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

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4.8 Solutions in Circular Cylindrical Coordinates 211

When ν is not an integer, the two independent solutions of the Bessel equation

are the Bessel functions [44] or Bessel functions of the ﬁrst kind with positive

and negative order:

(x) =

∞

m=0

(−1)

m!Γ (ν + m + 1)

ν+2m

, (4.176)

−ν

(x) =

∞

m=0

(−1)

m!Γ (−ν + m + 1)

−ν+2m

. (4.177)

The other possible solutions are the Bessel functions of the second kind or

the Neumann functions which are deﬁned as

(x) =

(x) cos νπ − J

−ν

(x)

sin νπ

. (4.178)

When ν is an integer or zero, ν = n, functions J

(x) and J

−n

(x) are not

linearly independent,

Γ (n + m + 1) = (n + m)! and J

−n

(x) = (−1)

(x).

In this case the two independent solutions of the Bessel equation become the

Bessel functions and the Neumann functions with integer order:

(x) =

∞

m=0

(−1)

m!(n + m)!

n+2m

, (4.179)

(x) = lim

ν→n

(x) = lim

ν→n

(x) cos νπ − J

−ν

(x)

sin νπ

= =

∂

∂ν

(x) − (−1)

∂

∂ν

−ν

(x)

ν=n

(4.180)

Finally, we have the solution of (4.171):

R(ρ) = a

(T ρ)+b

−ν

(T ρ), or R(ρ) = A

(T ρ)+B

(T ρ), (4.181)

when ν is not an integer, and

R(ρ) = A

(T ρ) + B

(T ρ), (4.182)

when ν is an integer, ν = n.

The following linear combinations of J

(x) and N

(x) are solutions of the

Bessel equation too,

(1)

(x) = J

(x) + jN

(x), H

(2)

(x) = J

(x) − jN

(x). (4.183)

Functions H

(1)

(x) and H

(2)

(x) are Hankel functions of the ﬁrst kind and the

second kind, respectively. Hence the solution of the Bessel equation can also

be the linear combinations of them,

R(ρ) = A

(1)

(T ρ) + B

(2)

(T ρ). (4.184)

212 4. Time-Varying Boundary Value Problems

In fact, any two of the functions J

(x), N

(x), H

(1)

(x), and H

(2)

(x) are

linearly independent, hence the linear combination of any two of them is the

complete solution of the Bessel equation. When ν is not an integer, J

−ν

(x)

is also independent.

If T

is negative, we replace T by jτ and x by jx, then equations (4.171)

and (4.175) become the modiﬁed Bessel equations

dρ

−

+ ν

R = 0, (4.185)

dR(x)

−

+ ν

R(x) = 0. (4.186)

The solution of (4.186 ) must be J

(jx) and N

(jx), but these two func-

tions are usually complex or imaginary. Construct the following two linearly

independent functions:

(x) = j

−ν

(jx), (4.187)

(x) = j

−ν+1

(1)

(jx) = j

−ν+1

(jx) + jN

(jx)]. (4.188)

Both I

(x) and K

(x) are real functions and are known as the modiﬁed Bessel

functions of the ﬁrst and the second kind, respectively. Hence the solution

of (4.185) is

R(ρ) = A

(τρ) + B

(τρ). (4.189)

The Bessel functions, Neumann functions, Hankel functions and modiﬁed

Bessel functions are cylindrical harmonics. Bessel functions and Neumann

functions are quasi-periodic functions with multiple zeros, see Fig. 4.5(a) and

(b). The modiﬁed Bessel functions are monotonic increasing or decreasing

functions, see Fig. 4.5(c). The Hankel functions are generally complex.

Comparing these plots with the plots of rectangular harmonics, we ﬁnd

that Bessel and Neumann functions are similar to sine and cosine functions,

which represent standing waves. See Fig. 4.6. The modiﬁed Bessel functions

are similar to hyperbolic functions and exponential functions with real ar-

guments, so the modiﬁed Bessel functions are also called hyperbolic Bessel

functions, which describe the decaying ﬁelds. The position of Hankel func-

tions in circular-cylinder coordinates is the same as that of the exponential

functions with imaginary arguments in the rectangular coordinates, which

describe the traveling waves.

For large arguments, the leading terms of the asymptotic series of cylin-

drical harmonics are listed in Appendix C, (C.10)–(C.12). We ﬁnd that the

asymptotic approximations of the Bessel functions and Neumann functions

are cosine and sine functions, the asymptotic approximations of the modi-

ﬁed Bessel functions are exponential functions with real arguments and the

asymptotic approximations of the Hankel functions are exponential functions

with imaginary arguments.

