2 Crystal Optics

2.1 Polarization
In this section, we review some basic properties of electromagnetism in crystalline me-
dia. In particular, we introduce the notion and classification of the polarization of an
EM-wave in a homogeneous medium. We consider monochromatic plane waves (time-
harmonic plane waves) that propagate in z-direction.


E (z, t) = Re Aei(kr−ωt) (2.1)
with complex amplitude A = Ax x̂ + Ay ŷ, Ax , Ay ∈ C. (2.2)

In the above equation, we have introduced the unit vectors a long the x- and y-direction,
x̂ and ŷ, respectively. Furthermore, we decompose the wave vector k into its magnitude
k and direction k̂. Without loss of generality, we assume that the wave vector points
along the z-axis (we can always choose the orientation of our coordinate system in that
way):


0
k = k · k̂ and k̂ =  0  = ẑ (2.3)
1
also Ai = ai eiϕi , for i = x, y. (2.4)
|{z}
∈R
⇒ Ex = ax cos (kz − ωt + ϕx ) (2.5)
Ey = ay cos (kz − ωt + ϕy ) (2.6)
E 2 Ey2 Ex Ey
⇒ 2x + 2 − 2 cos ϕ = sin2 ϕ (2.7)
ax ay ax ay
with ϕ = ϕx − ϕy . (2.8)

The above manipulations give the following picture: If we keep z fixed, then the direc-
tion of E rotates (as a function of time t) in the xy-plane and traces out the so-called
polarization ellipse. There are two special cases, which we discuss below:

21
2 Crystal Optics

Linearly polarized light :

ax = 0 or ay = 0
or (2.9)
ϕ = 0 or ϕ = π
| {z }
 
ay
⇒ Ey = ± Ex (2.10)
ax

In this case, the polarization ellipse degenerates into a line.

Circularly polarized light :

π
ϕ = ± , ax = ay = a0 (2.11)
2
⇒ Ex2 + Ey2 = a20 (2.12)

In this case, the polarization ellipse becomes a circle and can be traced out either in a
clockwise or counterclockwise fashion. These are the so-called Right- and Left-circularly
polarized light waves.

In all cases, there are (for a given wave vector) two linearly independent components
and one can combine this into a matrix description which is the so-called Jones matrix
representation that works with (2 × 2) matrices. This is somewhat analogous to the
description of electron spins in Quantum Mechanics.

2.2 Optics of (simple) anisotropic media

In order to understand the principal behavior of anisotropic materials, it is best to first
ignore any dispersive property (frequency dependence) of the material. Owing to the
Kramers-Kronig relation the material will then be lossless. This is what we mean by a
simple anisotropic material.
In mathematical terms, the meaning of ’simple’ is D = ǫ0 ǫ E , B = µ0 H. Here,
=
we have introduced the dielectric tensor ǫ , i.e., a (3 × 3) matrix that has the following
=
properties :

• Non-dispersiveness implies lossless medium (by Kramers-Kronig relation), i.e., ǫ

=
is Hermitian.

• For now, we assume that ǫ is real symmetric, i.e ǫij = ǫji . This assumption will
=
be lifted in Sec. 2.3.2.

22
 2.2 Optics of (simple) anisotropic media

• Because plane wave propagation is the only allowed form of propagation, ǫ is real
=
symmetric and positive definite.

• The above property implies that ǫ has 3 positive eigenvalues ǫ1 , ǫ2 , ǫ3 and corre-
=
sponding eigenvectors (principal axes) that are mutually orthogonal.

For a given material, the values ǫ1 , ǫ2 , ǫ3 and the corresponding principal axes are deter-
mined by the microscopic arrangements and symmetries of the material’s constituents
(atoms, molecules, etc). It is one of the tasks of solid-state physics to determine these
quantities. In optics we take these quantities as given input.
We consider time-harmonic plane waves, i.e., E, D, B, H depend on r and t through
the common factor ei(kk̂·r̂−ωt) as in E(r, t) = E0 ei(kk̂·r̂−ωt) with a constant vector E0
(same goes for D, B, and H). We also ignore (free) charges and (free) currents, hence
we set ρ = 0 and j = 0. Then we can deduce from the Maxwell equations (1.45) and
(1.47):

k k̂ × H + ωD = 0 ⇒ D ⊥ k̂, H (2.13)
k k̂ × E − µ0 ωH = 0 ⇒ H ⊥ k̂, E (2.14)
⇒ H ⊥ k̂, E, D (2.15)
| {z }
coplanar
k   ω
k̂ × H = k̂ × k̂ × E = − D (2.16)
µ0 ω | {z } k
=−E+(E·k̂)k̂
| {z }
2 h   i
k
⇒D= E − E · k̂ k̂ (2.17)
µ0 ω 2

Eq. (2.17) implies k̂ · D = 0, i.e., the divergence condition (2.13) for the displacement
field is consistent with (2.17).
Furthermore, E · k̂ can be non-zero now. If the dispersion relation is known, this
allows to determine the polarization (i.e., the direction) of D.

