2
MEANING OF VARIABLES IN THE FOURIER TRANSFORM
2
31
Meaning of variables in the Fourier transform
~ 1 , . . . , XN ) of the Fourier transform F (X)?
~ As the argument
What is the meaning of the variables X(X
~
of the kernel exp ik X · ~x of the Fourier transform must be dimensionless, it is evident, that the physical
~ depends on the choice of the constant k. It will be shown in the course of
or geometrical meaning of X
~ have
this chapter that if k is chosen to be dimensionless, e.g. k = 1, or better k = 2π, the variables X
the meanings of frequencies (temporal or spatial). But, if we choose k = 2π/λ, i.e. the wave number, as
~ 1 , X2 , X3 ) have the
it may be common in diffraction theory, we will see, that the Fourier variables X(X
meaning of differences of the directional cosines of diffracted and incident radiation.
2.1
The scattering of a plane wave by a three–dimensional object
The study of the structure of materials by scattering or diffraction of radiation persues usually in such
a way that a parallel beam of the radiation is incident on the specimen under study and, at a distance
R, large compared to the dimensions of the specimen, the diffraction or scattering pattern is observed.
From it, i.e. from the directional distribution of the diffracted radiation, we then try to draw conclusions
about the structure of the investigated material. If it is withal supposed that the diffraction is weak in
the sense that it does not influence (does not attenuate) the primary beam and that it is only a one–shot
diffraction, i.e. that only the primary radiation is scattered, i.e. that the diffraction of the radiation
already diffracted is negligible, we speak about the kinematical theory of diffraction. (Max v. Laue [1],
page XX, [2], page 123, used to employ the term geometrical diffraction. His designation is probably
more fitting but it did not become established.) If it is taken into account the attenuation of the primary
radiation in the course of the diffraction and the diffraction of already diffracted radiation, we speak
about the dynamical diffraction theory.
Mathematical basis of the kinematical theory of diffraction is the Fourier transform in EN . Independently on the branch of the diffraction theory — diffraction by one–dimensional gratings (E1 ), Fraunhofer
diffraction in optics (E2 ), structure analysis of materials (E3 ) including of quasicrystals (EN ), diffraction
of waves of various kind (acustic, electromagnetic, electrons, neutrons) and energies or wavelengths —
the Fourier transform formalism and its results are always used.
Let the three–dimensional specimen be characterized by a function f (~x) (transmission function in
optics, electron density in X–ray diffraction, electrostatic potential in electron diffraction, etc.), which
represents the scattering power of the material of the specimen. The parallel (i.e. collimated) beam of
radiation is characterized by the plane wave exp (ik~n0 · ~x), where ~n0 is the unit vector in the direction of
propagation and k = 2π/λ is the wave number. Every point M of the specimen causes elastic scattering
giving rise the spherical wave
exp(ikr)
,
kr
the amplitude of which is proportional to the scattering power f (~x) of the specimen on the one hand
and — on the other hand — to the incident wave exp(ik~n0 · ~x), as it is supposed in the kinematical
theory. The distance r between the point M with the position vector ~x and the point of observation P
~ is r = |R
~ − ~x| (cf. Figure 1). The radiation at the point P diffracted by the
with the position vector R
whole specimen is then characterized by the integral
ZZZ ∞
~ − ~x|
exp
ik|
R
~ =
d3 ~x.
(1)
Ψd (R)
f (~x) exp(ik~n0 · ~x)
~ − ~x|
k|R
−∞
f (~x) exp(ik~n0 · ~x)
Now we take advantage of the fact that the point of observation P is very far from the object. We
mean by that, that the function f (~x) takes values that are physically significant only if x R and is
equal to zero elsewhere. In other words, it is supposed that a linear extent of the scattering specimen
is very small compared to the distance between the specimen and the point of observation P . It is
commonly supposed that this allows us to use the approximation
~ − ~x|)
exp(ikR)
exp(ik|R
≈
exp(−ik~n · ~x),
~
kR
k|R − ~x|
(2)
~
in the integrand of (1), where ~n = R/R.
