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

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4.3 Solution of Vector Helmholtz Equations 191

By applying the conditions of (4.33), h

= 1 and (∂/∂u

)(h

) = 0, in

(4.44) and (4.45), gives

∂

∂u

) = −jω²

∂

∂u

= −

∂

∂u

jω²

∂

∂u

∂

∂u

) = jω²

∂

∂u

∂

∂u

jω²

∂

∂u

Integrating the two equations yields

= −jω²

∂

∂u

U, (4.49)

= jω²

∂

∂u

U. (4.50)

The integration constants provide solutions independent of u

, which are

included in the above solutions.

Substituting E

from (4.47) and H

from (4.49) into (4.28), and consid-

ering h

= 1 and k

= ω

µ², we have

∂E

∂u

∂

∂u

∂

∂u

+ k

Similarly,

∂E

∂u

∂

∂u

∂

∂u

+ k

We do not consider the constant ﬁeld along both u

and u

; hence we have

∂

∂u

+ k

U. (4.51)

Substituting (4.49), (4.50) and (4.51) into (4.32) yields

∂

∂u

∂U

∂u

∂

∂u

∂U

∂u

¶¸

∂

∂u

+ k

U = 0. (4.52)

This is just the equation (4.40).

(2) If E

= 0 and H

6= 0, when we apply a similar procedure, making an

auxiliary function V (x), we have

= −jωµ

∂V

∂u

, (4.53)

= jωµ

∂V

∂u

, (4.54)

∂

∂u

, (4.55)

192 4. Time-Varying Boundary Value Problems

∂

∂u

, (4.56)

and

∂

∂u

+ k

V. (4.57)

Function V satisﬁes the same equation (4.40).

According to the theorem of superposition, if E

6= 0 and H

6= 0, then

the sums of (4.47)–(4.51) and (4.53)–(4.57) are just expressions (4.34)–(4.39).

In this theorem, the condition h

= 1 means that at least one di-

rection of the coordinate system is a linear coordinate, and the condition

(∂/∂u

)(h

) = 0 means that the forms of functions h

and h

with re-

spect to u

are the same, so that their ratio is independent of u

The coordinate systems that satisfy the conditions of Theorem 1 include

all cylindrical coordinate systems and spherical coordinate systems.

Theorem 2

If the orthogonal coordinate system satisﬁes not only the conditions in The-

orem 1, (4.33), but also

∂

∂u

) = 0, (4.58)

then the Borgnis’ functions U and V satisfy the homogeneous scalar

Helmholtz equations:

∇

U + k

U = 0, ∇

V + k

V = 0. (4.59)

Proof

The expansion of the scalar Helmholtz equation (4.59) in an arbitrary or-

thogonal coordinate system is given by

∂

∂u

∂U

∂u

∂

∂u

∂U

∂u

∂

∂u

∂U

∂u

¶¸

+ k

U =0.

(4.60)

If the conditions h

= 1 and (∂/∂u

)(h

) = 0 are satisﬁed, the scalar

Helmholtz equation (4.60) becomes

∂

∂u

∂U

∂u

∂

∂u

∂U

∂u

¶¸

∂

∂u

+ k

U =0. (4.61)

The scalar Helmholtz equation (4.60) is the same as (4.40). So is the equation

for V .

The conditions of Theorem 2 mean that

= 1,

∂h

∂u

= 0,

∂h

∂u

= 0.

4.3 Solution of Vector Helmholtz Equations 193

Figure 4.3: Cylindrical coordinate system.

The Lame coeﬃcients h

and h

for two coordinates u

and u

are inde-

pendent of the third coordinate u

. The coordinate systems that satisfy the

conditions of Theorem 2 are cylindrical coordinate systems.

In an arbitrary cylindrical coordinate system, at least one axis is Cartesian

coordinates, which is recognized as the longitudinal axis, denoted by z. Let

z = u

, then h

= h

= 1, and the equal-z surfaces are parallel planes

perpendicular to the z axis. The other two coordinates are two-dimensional

orthogonal curve sets on the equal-z plane and are denoted by u

and u

, see

Fig. 4.3. This is just the equation (4.40).

Let u

= z, Laplacian operator ∇

may be expressed as

∇

= ∇

∂

∂u

= ∇

∂

∂z

, (4.62)

In cylindrical systems, when sinusoidal traveling waves propagate along

two opposite directions of the longitudinal axis z, the function of the ﬁeld

with respect to t and z is given as follows:

j(ωt∓βz)

and

∂

∂z

= −β

where β = k

denotes the longitudinal phase coeﬃcient. The expressions for

the longitudinal components of the ﬁelds (4.36) and (4.39) become:

− β

U = T

U, (4.63)

− β

V = T

V, (4.64)

where T

= k

− β

= k

+ (∂

/∂z

) denotes the transverse angular wave

number. If we are interested in the wave along the positive direction of the

axis +z, the function of the ﬁeld becomes e

−jβz

, and we have

∂

∂z

= −jβ.

