Electric field for HCVB with spiral phase distributions in cylindrical coordinates is given by [43]
$${E_{in}}=P(\theta )\exp (im\phi )\left\{ {\cos [(a - 1)\phi +{\varphi _0}]{e_r}+\sin [(a - 1)\phi +{\varphi _0}]{e_\phi }} \right\},$$
1
whereP(θ) is the pupil apodization function, m is the topological charge of the vortex, a is the polarization order or azimuthal index,erand eφ denotes the radial and azimuthal unit vectors.
The schematic diagram of the proposed system is shown in Fig. 1.Here HCVB is phase modulated by an annular Walsh filter and to tightly focused by high NA objective .
Under the tight focusing condition ,the electric field distribution in the cylindrical coordinate is expressed as[15, 43]
\(\left[ \begin{gathered} {E_r}(r,\psi ,z) \hfill \\ {E_\phi }(r,\psi ,z) \hfill \\ {E_z}(r,\psi ,z) \hfill \\ \end{gathered} \right]=\frac{{ - iA}}{2}\int\limits_{0}^{\alpha } {\int\limits_{0}^{{2\pi }} {\sin \theta \left( {\sqrt {\cos \theta } } \right)} } P(\theta )exp(im\phi ) \times exp\left[ {ik\left( {z\cos \theta +r\sin \theta \cos (\phi - \psi } \right)} \right]\)
$$\times \left[ \begin{gathered} - \sin [(a - 1)\phi +{\varphi _0}]\sin (\psi - \phi )+\cos [(a - 1)\phi +{\varphi _0}]\cos \theta \cos (\phi - \psi ) \hfill \\ \sin [(a - 1)\phi +{\varphi _0}]\cos (\psi - \phi )+\cos [(a - 1)\phi +{\varphi _0}]\cos \theta \sin (\phi - \psi ) \hfill \\ \cos [(a - 1)\phi +{\varphi _0}]\sin \theta \hfill \\ \end{gathered} \right]d\phi d\theta$$
2
whereA is a constant; ψ and θ denotes the azimuthal angle with respect to x axis and tangential angle with respect to the z axis, respectively. k is the wave number, α is the maximum converging semi angle given by α = arcsin(NA/n).NA is the numerical aperture of the lens and n is the refractive index of the surrounding medium. Here we consider Bessel Gaussian beam as an input beam, where P(θ) can be expressed as[43]
$$P(\theta )=\exp \left[ { - {\beta ^2}{{\left( {\frac{{\sin (\theta )}}{{\sin \alpha }}} \right)}^2}} \right]{J_1}\left( {2\beta \frac{{\sin (\theta )}}{{\sin \alpha }}} \right)$$
3
β is the ratio between pupil diameters to beam diameter and J1(x) is the first order Bessel function. Furthermore, the three components of electric field equations can be simplified as[15]
\({E_r}(r,\psi ,z)=\frac{{ - iA}}{2}\int\limits_{0}^{\alpha } {P(\theta )\sin \theta \sqrt {\cos \theta } } \exp (ikz\cos \theta )\)
$$\times \left\{ \begin{gathered} {i^{a+m}}\exp [i(a+m - 1)\psi +i{\varphi _0}]{J_{a+m}}(kr\sin \theta )(\cos \theta +1) \hfill \\ +{i^{m - a}}\exp [i(m - a+1)\psi - i{\varphi _0}]{J_{m - a}}(kr\sin \theta )(\cos \theta +1) \hfill \\ +{i^{m+a - 2}}\exp [i(m+a - 1)\psi +i{\varphi _0}]{J_{m+a - 2}}(kr\sin \theta )(\cos \theta - 1) \hfill \\ +{i^{m - a+2}}\exp [i(m - a+1)\psi - i{\varphi _0}]{J_{m - a+2}}(kr\sin \theta )(\cos \theta - 1) \hfill \\ \end{gathered} \right\}d(\theta )$$
4
\({E_\phi }(r,\psi ,z)=\frac{{ - A}}{2}\int\limits_{0}^{\alpha } {P(\theta )\sin \theta \sqrt {\cos \theta } } \exp (ikz\cos \theta )\)
