Chapter 12

GUIDED MICROWAVES ANDOPTICAL WAVES

12.1 Introduction

In communication engineering, the carrier frequency has been steadily increasing for the obvious

reason that a carrier wave with a higher frequency can accommodate more information. The

frequency band currently used in communication is vastly wide, ranging from MHz (AM radio)

to 1015 Hz (visible light). Electromagnetic waves with frequency higher than about 1GHz can be

con�ned in a waveguide. (At lower frequencies, the size of waveguides would be impractically large.

Exception is the natural, global waveguide formed by the ionospheric plasma and the earth, which

can be used for low frequency communication, such as short wave radio.) Microwaves (1 GHz f 300

GHz) can be propagated in a hollow conductor tube, while optical waves in a dielectric �ber. In both

cases, wave propagation mechanism can be qualitatively understood in terms of wave re�ection at

the waveguide wall. However, in contrast to electromagnetic waves in free space or in transmission

lines, waves con�ned in waveguides cannot be purely TEM. In microwave waveguides, either TE

(Transverse Electric) or TM (Transverse Magnetic) modes can exist, but not TEM modes. This

important deviation from TEM propagation mode is due to the boundary conditions imposed on

electromagnetic �elds. In optical waveguides practically used, even pure TE or TM mode is not

allowed, except for special propagation modes. Deviation from TEM nature not only complicates

�eld analysis, but also causes inevitable wave dispersion, that is, the propagation velocity becomes

dependent on the wave frequency. This in turn implies that the original waveform sent out is bound

to deform. Of course, the merit of guided waves is in eicient energy or signal transmission along a

desired path.

12.2 Waveguides

Microwave technology was greatly advanced during the World War II when radar detection became

practical in military applications. In general, the resolution of radar detection improves with the

1

frequency, and the invention of high power, high frequency microwave tubes, such as the magnetron,

played key roles in radar technology. Waveguides are used to transmit microwaves between various

microwave devices. They play the role of wire conductors in low frequency electric circuits. At high

frequency, an open transmission line will become an e¢ cient antenna, while closed transmissionn

lines (such as coaxial cables) are subject to strong dielectic losses. The only practical way to

transmit high frequency electromagnetic waves is to con�ne them in a hollow conductor tube. It is

desirable that the conductivity of the wall material is large to minimize wall losses. (Recall that the

skin depth and consequent wall loss are inversely proportional top!� where � is the conductivity.)

Waveguides most commonly used are either rectangular or circular, but the crosssection shape

can be arbitrary as long as it does not change abruptly along the waveguide. In either case, there is

a lower limit in the wave frequency allowed for propagation (cuto¤ frequency), similar to the case

of wave propagation in a plasma. Waves having frequencies lower than the cuto¤ frequency cannot

be propagated in a wave guide. The origin of the cuto¤ phenomena is in the boundary conditions

at the conductor wall that should be satis�ed by the electric and magnetic �elds, and consequent

deviation from the TEM nature.

If a waveguide is �lled with air, which is usually the case, the waveequation for the electromag-

netic �elds is still given by �r2 � 1

c2@2

@t2

�E (r; t) = 0 (12.1)�

r2 � 1

c2@2

@t2

�H (r; t) = 0 (12.2)

These are formally identical to the wave equation in free space. However, solutions allowed for

E and H as the electromagnetic �elds in a waveguide will be distinctly di¤erent from the TEM

mode we have been studying, because of the boundary conditions for the �elds which impose rather

stringent limitation for allowable solutions. Analysis of electromagnetic �elds in a waveguide boils

down to solving Eqs. (12.1) and (12.2) under appropriate boundary conditions.

Each equation has three components. For example, in the cartesian coordinates, Eq. (12.1) can

be decomposed into three scalar wave equations for Ex; Ey, and Ez. In a cylindrical waveguide, the

cylindrical coordinates (�; �; z) appear to be most convenient. Unlike the cartesian coordinates,

the vector wave equation in the cylindrical coordinates cannot be separated into three scalar wave

equations because of the complexity in the vector Laplacian. Fortunately, however, the axial

(z) component of the vector wave equation reduces to a scalar wave equation in the cylindrical

coordinates as well which we know how to handle. This suggests the possibility that the entire

electromagnetic �elds in a waveguide may be described by two axial components, Ez and Hz,

because both electric and magnetic �eld components are not entirely independent but constrained

through the Maxwell�s equations. Indeed, to describe electromagnetic �elds in a waveguide, it is

su¢ cient to solve the following two scalar wave equations�r2 � 1

c2@2

@t2

�Ez (r; t) = 0 (12.3)

2

�r2 � 1

c2@2

@t2

�Hz (r; t) = 0 (12.4)

as long as the waveguide crosssection does not change along the axis. The transverse components,

E? and H?, can be fully described by the axial �elds, Ez and Hz. Evidently, solving a scalar wave

equation is much simpler than solving a vector wave equation.

