Plane Electromagnetic Waves and Wave Propagation
1. Plane Monochromatic Waves in Nonconducting Media
In the absence of free charge and current densities the Maxwell equations are
𝛁⋅𝐃=0
𝛁⋅𝐁=0
𝜕𝐃
𝜕𝑡
𝜕𝐁
𝛁×𝐄= −
𝜕𝑡
𝛁×𝐇=
For uniform isotropic linear media we have 𝐃 = 𝜀𝐄 and 𝐁 = 𝜇𝐇, where 𝜀 and 𝜇 are in general
complex functions of frequency 𝜔. Then we obtain
𝜕 2𝐄
𝜕𝑡 2
𝜕 2𝐁
𝛁 × 𝛁 × 𝐁 = −𝜀𝜇 2
𝜕𝑡
𝛁 × 𝛁 × 𝐄 = −𝜀𝜇
Since 𝛁 × 𝛁 × 𝐄 = 𝛁(𝛁 ⋅ 𝐄) − 𝛁 𝟐 𝐄 = −𝛁 𝟐 𝐄 and, similarly, 𝛁 × 𝛁 × 𝐁 = −𝛁 𝟐 𝐁,
𝜕 2𝐄
𝜕𝑡 2
𝜕 2𝐁
𝛁 2 𝐁 = 𝜀𝜇 2
𝜕𝑡
𝛁 2 𝐄 = 𝜀𝜇
Monochromatic waves may be described as waves that are characterized by a single frequency.
Assuming the fields with harmonic time dependence 𝑒 −𝑖𝜔𝑡 , so that 𝐄(𝐱, 𝑡) = 𝐄(𝐱)𝑒 −𝑖𝜔𝑡 and
𝐁(𝐱, 𝑡) = 𝐁(𝐱)𝑒 −𝑖𝜔𝑡 we get the Helmholtz wave equations
𝛁 2 𝐄 + 𝜀𝜇𝜔2 𝐄 = 0
𝛁 2 𝐁 + 𝜀𝜇𝜔 2 𝐁 = 0
Electromagnetic plane wave of frequency 𝜔 and wave vector 𝐤
Suppose an electromagnetic plane wave with direction of propagation 𝐧 to be constructed, 𝐧
where is a unit vector. Then the variable 𝑧 in the exponent must be replaced by 𝐧 ⋅ 𝐱, the
projection of 𝐱 in the 𝐧 direction. Thus an electromagnetic plane wave with direction of
propagation 𝐧 is described by
𝐄(𝐱, 𝑡) = 𝓔𝑒 𝑖𝐤⋅𝐱−𝑖𝜔𝑡 = 𝓔𝑒 𝑖𝑘𝐧⋅𝐱−𝑖𝜔𝑡
𝐁(𝐱, 𝑡) = 𝓑𝑒 𝑖𝐤⋅𝐱−𝑖𝜔𝑡 = 𝓑𝑒 𝑖𝑘𝐧⋅𝐱−𝑖𝜔𝑡
where 𝓔 and 𝓑 are complex constant vector amplitudes of the plane wave. 𝐄 and 𝐁 satisfy the
wave equations, therefore the dispersion relation is given as
𝜔 2
𝜔
𝑘 2 = 𝜀𝜇𝜔2 = (𝑛 )
→ 𝑘=𝑛
𝑐
𝑐
1
Thus the Maxwell equations become
𝛁⋅𝐃=0
𝛁⋅𝐁=0
𝜕𝐃
𝜕𝑡
𝜕𝐁
𝛁×𝐄=−
𝜕𝑡
𝛁×𝐇 =
→
𝐤 ⋅ 𝓔 = 0 𝐤 × 𝓑 = −𝜀𝜇𝜔𝓔
𝐤⋅𝓑=0
𝐤 × 𝓔 = 𝜔𝓑
where 𝐤 = 𝑘𝐧. The direction 𝐧 and frequency 𝜔 are completely arbitrary. The divergence
equations demand that
𝐧 ⋅ 𝓔 = 0 and 𝐧 ⋅ 𝓑 = 0
This means that 𝐄 and 𝐁 are both perpendicular to the direction of propagation 𝐧. The
magnitude of 𝐤 is determined by the refractive index of the material
𝑘 = 𝑛(𝜔)
𝜔
𝑐
Then 𝓑 is completely determined in magnitude and direction
𝓑 = √𝜀𝜇 𝐧 × 𝓔 =
𝑛
𝐧×𝓔
𝑐
Note that in vacuum (𝑛 = 1), 𝐸 = 𝑐𝐵 in SI units. The phase velocity of the wave is 𝑣 = 𝑐/𝑛.
