Classical Electrodynamics by Konstantin K. Likharev - HTML preview
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EM: Classical Electrodynamics
2
2
E
L
E
0
0
.
(7.193)
z
2 fd 2
0
0
2
2 f
d
0
0
Now comparing this expression with Eq, (184), we see that remarkably, the ratio L z/ U does not depend on the shape and the width of the beam (and of course on the wave’s amplitude E 0), so these parameters are very simply and universally related:
Angular
U
momentum
L
.
(7.194)
z
at circular
polarization
Since this relation is valid in the plane-wave limit a , it may be attributed to plane waves as well, with the understanding that in reality, they always have some width (“aperture”) restriction.
As the reader certainly knows, in quantum mechanics the energy excitations of any harmonic oscillator of frequency are quantized in the units of , while the internal angular momentum of a particle is quantized in the units of s, where s is its spin. In this context, the classical relation (194) is used in quantum electrodynamics as the basis for treating the electromagnetic field excitation quanta ( photons) as some sort of quantum particles with spin s = 1. (Such integer spin also fits the Bose-Einstein statistics of the electromagnetic radiation.)
Unfortunately, I do not have time/space for a further discussion of the (very interesting) physics of paraxial beams but cannot help noticing, at least in passing, the very curious effect of helical waves –
the beams carrying not only the “spin” momentum (194), but also an additional “orbital” angular momentum. The distribution of their energy in space is not monotonic, as it is in the Gaussian beam (181), but reminds several threads twisted around the propagation axis – hence the term “helical”.77
Mathematically, their field structure is described by the associate Laguerre polynomials – the same special functions that are used for the quantum-mechanical description of hydrogen-like atoms.78
Presently, there are efforts to use such beams for the so-called orbital angular momentum (OAM) multiplexing for high-rate information transmission.79
7.8. Resonant cavities
Generally, resonators are structures that may sustain oscillations (in electrodynamics, of the electromagnetic field) even without an external source, until the oscillation amplitude slowly decreases in time due to unavoidable energy losses. If the resonator quality (described by the so-called Q-factor, which will be defined and discussed in the next section) is high, Q >> 1, this decay takes many oscillation periods. Alternatively, high- Q resonators may sustain high oscillating fields permanently, if driven by relatively weak incident waves. In contrast to lumped-element resonators, say, the well-known LC tank circuit, the subject of this section is resonant cavities (or “distributed resonators”) limited by either conducting or dielectric walls that contain distributed standing waves inside them.
77 Noticing such solutions of the Maxwell equations may be traced back to at least a 1943 theoretical work by J.
Humblet; however, this issue had not been discussed in literature too much until experiments carried out in 1992 –
see, e.g. L. Allen et al., Optical Angular Momentum; IOP, 2003.
78 See, e.g., QM Sec. 3.7.
79 See, e.g., J. Wang et al., Nature Photonics 6, 488 (2012).
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Conceptually the simplest resonant cavity is the Fabry-Pérot interferometer 80 that may be obtained by placing two well-conducting planes parallel to each other.81 Indeed, in Sec. 3 we have seen that if a plane wave is normally incident on such a “perfect mirror”, located at z = 0, its reflection, at negligible skin depth, results in a standing wave described by Eq. (61b):
E( z, t) Re2 E e i t
i / 2
sin kz.
(7.195)
This wave would not change if we place the second mirror (isolating the segment of length l from the external wave source) at any position z = l with sin kl = 0, i.e. with kl p ,
where p ,
1 ,....
2
(7.196)
This condition, which determines the spectrum of own (or resonance, or eigen-) frequencies of the resonator of fixed length l,
v
1
vk
p,
with v
,
(7.197)
p
p
a
1/2
has a simple physical sense: the resonator’s length l equals exactly p half-waves of the frequency p.
Though this is all very simple, please note a considerable change of philosophy from what we have been doing in the previous sections: the main task of the resonator’s analysis is finding its own frequencies
p, which are now determined by the system’s geometry rather than by an external wave source.
Before we move to cavities of more complex shapes, let us use Eq. (62) to represent the magnetic field in the Fabry-Pérot interferometer:
E
H ( z, t) Re2
e i t cos kz .
(7.198)
Z
Expressions (195) and (198) show that in contrast to traveling waves, each field of the standing wave changes simultaneously (proportionately) at all points of the Fabry-Pérot resonator, turning to zero everywhere twice a period. At these instants, the energy of the corresponding field vanishes, but the total energy of the two fields stays constant because the counterpart field oscillates with the phase shift
/2. Such behavior is typical for all electromagnetic resonators.
A more technical remark is that we can readily get the same results (195)-(198) by solving the Maxwell equations from scratch. For example, we already know that in the absence of dispersion, losses, and sources, they are reduced to wave equations (3) for any field components. For the Fabry-Pérot resonator’s analysis, we can use the 1D form of these equations, say, for the transverse component of the electric field:
2
1
2
E ,
0
(7.199)
2
2
2
z
v t
and solve it as a part of an eigenvalue problem with the corresponding boundary conditions. Indeed, by separating time and space variables as E( z, t) = Z( z) T( t), we obtain 80 This device, named after its inventors, Charles Fabry and Alfred Pérot; is also called the Fabry-P é rot etalon (meaning “gauge”), because of its initial usage for light wavelength measurements.
81 The resonators formed by well-conducting (usually, metallic) walls are frequently called resonant cavities.
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1 2
d Z
1 1
2
d
T 0.
(7.200)
2
2
2
Z dz
v T dt
Calling the separation constant k 2, we get two similar ordinary differential equations, 2
2
d Z
d
2
k Z ,
0
T
2 2
k v T ,
0
(7.201)
2
2
dz
dt
both with sinusoidal solutions, so the product Z( z) T( t) is a standing wave with the wave vector k and frequency = kv. (In this form, the equations are valid even in the presence of dispersion, but with a frequency-dependent wave speed: v 2 = 1/()().) Now using the boundary conditions E(0, t) = E( l, t)
= 0,82 we get the eigenvalue spectrum for kp and hence for p = vkp, given by Eqs. (196) and (197).
Lessons from this simple case study may be readily generalized to any cavity formed as a transmission line’s section:83 there are two approaches to finding the resonant frequency spectrum: (i) We may look at a traveling wave solution and find where reflecting mirrors may be inserted without affecting the wave’s structure.
(ii) We may solve the general 3D wave equations,
2
2
1
f (r, t) 0
,
(7.202)
2
2
v t
for field components, as an eigenvalue problem with appropriate boundary conditions. If the system’s parameters (and hence the coefficient v) do not change in time, the spatial and temporal variables of Eq.
(202) may be always separated by taking
f (r, t) f (r) T t() ,
(7.203)
k
k
k
where each function Tk( t) always obeys the same equation as in Eq. (201), having the sinusoidal solution of frequency k = vk. Plugging this solution back into Eqs. (202), for the spatial distribution of the field, we get the 3D Helmholtz equation,
3D
2
2
k f (r) 0 ,
(7.204) Helmholtz
k
equation
whose eigenfunctions fk(r) may be much more involved, especially for non-symmetric geometries.
Let us use these approaches to find the resonant frequency spectrum of a few simple, but practically important cavities. First of all, the first method is completely sufficient for the analysis of any resonator formed as a fragment of a uniform TEM transmission line (e.g., a coaxial cable), confined with two conducting lids normal to the line’s direction. Indeed, since in such lines kz = k = / v, and the electric field is perpendicular to the propagation axis, e.g., parallel to the lid surface, the boundary conditions are exactly the same as in the Fabry-Pérot resonator, and we again arrive at the eigenfrequency spectrum (197).
82 This is of course the expression of the first of the general boundary conditions (104). The second of these conditions (for the magnetic field) is satisfied automatically for the transverse waves we are considering.
83 The resonators may have different geometries as well, and in many cases, only the second approach may be used.
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Now let us analyze a slightly more complex system: a rectangular metallic-wall cavity of volume a b l – see Fig. 29. To use the first approach outlined above, let us consider the resonator as a finite-length (z = l) section of the rectangular waveguide extended along the z-axis, which was analyzed in detail in Sec. 6. As a reminder, at a < b, in the fundamental H 10 traveling wave mode, both vectors E and H do not depend on y, with E having only a y-component. In contrast, H has two components, Hx and Hz, with the phase shift /2 between them, and with Hx having the same phase as Ey – see Eqs. (131), (137), and (138). Hence, if a plane perpendicular to the z-axis, is placed so that the electric field vanishes on it, Hx also vanishes, so both boundary conditions (104), pertinent to a perfect metallic wall, are fulfilled simultaneously.
y
b
0
a
Fig. 7.29. Rectangular metallic-wall resonator
x
as a finite section of a waveguide with the
l
cross-section shown in Fig. 22.
z
As a result, the H 10 wave would not be perturbed by two metallic walls separated by an integer number of half-wavelengths z/2 corresponding to the wave number given by the combination of Eqs.
