Fisheye (optical system)

Maxwell’s beam path and fisheye imaging

Maxwell's Fisheye (1854) is an absolute optical system representing an inhomogeneous spherically symmetric medium characterized by a refractive index

$n(r)={\frac {n_{0}}{1+(r/a)^{2}}}$ ${\ displaystyle n (r) = {\ frac {n_ {0}} {1+ (r / a) ^ {2}}}}$ ${\ displaystyle n (r) = {\ frac {n_ {0}} {1+ (r / a) ^ {2}}}}$ ,

Where $r$ ${\ displaystyle r}$ $r$ - distance to the center of the system $O$ ${\ displaystyle O}$ $O$ , $n_{0}$ ${\ displaystyle n_ {0}}$ $n_ {0}$ and $a$ ${\ displaystyle a}$ $a$ - options.

Each ray represents a circle not passing through $O$ ${\ displaystyle O}$ $O$ , or a straight line passing through $O$ ${\ displaystyle O}$ $O$ . It is convenient to construct the image of a point created by the system on a direct ray: all rays from an arbitrary point $P_{0}$ ${\ displaystyle P_ {0}}$ $P_0$ gather at the point $P_{1}$ ${\ displaystyle P_ {1}}$ $P_1$ lying on a straight line that connects $P_{0}$ ${\ displaystyle P_ {0}}$ $P_0$ with $O$ ${\ displaystyle O}$ $O$ ; $P_{0}$ ${\ displaystyle P_ {0}}$ $P_0$ and $P_{1}$ ${\ displaystyle P_ {1}}$ $P_1$ located on opposite sides of $O$ ${\ displaystyle O}$ $O$ , and the product

$OP_{0}\cdot OP_{1}=a^{2}$ ${\ displaystyle OP_ {0} \ cdot OP_ {1} = a ^ {2}}$ ${\ displaystyle OP_ {0} \ cdot OP_ {1} = a ^ {2}}$ .

Therefore, the “fisheye” of Maxwell is an absolute optical system in which the mapping is performed by the inversion transform. Plane not passing through $O$ ${\ displaystyle O}$ $O$ is depicted by a sphere.

In this system, all aberrations are absent, except for distortion and curvature of the image field .

Literature

Born M. , Wolf E. Fundamentals of Optics. M.: 1973, S. 149-150.
JC Maxwell Solutions of Problems, problem no. 2. Cambridge and Dublin Math. Journal. Vol. 8, 1854. P. 188
The Scientific Papers of James Clerk Maxwell. Cambridge University Press, 1890, Dover, New York, 1965. P. 74-79

Fisheye (optical system)

See also

Literature

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