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 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 that 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 that 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 transfer of the wireless mobile technology to its next generations, with the frequencies moved up to the 28 GHz, 37-39 GHz, and possibly even the 64-71 GHz bands.
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 to EM waves 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 that 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 particle’s motion in the Born approximation is just m x = qE( t), so that 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 that 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 that 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 that 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 that the Sun looks just a bit yellowish during most of the day. However, an elementary geometry shows that on 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.
Chapter 8
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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 more clear 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 that the scattered wave field is proportional to
exp ik r ik r
( r ) eik r exp i k
( k ) r .
(8.54)
0
j
j
0
j
Since the first factor in the last expression does not depend on r j, to calculate the total scattering wave, it is sufficient to sum up the last phase factors, exp{– iqr j}, where the vector q k k
(8.55)
0
has the physical sense of the wave vector change at scattering.24 It may look like the phase factor depends on our choice of the reference frame. However, according to Eq. (7.42), the average intensity of the scattered wave is proportional to E
*
E , i.e. to the following real scalar function of the vector q:
*
Scattering
2
F(q)
function
exp{ iqr } exp{ iqr } exp{ iq(r r )} I(q) , (8.56)
j
j'
j
j'
j
j'
j, j'
where the complex function
Phase
I (q) exp{ i
q r }
(8.57)
j
sum
j
is called the phase sum, and may be calculated in any reference frame without affecting the final result given by Eq. (56).
So, besides the sin2 factor, the differential cross-section (49) of scattering by an extended object is also proportional to the scattering function (56). Its double-sum form is convenient to notice that for a system of many ( N >> 1) similar but randomly located scatterers, only the terms with j = j’
accumulate at summation, so that F(q), and hence d/ d, scale as N, rather than N 2 – thus justifying again our treatment of the Rayleigh scattering problem in the previous section.
Now let us apply Eq. (56) to a simple problem of just two similar small scatterers, separated by a fixed distance a:
2
q a
F(q) exp{ iq (r r )} 2 exp{ iq
}
a exp{ iq }
a
q a
a
, (8.58)
j
j'
a
a
2 1 cos a 4cos2
j, j' 1
2
where qa qa/ a is the component of the vector q along the vector a connecting the scatterers. The apparent simplicity of this result may be a bit misleading because the mutual plane of the vectors k and k0 (and hence of the vector q) does not necessarily coincide with the mutual plane of the vectors k0 and E, so that the scattering angle between k and k0 is generally different from (/2 – ) – see Fig. 5.
Moreover, the angle between the vectors q and a (within their common plane) is one more parameter independent of both and . As a result, the angular dependence of the scattered wave’s intensity (and hence d/ d), which depends on all three angles, may be rather involved, but some of its details are irrelevant for the basic physics of interference/diffraction.
24 In quantum mechanics, q has a very clear sense of the momentum transferred from the scattering object to the scattered particle (for example, a photon), and this terminology is sometimes smuggled even into classical electrodynamics texts.
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EM: Classical Electrodynamics
E
kx
n
r
k
q k k
0
Fig. 8.5. The angles important for the
0
k
k
k
z
0
general scattering problem.
y
This is why let me consider in detail only the simple cases when the vectors k, k0, and a all reside in the same plane, with k0 normal to a – see Fig. 6a.
(a)
x
x
(b)
a
k
k
q
a
2
q
q
q
a
a
Fig. 8.6. The
simplest cases of
k
z
k
z
0
a
0
(a) interference and
a sin
(b) diffraction.
a sin
2
In this case, qa = k sin, and Eq. (58) is reduced to
ka
2
sin
F(q) 4cos
.
(8.59)
2
This function always has two maxima, at = 0 and = , and, if the product ka is large enough, other maxima25 at the special angles n that satisfy the simple Bragg condition
Bragg
ka sin
2 n,
a
i.e. sin
n .
(8.60)
n
n
condition
As Fig. 6a shows, this condition may be readily understood as that of the in-phase addition (the constructive interference) of two coherent waves scattered from the two points, when the difference between their paths toward the observer, a sin, is equal to an integer number of wavelengths. At each such maximum, F = 4, due to the doubling of the wave amplitude and hence quadrupling its power.
If the distance between the point scatterers is large ( ka >> 1), the first maxima (60) correspond to small scattering angles, << 1. For this region, Eq. (59) is reduced to a simple periodic dependence of function F on the angle . Moreover, within the range of small , the wave polarization factor sin2 is virtually constant, so that the angular dependence of the scattered wave’s intensity, and hence of the differential cross-section are also very simple:
d
ka
Young’s
F(q) 4cos2
.
