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

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1.1 Maxwell’s Equations 11

Let

= I + χ

, ² = ²

= ²

(I + χ

), (1.47)

= I + χ

, µ = µ

= µ

(I + χ

). (1.48)

Then, the constitutive relations of linear, non-dispersive and anisotropic me-

dia become

D = ²

· E = ² · E, (1.49)

B = µ

· H = µ · H, (1.50)

where ² is the tensor permittivity and µ is the tensor permeability, they are

constitutive tensors or matrices, their elements are constitutive parameters,

² =









, (1.51)

µ =









. (1.52)

The anisotropic media characterized by symmetrical tensor permittivity

and permeability are reciprocal media, whereas the media characterized by

asymmetrical tensor permittivity and permeability are nonreciprocal media.

For lossless reciprocal media the permittivity and permeability are symmet-

rical real tensors. All isotropic media are undoubtedly recipro cal.

Detailed discussions of the reciprocal and nonreciprocal anisotropic media

and the electromagnetic ﬁelds and waves in them are given in Chapter 8.

(5) Bi-isotropic and Bi-anisotropic Media

Isotropic and anisotropic media become polarized when placed in an electric

ﬁeld and become magnetized when placed in a magnetic ﬁeld, without cross

coupling between the two ﬁelds. For such media, the constitutive relations

relate the two electric ﬁeld vector functions or the two magnetic ﬁeld vector

functions by either a scalar or a tensor. But in bi-isotropic and bi-anisotropic

media, cross coupling between the electric and magnetic ﬁelds exists. When

placed in an electric or magnetic ﬁeld, a bi-isotropic or bi-anisotropic medium

becomes both polarized and magnetized [53].

The general constitutive relations of bi-anisotropic medium are given by

D = ² · E + ξ · H, (1.53)

B = ζ · E + µ · H, (1.54)

where ², ξ, ζ, and µ are all 3 × 3 matrices or tensors. Generally, some of

them are tensors and some are scalars.

12 1. Basic Electromagnetic Theory

Figure 1.3: (a) Electric dipole, (b) magnetic dipole and (c) micro-helix.

If the above four tensors become scalars, the medium is bi-isotropic. The

constitutive relations of such a medium are

D = ²E + ξH, (1.55)

B = ζE + µH. (1.56)

This set of constitutive equations were conceived by Tellegen in 1948

for realizing the new circuit element, the “gyrator”, suggested by himself

[100]. He considered that the model of the medium had elements possessing

permanent electric and magnetic dipoles parallel or antiparallel to each other,

so that an applied electric ﬁeld that aligns the electric dipoles simultaneously

aligns the magnetic dipoles, and an applied magnetic ﬁeld that aligns the

magnetic dipoles simultaneously aligns the electric dipoles.

One of the physical models of bi-isotropic and bi-anisotropic media is the

micro-helix model. Molecules in isotropic and anisotropic media are consid-

ered to be a large amount of small electric dipoles and/or small current loops.

Molecules become aligned electric dipoles under the action of an electric ﬁeld,

and become aligned molecular current loops, i.e., aligned magnetic dipoles,

under the action of a magnetic ﬁeld, without cross coupling. In bi-isotropic

and bi-anisotropic media, molecules in the medium are considered to be a

large amount of conductive micro-helices shown in Fig. 1.3. Under the action

of a time-varying electric ﬁeld, not only do charges of opposite signs appear

at the two ends of the helix but also, according to the continuous equation,

current arises in the helix. On the other hand, under the action of a time-

varying magnetic ﬁeld, according to Lenz’s theorem, current arises in the

helix and again, according to the continuous equation, charges of opposite

signs appear at the two ends of the helix. The pair of charges at the two

ends form an electric dipole and the current in the helix forms a magnetic

dipole. In conclusion, aligned electric dipoles and magnetic dipoles appear

simultaneously under the action of an electric ﬁeld or a magnetic ﬁeld alone.

