Thin lens

 ( Lenses ) 

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.

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Focal length The focal length, f, of a lens in air is given by the lensmaker's equation:

$\backslash frac\{1\}\{f\}\; =\; (n-1)\; \backslash left,$

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

$\backslash frac\{1\}\{f\}\; \backslash approx\; \backslash left(n-1\backslash right)\backslash left.$

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Derivation using Snell's law Consider a thin lens with a first surface of radius $R$ and a flat rear surface, made of material with index of refraction $n$.

Applying Snell's law, light entering the first surface is refracted according to $\backslash sin\; i\; =\; n\; \backslash sin\; r\_1$, where $i$ is the angle of incidence on the interface and $r\_1$ is the angle of refraction.

For the second surface, $n\; \backslash sin\; r\_2\; =\; \backslash sin\; e$, where $r\_2$ is the angle of incidence and $e$ is the angle of refraction.

For small angles, $\backslash sin\; x\; \backslash approx\; x$. The geometry of the problem then gives:

If the incoming ray is parallel to the optical axis and distance $h$ from it, then $$\backslash sin\; i\; =\; \backslash frac\; hR\; \backslash implies\; i\; \backslash approx\; \backslash frac\; hR.$$

Substituting into the expression above, one gets $$e\; \backslash approx\; \backslash frac\; \{h\}\{R\}\; (n-1).$$

This ray crosses the optical axis at distance $f$, given by $$\backslash tan\; e\; =\; \backslash frac\; hf\; \backslash implies\; e\; \backslash approx\; \backslash frac\; hf$$

Combining the two expressions gives $\backslash frac\; 1f\; =\; \backslash frac\; 1R\; (n\; -\; 1)$.

It can be shown that if two such lenses of radii $R\_1$ and $-R\_2$ are placed close together, the inverses of the focal lengths can be added up giving the thin lens formula:

$$\backslash frac\; 1f\; =\; \backslash left(n-1\backslash right)\backslash left(\; \backslash frac\; 1\{R\_1\}\; -\; \backslash frac\; 1\{R\_2\}\; \backslash right)$$

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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 $f\_2$ on the other side.
• Any ray that arrives at the lens after passing through the focal point $f\_1$ 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.

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

$\{1\backslash over\; s\_o\}\; +\; \{1\backslash over\; s\_i\}\; =\; \{1\backslash over\; f\}$

which is known as the Gaussian thin lens equation, which sign convention is the following.

Hecht Eugene (2025). 9781292096933, Pearson. ISBN 9781292096933

+Sign convention for Gaussian lens equation !Parameter !Meaning !+ Sign !- Sign
so	The distance between an object and a lens.	Real object	Virtual object
si	The distance between an image and a lens.	Real image	Virtual image
f	The focal length of a lens.	Converging lens	Diverging lens
yo	The height of an object from the optical axis.	Erect object	Inverted object
yi	The height of an image from the optical axis	Erect image	Inverted image
MT	The transverse magnification in imaging (= the ratio of yi to yo).	Erect image	Inverted image

There are other sign conventions such as Cartesian sign convention where the thin lens equation is written as$$\{1\backslash over\; s\_o\}\; +\; \{1\backslash over\; f\}\; =\; \{1\backslash over\; s\_i\}.$$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.

Eugene Hecht (2025). 9781292096933, Pearson. ISBN 9781292096933

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Physical optics 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:

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

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