Zhang K., Li D. Electromagnetic Theory for Microwaves and Optoelectronics

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6.1 Metallic Waveguide with Diﬀerent Filling Media 321

Solving the above two equations for given ω and substituting the roots of k

and k

into (6.2) or (6.3), we get the longitudinal phase coeﬃcient of the

mode:

β =

− k

. (6.15)

When β = 0, the waveguide is cut oﬀ, ω = ω

. Then (6.2) and (6.3)

become

= ω

√

, k

= ω

√

Substituting these conditions into the characteristic equation (6.13), we ob-

tain

tan ω

√

h =

tan ω

√

(h − a). (6.16)

This is a transcendental equation and its mth root is the cutoﬀ angular

frequency of the TE

mode in a rectangular waveguide with two diﬀerent

ﬁlling media.

Supposing µ

> µ

, we may then write

< ω

, (6.17)

where ω

= mπ/a

√

, ω

= mπ/a

√

So, only the TE

modes can satisfy the boundary conditions by itself

alone. The TE

modes with n 6= 0 can not satisfy the boundary conditions,

i.e., cannot propagate in the waveguide alone. The modes with both m 6= 0

and n 6= 0 can exist in the waveguide only in the form of hybrid modes.

(2) TM Modes

For TM mo des, V = 0 and H

= 0. Applying the boundary conditions on

the conducting wall, we have the functions U

and U

for the two regions:

= A sin k

x sin k

−jβz

, 0 ≤ x ≤ h, (6.18)

= B sin k

(x − a) sin k

−jβz

, h ≤ x ≤ a, (6.19)

where k

= nπ/b.

The boundary conditions at x = h are such that the tangential compo-

nents of the electric and magnetic ﬁelds H

, E

, and E

on the two sides of

the boundary must be continuous, which gives us

y 1

(h) = H

y 2

(h) → B =

cos k

(h − a)

A, (6.20)

y 1

(h) = E

y 2

(h) → ²

cot k

h = ²

cot k

(h − a) (6.21)

and

(h) = E

(h) → (k

)²

cot k

h = (k

)²

cot k

(h−a).

(6.22)

322 6. Dielectric Waveguides and Resonators

As in our previous arguments for TE modes, the above two equations are

satisﬁed simultaneously only when the indices of refraction or the phase ve-

locities in the two media are equal, i.e.,

= µ

For n = 0, i.e., TM

modes, as we mentioned in the last chapter, all

the ﬁeld components are zero. This means that TM

modes cannot exist

in a rectangular waveguide. So, no TM mode alone can satisfy the boundary

conditions.

We come to the conclusion that only TE

modes, i.e., modes with uni-

form ﬁelds along the transverse tangential direction of the boundary of the

two media, can exist in the waveguide with two diﬀerent ﬁlling media. In

other words, the condition of existing TE or TM modes in the waveguide with

two diﬀerent ﬁlling media is that the phase co eﬃcient along the transverse

tangential direction must be zero. The ﬁelds of the other TE or TM modes

alone with nonuniform ﬁelds along the boundary cannot satisfy the boundary

conditions. Those other modes satisfying the boundary conditions must be

hybrid modes, i.e., HEM modes.

6.1.2 LSE and LSM Modes, HEM modes

We have mentioned in Section 4.7 that for rectangular geometry, we may

choose x or y rather than z as the special coordinate u

. In these choices, the

ﬁelds are expressed by LSE

(x)

and LSM

(x)

modes or by LSE

(y )

and LSM

(y )

modes, which are also denoted by TE

(x)

and TM

(x)

modes or TE

(y )

and

(y )

modes, respectively. We have analyzed this kind of modes in rectan-

gular waveguides given in Section 5.3.

Now we try to ﬁnd out whether LSE

(x)

and LSM

(x)

or LSE

(y )

and LSM

(y )

modes can satisfy the boundary conditions of the waveguide with two diﬀer-

ent ﬁlling media. We again deal with the rectangular waveguide with two

diﬀerent ﬁlling media aligned in x direction as shown in Fig. 6.1(a). The

relations of (6.2) and (6.3) are still valid.

