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

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7.8 Coupling of Modes 451

Let a(z) denote the complex quantity of any ﬁeld component of a desired

mode propagating in the z direction in a guided-wave system with spatial

dependence e

−jβ

. The normal mode form of the transmission-line equation

= −jβ

a. (7.204)

Consider two modes a(z) and b(z), which, in the absence of coupling, have

natural phase constants β

and β

, respectively. We have the normal mode

representation of the transmission line equations for the two modes:

= −jβ

b. (7.205)

The modes are assumed to be lossless, so that both β

and β

are real. If

and β

are positive, the phase velocities of both modes are in the +z

direction; if β

and β

are negative, the phase velocities of both modes

are in the −z direction; and if β

and β

are of opposite sign, the phase

velocities are in opposite directions.

We will see later that only the modes with analogous phase velocities can

have eﬀective coupling, so we can neglect all other modes and obtain sim-

ple coupled-wave equations that describe the interaction. If the two guided

modes with eﬀective coupling are a and b, some of the energy in mode a is

transferred to mode b and some of the energy in mode b is transferred to

mode a. Then the coupled-mode equations can be written as

= −jβ

a + κ

b, (7.206)

= −jβ

b + κ

a, (7.207)

where κ

and κ

, in general, are mutual coupling diﬀerential operators.

For weak coupling, κ

and κ

become complex coupling coeﬃcients and

and κ

are small compared with β

and β

, so the equations become

linear equations. The coupling is assumed to b e uniform over the length of

the coupling, so that κ

and κ

are independent of z.

For a source-free and lossless system, power conservation requires that

the total average power of the two modes must be independent of z, i.e.,

|a|

± |b|

= constant,

|a|

± |b|

= 0. (7.208)

If the waves carrying power travel in the same direction for the two uncoupled

modes, the plus sign is to be taken, whereas if the waves carrying power travel

in opposite directions for the two uncoupled modes, the minus sign is to be

taken. Then we have

d|a|

d|b|

daa

∗

dbb

∗

+ a

∗

± b

∗

± b

∗

= aκ

∗

+ a

∗

b ± bκ

∗

± b

∗

a = 0.

452 7. Periodic Structures and the Coupling of Modes

For strict energy conservation, this expression must be identically true for all

z. Since a and b are arbitrary complex quantities and the phases of a and b

are then arbitrary, the above equation is true if and only if

= ∓κ

∗

, (7.209)

where the upper sign, minus, is required for group velocities in the same

direction and the lower sign, plus, is required for group velocities in opposite

directions.

Applying the operator (d/dz + jβ

) to (7.206) then substituting (7.207)

into it, and applying the operator (d/dz + jβ

) to (7.207) then substituting

(7.206) into it, we have the wave equations in coupled-mode formalism:

+ j(β

+ β

)

− (β

+ κ

)a = 0, (7.210)

+ j(β

+ β

)

− (β

+ κ

)b = 0. (7.211)

These are homogeneous linear diﬀerential equations with constant coeﬃ-

cients. The solutions of the equations are in the form of e

−jβz

, where βs

denote the phase coeﬃcients for the coupled waves, which is diﬀerent from

those for the uncoupled waves β

and β

. Substituting the trial solutions

into the coupled-wave equations yields the determinant equation of the cou-

pled modes:

− (β

+ β

)β + (β

+ κ

) = 0. (7.212)

The two solutions of this equation are β

and β

1,2

= β ± B, B =

∆β

− κ

, (7.213)

where

β =

+ β

, ∆β =

− β

For the two uncoupled modes that have waves carrying power in the same

direction, κ

= −κ

∗

, κ

= −|κ

, and

B =

∆β

+ |κ

. (7.214)

In this case, β

and β

are always real.

For two uncoupled modes that have waves carrying power in opposite

directions, κ

= κ

∗

, κ

= |κ

, and

B =

∆β

− |κ

. (7.215)

In this case, if |∆β| > |κ

|, then β

and β

are still real, or if |∆β| < |κ

then B becomes imaginary and β

and β

become complex.

7.8 Coupling of Modes 453

Figure 7.29: Dispersion diagram of coupled modes for the case of power ﬂow

in the same direction (a), and in the opposite directions (b).

If the diﬀerence in the phase velocities of the two uncoupled modes is

suﬃciently large so that |∆β| À |κ

|, then

or β

≈ β

or β

The two modes propagate with their own natural phase velocities and are

free of coupling.