The medium is bi-isotropic if the above phenomena are isotropic, otherwise,

it is bi-anisotropic. This kind of media is also known as chiral media be-

cause such an object has the property of chirality or handiness and is either

right-handed or left-handed.

1.1 Maxwell’s Equations 13

The existence of bi-isotropic and bi-anisotropic media, or so-called magne-

toelectric materials, was theoretically predicted by Landau in 1957. Such ma-

terials were observed experimentally by Astrov in 1960 in anti-ferromagnetic

chromium oxide [9]. Now we know that a variety of magnetic crystal classes,

sugar arrays, amino acids, DNA, and organic polymers are among the natu-

ral chiral media, and wire helices and the M¨obius strip are considered to be

man-made chiral objects. An example of an artiﬁcial chiral medium is made

of randomly oriented and uniformly distributed lossless, small, wire helices.

1.1.3 Complex Maxwell’s Equations

The most important time-varying state of ﬁelds is the steady-state alternating

ﬁelds varying sinusoidally in time, that is, the time-harmonic ﬁelds [5, 38,

84, 96]. In linear media, the Maxwell’s equations are linear. A sinusoidal

excitation at a frequency ω produces a sinusoidal response, i.e., ﬁelds vary

sinusoidally with time. Any transient excitation and response, i.e., sources

and ﬁelds with time variations in arbitrary forms, may be considered as

a superposition of sinusoidal sources and ﬁelds of diﬀerent frequencies, by

means of the method of Fourier analysis.

(1) Complex Vectors

When the time variation is harmonic, that is, the ﬁelds are in a steady si-

nusoidal state or a-c state, the mathematical analysis can be simpliﬁed by

using complex quantities or so-called phasors, which have been well devel-

oped in the analysis of a-c circuits. Suppose the circular frequency of the

time harmonic ﬁelds is ω, then the instantaneous quantity of a sinusoidal

time-dependent scalar function A(x, t) can be written as the imaginary part

of a complex scalar function or a scalar phasor as follows:

A(x, t) = A(x) sin(ωt + φ) = =

A(x)e

jφ

jωt

= =

A(x)e

jωt

, (1.57)

where A(x) is the amplitude of the a-c scalar function, and

A(x) = A(x)e

jφ

is called the complex amplitude or phasor of the function, which explains

the amplitude as well as the phase of the ac scalar function. The complex

amplitude A(x) is a function of the space coordinate x only and the time-

dependence is explained in terms of e

jωt

A sinusoidal time-dependent vector function A(x, t) may be decomposed

into three components, as shown in (1.1). Each of the components is a

sinusoidal time-dependent scalar function, and can be expressed in the form

of (1.57). So that,

A (x, t) =

(x, t) +

(x, t)

(x) sin(ωt + φ

) +

(x) sin(ωt + φ

) +

(x) sin(ωt + φ

)

= =

(x)e

jφ

jωt

(x)e

jφ

jωt

(x)e

jφ

jωt

= =

©£

(x) +

(x)

jωt

, (1.58)

14 1. Basic Electromagnetic Theory

where

(x) = A

(x)e

jφ

(x) = A

(x)e

jφ

and

(x) = A

(x)e

jφ

are

the complex scalars of three components, A

(x), A

(x) and A

(x) are the

amplitudes and φ

, φ

and φ

are the phases of the corresponding complex

scalars.

A complex vector may be constructed by three complex scalar components

as follows:

A(x) =

(x) +

(x) = <

A(x)

+ j=

A(x)

. (1.59)

And the instantaneous vector A(x, t) is described as

A(x, t) = =

A(x)e

jωt

= <

A(x)

sin ωt + =

A(x)

cos ωt, (1.60)

where

A(x) denotes a vector phasor or a complex vector of the ﬁeld, which

is a vector function of x and is independent of t.

In the remainder of this book, we omit the dot on the expressions for

phasors except in some necessary cases. The physical ﬁeld is always obtained

by multiplying by a factor e

jωt

and taking the imaginary part. Note that

some authors use e

−jωt

instead of e

jωt

, and some authors take the real part

instead of the imaginary part. These diﬀerences will change the signs in some

equations but will not aﬀect the physical ﬁeld.

