## Abstract

An optical setup to achieve superresolution in microscopy using holographic recording is presented. The technique is based on off-axis illumination of the object and a simple optical image processing stage after the imaging system for the interferometric recording process. The superresolution effect can be obtained either in one step by combining a spatial multiplexing process and an incoherent addition of different holograms or it can be implemented sequentially. Each hologram holds the information of each different frequency bandpass of the object spectrum. We have optically implemented the approach for a low-numerical-aperture commercial microscope objective. The system is simple and robust because the holographic interferometric recording setup is done after the imaging lens.

© 2006 Optical Society of America

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### Equations (14)

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(1)
$$U(x,y)=\mathbf{\left(}f(-\frac{x}{M},-\frac{y}{M})\mathrm{exp}\left\{j\frac{k}{2d}[{x}^{2}+{y}^{2}]\right\}\mathbf{\right)}\otimes \mathrm{disk}\left(\mathrm{\Delta}\nu r\right),$$
(2)
$${U}_{m,n}^{I}(x,y)=\mathbf{\left(}f(-\frac{x}{M},-\frac{y}{M})\mathrm{exp}\left\{j\frac{k}{2d}[{(x-{M}_{s}{x}_{m})}^{2}+{(y-{M}_{s}{y}_{n})}^{2}]\right\}\mathbf{\right)}\otimes \mathrm{disk}\left(\mathrm{\Delta}\nu r\right),$$
(3)
$${d}^{\prime}=\frac{a(a-d)+{F}^{2}}{F-a}.$$
(4)
$${M}_{s}^{\prime}={M}_{s}\frac{F}{F-a}.$$
(5)
$${U}_{m,n}^{R}(x,y)=\mathrm{exp}\left\{j\frac{k}{2{d}^{\prime}}[{(x-{M}_{s}^{\prime}{x}_{m})}^{2}+{(y-{M}_{s}^{\prime}{y}_{n})}^{2}]\right\}.$$
(6)
$${I}_{m,n}(x,y)={\mid {U}_{m,n}^{I}(x,y)+{U}_{m,n}^{R}(x,y)\mathrm{exp}\left(j2\pi Qx\right)\mid}^{2}.$$
(7)
$${I}_{m,n}(x,y)=1+{\mid {U}_{m,n}^{I}(x,y)\mid}^{2}+{U}_{m,n}^{I}(x,y){\left[{U}_{m,n}^{R}(x,y)\right]}^{*}\mathrm{exp}(-j2\pi Qx)+{\left[{U}_{m,n}^{I}(x,y)\right]}^{*}{U}_{m,n}^{R}(x,y){e}^{j2\pi Qx}={T}_{1}(x,y)+{T}_{2}(x,y)+{T}_{3}(x,y)+{T}_{4}(x,y).$$
(8)
$${T}_{3}(x,y)=[\left(f(-\frac{x}{M},-\frac{y}{M})\mathrm{exp}\left\{j\frac{k}{2d}[{(x-{M}_{s}{x}_{m})}^{2}+{(y-{M}_{s}{y}_{n})}^{2}]\right\}\right)\otimes \mathrm{disk}\left(\mathrm{\Delta}\nu r\right)]\mathrm{exp}\{-j\frac{k}{2d}[{(x-{M}_{s}^{\prime}{x}_{m})}^{2}+{(y-{M}_{s}^{\prime}{y}_{n})}^{2}]\}\mathrm{exp}(-j2\pi Qx).$$
(9)
$${\stackrel{\u0303}{T}}_{3}(u,\nu )=K\left[(\stackrel{\u0303}{f}(Mu+\frac{M{M}_{s}}{\lambda d}{x}_{m},M\nu +\frac{M{M}_{s}}{\lambda d}{y}_{n})\otimes {\mathrm{FT}}^{-1}\left\{\mathrm{exp}\left[j\frac{k}{2d}({x}^{2}+{y}^{2})\right]\right\})\mathrm{circ}\left(\frac{\rho}{\mathrm{\Delta}\nu}\right)\right]\otimes {\mathrm{FT}}^{-1}\left\{\mathrm{exp}[-j\frac{k}{2{d}^{\prime}}({x}^{2}+{y}^{2})]\right\}\otimes \delta (u+Q-\frac{{M}_{s}^{\prime}}{\lambda {d}^{\prime}}{x}_{m},\nu -\frac{{M}_{s}^{\prime}}{\lambda {d}^{\prime}}{y}_{n}),$$
(10)
$${\mathrm{FT}}^{-1}\left\{\mathrm{exp}\left[j\frac{k}{2d}({x}^{2}+{y}^{2})\right]\right\}\otimes {\mathrm{FT}}^{-1}\left\{\mathrm{exp}[-j\frac{k}{2{d}^{\prime}}({x}^{2}+{y}^{2})]\right\}={\mathrm{FT}}^{-1}\left\{\mathrm{exp}\left[j\frac{k}{2d}\left(\frac{1}{d}\frac{1}{{d}^{\prime}}\right)({x}^{2}+{y}^{2})\right]\right\}.$$
(11)
$${\stackrel{\u0303}{T}}_{3}^{\prime}(u,\nu )=K\left\{\stackrel{\u0303}{f}(Mu+\frac{M{M}_{s}}{\lambda d}{x}_{m},M\nu +\frac{M{M}_{s}}{\lambda d}{y}_{n})\mathrm{circ}\left(\frac{\rho}{\mathrm{\Delta}\nu}\right)\right\}\otimes \delta (u+Q-\frac{{M}_{s}^{\prime}}{\lambda {d}^{\prime}}{x}_{m},\nu -\frac{{M}_{s}^{\prime}}{\lambda {d}^{\prime}}{y}_{n}).$$
(12)
$${\stackrel{\u0303}{{T}^{\prime}}}_{3}^{\mathrm{sum}}(u,\nu )=K\sum _{m,n}\{\left[\stackrel{\u0303}{f}(M(u+\frac{{M}_{s}}{\lambda d}{x}_{m}),M(\nu +\frac{{M}_{s}}{\lambda d}{y}_{n}))\mathrm{circ}\left(\frac{\rho}{\mathrm{\Delta}\nu}\right)\right]\otimes \delta (u-\frac{{M}_{s}^{\prime}}{\lambda {d}^{\prime}}{x}_{m},\nu -\frac{{M}_{s}^{\prime}}{\lambda {d}^{\prime}}{y}_{n})\}\otimes \delta (u+Q,\nu ).$$
(13)
$${\stackrel{\u0303}{{T}^{\u2033}}}_{3}^{\mathrm{sum}}(u,\nu )=K\stackrel{\u0303}{f}(Mu,M\nu )\mathrm{SA}(u,\nu )\otimes \delta (u+Q,\nu ),$$
(14)
$$\mathrm{SA}(u,\nu )=\sum _{m,n}\mathrm{circ}(\frac{u-\frac{{M}_{s}}{\lambda d}{x}_{m}}{\mathrm{\Delta}\nu},\frac{\nu -\frac{{M}_{s}}{\lambda d}{y}_{m}}{\mathrm{\Delta}\nu}).$$