(8.61) interference
d
2
pattern
25 In optics especially, such intensity maxima/minima patterns are called interference fringes.
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This simple interference pattern is well known from Young’s two-slit experiment.26 (As will be discussed in the next section, the theoretical description of the two-slit experiment is more complex than that of the Born scattering, but is preferable experimentally because, at such scattering, the wave of intensity (61) has to be observed on the backdrop of a much stronger incident wave that propagates in almost the same direction, = 0.)
A very similar analysis of scattering from N > 2 similar, equidistant scatterers, located along the same straight line shows that the positions (60) of the constructive interference maxima do not change (because the derivation of this condition is still applicable to each pair of adjacent scatterers), but the increase of N makes these peaks sharper and sharper. Leaving the quantitative analysis of this system for the reader’s exercise, let me jump immediately to the limit N 0, in which we may ignore the scatterers’ discreteness. The resulting pattern is similar to that at scattering by a continuous thin rod (see Fig. 6b), so let us first spell out the Born scattering formula for an arbitrary extended, continuous, uniform dielectric body. Transferring Eq. (56) from the sum to an integral, for the differential cross-section we get
d
k 2
2
k 2
2
( 2
2
)
1 F(q)sin
(
2
2
2
)
1 I (q) sin
,
(8.62)
d
4
4
where I(q) now becomes the phase integral,27
Phase
I (q)
iq ' r
integral
d 3
exp
r' ,
(8.63)
V
with the dimensionality of volume.
Now we may return to the particular case of a thin rod (with both dimensions of the cross-section’s area A much smaller than , but an arbitrary length a), otherwise keeping the same simple geometry as for two point scatterers – see Fig. 6b. In this case, the phase integral is just
a / 2
Fraunhofer
exp{ iq a / }
2 exp{ iq a / }
2
a
a
sin
diffraction
I q A exp{ iq x' ) dx' A
V
,
integral
(8.64)
a
iq
a / 2
where V = Aa is the volume of the rod, and is the dimensionless argument defined as q a
a
sin
ka
.
(8.65)
2
2
The fraction participating in the last form of Eq. (64) is met in physics so frequently that it has deserved the special name of the sinc (not “sync”, please!) function (see Fig. 7): 26 This experiment was described in 1803 by Thomas Young – one more universal genius of science, who also introduced the Young modulus in the elasticity theory (see, e.g., CM Chapter 7), besides numerous other achievements – including deciphering Egyptian hieroglyphs! It is fascinating that the first clear observation of wave interference was made as early as 1666 by another genius, Sir Isaac Newton, in the form of so-called Newton’s rings. Unbelievably, Newton failed to give the most natural explanation of his observations – perhaps because he was vehemently opposed to the very idea of light as a wave, which was promoted in his times by others, notably by Christian Huygens. Due to Newton’s enormous authority, only Young’s two-slit experiments more than a century later have firmly established the wave picture of light – to be replaced by the dualistic wave/photon picture formalized by quantum electrodynamics (see, e.g., QM Ch. 9), in one more century.
27 Since the observation point’s position r does not participate in this formula explicitly, the prime sign in r ’ could be dropped, but I keep it as a reminder that the integral is taken over points r ’ of the scattering object.
Chapter 8
Page 15 of 38
EM: Classical Electrodynamics
sin
sinc
.
(8.66) Sinc
function
It vanishes at all points n = n with integer n, besides such point with n = 0: sinc0 sinc 0 = 1.
1.5
1.25
1
sin 0.75
0.5
0.25
0
0.25
Fig. 8.7. The sinc function.
0.5 5 4 3 2 1 0
1
2
3
4
5
/
The function F(q) = V 2sinc2, given by Eq. (64) and plotted with the red line in Fig. 8, is called the Fraunhofer diffraction pattern.28
1
0.8
F(q) 0.6
V
0.4
Fig. 8.8. The Fraunhofer diffraction
pattern (solid red line) and its envelope
0.2
1/2 (dashed red line). For comparison,
0
the blue line shows the basic
3
2
1
0
1
2
3
interference pattern cos2 – cf. Eq. (61).
