ABSTRACT
Figure 8.2 Polarization orientations: (left) linear, (center) circular, (right) elliptical. If the plane wave travels in the z-direction, it can be written as E z t E z t e E z t ex x y y( , ) ( , ) ( , )= + (8.31) The electric field along the x-axis reads Ex (z, t) = Ex0 cos(kz – wt + fx) (8.32) The electric field along the y-axis reads Ey(z, t) = Ey0 cos(kz – wt + fy) (8.33) There are so many types of polarizations of light relying on the phase difference in wavelength in the x-and y-directions; the details
are summarized in Table 8.1. Note that polarization of natural light occurs in a random phase. Table 8.1 Polarization of light
ˆ ¯˜
2 1
2 mRight-circularly polarized light (negative helicity) d p= -ÊËÁ ˆ¯˜2 12mElliptically polarized light Other phase
An optical filter that allows light with a specific orientation to pass through it is called a polarizer. There are two types of polarizers: linear polarizer and circular polarizer. A linear polarizer generates linearly polarized light, whereas a circular polarizer generates circularly polarized light. Input polarizaon plane
Figure 8.3 Left: A half wave plate reduces about a half cycle the phase difference between the two components of polarized light traversing it. Right: Quarter wave plate is a birefringent material such that the light associated with the larger refractive index is retarded in phase by 90 degree with respect of that associated with the smaller refractive index. In addition to a polarizer, a wave plate is named according to the phase difference in wavelength in x-and y-directions, especially when nx is not equivalent to ny. The common types of wave plates are half wave plates (or o l/2 wave plates) and quarter wave plates (or l/4 wave plates), as shown in Fig. 8.3. A half wave plate rotates the
polarization, whereas a quarter wave plate generates the circularly polarized light out of the linearly polarized light. 8.4 Spectrum of Light
The spectrum of light is the power distribution of light fields as compared to energy. If a real electric field is written as E r t eE k r t( , ) cos( )= ◊ - +
0 w f (8.34)
we can write the real part of a complex electric field as E r t eE ei k r t( , ) ( )= ◊ -0 w (8.35) The spectrum of light is then the square of the Fouriertransformed amplitude of the electric fields (Fig. 8.4), such as E t eE e t i t( ) .= - -
I E( ) ( ) ( . )
w w w g
µ - +
8.5 Electromagnetic Waves in MediumWith charges and current present, Maxwell’s equations in MKS units are written as follows: —¥ = - ∑E B (8.38) —¥ = + ∑H J D (8.39) —◊ =D tr (8.40) —◊ =B 0 (8.41)Here, D E= e (8.42) D E P= +e0 (8.43) H B M B= - = m m0 (8.44) J E=s (8.45)where rt, s, m, and s are true charge density, dielectric function, magnetic permeability, and electric conductivity, respectively. J is the current density. For a wave equation in an anisotropic medium, we have the wave
equation for the electric field as
— = ∂ ∂
2 2 2E Etem (8.46) and the wave equation for the magnetic field as
— = ∂ ∂
2 2 2B Btem (8.47) Here, refractive index is n= em e m0 0 , and the velocity of light in the medium is c c
n ¢ = . k is the wave vector with magnitude k n
c = w . An electromagnetic wave is a transverse wave, and it is related to the Poynting vector S (a flow of electromagnetic energy) by S E H∫ ¥ (8.48)
Taking divergence of the Poynting vector S , we get —◊ = —◊ ¥S E H( ) (8.49) —◊ = ◊ — ¥ - ◊ —¥S H E E H( ) ( ) (8.50) —◊ = ◊ -Ê
ËÁ ˆ ¯˜ - ◊ + Ê ËÁ
ˆ ¯˜
∑ ∑ S H B E J D( ) ( ) (8.51)
—◊ = - ∂ ∂
◊ - ◊Ê ËÁ
ˆ ¯˜ - ◊S
t E D B H J E
