How Does a Circular Aperture Affect Light Diffraction?

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Diffraction occurs when light passes through a circular aperture, resulting in a specific pattern characterized by dark and bright rings. The irradiance in the far-field diffraction pattern is described by the equation I(θ) = I(0) (2 J_1(k r sinθ) / (k r sinθ))^2, where k is related to the wavelength. The first four dark rings appear at specific angular radii determined by the zeros of the Bessel function J_1. The Airy disk represents the central bright spot of this pattern, with its angular radius given by θ_Airy = 1.22 λ/d. Understanding these principles is crucial for applications in optics and imaging systems.
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Definition/Summary

This entry describes diffraction of a wave when it passes through a circular aperture.

Equations

The far-field (Fraunhofer) diffraction pattern for a circular aperture of radius r has a power per area (irradiance) given by:

<br /> I(\theta) = I(0) \left( \frac{2 J_1(k \ r \ \sin\theta)}{k \ r \ \sin\theta} \right)^2 <br />The angular radii of the first 4 dark rings in the diffraction pattern occur at

<br /> k \ r \ \sin\theta \ \approx \ 3.8317, \ \ 7.0156, \ \ 10.173, \ \ 13.324 <br />

where the numbers are the zeros of J_1

Equivalently, in terms of wavelength the zeroes are at

<br /> \sin\theta \ \approx \frac{\lambda}{r} \ \cdot \ 0.610, \ \ 1.12, \ \ 1.62, \ \ 2.12, \ \ 2.62, \ \ . . .<br />

Angular radius (angle between the central axis and the 1st dark ring) of Airy disk:

<br /> \theta_{Airy} \ = \ 1.22 \ \frac{\lambda}{d} <br />Airy disk radius for an imaging system:

<br /> r_{Airy} \ = \ 1.22 \ \lambda \ \frac{f}{d}<br />

Extended explanation

Definitions of terms
I = Power per area (irradiance) of the wave, with SI units of W/m2
I(0) = the irradiance at θ=0
r = the aperture radius
d = 2r = diameter of the aperture, lens, or mirror
λ = the wavelength of the wave
k = 2π/λ
θ = the angle at which the irradiance is evaluated
J1 = Bessel function of the first kind​

The Airy disk is the central bright spot of the diffraction pattern, within the 1st dark ring.

f and d are the focal length and diameter, respectively, of the lens or mirror in an imaging system.

The ratio f/d is the f/number of a lens or mirror. For example, an f/4 lens has f/d=4.

* This entry is from our old Library feature, and was originally created by Redbelly98.
 
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Thanks for the overview on circular apeture
 
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