A collimated beam from He-Ne laser is obtained using spatially filtered techniques as shown in Fig. 1. The collimated parallel beam is incident on the confocal arrangement of the microscope [2, 3, 6, 7]. In this confocal arrangement, the mechanical scanning synchronized with the electronic scanning in the detection plane allows to construct the image. The formation of images using scanned object (omicron corona virus) is summarized as follows:

Consider unit amplitude of coherent radiation incident upon the 1st spiky aperture P_{1} (u, v) composed of a circle surrounded by irregular spikes. Then, the transmitted complex amplitude is represented as follows:

$$\begin{aligned} {\text{P}}_{1} \left( {{\text{u}},v} \right) & = & 1{ };{\text{ for }}\frac{{\rho}}{\rho{_{{0{ }}} + \epsilon_{r} }} \le 1;{ \epsilon_{r}<< \rho{_{{0{ }}} }} \\ = {\text{ zero}};{\text{ otherwise}} \\ \end{aligned}$$

(1)

ρ: is the radial coordinate in the aperture plane,\({\uprho }_{0}\): the internal aperture radius surrounded by the spikes of average width \(_{r}\). Hence, the external radius is \({\uprho }_{{0{ }}} +\varepsilon_{r}\) and the ratio between the internal and external radii is \(\alpha= \frac{{\uprho }_{0}}{{\left({\uprho }_{{0{ }}} +\varepsilon_{r} \right)}}\).

In the front focal plane of the converging lens L_{1}, we get by applying the Fourier transform (F.T.) upon Eq. (1) the following:

$${\text{h}}_{1} \left( {{\text{x, y}}} \right) = {\text{F.T.}} \left\{ {{\text{P}}_{1} \left( {{\text{u, v}}} \right)} \right\} = \mathop \smallint \limits_{ -\infty\ }^{{\infty}} \mathop \smallint \limits_{ -\infty }^{{\infty}} {\text{P}}_{1} \left( {{\text{u, v}}} \right) \exp \left[ { - \frac{j2\pi}{f\lambda}\left( {ux + vy} \right)} \right] du dv$$

(2)

where \({\text{h}}_{1} \left( {{\text{x}},{\text{y}}} \right)\) is the PSF or amplitude impulse response corresponding to the irregular aperture of random spikes P_{1}.

The F.T. in Eq. (2), for the irregular spiky aperture, is rewritten in integral radial form as follows:

$${\text{h}}_{1} \left( {{\text{x, y}}} \right) = \mathop \smallint \limits_{0}^{2\pi}\ \mathop \smallint \limits_{0}^{{\infty}} {\text{P}}_{1} \left( {{\text{u, v}}} \right)\exp \left[ { - j\left( \frac{2\pi}{f\lambda} \right){{\rho}\text{r}}\cos \left(\theta -\phi \right)} \right] \rho{d\rho} {\text{ d}\theta}$$

(3)

Substitute from Eq. (1) in Eq. (3), we get:

$${\text{h}}_{1} \left( {{\text{x}},{\text{y}}} \right) = \mathop \smallint \limits_{0}^{2\pi} \mathop \smallint \limits_{0}^{{\rho_{{0{ }}} + \epsilon_{r} }} \exp \left[ { - j\left( \frac{2\pi}{f\lambda} \right){ \rho}{\text{ r}}\cos \left(\theta-\phi\right)}\right]\rho{d\rho}{\text{ d}\theta}$$

(4)

where \(x = r\cos\phi\), \(y = r\sin\phi\) , and \(r = \sqrt {x^{2} + y^{2} }\). The polar coordinates in the aperture plane (*ρ*, *θ*) are related to the cartesian coordinates (u, v) as follows:

\({\text{u}} = \rho\cos {\theta },\) \({\text{v}} = \rho\sin {\theta },{ }\) and \(\rho= \sqrt {u^{2} + v^{2} }\).

Assuming spherical symmetry of revolution for the aperture and hence using Bessel identity Eq. (4) becomes:

$${\text{h}}_{1} \left( {{\text{x}},{\text{y}}} \right) = 2\pi \mathop \smallint \limits_{0}^{{\rho_{0} + { }\epsilon_{r} }} { }J_{0} \left( {\frac{{2\pi\rho{\text{r}}}}{{\text{f}\lambda}}} \right) \rho{\text{ d}\rho}$$

(5)

where *J*_{0:} Bessel function of zero order.

The above integral is solved by separation into two parts as follows:

$${\text{h}}_{1} \left( {{\text{x}},{\text{y}}} \right) = 2\pi \mathop \smallint \limits_{0}^{{\rho_{0} }} { }J_{0} \left( {\frac{{2\pi{\rho}{\text{r}}}}{{\text{f}\lambda}}} \right) {\rho }{\text{ d }\rho} + { }2\pi \mathop \smallint \limits_{{\rho_{0} }}^{{\rho_{0} + { }\epsilon_{r} }} { }J_{0} \left( {\frac{{2\pi{\rho }{\text{r}}}}{{\text{f}\lambda}}} \right) {\rho }{\text{ d }\rho}$$

(6)

The solution of the above integral from zero to \(\rho_{0}\) gives the known Airy disc, and we get the following:

$${\text{h}}_{1} \left( {{\text{x}},{\text{y}}} \right) = \frac{{2J_{1} \left( w \right)}}{w} + { }2\pi \mathop \smallint \limits_{{\rho_{0} }}^{{\rho_{0} + { }\epsilon_{r} }} { }J_{0} \left( {\frac{{2\pi{\rho }{\text{r}}}}{{\text{f}\lambda}}} \right) {\rho }{\text{ d }\rho}$$

(7)

where *w* is the reduced coordinate given by \(w = \frac{{2{\pi }\rho_{{0{ }}} {\text{r}}}}{{\text{f}\lambda}}\).

