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

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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

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