Note:
k
H= k̂ × E (2.18)
µ0 ω

1
⇒S = Re(E × H∗ ) (2.19)
2
1 k h   i
= Re (E · E∗ ) k̂ − E · k̂ E∗ (2.20)
2 µ0 ω

23
2 Crystal Optics

⇒ The Poynting-vector S is not necessarily parallel to the wave vector k̂.

We now insert the constitutive relation D = ǫ0 ǫ E into Eq. (2.17). This gives
=

 
c2 k 2 c2 k 2
ǫ − 2 1 + 2 k̂ : k̂ E = 0 independent of Coordinate system (2.21)
= ω ω
| {z }
det=0 for non trivial solutions

Without loss of generality, we can assume that we are in a coordinate system that
coincides with the material’s principal axes:
   
ǫ1 k1
ǫ= ǫ2 , k̂ =  k2  . (2.22)
=
ǫ3 k3
Then, the requirement of the determinant being zero gives the celebrated Fresnel Equa-
tion:

  2 2 
c2 k 2 ǫ1 c k1 ǫ2 c2 k22 ǫ3 c2 k32
+ + (2.23)
ω2 ω2 ω2 ω2
 2 2 
c k1 c2 k22 c2 k32
− ǫ1 (ǫ2 + ǫ3 ) + 2 ǫ2 (ǫ1 + ǫ3 ) + 2 ǫ3 (ǫ1 + ǫ2 ) (2.24)
ω2 ω ω
+ ǫ1 ǫ2 ǫ3 = 0 (2.25)
where, k 2 = k12 + k22 + k32 .

2.3 Discussion of general properties

We proceed to discuss the properties of the Fresnel equations by first considering its
general properties and then consider successively more difficult special cases.

For given ǫ1 , ǫ2 , ǫ3 :
• For fixed ω and given direction k̂ there generally exist two distinct values for the
wave number k ∈ R that solve the Fresnel equation. Exceptions occur at so-called
double points.
• For fixed k and given direction of k̂ there exist two frequencies ω that solve the
Fresnel equation.
k
One can define a refractive index vector by n := c k̂ which is also called the slowness-
ω
vector for which sometimes also the notation s is used.
ck
One can define a refractive index n := |n| for the direction k̂ = n.
ω

24
 2.3 Discussion of general properties

Note:
• The index of refraction is a wave property, not a material property.
• ǫ1 , ǫ2 , ǫ3 and the principal axes are material properties and not wave properties.
• The dispersion relation ω = ω(k) (i.e., the solution of the Fresnel equation) char-
acterizes the wave completely, so that there is no real need to introduce a refractive
index.
However, if you do use the notion of a refractive index, be aware that a refractive index
is a wave and not a material property.

Note:
• Strictly speaking, the refractive index introduced above corresponds to the phase
ω
of the wave, i.e., (for given k̂) it relates the phase velocity to the vacuum speed
k
of light.
ck
Therefore, the correct statement should be that n = is a phase index, denoted as
ω
nph .
In general, we have in anisotropic media that
vgroup ∦ vphase (2.26)
and
|vgroup | =
6 |vphase | (2.27)
c
Therefore, one could equally well define a group index ngr := , but this would —
∂ω
∂k
in general — be for another direction !
In terms of the dispersion relation (recall that we are in lossless non-magnetic media),
we have less confusion with
ω (k)
vphase = k̂ (2.28)
k
and
∂ω(k) hSi
vgroup = = = vE , (2.29)
∂k hwi
where the vector of the energy transport velocity is sometimes vE called the ray vector.
The above property that the group velocity is identical to the energy transport veloc-
ity (and thus parallel to the Poynting vector) only applies to lossless and nonmagnetic
systems. However, if these criteria are satisfied for all constituent materials, the state-
ment also applies to periodically structured materials such as Bragg stacks or Photonic
Crystals:
1
vgroup k S = Re(E × H∗ ) (2.30)
2
??writeme: check reference The proof is found in the paper by P. Yeh JOSA 69, writeme
742 (1979) (see Fig. 2.13).