(2) Maxwell’s Equations in Complex Form

The partial derivative of the sinusoidal ﬁeld with resp ect to time can b e

expressed as follows,

∂

∂t

A(x, t) = =

A(x)

∂

∂t

jωt

= =

j ωA(x)e

jωt

. (1.61)

Thus Maxwell’s equations (1.25)–(1.28) can b e changed into complex form

by replacing ∂/∂t with j ω:

E · dl = −j ω

B · dS, ∇ × E = −j ωB, (1.62)

H · dl = j ω

D · dS +

J · dS, ∇ × H = j ωD + J , (1.63)

D · dS =

ρdV, ∇ · D = ρ, (1.64)

B · dS = 0, ∇ · B = 0. (1.65)

These are Maxwell’s equations in complex form or the so-called frequency-

domain Maxwell equations.

For uniform simple media,

D = ²E, B = µH,

1.1 Maxwell’s Equations 15

² and µ are constants, and the equations in derivative form become

∇ × E = −j ωµH, (1.66)

∇ × H = j ω²E + σE + J , (1.67)

∇ · E =

, (1.68)

∇ · H = 0. (1.69)

The equation of continuity in the complex form is

J · dS = −j ω

ρdV, ∇·J = −j ωρ. (1.70)

In source-free region, ρ = 0 and J = 0, Maxwell equations in complex

form become

∇ × E = −j ωµH, (1.71)

∇ × H = j ω²E + σE, (1.72)

∇ · E = 0, (1.73)

∇ · H = 0. (1.74)

In the medium with relatively low conductivity and at relatively high

frequency, so that σ ¿ ω², the source-free Maxwell equations in complex

form become

∇ × E = −j ωµH, (1.75)

∇ × H = j ω²E, (1.76)

∇ · E = 0, (1.77)

∇ · H = 0. (1.78)

These equations govern the behavior of steady-state sinusoidal electromag-

netic waves propagating in vacuum or nonconducting simple media.

1.1.4 Complex Permittivity and Permeability

For the steady-state sinusoidal time-dependent ﬁelds, by using the method of

complex phasor notation, the time derivatives of nth order in the constitutive

relations of dispersive media (1.39) and (1.40) can be replaced by the nth

power of j ω:

∂

∂t

→ j ω,

∂

∂t

→ −ω

∂

∂t

→ −j ω

, ···.

Then the equations (1.39) and (1.40) become [37]

D = ²E + j ω²

E − ω

E − j ω

E + ···,

B = µH + j ωµ

H − ω

H − j ω

H + ···.

16 1. Basic Electromagnetic Theory

Separating the real and the imaginary parts, we have

D =

² − ω

+ ···

E − j

−ω²

+ ω

− ···

E, (1.79)

B =

µ − ω

+ ···

H − j

−ωµ

+ ω

− ···

H. (1.80)

Let

(ω) = ² − ω

+ ···, ²

(ω) = −ω²

+ ω

− ···, (1.81)

(ω) = µ − ω

+ ···, µ

(ω) = −ωµ

+ ω

− ···. (1.82)

The complex permittivity ˙²(ω) and the complex permeability ˙µ(ω) are deﬁned

as follows

˙²(ω) = ²

(ω) − j ²

(ω), (1.83)

˙µ(ω) = µ

(ω) − j µ

(ω). (1.84)

They are both functions of frequency for dispersive media.

