Thin lens

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.

The focal length, f, of a lens in air is given by the lensmaker's equation: 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] Consider a thin lens with a first surface of radius

and a flat rear surface, made of material with index of refraction

Applying Snell's law, light entering the first surface is refracted according to

is the angle of incidence on the interface and

is the angle of refraction.

The geometry of the problem then gives:

If the incoming ray is parallel to the optical axis and distance

This ray crosses the optical axis at distance

Combining the two expressions gives

It can be shown that if two such lenses of radii

are placed close together, the inverses of the focal lengths can be added up giving the thin lens formula:

Certain rays follow simple rules when passing through a thin lens, in the paraxial ray approximation: If three such rays are traced from the same point on an object in front of the lens (such as the top), their intersection will mark the location of the corresponding point on the image of the object.

By following the paths of these rays, the relationship between the object distance so and the image distance si (these distances are with respect to the lens) can be shown to be which is known as the Gaussian thin lens equation, which sign convention is the following.

[2] There are other sign conventions such as Cartesian sign convention where the thin lens equation is written as

For a thick lens, the same form of lens equation is applicable with the modification that parameters in the equation are with respect to principal planes of the lens.

[3] In scalar wave optics, a lens is a part which shifts the phase of the wavefront.

Mathematically this can be understood as a multiplication of the wavefront with the following function:[4]