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Plane Waves, Polarization and the Poynting Vector

Hy =

Ex+ Exh h

Uniform Plane Wave in Free Space We have previously established the following properties of plane waves in free space: · Electric and magnetic field components in propagation direction are zero. r r r r · Electric and magnetic fields ( E x , By or E y , Bx ) are related. Each is the "source" of the other. · A set of three second order differential equations apply, one for each field component in rectangular coordinates. For example: r r 2 Ex m o e o 2 Ex , = z2 t 2 or for harmonic fields, in phasor notation: r 2 r Ex = - w 2 moe o Ex z2 A general solution to the Helmholz equation is written: r r r r E x = C 1 e - jb o z + C 2 e jb o z ( C i can be complex), where b o = w mo e o = w / c , c = speed of light. Propagation of Magnetic Field Suppose we have an electric field wave travelling in the positive z direction. r r Recall that curl E = - jw B , or for our travelling wave, r r r r Ex - jb z E x = E + xo e = - jw B y z r - jb o z E + xo e r = - jw B y z r r r b - j b o E + xo e - jb o z = - j w B y fi B y = o E + xo e - jb o z w w r = E + xo e - jb o z cw

o

d

i

r · Electric and magnetic ( B ) field are orthogonal (perpendicular, Right Hand Rule), in-phase, and the ratio of the field magnitudes is the impedance.

It isrvery useful to express this ratio of electric and magnetic fields in terms r of H , rather than B . r r r Ex = Ex moeo , mo Hy = c r mo mo Ex = or, after rearranging: r = m o c = e o mo eo Hy Dimensional analysis: or:

Henry / m = Farad / m

Joule / amp 2 volt = = Ohm!!! amp Joule / volt 2

r Electric fields are volts/meter and H fields are amps/meter.

r mo Ex r = h o Ohms, ho ª 120 p = 377 Ohms (impedance of freespace) r eo Hy E $ For a wave travelling in the - ez direction, r x = - h o . This just means that Hy r r $x direction (or equivalently with H y ). E x is oriented along the - e

Summary: r r · E and H are perpendicular to each other and propagation direction. Right Hand Rule gives direction. r r · Ratio of H to E is the intrinsic wave impedance, ho . The wavelength is the distance that the wave travels so that the phase changes by 2p radians.

b o l = 2p , or l =

2p 2p 2p c c = = = f bo w /c w

· Picture:

· Phase velocity, v p To understand the phase velocity, we must return to the real representation of the field with both space and time dependence; that is,

E = E + xo cos w t - b o z

b

g

Consider an observer moving along in z at the same point, say the peak, of the oscillating field. This means mathematically that:

w t - b o z = 0 or any constant. Thus, w dt = b o dz ,

or

dz w = vp , the phase velocity. dt b o

Polarization of Place Waves Consider the propagation characteristics of a plane wave in which the electric field has components in both the x and y directions:

r r r $ $ E = E x e x + E y e y e - jb o z ,

d

i

where the field components may be complex. That is,

r r E x = E x e ja and

r r E y = E y e jb

In phase a = b ,

r E =

eE

r

x

r - j b z-a $ $ ex + E y ey e b o g

j

or in real form:

r E =

eE

r

x

r e x + E y e y cos w t - b o z - a

j b

g

y

r Ey r Ex

r E

x

z

This is linear polarization. Now we consider a more general case. Elliptical Polarization In this case we allow arbitrary phase relationships a and b:

r r r E = e x E x e ja + e y E y e jb e - jb o z

e

j

It is easier to see what this means if we write each component out in real form: r r E x = E x cos w t + a - b z r r E y = E y cos w t + b - b z

a a

f f

Suppose we let a = 0 and b = p /2 . Then: r r E x = E x cos w t - b z r r E y = - E y sin w t - b z

a

f

a

f

r r r What can we say about E z ,t = E x e x + E y e y ?

a f

r r 1 Make a plot in the x-y plane with t as a parameter and E x = 1 , Ey = . 2

r Ex

5 4

6 3

7 2

1

wt 0 .707 p /4 p /2 p 5p / 4 3p / 2 7p / 4 .707

r Ey

1 2 3 4 5 6 7

0 -.707/ 2 0 - 1/ 2 -1 0 -.707 .707/ 2 0 1/ 2 .707/ 2

1

This clockwise rotation describes an ellipse, with major axes parallel to the x axis. As the wave propagates along z, the ellipse spreads out to an elliptical r r helix. This is called elliptical polarization. For the special case of E x = E y , we have circular polarization. If we had chosen b = -p /2 , we would have found that an identical ellipse would be formed except that the rotation would be counter-clockwise. Poynting Theorem We know that energy is propagated by waves, in general, and electromagnetic waves, in particular. We need to quantify the associated power flow. We easily obtain a hint of how to calculate power flow by recalling our circuit theory, where P = VI * , or by a dimensional analysis of the fields. Remember that the units of H are Amp/meter and the units of E are V/m. Their product Amp Volts/m2 has units of Watts/area, which is a power density, just what we want. The product must clearly have a direction associated with it, and it ought to somehow point in a reasonable direction. A reasonable direction for power flow in lossless, homogeneous, linear media, such as free space, would be in the direction of propagation. We might r r therefore guess that E ¥ H would be a reasonable definition of power density. We will consider the complex Poynting vector for time harmonic plane electromagnetic waves in phasor notation. This requires some thought because

r r of the nonlinear nature of E ¥ H .

$ Let E z , t = Re E z e jw t e x and

jw t

r a f af r r H a z , t f = Re H a z f e r

$ ey

We will write out the real and imaginary parts of E and H.

a f af r r H a z , t f = Re H a z f e

r

r r r $ E z , t = Re E r z e jw t + j E i z e jw t e x

jw t

af r + j H azf e

i

and

jw t

$ ey

Now expand the complex exponential and take the real part: r r r $ E z , t = Er z cos w t - Ei z sin w t ex r r r $ H z , t = H r z cos w t - Hi z sin w t e y

a f a f

af af

af af

r r r The Poynting vector is: P z, t = E ¥ H = r r r r r r r r $ Er H r cos 2 w t + Ei H i sin 2 w t - Er H i cos w t sin w t - Ei H r sin w t cos w t ez

a f

The time average Poynting vector is:

r 1 Pav z = T

af

z

T

0

r 1 r r 1 r r P z , t d t = E r H r + Ei H i 2 2

a f

The very same expression can be obtained directly from the phasors by the following rule:

r r r r r r r 1 1 $ $ P z , t = Re E ¥ H * = Re E r + j E i ex ¥ H r - j H i e y 2 2 r r 1 r r $ = Er H r + Ei H i e z 2

a f

d

i

d

i d

i

d

i

Example What is the time average Poynting vector for a plane wave propagating in free space with the following (phasor) fields:

r r E - jb o z $ H z = + 0 e - jb o z e y ? $ E z = E +0 e e x and ho

af

af

Then

r r r 1 Pav z , t = Re E ¥ H * = 2

a f

d

i

1 E $ $ Re E + 0 e - jb o z e x ¥ + 0 e - jb o z e y ho 2

FG H

IJ = 1 Re E K 2 h

2 +0 o

$ ez ,

or:

2 r E+0 $ Pav z, t = ez 2 ho

a f

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