194 4. Time-Varying Boundary Value Problems

The expressions for the ﬁeld comp onents (4.34), (4.35), (4.37), (4.38) become

= −

jβ

∂U

∂u

−

jωµ

∂V

∂u

, (4.65)

= −

jβ

∂U

∂u

jωµ

∂V

∂u

, (4.66)

= −

jβ

∂V

∂u

jω²

∂U

∂u

, (4.67)

= −

jβ

∂V

∂u

−

jω²

∂U

∂u

. (4.68)

4.3.2 Method of Hertz Vectors [103]

The time-dependent electromagnetic ﬁelds can be formulated by means of

Hertz vector potentials as shown in Section 1.6, Chapter 1. For the source-

free problem the electric Hertz vector Π

and magnetic Hertz vector Π

satisfy the vector Helmholtz equations (1.280) and (1.281):

∇

+ k

= 0, (4.69)

∇

+ k

= 0. (4.70)

The Hertz vectors may be used to solve the time-dependent ﬁeld problems

in cylindrical coordinates.

Assume that only the longitudinal component of the electric Hertz vector

exists in a cylindrical coordinates,

= ˆzΠ

. (4.71)

According to the expansions (B.10), the vector Helmholtz equation (4.69) for

is reduced to the scalar Helmholtz equation

∇

+ k

= 0. (4.72)

If only the longitudinal component of the magnetic Hertz vector exists in

a cylindrical coordinates,

= ˆzΠ

, (4.73)

the vector Helmholtz equation (4.70) for Π

is also reduced to a scalar

Helmholtz equation:

∇

+ k

= 0. (4.74)

We come to the conclusion that both longitudinal components of Hertz

vectors satisfy the scalar Helmholtz equations. Hence the solution of the vec-

tor Helmholtz equations of time-varying-ﬁeld problems can be reduced to the

solution of the scalar Helmholtz’s equations of the longitudinal components

of Hertz vectors.

4.3 Solution of Vector Helmholtz Equations 195

According to the principle of superposition, substituting Π

= ˆzΠ

and

= ˆzΠ

into (1.284) and (1.285) yields

∂

∂z∂u

− jωµ

∂Π

∂u

, (4.75)

∂

∂u

∂z

+ jωµ

∂Π

∂u

, (4.76)

∂

∂z

+ k

, (4.77)

∂

∂z∂u

+ jω²

∂Π

∂u

, (4.78)

∂

∂u

∂z

− jω²

∂Π

∂u

, (4.79)

∂

∂z

+ k

. (4.80)

Comparing the above six expressions with respect to Π

and Π

and those

with respect to U and V , (4.34) to (4.39), we ﬁnd that in the cylindrical

coordinates, Π

and Π

are identical to U and V , respectively,

= U, Π

= V.

4.3.3 Method of Longitudinal Components [84, 60]

In the cylindrical coordinate systems, the longitudinal components of the elec-

tric and magnetic ﬁelds satisfy the scalar Helmholtz equations. The vector

Helmholtz equations can be solved by starting with the longitudinal comp o-

nents, which is known as the longitudinal component method.

An arbitrary 3-dimensional vector function may be decomposed into a

transverse two-dimensional vector function and a longitudinal scalar function.

So the electric and magnetic ﬁeld vectors are expressed as follows:

E = E

+ ˆzE

, H = H

+ ˆzH

. (4.81)

According to (B.10), the expansion of the vector Laplacian operator in

any cylindrical coordinate system is given by

∇

A = ∇

+ ˆz∇

. (4.82)

Hence, the vector Helmholtz equations (4.21), (4.22) for E and H are de-

composed into the following two vector Helmholtz equations and two scalar

Helmholtz equations:

∇

+ k

= 0, ∇

+ k

= 0, (4.83)

196 4. Time-Varying Boundary Value Problems

∇

+ k

= 0, ∇

+ k

= 0. (4.84)

The equations for the longitudinal components are scalar Helmholtz equa-

tions. We can solve them ﬁrst and then ﬁnd the transverse components with

respect to the longitudinal components by means of Maxwell’s equations.

The nabla operator in cylindrical coordinates is expressed by

∇ = ∇

+ ˆz

∂

∂z

, where ∇

= ˆu

∂

∂u

+ ˆu

∂

∂u

. (4.85)

∇

denotes the transverse two-dimensional nabla operator.