The solution of the function V is the same as that of the function U.

4.8 Solutions in Circular Cylindrical Coordinates 213

Figure 4.5: Bessel functions (a), Neumann functions (b) and modiﬁed Bessel

functions (c).

Figure 4.6: Bessel functions and Neumann functions.

214 4. Time-Varying Boundary Value Problems

In circular cylindrical coordinates, the ﬁeld components may be expressed

in terms of U and V by (4.34)–(4.39) as follows:

∂

∂ρ ∂z

−

jωµ

∂V

∂φ

, (4.190)

∂

∂φ ∂z

+ jωµ

∂V

∂ρ

, (4.191)

∂

∂z

+ k

U =

− β

U = T

U = −τ

U, (4.192)

∂

∂ρ ∂z

jω²

∂U

∂φ

, (4.193)

∂

∂φ ∂z

− jω²

∂U

∂ρ

, (4.194)

∂

∂z

+ k

V =

− β

V = T

V = −τ

V. (4.195)

If we are interested in the wave along the positive direction of the axis

+z only, ∂/∂z = −jβ, and according to (4.63)–(4.68) the ﬁeld components

become:

= −jβ

∂U

∂ρ

−

jωµ

∂V

∂φ

, (4.196)

= −

jβ

∂U

∂φ

+ jωµ

∂V

∂ρ

, (4.197)

− β

U = T

U = −τ

U, (4.198)

= −jβ

∂V

∂ρ

jω²

∂U

∂φ

, (4.199)

= −

jβ

∂V

∂φ

− jω²

∂U

∂ρ

, (4.200)

− β

V = T

V = −τ

V. (4.201)

4.9 Solution of Helmholtz’s Equations in

Spherical Coordinates

For spherical coordinates,

= θ, u

= φ, u

= r, h

= r sin θ, h

= 1.

The conditions of the Borgnis’ Theorem 1 are satisﬁed but the conditions of

Theorem 2 are not satisﬁed in the spherical coordinates. So the equations

for functions U and V (4.40) are not scalar Helmholtz equations.

4.9 Solutions in Spherical Coordinates 215

In spherical coordinates the two-dimensional Laplacian operator (4.41)

becomes

∇

sin θ

∂

∂θ

sin θ

∂

∂θ

sin

∂

∂φ

Then, (4.40) for U (and similarly V ) becomes

sin θ

∂

∂θ

sin θ

∂U

∂θ

sin

∂

∂φ

∂

∂r

+ k

U = 0. (4.202)

Equation (4.202) is not a scalar Helmholtz equation. The diﬀerence between

(4.202) and the scalar Helmholtz equation is that the term ∂

/∂r

in (4.202)

must be replaced by (1/r

)∂/∂r(r

∂/∂r) in the scalar Helmholtz equation.

Making the function substitution

U = rF or V = rF, (4.203)

substituting (4.203) into (4.202), and dividing it by r, we ﬁnd the equation

for F becomes a scalar Helmholtz equation in spherical coordinates:

sin θ

∂

∂θ

sin θ

∂F

∂θ

sin

∂

∂φ

∂

∂r

∂F

∂r

+ k

F = 0. (4.204)

Applying the method of separation of variables, let

F (r, θ, φ) = R(r)Θ(θ)Φ(φ). (4.205)

Substituting it into (4.204) and being multiplying by r

sin

θ/RΘΦ, we have

sin θ

dθ

sin θ

dΘ

dθ

dφ

sin

+ k

sin

θ = 0. (4.206)

The left-hand side of the equation includes two parts, the ﬁrst part is the

second term, which is a function of φ only and the second part is the sum of

the rest of the terms, which are functions of r and θ. To satisfy the equation,

each part must be equal to a constant and the sum of the two constants must

be zero. Suppose the two constants are −m

and m

, respectively, we have

dφ

= −m

Φ, (4.207)

sin θ

dθ

sin θ

dΘ

dθ

−m

sin

+ k

sin

θ = 0. (4.208)

Dividing (4.208) by sin

θ, yields

Θ sin θ

dθ

sin θ

dΘ

dθ

−

sin

+ k

= 0. (4.209)

216 4. Time-Varying Boundary Value Problems

The ﬁrst and the second terms are function of θ only, and the third and

the fourth terms are functions of r only. They must be equal to constants

separately, and the sum of the two constants must be zero. Suppose they are

−χ

and χ

, respectively. This yields

sin θ

dθ

sin θ

dΘ

dθ

−

sin

Θ = 0 (4.210)

+ (k

− χ

)R = 0 (4.211)

The partial diﬀerential equation (4.204) is reduced to three ordinary diﬀer-

ential equations (4.207), (4.210), and (4.211).

