Based on the importance of spherical harmonics and their applicability in many physical problems, this research aimed to study the diffraction pattern of light by a circular aperture starting from the first Rayleigh–Sommerfeld diffraction equation and to expand the polar radius of a point on the surface of the circular aperture based on spherical harmonics. We depended on this theoretical framework in our paper. We calculated the optical intensity compounds

In the study of light, it is very important to understand the manner of diffraction of light passing through aperture, slit, or obstruction. Fresnel interpreted [

The calculation of optical intensity is an interesting subject among many features of studying the diffraction by a circular aperture. This topic has been a matter of interest to many researchers [

In this research, we studied the intensity distributions of spherical waves diffracted by a circular aperture in three cases as follows:

Along the optical axis

In the geometrical focal plane

Along the boundary of the geometrical shadow

We calculated the optical intensity compounds

If we assume that the light waves impinging on a screen contain a circular aperture, the diffused light diffracts only from the circular section of the aperture. It is clear that the same process occurs in the eye, microscope, telescope, and the camera lens. In order to calculate the resulting optical intensity, we started from the following diffraction equation [

Geometry of diffraction by a circular aperture.

The geometry of the experiment has a great importance in our study. Figure

A “

Spherical harmonics are a frequency-space basis for representing a series of special functions defined over the sphere. They have been applied in various fields ranging from the representation of gravitational and magnetic fields of planetary bodies in geodesy to atomic electron configurations in physical chemistry. They also appear in quantum mechanics as the solutions of the Schrödinger equation in spherical coordinates and in computer graphics [

In our research, each point of the wavefront diffracted by a circular aperture is a new secondary source that emits spherical wavelets according to the Huygens–Fresnel Principle [

For

To simplify the calculations and based on the mathematical properties of the spherical harmonic function [

Note that

Relationship (

Substituting relationships (

By deriving relationship (

We calculated the Bessel function

We expanded the exponential function

By solving the necessary integral, we obtain:

We also expanded the exponential function

Substituting relationships (

Relationship (

There are three cases of intensity study [

The intensity along the axis

The intensity in the geometrical focal plane

The intensity along the boundary of the geometrical shadow

The first case: the intensity along the axis

After substituting

We obtain “

The relationship of the optical coordinate

We calculated the values of the coefficients

We studied a special case (

Substituting relationships (

where

where 2

The second case: the intensity in the geometrical focal plane

Substituting

The third case: the intensity along the boundary of the geometrical shadow

Substituting

In this paper, we studied the diffraction pattern of light by a circular aperture starting from the first Rayleigh–Sommerfeld diffraction equation. We computed the optical intensity along the optical axis (

Since curve of the optical intensity distribution is a visual representation of the diffracted light by a circular aperture, we can explain the resulting curves with a set of the following points:

In Figures

By truncating a portion of the curves in Figures

We also observed that the intensity in relationships (

Figures

Figure

Optical intensity resulting from a plane wave diffraction by a circular aperture using spherical harmonics for

(a, b) Optical intensity resulting along the optical axis for

Three radiation regions for the field diffracted from a radiating aperture. Note that

Successive diffraction patterns. The three curves at the bottom represent Fresnel diffraction pattern (nearby), and the three curves at the top represent Fraunhofer diffraction pattern (faraway). Source Ref. [

Accordingly, Figures

Since the expansion coefficients are the analogs of Fourier coefficients (see [

As shown in Figure

In Figures

We started our study from the first Rayleigh–Sommerfeld diffraction formula that describes the Huygens–Fresnel principle. This formula is used to represent the spread of optical fields and it gives a physically realistic prediction for the axial intensity close to a circular aperture, whereas the second Rayleigh–Sommerfeld and Kirchhoff diffraction formulas do not [

Lommel succeeded in expressing the complex disturbance in terms of convergent series of Bessel functions based on the Huygens–Fresnel integral. Born and Wolf (see [

Intensity distribution along the optical axis is characterized by mathematical Sinc function square

This paper provides an insight into the effect of expansion coefficients on optical intensity distribution. It has been observed that there is a good agreement between our results and the previous results of Born and Wolf [

In this work, we studied the intensity distributions of a monochromatic light beam diffracted by a circular aperture in a homogeneous medium. The relationship of optical intensity was obtained using expansion of the spherical harmonics series in three-dimensional space for

We also calculated the optical wave amplitudes

Finally, we recommend extending spherical harmonics to higher orders for

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This study was supported by Tishreen University.