# An Analytical Study of the Diffraction of Light by a Circular Aperture Using Spherical Harmonics forn≤1

Authors: Mahmoud Ahmad; Najah Kabalan; Samar Omran
Journal: International Journal of Optics
 (2020)

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## Abstract

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 compoundsC00,C10,C11,C1−1 for n=0,1,m=−1,0,1. We studied the intensity distributions in three special cases (along the optical axis, at the geometrical focal plane, and along the boundary of the geometrical shadow). We presented numerical comparative examples to illustrate the variation of the intensity versus a ratio (Z/A is the ratio of the distance between the circular aperture and the observation plane to a radius of the circular aperture), and we used Maple program to represent these results. We noticed that the expansion we made using spherical harmonic analysis led to an increase in the number of fringes bright enough to be visible to the naked eye. We then concluded with a brief discussion of the results.


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## 1. Introduction

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 [1] the diffraction phenomenon according to the Huygens–Fresnel principle [2–5] with a wave-based approach, and the Huygens–Fresnel principle states that every point of a wavefront is a new secondary source that emits spherical wavelets. The Huygens–Fresnel principle has a close relationship with the diffraction by a circular aperture. The appropriate choice of the shape, size, and type of material from which the aperture is made has great scientific and practical importance; Basyigit et al. [6] studied various forms of apertures (rectangular, square, circular, hexagonal, and equilateral triangle shape apertures), and they ranked it from best performance to the worst with respect to electrical shielding effectiveness (ESE). They also studied polarization type of applied electromagnetic field. It is important to note that we neglected polarization in our study because the study is scalar. Basyigit and Dogan [7] investigated the effect of aperture size of metallic enclosure on ESE by using both Robinson’s analytical model and ANN methods (ANN is a universal approximation method used in electromagnetic research studies).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 [8–15]. Sheppard and Cooper [9] studied Fresnel diffraction by a circular aperture for off-axis illumination and various different levels of approximation. They obtained a new approximate equation for the shape of contours of equal intensity in the focal plane describing a distortion of the contours in comparison with those of elliptical shape predicted by Murty [10]. Acosta and Reino [14] presented a method for studying the intensity distribution of a beam diffracted by a circular aperture in terms of Bessel functions.In this research, we studied the intensity distributions of spherical waves diffracted by a circular aperture in three cases as follows:(i)
Along the optical axis(ii)
In the geometrical focal plane(iii)
Along the boundary of the geometrical shadowWe calculated the optical intensity compoundsC00,C10,C11,C1−1 based on the spherical harmonic analysis for n=0,1,m=−1,0,1. In addition, we discussed the special cases of intensity on graphs for a Rayleigh distance dRL. We presented numerical comparative examples to illustrate our analytical results, which greatly improve the design of optical devices that have a same circular aperture.


## 2. Theoretical Formulation

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 [8]:(1)Uρ,R=A2eiwRzRik+1R∫01ei/2uρ`2J0vρ`ρ`dρ`,where A is the radius of the circular aperture, k is the wave number, J0vρ` is the zero order of Bessel function, u,v,w are the optical coordinates, and ρ,ρ` are cylindrical radial coordinates in the observation and aperture planes, respectively (Figure 1).Figure 1
Geometry of diffraction by a circular aperture.The geometry of the experiment has a great importance in our study. Figure1 shows the geometry of diffraction for a uniform, monochromatic spherical wave diffracted by a circular aperture. There is a close relationship between the parameters of this figure and the diffraction phenomenon. The most important parameter is the size of circular aperture. The maximum limits of intensity become sharper and narrower with the increase in aperture size. Diffraction is the bending of waves around corners that occurs when a portion of a wavefront is cut off by a barrier or obstacle, and as a result of the differences in the diffraction angles, fringes form a specific pattern of intensity on the observation screen.A “Z” distance between circular aperture and observation screen is an important parameter in the diffraction phenomenon. It determines the conditions of transmission between the different patterns of diffraction (Rayleigh–Sommerfeld, Fresnel, and Fraunhofer), and this is what we explained later in the results.


