## Abstract

For applications in the domain of digital holographic microscopy, we present a fast algorithm to propagate scalar wave fields from a small source area to an extended, parallel target area of coarser sampling pitch, using the first Rayleigh-Sommerfeld diffraction formula. Our algorithm can take full advantage of the fast Fourier transform by decomposing the convolution kernel of the propagation into several convolution kernel patches. Using partial overlapping of the patches together with a soft blending function, the Fourier spectrum of these patches can be reduced to a low number of significant components, which can be stored in a compact sparse array structure. This allows for rapid evaluation of the partial convolution results by skipping over negligible components through the Fourier domain pointwise multiplication and direct mapping of the remaining multiplication results into a Fourier domain representation of the coarsly sampled target patch. The algorithm has been verified experimentally at a numerical aperture of 0.62, not showing any significant resolution limitations.

© 2010 OSA

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### Equations (17)

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(1)
$$u({x}^{\prime},{y}^{\prime})=\iint \text{d}x\text{d}yh({x}^{\prime}-x,{y}^{\prime}-y,z)o(x,y),$$
(2)
$${o}_{m,n}=o({x}_{0}+\Delta x\cdot m,{y}_{0}+\Delta y\cdot n)$$
(3)
$${u}_{p,q}=u({x}_{0}^{\prime}+\Delta {x}^{\prime}\cdot p,{{y}^{\prime}}_{0}+\Delta {y}^{\prime}\cdot q),$$
(4)
$${u}_{p,q}=\sum _{m}\sum _{n}h(p\Delta {x}^{\prime}-m\Delta x,q\Delta {y}^{\prime}-n\Delta y,z)\cdot {o}_{m,n}.$$
(5)
$${u}_{p,q}=\sum _{m}\sum _{n}{h}_{p-m,q-n}\cdot {o}_{m,n}\equiv {\{\mathbf{h}*\mathbf{o}\}}_{p,q},$$
(6)
$${h}_{p,q}=\frac{\Delta x\cdot \Delta y}{2\pi}\cdot \frac{z}{{r}_{p,q}^{2}}\cdot \left(\frac{1}{{r}_{p,q}}-ik\right)\cdot \text{exp}\left(-{ikr}_{p,q}\cdot \frac{z}{|z|}\right),$$
(7)
$${r}_{p,q}={\left[{(\Delta x\cdot p)}^{2}+{(\Delta y\cdot q)}^{2}+{z}^{2}\right]}^{1/2},$$
(8)
$$\Delta x\equiv \Delta y,$$
(9)
$$\mathbf{u}={\mathcal{F}}^{-1}[\mathcal{F}(\mathbf{h})\cdot \mathcal{F}(\mathbf{o})\equiv {\mathcal{F}}^{-1}[\mathbf{H}\cdot \mathbf{O}].$$
(10)
$${\left[\mathcal{F}(u)\right]}_{s,t}\equiv {U}_{s,t}=\sum _{p=0}^{N-1}\sum _{q=0}^{N-1}{u}_{p,q}\cdot \text{exp}\left(-2\pi i\cdot \frac{sp+tq}{N}\right).$$
(11)
$${h}_{p,q}=\sum _{\mu}\sum _{\nu}{h}_{p-S\cdot \mu ,q-S\cdot \nu}^{\mu \nu},\hspace{0.17em}(0\le \{p-S\cdot \mu ,q-S\cdot \nu \}<N),$$
(12)
$${u}_{p,q}=\sum _{\mu}\sum _{\nu}\left(\sum _{m=0}^{N-1}\sum _{n=0}^{N-1}{h}_{p-S\cdot \mu -m,q-S\cdot \nu -n}^{\mu \nu}\cdot {o}_{m,n}\right)=\sum _{\mu}\sum _{\nu}{\{{\mathbf{h}}^{\mu \nu}*\mathbf{o}\}}_{p-S\cdot \mu ,q-S\cdot \nu}.$$
(13)
$$b(d)={\mathit{sin}}^{2}\left(\frac{\pi}{2}\frac{d}{w}\right),$$
(14)
$${u}_{p}=\frac{1}{2N}\sum _{t=0}^{2N-1}{U}_{t}\cdot \mathit{exp}\left(2\pi i\cdot \frac{tp}{2N}\right).$$
(15)
$$\begin{array}{lll}{\widehat{u}}_{j}\hfill & \equiv \hfill & {u}_{Tj+s}\hfill \\ \hfill & =\hfill & \frac{1}{2N}\sum _{t=0}^{2N-1}{U}_{t}\cdot \mathit{exp}\left(2\pi i\cdot \frac{t\cdot (Tj+s)}{2N}\right)\hfill \\ \hfill & =\hfill & \frac{1}{2{N}^{\prime}}\sum _{l=0}^{2{N}^{\prime}-1}{\widehat{U}}_{l}\cdot exp\left(2\pi i\cdot \frac{jl}{2{N}^{\prime}}\right),\hfill \end{array}$$
(16)
$${\widehat{U}}_{l}\equiv \frac{1}{T}\sum _{m=0}^{T-1}{U}_{l+2{N}^{\prime}m}\cdot \mathit{exp}\left(2\pi i\cdot \frac{s\cdot (l+2{N}^{\prime}m)}{2N}\right).$$
(17)
$${\widehat{u}}_{j}={[{\mathcal{F}}^{-1}\widehat{\mathbf{U}}]}_{j}$$