Optics

Geometrical Optics
For a spherical plano convex lens, if the radius of the spherical surface is R, the thickness of the lens is h, the diameter of the lens is r:

R^2-r^2=(R-h)^2

R=(r^2+h^2)/2h=r^2/2h+h/2

If the refractive index of the lens material is n, then the focal length of the the lens is:

f=R/(n-1)

This is true only for the central part of the lens. The further away from the centre, the refracted rays would be more bent and intersect with the lens axis closer to the lens (spherical aberration).

If two thin lens (f1 and f2) are placed together with a distance d between them, the resulting focal length is

1/f=1/f1+1/f2-d/(f1*f2)

If d=0, then 1/f=1/f1+1/f2

Interception of a ray parallel with lens axis after passing a spherical plano convex lens
R: radius of the the spherical surface

h: distance between the ray and the axis of the lens

n: refractive index of the lens

f= [sqrt(R^2+h^2) + n*sqrt(R^2-n^2*h^2)]/(n^2-1)

Apparently, in order for the second sqrt to be real, R > n*h. In a real lens, this means that when the distance of the ray is farther than R/n, total inflexion will disallow the ray from entering or exiting the lens.

When h->0: f=(R+nR)/(n^2-1)=R/(n-1). This is the thin lens approximation.

Interesting links
http://www.physicsinsights.org/simple_optics_spherical_lenses-1.html

http://www.feynmanlectures.caltech.edu/II_29.html

https://link.springer.com/content/pdf/10.1007%2F0-387-26016-1_2.pdf