Due to its potential applications in optical fields such as axial multiple optical trapping and manipulating, optical fluorescence microscopy, high-density optical storage, optical lithography, and others, the generation of light sheet focal patterns has been extensively investigated in recent years. [1–14]. Many ways have recently been developed by researchers to construct such light sheet focal patterns by modifying the amplitude, phase, or polarization of incident vector beams at the input pupil plane using various types of pupil plane filters. Special types of novel light sheet structures, such as axial multiple sub wavelength scale longitudinally as well as transversely polarized spots/holes, axially splitted flattop profiles, multiple optical tunnels, and so on, are created by various algorithms such as 4pi configurations, phase filters such as binary, complex filters, and so on, in conjunction with different types of objective lenses such as high NA lenses, axicons, cylindrical lenses etc. In addition to objective lenses, various types of zone plates are proposed to generate axial light sheet patterns. [15–35]. In comparison to scalar fields (linear, circular, elliptical polarized beams), vector fields have recently demonstrated remarkable properties that can be used to create novel vector fields such as ultra sharp focal spots with extended focal depth. [36–39]. A radially polarized vector beam, for example, can produce a non-propagating strong axial electric field at the focus. [40, 41]. Azimuthally polarized vector beams, on the other hand, can produce hollow dark spots at the focus. [36, 40, 41]. Simultaneously, higher order vector beams focused through a high NA lens produce flower-like patterns[42–46]. With appropriate pupil mask engineering, all of the previously mentioned vector fields can also generate light sheet focal patterns [15–17, 21–24, 29, 31–35]. However, only the radial and azimuthal variant vector beams generate all of the above vector fields due to their greater degree of freedom in manipulating the states of polarization (SoPs) of the input beam. Many researchers have recently proposed theoretical and experimental generation of radial and azimuthal variant vector beams[47–54]. Z. man et al., for example, briefly examined the tight focusing properties of radial and azimuthal variant vector beams in order to generate linear, radial, azimuthal, higher order, and spatially variant (combination of both radial & azimuthal component) vector beams[55–59]. Unfortunately, the mutual effect of radial and azimuthal variant vector beams with pupil filters in generating light sheet focal structures is rarely used. To the best of our knowledge, no research has been conducted on the combination of radial and azimuthal variant vector beams with an annular Walsh filter to generate novel light sheet patterns at the focus of a high NA lens. J.L. Walsh defined the Walsh function in 1923 as a closed set of normal orthogonal functions over a finite interval that has a value of either + 1 or -1[60]. The order of the Walsh function is defined as the number of zero crossings within that finite interval. L.N. Hazra et al. investigated the focusing properties of Walsh filters in 1976-77 by incorporating the Walsh function into a pupil filter[61–64]. He also improved microscopic imaging resolution using an annular Walsh function filter[65]. P. Mukherjee and colleagues investigate the far-field properties of an annular Walsh function filter[66]. At the focus, the Walsh filters also generate self-similar focal patterns[67–69]. F. Machado et al. improved imaging capability in 2018 by using the Walsh function filter[70]. By tightly focusing an azimuthally polarized beam through a high Na lens, our group numerically generates an ultra-long multiple optical tube-like light sheet structure[71]. We numerically investigate the tight focusing behaviour of radial and azimuthal variant vector beams through an annular Walsh filter in the input pupil using Richard and Wolf's vector diffraction theory. The novel light sheet structures are generated numerically by carefully manipulating parameters such as radial and azimuthal index values, as well as the order and size of the annular obstruction values.

## Theory

## 1.1 Radial and azimuthal variant vector beam:

The field distribution for radial and azimuthal variant vector beams is given by [48, 52]

\(E(r,\phi )={E_r}{\hat {e}_r}+{E_\phi }{\hat {e}_\phi }=A(r)\left[ {\cos \left( {\left( {\frac{{2b\pi r}}{{{r_0}}}+a\phi } \right) - \phi +c} \right){e_r}+\sin \left( {\left( {\frac{{2b\pi r}}{{{r_0}}}+a\phi } \right) - \phi +c} \right){e_\phi }} \right]\)

where *A(r)* represent the amplitude constant, \({\hat {e}_r}\) and \({\hat {e}_\phi }\) are the unit vectors in the polar coordinate *(r,ϕ)*,*a* is the topological charge or the azimuthal index value and it denotes the number of rotation of SoPs around the beam axis(for the azimuthal angle *ϕ* from *0 to 2π* and *c* denotes the initial phase. Here *r**0* denotes the beam radius.