Appreciable coupling can occur only if |β

−β

| is of the order of |κ

which is small compared with β

and β

under the weak-coupling assump-

tion. Thus the condition for eﬀective coupling is

≈ β

i.e., the phase velocities of the two modes must be of the same sign and

approximately equal to each other. In the case of ∆β = 0, β

= β

we have

B =

|κ

|, for group velocities in the same direction,

j|κ

|, for group velocities in opposite directions.

This is known as the phase synchronous state or phase matching.

The ω–β diagrams for coupled modes are shown in Fig. 7.29. The group

velocities of the two uncoupled modes are diﬀerent and the unperturbed

dispersion curves of β

and β

may cross at the point of β

= β

shown by the dashed lines in the ﬁgure.

If the two uncoupled modes have group velocities in the same direction

as shown in Fig. 7.29(a), then the phase constants of the coupled waves β

and β

are always real. This is known as the co-directional mode coupling.

454 7. Periodic Structures and the Coupling of Modes

In this case, the appreciable coupling can occur only if the two modes are

both forward waves or both backward waves and the phase velocities of the

two modes are approximately equal. The coupled waves are two persistent

traveling waves with diﬀerent phase coeﬃcients. In consequence, the space

beat occurs for the composed wave. The two modes are said to be passively

coupled.

If the two uncoupled modes have group velocities in the opposite direc-

tions as shown in Fig. 7.29(a), this is known as the contra-directional mode

coupling. In this case, B becomes imaginary and β

, β

become complex

when |∆β| < |κ12|. It follows that a crossing of the β

and β

curves leads

to exponentially growing and decaying waves. In this case, the two modes

are said to be actively coupled. The condition of active coupling is that the

group velocities are in opposite directions and the phase velocities are in the

same direction and suﬃciently closed to each other. Accordingly, if one mode

is a forward-wave mode the other mode must be a backward-wave mode. The

other possibility is that the two modes with opposite group velocities cou-

pling to each other by means of a forward-wave space harmonic of one mode

and a backward-wave space harmonic of the other mode.

7.8.2 General Solutions for the Mode Coupling

The solutions to the coupled-wave equations (7.210) and (7.211) must be

linear combinations of the functions e

−jβ

and e

−jβ

a(z) = A

−jβ

+ A

−jβ

−jBz

+ A

jBz

−jβz

, (7.216)

b(z) = B

−jβ

+ B

−jβ

−jBz

+ B

jBz

−jβz

, (7.217)

where coeﬃcients A

, A

, B

are determined by the boundary conditions

at speciﬁc z.

Substituting the expressions for a(z) and b(z), (7.216) and (7.217), into

coupled-mode equation (7.206), we have

{j[β

−(β+B)]A

−κ

−jBz

+{j[β

−(β−B)]A

−κ

jBz

= 0. (7.218)

The two terms on the left-hand side must be equal to zero independently,

because this equation should be valid for arbitrary z. Hence we have

= j

∆β − B

, B

= j

∆β + B

. (7.219)

If both the uncoupled modes have group velocities in the same direction,

and suppose that the initial values of a and b at z = 0 are a(0) and b(0),

substituting (7.219) into (7.216) and (7.217), we have

+ A

= a(0), (7.220)

7.8 Coupling of Modes 455

+ B

= j

∆β − B

+ j

∆β + B

= b(0). (7.221)

The solutions of A

and A

are

∆β

a(0) + j

b(0), (7.222)

−

∆β

a(0) − j

b(0). (7.223)

Substituting the expressions for a(z) and b(z), (7.216) and (7.217), into

coupled-mode equation (7.207), by a procedure similar to the one used for

ﬁnding (7.219), we have

= −j

∆β + B

, A

= −j

∆β − B

. (7.224)

Suppose again that the initial values of a and b at z = 0 are a(0) and

b(0). By inserting (7.224) into (7.216) and (7.217), we have

+ A

= −j

∆β + B

− j

∆β − B

= a(0). (7.225)

+ B

= b(0). (7.226)

The solutions of B

and B

are

−

∆β

b(0) + j

a(0), (7.227)

∆β

b(0) − j

a(0). (7.228)

Substituting (7.222), (7.223), (7.227), and (7.228) into (7.216) and

(7.217), we have the general solutions of the coupled-wave equations for the

case of both of the uncoupled modes having group velocities in the same

direction, i.e., co-directional coupling:

a(z) =

·µ

cos Bz − j

∆β

sin Bz

a(0) +

sin Bz

b(0)

−jβz

, (7.229)

b(z) =

·µ

cos Bz + j

∆β

sin Bz

b(0) +

sin Bz

a(0)

−jβz

. (7.230)

It is remarkable that, in the case of co-directional coupling, each coupled

wave is to make up the beat of two persistent waves with diﬀerent phase

constants. The coupling waves become sine and cosine modulated traveling

waves. Power transfer takes place between the two modes and the average

power in each mode varies sinusoidally along z.