25
 2 Crystal Optics

writeme Note: Remember the assumptions: ??writeme: check this table!

lossless non-magnetic
ǫ is real symmetric µ=1
=
and positive definite
If any of the two does not apply, some or all of the above considerations do not hold.

Isotropic material (Cubic symmetry): For

ǫ1 = ǫ2 = ǫ 3 = ǫ (2.31)
the Fresnel Equation becomes
 2
c2 k 2
ǫ −ǫ = 0. (2.32)
ω2
Dispersion relation has shape of two identical spheres, one for each transverse polarization:??writeme:
writeme check next two lines
c √
vph = vgr = √ where ǫ = nph = ngr , (2.33)
ǫ
i.e. n k S.

Optically uni-axial materials (Trigonal, tetragonal, hexagonal symmetry): Here we

have
ǫ1 = ǫ2 = ǫ ⊥ . (2.34)
ǫ3 > ǫ⊥ : Positive uni-axial
ǫ3 < ǫ⊥ : Negative uni-axial
The Fresnel Equation becomes
 2 2  
ck c 2 k⊥
2
c2 k32
− ǫ⊥ ǫ⊥ 2 + ǫ3 2 − ǫ⊥ ǫ3 = 0. (2.35)
ω2 ω ω
In the above equation the solutions take on following shapes:
 2 2 
ck
− ǫ⊥ ⇒ one sphere ⇒ ordinary wave (2.36)
ω2
 
c 2 k⊥
2
c2 k32
ǫ⊥ 2 + ǫ3 2 − ǫ⊥ ǫ3 ⇒ one ellipsoid ⇒ extraordinary wave (2.37)
ω ω
The ellipsoids may be
positive uni-axial: circumscribed
negative uni-axial: inscribed
See Figs. 2.1, 2.2 and 2.3.

26
 2.3 Discussion of general properties

Figure 2.1: Wave vector surfaces for positive and negative optically uni-axial crys-
tals.

Special cases:
• k̂ k optical axis: Same phase and group velocity, same direction of k̂, S
For linearly polarized waves, the orientation of the polarization remains unchanged.
• k̂ ⊥ optical axis: Same direction of k̂, S, different magnitudes for phase and group
velocity. An initially linearly polarized wave will become elliptically polarized.

General case:
• k̂ ∦ S: Different magnitudes of phase and group velocity. Two rays.

Optically bi-axial materials (Orthorhombic, monoclinic, triclinic symmetry): We

order the the values as ǫ1 < ǫ2 < ǫ3 and have to consider the full Fresnel equation
 2 2 2 2 
ck ǫ1 c k1 ǫ2 c2 k22 ǫ3 c2 k32
+ +
ω2 ω2 ω2 ω2
 2 2 
c k1 c2 k22 c2 k32
− ǫ1 (ǫ2 + ǫ3 ) + 2 ǫ2 (ǫ1 + ǫ3 ) + 2 ǫ3 (ǫ1 + ǫ2 )
ω2 ω ω
+ ǫ1 ǫ2 ǫ3 = 0 (2.38)
To gain some insight, we first set k3 = 0. Then
 2 2  2 2 
ck c k1 c2 k22
− ǫ3 ǫ1 2 + ǫ2 2 − ǫ1 ǫ2 = 0. (2.39)
ω2 ω ω
In the above equation we have
 
c2 k 2
− ǫ3 ⇒ Circle (in k1 , k2 )
ω2
 2 2 
c k1 c2 k22
ǫ1 2 + ǫ2 2 − ǫ1 ǫ2 ⇒ Ellipse (in k1 , k2 )
ω ω

27
 2 Crystal Optics

Figure 2.2: Intersection of the k surface with the yz-plane for a uni-axial crystal.

Owing to the ordering of ǫ1 , ǫ2 , ǫ3 , the ellipse lies inside circle.

We next consider the analogous cases
k1 = 0: We obtain the same, except that the circle lies inside ellipse,
k2 = 0: We obtain that the circle and ellipse intersect at two opposite pairs of points
(see Fig. 2.4).
Those complicated dispersion surfaces have been considered by Sir Rowan Hamilton
shortly before the Maxwell equations were introduced. Sir Hamilton suggested that this
should lead to the phenomenon of inner and outer conical refraction, an effect which
subsequently has been confirmed by experiment (See Fig. 2.5) ??writeme: explain
writeme annotations in plot fig:ConicalRefraction

2.3.1 Reflection and Refraction at Interfaces

The general setup is shown in Fig. 2.6. We have materials with different dispersion
relations and consider the interface conditions:
Kinematics:
- Frequency conserved.
- Wave vector parallel to surface conserved.
Dynamics:
- reflected and transmitted waves have to take the incoming energy away from the
surface (valid for any dispersion relation).