(102) and (133):
1/ 2
2
2
k
.
(7.205)
z
2 k 2 kt
2
2
v
a
Using this expression, we see that the smallest of these distances, l = z/2 = / k z, gives the resonance frequency84
1/ 2
2
2
v ,
(7.206)
101
a
l
where the indices of show the numbers of half-waves along each dimension of the system, in the order
[ a, b, l]. This is the lowest (“fundamental”) frequency of the resonator (if b < a, l).
The field distribution in this mode is close to that in the corresponding waveguide mode H 10
(Fig. 22), with the important difference that the magnetic and electric fields are now shifted by phase /2
both in space and time, just as in the Fabry-Pérot resonator – see Eqs. (195) and (198). Such a time shift allows for a very simple interpretation of the H 101 mode, which is especially adequate for very flat resonators, with b << a, l. At the instant when the electric field reaches its maximum (Fig. 30a), i.e.
when the magnetic field vanishes in the whole volume, the surface electric charge of the broadest (in Fig. 30, horizontal) walls of the resonator is largest, being localized mostly near the centers of the walls.
At the immediate later times, the walls start to recharge via surface currents, whose density J is largest in the side walls, and reaches its maximal value in a quarter of the oscillation period T = 2/101 – see Fig. 30b. The currents generate the vortex magnetic field, with looped field lines in the plane of the 84 In most electrical engineering handbooks, the index corresponding to the shortest side of the resonator is listed last, so the fundamental mode is nominated as H 110 and its eigenfrequency as 110.
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broadest face of the resonator. The surface currents continue to flow in this direction until (in one more quarter period) the broader walls of the resonator are fully recharged in the polarity opposite to that shown in Fig. 30a. After that, the surface currents start to flow in the direction opposite to that shown in Fig. 30b. This process, which repeats again and again, is conceptually similar to the well-known oscillations in a lumped LC circuit, with the role of (now, distributed) capacitance played mostly by the broadest walls of the resonator, and that of (now, distributed) inductance, by its narrower walls.
(a)
(b)
b
-
-
-
J
J
Fig. 7.30. Fields, charges, and
E
currents in the fundamental ( H 101)
H
mode of a rectangular metallic
resonator, at two instants separated
0
+
+ +
a
by t = /2
101 – schematically.
In order to generalize Eq. (206) to higher oscillation modes, the second of the approaches discussed above is more prudent. Separating the variables in the Helmholtz equation (204) as R(r) =
X( x) Y( y) Z( z), we see that X, Y, and Z have to be either sinusoidal or cosinusoidal functions of their arguments, with the wave vector components satisfying the characteristic equation
2
2
2
2
2
k k k k
x
y
z
.
(7.207)
2
v
In contrast to the wave propagation problem, now we are dealing with standing waves along all three dimensions, and have to satisfy the macroscopic boundary conditions (104) on all sets of parallel walls.
It is straightforward to check that these conditions ( E = 0, Hn = 0) are fulfilled at the following field component distribution:
E E cos k x sin k y sin k z,
H H sin k x cos k y cos k z,
x
1
x
y
z
x
1
x
y
z
E E sin k x cos k y sin k z, H H cos k x sin k y cos k z, (7.208)
y
2
x
y
z
y
2
x
y
z
E E sin k x sin k y cos k z,
H H cos k x cos k y sin k z,
z
3
x
y
z
z
3
x
y
z
with each of the wave vector components having an equidistant spectrum similar to Eq. (196):
n
m
p
k
, k
, k
,
(7.209)
x
a
y
b
z
l
so the full spectrum of resonance frequencies is given by the following formula:
1/ 2
2
2
2
n
m
p
vk v
,
(7.210)
nmp
a
b
l
which is a natural generalization of Eq. (206). Note, however, that of the three integers m, n, and p, at least two have to be different from zero to keep the fields (208) from vanishing at all points.
We may use Eq. (210), in particular, to evaluate the number of different modes in a relatively small range d 3 k << k 3 of the wave vector space volume that is, on the other hand, much larger than the reciprocal volume, 1/ V = 1/ abl, of the cavity. Taking into account that each eigenfrequency (210), with Chapter 7
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nml 0, corresponds to two field modes with different polarizations,85 the argumentation absolutely similar to the one used for the 2D case at the end of Sec. 7, yields
Oscillation
3
d k
mode
dN 2 V
.
(7.211)
3
density
(2 )
This property, valid for resonators of arbitrary shape, is broadly used in classical and quantum statistical physics,86 in the following form. If some electromagnetic mode functional f(k) is a smooth function of the wave vector k, and the volume V is large enough, then Eq. (211) may be used to approximate the sum of the functional’s values over the modes by an integral:
dN
3
V
f k() f k() dN f k()
d k 2
f k
( ) d 3 k .
(7.212)
3
3
k
d k
2
N
k
k
Leaving similar analyses of resonant cavities of some other simple shapes for the reader’s exercises, let me finish this section by noting that low-loss resonators may be also formed by finite-length sections of not only metallic-wall waveguides of various cross-sections but also of dielectric waveguides. Moreover, even a simple slab of a dielectric material with a / ratio substantially different from that of its environment (say, of the free space) may be used as a high- Q Fabry-Pérot interferometer (Fig. 31), due to an effective wave reflection from its surfaces at the normal and especially an inclined incidence – see, respectively, Eqs. (68), and Eqs. (91) and (95).
0
Fig. 7.31. A dielectric Fabry-Pérot interferometer.
d ~
Actually, such dielectric Fabry-Pérot interferometers are frequently more convenient for practical purposes than metallic-wall resonators, not only due to possibly lower losses (especially in the optical range) but also due to a natural coupling to the environment, which offers a ready way of wave insertion and extraction – see Fig. 31 again. The backside of the same medal is that this coupling to the environment provides an additional mechanism of power losses, limiting the resonance’s quality factor –
see the next section.
7.9. Energy loss effects
The inevitable energy losses (“dissipation”) in passive media lead, in two basic situations, to two different effects. In a long transmission line fed by a constant wave source, the losses lead to a gradual 85 This fact becomes evident from plugging Eqs. (208) into the Maxwell equation E = 0. The resulting equation, kxE 1 + kyE 2 + kzE 3 =0, with the discrete, equidistant spectrum (209) for each wave vector component, may be always satisfied by two linearly independent sets of the constants E 1,2,3.
86 See, e.g., QM Sec. 1.1 and SM Sec. 2.6.
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attenuation of the wave, i.e. to a decrease of its amplitude, and hence its power P, with the growing distance z from the source. In linear materials, the power losses are proportional to the power P carried by the wave, so the energy balance on a small segment dz takes the form
d P
d P
.
(7.213)
loss d P
dz dz
P
dz
The coefficient participating in the last form of Eq. (213), and hence defined as d /
P dz
,
(7.214)
P
is called the attenuation constant.87 Comparing the solution of Eq. (213),
z
P ( z) P ( )
0 e
,
(7.215) Wave’s
attenuation
with Eq. (29), where k is replaced with kz, we see that may be expressed as
2 Im k ,
(7.216)
z
where kz is the component of the wave vector along the transmission line. In the most important limit when the losses are low in the sense << kz Re kz, its effects on the field distribution along the line’s cross-section are negligible, making the calculation of rather straightforward. In particular, in this limit, the contributions to attenuation from two major sources, the energy losses in the filling dielectric and the skin-effect losses in conducting walls, are independent and additive.
The dielectric losses are especially simple to describe. Indeed, a review of our calculations in Secs. 5-7 shows that all of them remain valid if either (), or (), or both, and hence k(), have small imaginary parts:
k"
Im 1/ 2 1/ 2 k' .
(7.217)
In TEM transmission lines, kz = k, and hence Eq. (216) yields
2 k"
2 Im
.
(7.218) Attenuation
filling
1/21/2
due to filling
For dielectric waveguides, in particular optical fibers, these losses are the main attenuation mechanism.
