• Tags: cut-lens, combination, focal-length
• Difficulty: Advanced

When a lens is cut:

• Along the principal axis (horizontally): Each half retains the same focal length but has half the intensity. The aperture reduces.
• Perpendicular to the principal axis (vertically into two plano-convex lenses): Each piece has focal length 2f (since one surface becomes flat, R→∞). Using lens maker's equation for the plano-convex piece: 1/f' = (n-1)(1/R - 0) = (n-1)/R, which is half of the original 1/f = (n-1)(1/$R_{1}$ - 1/$R_{2}$) = (n-1)(2/R) for a symmetric biconvex lens.

When two identical plano-convex lenses are placed in contact: 1/f_combination = 1/f_{1} + 1/f_{2}. If both have focal length 2f, the combination has focal length f (recovering the original biconvex lens).