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What is circular aperture

  1. Jul 24, 2014 #1
    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:

    [tex]
    I(\theta) = I(0) \left( \frac{2 J_1(k \ r \ \sin\theta)}{k \ r \ \sin\theta} \right)^2
    [/tex]


    The angular radii of the first 4 dark rings in the diffraction pattern occur at

    [tex]
    k \ r \ \sin\theta \ \approx \ 3.8317, \ \ 7.0156, \ \ 10.173, \ \ 13.324
    [/tex]

    where the numbers are the zeros of [itex]J_1[/itex]

    Equivalently, in terms of wavelength the zeroes are at

    [tex]
    \sin\theta \ \approx \frac{\lambda}{r} \ \cdot \ 0.610, \ \ 1.12, \ \ 1.62, \ \ 2.12, \ \ 2.62, \ \ . . .
    [/tex]

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

    [tex]
    \theta_{Airy} \ = \ 1.22 \ \frac{\lambda}{d}
    [/tex]


    Airy disk radius for an imaging system:

    [tex]
    r_{Airy} \ = \ 1.22 \ \lambda \ \frac{f}{d}
    [/tex]

    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.
     
    Last edited by a moderator: Jul 27, 2014
  2. jcsd
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