456 7. Periodic Structures and the Coupling of Modes

In the case of contra-directional coupling, when |∆β| > |κ12|, B is real,

the solutions are still (7.229) and (7.230). When |∆β| < |κ12| then B becomes

imaginary. Let B = jT , the solutions become

a(z) =

·µ

cosh T z−j

∆β

sinh T z

a(0)+

sinh T z

b(0)

−jβz

, (7.231)

b(z) =

·µ

cosh T z+j

∆β

sinh T z

b(0)+

sinh T z

a(0)

−jβz

. (7.232)

The coupling waves become hyperbolic-sine and hyperbolic-cosine modulated

traveling waves, i.e., decaying and growing waves along the directions of the

power ﬂow.

7.8.3 Co-Directional Mode Coupling

For co-directional coupling, the resulted coupled waves are sine and cosine

modulated traveling waves, and power transfer takes place between the two

modes. This is the theoretical basis of the waveguide coupler.

(1) Asynchronous State

If the uncoupled phase constants of the two modes are not equal to each

other, β

6= β

, this condition is said to be the asynchronous state. We

suppose that all the power is initially introduced to mode 1, i.e., a(0) 6= 0

and b(0) = 0, then according to (7.229) and (7.230), we can describe the

distributions of ﬁelds of the two modes as

a(z) = a(0)

cos Bz − j

∆β

sin Bz

−jβz

, (7.233)

b(z) = a(0)

sin Bze

−jβz

. (7.234)

The power-ﬂows of the two modes are P

and P

, which ar proportional to

|a(z)|

and |b(z)|

, respectively, and are given by

|a(z)|

= a(z)a

∗

(z) = |a(0)|

1 −

|κ

sin

, (7.235)

|b(z)|

= b(z)b

∗

(z) = |a(0)|

|κ

sin

Bz, (7.236)

and

|a(z)|

+ |b(z)|

= |a(0)|

The power fed in mode 1 at z = 0 will alternate back and forth between the

two modes, and the total power |a(z)|

+ |b(z)|

remain unchange along z

7.8 Coupling of Modes 457

Figure 7.30: Power ﬂow distributions of the co-directional coupled modes for

synchronous state (a) and asynchronous state (b).

and equals the power at the input, |a(0)|

. The maximum fraction of power

transfer is

F =

|κ

1 + (∆β/|κ

. (7.237)

This is illustrated in Fig. 7.30(a).

(2) Synchronous State

If both of the modes in the two modes are forward waves and the uncoupled

phase constants of the modes are equal to each other, ∆β = 0, β

= β

and B = |κ

| = |κ

|, this condition is said to be the synchronous state or

phase-matching state. Suppose again that the initial ﬁeld introduced to mode

1 at the input port is a(0) and there is no excitation in mode 2, b(0) = 0.

Then (7.233) and (7.234) becomes

a(z) = a(0) cos Bze

−jβ

, (7.238)

b(z) = a(0)

|κ

sin Bze

−jβ

. (7.239)

The power-ﬂow distributions along the two modes are

|a(z)|

= |a(0)|

cos

Bz =

|a(0)|

(1 + cos 2Bz), (7.240)

|b(z)|

= |a(0)|

sin

Bz =

|a(0)|

(1 − cos 2Bz), (7.241)

and

|a(z)|

+ |b(z)|

= |a(0)|

. (7.242)

Thus the power fed into mode 1 at z = 0 will completely alternate back and

forth between the two modes as long as they continue to be coupled together.

458 7. Periodic Structures and the Coupling of Modes

Figure 7.31: (a) Metallic waveguide coupler and (b) dielectric waveguide

coupler.

This is illustrated in Fig. 7.30(b). Furthermore, the transfer takes place in

exactly the same way if power is initially introduced to mode 2 as if p ower

were fed into mode 1.

For a synchronous state, complete power transfer from one mode to an-

other is possible. The minimum length for complete power transfer satisﬁes

Bl = κ

l =

This length is said to be the coupling length and this type of coupler is known

as a directional coupler.

(3) Waveguide Coupler

The waveguide coupler or the so-called directional coupler is made of two

parallel waveguides with a certain coupling mechanism between them [64, 75].

For metallic waveguides, the coupling is provided by means of permeated

ﬁelds through the slots or holes on the common wall of the two side-by-

side waveguides, and for dielectric waveguides, the coupling is provided by

means of the decaying ﬁelds outside the core, which can be realized simply by

making the two parallel waveguides close together on a common substrate.