(1) TE

(x)

Modes or LSE

(x)

Modes

For LSE

(x)

modes, U

(x)

= 0 and E

= 0. The general expressions for V

(x)

and V

(x)

are

(x)

= A sin(k

x + φ

) cos(k

y 1

y + ψ

−jβz

, 0 ≤ x ≤ h,

(x)

= B sin(k

x + φ

) cos(k

y 2

y + ψ

−jβz

, h ≤ x ≤ a,

6.1 Metallic Waveguide with Diﬀerent Filling Media 323

Applying the ﬁeld component expressions (4.153)–(4.158) and the boundary

conditions on the conducting wall, we have

y =0

= jωµ

∂V

(x)

∂y

y =0

= 0, sin ψ

= 0, ψ

= 0,

y =0

= jωµ

∂V

(x)

∂y

y =0

= 0, sin ψ

= 0, ψ

= 0,

y =b

= jωµ

∂V

(x)

∂y

y =b

= 0, sin k

y 1

b = 0, k

y 1

nπ

= k

y =b

= jωµ

∂V

(x)

∂y

y =b

= 0, sin k

y 2

b = 0, k

y 2

nπ

= k

x=0

= jωµ

∂V

(x)

∂y

x=0

= 0, sin φ

= 0, φ

= 0,

x=a

= jωµ

∂V

(x)

∂y

x=a

= 0, sin(k

a + φ

) = 0, φ

= −k

Functions V

(x)

and V

(x)

become accordingly

(x)

= A sin k

x cos k

−jβz

, 0 ≤ x ≤ h, (6.23)

(x)

= B sin k

(x − a) cos k

−jβz

, h ≤ x ≤ a. (6.24)

Substituting these expressions into (4.153)–(4.158), we obtain the ﬁeld-

component expressions in the two regions

Region 1, 0 ≤ x ≤ h:

y 1

= −jωµ

∂V

(x)

∂z

= −ωµ

βA sin k

x cos k

−jβz

, (6.25)

= jωµ

∂V

(x)

∂y

= −jωµ

A sin k

x sin k

−jβz

, (6.26)

− k

(x)

− k

A sin k

x cos k

−jβz

, (6.27)

y 1

∂

(x)

∂y ∂x

= −k

A cos k

x sin k

−jβz

, (6.28)

∂

(x)

∂z ∂x

= −jβk

A cos k

x cos k

−jβz

. (6.29)

Region 2, h ≤ x ≤ a:

y 2

= −jωµ

∂V

(x)

∂z

= −ωµ

βB sin k

(x − a) cos k

−jβz

, (6.30)

324 6. Dielectric Waveguides and Resonators

= jωµ

∂V

(x)

∂y

= −jωµ

B sin k

(x − a) sin k

−jβz

, (6.31)

− k

(x)

− k

B sin k

(x − a) cos k

−jβz

,(6.32)

y 2

∂

(x)

∂y ∂x

= −k

B cos k

(x − a) sin k

−jβz

, (6.33)

∂

(x)

∂z ∂x

= −jβk

B cos k

(x − a) cos k

−jβz

. (6.34)

Applying the boundary condition that states the tangential components

of ﬁelds must be continuous on the boundary x = h, we have

y 1

(h) = E

y 2

(h), E

(h) = E

(h),

which give

A sin k

h = µ

B sin k

(h − a), (6.35)

and

y 1

(h) = H

y 2

(h), H

(h) = H

(h),

which give

A cos k

h = k

B cos k

(h − a). (6.36)

Then we get the eigenvalue equation of the LSE

(x)

mode by combining (6.35)

and (6.36):

tan k

h =

tan k

(h − a). (6.37)

Solving this equation and (6.14) for a given ω and substituting the roots of

and k

into (6.2) or (6.3), we have the longitudinal phase coeﬃcient of

the TE

(x)

or LSE

(x)

mode:

β =

− k

−

nπ

− k

−

nπ

. (6.38)

For the cutoﬀ state, β = 0, ω = ω

, then

−

nπ

, k

−

nπ

. (6.39)

Substituting (6.39) into the above eigenvalue equation (6.37), we have

−

nπ

tan

−

nπ

−

nπ

tan

−

nπ

(h−a)

(6.40)

6.1 Metallic Waveguide with Diﬀerent Filling Media 325

The mth root of this transcendental equation is the cutoﬀ angular frequency

of the LSE

(x)

mode in a rectangular waveguide with two diﬀerent ﬁlling

media.