Figure 1. shows the spatial distribution of SoPs for different radial &azimuthal index values with respect to *c = 0**0*,*45**0*,*90**0*, respectively. From Fig. 1.(a-c),when *a,b = 0, c = 0**0*,*45**0*,*90**0* correspond to the well-known linearly polarized beams, respectively. Figure 1(d-f) shows the pure azimuthally variant vector beam(*a = 1,b = 0*) for *c = 0**0*,*45**0*,*90**0*.For *a = 1,b = 0,c = 0**0* is well known radially polarized beam. At the same time, *a = 1,b = 0,c = 90**0* corresponding to the pure azimuthally polarized beam. From Fig. 1(d-f),we conclude that the spatial distribution of the SoPs are always depend *r* and independent *ϕ*. If *a > 1*,will generates higher –order azimuthal –variant vector fields as depicted in Fig.(g-i & j-l) .From Fig. 1(g-i &j-l),we conclude that the higher order azimuthal index(*a > 1*) will breaking the cylindrical symmetry of the SoPs. Figure 1.(m-0& p-r) shows the pure radial variant(*a = 0*) distribution of SoPs for *b = 0.5 ,1* corresponding to *c* as *0**0*,*45**0*,*90**0*, respectively. The spatial distribution of SoPs are only depends *r* and independent the value of *ϕ*. Figure 1.(s-u &v-x) shows the spatial distribution of the SoPs for *a,b = 1* and *a,b = 2* corresponding to *c = 0**0*,*45**0*,*90**0*,respectively. The SoPs depends both *r* and *ϕ* values. The cylindrical symmetry breaks down both radial as well as azimuthal directions.

## 1.2 Tight focusing through a annular Walsh function filter

Fig.2. shows the schematic representation for radial and azimuthal variant vector beam focusing through a high NA lens and annular Walsh filter. The analytical model for electric field distribution in the cylindrical coordinate system is given as[40]

$$E(r,\psi ,z)=\left[ \begin{gathered} {E_r}(r,\phi ,z) \hfill \\ {E_\psi }(r,\phi ,z) \hfill \\ {E_z}(r,\phi ,z) \hfill \\ \end{gathered} \right]=\frac{{ - i{E_0}}}{2}\int\limits_{0}^{\alpha } {\int\limits_{0}^{{2\pi }} {\sin \theta \left( {\sqrt {\cos \theta } } \right)} } P(\theta )S(\theta ) \times exp\left[ {ik\left( {z\cos \theta +r\sin \theta \cos (\psi - \phi } \right)} \right]$$

\(\times \left[ \begin{gathered} \sin (\delta - \psi )\sin (\phi - \psi )+\cos (\delta - \psi )\cos \theta \cos (\psi - \phi ) \hfill \\ \sin (\delta - \psi )\cos (\phi - \psi )+\cos (\delta - \psi )\cos \theta \sin (\psi - \phi ) \hfill \\ - \cos (\delta - \psi )\sin \theta \hfill \\ \end{gathered} \right]d\psi d\theta\)

Where, *k* is the wave number in the image space, α = arcsin(NA/n).*NA* is the numerical aperture of the objective lens and *n* is the refractive index of the given medium.*E**0* is a constant; *ϕ* and *θ* represents the azimuthal angle and tangential angle, respectively. Here, *P(θ)* denotes the amplitude function of the Bessel-Gaussian beam, which is given by[40]

\(P(\theta )=\exp \left[ { - {\beta ^2}{{\left( {\frac{{\sin (\theta )}}{{\sin \alpha }}} \right)}^2}} \right]{J_1}\left( {2\beta \frac{{\sin (\theta )}}{{\sin \alpha }}} \right)\)

where, *J**1**(x)* is the first order Bessel function and *β* is the ratio between pupil diameters to beam diameter.

Here, *δ* describes the radial and azimuthal variant function, which is modified from equation one as given by [56]

\(\delta =a\psi +b\left( {\frac{{2\pi \sin \theta }}{{\sin \alpha }}} \right)+c\)

From above equation, *a & b* are the azimuthal and radial indices with respect to the azimuthal angle *ψ*

and angle *θ*. *c* denotes the initial phase.

The pupil function of the Annular Walsh filter *S(θ)* is given by [66]

\(S(\theta )=\left\{ \begin{gathered} \begin{array}{*{20}{c}} {0,}&{0 \leqslant \theta <\xi } \end{array} \hfill \\ \begin{array}{*{20}{c}} {\Psi _{m}^{\xi }(\theta ),}&{\xi \leqslant \theta <\alpha } \end{array} \hfill \\ \end{gathered} \right.\)

The Annular Walsh function \(\Psi _{m}^{\xi }(\theta )\) is represented as[66]

\(\Psi _{m}^{\xi }(\theta )=\prod\limits_{{n=0}}^{{v - 1}} {\operatorname{sgn} \left\{ {\cos \left[ {{K_n}{2^n}\pi \frac{{\left( {{\theta ^2} - {\xi ^2}} \right)}}{{\left( {1 - {\xi ^2}} \right)}}} \right]} \right\}}\)

*K* *n* are the bits,0 or 1 of the binary numerical for *m*, and (2v)is the power of 2 that just exceeds *m*, for all θ in (ξ,1)

The Walsh order *m* is

\(m=\sum\limits_{{n=0}}^{{\upsilon - 1}} {{K_n}{2^n}}\)

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 zero crossings for the functions\(\Psi _{m}^{\xi }(\theta )\),*m = 0,1,…,(N-1)* are given by[66]

\({\theta _i}=\sqrt {\frac{{[(N - i)]{\xi ^2}+i}}{N}} \times \alpha\)

The inner as well as outer angle of the filter is represented as *θ**0* *= ξ* and *θ**N* *= 1*.