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circular aperture
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Definition/Summary
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| This entry describes diffraction of a wave when it passes through a circular aperture. |
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Equations
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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] |
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Recent forum threads on circular aperture
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Breakdown
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Physics
> Classical Optics
>> Diffraction
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Extended explanation
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Definitions of termsI = 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. |
Commentary
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