Usually the approximation (2) is justified in the following way:
Consider the expansion
32
2
exp(i kn0·x )
MEANING OF VARIABLES IN THE FOURIER TRANSFORM
M
r
P
x
R
0
Figure 1: To the approximation (2).
s
~ − ~x| =
|R
q
("
~ · ~x + x2 = R
R 2 − 2R
1−
~ · ~x − x2
2R
= R − ~n · ~x + ε,
R2
(3)
where
ε
=
x2
2R
""
2 # 2
4
~n · ~x
~n · ~x
x2
~n · ~x
~n · ~x
1−
1+
−
1
−
6
+
5
+
x
R
4R2
x
x
"
)
2
4 ###
~n · ~x
~n · ~x
~n · ~x
+
3 − 10
+7
± · · · ≥ 0.
R
x
x
(4)
In the denominator of the expression on the left–hand side of (2) we may limit oneselves only to the first
term, because |~n ·~x| R and also ε R. On the other hand, in the argument of the exponential function
on the left–hand side of (2) we have to take the first two terms of the expansion (3) and to suppose that
the neglected part of the expansion, i.e. kε, is small compared to the period of the exponential function,
i.e.
kε 2π,
i.e.
x2
λ.
2R
Strictly speaking, we should write
~ − ~x|)
exp(ik|R
exp(ikR)
exp(ikε)
=
exp(−ik~n · ~x) ·
.
~
kR
1 − ~n·~xR−ε
k|R − ~x|
instead of (2). The approximation (2) then means that we put
exp(ikε)
1 − ~nR·~x + Rε
=
.
=
"
2 #
~n · ~x
x2
~n · ~x
1+
+ ik
1−
×
R
2R
x
(
2 )
~n · ~x
i
~n · ~x
~n · ~x
×
1+2
1+
1+
+2
± ···
R
kR
R
R
1.
It is surely possible if
~n · ~x R
(5)
and also
k
x2
πx2
=
1,
2R
λR
i.e.
π
x2
λ.
R
(6)
2
MEANING OF VARIABLES IN THE FOURIER TRANSFORM
33
Conditions (5) and (6) are usually satisfied at scattering of radiation by objects of atomic dimensions
(Born approximation, see e.g. [3], [4], chapter 5, §5, [5], §77, [6]). But in the diffraction theory the
fulfilment of the condition (6) need not be a commonplace. Especially at experiments where the lenses
are not used or cannot be used the condition (6) may be very severe. For example, in the diffraction of
.
.
X–rays by crystals the typical experimental parameters are R = 1 · 10−1 m, λ = 1 · 10−10 m and from the
condition (6) it follows
r
λR .
= 2 · 10−6 m.
x
π
The size of the crystals diffracting X-rays should be tenths of micrometer at most! In diffraction of light
or electrons it is possible to use a lens and to observe the diffraction pattern in its focal plane. Then the
distance R = ∞ and conditions (5) and (6) are fulfilled.
Using the approximation (2) and denoting by
~ = ~n − ~n0
X
(7)
the so called scattering vector, we rewrite integral (1) into the form
ZZZ ∞
exp (ikR)
~ · ~x d3 ~x.
f (~x) exp −ik X
Ψd =
kR
−∞
(8)
The integral in this expression has already the form of the Fourier integral.
~ in the integral (8) cannot take all
In contradistinction to the Fourier transform, the variable X
possible values in EN , but is limited by the condition (7). Diffraction theory comes to terms with this
difference by the so–called Ewald construction. Diffraction theory uses the Fourier transform but adds
~ satisfying the condition (7) are experimentally accessible, i.e. X
~ must be the
that only the values of X
difference of unit vectors.
Q
ρ
n
C
X
n0
O
~ (see (7)).
Figure 2: The scattering vector X
~
Hence, the essence of the Ewald construction is as follows (see Figure 2): In space of the variable X
of the Fourier transform, we draw the spherical surface ρ of the unit radius in such a way that it passes
through the origin O and its radius CO has the direction of the incident radiation, i.e. CO = ~n0 (Ewald’s
(also reflection) spherical surface). Then the amplitude of the radiation diffracted in the direction CQ =
~ is proportional to the Fourier transform F (X)
~ (of the function f (~x) characterizing the
~n = ~n0 + X
specimen) at the point Q of the Ewald spherical surface, the position vector of which is the scattering
~ i.e. X
~ = OQ.
vector X,
~ is accessible in experiments is, unfortunaThe fact, that only a part of the space of the variable X
tely, not the only problem which the structure analysis must face out. Another important fact is that
in experiments only the intensity of the diffracted radiation is registered, i.e. a quantity proportional
34
2
MEANING OF VARIABLES IN THE FOURIER TRANSFORM
~ 2 but not the amplitude which is proportional to the Fourier transform F (X).