2 ( ) (8.52) If the electromagnetic energy within a unit volume is U E D B H= ◊ + ◊1
(8.53)then Eq. 8.52 becomes —◊ = - - ◊∑S U J E( ) (8.54) Finally, we obtain the equation of continuity as (the sun is used for analogy in Fig. 8.5) —◊ + + ◊ =∑S U J E 0 (8.55)Here, Joule’s heat is J E E◊ =s 2 (8.56)
S⋅∇ EJ ⋅
U
8.6 Electromagnetic PotentialThe —◊ =B 0 in Maxwell’s equation will automatically satisfy if we define the scalar potential f( , )
r t and the vector potential A r t( , ) as B A= —¥ (8.57) From Faraday’s law (Eq. 8.38), we have —¥ = - ∂∂E Bt We substitute Eq. 8.57 in Eq. 8.38 as —¥ = - ∂
∂ —¥E
t A( ) (8.58)
—¥ + ∂∂ÊËÁ ˆ¯˜ =E At 0 (8.59) This is satisfied identically by a scalar potential if E A
t + ∂ ∂
= -—f (8.60)
or E A t
= -— - ∂ ∂
f (8.61)
There is a certain extent of arbitrariness in the definition of f( , )
r t and A r t( , ) as follows: A A u= +—0 (8.62) f f= - ∂
∂0 ut (8.63)where u is some scalar function. This transformation of potentials is called gauge transformation. This invariance of fields is called gauge invariance. From Eq. 8.61, we have
E A t
= -— - ∂ ∂
f Substituting Eqs. 8.62 and 8.63 in Eq. 8.61, we get E ut t A u= -— - ∂∂ÊËÁ ˆ¯˜ - ∂∂ +—f0 0( ) (8.64)-— - ∂∂ = -— - ∂∂ÊËÁ ˆ¯˜ - ∂∂ +— f fAt ut t A u0 0( ) (8.65)
E A t t
u A t t
u= -— - ∂ ∂
= -— + ∂ ∂
— - ∂ ∂
- ∂ ∂
—f f0 0 (8.66)
These show that the electric fields are left unchanged by the transformation of the potentials. From Eq. 8.39, we have —¥ = + ∂ ∂
B J E t
m me Substituting the gauge transformations from Eqs. 8.57 and 8.61 in Eq. 8.39, we get —¥ —¥ = + ∂
∂ -— - ∂
∂ Ê
ËÁ ˆ
¯˜ ( )A J
t A t
m me f (8.67)
— —◊ -— = - — ∂ ∂
- ∂ ∂
( )A A J c t c
A
1 1m f (8.68)
— - ∂ ∂
-— —◊ + ∂ ∂
Ê ËÁ
ˆ ¯˜ = -2
1 1 A
c
A
t A
c t J
f m (8.69)
If we define the Lorenz gauge condition frequently used for a system with electric charge and current as —◊ + ∂ ∂
=A c t 1
f (8.70) and apply the Lorenz gauge condition to Eq. 8.69, we get — - ∂
∂ = -2
1 A
c
A
t Jm (8.71) On the other hand, from Eq. 8.40, we have —◊ =E tr
e
Substituting Eq. 8.61 in Eq. 8.40, we get —◊ -— - ∂ ∂
Ê
ËÁ ˆ
¯˜ =f r e
A t
t (8.72) Df r e
+ ∂ ∂
—◊ = - t
A t( ) (8.73) And applying the Lorenz gauge condition to Eq. 8.73, we get Df f r
e - ∂
∂ = -1
2c t t (8.74) Use d’Alembert operator (represented by a box: ) as ∫ — - ∂
∂ ∫ - ∂
1 1
c t c t D (8.75)
Maxwell’s equations from Eqs. 8.71 and 8.74 will be A u J= - (8.76) f r
e = - t (8.77) Solution to these Maxwell’s equations will be ( , ) 1
4 f
pe r
r t r t R
drt= ¢ ¢
¢Ú ( , ) (8.78) ( , ) 1
A r t
J r t R
dr= ¢ ¢ ¢Úpe ( , ) (8.79)
Here, t t R c
¢ = ¢ and R r r∫ - ¢
When there is charge and current, together with dielectric, we can apply the Lorenz gauge condition to Eq. 8.69 and get [ — - ∂
∂ -— —◊ + ∂
∂ Ê ËÁ
ˆ ¯˜ = -2
1 1 A
c
A
t A
c t J
f m (which is Eq. 8.69)]
— - ∂ ∂
-——◊ - — ∂ ∂
1 1 A
c
A
t A
c t J
f m (8.80) And we define the Coulomb gauge condition normally used in field quantization as —◊ =A 0 (8.81) We apply the Coulomb gauge condition to Eq. 8.80 and get A
c t J-— ∂
∂ = -1
f m (8.82)
A
c t - — ∂
∂ = -1
f m - + — ∂ ∂
Ê ËÁ
ˆ ¯˜
-m e f1 * A
t (8.83)
If any vector field J is expressed as the sum of the longitudinal (irrotational) term and the transverse (solenoidal) term or J J J= +
Then, we will get the components of any vector field ase f — ∂ ∂
= t
(8.85)
A J= -m (8.86) On the other hand, applying the Coulomb gauge condition to Eq. 8.73, we get [ Df r
e + ∂ ∂
—◊ = - t
A t( )
(which is Eq. 8.73)]2-= -f r e
The quantization of electromagnetic fields is used to define elementary excitations, as shown in Fig. 8.6.