Energy density and flux
The time averaged energy density is
𝑢=
1
1
1
(𝐄 ⋅ 𝐃∗ + 𝐁 ⋅ 𝐇 ∗ ) = (𝜀𝐄 ⋅ 𝐄∗ + 𝐁 ⋅ 𝐁 ∗ )
4
4
𝜇
This gives
𝑢=
𝜀 2
|𝓔|
2
The time averaged energy flux is given by the real part of the complex Poyinting vector
𝐒=
1
(𝐄 × 𝐇 ∗ )
2
Thus the energy flow is
1 𝜀
1
𝐒 = √ |𝓔|2 𝐧 = 𝜀|𝓔|2 ⋅ 𝑣𝐧 = 𝑢𝐯
2 𝜇
2
where the speed of light in the uniform medium is
𝑣=
1
𝑐
=
√𝜀𝜇 𝑛
2
2. Polarization
There is more to be said about the complex vector amplitudes 𝓔 and 𝓑. We introduce a righthanded set of orthogonal unit vectors (𝛜𝟏 , 𝛜𝟐 , 𝐧), as shown in the figure below, where we take
𝐧 to be the propagation direction of the plane wave. In general, the electric field amplitude 𝓔
can be written as
𝓔 = 𝛜𝟏 𝐸1 + 𝛜𝟐 𝐸2
where the amplitudes 𝐸1 and 𝐸2 are arbitrary complex numbers. The two plane waves
𝐄𝟏 = 𝛜𝟏 𝐸1 𝑒 𝑖𝐤⋅𝐱−𝑖𝜔𝑡
𝐄𝟐 = 𝛜𝟐 𝐸2 𝑒 𝑖𝐤⋅𝐱−𝑖𝜔𝑡
with
𝑛
𝐁𝟏 = 𝛜𝟐 𝑐 𝐸1 𝑒 𝑖𝐤⋅𝐱−𝑖𝜔𝑡
𝑛
𝐁𝟐 = −𝛜𝟏 𝑐 𝐸2 𝑒 𝑖𝐤⋅𝐱−𝑖𝜔𝑡
(if the index of refraction 𝑛 is real, 𝓔 and 𝓑 have the same phase) are said to be linearly
polarized with polarization vectors 𝛜𝟏 and 𝛜𝟐 . Thus the most general homogeneous plane wave
propagating in the direction 𝐤 = 𝑘𝐧 is expressed as the superposition of two independent
plane waves of linear polarization:
𝐄(𝐱, 𝑡) = 𝓔𝑒 𝑖𝐤⋅𝐱−𝑖𝜔𝑡 = (𝛜𝟏 𝐸1 + 𝛜𝟐 𝐸2 )𝑒 𝑖𝐤⋅𝐱−𝑖𝜔𝑡
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3. Reflection and Refraction at a Plane Interface between Dielectrics
Reflection and refraction with polarization (a) perpendicular (s-polarization) and (b) parallel (ppolarization) to the plane of incidence
Phase matching on the boundary indicates that
(i)
All three vectors, 𝐤, 𝐤 ′ and 𝐤 ′′ , lie in a plane, i.e., 𝐤 ′ and 𝐤 ′′ lie in the plane of
incidence;
(ii)
Law of reflection: |𝐤 × 𝐧| = |𝐤 ′′ × 𝐧| → 𝑘 sin 𝜃𝑖 = 𝑘 sin 𝜃𝑟 , thus
𝜃𝑖 = 𝜃𝑟
′
(iii)
Snell’s Law: |𝐤 × 𝐧| = |𝐤 × 𝐧| → 𝑘 sin 𝜃𝑖 = 𝑘 ′ sin 𝜃𝑡 , thus
𝑛 sin 𝜃𝑖 = 𝑛′ sin 𝜃𝑡
Reflection coefficients for s- and p-polarization:
𝑟𝑠 =
𝐸0′′ 𝑛 cos 𝜃𝑖 − 𝑛′ cos 𝜃𝑡 𝑛 cos 𝜃𝑖 − √𝑛′2 − 𝑛2 sin2 𝜃𝑖
=
=
𝐸0 𝑛 cos 𝜃𝑖 + 𝑛′ cos 𝜃𝑡 𝑛 cos 𝜃𝑖 + √𝑛′2 − 𝑛2 sin2 𝜃𝑖
2
𝐸0′′ 𝑛′ cos 𝜃𝑖 − 𝑛 cos 𝜃𝑡 𝑛′ cos 𝜃𝑖 − 𝑛√𝑛′2 − 𝑛2 sin2 𝜃𝑖
𝑟𝑝 =
=
=
𝐸0 𝑛′ cos 𝜃𝑖 + 𝑛 cos 𝜃𝑡 𝑛′ 2 cos 𝜃𝑖 + 𝑛√𝑛′2 − 𝑛2 sin2 𝜃𝑖
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Brewster’s angle and total internal reflection
We consider the dependence of 𝑅 and 𝑇 on the angle of incidence, using the Fresnel
coefficients.