As was discussed in Sec. 7, in practical optical fibers tR >> 1, i.e. most of the field propagates (as an evanescent wave) in the cladding, with a field distribution very close to the TEM wave. This is why Eq.
(218) is approximately valid if it is applied to the cladding material alone. In waveguides with non-TEM waves, we can use the relations between kz and k, derived in the previous sections, to re-calculate k” into Im kz. (Note that at this recalculation, the values of kt have to be kept real, because they are just the eigenvalues of the Helmholtz equation (101), which does not include the filling media parameters.).
87 In engineering, wave attenuation is most frequently measured in decibels per meter, abbreviated as db/m (the term not to be confused with dBm standing for decibel-milliwatt):
P ( z)
1/ m
10
10 log
10 log e
m .
4 343 m .
db/m
10
1
10
1
P ( z m
1 )
ln10
Alternatively, it is sometimes measured in nepers per meter (Np/m) defined as np/m /2, so db/m 8.686 Np/m.
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In transmission lines and waveguides and with metallic walls, higher energy losses may come from the skin effect. If the wavelength is much larger than s, as it usually is,88 the losses may be readily evaluated using Eq. (6.36):
d P
d P
loss
2
s
H
,
(7.219)
wall
dA
dA
4
where H wall is the real amplitude of the tangential component of the magnetic field at the wall’s surface.
The total power loss Ploss/ dz per unit length of a waveguide, i.e. the right-hand side of Eq. (213), now may be calculated by the integration of this expression along the contour(s) limiting the cross-section of all conducting walls. Since our calculation is only valid for low losses, we may ignore their effect on the field distribution, so the unperturbed distributions may be used both in Eq. (219), i.e. in the numerator of Eq. (214), and also for the calculation of the average propagating power, i.e. the denominator of Eq.
(214) – as the integral of the Poynting vector over the cross-section of the waveguide.
Let us see how this approach works for the TEM mode in one of the simplest transmission lines, the coaxial cable (Fig. 20). As we already know from Sec. 5, in the coarse-grain approximation, implying negligible power loss, the TEM mode field distributions between the two conductors are the same as in statics, namely:
a
H ,
0
H ,
0
H () H
,
(7.220)
z
0
where H 0 is the field’s amplitude on the surface of the inner conductor, and 1/ 2
a
E ,
0
E () ZH () ZH
,
E ,
0
where Z
.
(7.221)
z
0
Neglecting the power losses for a minute, we may plug these expressions into Eq. (42) to calculate the time-averaged Poynting vector:
2
2
2
Z H ()
Z H
a
0
S
,
(7.222)
2
2
and from it, the total wave power flow through the cross-section:
2
2
Z H
a
b d
b
2
0
P Sd r
2
2
2
Z
H a ln .
(7.223)
2
2
0
a
A
a
Next, for this particular system (Fig. 20), the contours limiting the wall cross-section are circles of radii = a (where the surface field amplitude H walls equals, in our notation, H 0), and = b (where, according to Eq. (214), the field is a factor of b/ a lower). As a result, for the power loss per unit length, Eq. (219) yields
2
d P
d
P
a
a
2
2
loss
loss
0
s
dl 2 a
H
2 b
H
a 1 H
.
(7.224)
0
0
s
0
dz
dA
b
4
2
b
C C
a
b
88 As follows from Eq. (78), which may be used for crude estimates even in cases of arbitrary wave incidence, this condition is necessary for low attenuation: << k only if f << 1.
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Note that at a << b, the losses in the inner conductor dominate, despite its smaller surface, because of the higher surface field.
Now we may plug Eqs. (223) and (224) into the definition (214) of , to calculate the skin-effect contribution to the attenuation constant:
d P / dz
1
1 1
k
1 1
loss
s
s
.
(7.225)
skin
P
2ln( b / a) a b Z
2ln( b / a) a b
This result shows that the relative (dimensionless) attenuation, / k, scales approximately as the ratio
s/min[ a, b], in a semi-quantitative agreement with the plane-wave result (78).
Let us use this result to evaluate for the standard TV cable RG-6/U, with copper conductors of diameters 2 a = 1 mm, 2 b = 4.7 mm, and 2.20 and 0. According to Eq. (6.33), for f = 100 MHz (i.e. 6.3108 s-1) the skin depth of pure copper at room temperature (with 6.0107 S/m) is close to 6.510-6 m, while k = ()1/2 = (/0)1/2(/ c) 3.1 m–1. As a result, the attenuation is rather low: skin
0.016 m-1, so the attenuation length scale l d 1/ is about 60 m. Hence the attenuation in a cable connecting a roof TV antenna or a cable distribution box to a TV set is not a big problem, though using a worse conductor, e.g., steel, would make the losses rather noticeable. (Hence the current worldwide shortage of copper.) However, the use of such cable in the X-band ( f ~ 10 GHz) is more problematic.
Indeed, though the skin depth s –1/2 decreases with frequency, the wavelength drops, i.e. k increases, even faster ( k ), so the attenuation skin 1/2 becomes close to 0.16 m–1, i.e. l d to ~6 m.
This is why at such frequencies, it may be necessary to use rectangular waveguides, with their larger internal dimensions a, b ~ 1/ k, and hence lower attenuation. Let me leave the calculation of this attenuation, using Eq. (219) and the results derived in Sec. 7, for the reader’s exercise.
The main effect of dissipation on free oscillations in resonators is different: here it leads to a gradual decay of the oscillating fields’ energy U in time. A useful dimensionless measure of this decay, called the Q factor, is commonly defined by writing the following temporal analog of Eq. (213):89
dU
dt
Udt
loss
P
,
(7.226)
Q
where in the resonance frequency in the loss-free limit, and
loss
P
.
(7.227) Q-factor
Q
U
The solution of Eq. (226),
t
/
Q
Q /
2
U ( t) U ( )
0 e
,
with
T
Q
,
(7.228)
/
2
2
which is the temporal analog of Eq. (215), shows the physical meaning of the Q-factor: the characteristic time of the oscillation energy’s decay is ( Q/2) times longer than the oscillation period T = 2/.
(Another useful interpretation of Q comes from the universal relation90
89 As losses grow, the oscillation waveform deviates from the sinusoidal one, and the very notion of “oscillation frequency” becomes vague. As a result, the parameter Q is well-defined only if it is much higher than 1.
90 See, e.g., CM Sec. 5.1.
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Q
,
(7.229)
where is the so-called FWHM91 bandwidth of the resonance, namely the difference between the two values of the external signal frequency, one above and one below , at which the energy of the oscillations induced in the resonator by an input signal is twice lower than its resonance value.) In the important particular case of a resonant cavity formed by the insertion of metallic walls into a TEM transmission line of a small cross-section (with the linear size scale a much less than the wavelength ), there is no need to calculate the Q-factor directly, provided that the line attenuation coefficient is already known. In fact, as was discussed in Sec. 8 above, the standing waves in such a resonator, of the length given by Eq. (196): l = p(/2) with p = 1, 2,…, may be understood as an overlap of two TEM waves propagating in opposite directions, or in other words, a traveling wave plus its reflection from one of the ends, the whole roundtrip taking time t = 2 l/ v = p/ v = 2 p/ = pT.
According to Eq. (215), at this distance, the wave’s power drops by a factor of exp{ - 2 l} = exp{- p}.
On the other hand, the same decay may be viewed as taking place in time, and according to Eq. (228), results in the drop by a factor of exp{- t/} = exp{-( pT )/( Q/)} = exp{-2 p/ Q}. Comparing these two exponents, we get
k
Q 2
.
(7.230)
Q vs.
This simple relation neglects the losses at the wave reflection from the walls limiting the resonator length. This approximation is indeed legitimate at l >> ; if this relation is violated, or if we are dealing with more complex resonator modes (such as those based on the reflection of E or H waves), the Q-factor may be different from that given by Eq. (230), and needs to be calculated directly from Eq.
(227). A substantial relief for such a direct calculation is that, just at the calculation of small attenuation in waveguides, in the low-loss limit ( Q >> 1), both the numerator and denominator of the right-hand side of that formula may be calculated neglecting the effects of the energy loss on the field distribution in the resonator. I am leaving such a calculation, for a few simple resonant cavities, including the rectangular and the circular ones, for the reader’s exercise.