~ In this way
to |F (X)|
~
information about the phase of the complex function F (X) is lost. Here, the well known phase problem
of the structure analysis arises. There is infinite number of functions f (~x) the Fourier transform of which
~ 2.
has the same squared modulus |F (X)|
Thus, we should learn the function f (~x) only from the modulus of its Fourier transform known in a
~ Under these conditions, imposed by experiment, it is advantageous to
finite region of the space of X.
bear in mind theorems about the Fourier transform and with their relief to extract from the diffraction
pattern as much information about the function f (~x), i.e. about the structure of the investigated object,
as possible. Most of the present lectures, starting with the next chapter 3 deal with these theorems.
L0 n0·x M
-n·x
A0
x
n0
L
0
n
A
Figure 3: To the derivation of relation (9).
Let us now remind ourselves of a simple geometric reasoning which enables us to recognize that
the diffracted radiation is characterized by integral (8). Imagine again that plane wave exp(ik~n0 · ~x) is
incident in the direction ~n0 on the object and investigate the radiation scattered in the direction ~n. It
is evident from the wave surfaces OL0 and OL in Figure 3 that the path difference of the beams A0 OA
and A0 M A is the length L0 M L, i.e. (~n0 − ~n) · ~x. Hence, the phase delay between the waves scattered
by surroundings of the point M into the direction ~n and the waves scattered in the same direction by
surroundings of the origin O is −k(~n −~n0 )·~x. Taking into account the scattering power f (~x), it is evident,
that the wave function characterizing the radiation scattered by the whole specime in the direction ~n is
ZZZ ∞
ψd (~n − ~n0 ) ∼
f (~x) exp −ik(~n − ~n0 ) · ~x d3 ~x,
(9)
−∞
in agreement with (8).
In this section we have chosen the wave number k = 2π/λ as the parametr k in the definition of
~ 1 , X2 , X3 ) acquired the meaning of the difference
the Fourier transform. Consequently, the variable X(X
of unit vectors in the direction of diffracted and incident radiation (see (7)). In a Cartesian coordinate
~ are then the differences of direction cosines
system the coordinates of the vector X
X1 = cos α1 − cos α01 ,
X2 = cos α2 − cos α02 ,
X3 = cos α3 − cos α03
(10)
of the vectors ~n(cos α1 , cos α2 , cos α3 ) and ~n0 (cos α01 , cos α02 , cos α03 ) and it was expedient to construct
the Ewald spherical surface with the unit radius.
In crystallography and in structure analysis the scattering vector is not defined by relation (7) but
by the fraction
~ f = ~n − ~n0 /λ
X
(11)
2
MEANING OF VARIABLES IN THE FOURIER TRANSFORM
35
(cf. e.g. [7], p. 7, [8], § 8.4). It is related to the fact in these disciplines the value ±2π is taken as the
parameter k in the definition of the Fourier transform as mentioned in Section 1.1. The Ewald spherical
surface is then constructed with the radius 1/λ.
In solid state physics and in surface physics the parameter k in the definition of the Fourier transform
is chosen to be ±1, the scattering vector is defined by
~ ω = 2π ~n − ~n0 /λ
X
(12)
and the Ewald spherical surface is drawn with the radius 2π/λ (cf. e.g. [9], p. 37, [10], § 4.2). (The
reflection sphere of this radius has originally been used by P. Ewald himself [11]).
2.2
The Fraunhofer diffraction in optics and the Fourier transform
In laboratory parlance it is often said that the Fraunhofer diffraction is the Fourier transform, but it is
not specified the Fourier transform of what the Fraunhofer diffraction should be. In this section we will
try to shed light on this question.
The Fraunhofer diffraction – not specified up to now – is characterized by the Fourier transform
F (X1 , X2 ) which is a function of two variables. Hence, it is the Fourier transform of a function of
two variables f (x1 , x2 ). In optics we often have in hand objects which may be considered to be two–
dimensional, e.g. various stops, transparents (e.g. photographic films, biological slices used as specimens
for microscopical investigations) etc. These two–dimensional objects can be characterized by the so called
transmission function t(x1 , x2 ). We will investigate in this section the relation between the Fraunhofer diffraction and the Fourier transform of the transmission function t(x1 , x2 ), i.e. how the functions f (x1 , x2 )
and t(x1 , x2 ) are related.
