Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo

28	FUNDAMENTALS OF WAVE PROPAGATION

v is also known as the phase velocity and can be larger than the vacuum speed of light in the case of light waves.

More generally, a 1-D wave given by Eq. (3.2-8) may satisfy another partial differential equation with respect to x and t, in addition to Eq. (3.2-5). A general wave equation that has a simple harmonic solution can be written as

@

@

uðx; tÞ ¼ 0;

ð3:2-11Þ

f

;

@x

@t

where f is a polynomial function of

@

and

@

. For example,

@x

@t

@3u

þ a

@3u

B

@u

¼ 0

ð3:2-12Þ

@x3

@x2@t

@t

Using Eq. (3.2-8) for the simple

harmonic

solution allows

the substitution of

@

! jo, and Eq. (3.2-12) becomes

@t

k3 ak2o Bo ¼ 0

ð3:2-13Þ

Such an equation relating o and k is known as the dispersion equation. The phase velocity v in this case is given by

v ¼	o		¼	k2	ð3:2-14Þ
	k			ak2 þ B

A wave with a phase velocity, which is a function of wave number or wavelength other than Eq. (3.2-10), is known as a dispersive wave. Hence, Eq. (3.2-14) is a dispersive wave equation.

Some examples of waves are the following:

A.Mechanical waves, such as longitudinal sound waves in a compressible fluid, are governed by the wave equation

	@2uðx; tÞ	¼			@2uðx; tÞ	;	ð	3:2-15	Þ
	@x2		Kc @t2

where is the fluid density, and Kc is the compressibility. Hence, the phase velocity is given by

s

Kc

v ¼ ð3:2-16Þ

WAVES

29

B. Heat diffuses under steady-state conditions according to the wave equation

Kh	@2uðx; tÞ	¼		@uðx; tÞ	;	ð	3:2-17	Þ
s @x2
				@t

where Kh is the thermal conductivity, s is the specific heat per unit volume, and uðx; tÞ is the local temperature.

C. The Schrodinger’s wave equation in quantum mechanics is given by

h2

@2uðx; tÞ

þ

V x

u

x; t

Þ ¼

j

h

@uðx; tÞ

;

ð

3:2-18

Þ

4p2m @x2

2p @t

ð Þ

ð

where m is mass, V(x) is potential energy of the particle, h is Planck’s constant, and uðx; tÞ is the state of the particle. The dispersion relation can be written as

ho ¼	h2k2	ð3:2-19Þ
	2m þ VðxÞ;

where h equals h=2p.

D.Time-varying imagery can be considered to be a 3-D function gðx; y; tÞ. Its 3-D Fourier representation can be written as

1
gðx; y; tÞ ¼ ð ð ð Gðfx; fy; f Þej2pðfxxþfyyþftÞdfxdfydf ;	ð3:2-20Þ
1
where
1
Gðfx; fy; f Þ ¼ ð ð ð gðx; y; tÞe j2pðfxxþfyyþftÞdxdydt	ð	3:2-21	Þ
1

¼ Aðfx; fy; f Þej ðfx;fy;f Þ

For real gðx; y; tÞ, Gðfx; fy; f Þ ¼ G ð fx; fy; f Þ so that Eq. (3.2-21) can be written as

	ð	2	ð	ð		3
gðx; y; tÞ ¼ 2	1		1	1	Aðfx; fy; f Þ cosð2pðfxx þ fyy þ ftÞ þ ðfx; fy; f ÞÞdfxdfydf
	0	4 1 1				5

ð3:2-22Þ

30	FUNDAMENTALS OF WAVE PROPAGATION

Aðfx; fy; f Þ cosð2pðfxx þ fyy þ ftÞ þ ðfx; fy; f ÞÞ is a 2-D plane wave with amplitude Aðfx; fy; f Þ, phase ðfx; fy; f Þ, spatial frequencies fx and fy, and time frequency f. In this case, the wave vector k is given by

k ¼ 2pðfxex þ fyeyÞ ¼ kxex þ kyey;

ð3:2-23Þ

where ex and ey are the unit vectors along the x- and y-directions, respectively. The components kx and ky are the radian spatial frequencies along the x-direction and y-direction, respectively. fx and fy are the corresponding spatial frequencies.

