homogeneous, isotropic, non-magnetic media, the coefficients are given by: (For a derivation of the above, see Fresnel equations §Theory.) Now we suppose that the transmitted wave is evanescent. With the correct sign (+), substituting () into () gives where that is, is the index of the "internal" medium relative to the "external" one, or the index of the internal medium if the external one is vacuum. So the magnitude of is 1, and the argument of is which gives a phase advance of Making the same substitution in (), we find that has the same denominator as with a positive real numerator (instead of a complex conjugate numerator) and therefore has half the argument of so that the phase advance of the evanescent wave is half that of the reflected wave. With the same choice of sign, substituting () into () gives whose magnitude is 1, and whose argument is which gives a phase advance of Making the same substitution in (), we again find that the phase advance of the evanescent wave is half that of the reflected wave. Equations () and () apply when where θ is the angle of incidence and θc is the critical angle . These equations show that each phase advance is zero at the critical angle (for which the numerator is zero); each phase advance approaches 180° as ; and at intermediate values of θ because the factor is in the numerator of () and the denominator of (). For the reflection coefficients are given by equations () and (), and are real, so that the phase shift is either 0° (if the coefficient is positive) or 180° (if the coefficient is negative). In (), if we put (Snell's law) and multiply the numerator and denominator by , we obtain which is positive for all angles of incidence with a transmitted ray (since ), giving a phase shift of zero. If we do likewise with (), the result is easily shown to be equivalent to which is negative for small angles (that is, near normal incidence), but changes sign at Brewster's angle, where θ and θt are complementary. Thus the phase shift is 180° for small θ but switches to 0° at Brewster's angle. Combining the complementarity with Snell's law yields as Brewster's angle for dense-to-rare incidence. Equations () and () are known as Fresnel's sine law and Fresnel's tangent law. Both reduce