How Does a Circular Aperture Affect Light Diffraction?

[Greg Bernhardt]

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:

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

$$k \ r \ \sin\theta \ \approx \ 3.8317, \ \ 7.0156, \ \ 10.173, \ \ 13.324$$

where the numbers are the zeros of $J_1$

Equivalently, in terms of wavelength the zeroes are at

$$\sin\theta \ \approx \frac{\lambda}{r} \ \cdot \ 0.610, \ \ 1.12, \ \ 1.62, \ \ 2.12, \ \ 2.62, \ \ . . .$$

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

$$\theta_{Airy} \ = \ 1.22 \ \frac{\lambda}{d}$$Airy disk radius for an imaging system:

$$r_{Airy} \ = \ 1.22 \ \lambda \ \frac{f}{d}$$

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.

 

[Aaron Paulo]

Thanks for the overview on circular apeture