In optics, a thin lens is a lens with a thickness (distance along the optical axis between the two surfaces of the lens) that is negligible compared to the radii of curvature of the lens surfaces. Lenses whose thickness is not negligible are sometimes called thick lenses. The thin lens approximation ignores optical effects due to the thickness of lenses and simplifies ray tracing calculations. It is often combined with the paraxial approximation in techniques such as ray transfer matrix analysis.
Focal length The focal length, f, of a lens in air is given by the lensmaker's equation:
1 f = ( n − 1 ) [ 1 R 1 − 1 R 2 + ( n − 1 ) d n R 1 R 2 ] , {\displaystyle {\frac {1}{f}}=(n-1)\left[{\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}+{\frac {(n-1)d}{nR_{1}R_{2}}}\right],}
where n is the index of refraction of the lens material, R1 and R2 are the radii of curvature of the two surfaces, and d is the thickness of the lens. Here R1 is taken to be positive if the first surface is convex, and negative if the surface is concave. The signs are reversed for the back surface of the lens: R2 is positive if the surface is concave, and negative if it is convex. This is an arbitrary sign convention; some authors choose different signs for the radii, which changes the equation for the focal length. For a thin lens, d is much smaller than one of the radii of curvature (either R1 or R2). In these conditions, the last term of the Lensmaker's equation becomes negligible, and the focal length of a thin lens in air can be approximated by
1 f ≈ ( n − 1 ) [ 1 R 1 − 1 R 2 ] . {\displaystyle {\frac {1}{f}}\approx \left(n-1\right)\left[{\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}\right].}
Derivation using Snell's law
Consider a thin lens with a first surface of radius R {\textstyle R} and a flat rear surface, made of material with index of refraction n {\textstyle n} . Applying Snell's law, light entering the first surface is refracted according to sin i = n sin r 1 {\displaystyle \sin i=n\sin r_{1}} , where i {\displaystyle i} is the angle of incidence on the interface and r 1 {\displaystyle r_{1}} is the angle of refraction. For the second surface, n sin r 2 = sin e {\displaystyle n\sin r_{2}=\sin e} , where r 2 {\displaystyle r_{2}} is the angle of incidence and e {\displaystyle e} is the angle of refraction. For small angles, sin x ≈ x {\textstyle \sin x\approx x} . The geometry of the problem then gives:
e ≈ n r 2 = n ( i − r 1 ) ≈ n ( i − i n ) {\displaystyle {\begin{aligned}e&\approx nr_{2}\\&=n(i-r_{1})\\&\approx n(i-{\frac {i}{n}})\end{aligned}}}
If the incoming ray is parallel to the optical axis and distance h {\textstyle h} from it, then
sin i = h R ⟹ i ≈ h R . {\displaystyle \sin i={\frac {h}{R}}\implies i\approx {\frac {h}{R}}.}
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