The 2-D plane wave can be written as

uðr; tÞ ¼ Aðk; oÞ cosðk r þ ot þ ðk; oÞÞ;

ð3:2-24Þ

where

r ¼ xex þ yey

ð3:2-25Þ

If we assume uðr; tÞ moves in the direction k with velocity v, uðr; tÞ can be shown to be a solution to the 2-D nondispersive wave equation given by

	r	2u r; t		1	@2uðr; tÞ	;		3:2-26
			Þ ¼ v2 @t2				ð		Þ
		ð

where

r2	¼	@2		þ	@2	ð3:2-27Þ
			@x2		@y2

In conclusion, time-varying imagery can be represented as the sum of a number of 2-D plane waves. This conclusion can be extended to any multidimensional signal in which one of the dimensions is treated as time.

Equation (3.2-26) is the generalization of Eq. (3.2-5) to two spatial coordinates x and y. If three spatial coordinates x, y, and z are considered with the same wave property, Eqs. (3.2-23) and (3.2-25) remain the same, with the following definitions of r and r2:

r ¼ xex þ yey þ zez;

ð3:2-28Þ

where ez is the additional unit vector along the z-direction, and

r2	¼	@2	þ	@2	þ	@2	ð3:2-29Þ
		@x2		@y2		@z2

ELECTROMAGNETIC WAVES

31

The r2 operator is known as the Laplacian. The 3-D wave vector used in Eq. (3.2-24) is given by

k ¼ 2pðfxex þ fyey þ fzezÞ ¼ kxex þ kyey þ kzez

ð3:2-30Þ

Thus, kz is the additional radian frequency along the z-direction, with fz being the corresponding spatial frequency. The plane for which uðr; tÞ is constant is called the wavefront. It moves in the direction of k with velocity v. This is discussed further in Section 3.5.

In what follows, we will assume linear, homogeneous, and isotropic media unless otherwise specified. We will also assume that metric units are used.

EXAMPLE 3.1 Consider uðx1; t1Þ ¼ uðx2; t2Þ; x2x1; t2t1 for a right-traveling wave. Show how x1 and x2 are related.

Solution: uðx1; t1Þ can be considered to be moving to the right with velocity v. Letting t ¼ t2 t1, uðx1; t1Þ and uðx2; t2Þ will be the same when

x2 x1 ¼ v t

3.3ELECTROMAGNETIC WAVES

Electromagnetic waves are 3-D waves, with three space dimensions and one time dimension. There are four quantities D, B, H, and E, which are vectors in space coordinates and whose components are functions of x, y, z, and t. In other words, D, B, H, and E are directional in space. D is the electric displacement (flux) (vector)

field (C=m2), E is the electric (vector) field (V=m), B is the magnetic density (vector) field (Wb/m2), H is the magnetic (vector) field ðA=mÞ, is the charge density (C/ m3), and J is the current density (vector) field (A/m2).

Electromagnetic waves are governed by four Maxwell’s equations. In metric units, they are given by

r D ¼						ð3:3-1Þ
r B ¼ 0						ð3:3-2Þ
	@D
r H ¼				þ J		ð3:3-3Þ
	@t
r E ¼		@B			;	ð3:3-4Þ
			@t

where the divergence and curl of a vector A with components Ax, Ay, and Az are given by

r A ¼	@Ax	þ	@Ay	þ	@Az	ð3:3-5Þ
	@x		@y		@z

32

FUNDAMENTALS OF WAVE PROPAGATION

A

e

e

e

@x

@y

@z

r ¼

@x @y @z

Ax

Ay

Az

@Az

@Ay

@Ax

@Az

@Ay

@Ax

¼

ex þ

ey þ

ez

ð3:3-6Þ

@y

@z

@z

@x

@x

@y

Two more relations are

D ¼ eE

ð3:3-7Þ

B ¼ mH;

ð3:3-8Þ

where e is the permittivity, and m is the permeability. Their values in free space or vacuum are given by

10 9

e0 ¼ 36p F=m

m0 ¼ 4p 10 7 H=m

In dielectrics, e and m are greater than e0 and m0, respectively. In such media, we also consider dipole moment density P (C/m2), which is related to the electric field E by

P ¼ we0E;

ð3:3-9Þ

where w is called the electric susceptibility. w can be considered as a measure for electric dipoles in the medium to align themselves with the electric field.