/ ka sin /
2
Note that it oscillates with the same argument’s period ( ka sin) = 2/ ka as the interference pattern (61) from two-point scatterers (shown with the blue line in Fig. 8). However, at the interference, the scattered wave intensity vanishes at angles n’ that satisfy the condition
ka sin 'n
1
n ,
(8.67)
2
2
i.e. when the optical paths difference a sin is equal to a semi-integer number of wavelengths /2 = / k, and hence the two waves from the scatterers reach the observer in anti-phase – the so-called destructive interference. On the other hand, for the diffraction on a continuous rod the minima occur at a different set of scattering angles:
ka sin
n n ,
(8.68)
2
28 It is named after Joseph von Fraunhofer (1787-1826) – who has invented the spectroscope, developed the diffraction grating (see below), and also discovered the dark Fraunhofer lines in Sun’s spectrum.
Chapter 8
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i.e. exactly where the two-point interference pattern has its maxima – please have a look at Fig. 8 again.
The reason for this relation is that the wave diffraction on the rod may be considered as a simultaneous interference of waves from all its elementary fragments, and exactly at the observation angles when the rod edges give waves with phases shifted by 2 n, the interior points of the rod give waves with all phases within this difference, with their algebraic sum equal to zero. As even more visible in Fig. 8, at the diffraction, the intensity oscillations are limited by a rapidly decreasing envelope function 1/2 –
while at the two-point interference, the oscillations retain the same amplitude. The reason for this fast decrease is that with each Fraunhofer diffraction period, a smaller and smaller fraction of the road gives an unbalanced contribution to the scattered wave.
If the rod’s length is small ( ka << 1, i.e. a << ), then the sinc function’s argument is small at all scattering angles , so I(q) V, and Eq. (62) is reduced to Eq. (53). In the opposite limit, a >> , the first zeros of the function I(q) correspond to very small angles , for which sin 1, so that the differential cross-section is just
2
2
d
k
ka
( )
1 2 sinc2
,
(8.69)
d
4
2
i.e. Fig. 8 shows the scattering intensity as a function of the direction toward the observation point – if this point is within the plane containing the rod.
Finally, let us discuss a problem of large importance for applications: calculate the positions of the maxima of the interference pattern arising at the incidence of a plane wave on a very large 3D
periodic system of point scatterers. For that, first of all, let us quantify the notion of 3D periodicity. The periodicity in one dimension is simple: the system we are considering (say, the positions of point scatterers) should be invariant with respect to the linear translation by some period a, and hence by any multiple sa of this period, where s is any integer. Anticipating the 3D generalization, we may require any of the possible translation vectors R to that the system is invariant, to be equal sa, where the primitive vector a is directed along the (so far, the only) axis of the 1D system.
Now we are ready for the common definition of the 3D periodicity – as the invariance of the system with respect to the translation by any vector of the following set:
3
Bravais
R
lattice
s a ,
(8.70)
l
l
l 1
where sl are three independent integers, and {a l} is a set of three linearly-independent primitive vectors.
The set of geometric points described by Eq. (70) is called the Bravais lattice (first analyzed in detail, circa 1850, by Auguste Bravais). Perhaps the most nontrivial feature of this relation is that the vectors a l should not necessarily be orthogonal to each other. (That requirement would severely restrict the set of possible lattices and make it unsuitable for the description, for example, of many solid-state crystals.) For the scattering problem we are considering, we will assume that the position r j of each point scatterer coincides with one of the points R of some Bravais lattice, with a given set of primitive vectors a l, so that in the basic Eq. (57), the index j is coding the set of three integers { s 1, s 2, s 3}.
Now let us consider a similarly defined Bravais lattice, but in the reciprocal (wave-number) space, numbered by three independent integers { t1, t 2, t 3}:
3
a a
Reciprocal
m"
m'
Q t b ,
b
with
2
,
(8.71)
m
m
m
lattice
1
m
a
a
a
m
m" m'
Chapter 8
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where in the last expression, the indices m, m’, and m” are all different. This is the so-called reciprocal lattice, which plays an important role in all physics of periodic structures, in particular in the quantum energy-band theory.29 To reveal its most important property, and thus justify the above introduction of the primitive vectors b m, let us calculate the following scalar product:
3
3
a
3
a
a a
a
m"
m'
l
m"
m'
R Q s t a b 2
s t a
s t
. (8.72)
l m l
m
2
l m l
l k
l, m1
l, m1
a a
a
a
a
a
m
m"
m'
l, m1
m
m"
m'
Applying to the numerator of the last fraction the operand rotation rule of vector algebra,30 we see that it is equal to zero if l m, while for l = m the whole fraction is evidently equal to 1. Thus the double sum (72) is reduced to a single sum:
3
3
R Q 2 s t 2 n ,
(8.73)
l l
l
l 1
l 1
where each of the products nl sltl is an integer, and hence their sum, 3
n n s t s t s t ,
(8.74)
l
1 1
2 2
3 3
l 1
is an integer as well, so that the main property of the direct/reciprocal lattice couple is very simple: R Q 2 n
,
exp
hence
and
iR
Q 1 .
