The Lensmaker Equation
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The evolution of the derivation began with an assigned problem, namely to find the focal length BF for a plano-convex lens of index n if d, R (CE = CB), and n were given as numbers. When the law of sines showed up in the solution I thought about using it to find a general focal length and then to combine two such lenses to derive the lensmaker equation, as derivations are not provided in the standard texts. In the figure a is the angle of incidence on the curved surface and |
A light ray is incident in air along AE and is focused on the axis at F. CE is a normal to the curved surface, with C the center of the circular arc. |
b
the angle of refraction. <CEF, <ECF, and <CFE are gotten from geometry. Applying the law of sines to DCEF,
Solving for CF and using the expansion for 
plus the fact that sin(p - b) = sinb,
Since f = CF - CB = CF - R,
Applying Snell’s law, nsina = sinb, using sina = d/R and sinb = nd/R, and the identity
cosx = 
that results in 
, 
,
For thin lenses, d << R, so neglecting the second term in each radical as being "small" compared to the first,
.
Now the problem is to combine two such lenses with the same index and with radii R1 and R2, flat side to flat side, and derive the lensmaker equation. In this case the image formed by the first lens acts as a virtual object for the second. Using 1/si + 1/so = 1/f1 for the first and 
+
=1/f2 for the second (here so = -si, where si from the first is the virtual object distance for the second), adding the two yields
, or 
. Since 
and 
, substitution yields
.
