Notes on Random Topics
Tomoyuki Nakayama
July 13, 2011
1
Reflection and Transmission
Q. Why you cannot see an object if you immerse it in a liquid with the same
index of refraction?
1.1
1.1.1
Electromagnetic Waves
Boundary Conditions
Our starting point is Faraday’s law and Ampere-Maxwell’s Law
I
E · ds = Φ
I B
· ds = i + id
µ
(1)
(2)
where
Φ
id
=
I
B · dA
(3)
=
I
(ǫE) · dA
(4)
First, we apply Faraday’s law to the interface between two media with triangular
Amperian loop of dimensions l×w. If the width of the rectangle is small enough,
then flux is zero. We get
(E1,|| − E2,|| )l = 0.
(5)
The electric field component parallel to the interface is continuous, regardless
the direction of the travel of the wave.
Next, we apply Ampere-Maxwell’s law to the rectangular loop. The displacement current becomes zero as width w approaches zero. If there is no current
at the boundary, we have
B2,||
B1,||
l = 0.
(6)
−
µ1
µ2
Therefore, the parallel component of magnetic field strength (H-field), not magnetic field (B-field), is continuous at the boundary between two media.
1
1.1.2
Reflection and Transmission Coefficients
We consider a sinusoidal EM wave as the real part of the following complex
wave:
ẼI,x (z, t)
= ẼI eı(k1 x−ωt)
(7)
B̃I,y (z, t)
=
ẼI ı(k1 x−ωt)
e
v1
(8)
where ẼI = EI eıα . It contains the phase factor. This wave travels to the
positive z-direction. We assume that the interface is xy-plane. The reflected
wave is written as
ẼR,x (z, t)
= ẼR eı(−k1 x−ωt)
B̃R,y (z, t)
= −
ẼR −ı(k1 x−ωt)
e
v1
(9)
(10)
where ẼR = ER eıβ Note that the magnetic field is polarized in −y direction,
not +y. The transmitted wave is written as
ẼT,x (z, t)
= ẼT eı(k2 x−ωt)
(11)
B̃T,y (z, t)
=
ẼT ı(k2 x−ωt)
e
v2
(12)
where E˜T = ET eıγ Note the wave number or wavelengh depends on the materials, but the frequency does not. Also the polarization does not change after
reflection or transmission. Otherwise, they cannot satisfy the boundary condition at any moment.
The boundary condition at z = 0 must be satisfied at any moment. Hence
setting t = 0, we have
ẼI + ẼR
ẼI
ẼR
−
µ1 v1
µ1 v1
= ẼT
=
ẼT
µ2 v2
(13)
(14)
We rewrite the second equation as
ẼI − ẼR
= nẼT
µ1 n2
µ1 v1
=
n =
µ2 v2
µ2 n1
(15)
(16)
Solving the equations simultaneously, we obtain
ẼT
=
ẼR
=
2
ẼI
1+n
1−n
ẼI
1+n
2
(17)
(18)
Since eı(α−β) and eı(α−γ) are both real numbers, the relation between phases is
α = β + 2πm1 = γ + 2πm2
(19)
Setting α = 0, the magnitudes of electric field satisfy:
ET
=
ER
=
2
EI
1+n
1−n
EI
1+n
(20)
(21)
The second equation flips its sign according to the value of n. If n > 1, the
phase of the reflected wave changes 180◦ . The transmission wave is always in
phase with the incident wave. The intensity of EM waves is given by
1 2
ǫE v
2
We define transmission and reflection coefficients as
2
2
IT
4n
ǫ2 v2 ET
ǫ2 v2
2
T ≡
=
=
=
II
ǫ1 v1 EI
ǫ1 v1 1 + n
(1 + n)2
2
2 IR
ER
1−n
R ≡
=
=
II
EI
1+n
I=
(22)
(23)
(24)
where we used
ǫ2 v2
=
ǫ1 v1
1
µ2 v22
1
µ1 v12
v2
=n
(25)
v1
These coefficients satisfy
1=R+T
(26)
This guarantees the conservation of energy.
1.1.3
Reflection and Transimission in Diamagnetic/Paramagnetic Materials
Materials
Water
Diamond
Carbon dioxide (1atm)
Susceptibility ξ = µ/µ0 − 1
−9.0 × 10−6
−2.2 × 10−5
−1.2 × 10−8
Index of refraction
1.333
2.419
1.00
Most of the diamagnetic or paramagnetic materials have relative permeablility µ/µ0 very closed to 1. Thus we can safely set µ1 = µ2 for these materials,
and n = nn21 is now just a relative index of refraction. Therefore, if two materials
have the same index of refraction (n = 1), then reflection coefficient is
R=0
There is no reflection.
3
(27)