Thin lens

A lens can be considered a thin lens if d << R₁ or d << R₂.

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:

$\frac{1}{f} = (n-1) \left[ \frac{1}{R_1} - \frac{1}{R_2} + \frac{(n-1)d}{n R_1 R_2} \right],$

where n is the index of refraction of the lens material, and R₁ and R₂ are the radii of curvature of the two surfaces. 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]

${\frac {1}{f}}\approx \left(n-1\right)\left[{\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}\right].$

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.

Image formation

Certain rays follow simple rules when passing through a thin lens, in the paraxial ray approximation:

• Any ray that enters parallel to the axis on one side of the lens proceeds towards the focal point F on the other side.
• Any ray that arrives at the lens after passing through the focal point on the front side, comes out parallel to the axis on the other side.
• Any ray that passes through the center of the lens will not change its direction.

By tracing these rays, the relationship between the object distance s and the image distance s′ can be shown to be

${1\over s} + {1\over s'} = {1\over f}$.

Physical optics

In scalar wave optics a lens is a part which shifts the phase of the wave-front. Mathematically this can be understood as a multiplication of the wave-front with the following function:

$\exp\left(\frac{2\pi i}{\lambda} \frac{r^2}{2f}\right)$.

References

1. Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. § 5.2.3. ISBN 0-201-11609-X.