## Abstract

The characteristics of extended-range focal-line images generated in partially coherent light by annular-aperture logarithmic axicons are analyzed in terms of a certain generalized radiometric model. This radiometric description, which makes use of radiance propagation along geometric rays, is valid in the asymptotic short-wavelength limit, and it yields both the spectral density and the spatial coherence distributions of the images. The radiometrically obtained values on the image axis and in transverse planes are assessed against direct partially coherent wave-theoretical calculations. It is shown that the radiometric technique produces remarkably accurate results for the intensity and the spatial coherence profiles. Compared with diffraction calculations, the radiometric method is much faster, especially in conditions of relatively incoherent illumination.

© 1996 Optical Society of America

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### Equations (20)

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(1)
$$h(\mathbf{\rho})=t(\mathbf{\rho})\text{exp}\left[ik\mathrm{\phi}(\mathbf{\rho})\right],$$
(2)
$$\mathrm{\phi}(\mathbf{\rho})=-\frac{1}{2a}\text{log}\left[{d}_{1}+a({\mathrm{\rho}}^{2}-{{R}_{1}}^{2})\right],$$
(3)
$$t(\mathbf{\rho})=\{0.5+\hspace{0.17em}\mathrm{}\text{arctan}\left[\mathrm{\Delta}({R}_{2}-\mathrm{\rho})\right]/\mathrm{\pi}\}\times \{0.5+\text{arctan}[\mathrm{\Delta}(\mathrm{\rho}-{R}_{1})]/\mathrm{\pi}\},$$
(4)
$${W}_{0}({\mathbf{\rho}}_{1},{\mathbf{\rho}}_{2})={S}_{0}\hspace{0.17em}\text{exp}[-{({\mathbf{\rho}}_{1}-{\mathbf{\rho}}_{2})}^{2}/2{{\mathrm{\sigma}}_{g}}^{2}].$$
(5)
$$\begin{array}{l}W({\mathbf{r}}_{1},{\mathbf{r}}_{2})={(k/2\mathrm{\pi})}^{2}\text{exp}[-ik({z}_{1}-{z}_{2})]/{z}_{1}{z}_{2}\\ \times {\iint}_{\mathcal{A}}t({{\mathbf{\rho}}_{1}}^{\prime})t({{\mathbf{\rho}}_{2}}^{\prime}){W}_{0}({{\mathbf{\rho}}_{1}}^{\prime},{{\mathbf{\rho}}_{2}}^{\prime})\\ \times \hspace{0.17em}\text{exp}\{-ik[\mathrm{\phi}({{\mathbf{\rho}}_{1}}^{\prime})-\mathrm{\phi}({{\mathrm{\rho}}_{2}}^{\prime})]\}\text{exp}\{-ik[{({\mathbf{\rho}}_{1}-{{\mathbf{\rho}}_{1}}^{\prime})}^{2}\\ /2{z}_{1}-{({\mathbf{\rho}}_{2}-{{\mathbf{\rho}}_{2}}^{\prime})}^{2}/2{z}_{2}]\}{\text{d}}^{2}{{\mathrm{\rho}}_{1}}^{\prime}{\text{d}}^{2}{{\mathrm{\rho}}_{2}}^{\prime},\end{array}$$
(6)