We also have

D ¼ e0E þ P ¼ e0ð1 þ wÞE ¼ eE;	ð3:3-10Þ
where
e ¼ e0ð1 þ wÞ	ð3:3-11Þ

In a uniform isotropic dielectric medium in which space charge and current density J are zero, Maxwell’s equations become

r D ¼ er E ¼ 0									ð3:3-12Þ
r B ¼ mr H ¼ 0									ð3:3-13Þ
@D						@E
r H ¼			¼ e						ð3:3-14Þ
	@t					@t
r E ¼		@B			¼ m			@H	ð3:3-15Þ
		@t						@t

PHASOR REPRESENTATION

33

3.4PHASOR REPRESENTATION

The electric and magnetic fields we consider are usually sinusoidal with a timevarying dependence in the form

uðr; tÞ ¼ AðrÞ cosðk r wtÞ	ð3:4-1Þ
We can express uðr; tÞ as
uðr; tÞ ¼ Real½A0ðrÞejwt&;	ð3:4-2Þ
where
A0ðrÞ ¼ AðrÞejk r	ð3:4-3Þ

A0ðrÞ is called the phasor (representation) of uðr; tÞ. It is time independent. We note the following:

	d
		A0ðrÞejwt ¼ jwA0				ðrÞejwt	ð3:4-4Þ
	dt
ð A0			1		A0
			ðrÞejwtdt ¼			ðrÞejwt	ð3:4-5Þ
				jw

Hence, differentiation and integration are equivalent to multiplying A0ðrÞ by jw and 1=jw, respectively.

The electric and magnetic fields can be written in the phasor representation

as

	Eðr; tÞ ¼	~	jwt		&		ð3:4-6Þ
		Real½EðrÞe
	Hðr; tÞ ¼	~		jwt		&;	ð3:4-7Þ
		Real½HðrÞe
~	~
where EðrÞ and HðrÞ are the phasors. The corresponding phasors for D, B, and J are
defined similarly.
	~	~
Maxwell’s equations in terms of EðrÞ and HðrÞ can be written as
	~	¼					ð3:4-8Þ
	r D
	~	¼ 0					ð3:4-9Þ
	r B
	~	~	~				ð3:4-10Þ
	r H ¼ jweE		þ J
	~	~					ð3:4-11Þ
	r E ¼ jwB

34	FUNDAMENTALS OF WAVE PROPAGATION

3.5WAVE EQUATIONS IN A CHARGE-FREE MEDIUM

Taking the curl of both sides of Eq. (3.3-3) gives
r r H ¼ e	@	r E ¼ em	@2H
	@t		@t2
r r H can be expanded as

r r H ¼ rðr HÞ r rH;

where rH is the gradient of H. This means rHi; i ¼ x, y, z is given by

rHi ¼

@Hi

ex þ

@Hi

ey þ

@Hi

ez

@x

@y

@z

r rHi ¼ r2Hi is given by

@2Hi

@2Hi

@2Hi

r2Hi ¼

ex þ

ey þ

ez

@x2

@y2

@z2

ð3:5-1Þ

ð3:5-2Þ

ð3:5-3Þ

ð3:5-4Þ

Thus,		r rH	is a vector whose	components along the				three directions are
2
r	Hi; i ¼ x, y, z, respectively.
	It can be shown that rðr HÞ, which is the gradient vector of r H, equals zero.
Hence,
			r r H ¼ r rH ¼ r2H					ð3:5-5Þ
	Substituting this result in Eq. (3.5-1) gives
			r2H ¼ em		@2H			ð3:5-6Þ
					@t2
Similarly, it can be shown that
			r2E ¼ em		@2E			ð3:5-7Þ
					@t2

Equations (3.5-6) and (3.5-7) are called the homogeneous wave equations for E and H, respectively.

In conclusion, each component of the electric and magnetic field vectors satisfies p

the nondispersive wave equation with phase velocity v equal to 1= em. In free space, we get

1	3 108 m= sec	ð3:5-8Þ
c ¼ v ¼ pe0m0

WAVE EQUATIONS IN A CHARGE-FREE MEDIUM

35

Let us consider one such field component as uðr; tÞ. It satisfies

@2

@2

@2

1 @2u r; t

r2uðr; tÞ ¼

þ

þ

uðr; tÞ ¼

ð

Þ

ð3:5-9Þ

@x2

@y2

@z2

c2

@t2

A 3-D plane wave solution of this equation is given by

uðr; tÞ ¼ Aðk; oÞ cosðk r otÞ;

ð3:5-10Þ

where the wave vector k is given by

k ¼ kx þ kyey þ kzez

ð3:5-11Þ

and the phase velocity v is related to k and o by

v ¼

o

ð3:5-12Þ

jkj

A phase front is defined by

k r ot ¼ constant

ð3:5-13Þ

This is a plane whose normal is in the direction of k. When ot changes, the plane changes, and the wave propagates in the direction of k with velocity c. We reassert that cosðk r otÞ can also be chosen as cosðk r þ otÞ. Then, the wave travels in the direction of k.