(8.75)
Now returning to the scattering function (56) for a Bravais lattice of point scatters, we see that if the vector q k – k0 coincides with any vector Q of the reciprocal lattice, then all terms of the phase sum (57) take their largest possible values (equal to 1), and hence the sum as the whole is largest as well, giving a constructive interference maximum. This equality, q = Q, where Q is given by Eq. (71), is called the von Laue condition (named after Max von Laue) of the constructive interference; it is, in particular, the basis of the whole field of the X-ray crystallography of solids and polymers – the main tool for revealing their atomic/molecular structure.31
In order to recast the von Laue condition is a more vivid geometric form, let us consider one of the vectors Q of the reciprocal lattice, corresponding to a certain integer n in Eq. (75), and notice that if that relation is satisfied for one point R of the direct Bravais lattice (70), i.e. for one set of the integers
{ s 1, s 2, s 3}, it is also satisfied for a 2D system of other integer sets, which may be parameterized, for example, by two integers S 1 and S 2:
'
'
'
s s S t , s s S t , s s S t S t .
(8.76)
1
1
1 3
2
2
2 3
3
3
1 1
2 2
Indeed, each of these sets has the same value of the integer n, defined by Eq. (74), as the original one: n' s' t s' t s' t
.
(8.77)
1 1
2 2
3 3
s S t
1
1 3 t
1
s S t
2
2 3 t
2
s S t S t
3
1 1
2 2 t
n
3
Since according to Eq. (75), the vector of the distance between any pair of the corresponding points of the direct Bravais lattice (70),
29 See, e.g., QM Sec. 3.4, where several particular Bravais lattices R, and their reciprocals Q, are considered.
30 See, e.g., MA Eq. (7.6).
31 For more reading on this important topic, I can recommend, for example, the classical monograph by B.
Cullity , Elements of X-Ray Diffraction, 2nd ed., Addison-Wesley, 1978. (Note that its title uses the alternative name of the field, once again illustrating how blurry the boundary between the interference and diffraction is.) Chapter 8
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R
S
t a S
t a S
t S
t a ,
(8.78)
1 3 1
2 3 2
1 1
2 2 3
satisfies the condition RQ = 2 n = 0, this vector is normal to the (fixed) vector Q. Hence, all the points corresponding to the 2D set (76) with arbitrary integers S 1 and S 2, are located on one geometric plane, called the crystal (or “lattice”) plane. In a 3D system of N >> 1 scatterers (such as N ~1020 atoms in a ~1-mm3 solid crystal), with all linear dimensions comparable, such a plane contains ~ N 2/3 >> 1
points. As a result, the constructive interference peaks are very sharp.
Now rewriting Eq. (75) as a relation for the vector R’s component along the vector Q, 2
Q
R
n,
where R R n R
,
Q
and
Q ,
(8.79)
Q
Q
Q
Q
Q
we see that the parallel crystal planes corresponding to different numbers n (but the same Q) are located in space periodically, with the smallest distance
2
d
,
(8.80)
Q
so that the von Laue condition q = Q may be rewritten as the following rule for the possible magnitudes of the scattering vector q k – k0:
n
2
q
.
(8.81)
d
Figure 9a shows the diagram of the three wave vectors k, k0, and q, taking into account the elastic scattering condition k = k0 = k 2/. From the diagram, we immediately get the famous Bragg rule 32 for the (equal) angles /2 between the crystal plane and each of the vectors k and k0: Bragg
q
n
k sin
,
2
i.e. d sin
n .
(8.82)
rule
2
d
The physical sense of this relation is very simple – see Fig. 9b drawn in the “direct” space of the radius-vectors r, rather than in the reciprocal space of the wave vectors, as Fig. 9a. It shows that if the Bragg condition (82) is satisfied, the total difference 2 d sin of the optical paths of two waves, partly reflected from the adjacent crystal planes, is equal to an integer number of wavelengths, so these waves interfere constructively.
