Axially symmetric polarized beam is a kind of vector beams with vortex polarization profiles, and the beam propagation axis is its symmetric axis [23]. For ASPBs, the state of polarization (SoP) at any position (except the optical axis) on the beam cross-section is linearly polarized, and the polarization orientation angle \(\Phi \left( {r,\phi } \right)\) of the electric field only depends on the azimuthal angle as \(\Phi \left( {r,\phi } \right)=P \times \phi +{\phi _0}\), where *P* is the polarization order number, \(\phi\) is the azimuthal angle of the cylindrical coordinate system, and \({\phi _0}\) is the initial polarization orientation for \(\phi =0\). The phase distribution is uniform on the beam cross section (expect the optical axis). Because the intensity at optical axis is zero, the phase and polarization are uncertain. Its vector amplitude of the electric field can be expressed as,

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} \left( {r,\phi ,z} \right)={E_0}\left( {r,z} \right)\left[ {\cos \left( {P\phi +{\phi _0}} \right){{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {e} }_x}+\sin \left( {P\phi +{\phi _0}} \right){{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {e} }_y}} \right]$$

1

where *r* and *φ* are the cylindrical coordinates, *E*0 is the module of the vector amplitude, \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {e} _x}\) and \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {e} _y}\) are the unit vectors in the x axis and y axis.

The focusing properties of high-order axially symmetric polarized beams have been studied [7–9]. As shown in Fig. 1, in a cylindrical coordinated system, the focused fields can be expressed as,

$$\begin{gathered} \vec {E}\left( {{r_S},{\phi _S},{z_S}} \right)=\left[ {\begin{array}{*{20}{c}} {E_{r}^{{\left( S \right)}}} \\ {E_{\phi }^{{\left( S \right)}}} \\ {E_{z}^{{\left( S \right)}}} \end{array}} \right] \hfill \\ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{} \end{array}}&{}&{} \end{array}= - {i^{\left( {3P+1} \right)}}A\int\limits_{0}^{\alpha } {{l_0}\left( \theta \right)\sqrt {\cos \theta } \sin \theta \exp \left( {ik{z_S}\cos \theta } \right)} \hfill \\ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{} \end{array}}&{}&{\begin{array}{*{20}{c}} {}&{ \times \left[ {\begin{array}{*{20}{c}} {\cos \left[ {\left( {P - 1} \right){\phi _S}+{\phi _0}} \right]\left\{ {\cos \theta \left[ {{J_P}\left( {k{r_S}\sin \theta } \right) - {J_{P - 2}}\left( {k{r_S}\sin \theta } \right)} \right]+{J_P}\left( {k{r_S}\sin \theta } \right)+{J_{P - 2}}\left( {k{r_S}\sin \theta } \right)} \right\}} \\ {\sin \left[ {\left( {P - 1} \right){\phi _S}+{\phi _0}} \right]\left\{ {\cos \theta \left[ {{J_P}\left( {k{r_S}\sin \theta } \right)+{J_{P - 2}}\left( {k{r_S}\sin \theta } \right)} \right]+{J_P}\left( {k{r_S}\sin \theta } \right) - {J_{P - 2}}\left( {k{r_S}\sin \theta } \right)} \right\}} \\ {2i\cos \left[ {\left( {P - 1} \right){\phi _S}+{\phi _0}} \right]\sin \theta {J_{P - 1}}\left( {k{r_S}\sin \theta } \right)} \end{array}} \right]d\theta } \end{array}} \end{array} \hfill \\ \end{gathered}$$

2

where \(S\left( {{r_S},{\phi _S},{z_S}} \right)\) is an observation point near focus, \(E_{r}^{{\left( S \right)}}\), \(E_{\phi }^{{\left( S \right)}}\) and \(E_{z}^{{\left( S \right)}}\) are the amplitudes of the radial component, the azimuthal component and the longitudinal component at point *S*. *P* is the polarization order of the incident ASPBs, and \({J_P}\left( \bullet \right)\) is the Bessel function of the first kind of the order . is a constant related with the amplitude of incident beam. *k* is the wavelength number, and *θ* denotes the focusing angle (the angle between the optical axis and the propagation vector), so the relationship between the maximum of *θ* and NA of the objective lens is given by \(\alpha ={\sin ^{ - 1}}\left( {{{NA} \mathord{\left/ {\vphantom {{NA} n}} \right. \kern-0pt} n}} \right)\), where *n* is the refractive index of the surrounding medium. And \({l_0}\left( \theta \right)\) is the pupil apodization function which denotes the relative amplitude and phase of the incident beam.

Based on the equation, we can calculate the phase, amplitude and polarization distributions of focused fields. The intensity distributions are symmetric about the focal plane. As presented in previous studies [17–19], the focused fields of ASPBs show multi-focal spots patterns, the number of spots is 2×(P−1) when the polarization order number is P, and the intensities in the optical axis are always zero. Specifically, the intensity distributions of the radial, azimuthal and longitudinal components are periodical circularly, and there also exist multiple zero intensity planes with the number of 2×(P−1) along the azimuthal directions. Therefore, the phase singularities occur in the optical axis and on the zero-intensity planes.

Meanwhile, we can find the following relationship along the z axis,

\({E_r}\left( {{r_S},{\phi _S}, - {z_S}} \right)={\left( { - 1} \right)^{P+1}}E_{r}^{ * }\left( {{r_S},{\phi _S},{z_S}} \right)\) (3 − 1)

\({E_\phi }\left( {{r_S},{\phi _S}, - {z_S}} \right)={\left( { - 1} \right)^{P+1}}E_{\phi }^{ * }\left( {{r_S},{\phi _S},{z_S}} \right)\) (3 − 2)

\({E_z}\left( {{r_S},{\phi _S}, - {z_S}} \right)={\left( { - 1} \right)^P}E_{z}^{ * }\left( {{r_S},{\phi _S},{z_S}} \right)\) (3–3)

Obviously, if *P* is an even number, for radial and azimuthal field components, the focused fields about the focal plane are conjugate, so the signs of phase angles are opposite. For longitudinal field component, the sum of phase angles for the two symmetric points about the focal plane is π or −π. If *P* is an odd number, the situation is just opposite. Especially, on the focal plane (\({z_S}=0\)), the amplitudes of focused fields are pure real or imaginary numbers, which means the phases are binary.

Further, some phase singularity properties on the focused fields can be analyzed according to the Eq. 3. Phase singularities exist at the optical axis (\({r_S}=0\)) because the intensity is zero. And phase singularities also exist on the planes

$${\phi _S}=\left\{ \begin{gathered} \frac{{\left( {2m+1} \right)\pi - 2{\phi _0}}}{{2\left( {P - 1} \right)}}\begin{array}{*{20}{c}} {}&{\left( {\begin{array}{*{20}{c}} {radial}&{or}&{longitudinal}&{component} \end{array}} \right)} \end{array} \hfill \\ \frac{{\left( {m+1} \right)\pi - {\phi _0}}}{{\left( {P - 1} \right)}}\begin{array}{*{20}{c}} {}&{\left( {\begin{array}{*{20}{c}} {azimuthal}&{component} \end{array}} \right)} \end{array} \hfill \\ \end{gathered} \right.$$

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because the intensities are zero on these planes, where *m* is an integer, and the number of planes is 2⋅(*P*−1).Of course, the pupil apodization function and NA of objective lens also affect the distributions of phase singularities.