See Fig. 7.31. The coupling coeﬃcient for the former can be adjusted by

varying the size and the number of coupling holes, and for the latter by

varying the spacing between the two waveguides. The optimum length of the

coupler is the above indicated coupling length, i.e., Bl = κ

l =

(4) Waveguide Switch

For a directional coupler made from material capable of producing a large

electro-optic eﬀect designed so that β

= β

and Bl = κ

l = π/2, and

the power can be transferred completely from one waveguide to another if

oppositely oriented electric ﬁelds are applied to the two dielectric waveguides,

then the β of the waveguide can be shifted in opposite directions, so that

7.8 Coupling of Modes 459

Figure 7.32: Transverse ﬁeld pattern in two parallel dielectric waveguides.

6= β

. Hence the fraction of power transfer of the coupler can be adjusted

by means of the applied electric ﬁeld, i.e., the applied voltage on the electrodes

[51, 99]. When

∆β =

√

3|κ

we have

B = 2κ

, Bl = 2κ

l = π.

In this case, from (7.235) and (7.236), we have

|a(l)|

= |a(0)|

, |b(l)|

= 0,

and no power transfer occurs any more between the two waveguides. This is

the principle of the optical waveguide switch.

7.8.4 Coupling Coeﬃcient of Dielectric Waveguides

The coupling between two waveguides is caused by the power transfer from

one waveguide to another by the ﬁeld of one waveguide penetrating to the

other waveguide. Consider two parallel dielectric waveguides close together

on a common substrate, as shown in Fig. 7.31(b). The electric ﬁelds in the

two individual waveguides are

(x, y, z) = a(z)e

(x, y), E

(x, y, z) = b(z)e

(x, y),

where e

(x, y) and e

(x, y) are the normalized transverse ﬁeld patterns of

waveguides 1 and 2, respectively. For the ﬁrst-order approach, the total ﬁeld

in both waveguides is the superposition of the two ﬁeld patterns of waveguides

1 and 2:

E(x, y, z) = E

(x, y, z) + E

(x, y, z) = a(z)e

(x, y) + b(z)e

(x, y), (7.243)

where the ﬁeld a(z)e

, by deﬁnition, is the ﬁeld of waveguide 1 in the absence

of waveguide 2, this is to say that the inﬂuence of the permittivity increase

in waveguide 2 on the ﬁeld pattern of waveguide 1 is neglected. See Fig. 7.32.

460 7. Periodic Structures and the Coupling of Modes

The power transferred from waveguide 1 to waveguide 2 is caused by

the additional a.c. polarization current J

produced in waveguide 2 by the

electric ﬁeld of waveguide 1, as a result of the permittivity increment in the

core of waveguide 2:

d P

d t

= jωP

, P

=²

(χ

− χ

)a(z)e

(x, y)=(²

− ²

)a(z)e

(x, y),

= jω(²

− ²

)a(z)e

(x, y), (7.244)

where ²

and ²

are the permittivities for the core and the substrate, respec-

tively, and χ

and χ

are the corresponding susceptibilities.

The power transferred is

= −

∗

· (jωP

)dS+

· (jωP

)

∗

= −

jωab

∗

(²

− ²

· e

∗

dS−jωa

∗

(²

− ²

∗

· e

. (7.245)

From the coupled-mode formulation, we know that the power transfer is

d|b|

dbb

∗

= b

∗

+ b

∗

= κ

∗

+ κ

∗

b. (7.246)

Comparison of (7.245) and (7.246) gives

= −

jω

(²

− ²

· e

∗

dS. (7.247)

The same approach for the power transferred from waveguide 2 to waveguide

1 gives

= −

jω

(²

− ²

· e

∗

dS. (7.248)

7.8.5 Contra-Directional Mode Coupling

For contra-directional coupling, the coupling waves become hyperbolic-sine

and hyperbolic-cosine modulated traveling waves, i.e., decaying or growing

waves along the direction of the power ﬂow. For contra-directional coupling,

when |∆β| < |κ12| then B becomes imaginary. Let B = jT , the solutions

become 7.231 and 7.232. We suppose that all the power is initially introduced

to the input end of mode 1, z = 0, i.e., a(0) 6= 0 and there is no power

introduced to the input end of mo de 2, z = L, i.e., b(L) = 0, then according

to (7.231) and (7.232), we can describe the distributions of ﬁelds along the

two modes as

a(z) = a(0)

coshT z−

|κ

sinhTL sinhT(z−L)−j

∆β

sinhTz

1 +

|κ

sinh

T L

−jβz

, (7.249)