Thus, the constitutive relations (1.79) and (1.80) take the following com-

plex form:

D = ˙²E = (²

− j ²

)E, (1.85)

B = ˙µH = (µ

− j µ

)H. (1.86)

The complex Maxwell equations (1.66)–(1.69) become

∇ × E = −j ω ˙µ(ω)H = −j ωµ

H − ωµ

H, (1.87)

∇ × H = j ω ˙²(ω)E + σE + J = j ω²

E + ω²

E + σE + J, (1.88)

∇ · ˙²(ω)E = ∇ · (²

− j ²

)E = ρ, (1.89)

∇ · ˙µ(ω)H = ∇ · (µ

− j µ

)H = 0, (1.90)

In the above equations, ω²

E, and σE are terms with the same kind,which

describe the dissipation of the medium, σE expresses the dissipation caused

by conduction current namely Joule’s dissipation, and ω²

E expresses the dis-

sipation caused by the alternative polarization, i.e., polarization dissipation

or dielectric loss. Similarly, ωµ

H expresses the magnetization dissipation

or hysteresis loss.

The complex permittivity and permeability can also be expressed in polar

coordinate,

˙² = |˙²|e

−jδ

= |˙²|cos δ − j |˙²|sin δ, (1.91)

˙µ = |˙µ|e

−jθ

= |˙µ|cos θ − j |˙µ|sin θ, (1.92)

1.1 Maxwell’s Equations 17

where δ denotes the electric loss angle and θ denotes the magnetic loss angle.

The electric and magnetic loss tangents are deﬁned as

tan δ =

, (1.93)

tan θ =

. (1.94)

Note that the real parts of the complex permittivity and permeability

are generally diﬀerent from their static value. They are functions of the

frequency.

For dispersive lossless media, ²

= 0, µ

= 0, and ²

, µ

are functions

with respect to frequency. For some media, the complex permittivity and

permeability are approximately independent of frequency in certain frequency

band, they are non-dispersive lossy media.

For conductive media, σ and ω²

in (1.88) are factors of the same kind.

Equation (1.88) can be rewritten in the following form,

∇ × H = j ω ˙²E + J, (1.95)

where

˙²(ω) = ²

(ω) − j

(ω) +

. (1.96)

The loss tangent becomes

tan δ =

ω²

+ σ

ω²

, (1.97)

and the loss tangent due to the conductive dissipation is

tan δ =

ω²

. (1.98)

Note that in the divergence equation (1.89), ˙² is still deﬁned by (1.83),

instead of (1.96).

1.1.5 Complex Maxwell Equations in

Anisotropic Media

For sinusoidal time-dependent ﬁelds, the constitutive relations of anisotropic

media, (1.49) and (1.50), are written in complex form as follows:

D = ²

E + ²

· E = ²

(I +

) · E, (1.99)

B = µ

H + µ

· H = µ

(I +

) · H, (1.100)

D = ²

· E =

² · E, (1.101)

B = µ

· H =

µ · H, (1.102)

18 1. Basic Electromagnetic Theory

where

² and

µ are complex tensors of rank 2, or complex matrices. The

complex Maxwell equations (1.62)–(1.65) become

∇ × E = −j ω

µ · H, (1.103)

∇ × H = j ω

² · E + σE + J , (1.104)

∇ · (

² · E) = ρ, (1.105)

∇ · (

µ · H) = 0. (1.106)

For dispersive anisotropic media, the elements of the

² and

µ tensors are func-

tions of frequency [38]. Generally, a medium is either electrically anisotropic

or magnetically anisotropic. For the details of the solution of (1.103)–(1.106)

and the ﬁelds and waves in anisotropic media, please refer to Chapter 8.

1.1.6 Maxwell’s Equations in Duality form

Since P. Dirac theoretically argued the existence of magnetic monopoles, the

search for monopoles has been renewed whenever a new energy region has

been opened up in high-energy physics or a new source of matter, such as

rocks from the deep sea or from the moon, have become available. There

is still no universally acknowledged experimental evidence for the existence

of magnetic charges or monopoles in nature [43]. Nevertheless, in applied

electromagnetism, to introduce the ﬁctitious or equivalent magnetic charge

and the ﬁctitious or equivalent magnetic current is beneﬁcial for the analy-

sis of many engineering problems [24, 37]. For example, a small d-c current

loop can be seen as a magnetic dipole formed by two equal and opposite

equivalent magnetic charges, and an a-c current loop can be seen as an a-c

equivalent magnetic current element and two opposite a-c equivalent mag-

netic charges situated at the two ends of the equivalent magnetic current

element. When the equivalent magnetic charge and the equivalent magnetic

current are introduced, Maxwell’s equations become

E · dl = −

B · dS − I

, ∇×E = −

∂B

∂t

− J

, (1.107)

