Assume a plane wave is incident on a plane interface between two dielectric media.  Let the interface lie in the z = 0 plane.  Rotate the coordinate system so that the k vector lies in the y = constant planes.

At the boundary, the plane wave will be partially transmitted and partially reflected.  Let

E_r(r,t) = E_r exp(i(k_r×r-wt)) denote the reflected wave,

E_t(r,t) = E_t exp(i(k_t×r-wt)) denote the transmitted wave.

We want to find E_r, E_t, k_r, and k_t as a function of E_i and k_i.  The frequency of the incident, reflected and transmitted wave will be the same.

For dielectric-dielectric interfaces Maxwell’s equations yield the following boundary conditions.

Let us consider plane waves in an lih medium.  Then E_i ^ k_i.  We may write E_i = E_ip p + E_is s, where p is a unit vector perpendicular to k_i in the plane of incidence and s is a unit vector perpendicular to k_i and perpendicular to the plane of incidence.  The plane of incidence is the plane containing k_i and the vector normal to the interface.  In our example it is a y = constant plane.

The behavior of a plane wave at a boundary depends on its polarization.  We can treat p-polarization and s-polarization separately and then use the principle of superposition to treat arbitrary polarization.

p-polarization

A wave incident on an interface with p-polarization is called a transverse magnetic (TM) wave.  Assume the wave is incident on an interface between medium 1 and medium 2 from medium 1.  Then

E₁ = E_i + E_r,    E₂ = E_t.

Let E_i, E_r, and E_t denote the magnitudes of the field vectors and assume that they point into the directions shown in the figure above.

_______________________________________

The tangential component of E is continuous at z = 0.

Let . Then

E₂× = E_t×(-) = E_t exp(i(k_t×r-wt)) cos(q_t)

E₁× = E_i×(-) + E_r×(-) = E_i exp(i(k_i×r-wt)) cos(q_i) – E_r exp(i(k_r×r-wt)) cos(q_r)

At z = 0 we have E_tx = E_ix + E_rx, or

E_t exp(i(k_t×r)) cos(q_t) = E_i exp(i(k_i×r)) cos(q_i) – E_r exp(i(k_r×r)) cos(q_r),

which must hold for all r in the z = 0 plane.

We therefore need

exp(i(k_t×r)) = exp(i(k_i×r)) = exp(i(k_r×r)), or k_t×r = k_i×r = k_r×r  

for all r in the z = 0 plane.

For r = we then obtain k_i sinq_i = k_r sinq_r = k_t sinq_t.  Since k_i = w/v_i = wn₁/c, k_r = w/v_r = wn₁/c, k_t = w/v_t = wn₂/c, we obtain

sinq_i = sinq_r,    the law of reflection, and

n₁ sinq_i = n₂ sinq_t,    the law of refraction or Snell’s law.

_______________________________________

The tangential component of H is continuous at z = 0.   H_ty = H_iy + H_ry

From Maxwell’s equations we have

.

We therefore have

.

Let = -. Then, at z = 0,

or

.

This equation combined with E_t cos(q_t) = E_i cos(q_i) – E_r cos(q_r) at z = 0 yields the relation between the amplitudes E_t, E_i, and E_r.

r_12p is called the Fresnel reflection coefficient  for p-polarization.

t_12p is called the Fresnel transmission coefficient for p-polarization.

Note that the reflection coefficient can be negative.  E_i and E_r denote the magnitudes of the field vectors assuming that they point into the directions shown in the figure above.  A negative reflection coefficient denotes that given the direction of E_i as shown in the figure, the direction of E_r must be reversed.

For nonmagnetic materials m₁ = m₂ = m₀.

Defining a = cosq_t/cosq_i and b = n₂/n₁ we may then write

r_12p = (b-a)/(b+a) , t_12p = 2/(a+b).

_______________________________________

For any plane in a medium we define the characteristic wave impedance Z for a plane waves as the magnitude of the ratio of the transverse electric field E_|| to the transverse magnetic induction H_||.  For a TM wave and planes perpendicular to the z-axis, as shown above,  the wave impedance at position z is Z(z) = E_x/H_y.  (Here transverse refers to directions transverse to the normal and therefore parallel to the plane of the interface.)

Since the tangential components of E and H are continuous for a dielectric-dielectric interface, the wave impedance must be continuous.