To conclude this chapter, the last remark: in some distributed resonators (including certain dielectric resonators92 and metallic cavities with holes in their walls), additional energy losses due to the wave radiation into the environment are also possible. In some simple cases (say, the Fabry-Pérot interferometer shown in Fig. 31), the calculation of these radiative losses is straightforward, but sometimes it requires more elaborated approaches that will be discussed in the next chapter.
7.10. Exercise problems
7.1.* Find the temporal Green’s function of a medium whose complex permittivity () obeys the Lorentz oscillator model given by Eq. (32), by using:
(i) the Fourier transform of the underlying Eq. (7.30), and
91 FWHM is the acronym for Full Width at Half-Maximum.
92 Due to the absence of metallic walls and the associated skin-effect losses, some microwave dielectric resonators (e.g., those based on pure sapphire crystals cooled to helium temperatures) may feature Q-factors as high as 109 –
see, e.g., D. Creedon et al., Appl. Phys. 98, 222903 (2011).
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EM: Classical Electrodynamics
(ii) the direct solution of that equation.
Hint: For the Fourier-transform approach, you may like to use the Cauchy integral.93
7.2. The electric polarization of some material responds to an electric field step94 in the following way:
P t
,
0
for t
,
0
t /
E
e
E t
E
1
0 1
, if
0
,1
0
for t,
where > 0 and 1 are some constants. Calculate the complex permittivity () of this material, and discuss a possible simple physical model giving such dielectric response.
7.3. Calculate the complex permittivity () of a material whose dielectric-response Green’s function defined by Eq. (23), is
G G
/
1 e
,
0
with some positive constants G 0 and . What is the difference between this dielectric response and the apparently similar one considered in the previous problem?
7.4. Use the oscillator model of an atom, given by Eq. (30), to calculate its average potential energy in a uniform, sinusoidal ac electric field, and use the result to calculate the potential profile created for the atom by a standing electromagnetic wave with the electric field amplitude E(r).
7.5. The solution of the previous problem shows that a standing electromagnetic wave may exert a time-averaged force on an otherwise free non-relativistic charged particle. Reveal the physics of this force by writing and solving the equations of motion of such a particle in:
(i) a linearly-polarized monochromatic plane traveling wave, and
(ii) a similar but standing wave.
7.6. Use the first of Eqs. (54) to relate the integral
"
d to the plasma frequency for the
0
Lorentz oscillator model of a system of non-interacting particles.
7.7. Prove that Eq. (6.42) cannot be correct for all frequencies, and suggest its correction making the result compatible with both the causality principle and the physical model (6.39).
7.8. Calculate, sketch, and discuss the dispersion relation for electromagnetic waves propagating in a medium described by the Lorentz oscillator model (32), for the case of negligible damping.
7.9. As was briefly discussed in Sec. 2,95 a wave pulse of a finite but relatively large spatial extension z >> 2/ k may be formed as a wave packet – a sum of sinusoidal waves with wave 93 See, e.g., MA Eq. (15.2).
94 This function E( t) is of course proportional to the well-known Heaviside step function ( t) – see, e.g., MA Eq.
(14.3). I am not using this notion here just to avoid confusion between two different uses of the Greek letter .
95 For even more detail, see CM Sec. 5.3 and especially QM Sec. 2.2.
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vectors k within a relatively narrow interval. Consider an electromagnetic plane wave packet of this type, with the electric field distribution
i kz kt
E(r, t) Re E e
dk,
with k
,
k
k
k
k 1/ 2
propagating along the z-axis in an isotropic, linear, and dissipation-free (but not necessarily dispersion-free) medium. Express the full energy of the packet (per unit area of the wave’s front) via the complex amplitudes E k, and discuss its dependence on time.
7.10. Prove the Lorentz reciprocity relation (6.121) for a linear isotropic medium.
7.11.* A plane wave of frequency is normally incident, from free space, on a plane surface of a collision-free plasma with the electron density growing linearly and slowly with the distance from the surface: n = z for z 0, where > 0 is a small constant. Calculate the functional form of the resulting standing wave’s “tail” inside the plasma.
7.12.* Analyze the effect of a time-independent uniform magnetic field B0, parallel to the direction n of an electromagnetic wave propagation, on the wave’s dispersion in plasma, within the same simple model that was used in Sec. 2 for the derivation of Eq. (38). (Limit your analysis to relatively weak waves, whose magnetic field is much smaller than B0.)
Hint: You may like to represent the incident wave as a linear superposition of two circularly polarized waves, with opposite polarization directions.
7.13. A monochromatic plane electromagnetic wave is normally incident, from free space, on a uniform slab with electric permittivity and magnetic permeability , with the slab’s thickness d comparable with the wavelength.
(i) Calculate the power transmission coefficient T, i.e. the fraction of the incident wave’s power, that is transmitted through the slab.
(ii) Assuming that and are frequency-independent and positive, analyze in detail the frequency dependence of T. In particular, how does the function T () depend on the slab’s thickness d and the wave impedance Z (/)1/2 of its material?
7.14. A plane electromagnetic wave with a free-space wave number k 0 is normally incident on a planar conducting film of thickness d ~ s << 1/ k 0. Calculate the power transmission coefficient of the system and analyze the result in the limits of small and large values of the ratio d/s.
7.15. One of the results of the previous problem’s solution was the following expression for the coefficient of power transmission of a plane electromagnetic wave through a thin conducting film of thickness d << s, , at normal incidence:
1
T
1 Z / R ,
2
2
0
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where R 1/ d is the sheet resistance (“resistance per square”) of the film. Derive this formula in a simpler way, utilizing the smallness of d from the very beginning. Also, calculate the power reflection coefficient R, compare it with T, and comment.
,
,
'
, '
7.16. A plane wave of frequency is normally incident, from
0
0
free space, on a plane surface of a material with real electric
permittivity ’ and magnetic permeability ’. To minimize the wave’s reflection from the surface, it may be covered with a layer, of thickness
d, of another transparent material – see the figure on the right.
Calculate the optimal values of , , and d.
d
7.17. A monochromatic plane wave is incident from inside a medium with > 00 on its planar surface, at an incidence angle larger than the critical angle c = sin–1(00/)1/2. Calculate the depth
of the evanescent wave penetration into the free space, and analyze its dependence on . Does the result depend on the wave’s polarization?
7.18. Calculate the critical angle c for a wave of frequency , incident from free space upon a planar surface of a plasma with electron density n, and discuss the implications of the result for ultraviolet and X-ray optics.
7.19. Analyze the possibility of propagation of surface electromagnetic waves along a planar boundary between plasma and free space. In particular, calculate and analyze the dispersion relation of the waves.
Hint: Assume that the magnetic field of the wave is parallel to the boundary and perpendicular to the wave’s propagation direction. (After solving the problem, justify this mode choice.) z
7.20. Light from a very distant source arrives to an observer through a plane layer of nonuniform medium with a certain 1D gradient of its refraction
index n( z), at angle
i
0 – see the figure on the right. What is the genuine direction
n z
i to the source, if n( z) 1 at z ? (This problem is evidently important for
0
high-precision astronomical measurements at the Earth’s surface.)
7.21. Calculate the TEM impedance ZW of uniform TEM transmission lines with well-conducting electrodes and the cross-sections shown in the figure below:
(i)
(ii)
(iii)
w d
2 R
d
1
d R
d
d 2
2 R
w d ,12
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(i) two parallel round wires separated by distance d >> R,
(ii) a microstrip line of width w >> d,
(iii) a stripline with w >> d 1 ~ d 2,
in all cases using the coarse-grain boundary conditions on conductor surfaces. Assume that the conductors are embedded into a linear dielectric with constant and .
7.22. Modify the solution of Task (ii) of the previous problem for a superconductor microstrip line, taking into account the magnetic field’s penetration into both the strip and the ground plane.
7.23.* What lumped ac circuit would be equivalent to the TEM-line system shown in Fig. 19, with an incident wave’s power Pi? Assume that the wave reflected from the lumped load circuit does not return to it.
7.24. Find the lumped ac circuit equivalent to a loss-free
TEM transmission line of length l ~ , with a small cross-section
area A << 2, as “seen” (measured) from one end, if the line’s conductors are galvanically connected (“shortened”) at the other
l ~
end – see the figure on the right. Discuss the result’s dependence on
the signal frequency.
7.25. Represent the fundamental H 10 wave in a rectangular waveguide (Fig. 22) with a sum of two plane waves, and discuss the physics behind such a representation.
7.26.* For the coaxial cable (see, e.g., Fig. 20), find the lowest non-TEM mode and calculate its cutoff frequency.