2.2.1
The transmission function
Let us consider an object of the two–dimensional nature (i.e. sufficiently thin). Such an object may be
characterized by a transmission function t(x1 , x2 ) which is defined in the following way: Let us suppose
that a wave is incident on the object, passes through it and in the plane immediately behind the object is
specified by a function ψ(x1 , x2 ). Let ψ0 (x1 , x2 ) specify the same wave at the same plane in the absence
of any object. If the ratio
t(x1 , x2 ) =
ψ(x1 , x2 )
ψ0 (x1 , x2 )
(1)
does not depend under reasonable circumstances on the incident wave, we may consider the object to
be two–dimensional and specify it by the transmission function t(x1 , x2 ). (It is meant that the ratio (1)
should be the same for plane or spherical incident waves or should not depend on the angle of incidence
of a plane wave etc. The reasonable conditions exclude the cases like almost tangential incidence of a
plane wave or a spherical wave having its centre too close to the object, etc.)
It is evident that the transmission function is in the general case complex. In practice, however, two
cases are significant:
(i) Amplitude objects. They have transmission function of the form
t(x1 , x2 ) = τ (x1 , x2 ) exp(iε0 ),
(2)
where τ (x1 , x2 ) is a real function and ε0 is a real constant. For example, the transmission function
of objects consisting of empty apertures in a metal foil (e.g. a slit or, on the contrary, an opaque
strip) is modelled by the characteristic function of the openings, i.e. by function equal to one at
points of the openings and zero at point of opaque part of the screen.
(ii) Phase objects. They have at transparent part the transmission function of the form
t(x1 , x2 ) = τ0 exp[iε(x1 , x2 )],
(3)
where τ0 is a constant and ε(x1 , x2 ) is a real function. At opaque parts is t(x1 , x2 ) = 0. As examples
may serve ideally transparent optical elements which negligibly absorb the light but substantially
influence the phase of the light (e.g. thin lenses, phase gratings, etc.)
36
2
MEANING OF VARIABLES IN THE FOURIER TRANSFORM
The concept of the transmission function is so important that some authors try to keep it even for
objects which are not thin, i.e. even in the cases when the ratio at the right–hand side of (1) depends
on the incident wave [12], [13].
2.3
Alternative expression of the inverse Fourier transform in E1
The Fourier transform of a function of a single variable has been introduced in the form
Z ∞
f (x) exp(−ikxX)dx,
F (X) = A
Z
(1)
−∞
∞
f (x) = B
F (X) exp(ikxX)dX,
(2)
−∞
with the condition
|k|
.
(3)
2π
The inverse Fourier transform (2) expresses the function f (x) for all x ∈ (−∞, ∞) in terms of the function
F (X) defined also in the region X ∈ (−∞, ∞). Nevertheless, by rearrangements of the integrand in (2)
the function f (x) can be expressed also in the following ways:
AB =
∞
Z
f (x)
= B
[C(X) cos kxX + S(X) sin kxX] dX
(4)
D(X) cos [kxX + Φ(X)] dX
(5)
D(X) sin [kxX + Θ(X)] dX.
(6)
0
∞
Z
= B
Z0 ∞
= B
0
In expressions (4) to (6) the function f (x) is expressed in the whole region x ∈ (−∞, ∞) in terms of
couples of functions C(X), S(X), or D(X), Φ(X), or D(X), Θ(X), each of them being defined in the
interval x ∈ h0, ∞).
To get the expressions (4) to (6) we decompose the function F (X) into its even and odd part:
Fe (X) =
1
[F (X) + F (−X)] ,
2
Fo (X) =
1
[F (X) − F (−X)]
2
(7)
and rewrite the integral (2):
Z
f (x)
∞
= B
[Fe (X) exp(ikxX) + Fo (X) exp(ikxX)] dX
−∞
∞
Z
= B
[Fe (X) cos(kxX) + iFo (X) sin(kxX)] dX
−∞
∞
Z
=
2B
[Fe (X) cos(kxX) + iFo (X) sin(kxX)] dX.