28
 2.3 Discussion of general properties

Figure 2.3: (a) Ordinary wave, polarized perpendicular to the k-z plane. (b) Ex-
traordinary wave with a polarization vector in the k-z plane.

Note :

• No statement regarding the normal component of the wave vector is made.

In the absence of surface charges

 and surface
 currents it follows that
n̂ × (E1 − E2 ) = 0 n̂ · (D1 − D2 ) = 0
⇒
n̂ × (H1 − H2 ) = 0 n̂ · (B1 − B2 ) = 0
| {z } | {z }
Tangential components of E, H continuous Normal components of B, D continuous
⇓

-Implies normal component of S = E × H is continuous across the interface.

-Otherwise energy would be accumulated.

Based on the above discussion, we have to make the following ansatz for the media 1
and 2.

Ansatz medium 1:
2
X
Aβ êβ ei(kβ r−ωt)
′
i(kr−ωt)
E = êe + (2.40)
β=1

Ansatz medium 2:
2
X
Bγ êγ ei(kγ r−ωt)
′′
E= (2.41)
γ=1

The coefficients in these ansatzes have to be determined by, for instance, equating the
tangential components of the electric and magnetic field. See Figs. 2.7, 2.8 and 2.9.

29
2 Crystal Optics

Figure 2.4: Wave surfaces for optically bi-axial crystals.

2.3.2 A potpourri of optically anisotropic materials

Liquid Crystals
Liquid Crystals consist of cigar-shaped molecules (rotational ellipsoids) that have dif-
ferent dipole moments (or polarizabilities) along the various symmetry directions. For
sufficiently low temperatures these molecules order automatically in various types of
structures. Most common are nematic, smectic, cholesteric liquid crystals (see Fig. 2.15).
For instance, for a nematic liquid crystal the molecules point — on average — along a
certain direction. More generally, liquid crystals may be characterized by a local director
field n̂(r) that describes the average orientation of the molecules. The different dipole
moments lead — upon averaging — to a dielectric tensor of the form


ǫ = ǫ⊥ 1 + ǫk − ǫ⊥ n̂ : n̂. (2.42)
=

Optical Activity
A medium characterized by (harmonic time dependence assumed)
D = ǫ0 ǫE + iωǫ0 ξB (2.43)
= ǫ0 ǫE − ǫ0 ξ∇ × E (2.44)
is called optically active.
Such a relation arises in molecular structures with a helical character. In these struc-
tures, a time-varying magnetic flux density B induces a circulating current that sets up
an electric dipole moment (and hence polarization) proportional to iωB = −∇ × E.
The optically active medium is spatially dispersive, meaning that the relation between
D(r) and E(r) is non-local in space.

For a plane wave E(r) = E0 eikr we have D(r) = D0 eikr so that,

D0 = ǫ0 ǫE0 − iǫ0 G × E0 (2.45)

30
 2.3 Discussion of general properties

Figure 2.5: (a) Principle of inner conical refraction. (b) Principle of outer conical
refraction.

Figure 2.6: General setup for reflection at an interface.

Here, we have introduced the gyration vector G = ξk.

Note: As can be seen here, G changes sign under reversal of the propagation direc-
tion, i.e., change of the direction of the wave vector.

We can now express this as

D0 = ǫ 0 ǫ E 0 (2.46)
=

where
 
ǫ iGz −iGy
ǫ = −iGz ǫ iGx  . (2.47)
=
iGy −iGx ǫ

31
2 Crystal Optics

Figure 2.7: Determination of the angles of refraction by matching projections of

the k vectors in air and in a uni-axial crystal.

Figure 2.8: Double refraction at normal incidence.

32
 2.3 Discussion of general properties

Figure 2.9: Double refraction through an anisotropic plate. The plate serves as a
polarizing beam splitter.

Figure 2.10: An index ellipsoid where the principal axes are along the (x,y,z)-axes
and the principal refractive indices are given by n1 ,n2 ,n3 .

33
2 Crystal Optics

Figure 2.11: Normal modes determined from the index ellipsoid.

Figure 2.12: An octant of the k surface for (a) a bi-axial crystal, (b) a uni-axial
crystal and (c) an isotropic crystal.

Figure 2.13: Rays and wave fronts for (a) a spherical k surface and (b) a non-
spherical k surface.

34
 2.3 Discussion of general properties

Figure 2.14: (a)Variation of the refractive index n(θ), (b) direction of the D and
E vectors for the ordinary and extraordinary wave.