$$\mathrm{\mu}({\mathbf{r}}_{1},{\mathbf{r}}_{2})=W({\mathbf{r}}_{1},{\mathbf{r}}_{2})/{[S({\mathbf{r}}_{1})S({\mathbf{r}}_{2})]}^{1/2},$$
(7)
$$W({\mathbf{r}}_{1},{\mathbf{r}}_{2})=\int \mathcal{B}\left(\frac{{\mathbf{r}}_{1}+{\mathbf{r}}_{2}}{2},\mathbf{s}\right)\text{exp}[ik\mathbf{s}\xb7({\mathbf{r}}_{2}-{\mathbf{r}}_{1})]\text{d}\mathrm{\Omega},$$
(8)
$$\mathcal{B}\left(\frac{{\mathbf{r}}_{1}+{\mathbf{r}}_{2}}{2},\mathbf{s}\right)={B}^{(0)}[\mathbf{\rho}-({\mathbf{s}}_{\perp}z)/{s}_{z},\mathbf{s}],$$
(9)
$$\begin{array}{l}{s}_{x}(x,y)={s}_{0x}(x,y)-\frac{\partial \mathrm{\phi}(x,y)}{\partial x},\\ {s}_{y}(x,y)={s}_{0y}(x,y)-\frac{\partial \mathrm{\phi}(x,y)}{\partial y},\end{array}$$
(10)
$${B}_{0}(\mathbf{\rho},{\mathbf{s}}_{0})={S}_{0}\frac{{(k{\mathrm{\sigma}}_{g})}^{2}}{2\mathrm{\pi}}{s}_{0z}\hspace{0.17em}\text{exp}\left[-\frac{1}{2}{(k{\mathrm{\sigma}}_{g})}^{2}{{\mathbf{s}}_{0\perp}}^{2}\right],$$
(11)
$$\text{d}\mathrm{\Omega}=\frac{\mathrm{\rho}\text{d}\mathrm{\rho}\text{d}\mathrm{\varphi}\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\theta}}{({\mathrm{\rho}}^{2}+{z}^{2})},$$
(12)
$$W({z}_{1},{z}_{2})=2\mathrm{\pi}{\int}_{\mathcal{R}}{B}^{(0)}(\mathrm{\rho},\mathrm{\theta})\times \text{exp}[ik\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\theta}({z}_{2}-{z}_{1})]\frac{\mathrm{\rho}\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\theta}}{({\mathrm{\rho}}^{2}+{z}^{2})}\text{d}\mathrm{\rho},$$
(13)
$$\text{sin}\hspace{0.17em}{\mathrm{\theta}}_{0}=\hspace{0.17em}\text{sin}\hspace{0.17em}\mathrm{\theta}-\frac{\mathrm{\rho}}{{d}_{1}+a({\mathrm{\rho}}^{2}-{{R}_{1}}^{2})},$$
(14)
$${B}^{(0)}(\mathrm{\rho},\mathrm{\theta})={\mid t(\mathrm{\rho})\mid}^{2}{S}_{0}\frac{{(k{\mathrm{\sigma}}_{g})}^{2}}{2\mathrm{\pi}}\text{cos}\hspace{0.17em}{\mathrm{\theta}}_{0}\times \text{exp}\left[-\frac{1}{2}{(k{\mathrm{\sigma}}_{g})}^{2}\hspace{0.17em}{\text{sin}}^{2}\hspace{0.17em}{\mathrm{\theta}}_{0}\right],$$
(15)
$$\begin{array}{l}W({z}_{1},{z}_{2})={S}_{0}{k}^{2}\text{exp}[-ik({z}_{1}-{z}_{2})]/{z}_{1}{z}_{2}\\ \times \int {\int}_{\mathcal{R}}t({\mathrm{\rho}}_{1})t({\mathrm{\rho}}_{2})\text{exp}[-({{\mathrm{\rho}}_{1}}^{2}+{{\mathrm{\rho}}_{2}}^{2})/2{{\mathrm{\sigma}}_{g}}^{2}]\\ \times \hspace{0.17em}{I}_{0}({\mathrm{\rho}}_{1}{\mathrm{\rho}}_{2}/{{\mathrm{\sigma}}_{g}}^{2})\text{exp}\{-ik[\mathrm{\phi}({\mathrm{\rho}}_{1})-\mathrm{\phi}({\mathrm{\rho}}_{2})]\}\\ \times \hspace{0.17em}\text{exp}[-ik({{\mathrm{\rho}}_{1}}^{2}/2{z}_{1}-{{\mathrm{\rho}}_{2}}^{2}/2{z}_{2})]\\ \times \hspace{0.17em}{\mathrm{\rho}}_{1}{\mathrm{\rho}}_{2}{\text{d}\mathrm{\rho}}_{1}{\text{d}\mathrm{\rho}}_{2},\end{array}$$
