In the first model, we proposed five equal zones of higher-order polynomial *ρ*^{8} at the center ending with a linear function of *ρ* at the surface of the aperture as follows: *ρ*^{8}, *ρ*^{6}, *ρ*^{4}, *ρ*^{2}, and *ρ*.

The selection of five zones is presented to fulfill the arrangement assumed for the polynomial.

For the second model, this number is doubled since the center is assumed dark. In general, we can take any number of zones either even or odd depending on the proposed distribution. Hence, the first model has odd number of zones *N* = 5, while the second model has *N* = 10.

The assumed polynomial aperture has five equal zones of distributions, starting from the center, represented as *ρ*^{8}, *ρ*^{6}, *ρ*^{4}, *ρ*^{2}, and *ρ* as shown in Fig. 1. The corresponding line plot is shown as in Fig. 1c. In our case, the central zone has transmission intensity proportional to *ρ*^{8} instead of zero for the annular aperture.

Now, the polynomial aperture is written as follows:

$$\begin{aligned} P\left( \rho \right) & = a\rho ^{8 } , \quad {\text{for}}\; 0 \le \rho < 0.2\rho _{\max } \\ & = b \rho^{6 } , \quad {\text{for}}\; 0.2 \le \rho _{\max } < 0.4\rho _{\max } \\ & = c\rho ^{4 } , \quad {\text{for}}\; 0.4 \le \rho_{\max } < 0.6 \rho_{\max } \\ & = d \rho^{2 } , \quad {\text{for}}\; 0.6 \le \rho _{\max } < 0.8\rho _{\max } \\ & = e\rho , \quad {\text{for}}\; 0.8 \le \rho _{\max } < \rho _{\max } \\ \end{aligned}$$

(1)

*a*, *b*, *c*, *d*, *e*, constants are proportional to the cross-sectional areas of the corresponding zones.

In this model, referring to Eq. (1),

$$a = 0.04 \pi \rho_{\max }^{2} ,\;\; b = 0.12\pi \rho_{\max }^{2} ,\;\; c = 0.20 \pi \rho_{\max }^{2} ,\;\;d = 0.28\pi \rho_{\max }^{2} , \;\;e = 0.36\pi \rho_{\max }^{2}$$

Hence, \(a + b + c + d + e = \pi \rho_{\max }^{2}\) is the total area of open circular aperture of radius \(\rho_{\max }\).

*ρ* = (*u*, *v*) is the radial coordinate corresponding to the Cartesian coordinates (*u*, *v*) and \(\rho_{\max }\) is the total aperture radius.

The PSF corresponding to the polynomial aperture, described in Equ 1, is computed by operating the Fourier transform upon Eq. (1) considering coherent illumination emitted from spatially filtered Laser beam. Hence, the PSF is represented in integral form in polar coordinates as follows:

$$h\left( {r;\theta } \right) = \mathop \smallint \limits_{0}^{{\rho_{\max } }} \mathop \smallint \limits_{0}^{2\pi } P\left( \rho \right)\exp \left[ { - \frac{j2\pi }{{\lambda f}}\rho r\cos \left( {\Phi - \theta } \right)} \right] \rho {\text{d}}\rho {\text{d}} \Phi$$

(2)

where *u* = *ρ* cos *Φ*, *v* = *ρ* sin *Φ* are the Cartesian coordinates in the aperture plane corresponding to the polar coordinates (*ρ*, Φ), while *x* = *r* cos *θ*, *y* = *r* sin *θ* are the Cartesian coordinates in the Fourier or focal plane corresponding to the polar coordinates (*r*, *θ*). The Fourier transform lens has focal length = f.

