Electromagnetics P16-1

Edited by: Shang-Da Yang

Lesson 16 Plane Waves in Homogeneous Media

Introduction

By eq’s (15.5-7), E v

and H v

in a charge-free ( 0=ρ ), nonconducting ( 0=J v

), simple

(linear, homogeneous, isotropic) medium are governed by the vector wave equations:

01 2 2

2 2 =

∂ ∂

−∇ t E

u E

p

v v

, 01 2 2

2 2 =

∂ ∂

−∇ t H

u H

p

v v

, where µε 1

=pu .

The general solutions are waves propagating with phase velocity pu .

A plane wave is a particular solution to the vector wave equation, where every point on an

infinite plane perpendicular to the direction of propagation has the same electric field E v

and

magnetic field H v

(in terms of magnitude and direction, see Fig. 16-1).

Fig. 16-1. A general plane wave propagates in +z-direction illustrated at some time instant. The

fields E v

and H v

are constant throughout the plane 1zz = , but may differ from those on another plane 2zz = .

Time-harmonic (sinusoidal) plane waves are of particular importance because:

1) The mathematical treatments are greatly simplified.

2) Superposition of time-harmonic plane waves remains a solution to the (linear) vector

wave equation. They can be used as a basis to describe general electromagnetic waves.

Electromagnetics P16-2

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E.g. Superposition of time-harmonic plane waves of different frequencies and common

direction of propagation can describe plane wave packets. Superposition of time-

harmonic plane waves of the same frequency and different directions of propagation can

describe Gaussian beam.

3) It is a good approximation for far-field EM waves (e.g. sunlight received on the earth),

and waves whose cross-sections have a linear dimension much larger than the wavelength

(e.g. typical laser beam).

16.1 Plane Waves in Vacuum

Most simplified time-harmonic (sinusoidal) plane waves

The vector phasor E v

of a time-harmonic E-field is satisfied with:

022 =+∇ EkE vv

(15.13)

where puk ω= [eq. (15.15)] denotes the wavenumber. For simplicity, consider a

time-harmonic plane wave propagating in the z-direction and the electric field is polarized in

the x-direction, ⇒

)( zEaE xx vv = . (16.1)

Note that eq. (16.1) does represent a plane wave, for any point on an infinite plane 0zz =

(perpendicular to the direction of propagation za v ) must have the same electric field

)( 0zEa xx v . In this particular case, eq. (15.13) reduces to a scalar ordinary differential equation:

022 2

=+ x x Ek

dz Ed

(16.2)

In the absence of physical boundary, the general solution to eq. (16.2) consists of two

counter- propagating waves:

jkzjkz x eEeEzE

+−−+ += 00)( , (16.3)

Electromagnetics P16-3

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where +++ = φjeEE 00 and

−−− = φjeEE 00 are complex amplitudes corresponding to

time-harmonic waves propagating in the +z and −z directions, respectively.

1) As in transmission lines [eq’s (4.7), (4.8)], phasor jkzx eEzE −++ = 0)( describes a

time-harmonic wave with velocity kup ω= because:

{ } ( )++++ +−== φωω kztEezEtzE tjxx cos)(Re),( 0

is a function of variable ( )k zt

ω τ −= . By eq. (15.15),

µεµεω ω 1

==pu , which is the

same as eq. (15.7).

2) As in transmission lines, reflected wave jkzx eEzE +−− = 0)( can emerge if the medium is

nonhomogeneous along the z-direction (i.e. ε or µ changes with z ).

Characteristics of plane waves

The electrical field of a plane wave propagating in an arbitrary direction ka v is of the form

(Fig. 16-2):

( ) rkjzyxrkj eEaaaeEeE vvvv vvvvv ⋅−⋅− ++== 00 coscoscos γβα (16.4)

(1) The phase evolution term kz in eq. (16.3) is generalized by rk v v ⋅ , where

kakakakak kzzyyxx vvvvv =++=

is the wavevector, and zayaxar zyx vvvv ++= is the position vector of the observation point. (2)

The direction of electric field (polarization direction) xa v in eq. (16.1) is generalized by a

unit vector:

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Edited by: Shang-Da Yang

γβα coscoscos zyx aaae vvvv ++=

where { }γβα ,, are the included angles between ev and { }zyx aaa vvv ,, , respectively. For any

point rv on an infinite plane perpendicular to the direction of propagation ka v (intensity &

phase front, Fig. 16-2), constantcos ==⋅ θkrrk v v

, ⇒ rkjeEeE vvvv ⋅−= 0 is a constant vector.

Therefore, eq. (16.4) does represent a plane wave.

Fig. 16-2. A plane wave propagates in an arbitrary direction on the xz-plane.