## 3. Spherical Harmonics

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 [16].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 [2–5]. This makes the spherical harmonics a natural basis for our calculations. The main goals of the spherical harmonic analysis are to expand the polar radius of a point on the surface of the circular aperture that takes equation (2) and to calculate the associated coefficients of the spherical harmonic series [17]:(2)ρ`θ,φ=∑n=0∞∑m=−nnCnmYnmθ,φ,where ρ`θ,ϕ is the polar radius from the circular aperture center, φ,θ are the corresponding spherical coordinates, θ∈0,π is polar (colatitudinal) coordinate and represents the angle of diffraction, φ∈0,2π is azimuthal (longitudinal) coordinate, Cnm is the associated spherical harmonic coefficient, and Ynmθ,φ are the spherical harmonic functions that are a series of special functions defined on the surface of a sphere and are given by the following relationship [18, 19]:(3)Ynmθ,φ=2n+1n−m!4πn+m!Pnmcosθeimφ,where n is the degree of the associated Legendre function, m is the order of the associated Legendre function, and Pnmcosθ are the associated Legendre functions which are expressed by the following formula [18–21]:(4)Pnmx=1−x2m/2dmdxm12nn!dndxnx2−1n.Forn=0,m=0, we get an equation means that the zero degree spherical harmonic representation of all point on the surface of the circular aperture is a sphere, whose radius is determined by C00:(5)ρ0=C00Y00θ,φ=14πC00,where Y00θ,φ is spherically symmetric and represents the monopole moment of the function on a sphere and Ynmθ,φ represents the dipole moments of the function on a sphere.To simplify the calculations and based on the mathematical properties of the spherical harmonic function [19], we express the coefficients C10,C1−1,C11, which represent the optical intensity compounds for n=1,m=−1,0,1 with two real and imaginary parts as follows:(6)C10=c,(7)C11=a+ib,(8)C1−1=−a+ib.Note thatCn0 is always a real number.Ynmθ,φ are expressed from the standard spherical harmonic function table by the following formulas [18, 19]:(9)Y00θ,φ=14π,Y10θ,φ=34πcosθY11θ,φ=−38πsinθcosφ+isinθsinφY1−1θ,φ=38πsinθcosφ−isinθsinφ.Relationship (2) is expanded for n=0,1,m=−1,0,1 in the following formula:(10)ρ`θ,ϕ≈C00Y00+C10Y10+C1−1Y1−1+C11Y11.Substituting relationships (5)–(9) into relationship (10), we get the polar radius of a point on surface of the circular aperture:(11)ρ`θ,ϕ≈14πC00+34πccosθ+2sinθ−acosφ+bsinφ≈14πC00+34πccosθ−2sinθacosφ−bsinφ≈14πC00+34πccosθ+2sinθacosφ+bsinφρ`θ,ϕ≈14πC00+34πccosθ+32πa2+b2sinθ.By deriving relationship (11), we obtain:(12)dρ`=32πa2+b2cosθ−123πcsinθdθ.We calculated the Bessel functionJ0vρ` in relationship (1) based on the following rule (assuming x=vρ`):(13)J0vρ`=12π∫02πeivρ`sinθdθ.We expanded the exponential functioneivρ`sinθ in relationship (13) according to the following power series {assuming that a=ivρ`sinθ}:(14)ea=∑n=0∞ann!=1+a+a22!+⋯.By solving the necessary integral, we obtain:(15)J0vρ`=1+iv32πa2+b2.We also expanded the exponential functionei/2uρ`2 in relationship (1) according to power series (14) {assuming that a=i/2uρ`2}:(16)ei/2uρ`2≈1+iu8πC002+i3u8πc2cos2θ+i3u4πa2+b2sin2θ.Substituting relationships (11), (12), (15), and (16) into relationship (1), after solving the resulting integral, we obtain the following optical wave amplitude Uρ,R:(17)Uρ,R=A2ZRik+1ReiwR−32πcC00+3a2+b2162ππ2uvcC003+3a2+b2162ππ2uvc3C00−93a2+b2128ππuvc3+3a2+b2a2+b242ππ2uvcC00+93a2+b2264ππuvc−i3a2+b222ππvcC00+316π2ucC003+316π2uc3C00+3a2+b24π2ucC00+9a2+b2a2+b2322πuc+9a2+b2642πuc3.Relationship (17) expresses the optical field amplitude at any point behind the circular aperture, which represents the sum of spherical wavelets emitted from secondary sources and has the same primary wave frequency. After squaring the previous result (optical wave amplitude), we obtain the following optical intensity:(18)Iu,v=A4Z2R4k2+1R2⋅34π2c2C002+9a2+b28π3v2c2C002−3a2+b22π2uvc2C0034π2πC003+34π2πc2C00−9a2+b232πc2+3a2+b2π2πC00+9a2+b2a2+b216π+9a2+b216π2u2v2c2+C0034π2π+c2C004π2π−33a2+b232πc2+a2+b2π2πC00+33a2+b2a2+b216π2+u2c216π23C0034π+3c2C004π+3a2+b2πC00+9a2+b2a2+b282−9a2+b2162c22.