To see how this is done, let us go back to the Maxwell�s equations from which the wave equations

have been derived,

r�E = ��0@H

@t;r�H = "0

@E

@t(12.5)

Since the direction of wave (energy) propagation is obviously along the waveguide (z direction), we

may assume the phase �nction

ej(!t�kzz)

where kz is the axial wavenumber in the z direction. (kz is called the phase constant in engineerig

and often denoted by � which, however, is confusing with the normalized velocity � = v=c in

physics.) As will be shown, the ratio !=kz is not equal to c, as for TEM modes, but exceeds c. At

this stage, kz is yet to be determined. For the assumed phase function, the gradient operator along

the z axis and time derivative can be replaced by

@

@z! �jkz;

@

@t! j!:

Also, the electric and magnetic �elds may be decomposed into axial and transverse components as

E = E? +Ez; H = H? +Hz; (12.6)

Then, Eq. (12.5) becomes

r? �E? +r?Ez � ez � jkzez �E? = �j!�0 (H? +Hz) (12.7)

r? �H? +r?Hz � ez � jkzez �H? = j!"0 (E? +Ez) (12.8)

Note that r?� ez = 0 because ez is a unidirectional constant vector. From the transverse compo-

nents of Eq. (12.8), we obtain

E? = �j

!"0(r?Hz � ez � jkzez �H?) (12.9)

Substituting this into Eq. (12.7), we �nd the transverse magnetic �eld entirely in terms of the axial

�elds,

H? =�j

(!=c)2 � k2z(kzr?Hz � !"0r?Ez � ez) (12.10)

3

Similarly, the transverse electric �eld in terms of the axial �elds is given by

E? =�j

(!=c)2 � k2z(kzr?Ez + !�0r?Hz � ez) (12.11)

The results clearly indicate that if the axial �elds, Ez and Hz are known, the tranverse components

of the electric and magnetic �elds can be readily calculated. It is also apparent that there can be

no TEM modes in a waveguide. TEM modes are characterized by Ez = Hz = 0. Then, all �elds

must vanish according to Eqs. (12.10) and (12.11).

Although there are no TEM modes in a waveguide, modes with only transverse electric �elds

(TE) and modes with only tranverse magnetic (TM) �elds can exist independently. In fact, TE and

TM modes constitute basic independent modes in general electrodynamics, and such classi�cation

is not limited to waveguide modes. TEM mode is a rather idealized mode of propagation and as

long as the wave source is �nite, there can be no pure TEM mode. For example, an oscillating

electric dipole radiates TM modes and the radial (longitudinal) component of the electric �eld

does not completely vanishes. The radial component is responsible for the angular momentum �ux

which has been brie�y discussed in Chapter 11. Similarly, a magnetic dipole radiates TE modes

having a �nite radial component of the magnetic �eld.

Tranverse electric (TE) modes have no axial electric �eld, Ez = 0, and the electric �eld of TE

modes is determined by the axial magnetic �eld Hz from

E? =�j!�0

(!=c)2 � k2zr?Hz � ez (12.12)

Similarly, TM modes are determined by the axial electric �eld Ez from

H? =j!"0

(!=c)2 � k2zr?Ez � ez (12.13)

In the following Sections, TE and TM modes in rectangular and circular waveguidwes will be

discussed.

12.3 Rectangular Waveguides

We assume a rectangular waveguide having an inner crosssection a � b with a > b, as shown in

Fig. 12.1. (We do not lose generality by making this assumption.) The wall material is assumed

to have a su¢ cently large conductivity so that in the lowest order approximation we can regard

the wall material as an ideal conductor. (Otherwise, analysis will be rather complicated.) Such

approximation is not bad, as long as the skin depth is small enough and the �eld penetration

into the wall is negligible. Later, in evaluation of power loss due to the �nite (non-in�nite) wall

conductivity, we will remove this assumption, but for now, we assume that the wall is an ideal

conductor.

