Airy Disk - Obscured Airy Pattern

Obscured Airy Pattern

Similar equations can also be derived for the obscured Airy diffraction pattern which is the diffraction pattern from an annular aperture or beam, i.e. a uniform circular aperture (beam) obscured by a circular block at the center. This situation is relevant to many common reflector telescope designs that incorporate a secondary mirror, including Newtonian telescopes and Schmidt–Cassegrain telescopes.

where is the annular aperture obscuration ratio, or the ratio of the diameter of the obscuring disk and the diameter of the aperture (beam)., and x is defined as above: where is the radial distance in the focal plane from the optical axis, is the wavelength and is the f-number of the system. The fractional encircled energy (the fraction of the total energy contained within a circle of radius centered at the optical axis in the focal plane) is then given by:

E(R) = \frac{1}{ (1 - \epsilon ^2) } \left( 1 - J_0^2(x) - J_1^2(x) + \epsilon ^2 \left - 4 \epsilon \int_0^x \frac {J_1(t) J_1(\epsilon t)}{t}\,dt \right)

For the formulas reduce to the unobscured versions above.

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