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Surface waves
A semi-infinite, homogeneous medium fills the x > 0 region. The optical properties of the medium
are described by a dielectric function ε = ε(ω) having real values in the frequency range of interest.
A monochromatic EM wave propagates along the y-direction, parallel to the surface of the medium.
It is known that the magnetic field of the wave has a z-component only and it is given by
Bz = B0 e−qx cos(ky − ωt) = Re B0 e−qx eiky−iωt ,
(1)
where q is real and positive.
a) Using the wave equation for B inside the dielectric medium, find a relation between q, k and ω.
b) Write down the expression the the electric field E of the wave.
c) Calculate the Poynting vector S and specify the direction of the time-averaged EM energy flow.
Now consider two different media with dielectric functions ε1 and ε2
y
which fill the x < 0 and x > 0 regions, respectively. An EM wave propagates along the surface x = 0, with the magnetic field given by
ε1
ε2
(2)
B = Re Bz (x)ẑeiky−iωt ,
x
where
Bz (x) =
B1 e+q1 x (x < 0)
.
B2 e−q2 x (x > 0)
(3)
d) Using the boundary conditions for Bz at the x = 0 surface, find a relation between B1 and B2 .
e) Using the continuity of Ey at the x = 0 surface, find a relation between q1 and q2 . Show that ε1
and ε2 must have opposite sign in order to have q1,2 > 0, i.e. vanishing fields for |x| → ∞.
f ) From the results of points a) and e) find the dispersion relation ω = ω(k) as a function of ε1 and
ε2 , showing that wave propagation is possible if ε1 + ε2 < 0.
g) If the medium 1 is the vacuum (ε1 = 1), how should the medium 2 and the frequency be chosen
in order to fulfill the condition found at point f )?
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Solution
a) In a dielectric medium described by ε = ε(ω), a monochromatic EM field F of frequency ω satisfies
the Helmoltz equation (written down for Bz )
ω2
2
∇ +ε 2 F=0.
(4)
c
Inserting the expression (1) for Bz , we obtain
q2 − k2 +
ω2
ε=0.
c2
(5)
b) From the equation ∇ × B = µ0 (J + ε0 E) and the definition of ε we obtain (for monochromatic
waves in complex notation) ∇ × B = −iωεE/c2 . By inserting (1) we obtain
−iωεE = (x̂∂y − ŷ∂x )Bz c2 = (ikx̂ − qŷ)Bz c2 ,
(6)
which gives for the electric field
E=−
kx̂ + iqŷ
Bz .
εω
(7)
c) From the definition of S = ε0 c2 E × B we find that S has components both along x and y, given
by
qB02 2qx
e cos(ky − ωt) sin(ky − ωt) ,
εω
kB 2
= −ε0 c2 Ex Bz = ε0 c2 0 e2qx cos2 (ky − ωt).
εω
Sx = ε 0 c 2 E y B z = ε 0 c 2
(8)
Sy
(9)
Taking the average over an oscillation period, hSx i = 0 thus the net energy flux is in the y-direction
and is given by hSy i 6= 0.
d) The component of the magnetic field parallel to a discontinuity surface is continous. Posing
Bz (0− ) = Bz (0+ ) we find B1 = B2 .
e) The component of the electric field parallel to a discontinuity surface is also continous, thus
Ey (0− ) = Ey (0+ ). Using the results of points b) and d) we obtain
q1
q2
=− .
ε1
ε2
Since q1,2 > 0 must hold, we obtain that ε1 and ε2 have opposite sign.
f ) Using the relationship (q1 /ε1 )2 = (q2 /ε2 )2 and the result of point a) we obtain
ω2
ω2
2
2
2
2
ε2 k − 2 ε1 = ε1 k − 2 ε2 ,
c
c
2
(10)
(11)
from which it follows that
ω 2 = k 2 c2
ε22 − ε21
ε2 + ε1
= k 2 c2
.
2
2
ε2 ε1 − ε1 ε2
ε2 ε1
(12)
Propagation of the wave requires k 2 > 0 thus, being ε1 ε2 < 0, we get the additional condition
ε1 + ε2 < 0.
g) Since ε2 < −ε1 = −1 must hold, we may choose a metal√(a free electron gas, or an ideal plasma)
for which ε2 = 1 − ωp2 /ω 2 , and a frequency such that ωp > 2ω.
The above described EM modes are surface waves (also named surface plasmons. These waves
propagate along the surface of a conductor and are evanescent along the perpendicular direction, so
that the EM energy is confined in a narrow layer, thinner than the wavelength in vacuum. Surface
waves are a building block of plasmonics, a discipline oriented to develop optical and electronic
devices on a nanometric scale.1
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See e.g. W. L. Barnes et al, “Surface plasmon subwavelength optics”, Nature 424, 824 (2003); E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions”, Science 311, 189 (2006).
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