Equation (4.207) is the same as that for the cylindrical coordinates

(4.170), so the solutions of it are the same as (4.174):

Φ(φ) = F

cos mφ + G

sin mφ = f

jmφ

+ g

−jmφ

. (4.212)

Equation (4.210) belongs to a special kind of equations in spherical coor-

dinates. The solutions of it are generally power series, which become inﬁnite

when either θ = 0 or θ = π. These solutions would not be suitable for the

physical problems that include the positive and negative polar axes. For the

sake of avoiding this diﬃculty, let

= n(n + 1), n is an integer or zero, (4.213)

then equation (4.210) becomes

sin θ

dθ

sin θ

dΘ

dθ

n(n + 1) −

sin

Θ = 0. (4.214)

Let

x = cos θ,

the equation becomes

(1 − x

)

dΘ

n(n + 1) −

1 − x

Θ = 0. (4.215)

This is the associate Legendre equation. If m = 0, the function is independent

of the azimuth angle φ, the above equation becomes

(1 − x

)

dΘ

+ n(n + 1)Θ = 0. (4.216)

This is the standard Legendre equation.

When n is an integer or zero, the solutions of the Legendre equation

(4.216) are two sets of polynomials called Legendre functions or Legendre

polynomials [44]:

Θ = C

(x) + D

(x) = C

(cos θ) + D

(cos θ), (4.217)

4.9 Solutions in Spherical Coordinates 217

where

(x) =

− 1

, (4.218)

(x) =

(x) ln

1 + x

1 − x

−

l=1

l−1

(x)Pn − 1(x). (4.219)

Function P

(x) denotes the Legendre polynomial of the ﬁrst kind and Q

(x)

denotes the Legendre polynomial of the second kind, where n is the degree of

the functions, n is an arbitrary integer or zero. Equation (4.218) is the basic

expression of the Legendre function and is called Rodrigue’s formula.

The solutions of the associate Legendre equation (4.215) are also two

sets of polynomials called associate Legendre functions or associate Legendre

polynomials [44]:

Θ = C

(x) + D

(x) = C

(cos θ) + D

(cos θ), (4.220)

where

(x)=

− 1

m/2

(x)=

− 1

m/2

n+m

− 1

, (4.221)

and

(x) =

− 1

m/2

(x). (4.222)

Function P

(x) is the associate Legendre polynomial of the ﬁrst kind, and

(x) is the associate Legendre polynomial of the second kind, where n is

the degree and m is the order of the functions. The order m is an integer

between 1 and n. Functions P

(x) and Q

(x) are spherical harmonics.

The plots of the Legendre polynomials are shown in Figure 4.7. We see

that on the polar axes θ = 0 and θ = π,

(cos θ) → ∞, and Q

(cos θ) → ∞.

So, for the problem that involves positive and negative polar axes, the co-

eﬃcient of Q

(cos θ) must be zero, and function P

(cos θ) is the suitable

solution.

If n + m is not an integer, we rewrite n and m as ν and ω, respectively.

Then the two independent solutions of Legendre’s equation are P

(cos θ) and

(−cos θ).

Θ = C

νω

(cos θ) + D

νω

(−cos θ), (4.223)

and when ω = 0, n is not an integer,

Θ = C

(cos θ) + D

(−cos θ), (4.224)

When ν+ω ia an integer, n+m, P

(cos θ) and Q

(cos θ) are the two inde-

pendent solutions, and when ν + ω is not an integer, any two from P

(cos θ),

(−cos θ) and Q

(cos θ) can be the two independent solutions.

218 4. Time-Varying Boundary Value Problems

0.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

-5

-4

-3

-2

-1

n = 0

0.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

-1.0

-.8

-.6

-.4

-.2

0.0

1.0

n = 0

(cos

)

(cos

)

T

S

T

S

Figure 4.7: Legendre functions of the ﬁrst kind (a) and the second kind (b).

