Lens Focal Length Principle

The focal length of a lens is determined by the curvature of its optical surfaces and the refractive index of the material. It is independent of the lens's size or aperture, provided the light rays interact with the relevant curved surfaces.

Effect of Cutting Lens in Half

Consider a symmetrical biconvex lens with radii of curvature \(R\) and \(-R\), and focal length \(f\). The lens maker's formula gives:

\(\frac{1}{f} = (\mu - 1) \left( \frac{1}{R} - \frac{1}{-R} \right) = \frac{2(\mu-1)}{R}\)

When this lens is cut into two identical halves along its principal axis, each half becomes a plano-convex lens. This plano-convex lens has one flat surface (radius of curvature \(\infty\)) and one curved surface (radius of curvature \(R\)). Its focal length, \(f_{half}\), is given by:

\(\frac{1}{f_{half}} = (\mu - 1) \left( \frac{1}{R} - \frac{1}{\infty} \right) = \frac{\mu-1}{R}\)

Comparing the two equations:

\(\frac{1}{f_{half}} = \frac{1}{2} \left( \frac{2(\mu-1)}{R} \right) = \frac{1}{2} \left( \frac{1}{f} \right)\)

Therefore, the focal length of each half-lens is:

\(f_{half} = 2f\)

Focal Length of Quarter Pieces

The original lens is cut into four identical pieces along *and* perpendicular to its principal axis. This means the two halves (obtained by cutting along the axis) are further cut perpendicular to the axis, passing through the center.

• Each resulting piece is a smaller segment of the original lens.
• Crucially, each piece retains a portion of the original spherical curvature (with radius \(R\)).
• The cuts introduce flat surfaces, but these do not alter the fundamental refractive property determined by the original spherical curvature.
• Each quarter piece effectively behaves optically like the half-piece (plano-convex), maintaining the same relationship between curvature and focal length.

Thus, the focal length of each of the four identical pieces remains \(2f\).