Figure 2.15: Molecular orientation of various types of liquid crystals: (a) nematic,
(b) smectic and (c) cholesteric.

In particular, for circularly polarized light that propagates along the z-axis with
   
0 E0
k =  0  and E0 =  ±iE0  , (2.48)
k 0
we obtain  
D0
D0 =  ±iD0  (2.49)
0
where
D0 = ǫ0 (ǫ ± ξk)E0 (2.50)
| {z }
=n2±

In other words, we obtain different phase velocities for left- and right-circularly polarized
light. This phenomenon is called circular birefringence and should not be confused with
the case of linear birefringence. However, similar to linear birefringence, this leads to
rotation of plane of polarization for linear polarized light upon propagation. Note that
when the direction of propagation is reversed, the direction of polarization rotation is
also reversed.

35
2 Crystal Optics

Faraday Effect
Certain materials show for plane waves and for propagation in a given direction a similar
polarization rotation when placed in a static magnetic field Bstat.

D = ǫ0 ǫE + iωǫ0 γBstat × E. (2.51)

Here, γ is called the magneto-gyration coefficient.

The above may be rewritten as,

D = ǫ0 ǫE + iω GFaraday ×E = ǫ0 ǫ E (2.52)
| {z } =
γBstat

 
ǫ iωGz −iωGy
ǫ = −iωGz ǫ iωGx  (2.53)
=
iωGy −iωGx ǫ

Note:
1. Despite the apparent similarity to the case of optical activity, Faraday-Activity is
quite different. For instance, upon change of propagation direction, GFaraday does
not change sign!
a) The polarization rotation is the same direction for forward and backward
propagating modes. Therefore, Faraday-active materials break time-reversal
symmetry!
b) A Faraday medium is spatially non-dispersive.

2. Sometimes the Faraday-effect is described in terms of the Verdet constant V , given

as

πγ
V ≈− (2.54)
λ0 n 0
√ c
where n0 = ǫ and λ0 = 2π (2.55)
ω
Typical Faraday-active materials include

YIG (Yttrium-Iron-Garnet)

TGG (Terbium-Gallium-Garnet)

TbAlG (Terbium-Aluminum-Garnet)

36
 2.3 Discussion of general properties

In general, Faraday-active materials that contain rare-earth elements show the largest
effects, since the f-electrons of the rare-earth elements react best to magnetic fields.
For example:
min
V = −1.16 at λ0 = 500nm (2.56)
cmOe
Recall that in vacuum 1 Oe = 1 Gauss = 10−4 T.

Device Application:

- Optical Isolators (the optical analogon to an electrical diode, see Fig. 2.16)

- Circulators (see, e.g., Wikipedia)

Figure 2.16: Realization of an optical insulator by using a Faraday rotator. Light

is transmitted in one direction (a), while the propagation in the other
direction is blocked (b).

37
3 Diffraction Theory
This chapter is based largely on Goodman, Introduction to Fourier Optics [3] and Römer,
Theoretical Optics [4]. This chapter deals with monochromatic (only one single frequency
involved) scalar diffraction theory exclusively. Therefore, only the scalar complex valued
wave envelope U (r) in the stationary case is considered (i.e., U varies in space but not
in time), which can represent any electric or magnetic field component in the following,
since all these components obey the same differential equation here.

3.1 Phenomenology: The Principles of Huygens and

Fresnel
The simplest theory regarding light propagation is geometrical optics, where light is
treated as rays which propagate in straight lines in vacuum. However, this is just a crude
approximation and neglects the rich physical behavior due to the wave nature of light.
This wave character of light is responsible for effects like diffraction and interference
phenomena. Understanding diffraction in the wave picture is the subject of classical
diffraction theory.
According to Huygens’ principle, each point of the surface of a wave front (the surface
of constant phase F0 ) may be considered as the origin of a new elementary (spherical)
wave. New surfaces of constant phase at a later instant of time are obtained as the
enveloping surface of the family of phase surfaces of these elementary waves (see Fig. 3.1).
This enables one to construct the full wave pattern in all of space, if only one single
surface (of constant phase) of the wave is initially known.
However, this picture immediately raises the question: Why does light propagate
along lines in homogeneous materials then, which is properly described by the obviously
inferior geometrical optics approach? In fact, this puzzle has lead Newton to abandon
a wave theory for light in favor of a particle theory. The problem has been solved by
Augustin Jean Fresnel in 1818 (before Maxwell’s equations!): Consider a monochromatic
point source in Fig. 3.2, located at some point x0 in space and emitting a wave with
wavelength λ and corresponding wavenumber k = 2π λ
. We want to determine what
is observed at the point x? Therefore, imagine spherical shells around x with radii
rn = nλ + ρ. This decomposes the spherical wave emitted by the point source into
so-called Fresnel zones Zn .
All points in a given zone have the same distance ρ + (n − 1)λ ≤ rn (ρ) ≤ ρ + nλ from
x and can be considered as the origin of elementary waves of the form