7.27. Two coaxial cable sections are connected coaxially –
see the figure on the right, which shows the system’s cut along its
b
a' b'
symmetry axis. Relations (118) and (120) seem to imply that if the a ratios b/ a of these sections are equal, their impedance matching is perfect, i.e. a TEM wave incident from one side on the connection
would pass it without any reflection at all: R = 0. Is this statement correct?
7.28. Calculate the cutoff frequencies c of the fundamental mode
and the next lowest mode in the so-called ridge waveguide with the cross-
a
w
a
section shown in the figure on the right, in the limit t << a, b, w. Briefly b discuss possible advantages and drawbacks of such waveguides for signal
t
transfer and physical experiment.
7.29. Prove that TEM-like waves may propagate, in the radial direction, in the
free space between two coaxial, round, well-conducting cones – see the figure on the 0
right. Can this system be characterized by a certain transmission line impedance ZW, as defined by Eq. (115)?
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7.30. Use the recipe outlined in Sec. 7 to prove the characteristic equation (161) for the HE and EH waves in step-index optical fibers with a round cross-section.
7.31. Derive an approximate equation describing spatial variations of the complex amplitude of a general monochromatic paraxial beam propagating in a uniform medium, for the case when these variations are sufficiently slow. Is the Gaussian beam described by Eq. (181) one of the possible solutions of this equation? Give your interpretation of the last result.
z
7.32. Calculate the lowest resonance frequencies and the corresponding
l
field distributions of standing electromagnetic waves inside a round cylindrical
cavity with well-conducting walls (see the figure on the right), neglecting the 0
skin depth s in comparison with l and R.
R
7.33. Analyze electromagnetic waves that may propagate inside a relatively narrow gap between two well-conducting concentric spherical shells of radii R and R + d, in the limit d << R.
(i) Within the coarse-grain approximation, derive the 2D equation describing such waves with relatively large wavelengths ~ R >> d.
(ii) Calculate the lowest resonance frequencies of the system.
7.34. A molecule with an electric polarizability is placed inside an otherwise empty macroscopic cavity with well-conducting walls. Express the resulting shifts of its resonance frequencies via the unperturbed field distribution in the corresponding mode.
7.35. A plane monochromatic wave propagates through a medium with an Ohmic conductivity
and negligible electric and magnetic polarization effects. Calculate the wave’s attenuation and relate the result to a certain calculation carried out in Chapter 6.
7.36. Generalize the telegrapher’s equations (110)-(111) by accounting for small energy losses: (i) in the transmission line’s conductors, and
(ii) in the medium separating the conductors,
using their simplest (Ohmic) models. Formulate the conditions of validity of the resulting equations.
7.37. Calculate the skin-effect contribution to the attenuation constant of a TEM wave in the microstrip line discussed in Problem 21 (ii).
7.38. Calculate the skin-effect contribution to the attenuation coefficient defined by Eq. (214), for the fundamental ( H 10) mode propagating in a waveguide with well-conducting walls, of a rectangular cross-section – see Fig.22. Use the results to evaluate the wave decay length l d 1/ of a 10 GHz wave in the standard X-band waveguide WR-90 (with copper walls, a = 23 mm, b = 10 mm, and no dielectric filling), at room temperature. Compare the result with that (obtained in Sec. 9) for the standard TV
coaxial cable, at the same frequency.
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7.39. Calculate the skin-effect contribution to the attenuation coefficient of
(i) the fundamental ( H 11) wave, and
(ii) the H 01 wave,
in a conductor-wall waveguide with the circular cross-section (see Fig. 23a), and analyze the low-frequency ( c) and high-frequency ( >> c) behaviors of for each of these modes.
7.40. For a rectangular cavity of dimensions a b l, with b a, l, calculate the Q-factor of the fundamental oscillation mode, due to the skin-effect losses in its conducting walls. Evaluate the factor for a 232310 mm3 cavity with copper walls, at room temperature.
z
7.41.* Calculate the lowest eigenfrequency and the Q-factor (due to
r
the skin-effect losses) of the axially symmetric toroidal cavity with well-conducting walls and the interior’s cross-section shown in the figure on the
d
0
R
right, for the case d << r, R.96
7.42. Express the contribution to the damping coefficient (the reciprocal Q-factor) of a resonant cavity, by small energy losses in the dielectric that fills it, via the complex functions () and () of the material.
7.43. For the dielectric Fabry-Pérot resonator (Fig. 31) with the normal wave incidence, calculate the Q-factor due to radiation losses, in the limit of a strong impedance mismatch ( Z >> Z 0), by using two approaches:
(i) from the energy balance, using Eq. (227), and
(ii) from the frequency dependence of the power transmission coefficient, using Eq. (229).
Compare the results.
96 Such resonators are broadly used in particle accelerators and also in vacuum electron devices for high-power microwave amplification and generation (e.g., the so-called klystrons), where the electric field has to be concentrated in the region of charged particle passage – typically, along the symmetry axis (in the figure above, the z-axis), through a pair of small holes in the cavity’s walls, which do not affect the field distribution substantially.
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Chapter 8. Radiation, Scattering, Interference, and Diffraction
This chapter continues the discussion of electromagnetic wave propagation, now focusing on the results of wave incidence on various objects of more complex shapes. Depending on the shape, the resulting wave pattern is called either “scattering”, or “diffraction”, or “interference”. However, as the reader will see, the boundaries between these effects may be blurry, and their basic mathematical description may be conveniently based on the same key calculation – the electric-dipole radiation of a spherical wave by a localized source. Naturally, I will start the chapter from this calculation, deriving it from an even more general result – the “retarded-potential” solution of the Maxwell equations.
8.1. Retarded potentials
Let us start by finding the general solution of the macroscopic Maxwell equations (6.99) in a dispersion-free, linear, uniform, isotropic medium characterized by frequency-independent real and .1
The easiest way to perform this calculation is to use the scalar () and vector (A) potentials defined by Eqs. (6.7):
A
E
,
B A .
(8.1)
t
As was discussed in Sec. 6.8, by imposing upon the potentials the Lorenz gauge condition (6.117), 1
2
1
A
,
0
with v
,
(8.2)
2
v t
which does not affect the fields E and B, the Maxwell equations are reduced to a pair of very similar, simple equations (6.118) for the potentials:
2
1
2
,
(8.3a)
2
2
v t
2
1
2 A
A
. j
(8.3b)
2
2
v t
Let us find the general solution of these equations, for now thinking of the densities (r, t) and j(r, t) of the stand-alone charges and currents as known functions. (This will not prevent the results from being valid for the cases when (r, t) and j(r, t) should be calculated self-consistently.) The idea of such a solution may be borrowed from electro- and magnetostatics. Indeed, for the stationary case (/ t = 0), the solutions of Eqs. (8.3) are given by a ready generalization of, respectively, Eqs. (1.38) and (5.28) to a uniform, linear medium:
1
d 3 r'
r
( )
( ' r)
,
(8.4a)
4
r '
r
d 3 r'
A r
( )
j( ' r)
.
(8.4b)
4
r '
r
1 When necessary (e.g., at the discussion of the Cherenkov radiation in Sec. 10.5), it will be not too hard to generalize these results to a dispersive medium.
© K. Likharev
EM: Classical Electrodynamics
As we know, these expressions may be derived by, first, calculating the potential of a point source, and then using the linear superposition principle for a system of such sources.
Let us do the same for the time-dependent case, starting from the field induced by a time-dependent point charge at the origin:2
(r, t) q( t) (r) ,
(8.5)
In this case, Eq. (3a) is homogeneous everywhere but the origin:
2
1
2
,
0
for r 0 .
(8.6)
2
2
v t
Due to the spherical symmetry of the problem, it is natural to look for a spherically symmetric solution to this equation.3 Thus, we may simplify the Laplace operator correspondingly (as was repeatedly done earlier in this course), so Eq. (6) becomes
1
2
1
2
r
,
0
for r 0
.
(8.7)
2
2
2
r r
r v t
By introducing a new variable ( r, t) r( r, t), Eq. (7) is reduced to the 1D wave equation 2
1
2
,
0
for r 0
.