## 4. Particular Cases

There are three cases of intensity study [3, 4]:(i)
The intensity along the axisv=0(ii)
The intensity in the geometrical focal planeu=0(iii)
The intensity along the boundary of the geometrical shadowu=±vThe first case: the intensity along the axisv=0:After substitutingv=0 into relationship (18), the relationship of the optical intensity becomes(19)Iu,0=A4Z2R4k2+1R2⋅34π2c2C002+u2c216π23C0034π+3c2C004π+3a2+b2πC00+9a2+b2a2+b282−9a2+b2162C22.We obtain “R” from Figure 1:(20)R=Z1−sin2θ=Zcosθ.The relationship of the optical coordinateu [8] is given as follows:(21)u=2kA2Z1+A2/Z21/2+1.We calculated the values of the coefficientsCnm from the following renormalized spherical harmonic relationship [17, 22]:(22)Cnmθ,φ=4π2n+1Ynmθ,φ,C00=1C10=cosθC11=−12sinθcosφ+isinθsinφC1−1=12sinθcosφ−isinθsinφ.We studied a special case (φ = 0) where the studying point from the aperture is located on the x-axis. We obtained the following coefficients:(23)C00=1,C10=cosθ,C11=−12sinθ,C1−1=12sinθ.Substituting relationships (20), (21), and (23) into relationship (18), we obtain the optical intensity along the optical axis:(24)Iu,0=2dRLπ2A2λZ2cos4θ+A4cos6θZ4⋅34π2cos2θ+cos2θ16π22πdRLZ1+1+A2/Z223cos2θ+14π+3sin2θ2π+9sinθsin2θ−cos2θ322,wheredRL is a Rayleigh distance (which is an important parameter used in the theory of laser beams; it represents the axial distance from the radiating aperture to the observation plane) and is given by the following formula (see [23], pp.658):(25)dRL=22A2λ,where 2A is a diameter of the circular aperture.The second case: the intensity in the geometrical focal planeu=0:Substitutingu=0 and relationships (20) and (23) into relationship (18), we obtain the optical intensity in the geometrical focal plane:(26)I0,v=2dRLπ2A2λZ2cos4θ+A4cos6θZ4⋅34π2cos2θ+9v2sin2θcos2θ16π3.The third case: the intensity along the boundary of the geometrical shadowu=±v:Substitutingu=u and relationships (20), (21), and (23) into relationship (18), we obtain the optical intensity along the boundary of the geometrical shadow:(27)Iu,u=2dRLπ2A2λZ2cos4θ+A4cos6θZ4⋅34π2cos2θ+9sin2θcos2θ16π32πdRLZ1+1+A2/Z22−3sinθcos2θ22π22πdRLZ1+1+A2/Z2234π2π+34π2πcos2θ−9sinθcos2θ322π+3sin2θ2π2π+9sin3θ322π+9sin2θcos2θ32π22πdRLZ1+1+A2/Z2414π2π+cos2θ4π2π−33sinθcos2θ322π+sin2θ2π2π+33sin3θ322π2+cos2θ16π22πdRLZ1+1+A2/Z2234π+3cos2θ4π+3sin2θ2π+9sin3θ32−9sinθcos2θ322.