For TE modes, we need the solution for the axial magnetic �eld H which obeys the following

4

Figure 12-1: Rectangular waveguide.

scalar wave equation, �r2 � 1

c2@2

@t2

�Hz = 0

or more explicitly, �@2

@x2+@2

@y2+@2

@z2� 1

c2@2

@t2

�Hz = 0 (12.14)

The z and t dependence has already been assumed to be in the form ej(!t�kzz), that is,

Hz (x; y; z; t) = Hz (x; y) ej(!t�kzz) (12.15)

Noting @=@z = �jkz; @2=@z2 = �k2z ; @=@t = j!; @2=@t2 = �!2, we may rewrite Eq. (12.15) as�@2

@x2+@2

@y2� k2z +

�!c

�2�Hz (x; y) = 0 (12.16)

This is a two dimensional Helmholtz equation and can be solved by the method of separation of

variables as done for the Laplace equation. The boundary conditions for the electric and magnetic

�elds on the surface of an ideal conductor are

Et = 0; Hn = 0 (12.17)

where "t" and "n" indicate the tangential and normal component, respectively. For the geometry

5

assumed in Fig. 12.1, these boundary conditions can be translated into

Ex = 0 at y = 0; b (12.18)

Ey = 0 at x = 0; a (12.19)

Ez = 0 at x = 0; a and y = 0; b (12.20)

Hx = 0 at x = 0; a (12.21)

Hy = 0 at y = 0; b (12.22)

Let us assume that the solution for H(x; y) is separable as

H(x; y) = H0X(x)Y (y) (12.23)

where X(x) is a function of x only, and Y (y) is a function of y only. Then, Eq. (12.17) reduces to

1

X

d2X

dx2+1

Y

d2Y

dy2+�!c

�2� k2z = 0 (12.24)

Since (!=c)2 � k2z is a constant, the functions X and Y must be either sinusoidal or exponential

function. However, exponential functions cannot satisfy the boundary conditions, Eq. (12.19).

Therefore, solutions for X and Y must be sinusoidal.

Since the axial magnetic �eld Hz is tangential to the wall everywhere, the boundary condition

for the magnetic �eld, Eq. (12.20), is not useful. However, the boundary condition for the electric

�eld, Eq. (12.19), enables us to determine the magnetic �eld Hz(x; y) as follows. From the x

component of Eq. (12.10) wherein Ez = 0 for TE mode, we have

Ex =�j!�0

(!=c)2 � k2z@Hz@y

(12.25)

This should vanish at y = 0 and b. Since Hz(x; y) is sinusoidal, its derivative with respect to y is

still sinusoidal. Therefore, the solution for Ex should contain a function

sin�n�by�

(12.26)

where n is an integer. Integration of Eq. (12.23) thus determines the function Y (y),

Y (y) = cos�n�by�

(12.27)

Similarly, from the boundary condition for the y component of the electric �eld, we �nd

X (x) = cos�m�ax�

(12.28)

6

and the general solution for the axial magnetic �eld becomes

Hz (x; y; z; t) = H0 cos�m�ax�cos�n�by�ej(!t�kzz) (12.29)

The integers m and n cannot be zero simultaneously. (If so, both Ex and Ey identically vanish.)

Substitution of Eq. (12.27) into the original wave equation, Eq. (12.15), yields the following

relationship between the frequency ! and the axial wavenumber kz,�!c

�2= k2z +

�m�a

�2+�n�b

�2(12.30)

or

!2 = (ckz)2 + !2c

where !c is given by

!c = c

r�m�a

�2+�n�b

�2(12.31)

In Hz,

fc =c

2

r�ma

�2+�nb

�2(12.32)

Only waves having frequencies higher than !c can exist in the waveguide. The frequncy !c is thus

called the cuto¤ frequency. For a given generator frequency !, the dispersion relation determines

the value of kz uniquely. The lowest frequency allowed for wave propagation occurs at kz = 0. The

!�kz relationship is shown in Fig. 12.2. Note that the dispersion relation is identical to that of theelectromagnetic waves in a plasma, if the plasma frequency !p is replaced by the cuto¤ frequency.

Although the dispersion relations are formally identical, the plasma mode is still TEM as we have

seen in Chapter 10. Therefore, physics behind the waveguide mode and plasma mode is distinctly

dierent. p1 + x2

For a given crosssection of a rectangular waveguide, the smallest cuto¤ frequency of TEmnmodes occurs when m = 1; n = 0. (Recall the assumption a > b.) The cuto¤ frequency of the TE10mode is given by

fc10 =c

2a(12.33)

The second smallest cuto¤ frequency is that of TE20 or TE01 mode because most rectangular

waveguides have a ratio a=b ' 2. The cuto¤ frequency of these modes are given by

fc20 =c

a; fc01 =

c

2b(12.34)

The reason for the particular ratio a=b ' 2 is to avoid degeneracy between modes, that is, for a

given generator frequency, only a single mode can be excited in a waveguide. In practice, the TE10mode is most frequently used, and we will study more about this particular mode in the following

7

0 1 2 30

1

2

3

x

y

Figure 12-2: Dispersion relation ! =q(ckz)

2 + !2c : y axis: !=!c: x axis ckz=!c: The straight lineshows ! = ck (propagation in free space).

