Aplanatic points

Contributor: Stas Fainer

Aplanatic points of an optical system are special points on its optical axis, such that "rays proceeding from one of them will all converge to, or seen to diverge from the other point".

  • Ellipse: the two foci of the elliptical lens/mirror are aplanatic points, since light emitted from one focus will converge towards the other.
  • Sphere: a spherical lens has two aplanatic points, inside and outside the sphere - for more details see the simulation.
  • Hyperbola: the two foci of the Hyperbolic mirror simulation are also aplanatic points.

Given two points with horizontal coordinates x1 and x2, identical vertical coordinates, and given the refractive index outside and inside our optical element as n1 and n2 (respectively), for this two points to be aplanatic points, the boundary of our optical element must fulfillk1n1(xx1)2+y2+k2n2(xx2)2+y2=Esuch that ki=1 or 1 if the ray connecting xi and the boundary of our optical element is real or imaginary, respectively, and E is a constant for which this equation has a non-trivial solution. This equation (which can be derived using Fermat's principle) is an equation of a Cartesian oval, of which the conic sections are special cases.

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Aplanatic points