Since the aperture has circular symmetry of revolution, equation, (2) is reduced to a function of r only as follows [11]:

$$h_{{\text{model 1}}} \left( r \right) = 2\pi \mathop \smallint \limits_{0}^{{\rho_{\max } }} P\left( \rho \right) J_{0} \left( {\frac{2\pi }{{\lambda f}}\rho r} \right) \rho {\text{d}}\rho$$

(3)

where \(J_{0} \left( x \right)\) represents the Bessel function of zero order and the Bessel function of any order n \(J_{n} \left( x \right)\) is represented by the following summation:

$$J_{n} \left( x \right) = \mathop \sum \limits_{m = 0}^{\infty } \frac{{\left( { - 1} \right)^{m} }}{{m!\left( {m + n} \right)!}} \left( \frac{x}{2} \right)^{n + 2m} .$$

Substituting Eq. 1 in Equ 3, we get:

$$\begin{aligned} h_{{\text{model 1}}} \left( r \right) & = 2 \pi \left\{ {a \mathop \smallint \limits_{0}^{{0.2\rho_{\max } }} \rho^{8} J_{0} \left( {\frac{2\pi }{{\lambda f}} \rho r} \right) \rho {\text{d}}\rho + b\mathop \smallint \limits_{{0.2\rho_{\max } }}^{{0.4\rho_{\max } }} \rho^{6} J_{0} \left( {\frac{2\pi }{{\lambda f}}\rho r} \right) \rho {\text{d}}\rho } \right. \\ & \quad + c\mathop \smallint \limits_{{0.4\rho_{\max } }}^{{0.6\rho_{\max } }} \rho^{4} J_{0} \left( {\frac{2\pi }{{\lambda f}} \rho r} \right) \rho {\text{d}}\rho + d\mathop \smallint \limits_{{0.6\rho_{\max } }}^{{0.8\rho_{\max } }} \rho^{2} J_{0} \left( {\frac{2\pi }{{\lambda f}} \rho r} \right) \rho {\text{d}}\rho \\ & \quad \left. { + e \mathop \smallint \limits_{{0.8\rho_{\max } }}^{{\rho_{\max } }} \rho J_{0} \left( {\frac{2\pi }{{\lambda f}} \rho r} \right) \rho {\text{d}}\rho } \right\} \\ \end{aligned}$$

(4)

Solving Eq. (4), we finally get the corresponding result for the PSF as follows:

$$\begin{aligned} h_{{\text{model 1}}} \left( r \right) & = \frac{{J_{1} \left( {W_{5} } \right)}}{{W_{5} }} - 0.08\mathop \sum \limits_{i = 1}^{4} \frac{{J_{1} \left( {W_{i} } \right)}}{{W_{i} }} + 0.4 \frac{{J_{2} \left( {W_{1} } \right)}}{{W_{1}^{2} }} + 0.08 \frac{{J_{2} \left( {W_{2} } \right)}}{{W_{2}^{2} }} - 0.24 \frac{{J_{2} \left( {W_{3} } \right)}}{{W_{3}^{2} }} \\ & \quad - 0.56 \frac{{J_{2} \left( {W_{4} } \right)}}{{W_{4}^{2} }} + 0.36\left( { \frac{{J_{0} \left( {W_{5} } \right)}}{{W_{5}^{2} }} - \frac{{J_{0} \left( {W_{4} } \right)}}{{W_{4}^{2} }}} \right) - 0.96 \frac{{J_{3} \left( {W_{1} } \right)}}{{W_{1}^{3} }} + 1.28\frac{{J_{3} \left( {W_{2} } \right)}}{{W_{2}^{3} }} \\ & \quad + 1.6\frac{{J_{3} \left( {W_{3} } \right)}}{{W_{3}^{3} }} + 0.72 \mathop \sum \limits_{i = 1}^{N} \left( {\frac{{J_{i} \left( {W_{4} } \right)}}{{W_{4}^{3} }} - \frac{{J_{i} \left( {W_{5} } \right)}}{{W_{5}^{3} }} } \right) - 1.92 \frac{{J_{4} \left( {W_{1} } \right)}}{{W_{1}^{4} }} - 5.76\frac{{J_{4} \left( {W_{2} } \right)}}{{W_{2}^{4} }} \\ & \quad + 15.36 \frac{{J_{5} \left( {W_{1} } \right)}}{{W_{1}^{5} }} \\ \end{aligned}$$

(5)

where *i* = (1, 3, 5, …, *N*), \(W_{1} = \frac{2}{f} \left( {0.2\rho_{\max } } \right)r\), \(W_{2} = \frac{2}{f} \left( {0.4\rho_{\max } } \right)r\),

\(W_{3} = \frac{2}{f} \left( {0.6\rho_{\max } } \right)r\), \(W_{4} = \frac{2}{f} \left( {0.8\rho_{\max } } \right)r\), \(W_{5} = \frac{2}{f} \left( {\rho_{\max } } \right)r\).

