Fresnel Laws of reflection
Lightwaves are a form of electromagnetic oscillation for which the energy is shared equally
between the oscillating E and H fields
and so
and therefore
1
1
𝑤𝑀 = 2 𝜇𝐻 2 = 𝑤𝐸 = 2 𝜖𝐸 2
H 

E

(3-2.1)
 nE
(3-2.2)
   n or which n is the refractive index of the material, and so H  nE . The
refractive index of the material is the pointer that enables us to relate to lightwaves.
The factor
Lightwaves and RF waves must obey laws of continuity when they traverse a boundary. The
boundary will cause some waves to be reflected and some to be transmitted, as represented by
figure 3-2.1.
Figure 3-2.1 An incident electromagnetic wave separates into a reflected wave and a transmitted
wave when it encounters a boundary. The boundary is assumed to be planar and separates two
media of refractive indices n1 and n2, respectively. The E and H fields are transverse to each
other, and one is perpendicular to the plane of this diagram.
Courtesy of reflection at the boundary, the direction of propagation 𝑧̂ of the reflected wave is
from left to right. If the E-field is perpendicular to the plane of the diagram of figure e3-2.1 it is
transverse to 𝑧̂ and is then identified as a Transverse–Electric (TE) wave. Its transverse E-field
amplitudes must obey causality (cause = effect) and therefore
Ei  E r  Et
(3-2.3a)
And similarly for the magnetic field, the transverse amplitudes must also obey causality, for
which
H i cos 1   H r cos 1  H t cos 2
(3-2.3b)
Applying equation (3-2.2b) to equation (3-2.1b) we will have a second E-field equation at the
boundary,
n1 Ei cos 1  n1 Er cos 1  n2 Et cos 2
(3-2.4)
Eliminating Et between these equations (3-2.3a) and (3-2.4) we then have
n1 Ei cos 1  n2 Ei cos 2  n1 Er cos 1  n2 Er cos 2
For which we define an amplitude ration = reflectivity r12 = Er/Ei
r12 TE   
n1 cos 1  n2 cos  2
n1 cos 1  n2 cos 2
(3-2.5)
Similarly, if the electromagnetic wave and figure 3.2-1 are oriented so that the magnetic field is
perpendicular to the plane, or transverse the direction of propagation 𝑧̂ , then we identify it as a
Transverse–Magneic (TM) wave. Its transverse amplitudes must obey causality and therefore
Hi  Hr  Ht
or equivalently
n1 Ei  n1 Er  n2 Et
(3-2.6a)
Similarly for the electric field components, the transverse amplitudes must obey causality and
therefore,
Ei cos 1   Er cos 1  Et cos 2
(3-2.6b)
Eliminating Et between equations (3-2.6a) and (3-2.6b) we have
n2 Ei cos 1  n1 Ei cos 2  n2 Er cos 1  n1 Er cos 2
For which the reflectivity r12 = Er/Ei will be
r12 TM   
n2 cos 1  n1 cos 2
n2 cos 1  n1 cos 2
(3-2.7)
Equations (3-2.5) and (3-2.7) are Fresnel’s laws. If we apply the law of refraction
n1 sin 1  n2 sin 2
(3-2.8)
To the Fresnel laws then they can be rewritten as
r12 TE   
sin 2 cos 1  sin 1 cos 2
sin 2 cos 1  sin 1 cos 2

sin  2  1 
sin  2  1 
(3-2.9)
sin 1 cos 1  sin 2 cos 2
sin 1 cos 1  sin 2 cos 2

tan 2  1 
tan 2  1 
(3-2.10)
and
r12 TM   
where some trigonometry using the double angle formula 2 sin 1 cos 1  sin 21 and the halfangle formula
sin A  sin B tan  A  B  / 2

sin A  sin B tan  A  B  / 2
has been applied.
Equation (3-2.10) is informative because r12(TM) = 0 when (1 + 2) = /2. From the law of
refraction, the angle for which this condition is met is given by
n1 sin 1  n2 sin  / 2  1 
or
tan 1  tan  B1 
 n2 cos1
n2
n1
(3-2.11a)
Angle B1 is called the Brewster angle. It has a complementary angle
tan  2  tan  B 2 
n1
n2
(3-2.11b)
Using cos   1  sin 2  the Fresnel equations can also be written as
r12 TE   
n1 cos 1  n2 1  sin 2  2
n1 cos 1  n2 1  sin  2
2


1  n

sin
n1 cos 1  n2 1  n12 n22 sin 2 1
n1 cos 1  n2
2
1
n
2
2
2
1
(3-2.12a)
r12 TM   
n2 cos 1  n1 1  sin 2  2
n2 cos 1  n1 1  sin  2
2


1  n

sin
n2 cos 1  n1 1  n12 n22 sin 2 1
n2 cos 1  n1
2
1
n
2
2
2
1
(3-2.12b)
And if n1 > n2, then for sin1 > n2/n1 (note that the critical angle C is defined by sin C = n2/n1),
then these equations must be written in the form
r12 TE, 1  C   
n1 cos 1  jn2
n1 cos 1  jn2
r12 TM , 1  C   
n n sin   1
1  n n sin   1
2
1
2
2
2
1
n2 cos 1  jn1
n2 cos 1  jn1
n
n
2
1
2
2

sin
(3-2.13a)
2
1
2
1
n22 sin 2 1  1
2
1
n22
2
(3-2.13b)
1  1
Since the numerator is the complex conjugate of the denominator then the magnitudes of r12 (TE)
and r12(TM) for 1 > C both = 1, i.e
r12 TE , 1  C   e  (TE )
(3-2.14a)
r12 TM , 1  C   e  (TM )
(3-2.14b)
Where

 (TM )  2 tan n
 (TE )  2 tan 1 n2
1
1
n
2
1
n
2
1


  1 n cos  
n22 sin 2 1  1 n1 cos 1

n22 sin 2
1
2
1
(3-2.15a)
(3-2.15b)
Normally we express these in terms of the grazing angle  = /2 - and the relative refractive
index n12 = n1/n2, which simplifies these relationships considerably.

 (TM )  2 tan n

 1 sin  
 (TE )  2 tan 1 n122 cos 2 1  1 n12 sin 1
1
12
2
n12
cos 2 1
1
(3-2.16a)
(3-2.16b)