(a)
(b)
k
q
k
0
k
crystal
crystal
plane’s
plane
direction
k
d
0
crystal
plane
d sin
d sin
Fig. 8.9. Deriving the Bragg rule: (a) from the von Laue condition (in the reciprocal space), and (b) from a direct-space diagram. Note that the scattering angle equals 2.
32 Named after Sir William Bragg and his son, Sir William Lawrence Bragg, who were the first to demonstrate (in 1912) the X-ray diffraction by atoms in crystals. The Braggs’ experiments have made the existence of atoms (before that, a somewhat hypothetical notion ignored by many physicists) indisputable.
Chapter 8
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Finally, note that the von Laue and Bragg rules, as well as the similar condition (60) for the 1D
system of scatterers, are valid not only in the Born approximation but also follow from any adequate theory of scattering, because the phase sum (57) does not depend on the magnitude of the wave propagating from each elementary scatterer, provided that they are all equal.
8.5. The Huygens principle
As the reader could see, the Born approximation is very convenient for tracing the basic features of (and the difference between) the phenomena of interference and diffraction. Unfortunately, this approximation, based on the relative weakness of the scattered wave, cannot be used to describe more typical experimental implementations of these phenomena, for example, Young’s two-slit experiment, or diffraction on a single slit or orifice – see, e.g. Fig. 10. Indeed, at such experiments, the orifice size a is typically much larger than the light’s wavelength , and as a result, no clear decomposition of the fields to the “incident” and “scattered” waves is possible inside it.33
opaque
observation
r
screen
wave
angle
source
'
r
orifice
Fig. 8.10. Deriving the
Huygens principle.
However, another approximation called the Huygens (or “Huygens-Fresnel”) principle,34 is very instrumental for the description of such situations. In this approach, the wave beyond the screen is represented as a linear superposition of spherical waves of the type (17), as if they were emitted by every point of the incident wave’s front that has arrived at the orifice. This approximation is valid if the following strong conditions are satisfied:
a r ,
(8.83)
where r is the distance of the observation point from the orifice. In addition, as we have seen in the last section, at small / a the diffraction phenomena are confined to angles ~ 1/ ka ~ / a << 1. For observation at such small angles, the mathematical expression of the Huygens principle, for the complex amplitude f(r) of a monochromatic wave f(r, t) = Re[ f e-i t], is given by the following simple formula eikR
f (r) C
f ( '
r )
d 2 r'
.
(8.84)
R
orifice
33 Another complaint against the Born approximation is that it does not satisfy the so-called optical (or “forward scattering”) theorem relating to scattering with k = k0. This relation is especially important for quantum-mechanical description of particle scattering, and in this series, will be discussed in its QM part (Sec. 3.3).
34 Named after Christian Huygens (1629-1695) who had conjectured the wave nature of light (which remained controversial for more than a century, until T. Young’s experiments), and Augustin-Jean Fresnel (1788-1827) who developed a quantitative theory of diffraction, and in particular gave a mathematical formulation of the Huygens principle. (Note that Eq. (91), sufficient for the purposes of this course, is not its most general form.) Chapter 8
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Here f is any transverse component of any of the wave’s fields (either E or H),35 R is the distance between point r ’ at the orifice and the observation point r (i.e. the magnitude of vector R r – r ’), and C is a complex constant.
Before describing the proof of Eq. (84), let me carry out its sanity check – which also will give us the constant C. Let us see what the Huygens principle gives for the case when the field under the integral is a plane wave with the complex amplitude f( z), propagating along axis z, with an unlimited x-y front, (i.e. when there is no opaque screen at all), so that in Eq. (84) we should take the whole [ x, y]
plane, say with z’ = 0, as the integration area– see Fig. 11.
x'
“source”
observation
R r - '
r
point r ’
point r = {0, 0, z}
Fig. 8.11. Applying the
0
z
Huygens principle to a
y'
plane incident wave.
Then, for the observation point with coordinates x = 0, y = 0, and z > 0, Eq. (84) yields exp ik 2
2
2
x' y' z 1/ 2
f ( z) Cf (0) dx' dy'
.
(8.85)
2
2
2
x' y' z 1/2
Before specifying the integration limits, let us consider the range x’, y’ << z. In this range, the square root participating in Eq. (85) twice, may be approximated as
1/ 2
2
2
2
2
2
2
x' 2 y' 2 z 2 1/2
x'
y'
x'
y'
x'
y'
z 1
z 1
z
.
