## Abstract

Focal modulation microscopy (FMM) is a simple, yet efficient, method to preserve image quality in terms of signal-to-background ratio by selecting ballistic photons for image formation. The aim of this paper is to investigate the effect of the various aperture configurations of the spatial phase modulator on the modulation depth of the FMM signal. The definition of modulation depth in FMM and its calculation method are introduced. According to two brief principles of choosing aperture configuration, three types of configurations with different numbers of zones ranging from two to six (totaling eight aperture configurations) are selected, and their corresponding modulation depths and attainable spatial resolutions are simulated. The results show that the modulation depth increases significantly when the number of zones varies from two to six, with a slight or no sacrifice in resolution. In summary, the annular configuration is superior to the fan- and stripe-shaped configurations in modulation depth and spatial resolution.

© 2011 Optical Society of America

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### Equations (8)

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(1)
$$M=\frac{({I}_{\mathrm{max}}-{I}_{\mathrm{min}})/2}{({I}_{\mathrm{max}}+{I}_{\mathrm{min}})/2}=\frac{{I}_{\mathrm{max}}-{I}_{\mathrm{min}}}{{I}_{\mathrm{max}}+{I}_{\mathrm{min}}},$$
(2)
$$\mathrm{E}(x,y,z,t)=|\int {\int}_{A}P({x}^{\prime},{y}^{\prime},t)\mathrm{exp}\{\frac{ik}{f}({x}^{2}+{y}^{2})-\frac{ik}{2z}[(x-{x}^{\prime}{)}^{2}+(y-{y}^{\prime}{)}^{2}]\}\mathrm{d}{x}^{\prime}\mathrm{d}{y}^{\prime}{|}^{2},$$
(3)
$$I(t)=\int \int \int E(x,y,z,t)\times [|{h}_{D}(x,y,z){|}^{2}{\otimes}_{2}D(x,y)]\mathrm{d}x\mathrm{d}y\mathrm{d}z,$$
(4)
$${E}_{\mathrm{CM}}(x,y,z)=|\int {\int}_{A}\mathrm{exp}\{\frac{ik}{f}({x}^{2}+{y}^{2})-\frac{ik}{2z}[(x-{x}^{\prime}{)}^{2}+(y-{y}^{\prime}{)}^{2}]\}\mathrm{d}{x}^{\prime}\mathrm{d}{y}^{\prime}{|}^{2}\mathrm{.}$$
(5)
$${I}_{\mathrm{max}}=\int \int \int {\mathrm{E}}_{\mathrm{CM}}(x,y,z)\times [|{h}_{D}(x,y,z){|}^{2}\phantom{\rule{0ex}{0ex}}{\otimes}_{2}D(x,y)]\mathrm{d}x\mathrm{d}y\mathrm{d}z.$$
(6)
$${\mathrm{E}}_{\mathrm{FMM}}(x,y,z)={\mathrm{E}}_{\mathrm{CM}}(x,y,z)-|\int {\int}_{A}P({x}^{\prime},{y}^{\prime},t){|}_{\phi (t)=\pi}\phantom{\rule{0ex}{0ex}}\mathrm{exp}\{\frac{ik}{f}({x}^{2}+{y}^{2})-\frac{ik}{2z}[(x-{x}^{\prime}{)}^{2}\phantom{\rule{0ex}{0ex}}+(y-{y}^{\prime}{)}^{2}]\}\mathrm{d}{x}^{\prime}\mathrm{d}{y}^{\prime}{|}^{2},$$
(7)
$${I}_{\mathrm{max}}-{I}_{\mathrm{min}}=\int \int \int {E}_{\mathrm{FMM}}(x,y,z)\times [|{h}_{D}(x,y,z){|}^{2}\phantom{\rule{0ex}{0ex}}{\otimes}_{2}D(x,y)]\mathrm{d}x\mathrm{d}y\mathrm{d}z\mathrm{.}$$
(8)
$${I}_{\text{Total}}={I}_{\text{non}}+{I}_{\text{mod}}+2\sqrt{{I}_{\text{non}}{I}_{\text{mod}}}\mathrm{cos}(\mathrm{\Delta}{\varphi}_{\text{mod}}),$$