(8.8)
2
2
2
r
v t
From discussions in Chapter 7,4 we know that its general solution may be represented as
r
r
( r, t)
,
(8.9)
out t
in t
v
v
where in and out are (so far) arbitrary functions of one variable. The physical sense of out = out/ r is a spherical wave propagating from our source (located at r = 0) to outer space, i.e. exactly the solution we are looking for. On the other hand, in = in/ r describes a spherical wave that could be created by some distant spherically-symmetric source, that converged exactly on our charge located at the origin –
evidently not the effect we want to consider here. Discarding this term, and returning to = / r, we get 1
r
( r, t) t
,
for r 0 .
(8.10)
out
r
v
In order to calculate the function out, let us consider the solution (10) at distances r so small ( r
<< vt) that the time-derivative term in Eq. (3a), with the right-hand side (5),
2
1
2
q( )
t
(r) ,
(8.11)
2
2
v
t
2 Admittedly, this expression does not satisfy the continuity equation (4.5), but this deficiency will be corrected imminently, at the linear superposition stage – see Eq. (17) below.
3 Let me emphasize that this is not the general solution to Eq. (6). For example, it does not describe the possible waves created by other sources, that pass by the considered charge q( t). However, such fields are irrelevant to our current task: to calculate the field induced by the charge q( t). The solution becomes general when it is integrated (as it will be) over all relevant charges.
4 See also CM Sec. 6.3.
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is much smaller than the spatial derivative term (which diverges at r 0) . Then Eq. (11) is reduced to the Poisson equation, whose solution (4a), for the source (5), is
q t
( )
( r ,
0 t)
.
(8.12)
4
r
Now requiring the two solutions, Eqs. (10) and (12), to coincide at r << vt, we get out( t) = q( t)/4 r, so Eq. (10) becomes
1
r
( r, t)
q t .
(8.13)
4
r
v
Just as was repeatedly done in statics, this result may be readily generalized for the arbitrary position r ’ of the point charge:
(r, t) q( t) (r - r ' ) q( t) (R) , (8.14)
where R is the distance between the field observation point r and the source position point r ’, i.e. the length of the vector,
R r '
r ,
(8.15)
connecting these points – see Fig. 1.
R
'
r
r
a
0
Fig. 8.1. Calculating the retarded
n
potentials of a localized source.
Obviously, now Eq. (13) becomes
1
R
(r, t)
q t .
(8.16)
4
R
v
Finally, we may use the linear superposition principle to write, for the arbitrary charge distribution, Retarded
1
R d 3 r'
scalar
(r, t)
potential
' r, t
,
(8.17a)
4
v R
where the integration is extended over all charges of the system under analysis. Solving Eq. (4b) absolutely similarly, for the vector potential we get5
Retarded
R d 3 r'
vector
A(r, t)
potential
j ' r, t
.
(8.17b)
4
v R
5 As should be clear from the analogy of Eqs. (17) with their stationary forms (4), which were discussed, respectively, in Chapters 1 and 5, in the Gaussian units the retarded potential formulas are valid with the coefficient 1/4 dropped in Eq. (17a), and replaced with 1/ c in Eq. (17b).
Chapter 8
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(Now nothing prevents the functions (r, t) and j(r, t) from satisfying the continuity equation.) The solutions expressed by Eqs. (17) are traditionally called the retarded potentials, the name signifying the fact that the observed fields are “retarded” (in the meaning “delayed”) in time by t = R/ v relative to the source variations – physically, because of the finite speed v of the electromagnetic wave propagation. Note that, very remarkably, these simple expressions are exact solutions of the macroscopic Maxwell equations (again, in a uniform, linear, dispersion-free medium) for an arbitrary distribution of stand-alone charges and currents. They also may be considered as the general solutions of these equations, provided that the integration has been extended over all field sources in the Universe
– or at least over those ones that affect our observations.
Note also that due to the mathematical similarity of the microscopic and macroscopic Maxwell equations, Eqs. (17) are valid, with the coefficient replacement 0 and 0, for the exact, rather than the macroscopic fields, provided that the functions (r, t) and j(r, t) describe not only stand-alone but all charges and currents in the system. (Alternatively, this statement may be formulated as the validity of Eqs. (17), with the same coefficient replacement, in free space.)
Finally, note that Eqs. (17) may be plugged into Eqs. (1), giving (after an explicit differentiation) the so-called Jefimenko equations 6 for fields E and B – similar in structure to Eqs. (17), but more cumbersome. Conceptually, the existence of such equations is good news, because they are free from the gauge ambiguity pertinent to the potentials and A. However, the practical value of these explicit expressions for the fields is not overly high: for all applications I am aware of, it is easier to use Eqs.
(17) to calculate the particular expressions for the potentials first, and only then calculate the fields from Eqs. (1). Let me now present an (arguably, the most important) example of this approach.
8.2. Electric dipole radiation
Consider again the problem that was discussed in electrostatics (Sec. 3.1), namely the field of a localized source with linear dimensions a << r (see Fig. 1 again), but now with time-dependent charge and/or current distributions. Using all the arguments of that discussion, in particular the condition expressed by Eq. (3.1), r’ << r, we may apply the Taylor expansion (3.3), truncated to two leading terms,
f (R) f (r) r ' f (r) ... , (8.18)
to the scalar function f(R) R (for which f(r) = R = n, where n r/ r is the unit vector directed toward the observation point – see Fig. 1) to approximate the distance R as R r r ' n .
(8.19)
In each of the retarded potential formulas (17), R participates in two places: in the denominator and in the source’s time argument. If and j change in time on the scale ~1/, where is some characteristic frequency, then any change of the argument ( t – R/ v) on that time scale, for example due to a change of R on the spatial scale ~ v/ = 1/ k, may substantially change these functions. Thus, the expansion (19) may be applied to R in the argument ( t – R/ v) only if ka << 1, i.e. if the system’s size a 6 They were published by O. D. Jefimenko only in 1966, but the Fourier representation of the same result was obtained much earlier (in 1912) by G. A. Scott.
Chapter 8
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is much smaller than the radiation wavelength = 2/ k. On the other hand, the function 1/ R changes relatively slowly, and for it even the first term of the expansion (19) gives a good approximation as soon as a << r, R. In the latter approximation alone, Eq. (17a) yields 1
R 3
1
R
(r, t)
' r, t d r'
Q t ,
(8.20)
4
r
v
4
r
v
where Q( t) is the net electric charge of the localized system. Due to the charge conservation, this charge cannot change with time, so the approximation (20) describes just a static Coulomb field of our localized source, rather than a radiated wave.
Let us, however, apply a similar approximation to the vector potential (17b):
R
A(r, t)
j ' r, t 3
d r' .
(8.21)
4 r
v
According to Eq. (5.87), the right-hand side of this expression vanishes in statics, but in dynamics, this is no longer true. For example, if the current is due to some non-relativistic motion7 of a system of point charges qk, we can write
d
j( '
r , t d 'r
3
)
q r r p ,
(8.22)
k k t
qk k t t
k
dt k
where p( t) is the dipole moment of the localized system, defined by Eq. (3.6). Now, after the integration, we may keep only the first term of the approximation (19) in the argument ( t – R/ v) as well, getting
r
1
A(r, t)
p t
,
a
for R, .
(8.23)
4 r
v
k
Let us analyze what exactly this result describes. The second of Eqs. (1) allows us to calculate the magnetic field by the spatial differentiation of A. At large distances r >> (i.e. in the so-called far-field zone), where Eq. (23) describes a locally-plane wave, the dominating contribution to this derivative is given by the dipole moment factor:
Far-field
r
r
wave
B r
( , t)
p t
n p
t .
(8.24)
4 r
v
4 rv
v
This expression means that the magnetic field, at the observation point, is perpendicular to the vectors n and (the retarded value of) p , and its magnitude is
r
1
r
B
p
t sin ,
i.e. H
p
t sin ,
(8.25)
4 rv
v
4 rv
v
where is the angle between those two vectors – see Fig. 2.8
7 For relativistic particles, moving with velocities of the order of speed of light, one has to be more careful. As the result, I will postpone the discussion of their radiation until Chapter 10, i.e. until after the detailed discussion of special relativity in Chapter 9.
8 From the first of Eqs. (1) for the electric field, in the first approximation (23), we would get –A/ t = –(1/4 vr) p ( t – r/ v) = –( Z/ 4 r)p ( t – r/ v). The transverse component of this vector (see Fig. 2) is the proper electric field E
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n
1
r
H A
p t
c
r
E
A
Fig. 8.2. Far-fields of a localized source,
-
0
contributing to its electric dipole radiation.
t
The most important feature of this result is that the time-dependent field decreases very slowly (only as 1/ r) with the distance from the source, so the radial component of the corresponding Poynting vector (7.9b),9
Instant
2
Z
r 2
S ZH
p
,
(8.26) power
r
t
2
sin
2
(4 vr)
v
density
drops only as 1/ r 2, i.e. the full instant power P of the emitted wave,
Z
Z
2
2
3
2
Larmor
P
S r d
p
2 sin d
p .