(8.86)
z 2
2 z 2
2 z
At z >> , the denominator of Eq. (85) is a much slower function of x’ and y’ than the exponent in the nominator, and in the former case, it is sufficient (as we will check a posteriori) to keep just the main, first term of expansion (86). With that, Eq. (85) becomes
eikz
ik x' 2 y' 2
eikz
f ( z) Cf ( )
0
dx' dy' exp
Cf ( )
0
I I
,
(8.87)
x y
z
2 z
z
where Ix and Iy are two similar integrals; for example,
ikx' 2
2 z 1/2
2
2
z 1/ 2
I
2
2
exp
exp
cos
sin
, (8.88)
x
dx'
i
d
d i
d
2 z
k
k
where ( k/2 z)1/2 x’. These are the so-called Fresnel integrals. I will discuss them in more detail in the next section (see, in particular, Fig. 13), and for now, only one property36 of these integrals is important 35 The fact that the Huygens principle is valid for any field component should not too surprising. In the limit a >>
, the real boundary conditions at the orifice edges are not important; it is only important for the screen that limits the orifice, to be opaque. Because of this, the Huygens principle (84) is a part of the so-called scalar theory of diffraction. (I will not have time to discuss the vector theory of these effects, which is more accurate at smaller a –
see, e.g., Chapter 11 of the monograph by M. Born and E. Wolf, cited at the end of Sec. 7.1.) Chapter 8
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for us: if taken in symmetric limits [-0, +0], both of them rapidly converge to the same value, (/2)1/2, as soon as 0 becomes much larger than 1. This means that even if we do not impose any exact limits on the integration area in Eq. (85), this integral converges to the value
ikz
2
e
2 z 1/2
1/ 2
1/ 2
2 i
ikz
f ( z) Cf ( )
0
i C
f ( )
0 e
,
(8.89)
z k 2
2
k
due to contributions from the central area with a linear size corresponding to ~ 1, i.e. to 1/ 2
z
x
~ y
~
~ z1/2 , .
(8.90)
k
so that the net contribution from the front points r ’ well beyond the range (90) is negligible.37 (Within our assumptions (83), which in particular require to be much less than z, the diffraction angle x/ z ~
y/ z ~ (/ z)1/2, corresponding to the important area of the front, is small.) According to Eq. (89), to sustain the unperturbed plane wave propagation, f( z) = f(0) eikz, the constant C has to be taken equal to k/2 i. Thus, the Huygens principle’s prediction (84), in its final form, reads k
eikR
f (r)
f ( '
r )
d 2 r'
Huygens
,
(8.91)
2 i
R
principle
orifice
and describes, in particular, the straight propagation of the plane wave (in a uniform medium).
Let me pause to emphasize how nontrivial this result is. It would be a natural corollary of Eqs.
(25) (and the linear superposition principle) if all points of the orifice were filled with point scatterers that re-emit all the incident waves as spherical waves. However, as it follows from the above example, the Huygens principle also works if there is nothing in the orifice but the free space!
This is why let us discuss a proof of this principle,38 based on Green’s theorem (2.207). Let us apply it to the function f = f where f is the complex amplitude of a scalar component of one of the wave’s fields, which satisfies the Helmholtz equation (7.204),
2
2
k f (r) 0
,
(8.92)
and the function g = G is the temporal Fourier image of the corresponding Green’s function. The latter function may be defined, as usual, as the solution of the same equation with the added delta-functional right-hand side with an arbitrary coefficient, for example,
2
2
k G (r, ' r) 4
(r '
r )
.
(8.93)
Using Eqs. (92) and (93) to express the Laplace operators of the functions f and G, we may rewrite Eq. (2.207) as
36 See, e.g., MA Eq. (6.10).
37 This result very is natural, because the function exp{ ikR} oscillates fast with the change of r ’, so that the contributions from various front points are averaged out. Indeed, the only reason why the central part of the plane
[ x’, y’] gives a non-zero contribution (89) to f( z) is that the phase exponents stop oscillating as ( x’ 2 + y’ 2) is reduced below ~ z/ k – see Eq. (86).
38 This proof was given in 1882 by the same G. Kirchhoff whose circuit rules were discussed in Sec. 4.1 and 6.6.
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r, r
2
2
3
G
'
f
f
r r
k G , ' 4
(r '
r ) G r, '
r k f
d r
f
G
r, '
r
2
d r , (8.94)
n
n
V
S
where n is the outward normal to the surface S limiting the integration volume V. Two terms on the left-hand side of this relation cancel, so that after swapping the arguments r and r ’, we get
G
' r, r
G
( '
r )
4 f (r)
f ( '
r )
g
'
r , r
d 2 'r
.