(16)
$$\text{d}\mathrm{\Omega}=\frac{\mathrm{\rho}\text{d}\mathrm{\rho}\text{d}\mathrm{\varphi}\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\theta}}{({\mathrm{\rho}}^{2}+{z}^{2}+{l}^{2}-2l\mathrm{\rho}\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\varphi})},$$
(17)
$$W({l}_{1},{l}_{2},z)={\int}_{\mathcal{A}}{B}^{(0)}(\mathrm{\rho},\mathrm{\varphi},{s}_{x},{s}_{y})\text{exp}[ik{s}_{x}({l}_{2}-{l}_{1})]\times \frac{\mathrm{\rho}{s}_{z}}{({\mathrm{\rho}}^{2}+{z}^{2}+{l}^{2}-2l\mathrm{\rho}\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\varphi})}\text{d}\mathrm{\rho}\text{d}\mathrm{\varphi},$$
(18)
$$\begin{array}{l}{s}_{0x}={s}_{x}-\hspace{0.17em}\text{cos}\hspace{0.17em}\mathrm{\varphi}\frac{\mathrm{\rho}}{{d}_{1}+a({\mathrm{\rho}}^{2}-{{R}_{1}}^{2})},\\ {s}_{0y}={s}_{y}-\hspace{0.17em}\text{sin}\hspace{0.17em}\mathrm{\varphi}\frac{\mathrm{\rho}}{{d}_{1}+a({\mathrm{\rho}}^{2}-{{R}_{1}}^{2})}.\end{array}$$
(19)
$$\begin{array}{l}W({l}_{1},{l}_{2},z)={S}_{0}{(k/2\mathrm{\pi}z)}^{2}\hspace{0.17em}\text{exp}[-ik({{l}_{1}}^{2}-{{l}_{2}}^{2})/2z]\\ \times \int {\int}_{\mathcal{R}}t({\mathrm{\rho}}_{1})t({\mathrm{\rho}}_{2})\text{exp}[-{{\mathrm{\rho}}_{1}}^{2}+{{\mathrm{\rho}}_{2}}^{2})/2{{\mathrm{\sigma}}_{g}}^{2}]\\ \times \hspace{0.17em}C({\mathrm{\rho}}_{1},{\mathrm{\rho}}_{2};{l}_{1},{l}_{2},z)\text{exp}\{-ik[\mathrm{\phi}({\mathrm{\rho}}_{1})-\mathrm{\phi}({\mathrm{\rho}}_{2})]\}\\ \times \hspace{0.17em}\text{exp}[-ik({{\mathrm{\rho}}_{1}}^{2}-{{\mathrm{\rho}}_{2}}^{2})/2z)]{\mathrm{\rho}}_{1}{\mathrm{\rho}}_{2}{\text{d}\mathrm{\rho}}_{1}{\text{d}\mathrm{\rho}}_{2},\end{array}$$
(20)
$$C({\mathrm{\rho}}_{1},{\mathrm{\rho}}_{2};{l}_{1},{l}_{2},z)=\int {\int}_{0}^{2\mathrm{\pi}}\text{exp}[{\mathrm{\rho}}_{1}{\mathrm{\rho}}_{2}\hspace{0.17em}\text{cos}({\mathrm{\varphi}}_{1}-{\mathrm{\varphi}}_{2})/{{\mathrm{\sigma}}_{g}}^{2}]\times \hspace{0.17em}\text{exp}[ik({l}_{1}{\mathrm{\rho}}_{1}\hspace{0.17em}\text{cos}\hspace{0.17em}{\mathrm{\varphi}}_{1}-{l}_{2}{\mathrm{\rho}}_{2}\hspace{0.17em}\text{cos}\hspace{0.17em}{\mathrm{\varphi}}_{2})/z]\times \hspace{0.17em}\text{d}{\mathrm{\varphi}}_{1}\text{d}{\mathrm{\varphi}}_{2}.$$