For the LSE

(x)

mode, n = 0, i.e., k

= 0, (6.40) reduces to the same form

as (6.16) for the cutoﬀ frequency for the TE

mode, and equation (6.37)

of k

for LSE

(x)

is also the same as the corresponding equation (6.13) for

the TE

mode. Furthermore, we easily see that the ﬁeld components of

the two modes are identical too. All the arguments put together mean that

the LSE

(x)

mode is identical with the TE

(z)

mode. The cutoﬀ frequency of

the LSE

(x)

mode is between the cutoﬀ frequencies of the TE

mode in the

waveguide ﬁlled with uniform medium 1 and the TE

mode in the waveguide

ﬁlled with uniform medium 2.

The lowest LSE

(x)

mode is the LSE

(x)

mode, which is identical with TE

(z)

for which

√

< ω

√

if µ

< µ

The LSE

(x)

or TE

(x)

mode is a mixture of TE

(z)

and TM

(z)

modes, i.e.,

hybrid modes or HEM modes.

(2) TM

(x)

Modes or LSM

(x)

Modes

For LSM

(x)

modes, V

(x)

= 0 and H

= 0. Applying (4.153)–(4.158) and the

boundary conditions on the conducting wall, we get the expressions for U

(x)

and U

(x)

= A cos k

x sin k

−jβz

, 0 ≤ x ≤ h, (6.41)

(x)

= B cos k

(x − a) sin k

−jβz

, h ≤ x ≤ a, (6.42)

Substituting them into (4.153)–(4.158), we may obtain the ﬁeld component

expressions in the two regions, which the reader can derive by the method

used previously.

Applying the boundary conditions that states the tangential components

of ﬁelds must be continuous on the boundary x = h, we have

y 1

(h) = E

y 2

(h), i.e.,

∂

(x)

∂y ∂x

x=h

∂

(x)

∂y ∂x

x=h

(h) = E

(h), i.e.,

∂

(x)

∂z ∂x

x=h

∂

(x)

∂z ∂x

x=h

which give

A sin k

h = k

B sin k

(h − a), (6.43)

326 6. Dielectric Waveguides and Resonators

and

y 1

(h) = H

y 2

(h), H

(h) = H

(h),

which give

A cos k

h = ²

B cos k

(h − a). (6.44)

Finally we obtain the eigenvalue equation of the LSE

(x)

mode from combining

the above two equations:

tan k

h =

tan k

(h − a). (6.45)

The longitudinal phase coeﬃcient β is determined by this equation and (6.14)

and (6.2) or (6.3) in a manner similar to the previous derivation, which the

reader can work out as an exercise.

When β = 0, ω = ω

, substituting (6.39) into the eigenvalue equation

(6.45) gives

−

nπ

tan

−

nπ

−

nπ

tan

−

nπ

(h−a)

(6.46)

The mth root of this transcendental equation is the cutoﬀ angular frequency

of the TM

(x)

or LSM

(x)

mode in a rectangular waveguide with two diﬀerent

ﬁlling media.

By investigating the ﬁeld-component expressions, we ﬁnd that if n = 0,

i.e., k

= 0, all the ﬁeld components would b e zero. So the LSM

(x)

mode

cannot exist here.

We ﬁnd that unlike (6.37), equation (6.45) has a set of near-zero roots,

≈ 0 and k

≈ 0. This mode is denoted by m = 0, i.e., by LSM

(x)

modes.

For these modes,

tan x ≈ x, x ¿ 1,

and the eigenvalue equation (6.45) becomes

h ≈

(h − a). (6.47)

The equation for the cutoﬀ frequency (6.46) becomes

−

nπ

h ≈

−

nπ

(h − a). (6.48)

6.2 Symmetrical Planar Dielectric Waveguides 327

So the cutoﬀ angular frequency of the LSM

(x)

mode can be solved as

≈

nπ

√

(²

/²

)h + (a − h)

(µ

/µ

)h + (a − h)

nπ

√

h +

(²

/²

)(a − h)

h + (µ

/µ

)(a − h)

(6.49)

The cutoﬀ frequency of the LSM

(x)

mode lies between the cutoﬀ frequen-

cies of the TE

(z)

mode in the waveguide ﬁlled with uniform medium 1 and

the TE

(z)

mode in the waveguide ﬁlled with uniform medium 2, i.e.,

nπ

√

< ω

nπ

√

For the LSM

(x)

mode, both k

and k

are close but not equal to zero.