## 5. Numerical Examples and Results

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 (25), intensity in the geometrical focal plane (26), and intensity along the boundary of the geometrical shadow (27). We obtained the graphs using Maple program [24] for the following numerical values: {A=1×10−4m (radius of the circular aperture), λ=6.328×10−7m wavelength, dRL=12.6×10−2m Rayleigh distance, Z=135,115,197×10−4m distance between the circular aperture and the observation plane, and the optical coordinate value (v = 3.832), which represents the first zero in intensity [9]}. We got the following graphs.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:(i)
In Figures2(a), 2(c), and 2(e), we observe that the presence of bands of frequencies appears clearly in specific and repeated regularly domains. This is due to the introduction of spherical harmonics in calculation of the optical intensity, where spherical harmonics represent a complete set of orthogonal and regular functions defined on the surface of a sphere. They are very similar to Fourier series where the frequencies appear intermittently on the observation screen.(ii)
By truncating a portion of the curves in Figures2(a), 2(c), and 2(e), we obtain Figures 2(b), 2(d), and 2(f), where the intensity levels appear close to each other at the beginning, and then these levels spaced from each other with an increase in the value of Z. It is worthy to note that the obtained curves in Figures 2(b), 2(d), and 2(f)) are identical to those obtained by Sheppard and Hrynevych (see [8], figure 2).(iii)
We also observed that the intensity in relationships (24), (26), and (27) is a sinusoidal change. The resulting relationship contains the functions sin (θ) and cos (θ), which are important for studying periodic phenomena such as waves. The graphs show that the intensity oscillates to reach maximum and minimum values appear as successive bright and dark fringes because of constructive and destructive interference of waves.(iv)
Figures2 and 3 illustrate the intensity variation versus the ratio (Z/A: the ratio of the distance between the circular aperture and the observation plane to the radius of the circular aperture).(v)
Figure4 represents the diffraction of a monochromatic light wave by a circular aperture, whose illumination phase is almost constant [23] (where D = 2A is diameter of the circular aperture and its unit is meter, λ is the wavelength and its unit is meter, Z is the distance between the circular aperture and the observation plane and its unit is meter, dRL is Rayleigh distance and its unit is meter, and θ is the diffraction angle and its unit is radian). We can distinguish between Fresnel and Fraunhofer diffraction according to the relationship between a Rayleigh distance and the distance Z, if:(28)Z<dRLFresnel diffractionZ>dRLFraunhofer diffraction.Figure 2
Optical intensity resulting from a plane wave diffraction by a circular aperture using spherical harmonics forn≤1. (a, b) Optical intensity resulting along the optical axis for Z=135×10−4m. (c, d) Optical intensity resulting in the geometrical focal plane for Z=115×10−4m. (e, f) Optical intensity resulting along the boundary of the geometrical shadow for Z=197×10−4m.
(a)(b)(c)(d)(e)(f)Figure 3
(a, b) Optical intensity resulting along the optical axis forZ=655×10−3m. (c, d) Optical intensity resulting in the geometrical focal plane for Z=655×10−3m. (e, f) Optical intensity resulting along the boundary of the geometrical shadow for Z=655×10−3m.
(a)(b)(c)(d)(e)(f)Figure 4
Three radiation regions for the field diffracted from a radiating aperture. Note thatD is the radius of the aperture (D = 2A). Source: Ref. [23], figure 7.42.Figure 5
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. [26], figure 10.2.Accordingly, Figures2(a), 2(c), and 2(e) express Fresnel diffraction at the following distances Z=135,115,197×10−4m and Figures 3(a), 3(c), and 3(e) express Fraunhofer diffraction at a distance Z=655×10−3m, which achieves condition (28). We also mention that each band in Figures 2(a), 2(c), 2(e), 3(a), 3(c), and 3(e) is a set of the successive bright and dark fringes. By truncating a portion of the drawing field, we obtain Figures 3(b), 3(d), and 3(f) which show the shape of any band in curves of Figures 3(a), 3(c), and 3(e).(i)
Since the expansion coefficients are the analogs of Fourier coefficients (see [25], pp. 277–278) and as a result of the expansion of the optical intensity compounds, we obtained more complicated shapes, where the plots in Figures 2 and 3 show that the number of resulting intensity peaks increase in comparison with the intensity in the previously studied case [8].(ii)
As shown in Figure5, the curves in Figures 2(a), 2(c), 2(e), 3(a), 3(c), and 3(e) are succession of diffraction patterns at increasing distance from aperture (Fresnel diffraction pattern is shown in Figure 2; Fraunhofer diffraction pattern is shown in Figure 3).(iii)
In Figures2(c), 2(d), 3(c), and 3(d) (which represent the optical intensity resulting in the geometrical focal plane), there is a division in the maximum peaks of intensity. This may be due to the interference or diffraction of the higher orders in the optical intensity expansion.(iv)
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 [27]. We also chose the first Rayleigh–Sommerfeld diffraction formula because it is accurate in our study, and importantly, it can be solved readily analytically, whereas Kirchhoff theory is not mathematically consistent, despite its success [28]. In addition, we expanded our study to include the Fresnel and Fraunhofer diffraction regions based on a Rayleigh distance. We obtained a set of results explained above.(v)
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 [4], pp. 435–442) studied Fraunhofer diffraction by a circular aperture. They discussed the three special cases of light intensity (the intensity along the axis, the intensity in the geometrical focal plane, and the intensity along the boundary of the geometrical shadow). They obtained a set of results as follows.Intensity distribution along the optical axis is characterized by mathematical Sinc function squaresinc2u/4. Intensity distribution in the geometrical focal plane is characterized by square of Bessel function of first order J1v/v2, while the intensity along the boundary of the geometrical shadow is characterized by the Bessel function of zero order J0u and its square J0u2 and cosine function. From these results, they obtained the curves of intensity that have a principal maximum in the center, and then the intensity oscillates and gradually amplitude diminishes with increasing distance (z). In other words, they obtained a bright disk surrounded by concentric bright and dark rings. Then, the intensity of the bright rings decreases rapidly with their radius. Normally, only the first or second rings being bright enough are visible to the naked eye. So, the total incident energy is contained within the central core of the diffraction pattern. In this paper, we expanded the polar radius of a point on the surface of the circular aperture based on spherical harmonics. We obtained intensity containing sine and cosine functions of higher orders {relationships (24), (26) and (27)}. Because of this expansion, we obtained intensity distribution curves versus the ratio Z/A (Figures 2 and 3), in which the successive bright and dark fringes are distributed larger and more clearly, which provides a better view of the image to the naked eye. In case of studying without taking into account the mathematical expansion process of the polar radius (relationship (2)) which we did, the relationship of intensity will be reduced to a simpler relationship (see Ref. [6], equation (21)).


