## Abstract

A simple and rigorous analytical expression of the propagating field behind an axicon illuminated by an azimuthally polarized beam has been deduced by use of the vector interference theory.
This analytical expression can easily be used to calculate accurately the propagation field distribution of azimuthally polarized beams throughout the whole space behind an axicon with any size base angle,
not just restricted inside the geometric focal region as does the Fresnel diffraction integral. The numerical results show that the pattern of the beam produced by the azimuthally polarized Gaussian beam that passes through an axicon is a multiring, almost-equal-intensity, and propagation-invariant interference beam in the geometric focal region. The number of bright rings increases with the propagation distance, reaching its maximum at half of the geometric focal length and then decreasing. The intensity of bright rings gradually decreases with the propagation distance in the geometric focal region. However, in the far-field
(noninterference) region, only one single-ring pattern is produced and the dark spot size expands rapidly with propagation distance.

© 2007 Optical Society of America

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### Equations (12)

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(1)
$${p}_{0}=\left[\begin{array}{c}a\left(\phi \right)\\ b\left(\phi \right)\\ 0\end{array}\right].$$
(2)
$$E\prime ={E}_{1}\prime +{E}_{2}\prime \mathrm{.}$$
(3)
$${I}_{\phi}\left(\rho ,\phi ,z\right)={{t}_{s}}^{2}\left[{{E}_{1}}^{2}+{{E}_{2}}^{2}-2{E}_{1}{E}_{2}\text{\hspace{0.17em}}\mathrm{cos}\left(k\Delta \right)\right],$$
(4)
$${E}_{1}=\mathrm{exp}(-{{w}_{1}}^{2}/{{w}_{0}}^{2}),$$
(5)
$${E}_{2}=\mathrm{exp}\left(-{{w}_{2}}^{2}/{{w}_{0}}^{2}\right),$$
(6)
$$\Delta =n\left({d}_{1}-{d}_{2}\right)+\left({r}_{1}-{r}_{2}\right),$$
(7)
$$\text{}{w}_{1}=\frac{z\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\sigma +\rho}{1-\mathrm{tan}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\sigma},$$
(8)
$${w}_{2}=\frac{z\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\sigma -\rho}{1-\mathrm{tan}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\sigma},$$
(9)
$${r}_{1}=\frac{1}{\mathrm{cos}\text{\hspace{0.17em}}\sigma}\text{\hspace{0.17em}}\frac{z+\rho \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\alpha}{1-\mathrm{tan}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\sigma},$$
(10)
$${r}_{2}=\frac{1}{\mathrm{cos}\text{\hspace{0.17em}}\sigma}\text{\hspace{0.17em}}\frac{z-\rho \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\alpha}{1-\mathrm{tan}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\sigma},$$
(11)
$${d}_{1}=\frac{d\left(1-\mathrm{tan}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\sigma \right)-\mathrm{tan}\text{\hspace{0.17em}}\alpha \left(z+\rho \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\alpha \right)}{1-\mathrm{tan}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\sigma},$$
(12)
$${d}_{2}=\frac{d\left(1-\mathrm{tan}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\sigma \right)+\mathrm{tan}\text{\hspace{0.17em}}\alpha \left(z-\rho \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\alpha \right)}{1-\mathrm{tan}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\sigma}.$$