39
3 Diffraction Theory

Figure 3.1: Elementary waves originating at each point of a non-trivial surface

defined by the phase value F0 of a wave. As these elementary waves
propagate, a new non-trivial wave front forms as the envelope defining
the surface of equal phase from the superposition of elementary waves.
Thus the whole wave front (of complicated shape) propagates in space
in terms of simple elementary waves.

Figure 3.2: Construction of Fresnel zones

eik|x−y|
Cκ(χ) y ∈ Zn , (3.1)
|x − y|
where the terms mean
C: some normalization constant,
κ(χ): Inclination factor that allows that the strength of the elementary wave emitted
in y may depend on the angle χ between the normal direction of the wave surface
and the direction x − y.
In each zone, the phase changes by 2π and for small wavelengths λ (compared to the
distance |x − x0 |) these zones are small.

Fresnel’s Argument:
Elementary waves from the outer half of each zone cancel with the elementary waves
from the inner half of the next zone.

40
 3.2 Scalar Diffraction Theory

• Only the contributions from the inner half of the very first zone and the outer half
of the very last zone remain.

• Both regions are small (for small λ) which explains the almost straight propagation
of light.

The prerequisite of small wavelengths λ is exactly the justification for the use of
geometrical optics. Hence, the theories are compatible and do not contradict each other,
as geometrical optics is contained as a special case in wave optics.
However, one problem remains in this argument: We do not observe the backward
wave emanating from the last zone. Fresnel’s ad-hoc solution was the ansatz
π
κ(χ) = 0 for χ ≥ . (3.2)
2
If we generalize this to the case of arbitrary wave fronts F0 of wave fields U (r), we see
that the principles of Huygens and Fresnel correspond to an identity of the form
Z ik|r−r′ |
′ ′ e
U (r) = C ds κ(χ)U (r ) , (3.3)
F0 |r − r′ |
which is a surface integral with χ depending both on r and r′ . The integration range F0
in the above integral represents the surface of the wavefront. It is the task of diffraction
theory to put the above phenomenology on a solid foundation and to derive consequences.

Remark: Fresnel presented his paper to a prize committee of the French Academy of
Science. His theory was strongly disputed by Simeon Denis Poisson (member of the
committee) who showed, that it predicted the existence of a bright spot at the center
of the shadow of an opaque disk (absurd in geometrical optics). Francois Arago (chair
of the prize committee) performed such an experiment and found the predicted spot.
Fresnel got the prize, and the effect has since been known as ‘Poisson’s spot’.

3.2 Scalar Diffraction Theory

The basic problem we are going to investigate is this: Given the wave distribution (am-
plitude and phase) on some surface (e.g., an illuminated aperture or grating), what is
the wave distribution at some other point in space (i.e., which interference pattern will
we see on a screen behind that aperture/grating)? Therefore, we need to establish a
mathematical relationship between the values of a function (obeying a particular dif-
ferential equation) on a surface of a volume and the values of that function inside the
volume.
In the following, we will consider only scalar diffraction theory, i.e. we neglect the
vector nature of light. This is an excellent approximation if the coupling between vector
components of the EM-field can be ignored. We further assume
• homogeneous material: ǫ(r) ≡ ǫ (constant),

41
3 Diffraction Theory

• isotropic material: ǫ ≡ ǫ1,

=

• no strong coupling from boundaries.

The last assumption means that we look at the field far away from boundaries and that
the influence of these boundaries on the diffraction process is so weak, such that no
coupling between the vector components is introduced and thus the scalar theory stays
valid.
We will restrict our discussion to monochromatic waves, i.e. the scalar EM-field of our
waves is written as
U (r, t) = U (r)e−iωt . (3.4)

This field then obeys the following scalar wave equation

ǫ 2
△U (r, t) − ∂ U (r, t) = 0, (3.5)
c2 t

where △ = ∂x2 +∂y2 +∂z2 is the Laplacian (i.e., the Laplace operator). Evaluating the time
derivative by entering the above form of the scalar wave, we get the (scalar) Helmholtz
equation
(△ + k 2 )U (r) = 0 (3.6)

with
ω√ 2π
k := ǫ= . (3.7)
c λ
Here, λ denotes the wavelength in the dielectric medium. This is the equation the
envelope function U (r) has to obey in the stationary case along with the boundary
conditions that are particular to each physical setup.
Now we will derive the connection between function values at different positions in
space for this envelope function U (r).

