Optics and Photonics Journal, 2013, 3, 192-196
doi:10.4236/opj.2013.32B046 Published Online June 2013 (http://www.scirp.org/journal/opj)
Evanescent Fields Inside a Cut-off Waveguide as Near
Fields
Zhi-Yong Wang, Jun Gou, Shuangjin Shi, Qi Qiu
School of Optoelectronic Information, University of Electronic Science and Technology of China, Chengdu, China
Received 2013
ABSTRACT
Usually, electromagnetic evanescent waves are some kinds of near fields. However, it looks as if the evanescent waves
inside a cut-off waveguide had nothing to do with any near field. In this paper, we will show that the evanescent waves
inside a cut-off waveguide can also be regarded as the near fields of an aerial array.
Keywords: Evanescent Fields; Cut-off Waveguide; Near Fields
1. Introduction
From the point of view of classical field theory, an eva-
nescent wave is a standing wave with an intensity that
exhibits exponential decay with distance from the
boundary at which the wave was formed. As far as elec-
tromagnetic evanescent waves are concerned, for exam-
ple, they are formed when waves travelling in a medium
undergo total internal reflection at its boundary; they also
are found in the near field region of an antenna, where
the antenna emits electromagnetic fields into the sur-
rounding near field region, and a portion of the field en-
ergy is re-absorbed provided that there is no receiver,
while the remainder is radiated as electromagnetic waves.
In quantum mechanics, the evanescent-wave solutions of
the Schrödinger equation give rise to the phenomenon of
quantum tunneling. In optics, evanescent wave coupling
is a process by which electromagnetic waves are trans-
mitted from one medium to another by means of the
evanescent, exponentially decaying electromagnetic field.
Mathematically, the process is the same as that of quan-
tum tunneling, except with electromagnetic waves in-
stead of quantum-mechanical wavefunctions. As a result,
people call the process photonic quantum tunneling.
Usually, electromagnetic evanescent waves are some
kinds of near fields (e.g., the ones in total internal reflec-
tion). However, it looks as if the evanescent waves inside
a cut-off waveguide had nothing to do with any near field.
In this paper, by means of another way of looking at the
guided waves, we will show that the evanescent fields
inside a cut-off waveguide can be regarded as the near
fields of an aerial array.
2. Another Way of Looking at the Guided
Waves
Let us assume that a hollow rectangular waveguide is
placed along the direction of z-axis, and the waveguide is
a straight perfect metal pipe with the transversal dimen-
sions a and b (a>b, the cross-section of the waveguide
lies in 0≤x≤a and 0≤y≤b).
For convenience let us just consider the TE10 mode, in
which the transverse electric field is perpendicular to
z-axis and with only a y-component Ey that will vary with
x and z. In terms of the frequency ω and wave-number
vector (, ,)
x
yz
kkk

k (with ), the transverse
electric field Ey can be written as
0
y
k
0sin exp[i()], yx z
EEkx tkz


 (1)
where E0 is a constant factor. For the TE10 mode,
π
x
ka

and 222 2 π
z
kc  a are the wavenum-
bers along the x- and z-axis directions, respectively,
where c is the velocity of light in vacuum. The cutoff
frequency of the TE10 mode is cπ
x
kc ca

 . There
are no charges in the free space inside the waveguide,
such that Ey must satisfy the wave equation
2222
22222
0.
yyyy
EEEE
xyzct


 
 (2)
According to the traditional waveguide theory one has:
1) for c,

  the electromagnetic field inside the wave-
guide is the propagation mode (i.e., the travelling wave),
and its phase and group velocities are
2
pc
1( )
z
vkc

 
and 2
gc
1( )
z
vkc

   ,

Z.-Y. WANG ET AL. 193
respectively. For the moment, one has 2π
z
g
k


,
where 2
00
1( 2)
ga
   is the wavelength of the
oscillation along the z-direction (i.e., the guide wave-
length), and it is different from the free-space wavelength
02πc

  of electromagnetic waves of the same fre-
quency. 2) for c

 , i.e., 02a

, the wave number
z
k (and also
g

) becomes imaginary, it follows from Eq.
(1) that EE
0sin e yx
kxxp( ),t)exp(iz

  where
22
c,c
 
then the electromagnetic field inside the waveguide oscil-
lates with time as exp(i )t

