Sagitta (optics)

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Deep blue ray refers the radius of curvature and the red line segment is the sagitta of the curve (black).

In optics and especially telescope making, sagitta or sag is a measure of the glass removed to yield an optical curve. It is approximated by the formula

S ( r ) ≈ r 2 2 × R {\displaystyle S(r)\approx {\frac {r^{2}}{2\times R}}} {\displaystyle S(r)\approx {\frac {r^{2}}{2\times R}}},

where R is the radius of curvature of the optical surface. The sag S(r) is the displacement along the optic axis of the surface from the vertex, at distance r {\displaystyle r} {\displaystyle r} from the axis.

A good explanation of both this approximate formula and the exact formula can be found here.

Aspheric surfaces

Optical surfaces with non-spherical profiles, such as the surfaces of aspheric lenses, are typically designed such that their sag is described by the equation

S ( r ) = r 2 R ( 1 + 1 − ( 1 + K ) r 2 R 2 ) + α 1 r 2 + α 2 r 4 + α 3 r 6 + ⋯ . {\displaystyle S(r)={\frac {r^{2}}{R\left(1+{\sqrt {1-(1+K){\frac {r^{2}}{R^{2}}}}}\right)}}+\alpha _{1}r^{2}+\alpha _{2}r^{4}+\alpha _{3}r^{6}+\cdots .} {\displaystyle S(r)={\frac {r^{2}}{R\left(1+{\sqrt {1-(1+K){\frac {r^{2}}{R^{2}}}}}\right)}}+\alpha _{1}r^{2}+\alpha _{2}r^{4}+\alpha _{3}r^{6}+\cdots .}

Here, K {\displaystyle K} {\displaystyle K} is the conic constant as measured at the vertex (where r = 0 {\displaystyle r=0} {\displaystyle r=0}). The coefficients α i {\displaystyle \alpha _{i}} {\displaystyle \alpha _{i}} describe the deviation of the surface from the axially symmetric quadric surface specified by R {\displaystyle R} {\displaystyle R} and K {\displaystyle K} {\displaystyle K}.[1]

See also

References

  1. Barbastathis, George; Sheppard, Colin. "Real and Virtual Images" (PDF). MIT OpenCourseWare. Massachusetts Institute of Technology. p. 4. Retrieved 8 August 2017.