Applying (4.81) and (4.85) in Maxwell’s equations (4.25) and (4.26), we

have

∇

+ ˆz

∂

∂z

× (E

+ ˆzE

) = −jωµ(H

+ ˆzH

), (4.86)

∇

+ ˆz

∂

∂z

× (H

+ ˆzH

) = jω²(E

+ ˆzE

). (4.87)

The transverse vector and the longitudinal component must satisfy the fol-

lowing equations separately,

∇

× E

= −jωµH

ˆz, (4.88)

∇

× ˆzE

+ ˆz ×

∂E

∂z

= −jωµH

, (4.89)

∇

× H

= jω²E

ˆz, (4.90)

∇

× ˆzH

+ ˆz ×

∂H

∂z

= jω²E

. (4.91)

Applying the operator ˆz × (∂/∂z) to (4.89) and multiplying (4.91) by

−jωµ, then adding up them and canceling H

, we have

µ²E

−ˆz ×

∂

∂z

ˆz ×

∂E

∂z

= ˆz ×

∂

∂z

(∇

× ˆzE

)−jωµ∇

×ˆzH

. (4.92)

Similarly, canceling E

from (4.91) and (4.89), we have

µ²H

−ˆz ×

∂

∂z

ˆz ×

∂H

∂z

= ˆz×

∂

∂z

(∇

× ˆzH

)+jω²∇

×ˆzE

. (4.93)

Applying the following vector formulas,

ˆz ×

∂

∂z

ˆz ×

∂A

∂z

= ˆz × ˆz ×

∂

∂z

= −

∂

∂z

ˆz ×

∂

∂z

(∇

× ˆzA

)=−ˆz ×

∂

∂z

(ˆz ×∇

)=−ˆz × ˆz ×

∂

∂z

∇

∂

∂z

∇

4.3 Solution of Vector Helmholtz Equations 197

(4.92) and (4.93) become

∂

∂z

∂

∂z

∇

+ jωµˆz × ∇

, (4.94)

∂

∂z

∂

∂z

∇

− jω²ˆz × ∇

. (4.95)

These are the expressions of the transverse components of the ﬁelds in terms

of the longitudinal components.

For sinusoidal traveling waves propagating in two opposite directions

along the longitudinal axis z,

∓jβz

∂

∂z

= −β

, and T

= k

− β

= k

∂

∂z

With this notation and the notation of the transverse Laplacian operator

(4.62), the equations of the transverse ﬁeld vectors (4.83) become

∇

+ T

= 0, ∇

+ T

= 0. (4.96)

The expressions for E

, H

, (4.94) and (4.95) become:

∂

∂z

∇

− jωµˆz × ∇

, (4.97)

∂

∂z

∇

− jω²ˆz × ∇

. (4.98)

Each of the transverse vectors may be decomposed into two components:

= ˆu

+ ˆu

, H

= ˆu

+ ˆu

. (4.99)

Applying the expansions of ∇

in two-dimensional coordinates u

, u

, (4.85),

we have

∂

∂u

∂z

− jωµ

∂H

∂u

, (4.100)

∂

∂u

∂z

+ jωµ

∂H

∂u

, (4.101)

∂

∂u

∂z

+ jω²

∂E

∂u

, (4.102)

∂

∂u

∂z

− jω²

∂E

∂u

. (4.103)

Comparing these expressions with the expressions with respect to U, V ,

(4.34), (4.35), (4.37), and (4.38) and the expressions with respect to Π

, (4.75), (4.76), (4.78), and (4.79), we see that

= T

U = T

, H

= T

V = T

. (4.104)

198 4. Time-Varying Boundary Value Problems

These two expressions are the same as (4.36), (4.39) and (4.77), (4.80), re-

spectively.

We come to the conclusion that the Borgnis’ potentials are identical to the

longitudinal components of the Hertzian vectors and the diﬀerence between

them and the longitudinal ﬁeld components are a multiplying factor T

only.

4.4 Boundary Conditions of

Helmholtz’s Equations

The general boundary conditions of time-dependent ﬁelds are given in Sec-

tion 1.2. Now, the boundary conditions of the Borgnis’ potentials U , V , the

longitudinal components of the Hertzian vectors Π

, Π

, and the longitu-

dinal ﬁeld components E

, H

are to be investigated.

In arbitrary orthogonal curvilinear coordinates, the boundaries are diﬀer-

entiated into two sorts, the boundary p erpendicular to u

and the boundaries

parallel to u

, The former is the u

surface or transverse cross section and

the later are the u

and u

surfaces or longitudinal boundary surfaces.

We now look into the short-circuit boundaries. At the longitudinal short-

circuit boundary surfaces u

= a, denoted by S

, the tangential components

of the electric ﬁeld must be zero, i.e.,

= 0 and E

= 0.

