Circle of least confusion

Circle of least confusion: because of chromatic aberration, a lens focuses violet light at F_V and red light at F_R. A screen placed anywhere between them shows a coloured disc rather than a point. The smallest such disc is the circle of least confusion.

 marginal rays cross at X, between F_V and F_R

  lens        F_V      X      F_R
   |           .      ||      .
 --|-----------\-----||-----/---> axis
   |            (circle of
   |             diameter d at X)

Derivation: let the lens aperture diameter be D = 2h, with focal lengths f_V (violet) and f_R (red), where f_R − f_V = ω f.

1. The marginal violet ray starts at height h and converges to F_V. At distance x from the lens its height is h(1 − x/f_V). Beyond F_V this is negative (the ray has crossed the axis).
2. The marginal red ray's height at distance x is h(1 − x/f_R), still positive for x < f_R.
3. The narrowest section lies where these two marginal rays have equal and opposite heights: −h(1 − x/f_V) = h(1 − x/f_R) x (1/f_V + 1/f_R) = 2, so x = 2 f_V f_R / (f_V + f_R).
4. Radius of the circle at this place: r = h(1 − x/f_R) = h (1 − 2f_V/(f_V + f_R)) = h (f_R − f_V)/(f_R + f_V).
5. Diameter: d = 2r = D (f_R − f_V)/(f_R + f_V).
6. Substituting f_R − f_V = ω f and f_R + f_V ≈ 2f:

d = D ω f / 2f = ω D / 2

The focal length cancels. The diameter of the circle of least confusion depends only on the aperture D of the lens and the dispersive power ω of its material, and not on the focal length. It is reduced by using a smaller aperture (stop) or a material of lower dispersive power.