Refraction:

Consider a plane wave with k i that is incident on a dielectric-dielectric interface.  The wave will be reflected and refracted.  The interface has media on the two sides with indices n1 and n2, respectively, and is assumed to be smooth and have no charge or current.  The refracted wave is the transmitted part and will propagate in media #2 with a propagation vector k t.  In general, the direction of the refracted wave is different from the incident wave, but it will lie in the plane of incidence.  The p-polarized component of the incident wave will produce the p-polarized component of the refracted wave and the s-polarized component of the incident wave will produce the like component of the refracted wave.

 

Direction - Snell’s Law

n1 sin ¿ 1 = n2 sin ¿ 2

 

where n1 and n2 are the indices of the media and ¿ 1 and ¿ 2 are angles of k i and k t, respectively, with respect to the interface normal.

 

Magnitude

The complex magnitude of the phasors for the fields will be determined by the application of electromagnetic boundary conditions.  Consider an incident electric field of 

E i(r) = |Ei| exp(+jk i•r + jT i ) ae  
with phasor
At = |Ei| exp( + jT i )

and a refracted field of

E t (r) = |Et| exp(+jk t•r + jT t ) ae 
 with phasor
At = |Et| exp( + jTt).

 

The refraction coefficients are

tP = AtP/AiP = (2n1 cos¿ 1) / (n2 cos¿ 1 + n1 cos¿ 2)

tS = AtS/AiS = (2n1 cos¿ 1) / (n1 cos¿ 1 + n2 cos ¿ 2 )

Note that the refraction coefficients would take a different form if the ratio of irradiance values were desired.

 

Reflection:

Again consider a plane wave with k i that is incident on a dielectric-dielectric interface.  In addition to refraction, the wave will be reflected in a specular or mirror-like fashion.  The reflected wave will propagate in media #1 with a propagation vector k r.  It will lie in the plane of incidence. The p-polarized component of the incident wave will produce the p-polarized component of the reflected wave and the s-polarized component of the incident wave will produce the like component of the reflected wave.

 

Direction

 

¿ 1 = ¿ incidence = ¿ reflection

 

where ¿ 1 is the angle of k i and k r with respect to the interface normal.

 

Magnitude

The complex magnitude of the phasors for the fields will be determined by the application of electromagnetic boundary conditions.  Consider an incident electric field of 

E i (r) = |Ei| exp(+jk i•r + jT i ) ae 
 with phasor
Ai = |Ei| exp( + jT i )

and a reflected field of

E r (r) = |Er| exp(+jk r•r + jTr) ae 
 with phasor
Ar = |Er| exp( + jT r).

 

The reflection coefficients are

rP = ArP/AiP = (n2 cos ¿ 1 – n1 cos¿ 2) / (n2 cos ¿ 1 + n1 cos¿ 2)

rS = ArS/AiS = (n1 cos¿ 1 – n2 cos ¿ 2 ) / (n1 cos¿ 1 + n2 cos¿ 2)

Note that the reflection coefficients would take a different form if the ratio of irradiance values were desired.

Special Cases:

Inspection of the reflection and refraction coefficients in combination with Snell’s law reveals that total transmission and total reflection are possible.

 

Brewster’s Angle

Total transmission in which rP = 0 is possible for p-polarization. The incidence angle ¿ 1 for which this occurs is dependent on the indices of the media.  This Brewster’s angle ¿ 1 = ¿ B is obtained from the simultaneous solution of 

            n1 sin¿ B = n2 sin¿ 2    (Snell’s law)

            n2 cos ¿ B = n1 cos¿ 2   (from rP = 0).

 

Total transmission with no reflected part is not possible for s-polarization.

 

Total Internal Reflection

Total internal reflection in which |rP| = 1 and |rS| = 1 may occur only when n1 > n2.  It occurs for all incidence angles ¿ 1 for which

            n1 sin ¿ 1 > n2

(i.e. Snell’s law cannot be satisfied with a real angle).

 

Letting cos¿ 2 = v[1 – (n1 sin¿ 1 / n2 )2] (an imaginary quantity), the reflection coefficients become

rP = ArP/AiP
 = exp{-j2 tan-1 [(n1 2 / n2 2) v( n1 2 sin2 ¿ 1 - n2 2)/ v( n1 2– n1 2 sin2 ¿ 1)]}

rS = ArS/AiS
 = exp{-j2 tan-1 [v( n1 2 sin2 ¿ 1 - n2 2)/ v( n1 2– n1 2 sin2 ¿ 1)]}

 

Reflection and transmission can be greatly manipulated at an interface by thin films that form interference filters.  The preceding analysis applies for a bare interface.