The PSF corresponding to the second model is computed by following the above analysis; Eq. (4) except the integral limits changed following the new intervals between the ten concentric equal zones of different distributions. Hence, we write the PSF as follows:

$$\begin{aligned} h_{{\text{model 2}}} \left( r \right) & = 2\pi \left\{ {a\mathop \smallint \limits_{{0.1\rho_{\max } }}^{{0.2\rho_{\max } }} \rho^{8 } J_{0} \left( {\frac{2\pi }{{\lambda f}} \rho r} \right) \rho {\text{d}}\rho + b \mathop \smallint \limits_{{0.3\rho_{\max } }}^{{0.4\rho_{\max } }} \rho^{6 } J_{0} \left( {\frac{2\pi }{{\lambda f}} \rho r} \right)\rho {\text{d}}\rho } \right. \\ & \quad + c\mathop \smallint \limits_{{0.5_{\max } }}^{{0.6_{\max } }} \rho^{4 } J_{0} \left( {\frac{2\pi }{{\lambda f}} \rho r} \right) \rho {\text{d}}\rho + d\mathop \smallint \limits_{{0.7\rho_{\max } }}^{{0.8\rho_{\max } }} \rho^{2 } J_{0} \left( {\frac{2\pi }{{\lambda f}} \rho r} \right) \rho {\text{d}}\rho \\ & \quad \left. { + e\mathop \smallint \limits_{{0.9\rho_{\max } }}^{{\rho_{\max } }} \rho J_{0} \left( {\frac{2\pi }{{\lambda f}}\rho r} \right) \rho {\text{d}}\rho } \right\} \\ \end{aligned}$$

(6)

It is noted that the other five integrals are set equal to zero for the dark zones in the (B/W_{polynomial}) aperture. The cross-sectional areas corresponding to the transparent zones have the values:

$$a = 0.03 \pi \rho_{\max }^{2} , \;\;b = 0.07\pi \rho_{\max }^{2} ,\;\; c = 0.11\pi \rho_{\max }^{2} ,\;\;d = 0.15\pi \rho_{\max }^{2} ,\;\; e = 0.19\pi \rho_{\max }^{2} .$$

We finally get the PSF corresponding to the second model of polynomial aperture as follows:

