Math 2.0

Optics, electromagnetic theory Geometrical optics Theorem

Malus–Dupin theorem

Builds on Optical path lengthNormal vectorSnell's lawLaw of reflection

Statement

Let a smooth family of light rays be orthogonal to a smooth surface, so that the rays are the normals of this surface. Let the rays undergo finitely many reflections and refractions at smooth surfaces between homogeneous isotropic media. These follow the law of reflection and Snell's law. Then the resulting family of rays is again orthogonal to a family of surfaces.

Remarks

The surfaces can be described explicitly. Mark on each ray the point at which a fixed optical path length from the original surface is reached. These points form a surface, and it is orthogonal to the rays. The proof shows that the first variation of the optical path length across neighbouring rays vanishes, using the laws of reflection and refraction at each interface.

Example: rays from a point source are normal to the spheres centred at the source. After reflection in a parabolic mirror with the source at the focus, they become parallel, and they are normal to the planes perpendicular to the axis.

A family of lines that are the normals of a surface is called a normal congruence. Not every two-parameter family of lines is a normal congruence, so the theorem has content.

Going deeper

For a student

The theorem is the ray-optics form of the fact that the rays are the gradient lines of a solution of the eikonal equation. It allows one to speak of wave fronts after an optical system and to define the wave aberration as the deviation of these fronts from spheres. In symplectic terms, a normal congruence corresponds to a Lagrangian submanifold of the space of rays, and optical systems act by symplectic maps that preserve this property.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Calculus of variations, concept, no evidence levelCalculus of variationsconceptEuler–Lagrange equation, concept, no evidence levelEuler–Lagrange equati…conceptGeodesic, concept, no evidence levelGeodesicconceptNormal vector, concept, no evidence levelNormal vectorconceptOptical path length, concept, no evidence levelOptical path lengthconceptReflection, concept, no evidence levelReflectionconceptRefractive index, concept, no evidence levelRefractive indexconceptFermat's principle, statement, no evidence levelFermat's principlestatementLaw of reflection, statement, no evidence levelLaw of reflectionstatementSnell's law, statement, no evidence levelSnell's lawstatementMalus–Dupin theorem, statement, no evidence levelMalus–Dupin theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Calculus of variations concept
  2. Euler–Lagrange equation concept
  3. Geodesic concept
  4. Normal vector concept
  5. Optical path length concept
  6. Reflection concept
  7. Refractive index concept
  8. Fermat's principle statement
  9. Law of reflection statement
  10. Snell's law statement
  11. Malus–Dupin theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Malus%E2%80%93Dupin_theorem
About this page
Tag83FKCHSY02
URIm2://83FKCHSY02
Kindstatement, theorem
WritingW0 stub
MSC78A05, 53A05
StatusThis copy of the site has no build data. Run m2 status 83FKCHSY02.
AboutOptical path length
UsesOptical path length, Normal vector, Snell's law, Law of reflection
Same as
Changed3 November 2026

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