We assume that the electric and magnetic fields across the boundary are not dis-continuous, but merely have a sharp gradient. If we construct a pill box that
in-n
Figure 2.21.Pill box intersecting the boundary between two media.
tersects the boundary between two dielectric materials, as shown in Figure 2.21, then the fields gradually change from the top of the volume to the bottom, which is located on the other side of the boundary.
By Gauss’ theorem (see Appendix A.5), the divergence of B integrated over the volume of the pill box is related to the integral of the normal component of B, when integrated over the surface of the pill box:
v∇·B dv =
s
B· n ds = 0. (2.103)
In the limit that the height of the pill box goes to zero, contribution of the cylin-drical side of the pill box also goes to zero. If the areas of the top and bottom of the pill box (δA1andδA2) are small and there is no flux or induction at the
surface, then the above surface integral simplifies to
B(1)· nδA1− B(2)· nδA2= 0, (2.104a)
B(1)− B(2)
· n = 0, (2.104b)
whereδA=δA1=δA2. This equation shows that the component of the magnetic induction B normal to the boundary between the two media is continuous across the media.
The normal component of the electric displacement D is analyzed similarly:
vD dv=
sD· n ds =
vρdv. (2.105)
In the limit that the pill box is shrunk to zero height, the charge densityρ, i.e., the charge per unit of volume, goes to infinity. For this reason, the concept of surface
2.5. Reflection and Refraction 55
Figure 2.22.Plane intersecting the boundary between two media.
charge density ˆρis introduced as follows:
δ h→0lim
vρ dv=
s
ρˆdA. (2.106)
The normal component of the electric displacement D may therefore be derived
from
D(1)− D(2)
· n = ˆρ. (2.107)
This means that upon crossing a boundary between two dielectrics, the surface charge density causes the normal component of D to be discontinuous and jump by ˆρ.
To analyze the tangential components of E and H, we can make use of Stokes’
theorem (Appendix A.7), which relates line integrals to surface integrals. We place a plane perpendicular to the surface boundary that intersects the bound-ary, as shown in Figure 2.22. Note also that the incident, reflected and refracted Poynting vectors are assumed to lie in this plane, as shown in Figure 2.23.
In Figure 2.22, n is the normal of the boundary between the two dielectrics, b is the normal of the newly created plane, and t= b×n is a vector that lies both
s(t)
Figure 2.23. Plane (green) intersecting the boundary between two media. The Poynting vectors of incident, reflected, and refracted TEM waves all lie in this plane.
in the new plane and in the plane of the surface boundary. All three vectors are assumed to be unit length. Stokes’ theorem then gives
where the middle integral is a line integral over the contour c that surrounds the plane. The other two integrals are surface integrals over the area of the plane. If δs1andδs2are small, (2.108) simplifies as follows:
E(2)· tδs2− E(1)· tδs1= −∂B
∂t · bδsδh. (2.109) In the limit that the heightδhgoes to zero, we get
E(2)− E(1)
· tδs= 0. (2.110)
By noting that t= b×n and −t = −b×n, this equation can be rewritten as b·
n×
E(2)− E(1)
= 0. (2.111)
This result indicates that at the boundary between two dielectrics, the tangential component of the electric vector E is continuous. A similar derivation can be made for the magnetic vector, which yields
n×
H(2)− H(2)
= ˆj, (2.112)
where ˆj is the surface current density (introduced for the same reason ˆρ was above). Thus, the tangential component of the magnetic vector H is discontin-uous across a surface and jumps by ˆj. Of course, in the absence of a surface current density, the tangential component of vector H becomes continuous.