The three vectors E v

, H v

, k v

of a time-harmonic plane wave are mutually orthogonal in a

charge-free simple medium.

Proof:

(1) In a charge-free ( 0=ρ ) simple medium, eq. (7.8) reduces to:

0=⋅∇ E v

.

(2) By the vector identity ( ) ( ) ( )ψψψ ∇⋅+⋅∇=⋅∇ AAA vvv (where ψ , Av are arbitrary scalar and vector fields, respectively), eq. (16.4) reduces to:

( ) ( ) ( ) ( )[ ]rkjrkjrkjrkj eEeeEeeeEeEeE vvvvvvvv vvvvv ⋅−⋅−⋅−⋅− ∇⋅=∇⋅+⋅∇=⋅∇=⋅∇ 0000 . (3) By the formula of gradient in Cartesian coordinate system,

( ) ( ) rkjrkjzzyyxxzkykxkjzyxrkj ekjekakakajezayaxae zyx

vvvvvv vvvvvvv ⋅−⋅−++−⋅− −=++−= 

  

 ∂ ∂

+ ∂ ∂

+ ∂ ∂

=∇ )( .

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(4) ( )[ ] ( ) 000 =⋅−=−⋅=⋅∇ ⋅−⋅− krkjrkj aeejkEekjEeE vvvvv v vvv

, ⇒ kae vv ⊥ , i.e.,

kE vv

⊥ (16.5)

If E v

has been known, one can derive H v

by HjE vv

ωµ−=×∇ [eq. (15.9)]:

(5) By the formula of curl in Cartesian coordinate system,

( ) γβα

ωµωµ coscoscos

11

000 rkjrkjrkj

zyx

eEeEeE zyx

aaa

j E

j H

vvvvvv

vvv

vv

⋅−⋅−⋅−

∂∂∂∂∂∂ −

=×∇ −

=

γβα ωµ

coscoscos

1

000 rkjrkjrkj zyx

zyx

eEeEeE jkjkjk aaa

j vvvvvv

vvv

⋅−⋅−⋅−

−−− −

=

( ) ( )k EaEkEkj

j k

ωµωµωµ

vvvvvv × =

× =×−

− =

1 , ⇒

η EaH k vvv ×

= , (16.6)

where ε µ

µεω ωµωµη ===

k is the intrinsic impedance of the medium:

ε µη = (Ω) (16.7)

For vacuum, ( )Ω== 377000 εµη . Eq. (16.6) implies that kH vv

⊥ and EH vv

⊥ .

Combined with eq. (16.5), ⇒ E v

, H v

, k v

of a time-harmonic plane wave are mutually

orthogonal in a charge-free simple medium. The fact that both E v

and H v

are perpendicular

to the direction of propagation ( ka v ) makes the plane wave a transverse electromagnetic

(TEM) wave.

Example 16-1: (1) If jkzxxx eEazEaE −++ == 0)(

vvv , ⇒ kak z vv = . By eq. (16.6), ⇒

ηη )()( zEazEaaH xyxxz

++

= ×

= v vvv

, where η

)()( zEzH xy +

+ = . (2) If jkzxxx eEazEaE +−− == 0)(

vvv , ⇒

Electromagnetics P16-6

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kak z vv −= ,

ηη − =

×− =

−− )()( zEazEaaH xyxxz v

vvv , where

η− =

− − )()( zEzH xy . (3) If

jkz yyy eEazEaE

−++ == 0)( vvv , ⇒ kak z

vv = , ηη −

= ×

= ++ )()( zE

a zEaa

H yx yyz v

vv v

, where

η− =

+ + )()(

zE zH yx .

1) Intrinsic impedance η [eq. (16.7)] of plane wave is analogous to the characteristic

impedance 0Z [eq. (2.8)] in transmission lines:

0Z VIEaH k

+ + =↔

× =

η

vvv implies { vE↔

v , iH ↔

v }.

C LZ =↔= 0ε

µη implies { L↔µ , C↔ε }.

2) Not all TEM waves are plane waves. E.g. Coaxial transmission lines support TEM waves

( raE vv // , φaH

vv // , zak vv // ) with −r dependent (non-uniform) fields.

What is the state of polarization of time-harmonic plane waves?

By eq. (16.4), the electric field phasor of a time-harmonic plane wave propagating in the

+z-direction is:

( ) jkzyxrkj eEaaeEeE −⋅− +== 00 coscos βα vvv v vv

.

At 0zz = , the time-dependent electric field vector:

{ } ( )0000 cos)(Re),( kztEeezEtzE tj −== ωω v vv

will traverse a line of length 02E along e v in one period ωπ2=T (Fig. 16-3a). In

general, the electric field of a time-harm