Section.

12.4 Field Pro�les of TE10 Mode

In a waveguide, the direction of energy propagation is evidently along the wageguide, that is, in

the z direction. The Poynting vector is therefore expected to be directed in the z direction. In the

cartesian coordinates, the z component of the Poynting vector is given by

Sz = (E�H�)z = ExH�y � EyH�

x; W=m2 (12.35)

For the TE10 mode, Ex = 0 (because n = 0), and the Poynting vector reduces to

Sz = �EyH�x (12.36)

If the axial magnetic �elds of the TE10 mode is assumed to be

Hz (x; z; t) = H0 cos��ax�ej(!t�kzz) (12.37)

the y component of the electric �eld can be found from Eq. (12.11)

Ey (y; z; t) = �j!�0�=a

H0 sin��ax�ej(!t�kzz) (12.38)

and the x component of the magnetic �eld from either Eq. (12.10) or more directly from r�H = 0,

Hy =jkz�=a

H0 sin��ax�ej(!t�kzz) (12.39)

8

Other components are zero. If we introduce a complex amplitude for the electric �eld through

E0 = �!�0�=a

H0 (12.40)

the �eld components can be rewritten as

Ey = E0 sin��ax�ej(!t�kzz) (12.41)

Hx = �kz!�0

E0 sin��ax�ej(!t�kzz) (12.42)

Hz = j�=a

!�0E0 cos

��ax�ej(!t�kzz) (12.43)

The real parts of these �elds are

ReEy = E0 sin��ax�cos (!t� kzz) (12.44)

ReHz = �kz!�0

E0 sin��ax�cos (!t� kzz) (12.45)

ReHz = ��=a

!�0E0 cos

��ax�sin (!t� kzz) (12.46)

which allow us to plot the �eld pro�les (snap-shot) at a given instant, say t = 0. This is shown

in Fig. 12.3 over one axial wavelength, �z = 2�=kz. The pro�les shown propagate with the phase

velocity !=kz in the z direction.

Observe that the boundary conditions for the �elds E and H are satis�ed on the waveguide

wall. The electric �eld is normal to the wall and the magnetic �eld is tangential. The induced

surface charge � (C/m2) and the electric �eld at the wall surface are related through

En =�

"0(12.47)

as we learned in Chapter 3. Since the induced charge is varying with time, a surface current must

�ow according to the charge conservation

@�

@t+r � Js = 0 (12.48)

where Js (A/m) is the surface (sheet) current density on the wall surface. The surface current andthe tangential component of the magnetic �eld at the wall are in turn related through

n�Ht = Js (12.49)

where n is the normal unit vector directed on the inner wall surface. The surface current pro�lesare shown in Fig. 12.5. Note that the surface current at the midway, x = a=2, �ows along the z

9

Ey

Ey

Ey

λz

a

b

Figure 12-3: Electric �eld pro�le of the TE10 mode Ey (x; z) = E0 sin��ax�cos (kzz)

axis. A thin slit cut along the waveguide at this location should not very much disturb the current

pro�le and thus the whole electromagnetic �eld pro�le. This fact is exploited in microwave standing

wave meters. A probe can be inserted through the slit to measure the electric �eld intensity along

the waveguide.

x 10.80.60.40.20

z

2

1.5

1

0.5

0

Figure 12-4: Magnetic �eld pro�le of the TE10 mode in a rectangular waveguide.

12.5 Power Associated with TE10 Mode

The concrete expressions for the relevant �elds of TE10 mode in the preceding Section allow us to

calculate the amount of power carried by the TE10 mode. The power going through the crosssection

10

x 10.80.60.40.20

z

2

1.5

1

0.5

0

Figure 12-5: Surface current density pattern on the upper inner surface.

of the waveguide can be evaluated by integrating the Poynting �ux,

P =

Z a

0dx

Z b

0Sz (12.50)

where the z component of the Poynting �ux is

Sz = �EyH�x

=kz!�0

E20 sin2��ax�

(12.51)

Then

P =kz!�0

E20

Z a

0sin2

��ax�dx� b

=kz!�0

E20ab

2(W) (12.52)

Noting

kz!�0

=

r"0�0

s1�

�fcf

�2we can rewrite

Prms =

r"0�0

s1�

�fcf

�2E20ab

4(12.53)

Example: A rectangular waveguide having an inner crosssection 1� 2 cm2 (a = 2 cm, b = 1 cm)is excited by a klystron at a frequency 9 GHz and an RMS power of 2 W. Estimate the peak

11

electric �eld in the waveguide. What is the phase velocity? Group velocity?

