{}r),$$
(2)
$${I}_{m\times n}^{\mathrm{CCD}}\text{\hspace{0.17em}}(x,y)=|\mathbf{\left(}f(-\frac{x}{M},-\frac{y}{M})\mathrm{exp}\{j\text{\hspace{0.17em}}{{\frac{k}{2Md}\text{\hspace{0.17em}}[(x-{x}_{m})}^{2}+(y-{y}_{n})}^{2}]\}\mathbf{\right)}\otimes \text{disk}(\mathrm{\Delta \nu}\mathrm{}r)+\mathrm{exp}{\{j\text{\hspace{0.17em}}{{\frac{k}{{2d}^{\prime}}\text{\hspace{0.17em}}[(x-{x}_{m})}^{2}+(y-{y}_{n})}^{2}]+j2\pi Qx\left\}\right|}^{2},$$
(3)
$${I}_{m,n}^{\mathrm{CCD}}(x,y)=1+\left|\mathbf{\right(}f(-\frac{x}{M},-\frac{y}{M})\mathrm{exp}\{j\text{\hspace{0.17em}}\frac{k}{2\mathit{Md}}[(x-{x}_{m}{)}^{2}+(y-{y}_{n}{)}^{2}]\}\mathbf{)}\otimes \mathrm{disk}(\mathit{\Delta vr}){|}^{2}+\mathbf{[}\mathbf{\left(}f\right(-\frac{x}{M},-\frac{y}{M}\left)\mathrm{exp}\right\{j\text{\hspace{0.17em}}\frac{k}{2\mathit{Md}}[(x-{x}_{m}{)}^{2}+(y-{y}_{n}{)}^{2}]\left\}\mathbf{\right)}\otimes \mathrm{disk}(\mathit{\Delta vr})\mathbf{\left]}\mathrm{exp}\right\{-j\text{\hspace{0.17em}}\frac{k}{2d\prime}[(x-{x}_{m}{)}^{2}+(y-{y}_{n}{)}^{2}]-j2\mathit{\pi Qx}\}+\mathbf{[}\mathbf{(}f*(-\frac{x}{M},-\frac{y}{M}\left)\mathrm{exp}\right\{-j\text{\hspace{0.17em}}\frac{k}{2\mathit{Md}}[(x-{x}_{m}{)}^{2}+(y-{y}_{n}{)}^{2}]\left\}\mathbf{\right)}\otimes \mathrm{disk}(\mathit{\Delta vr})\mathbf{\left]}\mathrm{exp}\right\{j\text{\hspace{0.17em}}\frac{k}{2d\prime}[(x-{x}_{m}{)}^{2}+(y-{y}_{n}{)}^{2}]+j2\mathit{\pi Qx}\}\mathrm{.}$$
(4)
$${\tilde{P}}_{3}(u,v)=K\mathbf{\left\{}\mathbf{[}\tilde{f}(Mu+\frac{{x}_{m}}{\lambda d},\mathit{M\nu}+\frac{{y}_{n}}{{\lambda}_{d}})\otimes {\mathrm{FT}}^{-1}\right\{\mathrm{exp}[j\text{\hspace{0.17em}}\frac{k}{2Md}\text{\hspace{0.17em}}({x}^{2}+{y}^{2})]\left\}\mathbf{\right]}\text{}\mathrm{circ}\left(\frac{\rho}{\mathrm{\Delta \nu}}\right)\mathbf{\}}\otimes {\mathrm{FT}}^{-1}\{\mathrm{exp}[-j\text{\hspace{0.17em}}\frac{k}{{2d}^{\prime}}\text{\hspace{0.17em}}({x}^{2}+{y}^{2})]\mathrm{exp}[j\text{\hspace{0.17em}}\frac{k}{{d}^{\prime}}({xx}_{m}+y{y}_{n})]\}\otimes \delta (u+Q,\nu ),\text{}$$
(5)
$${\tilde{P}}_{3}(u,v)=K\left\{\right[\tilde{f}(Mu+\frac{{x}_{m}}{\lambda d},\text{\hspace{0.17em}}M\nu +\frac{{y}_{n}}{\lambda d})\left]\mathrm{circ}\left(\frac{\rho}{\mathrm{\Delta \nu}}\right)\right\}\otimes {\mathrm{FT}}^{-1}\left\{\mathrm{exp}\right[j\text{\hspace{0.17em}}\frac{k}{{d}^{\prime}}\text{\hspace{0.17em}}({xx}_{m}+{yy}_{n})\left]\right\}\otimes \delta (u+Q,\nu )\mathrm{.}$$
(6)
$${\tilde{P}}_{3}^{\mathrm{sum}}(u,v)=K\left\{{\sum}_{m,n}\text{\hspace{0.17em}}\right[\tilde{f}(Mu+\frac{{x}_{m}}{\lambda d},\text{\hspace{0.17em}}M\nu +\frac{{y}_{n}}{\lambda d})\mathrm{circ}\left(\frac{\rho}{\mathrm{\Delta \nu}}\right)]\otimes \delta (u+\frac{{x}_{m}}{\lambda {d}^{\prime}},\text{\hspace{0.17em}}\nu +\frac{{y}_{n}}{\lambda {d}^{\prime}})\}\otimes \delta (u+Q,\nu )\mathrm{.}$$