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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

, which is

the distance from A to C. That distance is related to θ by

0

cos

, 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

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