$$\begin{aligned} h_{{\text{model 2}}} \left( r \right) & = 0.19 \left[ { \frac{{J_{0} \left( {W_{10} } \right)}}{{W_{10}^{2} }} - \frac{{J_{0} \left( {W_{9} } \right)}}{{W_{9}^{2} }}} \right] - 0.38\mathop \sum \limits_{i = 1,3,5, \ldots } \left[ { \frac{{J_{i} \left( {W_{10} } \right)}}{{W_{10}^{3} }} - \frac{{J_{i} \left( {W_{9} } \right)}}{{W_{9}^{3} }}} \right] \\ & \quad + \mathop \sum \limits_{i = 1}^{5} \left[ { \frac{{J_{1} \left( {W_{2i} } \right)}}{{W_{2i} }} - \frac{{J_{1} \left( {W_{2i - 1} } \right)}}{{W_{2i - 1} }}} \right] - 0.24 \left[ {\frac{{J_{2} \left( {W_{2} } \right)}}{{W_{2}^{2} }} - \frac{{J_{2} \left( {W_{1} } \right)}}{{W_{1}^{2} }}} \right] \\ & \quad - 0.42 \left[ {\frac{{J_{2} \left( {W_{4} } \right)}}{{W_{4}^{2} }} - \frac{{J_{2} \left( {W_{3} } \right)}}{{W_{3}^{2} }}} \right] - 0.44 \left[ {\frac{{J_{2} \left( {W_{6} } \right)}}{{W_{6}^{2} }} - \frac{{J_{2} \left( {W_{5} } \right)}}{{W_{5}^{2} }}} \right] \\ & \quad - 0.3 \left[ {\frac{{J_{2} \left( {W_{8} } \right)}}{{W_{8}^{2} }} - \frac{{J_{2} \left( {W_{7} } \right)}}{{W_{7}^{2} }}} \right] + 1.44\left[ { \frac{{J_{3} \left( {W_{2} } \right)}}{{W_{2}^{3} }} - \frac{{J_{3} \left( {W_{1} } \right)}}{{W_{1}^{3} }}} \right] + 1.68\left[ { \frac{{J_{3} \left( {W_{4} } \right)}}{{W_{4}^{3} }} - \frac{{J_{3} \left( {W_{3} } \right)}}{{W_{3}^{3} }}} \right] \\ & \quad + 0.88\left[ { \frac{{J_{3} \left( {W_{6} } \right)}}{{W_{6}^{3} }} - \frac{{J_{3} \left( {W_{5} } \right)}}{{W_{5}^{3} }}} \right] - 5.76\left[ { \frac{{J_{4} \left( {W_{2} } \right)}}{{W_{2}^{4} }} - \frac{{J_{4} \left( {W_{1} } \right)}}{{W_{1}^{4} }}} \right] \\ & \quad - 3.36\left[ { \frac{{J_{4} \left( {W_{4} } \right)}}{{W_{4}^{4} }} - \frac{{J_{4} \left( {W_{3} } \right)}}{{W_{3}^{4} }}} \right] + 11.52\left[ {\frac{{J_{5} \left( {W_{2} } \right)}}{{W_{2}^{5} }} - \frac{{J_{5} \left( {W_{1} } \right)}}{{W_{1}^{5} }}} \right] \\ \end{aligned}$$

(7)

An application in microscopy is given, particularly in the case of the CSLM [1,2,3,4,5], provided with polynomial apertures of type 1 or type 2 described above, and the obtained image is computed from Eq. (8), where the polynomial aperture for both microscope objectives is given in Eq. (1) for the first model:

$$I\left( {x,y} \right) = \left| {\iint\limits_{ - \infty }^{\infty } {h_{{{\text{polynomial}}}} \left( {x,y} \right) \cdot h_{{{\text{polynomial}}}} \left( {x,y} \right) \cdot g\left( {x - x^{\prime } ,y - y^{\prime } } \right) {\text{d}}x^{\prime } {\text{d}}y^{\prime } } } \right|^{2}$$

(8)

Consequently, the formed image is the modulus square of the convolution product of the resultant point spread function and the complex amplitude of the object. It is written symbolically as:

$$I\left( {x,y} \right) = \left| { h_{r} \left( {x,y} \right) \otimes g\left( {x,y} \right) } \right|^{2}$$

\({\text{h}}_{r} \left( {x,y} \right) = [h_{{{\text{polynomial}}}} \left( {x,y} \right)]^{2} .\); for two symmetric objectives of polynomial apertures.

Here, \(h_{{{\text{polynomial}}}} \left( r \right)\) is computed from Eq. (5) for the first model and computed from Eq. (7) for the second model. The image used in the processing is the Siemen’s test chart.

For a point object, the above convolution is reduced to the resultant PSF squared computed as follows:

$$\begin{aligned} I\left( {x,y} \right) & = \left| {\iint\limits_{ - \infty }^{\infty } {h_{{{\text{polynomial}}}} \left( {x,y} \right) \cdot h_{{{\text{polynomial}}}} \left( {x,y} \right) \cdot \delta \left( {x - x^{\prime } ,y - y^{\prime } } \right) {\text{d}}x^{\prime } {\text{d}}y^{\prime } } } \right|^{2} \\ & = \left| { h^{2}_{{{\text{polynomial}}}} \left( {x,y} \right) } \right|^{2} = \left[ { \frac{{2J_{1} \left( {W_{5} } \right)}}{{W_{5} }}} \right]^{4} \\ \end{aligned}$$

when the polynomial aperture is replaced by open circular aperture [1].