and varies with z as exp( ),z


and is called the evanescent field.
For our purpose, let us discuss another way of looking
at the guided waves [1]. For the TE10 mode described
above, the vertical dimension (in y) had no effect, so we
can ignore the top and bottom of the waveguide and
imagine that the waveguide is extended indefinitely in
the vertical direction. Then the waveguide can be imag-
ined as just consisting of two vertical plates with the
separation a. Let’s say that the source of the fields is a
vertical wires placed in the middle of the waveguide,
with the wire carrying a current that oscillates at the fre-
quency ω. In the absence of the waveguide walls such a
wire would radiate cylindrical waves. Consider that the
waveguide walls are perfect conductors, the conditions at
the surface will be correct if we add to the field of the
wire the field of one or more suitable image wires. The
image idea works just as well for electrodynamics as it
does for electrostatics, provided that we also include the
retardations. Now let’s take a horizontal cross section, as
shown in Figure 1, where W1 and W2 are the two guide
a
S
Line
source
S
S
S
S
S
S
Image
sources
Image
sources
W1
W2
Waveguide
Figure 1. The line source S0 between the conducting plane
walls W1 and W2.
walls and S0 is the source wire. Assume that the direction
of the current in the wire is positive. Now if there were
only one wall, say W1, one could remove it if an image
source (with opposite polarity) were placed at the posi-
tion marked S1. But with both walls in place there will
also be an image of S0 in the wall W2, which is shown as
the image S2. This source, too, will have an image in W1,
which is called S3. Now both S1 and S3 will have images
in W2 at the positions marked S4 and S6, and so on. For
our two plane conductors with the source halfway be-
tween, the fields are the same as those produced by an
infinite line of sources, all separated by the distance a.
For the fields to be zero at the walls, the polarity of the
current in the images must alternate from one image to the
next. In other words, they oscillate 180 out of phase.
The waveguide field is, then, just the superposition of the
fields of such an infinite set of line sources.

The walls can be replaced by the infinite sequence of
image sources.
Let us look at the fields which arrive at a large dis-
tance from the array of image sources. The fields will be
strong only in certain directions which depend on the
frequency—only in those directions for which the fields
from all the sources add in phase. At a reasonable dis-
tance from the source the field propagates in these spe-
cial directions as plane waves. Such a wave is sketched
in Figure 2, where the solid lines represent the wave
0

5
s
3
s
1
s
0
s
2
s
4
s
6
s
a

0
2

vc
0
r
2
r
Figure 2. One set of coherent waves from an array of
line sources.

Z.-Y. WANG ET AL.
194
crests and dashed lines represent the troughs. The wave
direction will be the one for which the difference in the
retardation for two neighboring sources to the crest of a
wave corresponds to one-half a period of oscillation. In
other words, the difference between r2 and r0 in the fig-
ure is one-half of the free-space wavelength: 20 rr


02

. The angle θ in then given by 0 sin 2a

 .
There is, of course, another set of waves traveling
downward at the symmetric angle with respect to the
array of sources. The complete waveguide field (not too
close to the source) in the superposition of these two sets
of waves, as shown in Figure 3. The actual fields are
really like this, of course, only between the two walls of
the waveguide.
At points like A and C, the crests of the two wave pat-
terns coincide, and the field will have maximum, and
points like B, both waves have their peak negative value,
and the field has its minimum (largest negative) value.
As time goes on the field in the guide appears to be trav-
eling along the waveguide with wavelength
g

, which is
the distance from A to C. That distance is related to θ by
0
cos
g

 , using 0
sin 2a

 , one has
2
000
cos1(2 ),
ga
   (3)
which is just what we found before.
Now we see why there is only wave propagation above
the cutoff frequency c

. If the free-space wavelength is
longer than 2a, there is no angle where the waves shown
in Figure 2 can appear. The necessary constructive in-
terference appears suddenly when λ0 drops below 2a, or
when ω goes above cπca

. If the frequency is high
enough, there can be two or more possible directions in
which the waves will appear. In general, it could happen
when 0a


sin
. These additional waves correspond to the
higher guide modes. It has also been made evident by our
analysis why the phase velocity of the guided waves is
greater than c and why this velocity depends on ω. As ω
is changed, the angle of the free waves of Figure 2
changes, and therefore so does the velocity along the
waveguide. Although we have described the guided wave
as the superposition of the fields of an infinite array of
line sources, one can see that we could arrive at the same
result if we imagined two sets of free-space waves being
continually reflected back and forth between perfect
mirrors—remembering that a reflection means reversal
of phase. These sets of reflecting waves would all cancel
each other unless they were going at just the angle θ
given in 0/ 2a

 .
3. Evanescent Fields Inside A Cut-Off
Waveguide as Near Fields
As shown in Figures 1-3, if we are close to the sources,
the field is very much like the static fields. Here the av-
erage source strength is zero, because the sign alternates
Figure 3. The waveguide field can be viewed as the super-
position of two trains of plane waves.
from one source to the next. In other words, close to the
source, we see the field mainly of the nearest source; at
large distances, many sources contribute and their aver-
age effect is zero. So now we see why the waveguide
below cutoff frequency gives an exponentially decreas-
ing field. At low frequency, in particular, the static ap-
proximation is good, and it predicts a rapid attenuation of
the fields with distance (on the other hand, at high fre-
quencies the retardation of the fields can introduce addi-
tional changes in phase which can cause the fields of the
out-of-phase sources to add instead of canceling, such
that the waves can propagate, just as discussed in Section
2). This implies that evanescent fields inside the cutoff
waveguide have a close relationship with near fields.
In fact, in frustrated total internal reflection, evanes-
cent fields are directly identical with near fields consist-
ing of virtual photons [2,3], these virtual photons corre-
spond to the elementary excitations of electromagnetic
interactions. Now we will show that evanescent fields
inside an undersized waveguide are also identical with
near fields and then consist of virtual photons. As we
know, the near fields of a dipole antenna fall off with the
distance r from the antenna like 1n r (). However,
if we assume that an aerial array formed by an infinite set
of infinite-length line sources arranging in a periodic
manner, then the near fields of the aerial array falls off
like
2n
exp( )r