First, let us calculate the phase and group velocities. since the dispersion relation of waveguide

modes and that of the plasma mode are identical, we can use the results in Chapter 10,

vp =cp

1� (fc=f)2= 5:43� 108m=sec;

vg =p1� (fc=f)2c = 1:66� 108m=sec;

where the cuto¤ frequency is fc = c=2a = 7:5� 109Hz. The peak electric �eld, which occursat x = a=2, can be estimated from Eq. (12.50) by equating the RMS power to 2 W. The

result is E0 = 5:2� 103 V/m.

Often the characteristic impedance of the TE10 mode is de�ned by

ZTE = �EyHx

=

r�0"0

1s1�

�fcf

�2Note that the negative sign here is related to that of the Poynting vector in Eq. (12.34). In terms

of the impedance, the power may be rewritten as

It should be cautioned that the impedance de�ned in Eq. (12.51) is meaningful only for TE

modes. For TM modes, the impedance takes a rather dierent form

ZTM =

r�0"0

s1�

�fcf

�2

12.6 Circular Waveguides

Electromagnetic waves in a conducting cylinder with circular crosssection can also be divided into

TE and TM modes, as in rectangular wave guides. The axial components Ez and Hz satisfy the

scalar wave equation �r2 � 1

c2@2

@t2

�Ez = 0;

�r2 � 1

c2@2

@t2

�Hz = 0 (12.54)

where the Laplacian r2 in the cylindrical coordinates is

r2 = @2

@�2+1

�

@

@�+1

�2@2

@�2+@2

@z2(12.55)

We consider a cylindrical waveguide with inner wall radius a as shown in Fig. 12.6. The boundary

conditions for the electric and magnetic �elds are

E� = Ez = 0 at � = 0

12

a z

xρ

φ

Figure 12-6: Circular waveguide with radius a and the cylyndrical coordinates (�; �; z) :

H� = 0 at � = a

For TE modes, solutions of the wave equation for Hz(�; �; z; t) are required. Separating the z and

t dependence as

Hz(�; �; z; t) = Hz(�; �)ej(!t�kzz) (12.56)

we can reduce the original wave equation to�@2

@�2+1

�

@

@�+1

�2@2

@�2� k2z +

�!c

�2�Hz (�; �) = 0 (12.57)

We have encountered this type of equation in Chapter 3 on electrostatic boundary value problems.

Since Hz(�; �) should be a periodic function of �, the � dependence may be assumed either sin(n�)

or cos(n�) where n is an integer. Then, noting

@2

@�2

sin (n�)

cos (n�)

!= �n2

sin (n�)

cos (n�)

!(12.58)

we can further reduce Eq. (12.73) to an ordinary dierential equation with respect to �,�d2

d�2+1

�

d

d�� k2z +

�!c

�2� n

2

�2

�R (�) = 0 (12.59)

where Hz(�; �) = R(�)F (�) with F (�) being either sin(n�) or cos(n�). Introducing k de�ned by

k2 =�!c

�2� k2z (12.60)

we �nally reduce Eq. (12.75) to the standard form of the Bessel�s equation�d2

d�2+1

�

d

d�+ k2 � n

2

�2

�R (�) = 0 (12.61)

whose solution is the n-th order Bessel function Jn(k�). J0 (x) and J1 (x) are shown in Fig. 12.5.

J0 (x) ; J1 (x)

13

2 4 6 8 10 12 14 16 18

­0.4­0.2

0.20.40.60.81.0

x

y

Bessel functions J0 (x) (solid) and J1 (x) (dashed). J0 (0) = 1:0; J1 (0) = 0:

(The Bessel function of the second kind Nn(k�) should be discarded because it diverges at � = 0.)

Therefore, the general solution for the axial magnetic �eld of circular TE modes may be written as

Hz (�; �; z; t) = H0Jn (k�) ejn�ej(!t�kzz) (12.62)

where sin(n�) and cos(n�) have been replaced by their equivalents, e�jn�. At this stage, the

axial wavenumber kz is yet to be determined. It can be uniquely determined from the boundary

conditions as follows.

In TE modes we are considering, the axial electric �eld is evidently zero, and Ez = 0 at � = a is

automatically satis�ed. The useful boundary condition for TE modes is therefore E� = 0 at � = a.

