many droplets with the net appearance of the rainbow with red light lying around the outer rim.
In fact, if the viewer was at a high enough altitude (in practice, in an aircraft), the rainbow would appear as a complete circle. On ground level, however, the lower portion of the circle is cut off and we just see a ‘bow’.
6.10 The Poynting vector
The conservation of electromagnetic energy may be stated in terms of Poynting’s theorem
−∂u
∂t = ∇ · S + jf · E, (6.101)
where u is the energy density
u = 12(E · D + B · H) , (6.102) and S is the Poynting vector giving the flow of energy crossing unit area.
Hence, −∂u/∂t is the rate at which the energy density decreases.
Taking the time derivative of Eq. (6.102), ∇ · S is the divergence of the energy flow (the volume integral of this being the energy flux through the surface of the volume) and jf · E is the work done on any free charges by the electric field.
The time derivative of Eq. (6.102) is
∂u
We shall be assuming a linear and homogeneous medium but allowing it to be anisotropic. We shall therefore need to make use of the constitutive equations, Eqs. (6.44) and (6.45) for the electric displacement D and field vector H respectively.
Since the susceptibility tensors have no time dependence we have, from Eq. (6.44),
∂D
∂t = ε0(I + χE)∂E
∂t. (6.104)
Now, the ith component of the matrix product of χE and the time derivative of E is
E · χE∂E
In Section 6.8, we found that in a loss-less medium, χE is a symmetric matrix, i.e. χij = χji. So, since the scalar components commute,
E · χE∂E Taking the dot product of E and Eq. (6.105), we therefore find
E ·∂D
A similar argument holds for B and H yielding B ·∂H
∂t = ∂B
∂t · H. (6.109)
Using Eqs. (6.108) and (6.109), Eq. (6.103) for the time derivative of the energy density becomes
∂u
∂t = E ·∂D
∂t + ∂B
∂t · H. (6.110)
We now apply Faraday’s law, Eq. (6.3), and Maxwell’s modified form of Ampere’s law, Eq. (6.4), to give
−∂u
∂t = (∇ × E) · H − E · (∇ × H − jf) ,
= (∇ × E) · H − E · (∇ × H) + jf · E. (6.111) Equating this to the right-hand-side of Eq. (6.101), we have
(∇ × E) · H − E · (∇ × H) = ∇ · S. (6.112) We now make use of the identity
∇ · (E × H) ≡ (∇ × E) · H − E · (∇ × H) (6.113) and hence arrive at the result
S = E × H. (6.114)
This is the instantaneous Poynting vector, being the energy flowing across unit area per unit time.
6.11. SUMMARY 127 Isotropic medium
In an isotropic medium, S is in the direction of the wave-vector k (although this is not true in the anisotropic case). It may then be shown that
S = ˆkE2 εε0
µµ0
1/2
, (6.115)
where ˆkis the unit vector in the k-direction. After time-averaging, we then have
hSi = 12kEˆ 02 εε0 µµ0
1/2
, (6.116)
where E0 is the amplitude of the electric field component of the wave.
6.11 Summary
• The speed of light
Maxwell’s equations yield electromagnetic wave solutions. In free space, the wave speed is a universal constant
c = (ε0µ0)−1/2. (6.117)
• The electric and magnetic susceptibility tensors
The electric and magnetic susceptibility tensors, χE and χB, give the response of a medium to applied electric and magnetic fields respec-tively.
• The relative permittivity ε and permeability µ
The relative permittivity ε and permeability µ of a medium is given in terms of the electric and magnetic susceptibilities
(I + χE) = ε (6.118)
and
(I − χB) = µ−1. (6.119)
• The wave speed and refractive index
The wave speed in a dielectric becomes modified according to v = c
n, (6.120)
where
n = (εµ)1/2 (6.121)
is therefractive index.
• Frequency dependence of the electric susceptibility
The electric susceptibility tensor χEhas a frequency dependence and is complex. This means that the refractive index becomes frequency dependent and complex. The real part modifies the wave speed lead-ing to dispersion. The dispersion is said to be normal if the condition
dn
dλ < 0 (6.122)
is satisfied.
• Optical loss in media
The imaginary part of the refractive index implies a loss of energy from the optical field. The intensity of the field then decays exponen-tially as
I (z) = I (0) e−αz, (6.123) where
α = 2ωη
c (6.124)
is theabsorption coefficient.
This can be related to a transition between electronic states due to the absorption photons of the required energies.
• Poynting vector
The electromagnetic power intensity (energy flow across unit area) is given by the Poynting vector S by
S = E × H. (6.125)
This result holds for an anisotropic medium.
6.12. REFERENCES 129
6.12 References
[1] James Clerk Maxwell, A Dynamical Theory of the Electromagnetic Field, Philosophical Transactions of the Royal Society of London155, 459-512 (1865)
[2] Heinrich Hertz, Untersuchungen ¨uber die Ausbreitung der elektrischen Kraft, Johann Ambrosius Barth, Leipzig (1892)
[3] Albert Einstein, Zur Elektrodynamik bewegter K ¨orper, Annalen der Physik17, 891 (1905)
7. Polarisation
7.1 General remarks
In Chapter 6, we saw how Maxwell’s equations predict electromagnetic wave propagation. The polarisation of such radiation is characterised by the direction of the electric field. Although we have seen that the electric field directions must be transverse to the propagation direction, there is still a wealth of different possibilities to explore. In this Chapter we discuss the polarisation of EM waves in a fairly formal manner in terms of Jones vectors. The advantage of using Jones vectors is that
• the complex phase factor involving the spatial and temporal depen-dence (known as the propagator may often be abstracted from the formal presentation of the polarisation
• optical elements may be modelled by Jones matrices operating on the vectors
The physics of the optical elements is only dealt with in passing, since such materials are generally anisotropic and will be covered in more depth in Part V.
7.2 Learning objectives
The objectives of this section are to understand
• Linear polarisation – Linearly x-polarised – Linearly y-polarised
• Linear polarisers – Dichroism – Polaroid sheets
• Retardation
131
• Circular polarisation
– Γ = π/2 right circularly polarised – Γ = −π/2 left circularly polarised
• Elliptical polarisation
• Jones matrix – Linear polariser
– Rotation of a state of polarisation by an angle θ
• Wave plates – Birefringence – Half-wave plate – Quarter-wave plate
– General retardation plate - phase shift = Γ
• Analysis of polarised light
• Malus’ Law