Fig. 1.13 Ray propagation through a plane surface separating two media: Snell's law
Fig. 1.13 Ray propagation through a plane surface separating two media: Snell's law n sin i = n sin i. (1.6)
If n' < n and sini' = 1, we find i = arcsin (n'/n), that is i = 41.8° for n = 1.5 and n' = 1. For larger value of i, (1.6) cannot be satisfied, no refracted ray exists, the ray is reflected at the separating surface in the medium of higher refractive index; this total internal reflection has many applications in the design of prisms.
Denoting v and v' the velocity of light in these respective media, the sine ratio of the angles is equal to the velocity ratio (cf. Sect. 1.3), i.e.
v' v so we obtain n'v' = nv. By setting n = 1 for a vacuum medium, the velocities in each medium are cc v =-, and v' = —, (1.8)
and the identification with (1.5) gives Maxwell's formula for the refractive index: n = y/e]I, where n > 1 for any transparent medium, and conventionally n = 1 in a vacuum.
The dependance of the refractive index of glass on the wavelength has been represented by various possible functions. Some of them - known as Schott or Hertzberger formulas - are directly related to power series of the wavelength X. The Sellmeier-1 dispersion formula i ' K X2
with three couples (K, Li) of constant parameters, is used by Schott and some other glass manufacturers. Because the Sellmeier-1 formula does not have linear coefficients, the dispersion set up is iterative and thus, takes more computer time to fit the data than for a power series formula. Given a wavelength, an accuracy better than An/n = 10~5 is always achieved into a large spectral range. The Sellmeier-3 formula has a supplementary (K, L) couple if necessary.
Whatever the slope of the normal unit vector n at the intersection point of the surface boundary, if r and r' are the unit vectors along the incident and refracted rays, a vector product representation of Snell's law is
This relation ensures the local coplanarity between the two rays and the normal. For analytic ray tracing, one generally takes the convention with rays in the first medium coming from left to right towards a positive z axis. An additional sign convention is to consider that incident and refracted rays have their z direction cosines both represented by the same sign, whatever the sign of x, y direction cosines. Using a convenient convention, one can also treat the case of a reflected ray with (1.10) by replacing n' by -n.
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