Reflectance for s- and p-polarization at an air-glass interface.
Brewster’s angle is 𝜽𝑩 = 𝟓𝟔∘
𝒏=𝟏
𝒏′ = 𝟏. 𝟓
Brewster angle
Brewster’s angle 𝜃𝐵 = 𝜃𝑖 at which the p-polarized reflected wave is zero:
𝜋
𝑛 sin 𝜃𝐵 = 𝑛′ sin ( − 𝜃𝐵 ) = 𝑛′ cos 𝜃𝐵
2
or
𝑛′
tan 𝜃𝐵 =
𝑛
Polarization at the Brewster angle is a practical means of producing polarized radiation. If a
plane wave of mixed polarization is incident on a plane interface at the Brewster angle, the
reflected radiation is completely s-polarized. The generally lower reflectance for p-polarized
lights accounts for the usefulness of polarized sunglasses. Since most outdoor reflecting
surfaces are horizontal, the plane of incidence for most reflected glare reaching the eyes is
vertical. The polarized lenses are oriented to eliminate the strongly reflected s-component. The
figure above shows 𝑅𝑠 and 𝑅𝑝 as a function of 𝜃𝑖 with 𝑛 = 1 and 𝑛′ = 1.5, as for an air-glass
interface. The Brewster angle is 𝜃𝐵 = 56∘ for this case.
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Total internal reflection
There is another case in which 𝑅𝑠 = 𝑅𝑝 = 1. Perfect reflection occurs for 𝜃𝑡 = 𝜋/2. The
incident angle for which 𝜃𝑡 = 𝜋/2 is called the critical angle, 𝜃𝑖 = 𝜃𝑐 . From Snell’s law
𝑛′
sin 𝜃𝑐 =
𝑛
𝜃𝑐 can exist only if 𝑛 > 𝑛′, i.e., the incident and reflected waves are in a medium of larger index
of refraction than the refracted wave.
Reflectance for s- and p-polarzation
at an air-glass interface. Brewster’s
angle is 𝜽𝑩 = 𝟑𝟒∘ and the critical
angle is 𝜽𝒄 = 𝟒𝟐∘
𝒏 = 𝟏. 𝟓
𝒏′ = 𝟏
For waves incident at 𝜃𝑐 , the refracted wave is propagated parallel to the surface. There can be
no energy flow across the surface. Hence at that angle of incidence there must be total
reflection. For incident angles greater than the critical angle 𝜃𝑖 > 𝜃𝑐 , Snell’s law gives
sin 𝜃𝑡 =
𝑛
𝑛
sin
𝜃
>
sin 𝜃𝑐 = 1
𝑖
𝑛′
𝑛′
This means that 𝜃𝑡 is a complex angle with a purely imaginary cosine.
sin 𝜃𝑖 2
√
cos 𝜃𝑡 = 𝑖 (
) −1
sin 𝜃𝑐
Then the reflection coefficients 𝑟𝑠 and 𝑟𝑝 both take the form
𝑟=
𝑎 − 𝑖𝑏
𝑎 + 𝑖𝑏
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where 𝑎 and 𝑏 are real, therefore,
𝑅=
|𝑟|2
𝑎 − 𝑖𝑏 2
=|
| =1
𝑎 + 𝑖𝑏
The result is that 𝑅𝑠 = 𝑅𝑝 = 1 for all 𝜃𝑖 > 𝜃𝑐 . This perfect reflection is called total internal
reflection. The meaning of this total internal reflection becomes clear when we consider the
propagation factor for the refracted wave:
𝑒 𝑖𝐤
′ ⋅𝐱
= 𝑒 𝑖𝑘
′ (𝑥 sin 𝜃 +𝑧 cos 𝜃 )
𝑡
𝑡
𝑧 𝑖𝑘 ′ (sin 𝜃𝑖 )𝑥
sin 𝜃𝑐
= 𝑒− 𝛿𝑒
where
1
2𝜋𝑛
√sin2 𝜃𝑖 − sin2 𝜃𝑐
= −𝑖𝑘 ′ cos 𝜃𝑡 = 𝑘√sin2 𝜃𝑖 − sin2 𝜃𝑐 =
𝛿
𝜆
with the wavelength of the radiation 𝜆 in vacuum. This shows that, for 𝜃𝑖 > 𝜃𝑐 , the refracted
wave is propagating only parallel to the surface and is attenuated exponentially beyond the
interface. The attenuation occurs within a few wavelengths of the boundary except for 𝜃𝑖 ≈ 𝜃𝑐 .
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