Substituting (4.213) into (4.211), we have the equation for R as

− n(n + 1)

R = 0. (4.225)

With the substitution,

r =

, R =

√

f, (4.226)

(4.225) becomes

−

n +

f = 0. (4.227)

This is a Bessel equation, and the solutions are Bessel functions of order

n + 1/2,

f(u) = A

(u) + B

(u)=a

(1)

(u) + b

(2)

(u). (4.228)

Considering (4.226), we have

R(r) = A

√

(kr) + B

√

(kr), (4.229)

R(r) = a

√

(1)

(kr) + b

√

(2)

(kr). (4.230)

If n is an integer, these Bessel functions of order n + 1/2 reduce to alge-

braic combinations of sinusoids, see Appendix C.5.1, which represent spheri-

cal waves and explain the wave nature of the solutions of Helmholtz’s equa-

tions in spherical coordinates.

4.10 Vector Eigenfunctions and Normal Modes 219

In diﬀerent literatures, a sets of spherical Bessel functions [37, 96]

(x), n

(x), h

(1)

(x), h

(2)

(x);

and a set of spherical Bessel functions deﬁned by S. A. Schelkunoﬀ [86]

(x),

(1)

(x),

(2)

(x)

are introduced, refer to Appendix C.5.2 and C.5.3.

Finally, we have the 3-D solutions in spherical coordinates

U (or V ) = rF = rR ΘΦ =

√

(kr) + b

√

(kr)

·[C

(cos θ)+D

(cos θ)]

·[F

cos mφ + G

sin mφ] , (4.231)

U (or V ) = rF = rR ΘΦ =

√

(1)

(kr) + B

√

(2)

(kr)

·[C

(cos θ)+D

(cos θ)]

·[F

cos mφ + G

sin mφ] . (4.232)

In spherical coordinates, the ﬁeld components may be expressed in terms

of U and V by (4.34)–(4.39) as follows:

∂

∂θ ∂r

−

jωµ

r sin θ

∂V

∂φ

, (4.233)

r sin θ

∂

∂φ ∂r

jωµ

∂V

∂θ

, (4.234)

∂

∂r

+ k

U, (4.235)

∂

∂θ ∂r

jω²

r sin θ

∂U

∂φ

, (4.236)

r sin θ

∂

∂φ ∂r

−

jω²

∂U

∂θ

, (4.237)

∂

∂r

+ k

V. (4.238)

In most cases, the waves in spherical coordinates may also be classiﬁed as

a mode with U = 0 and E

= 0 and a mode with V = 0 and H

= 0. They

are spherical TE and spherical TM modes, respectively

4.10 Vector Eigenfunctions and

Normal Modes

In the previous sections, the vector Helmholtz equation is separated into

three ordinary diﬀerential equations. The problems of solving these equations

with given boundary conditions are eigenvalue problem or so-called Sturm–

Liouville problems.

220 4. Time-Varying Boundary Value Problems

4.10.1 Eigenvalue Problems and Orthogonal Expansions

(1) Sturm–Liouville Problems

By separation of variables in an appropriate coordinate system, the Laplace

equation or Helmholtz equation reduces to ordinary diﬀerential equations in

the following general form

p(x)

dy(x)

− [q(x) − λρ(x)] y(x) = 0. (4.239)

This equation is known as the Sturm–Liouville equation. The function y(x)

satisﬁes the constant boundary conditions of the ﬁrst or the second kind at

x = a and x = b,

y(x)|

x=a

= C

, y(x)|

x=b

= C

, or

dy(x)

x=a

= C

dy(x)

x=b

= C

or the more general mixed boundary conditions:

dy(x)

− βy(x)

x=a, x=b

= C

1, 2

, (4.240)

where α and β are two constants, including α = 0 for the boundary condition

of the ﬁrst kind and β = 0 for that of the second kind.

The problem of solving Sturm–Liouville equation (4.239) with the bound-

ary conditions (4.240) is known as the Sturm–Liouville problem or eigenvalue

problem. The following are four basic theorems about the Sturm–Liouville

problems.

Theorem 1 Only speciﬁc or discrete real values of λ are allowed for a

nontrivial solution of the Sturm–Liouville equation satisfying the speciﬁc set

of boundary conditions. These allowed λ values are called eigenvalues. The

eigenvalues, ordered with respect to magnitude form a denumerable sequence

, λ

, ···, λ

, ···.

For each speciﬁc eigenvalue λ

, there is a corresponding function y

(x) that

satisﬁes the diﬀerential equation (4.239) and the boundary conditions (4.240).

These functions y

(x) are called the eigenfunctions of the problem. The

complete set of eigenfunctions is

(x), y

(x), ···, y

(x), ···.

Each eigenvalue corresponds to one eigenfunction or a number of linearly

independent eigenfunctions. If more than one eigenfunction is allowed for a

particular eigenvalue the problem is said to be degenerate.