The ﬁelds of the LSM

(x)

mode are similar to but not identical to those of

the TE

(z)

mode, while the longitudinal electric ﬁeld component in the TM

(x)

mode is very small yet not zero. The lowest LSM

(x)

mode is the LSM

(x)

mode.

The waveguide with two diﬀerent ﬁlling media aligned in the y direc-

tion as shown in Fig. 6.1(b) can also be analyzed by means of the same

method. The complete set of modes in this waveguide are LSE

(y )

modes

(m = 0, 1, 2, 3, ···, n = 1, 2, 3, ···) and LSM

(y )

modes (m = 1, 2, 3, ···, n =

0, 1, 2, 3, ···). The LSE

(y )

modes are identical to the TE

(z)

modes and the

LSM

(y )

modes are similar to but not identical to the TE

(z)

modes.

The most important conclusion drown in this section is that only the

modes with a uniform ﬁeld in the transverse direction tangential to the

boundary between the media, in other words, with zero phase coeﬃcient

along the transverse tangential direction, can be decomposed into TE and

TM modes. Otherwise they can only be hybrid modes.

6.2 Symmetrical Planar Dielectric

Waveguides

The slab waveguide is the simplest dielectric waveguide for millimeter wave

and optical wave transmission; it is also known as a planar dielectric waveg-

uide. A symmetrical planar dielectric waveguide is a dielectric slab of re-

fractive index n

√

immersed in another medium of refractive index

√

, as shown schematically in Fig. 6.2. The slab has a thick-

ness of 2h in the x direction and extends to inﬁnity in the y and z direc-

tions. The whole space is divided into three regions, the slab or core region 1

(−h ≥ x ≥ h), the lower cladding region 2 (x ≤ −h), and the upper cladding

region 3 (x ≥ h).

The slab waveguide is a one-dimensional conﬁned waveguide. We consider

the two-dimensional modes where ﬁelds are uniform along axis y and are

328 6. Dielectric Waveguides and Resonators

Figure 6.2: Symmetrical planar dielectric waveguide.

traveling waves along axis z. In the last section, we argued that, if the

ﬁelds are uniform along the transverse tangential direction of the boundary

of media, the ﬁelds of a single TE or TM mode alone can satisfy the boundary

conditions.

6.2.1 TM Modes

For TM modes, V = 0, U is a function of x and z only and is independent of

y, i.e., k

= 0. To satisfy the ﬁeld-matching conditions on the slab surface,

ﬁelds in all the three regions must be traveling waves along z with the same

longitudinal phase coeﬃcient k

= β.

In region 1, the core, −h ≤ x ≤ h, function U must be a standing waves

along x, and T denotes the transverse angular wave number.

= (A cos T x + B sin T x) e

−jβz

. (6.50)

Following the expressions of ﬁeld components, (4.147)–(4.152), we have

= jβT (A sin T x − B cos T x) e

−jβz

, (6.51)

= T

(A cos T x + B sin T x) e

−jβz

, (6.52)

y 1

= jω²

T (A sin T x − B cos T x) e

−jβz

. (6.53)

In regions 2 and 3, for guided modes, function U must be a decaying

function along −x and +x, and τ denotes the transverse decaying factor.