## 6. Discussion

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 [4] and Sheppard and Hrynevych [8], if we neglect the mathematical expansion process in relationship (2) and calculate the integral in relationship (1) in terms of Lommel functions. In addition, we studied the special intensity cases. We provided numerical comparison examples, and we used Maple program to represent these results. We noticed that the expansion we made using spherical harmonic analysis led to an increase in the number of fringes bright enough to be visible to the naked eye. Bands of frequencies also appeared in the curves of Figures 2(a), 2(c), 2(e), 3(a), 3(c), and 3(e) due to the introduction of spherical harmonics in the calculations. Finally, as practical applications on optical devices such as the camera or telescope, we can obtain more accurate and clear images, and this is what modern science seeks to accomplish.


## 7. Conclusion

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 forn≤1, where spherical harmonics form an orthonormal basis on the unit sphere. They can be considered as signal-processing tools on the unit sphere, analogous to the Fourier series or sines and cosines on the line or circle, while the Zernike polynomials (which are closely related to the general spherical harmonics) are especially useful in numerical calculations, but any wavefront aberration can be decomposed in terms of them, and advantages of orthogonality are lost in noncircular cases [29].We also calculated the optical wave amplitudesC00,C10,C11,C1−1. We discussed these results for Rayleigh distance dRL. We used the correction [8] of the optical coordinate u in order to calculate the optical intensity along the optical axis and along the boundary of the geometrical shadow. We also used a specific value for the optical coordinate v [9] in order to calculate the optical intensity in the geometrical focal plane. We performed a set of calculations using spherical harmonics expansion and applied this method in the illuminated region from the near field to the far field.Finally, we recommend extending spherical harmonics to higher orders forn>1 to obtain more accurate results and changing the geometry shape of the used aperture and recalculating again.



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