For a single TM plane wave propagating in a medium with a wave vector k making an angle q with the z-axis, we have 

Z(z) = E_x(z) /H_y(z)  = (mc/n) cosq = (k/ew) cosq = (k_z/ew) º Z₀.

[mc/n = mv = mev/e = 1/(ve) = k/(ew), since v = (me)^-1/2 = w/k.]

Z₀ is independent of position.  It depends only on the properties of the medium and the angel q.

At z = 0 in medium 2 we have, for the TM wave shown above,

Z(0)⁽²⁾ = (m₂c/n₂) cosq_t = (k_t/e₂w) cosq_t = (k_tz/e₂w) = Z₀⁽²⁾.

In medium 1 an incident and a reflected wave propagate.

At z = 0 in medium 1 we have, for the TM waves shown above, 

Z(0)⁽¹⁾ = (m₁c/n₁) cosq_i(H_iy - H_ry)/(H_iy + H_ry) = Z₀⁽¹⁾(H_iy - H_ry)/(H_iy + H_ry).

Here H is the magnetic induction at z = 0.

Since the wave impedance is continuous we have

Z₀⁽¹⁾(H_iy - H_ry)/(H_iy + H_ry) = Z₀⁽²⁾.

The reflection coefficient r_12p is equal to

r_12p = H_ry/H_iy = -(Z₀⁽²⁾ - Z₀⁽¹⁾)/( Z₀⁽²⁾ + Z₀⁽¹⁾)

This is an expression of the reflection coefficient r_12p in terms of the wave impedance.

s-polarization

A wave incident on an interface with s-polarization is called a transverse magnetic (TE) wave.

We analyze its behavior at the interface in a similar way as the behavior of a wave with  p-polarization.

The laws of reflection and refraction are unchanged.  For the reflection and transmission coefficients we now find

 .

r_12s is called the Fresnel reflection coefficient  for s-polarization.

 .

t_12s is called the Fresnel transmission coefficient for s-polarization.

If m₁ = m₂ = m₀, then

r_12s = (1 - ba)/(1 + ba) ,    t_12s = 2/(1 + ab).

_______________________________________

For a TE wave and planes perpendicular to the z-axis, the wave impedance at position z is Z(z) = -E_y(z)/H_x(z).

Since the tangential components of E and H are continuous for a dielectric-dielectric interface, the wave impedance must be continuous.

For a single TE plane wave propagating in a medium with a wave vector k making an angle q with the z-axis, we have 

Z(z) = -E_y(z)/H_x(z) = (wm/k_z) º Z₀.

[wE = -k ´ B .  E = (cm/n) H().  -E_y/H_x = cm/(n cosq) = vm/(cosq) = wm/(k cosq).]

At z = 0 in medium 2 we have for the transmitted TE wave

Z(0)⁽²⁾ = (wm₂/k_tz) = Z₀⁽²⁾.

At z = 0 in medium 1 we have for the incident and reflected TE wave

Z(0)⁽¹⁾ = Z₀⁽¹⁾(E_iy + E_ry)/(E_iy - E_ry)

Here E is the electric field at z = 0.

Since the wave impedance is continuous we have

Z₀⁽¹⁾(E_iy + E_ry)/(E_iy - E_ry) = Z₀⁽²⁾.

The reflection coefficient r_12s is equal to

r_12s = E_ry/E_iy = (Z₀⁽²⁾ - Z₀⁽¹⁾)/( Z₀⁽²⁾ + Z₀⁽¹⁾)

This is an expression of the reflection coefficient r_12s in terms of the wave impedance.

The ratio of the reflected intensity to the incident intensity is obtained by squaring the Fresnel reflection coefficient.  The reflected and incident wave have the same speed and their wave vectors make the same angle with the normal to the interface.

R = |r₁₂|² = <S>_r×/<S>_i× º reflectance

is a unit vector normal to the interface.

The ratio of the transmitted intensity to the incident intensity is 

T = (cosq_t n₁)/(cosq_i n₂) ´ |t₁₂|² = <S>_t×/<S>_i× º transmittance

We compare the incident and transmitted energy per unit area perpendicular to the z-axis per unit time.  Energy conservation requires that in  lossless media

T + R = 1.

Link:

Excel Spreadsheet: Reflectance vs. q_i (s and p polariztion)