Tuesday, March 13, 2012

A Thin Spherical Lens

To understand image formation by a lens one needs to know how the refractive index of glass bends the rays of light along an optical path. Snell's law, n sin(θ) = n' sin(θ'), gives the relation between the angles of the path relative to the surface normal as light goes from one medium to another. Below a set of parallel rays pass from the left through a lens and come to a focus on the right. The lens consists of two identical spherical surfaces with a radius of curvature, ρ = 5 cm, with a thickness, t = 3 mm, which is the maximum distance between the surfaces. The light comes to a focus just beyond the center of curvature to the right marked by the blue dot.

The focal distance, f, computed from the Lensmaker's equation is marked by the cyan dot and the focal point of our thin lens is close to the theoretical value. Looking more closely at the rays coming from the lens one sees that the rays farther from the center converge more closely to the lens than those passing near its center showing what is known as spherical aberration. The majority of the rays come to a focus within 1 mm of the value computed from the formula and the flux of light is greatest there.

The vertical scale is exaggerated somewhat since the lens is a little over 2 cm in diameter and the width of the focal point is actually just a few tenths of a millimeter in diameter. Decreasing the aperature of the lens, the opening about the center through which light is allowed to pass, sharpens the focal point but decreases the amount of light going through it. P = 1/f is known as the converging power of the lens. In this case P = 20.0 diopters (1/#meters) since we were measuring distances in centimeters.*

*edit (Mar 14): Set n = 1.00 which is a better value for air and simplifies the results. Note that in this case the focal length equals the radius of curvature. The difference in position is due to the thickness of the lens.

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