Editing Refraction (section)

===Bending of light===

Consider a wave going from one material to another where its speed is slower as in the figure. If it reaches the interface between the materials at an angle one side of the wave will reach the second material first, and therefore slow down earlier. With one side of the wave going slower the whole wave will pivot towards that side. This is why a wave will bend away from the surface or toward the [[Normal (geometry)|normal]] when going into a slower material. In the opposite case of a wave reaching a material where the speed is higher, one side of the wave will speed up and the wave will pivot away from that side.

Another way of understanding the same thing is to consider the change in wavelength at the interface. When the wave goes from one material to another where the wave has a different speed {{mvar|v}}, the [[frequency]] {{mvar|f}} of the wave will stay the same, but the distance between [[wavefront]]s or [[wavelength]] {{math|1= ''λ'' = ''v''/''f''}} will change. If the speed is decreased, such as in the figure to the right, the wavelength will also decrease. With an angle between the wave fronts and the interface and change in distance between the wave fronts the angle must change over the interface to keep the wave fronts intact. From these considerations the relationship between the [[Angle of incidence (optics)|angle of incidence]] {{math|''θ''{{sub|1}}}}, angle of transmission {{math|''θ''{{sub|2}}}} and the wave speeds {{math|''v''{{sub|1}}}} and {{math|''v''{{sub|2}}}} in the two materials can be derived. This is the [[law of refraction]] or Snell's law and can be written as<ref name="Hecht">{{cite book|author=Hecht, Eugene|title=Optics|publisher=Addison-Wesley|year=2002|isbn=0-321-18878-0|page=101}}</ref>

<math display="block">\frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} \,.</math>

The phenomenon of refraction can in a more fundamental way be derived from the 2 or 3-dimensional [[wave equation]]. The boundary condition at the interface will then require the tangential component of the [[wave vector]] to be identical on the two sides of the interface.<ref>{{cite web|url=https://www.rp-photonics.com/refraction.html|title=Refraction|author=<!--Not stated-->|website=RP Photonics Encyclopedia|publisher=RP Photonics Consulting GmbH, Dr. Rüdiger Paschotta|access-date=2018-10-23|quote=It results from the boundary conditions which the incoming and the transmitted wave need to fulfill at the boundary between the two media. Essentially, the tangential components of the wave vectors need to be identical, as otherwise the phase difference between the waves at the boundary would be position-dependent, and the wavefronts could not be continuous. As the magnitude of the wave vector depends on the refractive index of the medium, the said condition can in general only be fulfilled with different propagation directions.}}</ref> Since the magnitude of the wave vector depend on the wave speed this requires a change in direction of the wave vector.

The relevant wave speed in the discussion above is the [[phase velocity]] of the wave. This is typically close to the [[group velocity]] which can be seen as the truer speed of a wave, but when they differ it is important to use the phase velocity in all calculations relating to refraction.

A wave traveling perpendicular to a boundary, i.e. having its wavefronts parallel to the boundary, will not change direction even if the speed of the wave changes.