3.2.1 Mathematical Interlude

Green’s Theorem

Let u(r), g(r) be complex valued functions with single-valued and continuous first and
second derivatives. V denotes a volume that is bounded by the closed surface S. Then
y x
d3 r (u△g − g△u) = d2 r (u∂n g − g∂n u), (3.8)
V S

where ∂n u is the normal derivative in the outward normal direction ∂n u := n̂ · ∇u and

n̂ is the outward normal direction at each point of the surface S.

42
 3.2 Scalar Diffraction Theory

Green’s Function
Consider the differential equation

a2 ∂x2 f + a1 ∂x f + a0 f = V (x) (3.9)

with certain boundary conditions for a given inhomogeneity V (x). The Green’s function
of the above ODE is defined as the response of the system (whose dynamics are described
by the differential equation above) to an impulse δ(x − x′ ), that is subject to the same
boundary conditions:

a2 ∂x2 G(x, x′ ) + a1 ∂x G(x, x′ ) + a0 G(x, x′ ) = δ(x − x′ ). (3.10)

Then, for any given right hand side V (x), the solution of the initial problem can simply
be obtained by evaluating the integral
Z
f (x) = dx′ G(x, x′ )V (x′ ). (3.11)

Note that there are usually various Green’s functions G(x, x′ ) that solve the above
differential equation, but only one (or a few) that solve the boundary value problem (i.e.,
solve the differential equation and are compatible with the given boundary conditions).
Only then does the integral also yield a solution to the boundary value problem for f (x).
In the following, we derive results corresponding to different assumptions about the
Green’s function of the problem. For the Helmholtz equation in free space (the differ-
ential equation) and the field being zero at infinity (the boundary condition), we have
′
′ eik|r−r |
G(r, r ) = . (3.12)
|r − r′ |
In optics, Green’s function is known under various names:
• Impulse Response

• Point Spread Function

• Green’s Function

3.2.2 Integral Theorem of Helmholtz and Kirchhoff

The aim here is a rigorous derivation of the Huygens-Fresnel principle. We use the
Green’s function defined above, but have to exclude a tiny volume around the point of
singularity r (where G has a pole) since otherwise we could not apply Green’s theorem
(see Fig. 3.3). Hence, instead of the volume V bounded by the surface S of interest, we
look at V ′ bounded by S ′ , which is basically V without a small sphere of radius ǫ. At
the end, however, we will consider the limit of this tiny volume going to zero (i.e., take
the limit ǫ → 0) which does indeed exist and we arrive at an expression for the field
valid in all of V .

43
3 Diffraction Theory

Figure 3.3: The surfaces S and Sǫ enclosing the volume V ′ of integration with
exemplary outward surface normals n̂.

So, within the volume V ′ , the field U and the Green’s function G obey the Helmholtz
equation:
(△ + k 2 )G(r, r′ ) = 0, (3.13)
(△ + k 2 )U (r) = 0, r∈V .′
(3.14)
Applying Green’s theorem in V ′ for these two functions yields (Note that the integration
as well as the differentiation ∂n is with respect to the primed spatial coordinates here
and in the following!)
x  
ds′ U (r′ )∂n G(r, r′ ) − G(r, r′ )∂n U (r′ ) = 0, (3.15)
S′
x x
′
=⇒ − ds′ (U ∂n G − G∂n U ) = ds′ (U ∂n G − G∂n U ). (3.16)
S =S∪Sǫ
Sǫ S

Here and in the following, we write ds′ = d2 r′ for the surface integral (this is just
notation, no additional math is involved). This has nothing to do in particular with the
surface S ′ .

On S: The normal derivative with respect to r′ of the Green’s function is given as

′
eik|r−r |
G(r, r′ ) = , (3.17)
|r − r′ |
  ik|r−r′ |
′ ′

1 e
∂n G(r, r ) = cos ∠(n̂, r − r ) ik − . (3.18)
|r − r′ | |r − r′ |

On Sǫ : Here we can exploit the explicit spherical shape of Sǫ with constant radius
ǫ = |r − r′ | and get
eikǫ
G(r, r′ ) = , (3.19)
ǫ  
eikǫ 1
∂n G(r, r′ ) = − ik . (3.20)
|{z}
ǫ ǫ
cos ∠(n̂,r−r′ ) =−1