Region 2, lower cladding, x ≤ −h:

= Ce

τx

−jβz

, (6.54)

= −jβτ Ce

τx

−jβz

, (6.55)

= −τ

τx

−jβz

, (6.56)

y 2

= −jω²

τCe

τx

−jβz

. (6.57)

Region 3, upper cladding, x ≥ h:

= De

−τx

−jβz

, (6.58)

6.2 Symmetrical Planar Dielectric Waveguides 329

= jβτ De

−τx

−jβz

, (6.59)

= −τ

−τx

−jβz

, (6.60)

y 3

= jω²

τDe

−τx

−jβz

. (6.61)

The relations among T , τ, and β are

+ T

= k

= ω

, β

− τ

= k

= ω

. (6.62)

The ﬁeld-matching conditions on the boundary of the media are

(−h, z)=E

(−h, z) → −τ

−τh

(A cos T h−B sin T h), (6.63)

(h, z)=E

(h, z) → −τ

−τh

(A cos T h+B sin T h), (6.64)

y 2

(−h, z)=H

y 1

(−h, z) → ²

τCe

−τh

=²

T (A sin T h+B cos T h), (6.65)

y 3

(h, z)=H

y 1

(h, z) → ²

τDe

−τh

=²

T (A sin T h−B cos T h). (6.66)

They yield

A cos T h + B sin T h

A cos T h − B sin T h

A sin T h − B cos T h

A sin T h + B cos T h

. (6.67)

From (6.63) and (6.65), we obtain

τT

(A cos T h − B sin T h) = −²

T τ

(A sin T h + B cos T h), (6.68)

and from (6.64) and (6.66) we obtain

τT

(A cos T h + B sin T h) = −²

T τ

(A sin T h − B cos T h), (6.69)

The normal modes in the waveguide are classiﬁed as even modes and

odd modes. Whether a mode is even or odd is determined by the symmetry

property of the transverse ﬁeld components which provide the power ﬂow in

the longitudinal direction.

For even modes, A = 0 and B 6= 0, the ab ove two equations (6.68) and

(6.69) become

T h tan T h = ²

τh, (6.70)

and for odd modes, B = 0 and A 6= 0, (6.68) and (6.69) become

−²

T h cot T h = ²

τh. (6.71)

These are the eigenvalue equations for the even modes and the odd modes.

Since

tan

θ −

mπ

tan θ, m = 0, 2, 4, ···,

−cot θ, m = 1, 3, 5, ···,

equations (6.70) and (6.71) can be combined as one equation

T h tan

T h −

mπ

= ²

τh, (6.72)

330 6. Dielectric Waveguides and Resonators

where m is 0 or an even number for even modes and is an odd number for

odd modes.

From (6.62) we obtain

(T h)

+ (τh)

= (ωh)

(µ

− µ

). (6.73)

The transverse wave numbers T and τ and the longitudinal wave number β

are determined by (6.72), (6.73), and (6.62).

The coeﬃcients for even modes and for odd modes can be obtained from

the boundary equations. For even modes:

A = 0, B =

−τh

sin T h

C, D =−C. (6.74)

For odd modes:

A=−

−τh

cos T h

C, B = 0, D =C. (6.75)

Substituting (6.74) or (6.75) into (6.51) to (6.61), we get the expressions

for the ﬁeld components of the even TM modes or odd TM modes, respec-

tively, inside and outside the slab. This work is left to the reader. The

ﬁelds are standing waves in the core and are decaying in the cladding, along

the transverse direction x. They are traveling waves with the same phase

coeﬃcient β along the longitudinal direction z, both in the core and in the

cladding. This kind of mode is known as a surface wave mode, since the ﬁelds

outside the dielectric slab are gathered in regions near the surface of the slab.

The impedance at the surface of the slab is deﬁned as the ratio of the

tangential electric ﬁeld to the tangential magnetic ﬁeld. For TM modes,

(TM)

(h)

= −

(−h)

= j

ω²

. (6.76)

This is an inductive reactance. We can see that an arbitrary cylindrical

system enclosed by an inductive surface can support TM surface waves.

6.2.2 TE Modes

For TE modes, U = 0, k

= 0, k

= β, and we have the V functions and the

expressions for ﬁeld components in the three regions as follows:

Region 1, −h ≤ x ≤ h:

= (A cos T x + B sin T x) e

−jβz

, (6.77)

y 1

= −jωµ

T (A sin T x − B cos T x) e

−jβz

, (6.78)

= jβT (A sin T x − B cos T x) e

−jβz

, (6.79)

= T

(A cos T x + B sin T x) e

−jβz

; (6.80)