1. Electromagnetic (E-M) theory of waves at a dielectric interface
While it is possible to understand reflection and refraction from Fermat’s principle, we need to use E-M theory in order to understand quantitatively the relationship between the incident, reflected, and transmitted radiant flux densities:
We can accomplish this treatment by assuming incident monochromatic light waves which form plane waves with well defined k-vectors as shown in the diagram. The interface is shown with an origin and coordinates (x,y,z). Ir i r t ni nt ûn x y . z b Ii It
We will consider E-field polarizations which are (i) in the plane of incidence and (ii) perpendicular to the plane of incidence, as shown below.

2. E-field is perpendicular to the plane-of incidence
E-field is parallel to the plane of incidence

3. Maxwell’s Equations for time-dependent fields in matter
D – Displacement field
H – Magnetic Intensity
P – Polarization
M – Magnetization
 - Magnetic permeability
 - Permittivity
e - Dielectric Susceptibility
m - Magnetic Susc.
g – Conductivity
j – Current density

4. Summary of the boundary conditions for fields at an interface
Side 1
Maxwell’s equations in integral form allow for the derivation of the boundary conditions for the total fields on both sides of a boundary.
Boundary
Side 2
Normal component of D is discontinuous by the free surface charge density
Tangential components of E are continuous
Normal components of B are continuous
Tangential components of H are discontinuous by the free surface current density

5. For dielectrics, j = 0. Therefore, the components of E and H that are tangent to the interface must be continuous across it. Since we have Ei, Er, and Et the continuity of E components yield:  
Note that i r ûn ni x y . z nt b t

6. Consider the expression on the interface (y = b) for all x, z and t
Consider the expression on the interface (y = b) for all x, z and t. The above relationship must hold at all points and at any instant in time on the interface. Therefore
Since then we have

7. Thus, at the interface plane
r cos 
which is again Snell’s Law

8. Case 1: E  Plane of incidence
Continuity of the tangential components of E and H give
Cosines cancel
Using H = -1 B, the tangential components are

9. The last two equations give
The symbol  means E  Plane of incidence. These are called the Fresnel equations; most often i  t  o.
Let r = amplitude reflection coefficient and t = amplitude transmission coefficient. Then, the Fresnel equations appear as
Note that t - r = 1

10. Case 2: E || Plane of Incidence
Continuity of the tangential components of E:
Continuity of the tangential components of -1 B:

11. If both media forming the interface are non-magnetic i  t  o then the amplitude coefficients become
Using Snell’s law
the Fresnel Equations for dielectric media become
Note that t - r = 1 holds for all i , whereas t|| + r|| = 1 is only true for normal incidence, i.e., i = 0.

12. Consider limiting cases for nearly normal incidence: i  0.
In which case, we have:
since
Also, using the following identity with Snell’s law
Therefore, the amplitude reflection coefficient can be written as:

13. In the limiting case for normal incidence i=t = 0, we have :
Note that these equalities occur for near normal incidence as a consequence of the fact that the plane of incidence is no longer specified when i  t  0.
Consider a specific example of an air-glass interface: i
ni = 1
We will consider a particular angle called the Brewster’s angle: p + t = 90 t
nt = 1.5
External reflection nt > ni Internal reflection ni > nt
At the polarization angle p, only the component of light polarized normal to the incident plane and therefore parallel to the surface will be reflected.

14. External Reflection (nt > ni) Internal Reflection (nt < ni)

15. Concept of Phase Shifts () in E-M waves:
Since
when nt > ni and t < i
as in the Air  Glass interface,
we expect a reversal of sign in the electric field for the Ecase when
We need to define phase shift for two cases:
When two fields E or B are  to the plane of incidence, they are said to be (i) in-phase (=0) if the two E or B fields are parallel and (ii) out-of-phase ( = ) if the fields are anti-parallel.
When two fields E or B are parallel to the plane of incidence, the fields are (i) in-phase if the y-components of the field are parallel and (ii) out-of-phase if the y-components of the field are anti-parallel.

16. Examples of Phase shifts for two particular cases:
(b)

17. Analogy between a wave on a string and an E-M wave traversing the air-glass interface.
Glass (n = 1.5) Air (n = 1)
 =  = 0
Air (n = 1) Glass (n = 1.5)
 =   = 0
Compare with the case of

18. Examples of phase-shifts using our air-glass interface:
In order to understand these phase shifts, it’s important to understand the definition of .

19. Reflected E-field orientations at various angles for the case of External Reflection (ni < nt). It is worth checking and comparing with the various plots for the phase shift  on the previous slides.

20. Reflected E-field orientations at various angles for the case of Internal Reflection (ni > nt). It is worth checking and comparing with the various plots for the phase shift  on the previous slides.

21. Reflectance and Transmittance
Remember that the power/area crossing a surface in vacuum (whose normal is parallel to the Poynting vector) is given by
The radiant flux density or irradiance (W/m2) is
Phase velocity
From the geometry and total area A of the beam at the interface, the power (P) for the (i) incident, (ii) reflected and (ii) transmitted beams are:

22. Define Reflectance and Transmittance:
Note that
Conservation of Energy at the interface yields:

23. Therefore,
We can write this expression in the form of componets  and ||:
We must use the previously calculated values for

24. It’s possible to verify for the special case of normal incidence:
Consider the case of Total Internal Reflection (TIR): t
nt = 1
ni = 1.5 i