Math 2.0

Optics, electromagnetic theory Definition

Negative refraction

Builds on Refractive indexPermittivityPermeabilitySnell's lawPoynting vectorMetamaterial

Definition

Let a plane wave pass through a flat interface from a medium of refractive index n1>0n_1 > 0 into a second medium. Let the second medium have relative permittivity ε2\varepsilon_2 and relative permeability μ2\mu_2. If ε2<0\varepsilon_2 < 0 and μ2<0\mu_2 < 0, the refractive index of the second medium is n2=−ε2μ2<0n_2 = -\sqrt{\varepsilon_2 \mu_2} < 0. The angles of incidence θ1\theta_1 and of refraction θ2\theta_2, measured from the normal and signed by the side, obey Snell's law n1sin⁡θ1=n2sin⁡θ2n_1 \sin\theta_1 = n_2 \sin\theta_2. Negative refraction is the case θ2<0\theta_2 < 0 for θ1>0\theta_1 > 0. Then the refracted ray, the direction of the Poynting vector, lies on the same side of the normal as the incident ray.

Remarks

In a medium with ε<0\varepsilon < 0 and μ<0\mu < 0, the vectors 𝐄\mathbf{E}, 𝐇\mathbf{H}, 𝐤\mathbf{k} form a left-handed triple, so the wave vector points against the Poynting vector. The choice of the negative square root for n2n_2 follows from this and from causality.

Example: with n1=1n_1 = 1, ε2=μ2=−1\varepsilon_2 = \mu_2 = -1 and θ1=30∘\theta_1 = 30^\circ, one gets n2=−1n_2 = -1 and sin⁡θ2=−1/2\sin\theta_2 = -1/2, so θ2=−30∘\theta_2 = -30^\circ. The refracted ray is the mirror image of the continued incident ray in the normal.

Non-example: a medium with ε2<0\varepsilon_2 < 0 and μ2>0\mu_2 > 0, such as a metal below its plasma frequency, has imaginary index and supports no propagating wave. It fails the condition that both parameters are negative.

Such media are made as metamaterials.

Reading path

To understand this page from the ground up, read these 12 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 levelGeodesicconceptMetamaterial, concept, no evidence levelMetamaterialconceptNormal vector, concept, no evidence levelNormal vectorconceptPermeability, concept, no evidence levelPermeabilityconceptPermittivity, concept, no evidence levelPermittivityconceptPoynting vector, concept, no evidence levelPoynting vectorconceptRefractive index, concept, no evidence levelRefractive indexconceptFermat's principle, statement, no evidence levelFermat's principlestatementSnell's law, statement, no evidence levelSnell's lawstatementNegative refraction, definition, no evidence levelNegative refractiondefinition
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. Metamaterial concept
  5. Normal vector concept
  6. Permeability concept
  7. Permittivity concept
  8. Poynting vector concept
  9. Refractive index concept
  10. Fermat's principle statement
  11. Snell's law statement
  12. Negative refraction definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Negative_refraction
About this page
Tag5B78SXJPTN
URIm2://5B78SXJPTN
Kinddefinition
WritingW0 stub
MSC78A48, 78A10
DefinesNegative refraction
UsesRefractive index, Permittivity, Permeability, Snell's law, Poynting vector, Metamaterial
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.