The 2nd integral is approximated by annulus of random surface; hence, we finally get for the PSF the following formula:

$${\text{h}}_{1} \left( {\text{w}} \right) = \frac{{2J_{1} \left( w \right)}}{w} + {\beta }J_{0} \left( w \right); \beta< 1$$

(8)

It is assumed that for the spiky pattern it is approximated by annular distribution where the value of the parameter \(= 0.5\).

Accurate solution obtained for the second integral corresponding to the spiky part is computed from the difference between the computed values corresponding to the external and internal radii. Hence, from Eq. (7), we get the following result for the PSF [21].

$${\text{h}}_{1} \left( {\text{w}} \right) = \frac{{2J_{1} \left( w \right)}}{w} + \frac{2}{{1 -\alpha^{2} }} \left[ {\frac{{J_{1} \left( {w_{1} } \right)}}{{w_{1} }} -\alpha \frac{{J_{1} \left( w \right)}}{w}} \right]$$

(9)

The reduced coordinate w_{1} is related to the reduced coordinate w by the formula:

$$w_{1} = \frac{{2{\pi }(\rho_{{0{ }}}+ { \epsilon}_{r} ){\text{r}}}}{{\lambda}\text{f}}= w(1+\frac{{\epsilon}_{r}}{\rho_{{0 }}})$$

(10)

The omicron corona virus image is placed in the object plane of transmitted amplitude \(\mathrm{g}(\mathrm{x},\mathrm{y})\).

The 2nd lens shown conjugate to the 1^{st} lens where the pupil aperture \({P}_{2}\left(u,v\right)\) are placed in front of the 2nd lens. The point spread function is equal \({\mathrm{h}}_{2}\left(\mathrm{x},\mathrm{y}\right)\) for uniform circular aperture represented as:

$$\begin{aligned} {\text{P}}_{2} \left( {{\text{u}},{\text{ v}}} \right) & = 1{ };{\text{ for }}\mid\frac{{\rho}}{\rho{_{{0{ }}} }}\mid \le 1 \\ = {\text{ zero}};{\text{ otherwise}} \\ \end{aligned}$$

(11)

Hence, operating F.T. upon Eq. (11), we get the known result of Airy disc represented as follows:

$${\text{h}}_{2} \left( {\text{w}} \right) = \frac{{2J_{1} \left( w \right)}}{w}$$

(12)

where *w* is the reduced coordinate as given in Eq. (7).

The object \({\text{g}}\left( {{\text{x}},{\text{y}}} \right)\) is placed in the common short focus corresponding to the two microscope lenses. Consequently, the scanned object transparency g (x, y) is convoluted by both the PSF’s. Hence, we get the following convolution in the object plane of transmitted complex amplitude C (x, y) written in integral form as follows:

$${\text{C}}\left( {{\text{x}},{\text{y}}} \right) = { }\mathop \smallint \limits_{ - }^{{}} \mathop \smallint \limits_{ - }^{{}} {\text{g}}\left( {{\text{x}}_{s} , {\text{y}}_{s} } \right) {\text{h}}_{1} \left( {{\text{x}} - {\text{x}}_{s} { },{\text{y}} - {\text{ y}}_{s} { }} \right).{\text{h}}_{2} \left( {{\text{x}} - {\text{ x}}_{s} ,{\text{y}} - {\text{ y}}_{s} } \right) {\text{dx}}_{s} {\text{dy}}_{s}$$

(13)

The mechanical scanning in the object plane is represented by (\(x_{s} ,y_{s} ).\)

Equation (13) is written symbolically as follows:

$${\text{C}}\left( {{\text{x}},{\text{y}}} \right) = { }\{ {\text{h}}_{1} \left( {{\text{x}},{\text{y}}} \right).{\text{h}}_{2} \left( {{\text{x}},{\text{y}}} \right)\} {\text{ g}}\left( {{\text{x}},{\text{y}}} \right)$$

(14)

where \(symbol for convolution product.\)

The multiplication of

$${\text{ h}}_{1} \left( {{\text{x}},{\text{y}}} \right).{\text{h}}_{2} \left( {{\text{x}},{\text{y}}} \right) = {\text{h}}_{{{\text{eff}}}} \left( {{\text{x}},{\text{y}}} \right)$$

(15)

Substitute from Eqs. (8, 12), we get the following resultant PSF:

$${\text{h}}_{{{\text{eff}}}} \left( {\text{w}} \right) = \left[ {\frac{{2J_{1} \left( w \right)}}{w}} \right]^{2} +\beta J_{0} \left( {\text{w}} \right) \frac{{2J_{1} \left( w \right)}}{w}$$

(16)

\({\text{h}}_{{{\text{eff}}}} \left( {{\text{x}},{\text{y}}} \right)\) is the effective or resultant PSF in confocal microscope using the illuminating aperture P_{1} of irregular spiky shape while the 2nd aperture corresponding the detector has uniform circular shape. It is shown that the 2nd term in the R.H.S. of Eq. (16) is the contribution of the spiky part of the aperture upon the \({\text{h}}_{{{\text{eff}}}} \left( {\text{w}} \right).\)

The detected intensity is represented as the modulus square of the complex amplitude \(C\left( {x,y} \right),\) represented as follows:

$${\text{I}}\left( {{\text{x}},{\text{y}}} \right) = \mid{\text{ C}}\left( {{\text{x}},{\text{y}}} \right){ }\mid^{2} = \mid{\text{ h}}_{{{\text{eff}}}} \left( {{\text{x}},{\text{y}}}\right){\text{ g}}\left( {{\text{x}},{\text{y}}} \right){ }\mid^{2}$$

(17)

Equation (17) is the intensity distribution in case of Confocal Scanning Laser Microscope (CSLM).