44
 3.3 Kirchhoff Formulation of Diffraction by a Planar Screen

Plugging these relations into (3.16), we find for the left hand side
x x  ikǫ

1
 ikǫ

′ ′ ′ e ′ e
ds (U ∂n G − G∂n U ) = ds U (r ) − ik − ∂n U (r ) (3.21)
ǫ ǫ ǫ
Sǫ Sǫ

Now, only U (r′ ) and ∂n U (r′ ) still depend on the integration variable r′ , the remaining
terms are constant. The sphere with radius ǫ is centered around the position r. When
choosing ǫ smaller and smaller, the integrals over these functions essentially become
area of sphere × U (r) and area of sphere × ∂n U (r) by the first mean value theorem for
integration. Hence, we get
x ǫ≪1
 
eikǫ 1

eikǫ

′ 2
ds (U ∂n G − G∂n U ) ≃ 4πǫ U (r) − ik − ∂n U (r) (3.22)
ǫ ǫ ǫ
Sǫ
ǫ→0
−→ 4πU (r). (3.23)
Thus, after taking the limit ǫ → 0, (3.16) becomes the following statement valid for all
points r ∈ V :
1 x ′h
′ ′
eik|r−r | eik|r−r | i
U (r) = ds ∂n U (r′ ) − U (r ′
)∂ n . (3.24)
4π |r − r′ | |r − r′ |
S

Note the remarkable thing, that we have accomplished now. We can compute the func-
tion values U (r) for r ∈ V when all we know are the function values U (r′ ) on the
boundary surface S of V , i.e. for r′ ∈ S. The connection between the function values at
these two positions in space is governed by the Green’s function G that belongs to the
differential equation U obeys.
This theorem of Helmholtz and Kirchhoff is the central result of scalar diffraction the-
ory on which we base our subsequent discussions. If we further impose some restrictions
on U , we can manage to shrink the large surface area S to just some relevant portion
of space (some small finite aperture, e.g.) and ignore the remaining part of the integral
that contributes nothing to the values of U (r), making the evaluation of the integral
feasible. These restrictions will be discussed in the following.

3.3 Kirchhoff Formulation of Diffraction by a Planar

Screen
We consider now the situation of Fig. 3.4, where a monochromatic plane wave is incident
from the left side of an opaque screen with an aperture. We want to use (3.24) to compute
the resulting wave envelope U (r) on the right side of the aperture, where the surface
S = S1 ∪ S2 consists of a plane S1 and a spherical shell S2 , much like a bowl.
As alluded to above, the problem now is that the Helmholtz-Kirchhoff theorem requires
a closed surface. It would be way more convenient, if we just specified the value of U
and its normal derivative on the screen S1 . Hence, we have to consider the infinite half-
sphere S2 on the r.h.s of the screen and derive appropriate boundary conditions which
make the contribution from this part of the surface vanish.

45
3 Diffraction Theory

Figure 3.4: Kirchhoff formulation of diffraction by a plane screen. A monochro-

matic plane wave impinges from the left side of the screen with the
aperture area Σ. We are interested in the resulting field U (r) on the
right side of the screen due to diffraction. It will be shown, that the
contributions to the integral (3.24) from the sphere surface S2 vanish.

Consider S2 : Here’s the idea: Let’s make the spherical shell infinitely large, i.e., take
the limit R → ∞. Since the sphere surface area is of the order R2 , we then may neglect
all contributions of the integrands that decay faster than R−2 (but retain contributions of
exactly the order of R−2 ). The arguments here are motivated by physical considerations1
as well as mathematically rigorous derivations and can be looked up also in Goodman
[3]. Let’s first have a look at the functions U and G that occur as integrands in (3.24).
For the modulus of G we have
′
eik|r−r |
|G| = (3.25)
|r − r′ |
1
= (3.26)
|r − r′ |
1
= , (3.27)
R
because the modulus of the complex exponential is exactly 1 and by Fig. 3.4 the distance
from r is defined as the radius R of the spherical surface S2 . G itself is an outward
propagating spherical wave. From physical intuition (i.e., energy conservation) it is
sensible to assume, that the wave part U (r) also decays as fast as a spherical wave with
the distance from the aperture, which is basically R when R becomes large (i.e., much
larger than r, the distance of the point of observation from Σ). This will additionally
be justified a posteriori below. Thus, we know that the functions G and U decay as
R−1 , which allows us to throw away those parts of the outward normal derivatives that
decay faster than R−1 . The product will then be of the order R−2 and constitute the
only significant contribution to the surface integral on S2 .
1
Also known as handwaving.

46