Recalling Eq. (12.11),

E? _ rHz � ez =�e�@

@�+ e�

1

�

@

@�

�Hz � ez

= e�1

�

@Hz@�

� e�@Hz@�

(12.63)

we see that the � component of the electric �eld is generated by the radial derivative of the axial

magnetic �eld. Therefore, the boundary condition E� = 0 at � = 0 requires that

dJn (k�)

d�= 0 at � = a (12.64)

This is the basic condition to determine the dispersion relation of TE modes in a circular waveguide.

Introducing a dimensionless variable x = k�, we seek roots of the following equation

dJn (x)

dx= 0 (12.65)

Since the Bessel function Jn(x) oscillates, there are in fact in�nitely many roots satisfying Eq.

(12.81). In the Table, some low order solutions are listed. x0mn means the m-th root of dJn(x)=dx =

14

0.n = 0 n = 1 n = 2

m = 1 3:83 1:84 3:05

m = 2 7:02 5:33 8:54

m = 3 10:17 8:54 9:97

Table 12.1 x0mn (m-th root of dJn(x)=dx = 0)

The solution for k is therefore given by

kmn =x0mna

(12.66)

and the desired dispersion relation by

!2 = (ckz)2 + c2

�x0mna

�2= (ckz)

2 + !2c (12.67)

with the cuto¤ frequency de�ned by

!c = cx0mna

(12.68)

The lowest order TE mode (having the smallest cuto¤ frequency) corresponds to the smallest root

of x0mn, which occurs when m = 1; n = 1, x011 ' 1:84. As an example, consider a circular waveguidehaving a radius of 5 mm. The cuto¤ frequency of the TE11 mode in the waveguide is

fc =c

2�a� 1:84 ' 17:6GHz:

Note that the indices m and n of the circular waveguide modes have entirely di¤erent meanings

from those in rectangular modes.

Let us assume that the axial magnetic �eld of the TE11 mode is of the form

Hz (�; �; z; t) = H0J1 (k11�) cos�ej(!t�kzz) (12.69)

We have chosen cos� function because sin� function corresponds to the rotation by an angle �=2

in � direction. In so doing, we do not lose generality. The transverse electric and magnetic �elds

can be found by referring to the general forlumae Eqs. (12.10) and (12.11),

E? =�j!�0

(!=c)2 � k2zr?Hz � ez

=j!�0k2

H0

�1

�J1 (k�) sin�e� +

dJ1 (k�)

d�cos�e�

�(12.70)

15

rectangularTE 10 mode

Taper

circularTE 11 mode

Figure 12-7: Circular TE11 mode and rectangular TE10 mode are topologically in the same family.One mode can be converted to another through a taper with gradual change in the cross-section.

H? =�j!�0

(!=c)2 � k2zr?Hz

=�jkzH0

(!=c)2 � k2zH0

�dJ1 (k�)

d�cos�e� �

1

�J1 (k�) sin�e�

�(12.71)

A qualitative sketch of the �eld pro�les is shown in Fig. 12.7(a). The rectangular TE10 mode

and circular TE11 mode are in fact in the same family and mutually convertible when the sizes of

both waveguides are not vastly dierent. This is illustrated in Fig. 12. 7(b). Conversion can be

achieved by gradual tapering. (Abrupt change in the cross section shape causes large wave re�ection

and should be avoided.) Such mutual conversion between rectangular and circular modes is often

required because many microwave components are based on circular waveguide modes. Typical

examples are attenuators and isolators. (See "Introduction to Microwave Technology".)

12.7 TM Mode in Circular Slow Waveguide

Waveguides used in linear electron accelerators must accommodate TM modes having a phase

velocity close to c. Waveguides with smooth inner walls can only accommodate modes having

phase velocities larger than c and thus cannot be used for this purpose. Modes must be TM

because TE modes have no electric �eld in the axial direction needed to accelerate electrons.

Slow waveguides have conductor diaphragms placed periodically along the axis as shown in

Fig.12-10. The purpose of the diaphragms is to increase the capacitance per unit length of the

waveguide which contributes to slowing down the phase velocity of electromagnetic waves. We

consider modes symmetric about the axis, m = 0: The waveguide has a radius a and diaphragms

have holes of radius b: In the region � < b; the Helmholtz equation for the axial electric �eld is�@2

@�2+1

�

@

@�+�!c

�2� k2z

�Ez(�) = 0; � < b: (12.72)

16

Figure 12-8: Circular slow waveguide of radius a with diaphragms of radius b. The spacing betweendiaphragms is much smaller than the axial wavelength �z:

b

a

Figure 12-9: In the region b < � < a; the eelctric �eld lines are straight, @Ez=@z = 0: In the region� < b; the �eld lines are curved and @Ez=@z = �jkzEz:

Since we are interested in modes having an axial phase velocity slightly smaller than c; that is,

!