(8.95)
n'
n'
S
This relation is only correct if the selected volume V includes the point r (otherwise we would not get its left-hand side from the integration of the delta function), but does not include the genuine source of the wave – otherwise, Eq. (92) would have a non-zero right-hand side. Now let r be the field observation point, V be all the source-free half-space (for example, the space right of the screen in Fig.
10), so that S is the surface of the screen, including the orifice. Then the right-hand side of Eq. (95) describes the field (at the observation point r) induced by the wave passing through the orifice points r ’.
Since no waves are emitted by the opaque parts of the screen, we can limit the integration by the orifice area.39 Assuming also that the opaque parts of the screen do not re-emit the waves “radiated” by the orifice, we can take the solution of Eq. (93) to be the retarded potential for the free space:40
eikR
G (r, '
r )
.
(8.96)
R
Plugging this expression into Eq. (82), we get
Kirchhoff
eikR eikR
f ( '
r )
integral
4 f (r)
f ( ' r)
2
d r' .
(8.97)
n'
R
R
n'
orifice
This is the so-called Kirchhoff (or “Fresnel-Kirchhoff”) integral. (Again, with the integration extended over all boundaries of the volume V, this would be an exact mathematical result.) Now, let us make two additional approximations. The first of them stems from Eq. (83): at ka >> 1, the wave’s spatial dependence in the orifice area may be represented as
f ( '
r ) (a
of
function
slow
'
r ) exp{ ik '
r }
,
(8.98)
0
where “slow” means a function that changes on the scale of a rather than . If, also, kR >> 1, then the differentiation in Eq. (97) may be, in both instances, limited to the rapidly changing exponents, giving eikR
4 f (r)
i
k k
(8.99)
0
2
n '
f ( '
r ) d r' .
R
orifice
Second, if all observation angles are small, we may take kn ’ k0n ’ – k. With that, Eq. (99) is reduced to the Huygens principle in its form (91).
39 Actually, this is a nontrivial point of the proof. Indeed, it may be shown that the exact solution of Eq. (94) identically is equal to zero if f(r ’) and f(r ’)/ n’ vanish together at any part of the boundary, of a non-zero area. A more careful analysis of this issue (it is the task of the formal vector theory of diffraction, which I will not have time to pursue) confirms the validity of the described intuition-based approach at a >> .
40 It follows, e.g., from Eq. (16) with a monochromatic source q( t) = qexp{- i t}, with the amplitude q = 4
that fits the right-hand side of Eq. (93).
Chapter 8
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It is clear that the principle immediately gives a very simple description of the interference of waves passing through two small holes in the screen. Indeed, if the holes’ sizes are negligible in comparison with the distance a between them (though still are much larger than the wavelength!), Eq.
(91) yield
f (r)
ikR
ikR
1
2
c e
c e
, with c
kf A
2 iR ,
(8.100)
1
2
,
1 2
,
1 2
,
1 2
,
1 2
where R 1,2 are the distances between the holes and the observation point, and A 1,2 are the hole areas. For the wave intensity, Eq. (100) gives
2
2
*
S f f c
c
2 c c cos k R R ,
where arg c arg c
. (8.101)
1
2
1
2
1 2
1
2
The first two terms in the last expression clearly represent the intensities of the partial waves passed through each hole, while the last one is the result of their interference. The interference pattern’s contrast ratio
2
S
c c
max
1
2
,
(8.102)
S
c
min
c
1
2
is the largest (infinite) when both waves have equal amplitudes.
The analysis of the interference pattern is simple if the line connecting the holes is perpendicular to wave vector k k0 – see Fig. 6a. Selecting the coordinate axes as shown in that figure, and using for the distances R 1,2 the same expansion as in Eq. (86), for the interference term in Eq. (101) we get
kxa
cos k R R
.
(8.103)
1
2
cos
z
This means that the term does not depend on y, i.e. the interference pattern in the plane of constant z is a set of straight, parallel strips, perpendicular to the vector a, with the period given by Eq. (60), i.e. by the Bragg law.41 This result is strictly valid only at y 2 << z 2; it is straightforward to use the next term in the Taylor expansion (73) to show that farther on from the interference plane y = 0, the strips start to diverge.
