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 |
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uðr; tÞ ¼ Real½A0ðrÞejwt&; |
ð3:4-2Þ |
where |
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A0ðrÞ ¼ AðrÞejk r |
ð3:4-3Þ |
A0ðrÞ is called the phasor (representation) of uðr; tÞ. It is time independent. We note the following:
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A0ðrÞejwt ¼ jwA0 |
ðrÞejwt |
ð3:4-4Þ |
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dt |
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ð A0 |
1 |
A0 |
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ðrÞejwtdt ¼ |
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ðrÞejwt |
ð3:4-5Þ |
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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
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Eðr; tÞ ¼ |
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jwt |
& |
ð3:4-6Þ |
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Real½EðrÞe |
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Hðr; tÞ ¼ |
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jwt |
&; |
ð3:4-7Þ |
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Real½HðrÞe |
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~ |
~ |
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where EðrÞ and HðrÞ are the phasors. The corresponding phasors for D, B, and J are |
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defined similarly. |
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~ |
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Maxwell’s equations in terms of EðrÞ and HðrÞ can be written as |
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¼ |
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ð3:4-8Þ |
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r D |
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¼ 0 |
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ð3:4-9Þ |
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r B |
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~ |
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ð3:4-10Þ |
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r H ¼ jweE |
þ J |
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~ |
~ |
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ð3:4-11Þ |
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r E ¼ jwB |
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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 |
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r r H ¼ e |
@ |
r E ¼ em |
@2H |
@t |
@t2 |
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r r H can be expanded as |
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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 |
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@y |
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@z |
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r rHi ¼ r2Hi is given by |
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@2Hi |
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@2Hi |
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@2Hi |
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r2Hi ¼ |
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ex þ |
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ey þ |
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ez |
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@x2 |
@y2 |
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@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 |
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r |
Hi; i ¼ x, y, z, respectively. |
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It can be shown that rðr HÞ, which is the gradient vector of r H, equals zero. |
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Hence, |
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r r H ¼ r rH ¼ r2H |
ð3:5-5Þ |
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Substituting this result in Eq. (3.5-1) gives |
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r2H ¼ em |
@2H |
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ð3:5-6Þ |
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@t2 |
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Similarly, it can be shown that |
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r2E ¼ em |
@2E |
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ð3:5-7Þ |
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@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 |
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WAVE EQUATIONS IN A CHARGE-FREE MEDIUM |
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Let us consider one such field component as uðr; tÞ. It satisfies |
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@2 |
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@2 |
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1 @2u r; t |
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r2uðr; tÞ ¼ |
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þ |
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uðr; tÞ ¼ |
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ð |
Þ |
ð3:5-9Þ |
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@x2 |
@y2 |
@z2 |
c2 |
@t2 |
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A 3-D plane wave solution of this equation is given by |
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uðr; tÞ ¼ Aðk; oÞ cosðk r otÞ; |
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ð3:5-10Þ |
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where the wave vector k is given by |
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k ¼ kx þ kyey þ kzez |
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ð3:5-11Þ |
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and the phase velocity v is related to k and o by |
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v ¼ |
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ð3:5-12Þ |
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jkj |
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A phase front is defined by |
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k r ot ¼ constant |
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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 |
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ay ¼ lfy |
ð3:5-16Þ |
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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):
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ð3:6-1Þ |
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r r E |
¼ jwmH; |
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¼ 0 in Eq. (3.6-1) yields |
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where B ¼ mH is used. Utilizing Eq. (3.4-10) with J |
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ð3:6-2Þ |
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r r E ¼ w |
meE |
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As in Section 3.4, we have |
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ð3:6-3Þ |
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r r E ¼ rðr EÞ r r~E |
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As r E ¼ 0 by Eq. (3.4-8), we have |
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ð3:6-4Þ |
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r |
E þ w |
meE ¼ 0 |
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We define the wave number k by |
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k |
¼ |
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ð |
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wpme |
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3:6-5 |
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Then, Eq. (3.6-4) becomes |
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2 ~ |
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þ k |
2 ~ |
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ð3:6-6Þ |
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r |
E |
E ¼ 0 |
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This is called the homogeneous wave equation for |
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E. The homogeneous wave |
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equation for H can be similarly derived as |
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2 |
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þ k |
2 ~ |
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ð3:6-7Þ |
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r |
H |
H ¼ 0 |
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Let E be written as |
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ð3:6-8Þ |
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E ¼ Exex þ Eyey þ Ezez |
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Equation (3.6-6) can now be written for each component Ei of E as |
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@2 |
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þ |
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þ |
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þ k2 E~i ¼ 0 |
ð3:6-9Þ |
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@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 |
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Solution: (a) A uniform plane wave is characterized by |
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dEi |
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dEi |
dHi |
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dHi |
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¼ |
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¼ |
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¼ |
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¼ 0 |
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dx |
dy |
dx |
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Hence, Eq. (3.6-9) simplifies to
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@ |
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2 |
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@z2 þ k |
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¼ 0 |
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(b) Consider the z-component of Eq. (3.4-10): |
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~ |
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dHy |
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dHx |
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dx |
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¼ jweEz |
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dHy |
dHx |
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As |
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¼ dy |
¼ 0, Ez ¼ 0. We can similarly show that Hz ¼ 0 by using Eq. (3.4-11). |
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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