kz. c; (12.73)

the quantity (!=c)2 � k2z must be negative, and thus solution for Ez(�) may be assumed to be

Ez(�) = AI0(k�); � < b (12.74)

where

k2 = k2z ��!c

�2> 0: (12.75)

In the diaphragm region b < � < a; the electric �elds lines are essentially straight provided the

axial period of the diaphragms is su¢ ciently smaller than the axial wavelength. We assume that

this condition is met. Then, the wave equation in the region b < � < a may be approximated by�@2

@�2+1

�

@

@�+�!c

�2�Ez(�) = 0; b < � < a: (12.76)

General solutions are

Ez(�) = BJ0

�!c��+ CN0

�!c��; b < � < a: (12.77)

17

The boundary conditions are:

Ez(� = a) = 0; (12.78)

and

Ez and H� be continuous at � = b: (12.79)

These boundary conditions yield

BJ0

�!ca�+ CN0

�!ca�= 0; (12.80)

AI0(kb) = BJ0

�!cb�+ CN0

�!cb�; (12.81)

A

kI1(kb) = �

c

!

hBJ1

�!cb�+ CN1

�!cb�i; (12.82)

where J 00(x) = �J1(x); N 00(x) = �N1(x); I 00(x) = I1(x) are noted. Eqs. (12.80) through (12.82)

give the following dispersion relation

ck

!

I0(kb)

I1(kb)=J0

�!ca�N0

�!cb�� J0

�!cb�N0

�!ca�

J1

�!cb�N0

�!ca�� J0

�!ca�N1

�!cb� : (12.83)

For the purpose of accelerating highly relativistic electrons, the axial phase velocity !=kz must be

close to c or k ' 0: Then I0(kb) ' 1; I1(kb) ' kb=2; and the LHS of Eq. (12.83) reduces to 2c=!b:For a given rf frequency ! and the size of the waveguide a; the dispersion relation

2c

!b'J0

�!ca�N0

�!cb�� J0

�!cb�N0

�!ca�

J1

�!cb�N0

�!ca�� J0

�!ca�N1

�!cb� ; (12.84)

can be solved numerically to determine the aspect ratio a=b of a slow wave circular waveguide.

Fig.12-10 shows the function

f(x) =2a

xb�J0 (x)N0

�b

ax

�� J0

�b

ax

�N0 (x)

J1

�b

ax

�N0 (x)� J0 (x)N1

�b

ax

� ; x = !a

c; (12.85)

when a=b = 2:5: The �rst root occurs at x ' 3:89 and for a given rf frequency !; the outer radiusa can thus be determined.

12.8 Dielectric Waveguides

An optical �ber can con�ne light waves because of total re�ection at the surface. In contrast to

conductor waveguides, light waves in optical waveguides cannot be pure TE or TM modes. This

is because electromagnetic �elds outside, as well as inside, the optical �ber must be considered

18

x 43.93.83.73.63.5

1

0.5

0

­0.5

­1

Figure 12-10: Root of f(x) = 0 when a=b = 2:5:

simultaneously. Although the outer �elds are evanescent (otherwise waves cannot be con�ned), the

�elds near the surface do a¤ect those inside.

We �rst consider a simple case of step change in the index of refraction,

n(�) =

(n; � < a;

1; � > a:

Such an optical �ber is of no practical interest, for �bers used in optical communication all have

graded index of refraction with a gradual change with the radius �: The axial electric �eld Ez(r; t)

satis�es the wave equations in both regions,�@2

@�2+1

�

@

@�+1

�2@2

@�2+@2

@z2+ �0"!

2

�E<z (r) = 0; � < a; (12.86)

�@2

@�2+1

�

@

@�+1

�2@2

@�2+@2

@z2+ �0"0!

2

�E>z (r) = 0; � > a: (12.87)

The azimuthal dependence may be assumed to be eim� and the axial dependence eikzz;

E(r) = E(�)eim�+ikzz:

Then, �d2

d�2+1

�

d

d�� m

2

�2+ �0"!

2 � k2z�E<z (�) = 0; � < a; (12.88)�

d2

d�2+1

�

d

d�� m

2

�2+ �0"0!

2 � k2z�E>z (�) = 0; � > a; (12.89)

19

which admit the following bounded solutions,

E<z (�) = AJm(k1�); � < a; (12.90)

E>z (�) = BKm(k2�); � > a: (12.91)

Here

k1 =p�0"!

2 � k2z ; k2 =pk2z � �0"0!2: (12.92)

Note that the outer �eld should be evanescent for the waveguide to con�ne light waves.