0
Having used (7) we may rewrite the last integral into the form
Z ∞
f (x) = B
{[F (X) + F (−X)] cos(kxX) + i [F (X) − F (−X)] sin(kxX)} dX.
(8)
0
By comparing (4) and (8) we see that
C(X) = F (X) + F (−X),
S(X) = i [F (X) − F (−X)] .
(9)
(10)
Thereof
1
[C(X) − iS(X)]
2
= F (X),
if X ≥ 0,
(11)
1
[C(|X|) + iS(|X|)]
2
= F (X),
if X ≤ 0.
(12)
2
MEANING OF VARIABLES IN THE FOURIER TRANSFORM
37
By a manipulation with the trigonometric functions in (5) and (6) and by comparison with (8) we get
p
C 2 (X) + S 2 (X) = 2 F (X)F (−X),
S(X)
F (X) − F (−X)
tg Φ(X) = −
= −i
,
C(X)
F (X) + F (−X)
C(X)
F (X) + F (−X)
tg Θ(X) =
= −i
.
S(X)
F (X) − F (−X)
D(X)
=
p
(13)
(14)
(15)
If the function f (x) is real, F (−X) = F ∗ (X) (cf. Section 5.2 below). Functions (9), (10), (13), (14) and
(15) are then also real and have more simple shape
C(X) = 2ReF (X),
S(X) = −2ImF (X),
p
D(X) = 2 F (X)F ∗ (X),
ImF (X)
tg Φ(X) =
,
ReF (X)
ReF (X)
tg Θ(X) = −
.
ImF (X)
(16)
(17)
(18)
(19)
(20)
In the conclusion of this section we emphasize that to specify a function f (x) defined in the interval
x ∈ (−∞, ∞) by the inverse Fourier transform we can do it by means of the single function F (X) defined
in the interval X ∈ (−∞, ∞) or by couple of functions C(X), S(X), or D(X), Φ(X), or D(X), Θ(X)
defined in the interval X ∈ h0, ∞). (Of course, in special cases one of the functions of the couple may
be zero. For example, if f (x) is an even function, S(X) = 0, Φ(X) = 0; if f (x) is an odd function then
C(X) = 0, Θ(X) = 0 as it follows from the symmetry of the Fourier transform dealt with in chapter 6.)
References
[1] Laue M. v.: Materiewellen und ihre Interferenzen. 2. Auflage. Akademische Verlagsgesellschaft,
Geest & Portig, Leipzig 1948.
[2] Laue M. v.: Röntgenstrahlinterferenzen. 3. Auflage. Akademische Verlagsgesellschaft, Frankfurt am
Main 1960.
[3] Born M.: Quantenmechanik der Stoßvorga̋nge. Zeitschrift fűr Physik 38 (1926), 803–827.
[4] Sommerfeld A.: Atombau und Spektrallinien. II. Band. Friedr. Vieweg & Sohn, Braunschweig 1951.
[5] Blochincev D. I.: Základy kvantové mechaniky. Nakladatelství Československé akademie věd, Praha
1956.
[6] Landau L.D., Lifšic E.M.: Quantum Mechanics. Pergamon Press, Oxford 1965, §125.
[7] Guinier A.: X–Ray Diffraction In Crystals, Imperfect Crystals, and Amorphous Bodies. W. H. Freeman and Co., San Francisco 1963.
[8] Hammond Ch.: The Basis of Crystallography and Diffraction. International Union of Crystallography. Oxford University Press, Oxford 1997.
[9] Kittel Ch.: Introduction to Solid State Physics. 7th ed., John Wiley, Inc., New York 1996.
[10] Lüth H.: Surfaces and Interfaces of Solid Materials. 3rd ed. Springer Verlag, Berlin 1995.
[11] Ewald P. P.: Zur Theorie der Interferenzen der Röntgenstrahlen in Kristallen. Physikalische Zeitschrift 14 (1913), 465 - 472.
[12] Richter I., Ryzí Z., Fiala P.: Analysis of binary diffraction gratings: comparison of different approaches. Journal of Modern Optics 45 (1998), 1335–1355.
[13] Fiala P., Richter I., Ryzí Z.: Analysis of diffraction process in gratings. SPIE Proc. 3820 (1999),
131–143.