Monday, 30 September 2019

refraction - Snell's law in vector form


Snell's law of refraction at the interface between 2 isotropic media is given by the equation: n1sinθ1=n2sinθ2

where θ1 is the angle of incidence and θ2 the angle of refraction. n1 is the refractive index of the optical medium in front of the interface and n2 is the refractive index of the optical medium behind the interface.



How can this be expressed in vector form: n1(i×n)=n2(t×n)

where i(ix,iy and iz) and t(tx,ty,tz) are the unit directional vector of the incident and transmitted ray respectively. n(nx,ny,nn) is the unit normal vector to the interface between the two media pointing from medium 1 with refractive index n1 into medium 2 with refractive index n2.


Further more how can the Snell's law of refraction be expressed1 in the following way? t=μi+n1μ2[1(ni)2]μn(ni)

Here μ=n1n2 and ni=nxix+nyiy+nziz denotes the dot (scalar) product of vectors n and i.


References:



  1. Antonín Mikš and Pavel Novák, Determination of unit normal vectors of aspherical surfaces given unit directional vectors of incoming and outgoing rays: comment, 2012 Optical Society of America, page 1356




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