$$\times \left\{ \begin{gathered} {i^{a+m}}\exp [i(a+m - 1)\psi +i{\varphi _0}]{J_{a+m}}(kr\sin \theta )(\cos \theta +1) \hfill \\ - {i^{m - a}}\exp [i(m - a+1)\psi - i{\varphi _0}]{J_{m - a}}(kr\sin \theta )(\cos \theta +1) \hfill \\ +{i^{m+a - 2}}\exp [i(m+a - 1)\psi +i{\varphi _0}]{J_{m+a - 2}}(kr\sin \theta )(1 - \cos \theta ) \hfill \\ - {i^{m - a+2}}\exp [i(m - a+1)\psi - i{\varphi _0}]{J_{m - a+2}}(kr\sin \theta )(1 - \cos \theta ) \hfill \\ \end{gathered} \right\}d(\theta )$$
5
\({E_z}(r,\psi ,z)= - A\int\limits_{0}^{\alpha } {P(\theta ){{\sin }^2}\theta \sqrt {\cos \theta } } \exp (ikz\cos \theta )\)
$$\times \left\{ \begin{gathered} {i^{m+a - 1}}\exp [i(m+a - 1)\psi +i{\varphi _0}]{J_{m+a - 1}}(kr\sin \theta ) \hfill \\ +{i^{m - a+1}}\exp [i(m - a+1)\psi - i{\varphi _0}]{J_{m - a+1}}(kr\sin \theta ) \hfill \\ \end{gathered} \right\}d(\theta )$$
6
Figure 2 shows that the HCVB(a > 1) generates petal shaped or flower like structures in the xy-plane perpendicular to the optic axis. The number of petals as well as size of the centre annular region depends on the value of polarization order a, the number of petals structures are equal to 2(a-1).It is noted that, spiral phase added on the input pupil turns the petal structure to a confined spot in the xy plane.
When the annular Walsh filter is placed at input pupil, the pupil function P(θ) is replaced by P(θ)T(θ) and is given by[34]
$$T(\theta )=\left\{ \begin{gathered} \begin{array}{*{20}{c}} {0,}&{0 \leqslant \theta <\varepsilon } \end{array} \hfill \\ \begin{array}{*{20}{c}} {\Psi _{n}^{\varepsilon }(\theta ),}&{\varepsilon \leqslant \theta <\alpha } \end{array} \hfill \\ \end{gathered} \right.$$
7
Here, \(\Psi _{n}^{\varepsilon }(\theta )\) is the Walsh function that depends on the Walsh order n and annular obstruction ε and is given by[34]
$$\Psi _{n}^{\varepsilon }(\theta )=\prod\limits_{{m=0}}^{{v - 1}} {\operatorname{sgn} \left\{ {\cos \left[ {{K_m}{2^m}\pi \frac{{\left( {{\theta ^2} - {\varepsilon ^2}} \right)}}{{\left( {1 - {\varepsilon ^2}} \right)}}} \right]} \right\}}$$
8
Km are the bits,0 or 1 of the binary numerical for n, and (2v)is the power of 2 that just exceeds n, for all θ in (ε,1)
The Walsh order n is expressed as
\(n=\sum\limits_{{m=0}}^{{\upsilon - 1}} {{K_m}{2^m}}\)
Where
\(\operatorname{sgn} (x)=\left\{ \begin{gathered} \begin{array}{*{20}{c}} {+1,}&{x>0} \end{array} \hfill \\ \begin{array}{*{20}{c}} {0,}&{x=0} \end{array} \hfill \\ \begin{array}{*{20}{c}} { - 1,}&{x<0} \end{array} \hfill \\ \end{gathered} \right.\)
The locations of the points of zero crossings for members of the set of functions\(\Psi _{n}^{\varepsilon }(\theta )\),n = 0,1,…,(N-1) are given by
$${\theta _i}=\sqrt {\frac{{[(N - i)]{\varepsilon ^2}+i}}{N}} \times \alpha$$
9
The inner and outer angle of the filter is θ0 = ε and θN = 1. The set of (N-1) phase or zero crossing locations, θi,i = 1,2,…(N-1) consist of all phase transiting locations required for specifying zones of this particular set of Walsh functions. It is noted that an individual zone of this set of Walsh functions will have the same number of phase transition as its order.
Annular Walsh function \(\Psi _{n}^{\varepsilon }(\theta )\)of n ≥ 0 & θ over an annular region with ε and 1 as inner and outer radii is depicted in Fig. 3.In Fig. 3, black color represents the central annular obstruction, yellow and orange colors represents the phase transition value of + π and –πradians, respectively.