(8.27)
r
2
2
(4 v)
6
formula
4
0
v
does not depend on the distance from the source – as it should for radiation. 10
This is the famous Larmor formula 11 for the electric dipole radiation; it is the dominating component of radiation by a localized system of charges – unless p = 0. Please notice its angular dependence: the radiation vanishes at the axis of the retarded vector p (where = 0), and reaches its maximum in the plane normal to that axis.
In order to find the average power, Eq. (27) has to be averaged over a sufficiently long time. In particular, if the source is monochromatic, p( t) = Re[p exp{– i t}], with a time-independent vector amplitude p, such averaging may be carried out just over one period, giving an extra factor 2 in the denominator:
4
Z
Average
2
P
p
.
(8.28) radiation
12
2
v
power
The easiest application of this formula is to a point charge oscillating, with frequency , along a straight line (which we may take for the z- axis), with amplitude a. In this case, p = qz( t)n z = qa Re[exp{– i t}]n z, and if the charge velocity amplitude, a, is much less than the electromagnetic wave’s speed v, we may use Eq. (28) with p = qa, giving
= ZHn of the radiated wave, while its longitudinal component is exactly compensated by (–) in the next term of the Taylor expansion of Eq. (17a) in small parameter ka ~ a/ << 1.
9 Note the “doughnut” dependence of Sr on the direction n, frequently used to visualize the dipole radiation.
10
3
2
In the Gaussian units, for free space ( v = c), Eq. (27) reads P (2 / 3 c ) p .
11 Named after Joseph Larmor, who was the first to derive this formula (in 1897) for the particular case of a single point charge q moving with acceleration r, when p r q .
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2
2
4
Zq a
P
.
(8.29)
12
2
v
Applied to a classical picture of an electron (with q = – e 1.610–19C), initially rotating about an atom’s nucleus at an atomic distance a ~ 10–10 m, Eq. (29) shows12 that the energy loss due to the dipole radiation is so large that it would cause the electron to collapse on the nucleus in just ~10-11 s. In the beginning of the 1900s, this result was one of the main arguments for the development of quantum mechanics, which prevents such a collapse of electrons for their lowest-energy (ground) quantum state.
Another useful application of Eq. (28) is the radio wave radiation by a short, straight, symmetric antenna which is fed, for example, by a TEM transmission line such as a coaxial cable – see Fig. 3.
z
l / 2
I (0)
in
P
0
P
Fig. 8.3. The dipole antenna.
l / 2
The exact solution of this problem is rather complicated because the law I( z) of the current variation along the antenna’s length should be calculated self-consistently with the distribution of the electromagnetic field induced by the current in the surrounding space. (Unfortunately, this fact is not mentioned in some textbooks.) However, the current should be largest in the feeding point (in Fig. 3, taken for z = 0) and vanish at the antenna’s ends ( z = l/2), and hence we may guess that at l << , the linear function
2
I ( z) I (
)
0 1
z ,
(8.30)
l
should be a good approximation of the actual distribution – as it indeed is. Now we can use the continuity equation Q/ t = I, i.e. – i Q = I, to calculate the complex amplitude Q( z) = iI( z)sgn( z)/
of the electric charge Q( z, t) = Re[ Qexp{– i t}] of the wire’s segment [0, z], and from it, the amplitude of the charge’s linear density:
dQ ( z)
2 I ( )
0
( z )
i
sgn z
.
(8.31)
d z
l
From here, the dipole moment’s amplitude is
l / 2
I (0)
p 2 ( z) zdz i
l
,
(8.32)
2
0
so Eq. (28) yields
12 Actually, the formula needs a numerical coefficient adjustment to account for the electron’s orbital (rather than linear) motion – the task left for the reader’s exercise. However, this adjustment does not affect the order-of-magnitude estimate given above.
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4
I ( )
0 2
2
Z kl
I
2
( )2
( )
0
P Z
l
,
(8.33)
12
2
v
4 2
24
2
where k = / v. The analogy between this result and the dissipation power P = Re Z I 2/2 in a lumped linear circuit element, enables the interpretation of the first fraction in the last form of Eq. (33) as the real part of the antenna’s impedance:
( )2
kl
Re Z Z
,
(8.34)
A
24
as felt by the transmission line.
According to Eq. (7.118), the wave traveling along the line toward the antenna is fully radiated, i.e. not reflected back, only if ZA equals to ZW of the line. As we know from Sec. 7.5 (and the solution of the related problems), for typical TEM lines, ZW ~ Z 0, while Eq. (34), which is only valid in the limit kl
<< 1, shows that for the radiation into free space ( Z = Z0), Re ZA is much less than Z0. Hence to reach the impedance matching condition ZW = ZA, the antenna’s length should be increased – as a more involved theory shows, to l /2. However, in many cases, practical considerations make short antennas necessary. The example most often met nowadays is the cell phone antennas, which use frequencies close to 1 or 2 GHz, with free-space wavelengths between 15 and 30 cm, i.e. much larger than the phone size.13 The quadratic dependence of the antenna’s efficiency on l, following from Eq. (34), explains why every millimeter counts in the design of such antennas, and why the designs are carefully optimized using software packages for the (virtually exact) numerical solution of the Maxwell equations for the specific shape of the antenna and other phone parts.14
To conclude this section, let me note that if the wave source is not monochromatic, so p( t) should be represented as a Fourier series,
p t
( ) Re
i t
p e
,
(8.35)
the terms corresponding to the interference of spectral components with different frequencies are averaged out at the time averaging of the Poynting vector, and the average radiated power is just a sum of contributions (28) from all substantial frequency components.
8.3. Wave scattering
The Larmor formula may be used as the basis of the theory of scattering – the phenomenon illustrated by Fig. 4. Generally, scattering is a complex problem. However, in many cases it allows the so-called Born approximation,15 in which the scattered wave field’s effect on the scattering object is assumed to be much weaker than that of the incident wave, and is neglected.
13 The situation will be partly remedied by the planned transfer of wireless mobile technology to the next generations, with the signal frequencies gradually moving up.
14 A partial list of popular software packages of this kind includes both publicly available codes such as Nec2
(whose various versions are available online, e.g., at http://www.qsl.net/4nec2/), and proprietary packages – such as Momentum from Agilent Technologies (now owned by Hewlett-Packard), FEKO from EM Software & Systems, and XFdtd from Remcom.
15 Named after Max Born, one of the founding fathers of quantum mechanics. However, the basic idea of this approach was developed much earlier (in 1881) by Lord Rayleigh – born John William Strutt.
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incident
scattered
wave
wave
scattering
object
Fig. 8.4. Wave scattering (schematically).
As the first example of this approach, let us consider the scattering of a plane wave, propagating in free space ( Z = Z 0, v = c), by an otherwise free16 charged particle whose motion may be described by non-relativistic classical mechanics. (This requires, in particular, the incident wave to be not too powerful, so the speed of the charge’s motion induced by the wave remains much lower than c.) As was already discussed at the derivation of Eq. (7.32), in this case, the magnetic component of the Lorentz force (5.10) is negligible in comparison with the force Fe = qE exerted by the wave’s electric field.
Thus, assuming that the incident wave is linearly polarized along some axis x, the equation of the particle’s motion in the Born approximation is just m x = qE( t), so for the x-component px = qx of its dipole moment we can write
2
q
p x
q
E( t) .
(8.36)
m
As we already know from Sec. 2, oscillations of the dipole moment lead to radiation of a wave with a wide angular distribution of intensity; in our case, this is the scattered wave – see Fig. 4. Its full power may be found by plugging Eq. (36) into Eq. (27):
4
Z
Z q
0
2
0
2
P
p
E ( t) ,
(8.37)
6
2
c
6
2
2
c m
so for the average power, we get
4
Z q
2
0
P
E
.
(8.38)
2
2
12
c m
Since this power is proportional to the incident wave’s intensity S, it is customary to characterize the scattering ability of an object by the ratio,
Full
cross-
P
P
,
(8.39)
2
section:
S incident
E
/ 2 Z
definition
0
which has the dimensionality of area, and is called the total cross-section of scattering.17 For this measure, Eq. (38) yields the famous result
16 As Eq. (7.30) shows, this calculation is also valid for an oscillator with a low own frequency, 0 << .
17 This definition parallels those accepted in the classical and quantum theories of particle scattering – see, e.g., respectively, CM Sec. 3.5 and QM Sec. 3.3.