The above results for the continuity and discontinuity of the various fields will now be used to help derive the Fresnel equations, which predict the amplitudes of the transmitted and reflected waves. To facilitate the derivation, the vector E at the boundary between the two dielectrics may be decomposed into a component parallel to the plane of incidence, and a component perpendicular to the plane of incidence, as indicated in Figure 2.24. The magnitudes of these components are termed A and A⊥. The x and y components of the incident electric field are then given by
Ex(i)= −A cos(Θi)e−iτi, (2.113a)
Ey(i)= A⊥e−iτi, (2.113b)
Ez(i)= A sin(Θi)e−iτi, (2.113c)
2.5. Reflection and Refraction 57
Figure 2.24. The electric vector E can be projected onto the plane of incidence as well as onto a perpendicular plane. The magnitudes of the projected vector are indicated by A and A⊥.
withτiequal to the variable part of the wave function:
τi=ω
As for dielectric materials, the magnetic permeability is close to one (μ = 1), and therefore we can derive the components of the magnetic vector H from E by applying (2.46b):
and R and T the complex amplitudes of the reflected and refracted waves, ex-pressions for the reflected and transmitted fields may be derived analogously.
The reflected field is then given by
Ex(r)= −R cos(Θr)e−iτr Hx(r)= −R⊥cos(Θr)√
ε1,e−iτr, (2.118a) Ey(r)= R⊥e−iτr Hy(r)= −R √
ε1e−iτr, (2.118b) Ez(r)= R sin(Θr)e−iτr Hz(r)= R⊥sin(Θr)√
ε1e−iτr, (2.118c)
and the transmitted field is
Ex(t)= −T cos(Θt)e−iτt Hx(t)= −T⊥cos(Θt)√
ε2,e−iτt, (2.119a) Ey(t)= T⊥e−iτt Hy(t)= −T √
ε2e−iτt, (2.119b) Ez(t)= T sin(Θt)e−iτt Hz(t)= T⊥sin(Θt)√
ε2e−iτt. (2.119c)
According to (2.111) and (2.112) and under the assumption that the surface cur-rent density is zero7, the tangential components of E and H should be zero at the surface boundary. We may therefore write
Ex(i)+ Ex(r)= Ex(t) Hx(i)+ Hx(r)= Hx(t), (2.120a) Ey(i)+ Ey(r)= Ey(t) Hy(i)+ Hy(r)= Hy(t). (2.120b)
By substitution of this equation, and using the fact that cos(Θr) = cos(π− Θr) =
−cos(Θr), we get
cos(Θi)
A − R
= cos(Θt) T , (2.121a) A + R =
ε2
ε1T , (2.121b)
cos(Θi)(A⊥− R⊥) =
ε2
ε1cos(Θt) T⊥, (2.121c)
A⊥+ R⊥= T⊥. (2.121d)
7This is a reasonable assumption for dielectric materials.
2.5. Reflection and Refraction 59
We use Maxwell’s relation n= √μ ε=√
εand solve the above set of equations for the components of the reflected and transmitted waves, yielding
R = A n2cos(Θi) − n1cos(Θt) These equations are known as the Fresnel equations; they allow us to compute the transmission coefficients t and t⊥as follows:
t =T The reflection coefficients r and r⊥are computed similarly:
r =R We may rewrite both the reflection and the transmission coefficients in terms of the angle of incidence alone. This can be achieved by applying Snell’s law (2.102), so that
sin(Θt) =n1
n2
sin(Θi) (2.125)
and therefore, using Equation (B.11), we have
cos(Θt) =
By substitution into (2.123) and (2.124) expressions in n1, n2, and cos(Θi) can be obtained. The reflection and transmission coefficients give the percentage of incident light that is reflected or refracted:
r= E0,r
E0,i (2.127a)
t= E0,t
E0,i (2.127b)
In the case of a wave incident upon a conducting (metal) surface, light is both reflected and refracted as above, although the refracted wave is strongly attenuated by the metal. This attenuation causes the material to appear opaque.
A water surface is an example of a reflecting and refracting boundary that occurs in nature. While all water reflects and refracts, on a wind-still day the
Figure 2.25. On a wind-still morning, the water surface is smooth enough to reflect light specularly on the macroscopic scale; Konstanz, Bodensee, Germany, June 2005.
2.5. Reflection and Refraction 61
boundary between air and water becomes smooth on the macroscopic scale, so that the effects discussed in this section may be observed with the naked eye, as shown in Figure 2.25.