8.6. Fresnel and Fraunhofer diffraction patterns
Now let us use the Huygens principle to analyze a slightly more complex problem: plane wave’s diffraction on a long, straight slit of a constant width a (Fig. 12). According to Eq. (83), to use the Huygens principle for the problem’s analysis we need to have << a << z. Moreover, the simple version (91) of the principle is only valid for small observation angles, x << z. Note, however, that the relation between two dimensionless parameters of the problem, z/ a and a/, which are both much less than 1, is so far arbitrary; as we will see in a minute, this relation determines the type of the observed diffraction pattern.
41 The phase shift vanishes at the normal incidence of a plane wave on the holes. Note, however, that the spatial shift of the interference pattern following from Eq. (103), x = –( z/ ka), is extremely convenient for the experimental measurement of the phase shift between two waves, especially if it is induced by some factor (such as insertion of a transparent object into one of the interferometer’s arms) that may be turned on/off at will.
Chapter 8
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incident
x'
wave
diffracted
screen with
wave
x
observation
a slit
plane
a / 2
0
z
a / 2
Fig. 8.12. Diffraction on a slit.
Let us apply Eq. (91) to our current problem (Fig. 12), for the sake of simplicity assuming the normal wave incidence, and taking z’ = 0 at the screen plane:
a
1/ 2
k
exp ik x x' 2 y' 2 z 2
f ( x, z) f
'
,
(8.104)
0
dx dy'
2 i
1/ 2
2
2
2
a
x x' y' z
where f 0 f( x’, 0) = const is the incident wave’s amplitude. This is the same integral as in Eq. (85), except for the finite limits for the integration variable x’, and may be simplified similarly, using the small-angle condition ( x – x’)2 + y’ 2 << z 2:
ikz a / 2
k e
ik
x x'2 2 y'
k eikz
f ( x, z) f
dx' dy' exp
f
I I
.
(8.105)
0
0
2 i z
2 z
2
x y
i z
a / 2
The integral over y’ is the same as in the last section:
1/ 2
2
iky'
2 iz
I exp
dy'
,
(8.106)
y
2 z
k
but the integral over x’ is more general, because of its finite limits:
a / 2
ik x x' 2
I
dx' .
(8.107)
x
exp 2 z
a / 2
It may be simplified in the following two (opposite) limits.
(i) Fraunhofer diffraction takes place when z/ a >> a/ – the relation which may be rewritten either as a << ( z)1/2, or as ka 2 << z. In this limit, the ratio kx’ 2/ z is negligibly small for all values of x’
under the integral, and we can approximate it as
a / 2
ik 2
x 2
2
xx' x'
a / 2
ik 2
x 2 xx'
I
exp
dx'
exp
dx'
x
2 z
2 z
a / 2
a / 2
(8.108)
2 a / 2
ikx
ikxx'
2
2
z
ikx
kxa
exp
exp
dx'
exp
sin
,
2 z
z
kx
2 z
2 z
a / 2
so that Eq. (105) yields
Chapter 8
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EM: Classical Electrodynamics
k eikz 2 z 2 iz 1/ 2
ikx 2
kxa
f ( x, z) f
exp
sin
,
(8.109)
0 2 i z kx k
2 z
2 z
and hence the relative wave intensity is
2
S( x, z)
f ( x, z)
8 z
kxa
2
2
ka
ka
Fraunhofer
2
2
sin
sinc
,
(8.110) diffraction
2
S
f
kx
2 z
z
0
0
2
pattern
where S 0 is the intensity of the incident wave, and x/ z << 1 is the observation angle. Comparing this expression with Eq. (69), we see that this diffraction pattern is exactly the same as that for a similar (uniform, 1D) object in the Born approximation – see the red line in Fig. 8. Note again that the angular width of the Fraunhofer pattern is of the order of 1/ ka, so that its linear width x = z is of the order of z/ ka ~ z/ a.42 Hence the condition of the Fraunhofer approximation’s validity may be also represented as a << x.
(ii) Fresnel diffraction. In the opposite limit of a relatively wide slit, with a >> x = z ~ z/ ka ~
z/ a, i.e. ka 2 >> z, the diffraction patterns at two edges of the slit are well separated. Hence, near each edge (for example, near x’ = – a/2) we may simplify Eq. (107) as
ik x x' 2
1/ 2
2 z
I ( x)
,
(8.111)
x
exp
dx'
2
exp i
d
2 z
k
a / 2
k/2 z1/2( x a/2)
and express it via the special functions called the Fresnel integrals:43
2 1/2
2