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2
4
2
4
Z q
q
0
0
,
(8.40)
2
2
2
6 c m
6 m
which is called the Thomson scattering formula,18 especially when applied to an electron. This relation is most frequently represented in the form19
2
8
q
1
2
r ,
with r
.
(8.41) Thomson
c
c
2
3
4
mc
scattering
0
This constant r c is called the classical radius of the particle (or sometimes the “Thomson scattering length”); for the electron ( q = – e, m = m e) it is close to 2.8210-15 m. Its possible interpretation is evident from Eq. (41) for r c: at that distance between two similar particles, the potential energy q 2/40 r of their electrostatic interaction is equal to the particle’s rest-mass energy mc 2.20
Now we have to go back and establish the conditions at which the Born approximation, when the field of the scattered wave is negligible, is indeed valid for a point-object scattering. Since the scattered wave’s intensity described by Eq. (26) diverges at r 0 as 1/ r2, according to the definition (39) of the cross-section, it may become comparable to S incident at r 2 ~ . However, Eq. (38) itself is only valid if r
>> , so the Born approximation does not lead to a contradiction only if
2
.
(8.42)
For the Thompson scattering by an electron, this condition means >> r c ~ 310-15 m and is fulfilled for all frequencies up to very hard -rays with photon energies ~100 MeV.
Possibly the most notable feature of the result (40) is its independence of the wave frequency. As it follows from its derivation, particularly from Eq. (37), this independence is intimately related to the unbound character of charge motion. For bound charges, say for electrons in gas molecules, this result is only valid if the wave frequency is much higher than the frequencies j of most important quantum transitions. In the opposite limit, << j, the result is dramatically different. Indeed, in this limit we may approximate the molecule’s dipole moment by its static value (3.48):
p E .
(8.43)
In the Born approximation, and in the absence of the molecular field effects mentioned in Sec. 3.3, E in this expression is just the incident wave’s field, and we can use Eq. (28) to calculate the power of the wave scattered by a single molecule:
18 Named after Sir Joseph John (“JJ”) Thomson, the discoverer of the electron – and isotopes as well! He should not be confused with his son, G. P. Thomson, who discovered (simultaneously with C. Davisson and L. Germer) quantum-mechanical wave properties of the same electron.
19 In the Gaussian units, this formula looks like r c = q 2/ mc 2 (giving, of course, the same numerical values: for the electron, r c 2.8210-13 cm). This classical quantity should not be confused with the particle’s Compton wavelength C 2/ mc (for the electron, close to 2.2410-12m), which naturally arises in quantum electrodynamics – see a brief discussion in the next chapter, and also QM Sec. 1.1.
20 It is fascinating how smartly the relativistic expression mc 2 sneaked into the result (40)-(41), which was obtained using the non-relativistic equation (36) of the particle motion. This was possible because the calculation engaged electromagnetic waves, which propagate with the speed of light, and whose quanta ( photons), as a result, may be frequently treated as relativistic (moreover, ultra-relativistic) particles – see the next chapter.
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4
Z
2
0
2
P
E .
(8.44)
2
4
c
Now, using the last form of the definition (39) of the cross-section, we get a very simple result, 2
4
Z
0
2
,
(8.45)
2
6 c
showing that in contrast to Eq. (40), at low frequencies changes as fast as 4.
Now let us explore the effect of such Rayleigh scattering on wave propagation in a gas, with a relatively low volumic density n. We may expect (and will prove in the next section) that due to the randomness of molecule positions, the waves scattered by individual molecules may be treated as incoherent ones, so the total scattering power may be calculated just as the sum of those scattered by each molecule. We can use this fact to write the balance of the incident’s wave intensity in a small volume dV of length (along the incident wave direction) dz, and area A across it. Since such a segment includes ndV = nAdz molecules, and according to Eq. (39), each of them scatters power S = P/ A, the total scattered power is n P dz; hence the incident power’s change is d P n P
dz .
(8.46)
Comparing this equation with the definition (7.213) of the wave attenuation constant, applied to the scattering, 21
d P
.
(8.47)
scat P dz
we see that this effect gives the following contribution to attenuation: scat = n. From here, using Eq.
(3.50) to write = 0( – 1)/ n, where is the dielectric constant, and Eq. (45) for , we get Rayleigh
k 4
2
2
1 ,
where
.
(8.48)
scattering
scat
k
6 n
c
0
This is the famous Rayleigh scattering formula, which in particular explains the colors of blue sky and red sunsets. Indeed, through the visible light spectrum, changes almost two-fold; as a result, the scattering of blue components of sunlight is an order of magnitude higher than that of its red components. For the air near the Earth’s surface, – 1 610-4, and n ~ 2.51025 m-3 – see Sec. 3.3.
Plugging these numbers into Eq. (48), we see that the effective length l scat 1/scat of scattering is ~30
km for the blue light and ~200 km for the red light.22 The effective thickness h of the Earth’s atmosphere is ~10 km, so the Sun looks just a bit yellowish during most of the day. However, an elementary geometry shows that at sunset, the light has to pass the length l ~ ( R E h)1/2 300 km to reach an Earth-surface observer; as a result, the blue components of the Sun’s light spectrum are almost completely scattered out, and even the red components are weakened very substantially.
21 I am sorry for using the same letter () for both the molecular polarizability and the wave attenuation, but both notations are traditional. Hopefully, the subscript “scat” marking in the latter meaning minimizes the possibility of confusion.
22 These values are approximate because both n and ( – 1) vary through the atmosphere’s thickness.
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8.4. Interference and diffraction
Now let us discuss scattering by objects with a size of the order of, or even larger than . For such extended objects, the phase difference factors (neglected above) step in, leading in particular to the important effects of interference and diffraction. These effects show up not as much in the total power of the scattered radiation, as in its angular distribution. It is common to characterize this distribution by the differential cross-section defined as
2
d
S r r
Differential
,
(8.49) cross-
Ω
d
S
section
incident
where r is the distance from the scatterer, at which the scattered wave is observed.23 Both the definition and the notation may become clearer if we notice that according to Eq. (26), at large distances ( r >> a), the numerator of the right-hand side of Eq. (49), and hence the differential cross-section as a whole, do not depend on r, and that its integral over the total solid angle = 4 coincides with the total cross-section defined by Eq. (39):
d
1
1
Ω
d
2
r
S
P
r Ω
d
2
S d r
.
(8.50)
Ω
d
S
S
r
S
4
incident
4
incident r const
incident
For example, according to Eq. (26), the angular distribution of the radiation scattered by a single dipole is rather broad; in particular, in the quasistatic case (43), within the Born approximation,
d
2
k 2
2
sin
.
(8.51)
Ω
d
40
If the wave is scattered by a small dielectric body, with a characteristic size a << (i.e., ka << 1), then all its parts re-radiate the incident wave coherently. Hence, we can calculate it similarly, just replacing the molecular dipole moment (43) with the total dipole moment of the object – see Eq. (3.45): p P V
1 E V ,
(8.52)
0
where V ~ a 3 is the body’s volume. As a result, the differential cross-section may be obtained from Eq.
(51) with the replacement mol ( – 1)0 V:
d
k V 2
2
( 2
2
)
1 sin
,
(8.53)
d
4
i.e. follows the same sin2 law.
The situation for extended objects, with at least one dimension of the order of (or larger than) the wavelength, is different: here we have to take into account the phase shifts between the wave’s re-radiation by various parts of the body. Let us analyze this issue first for an arbitrary collection of similar point scatterers located at points r j. If the wave vector of the incident plane wave is k0, the wave’s field has the phase factor exp{ ik0r} – see Eq. (7.79). At the location r j of the j th scattering center, this factor equals exp{ ik0r j}, defining the time dependence of the dipole vector p, and hence of the scattered wave.
23 Just as in the case of the total cross-section, this definition is also similar to that accepted at the particle scattering – see, e.g., CM Sec. 3.5 and QM Sec. 3.3.
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According to Eq. (17), the scattered wave with a wave vector k (with k = k 0) acquires, on its way from the source point r j to the observation point r, an additional phase factor exp{ ik(r – r j)}, so the scattered wave field is proportional to