H · dl =

D · dS + I, ∇ × H =

∂D

∂t

+ σE + J , (1.108)

D · dS = q, ∇ · D = %, (1.109)

B · dS = q

, ∇·B = %

. (1.110)

where %

denotes the equivalent magnetic charge density and J

denotes the

equivalent magnetic current density. Note that the equivalent magnetic cur-

rent density J

, the result of the motion of the equivalent magnetic charge, is

1.2 Boundary Conditions 19

entirely diﬀerent from the molecular current density J

, the electric current

due to magnetization.

The complex Maxwell equations become

E ·dl = −j ω

B ·dS −

·dS, ∇×E = −j ωB −J

, (1.111)

H ·dl = j ω

D ·dS +

J ·dS, ∇× H = j ωD + σE + J , (1.112)

D · dS =

ρdV, ∇ · D = ρ, (1.113)

B · dS =

dV, ∇·B = ρ

, (1.114)

In nonconducting media the above equations for an electric ﬁeld are all

the same as those for a magnetic ﬁeld. Fields E and H are therefore dual

quantities, and their equations are in duality form, see Section 1.7.

1.2 Boundary Conditions

The behavior of electromagnetic ﬁelds on the boundary or interface between

media is important in the solution of electromagnetic problems. In macro-

scopic theory, the boundary is considered as a geometrical surface. Maxwell’s

equations in derivative form are applicable only for the ﬁelds that are con-

tinuous and diﬀerentiable functions. The ﬁeld functions and their derivatives

are discontinuous across the boundary between two media, whether they are

simple media or not. At this case, Maxwell’s equations in integral form

must be considered. When account is taken of the eﬀects of polarization and

magnetization, the instantaneous Maxwell equations, the complex Maxwell

equations in integral form and the integral Maxwell equations in duality form

are given by (1.25)–(1.28), (1.62)–(1.65) and (1.111)–(1.114) (left column),

respectively.

1.2.1 General Boundary Conditions

By applying Maxwell’s equations in integral form, (1.62)–(1.65), on the

boundary between media, refereing to Fig. 1.4(a), we have the following

boundary equations:

n × (E

− E

) = 0, (1.115)

n × (H

− H

) = J

, (1.116)

n · (D

− D

) = ρ

, (1.117)

n · (B

− B

) = 0, (1.118)

20 1. Basic Electromagnetic Theory

Figure 1.4: (a) Boundary between two media and (b) boundary between a

perfect conductor and an insulator.

where n is the unit vector normal to the boundary and pointing from medium

1 to medium 2, J

is the surface current density in A/m; ρ

is the surface

charge density in C/m. Note that only free surface charge and true surface

current are included in the ρ

and J

, respectively, the bound charge and the

molecular current on the surface are not included.

By using (1.111)–(1.114), the boundary equations in duality form become

n × (E

− E

) = −J

, (1.119)

n × (H

− H

) = J

, (1.120)

n · (D

− D

) = ρ

, (1.121)

n · (B

− B

) = ρ

, (1.122)

where J

is the equivalent surface magnetic current density and ρ

is the

equivalent surface magnetic charge density [24, 37].

At the two sides of a boundary between conductive media with diﬀerent

conductivities, from the equation of continuity in integral form (1.70), the

following boundary equation is obtained:

n · (J

− J

) = −j ωρ

. (1.123)

The instantaneous form of the above equation is

n · (J

− J

) = −

∂ρ

∂t

. (1.124)

Equations (1.119)–(1.122) and (1.123) are the general boundary equa-

tions. We must determine whether the surface charge and the surface current

exist or not by knowledge of the physical situation of the boundary. Here are

the boundary conditions for some special cases.