Download 10 Electromagnetic wave propagation: Superposition and their types

10 Electromagnetic wave propagation:
Superposition and their types
Contents
10.1 Superposition of Plane waves
10.2 Linear Polarisation
10.3 Circular and Elliptical Polarisation
Keywords: Superposition, Linear, Circular and Elliptical Polarisation
Ref: J. D. Jackson: Classical Electrodynamics; A. Sommerfeld: Electrodynamics.
10.1 Superposition of Plane waves
Mathematically it is convenient to write the plane wave solutions in terms of
exponentials (due to the simplicity in differentiation and integration) and for
an electric field or a magnetic fields one can take either the real part or the
imaginary part of the exponential solution. Since the equation is linear so a
linear combination of two or more solutions will also be a solution of the wave
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10 Electromagnetic wave propagation: Superposition and their types
equation. Let us take two solutions of the wave equation as the following,
Ex = Aei(kz−ωt)
A = |A|eiφ1 ,
(10.1)
Ey = Bei(kz−ωt)
B = |B|eiφ2 .
(10.2)
Notice that phases are absorbed in complex amplitudes. The solution (10.1)
is said to be x-polarised since its electric field is along the x-direction whereas
the (10.2) is y-polarised for a similar reason. Different superpositions of the
above two solutions will make different types of electromagnetic waves. For
example, φ1 − φ2 = 0 or π will produce linearly polarised wave where again
the electric vibrations are always along a particular direction. The combination φ1 − φ2 = π/2 or 3π/2 in general will produce elliptically polarised waves
and for the particular case of |A| = |B| one will have a circularly polarised
wave. For linearly polarised wave if we focus our attention to any particular point in space we would observe that the electric(or magnetic) field will
start growing in a particular direction, reaches a maximum value and then
starts diminishing till the field becomes zero and then starts growing in the
opposite direction in a similar fashion, attains a maximum and then reduces
to zero periodically. The tip of the electric(or magnetic) field at any point in
space will trace an ellipse or a circle with time for an elliptically or circularly
polarised wave respectively. For any other arbitrary value of φ1 − φ2 one has
an elliptically polarised wave. The nomenclature is easy to appreciate by
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10.2 Linear Polarisation
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constructions of Lissajous’s’ figures for two sinusoidal vibration of the same
frequency in orthogonal directions with different phase difference.
Problem 1: Calculate the magnetic field components of the above wave for
which the electric field is given by (10.1) and (10.2).
10.2 Linear Polarisation
The generalisation of the wave considered in the previous section will be a
plane wave. For plane waves at any instant the points in the space having
equal phase lie in a two dimensional flat plane. For the plane waves we first
separate out the temporal part of the solution.
E = Ee−iωt ,
(10.3)
B = Be−iωt .
(10.4)
E = E0 eik·r ,
(10.5)
B = B0 eik·r .
(10.6)
Further we choose,
Substituting the above ansatz in the wave equation we get the constraints,
(−k · k + µǫω 2 )E0 ,
(10.7)
(−k · k + µǫω 2 )B0 .
(10.8)
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10 Electromagnetic wave propagation: Superposition and their types
4
That shows the same dispersion relation as (9.6)
c2 k · k = c2 (kx2 + ky2 + kz2 ) = c2 k 2 = ω 2 .
(10.9)
Now substituting the above solution in Maxwell’s equations (3.1) and (3.3)
we get,
k·E =0
and
k · B = 0.
(10.10)
The condition (10.11) is known as transversality condition, i.e. the direction
of the wave vector (or the direction of the propagation of the wave) is orthogonal to the field directions. Further substitution in the rest of the Maxwell’s
equation demands
k × E = ωB
and
1
k × B = −ωǫE.
µ
(10.11)
Problem 2: Show that E, cB and k̂ = k/k define a right-handed co-ordinate
system.
Since the wave equation is linear so a linear superposition of two solutions
is again a solution. Let us take a solution where the electric filed is given
by, E1 = ε1 E0 eik·r , where ε1 the unit vector along E1 . In this case then the
magnetic field will be given by cB1 = ε2 E0 eik·r where ε2 is unit normal to
the plane defined by k̂ and ε1 . The three unit vectors ε1 , ε2 and k̂ make
a right handed co-ordinate system. The two vectors ε1 and ε2 are taken as
the basis vectors to define the polarisation of a general plane wave. Now if
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10.3 Circular and Elliptical Polarisation
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we take a second solution E2 = ε2 E0′ eik·r with corresponding magnetic field
cB2 = −ε1 E0′ eik·r and add to the earlier solution, we get the resultant solution
as, E = (ε1 E0 +ε2 E0′ )ei(k·r−ωt) and cB = (ε2 E0 −ε1 E0′ )ei(k·r−ωt) . The resultant
solution is again linearly polarised.
E’0
E’0
^
ε2
^
ε2
E
^
ε1
E0
E
^
ε1
E0
Fig. 10.1: Linearly polarised and right-elliptically polarised waves
10.3 Circular and Elliptical Polarisation
Had we added the two solutions with a phase difference of π/2 we would have
produced a right-elliptically polarised wave.
E = (ε1 E0 + iε2 E0′ )ei(k·r−ωt)
(10.12)
Note that the i could be written as eiπ/2 . Or for a phase shift of 3π/2 we
could produce a left-elliptically polarised
E = (ε1 E0 − iε2 E0′ )ei(k·r−ωt)
(10.13)
For the special case of E0 = E0′ the above superposition would make rightcircularly and left circularly polarised waves respectively. Again it is easy to
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10 Electromagnetic wave propagation: Superposition and their types
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E 0’
E
^
ε2
^
ε
1
E’0
E 0’
E
^
ε2
^
ε
1
E0
Fig. 10.2: Left-circularly polarised and left-elliptically polarised waves
see that a superposition of one left-circularly polarised wave and one rightcircularly polarised wave will produce a linearly polarised wave if the amplitudes of superposed waves are same. Arbitrary superpositions in general
produce an elliptically polarised wave. Hence the left-circular polarisation
and the right-circular polarisation can be used as the basis vectors to show a
general polarisation as well.
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