Similarly, the axial magnetic �eld Hz(r) may be assumed to be

H<z (�) = CJm(k1�); � < a; (12.93)

H>z (�) = DKm(k2�); � > a: (12.94)

The transverse �elds E? and H? can then be calculated by referring to Eqs. (??) and (??). Theazimuthal components of the �elds are

E<� (�) =�ik21

�kzim

�AJm(k1�)� !�0k1CJ 0m(k1�)

�; (12.95)

E>� (�) =i

k22

�kzim

�BJm(k2�)� !�0k2DK 0

m(k2�)

�; (12.96)

H<� (�) =

i

k21

�kzim

�CJm(k1�) + !"k1AJ

0m(k1�)

�; (12.97)

and

H>� (�) =

�ik21

�kzim

�DKm(k2�) + !"0k2BK

0m(k1�)

�: (12.98)

The continuity of Ez; E�;Hz and H� yields

AJm(k1a) = BKm(k2a); (12.99)

CJm(k1a) = DKm(k2a); (12.100)

� 1

k21

�kzim

aAJm(k1a)� !�0k1CJ 0m(k1a)

�=1

k22

�kzim

aBKm(k2a)� !�0k2DK 0

m(k2a)

�;

(12.101)1

k21

�kzim

aCJm(k1a) + !"k1AJ

0m(k1a)

�= � 1

k22

�kzim

aDKm(k2a) + !"0k2BK

0m(k1a)

�: (12.102)

20

Then the determinantal dispersion relation is�1

k21+1

k22

�2�mkza

�2=

�!

k1c1

�2�J 0m(k1a)Jm(k1a)

�2+

�!

k2c2

�2�K 0m(k2a)

Km(k2a)

�2+!2

k1k2

�1

c21+1

c22

�J 0m(k1a)K

0m(k2a)

Jm(k1a)Km(k2a): (12.103)

Figure 12-11 shows the dispersion relation, namely, axial wavenumber kz normalized by k0 = !=c

as a function of the normalized frequency k0a = !a=c when m = 1 and n = 1:1: It can be seen

that a cuto¤ occurs at !a=c ' 1:386: Fig.12-12 shows the case when m = 2: The cuto¤ frequency

increases to !a=c ' 5:43:

x 302520151050

1.1

1.08

1.06

1.04

1.02

1

Figure 12-11: ckz=! vs. !a=c when n = 1:1; m = 1: The cuto¤ frequency is !ca=c ' 1:386:

x 302520151050

1.1

1.08

1.06

1.04

1.02

1

Figure 12-12: ckz=! vs. !a=c when n = 1:1;m = 2: The cuto¤ frequency is !ca=c ' 5:43:

21

12.9 Graded Index Fibers

In optical �bers used in practical communication, the index of refraction is designed to have gradual,

rather than step, variation with the radius. Quadratic variation is commonly employed,

n(�) = n0�1� �2�2

�; � = constant (12.104)

because the electromagnetic �elds are then well con�ned with a Gaussian pro�le e�a2�2 . The

corresponding permittivity is

"(�) = "0n20

�1� �2�2

�2: (12.105)

Since

r � ["(r)E] = "(r)r �E+E � r" = 0; (12.106)

the wave equation

r�r�E = !2�0"(r)E; (12.107)

reduces to

r2E+ !2�0"(r)E+r�E � r""

�= 0: (12.108)

If the change in the permittivity is small, the last term can be ignored in the lowest order approx-

imation, and we obtain a simple wave equation with an inhomogeneous permittivity,

r2E+ !2�0"(r)E ' 0: (12.109)

In the cylindrical geometry, a cartesian component of the transverse electric �eld satis�es�@2

@�2+1

�

@

@�+1

�2@2

@�2+@2

@z2+ !2�0"(�)

�Ei = 0: (12.110)

For weak variation of "(�);

"(�) = "0n20(1� �2�2)2 ' "0n20

�1� 2�2�2

�;

and axially symmetric mode @=@� = 0; Eq. (12.110) reduces to�d2

d�2+1

�

d

d�� k2z + k20

�1� 2�2�2

��E(�) = 0; (12.111)

where

k20 = !2"0�0n

20: (12.112)

Assuming

E(�) = E0e�a2�2 ;

22

we �nd

a2 =1p2k0�; (12.113)

k2z = k20 �

4p2k0�: (12.114)

The electric �eld is con�ned with a Gaussian pro�le in the radial direction. The e-folding radial

distance is

w =1

a=

4p2pk0�

; (12.115)

which is called beam radius. A constant beam radius is maintained only for appropriate injection

of light wave at the input end. If not, the beam radius varies with the axial distance accompanied

by periodic focusing and defocusing as intuitively expected from the picture of

23