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, and R_{1} and R_{2} are the radii of curvature of the two surfaces. Here R_{1} 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: R_{2} 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 R_{1} or R_{2}). 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 , where is the angle of incidence on the interface and is the angle of refraction.
For the second surface, , where is the angle of incidence and is the angle of refraction.
For small angles, . The geometry of the problem then gives:
If the incoming ray is parallel to the optical axis and distance from it, then
Substituting into the expression above, one gets
This ray crosses the optical axis at distance , given by
Combining the two expressions gives .
It can be shown that if two such lenses of radii and 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 s_{o} and the image distance s_{i} (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]}
Parameter | Meaning | + Sign | - Sign |
---|---|---|---|
s_{o} | The distance between an object and a lens. | Real object | Virtual object |
s_{i} | The distance between an image and a lens. | Real image | Virtual image |
f | The focal length of a lens. | Conversing lens | Diverging lens |
y_{o} | The height of an object from the optical axis. | Erect object | Inverted object |
y_{i} | The height of an image from the optical axis | Erect image | Inverted image |
M_{T} | The transverse magnification in imaging (= the ratio of y_{i} to y_{o}). | Erect image | Inverted image |
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]}