Total internal reflection

Total internal reflection

The formulae describing the reflectance and transmittance whether for TM incidence
or TE incidence, were obtained by assuming that the light is incident from a less dense
medium (1) to a denser medium (2), or that the refractive index of medium (1) is
lower that the refractive index of medium (2) (n1 < n2). This is called soft incidence
(formulae (2.110), (2.137), (2.138)). From Snell’s law (2.110), if n1 < n2 holds, it is
easy to show that, regardless of the incident angle θi, the refracted angle always will
exist, or in other words, the refracted angle θt, will always be a real number.
In contrast, if the plane wave is incident from a denser to a less dense medium
(n1 > n2, hard incidence) an exceptional phenomenon takes place for a certain range
of incident angles, for which the formulae formerly given for R and T can no longer
be applied. For hard incidence (n1 > n2) there exists an incident angle θi for which
the refracted angle θt takes the value of π/2 radians. This angle is called the critical
angle θc, and its value, calculated directly from Snell’s law is:
θc ≡ sin−1(n2/n1) (2.140)
For incident angles higher than the critical angle, the sine of the refracted angle will
reach values greater than 1, thus the refracted angle is no longer a real number according
to Snell’s law. Nevertheless, this does not imply that in medium (2) there is no
transmitted wave, as we will show.
In order to calculate the reflectivity in a case of hard incidence, it is necessary to
evaluate cos θt included in the formulae for the reflection and transmission coefficients.
According to Snell’s law, it follows that:
cos θt = −(1 − sin2 θt)1/2 (2.141)
where the negative sign of the squared root has been chosen so that the complete
expression for the electric field of the transmitted wave has a correct physical meaning.
Taking into account that now sin θt > 1, the last formula can be expressed as:
cos θt = −i(sin2 θt − 1)1/2 = −iB (2.142)
where the magnitude B has been defined as a real number by:
B ≡ (sin2 θt − 1)1/2 = (n21sin2 θi/n22− 1)1/2