, from which one can show another way
of understanding why a waveguide attenuates the fields
exponentially for frequencies below the cutoff frequency.
As mentioned in Section 2, the guided wave can be
described as the superposition of the fields of an aerial
array formed by an infinite set of infinite-length line
sources arranging in a periodic manner (with the period
of 2a), there is a out-of-phase between two neighboring
line sources (because the sign alternates from one source
to the next). In view of the fact that if we are close to
these line sources, the field is very much like the static
fields, let us firstly study the static field of a grid of line
sources. To associate the static field with the above

Z.-Y. WANG ET AL. 195
guided wave in the limit of 0

, the grid is taken as
an array of parallel wires lying in a plane, where the
wires are infinitely long and with a uniform spacing of a
between them, and carry uniform charge with the sign
alternates from one source to the next, such that the
grid’s period is 2a. At a large distance above the plane of
the wires, the electric field vanishes because of the grid
being neutral in total. While, as we approach the grid of
wires, the field begins to deviate from which one found
at large distances from the grid. The closer one gets to
the grid, the larger the variations. Traveling parallel to
the grid, one observes that the field fluctuates in a peri-
odic manner. Any periodic quantity can be expressed a
sum of sine waves (Fourier’s theorem). If the wires lie in
the xy-plane and run parallel to the y-axis, one has
(,) ()sinπ,
yn n
ExzFznxa (4)
where n is the harmonic number (we have assumed long
wires, so there should be no variation with y). A com-
plete solution would be made up of a sum of such terms
for . Eq. (4) must satisfy Laplace’s equation
in the region above the wires (there are no charges), i.e.,
1, 2,3...n
2222
0
yn yn
Ex Ez, using Eq. (4) one has
222
22
dπ,
d
n
n
Fn
F
za
 (5)
it follows that ()exp()
nnn
F
zE z

, where π
nna


,
, and are the constant coefficients, then Eq. (4)
becomes
1nn E
(,) sin(π)exp( ).
yn nn
ExzEnxaz


(6)
That is, each Fourier component of the field will de-
crease exponentially with a characteristic distance 1n

πan. In other words, the near fields of the aerial array
falls off like exp( )
nz

. Comparing Eq. (1) with Eq. (6),
one can find that the static field with n=1 is equivalent to
the guided wave of the TE10 mode in the limit of 0

,
such that the evanescent field inside the cut-off
waveguide is equivalent to the near field of the aerial
array. To show that the TE10 mode with the frequency
c
0

  is equivalent to the near field of the aerial
array, that is, to obtain Eq. (1) with c 0

  from Eq.
(6) with n=1, one ought to make the replacement of
c
00

  , such that one has the following
replacements:
c
00 ,

   (7)
1exp(i0)exp(i ),tt

 (8)
222 2
cc
π11
0i,
z
k
ac c
    (9)
(,) (,,)
yy yy
EExz EExzt (10)
2222 2
22 2222
()0( )
y
E
xz xzct
 
 
 0,
y
E (11)
0
0
π
(,)sin( )exp()
π
(,,)sin( )exp(i),
y
y
x
ExzEz
a
x
Exzt Etz
a





(12)
where 1


, 1
y
y
EE

, 222 2 π
z
kca  . As
shown in presence of the decay factor
Fi
)z
gures 2-3, the
exp(


impli field y ex-
ists in the neighborhood of the aerial array, and then is
the neaield of the aerial array. , evanescent
fields inside an undersized waveguide can also be de-
scribed as near fields, such that the field quanta of eva-
nescent fields inside the undersized waveguide are also
virtual photons.
It is important to note that, from the point of view of
quantum mechanics, as a wavepacket falls off with a
di
es that the(,,)
y
Exzt
mainl
r fTherefore
stance L, it is the probability of photons propagating
the distance of L that decays with L. Therefore, though a
wavepacket inside a barrier contains an exponentially
decay factor such as exp( )r


, it is not implies that
inside the barrier the wavepacket exponentially decays
with the propagation d That is, the particle or
wave packet which has entered the barrier is not attenu-
ated, because the reflection takes place at the barrier
front, where the barrier length only determines exponen-
tially how many photons are reflected at the front already.
All discussions here are similar for any evanescent field
inside a cut-off waveguide.
4. Conclusions and Discusses
istance r.
fields inside a cut-
near fields, owing
the Fundamental Research
ersities (Grant No: ZYGX
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causality.
5. Acknowledgements
This work was supported by
Funds for the Central Univ
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