The components of the wave vector k can be written as

ki ¼	2p	ai ¼ kai	i ¼ x; y; z;	ð3:5-14Þ
	l

where k ¼ jkj, and ai is the direction cosine in the ith direction. If the spatial frequency along the ith direction is denoted by fi equal to ki=2p, then the direction cosines can be written as

ax ¼ lfx		ð3:5-15Þ
ay ¼ lfy		ð3:5-16Þ
	¼ q	ð3:5-17Þ
az	1 l2fx2 l2fy2

as

ax2 þ ay2 þ az2 ¼ 1

ð3:5-18Þ

36	FUNDAMENTALS OF WAVE PROPAGATION

3.6 WAVE EQUATIONS IN PHASOR REPRESENTATION IN A CHARGE-FREE MEDIUM

This time let us start with the curl of Eq. (3.4-11):

~

~

ð3:6-1Þ

r r E

¼ jwmH;

~

~

~

¼ 0 in Eq. (3.6-1) yields

where B ¼ mH is used. Utilizing Eq. (3.4-10) with J

~

2

~

ð3:6-2Þ

r r E ¼ w

meE

As in Section 3.4, we have

~

~

ð3:6-3Þ

r r E ¼ rðr EÞ r r~E

~

As r E ¼ 0 by Eq. (3.4-8), we have

2

~

2

~

ð3:6-4Þ

r

E þ w

meE ¼ 0

We define the wave number k by

k

¼

ð

Þ

wpme

3:6-5

Then, Eq. (3.6-4) becomes

2 ~

þ k

2 ~

ð3:6-6Þ

r

E

E ¼ 0

This is called the homogeneous wave equation for

~

E. The homogeneous wave

~

equation for H can be similarly derived as

2

~

þ k

2 ~

ð3:6-7Þ

r

H

H ¼ 0

~

Let E be written as

~

~

~

~

ð3:6-8Þ

E ¼ Exex þ Eyey þ Ezez

~

~

Equation (3.6-6) can now be written for each component Ei of E as

@2

@2

@2

þ

þ

þ k2 E~i ¼ 0

ð3:6-9Þ

@x2

@y2

@z2

EXAMPLE 3.2 (a) Simplify Eq. (3.6-9) for a uniform plane wave moving in the

~ ~

z-direction, (b) show that the z-component of E and H of a uniform plane wave equals zero, using the phasor representation, (c ) repeat part (b) using Maxwell’s equations.

PLANE EM WAVES								37
Solution: (a) A uniform plane wave is characterized by
~		~		~		~
dEi			dEi	dHi			dHi
		¼		¼		¼		¼ 0
	dx		dy		dx		dy

Hence, Eq. (3.6-9) simplifies to

2

@

2

~

@z2 þ k

Ei

¼ 0

(b) Consider the z-component of Eq. (3.4-10):

~

~

dHy

dHx

~

dx

dy

¼ jweEz

~

~

~

~

dHy

dHx

As

¼ dy

¼ 0, Ez ¼ 0. We can similarly show that Hz ¼ 0 by using Eq. (3.4-11).

dx

(c) We write

E ¼ ExejðkzþwtÞex þ EyejðkzþwtÞey þ EzejðkzþwtÞez

Substituting E in r E ¼ 0, we get

@E ejðkzþwtÞ ¼ 0

@z z

implying Ez ¼ 0. We can similarly consider the magnetic field H as

H ¼ HxejðkzþwtÞex þ HyejðkzþwtÞey þ HzejðkzþwtÞez

Substituting H in r H ¼ 0, we get

@H ejðkzþwtÞ ¼ 0

@z z

implying Hz ¼ 0.

3.7PLANE EM WAVES

Consider a plane wave propagating along the z-direction. The electric and magnetic fields can be written as

E ¼ Exejðkz wtÞ^ex þ Eyejðkz wtÞ^ey

H ¼ Hxejðkz wtÞ^ex þ Hyejðkz wtÞ^ey;

ð3:7-1Þ