In cylindrical or spherical coordinates, according to (4.36) and (4.35), the

conditions are satisﬁed when

= 0 and

∂V

∂u

= 0.

In cylindrical coordinates, according to (4.77) and (4.76), or (4.101), the

conditions are satisﬁed when

= 0 and

∂Π

∂u

= 0 or E

= 0 and

∂H

∂u

= 0.

Similarly, At the longitudinal short-circuit boundary surfaces u

= b, denoted

by S

, the tangential components of the electric ﬁeld must also be zero, i.e.,

= 0 and E

= 0.

According to (4.36) and (4.34), the conditions are satisﬁed when

= 0 and

∂V

∂u

= 0.

According to (4.77) and (4.75), or (4.100), the conditions are satisﬁed when

= 0 and

∂Π

∂u

= 0 or E

= 0 and

∂H

∂u

= 0.

4.5 Separation of Variables 199

The above conditions can be summarized as follows:

1,2

= 0 or Π

1,2

= 0 or E

1,2

= 0, (4.105)

and

∂V

∂n

1,2

= 0 or

∂Π

∂n

1,2

= 0 or

∂H

∂n

1,2

= 0, (4.106)

where n denotes the normal of the boundary surface. The conclusion is that

the functions U , Π

, and E

are equal to zero and the normal derivatives of

V , Π

, and H

are equal to zero at the longitudinal short-circuit boundaries.

At the transverse cross-section short-circuit boundaries, u

= c, denoted

by S

, the tangential components of the electric ﬁeld are E

and E

, and we

have

= 0 and E

= 0.

According to (4.34), (4.35), (4.75), (4.76), (4.100), and (4.101), the boundary

conditions of U, V , Π

, Π

, E

, and H

become

V |

= 0 or Π

= 0 or H

= 0, (4.107)

and

∂U

∂n

= 0 or

∂Π

∂n

= 0 or

∂E

∂n

= 0. (4.108)

Functions V , Π

, and H

are equal to zero and the normal derivatives of

U, Π

, and E

are equal to zero at the transverse cross-section short circuit

boundaries.

For the open-circuit boundaries, the conditions are dual to those for the

short-circuit boundaries.

4.5 Separation of Variables

Using the methods given in the previous two sections, the boundary-value

problems of vector Helmholtz equations may reduce to the problems of solving

the scalar Helmholtz equation

∇

U + k

U = 0

with certain boundary conditions,

= 0 or

∂U

∂n

= 0,

where U can also be one of V , Π

, Π

, E

, and H

200 4. Time-Varying Boundary Value Problems

The method of separation of variables is an important and convenient

way to solve scalar partial diﬀerential equations in mathematical physics. By

choosing an appropriate orthogonal coordinate system, we can represent the

solution by a product of three functions, one for each coordinate, and the

three-dimensional partial diﬀerential equation is reduced to three ordinary

diﬀerential equations. The functions that satisfy these ordinary diﬀerential

equations are orthogonal function sets called harmonics. The solution of the

diﬀerential equation with speciﬁc boundary conditions is usually a series of

the speciﬁc harmonics set.

Equations involving the three-dimensional Laplacian operator, for exam-

ple Laplace’s equation and Helmholtz’s equation, are known to be separable

in eleven diﬀerent orthogonal coordinate systems, included in the following

three groups:

Cylindrical

1. Rectangular coordinates: Consists of three sets of mutual or-

thogonal parallel planes.

2. Circular-cylinder coordinates: Consists of a set of coaxial circu-

lar cylinders, a set of half planes rotated around the axis, and a set

of parallel planes perpendicular to the axis. The circular-cylinder

coordinate system is also a rotational coordinate system.

3. Elliptic-cylinder coordinates: Consists of a set of confo cal ellip-

tic cylinders, a set of confocal hyperbolic cylinders perpendicular

to the elliptic cylinders, and a set of parallel planes perpendicular

to the axis.

4. Parabolic-cylinder coordinates: Consists of two sets of mutual

orthogonal parabolic cylinders and a set of parallel planes perpen-

dicular to the axis.

Rotational

5. Spherical coordinates: Consists of a set of concentric spheres, a

set of cones perpendicular to the spheres, and a set of half planes

rotated around the polar axis.

6. Prolate spheroidal coordinates: Consists of a set of confocal

prolate spheroids, a set of confocal hyperboloids of two sheets,

and a set of half planes rotated around the polar axis.

7. Oblate spheroidal coordinates: Consists of a set of confocal

oblate spheroids, a set of confocal hyperboloids of one sheet and

a set of half planes rotated around the polar axis.

8. Parabolic coordinates: Consists of two sets of mutual orthogonal

circular paraboloids and a set of half planes rotated around the

polar axis.