CUST 2023-2024

Physical Optics

Session 9

Diffraction
Summary

In this session we discuss the notion of diffraction, which is deeply connected with
the notion of interference. We introduce the Huygens-Fresnel principle, as well as its
mathematical formulation in the form of the Huygens-Fresnel diffraction integral (or
formula). We illustrate how this diffraction integral allows to predict the existence
of a bright spot behind a circular obstacle: this is the so-called Poisson spot, which
is a historically important problem. We also discuss how the diffraction integral
can be simplified under certain approximations, namely the Fresnel and Fraunhofer
approximations.

i
Contents

1 The Huygens-Fresnel principle and diffraction integral 1

1.1 The principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Huygens-Fresnel diffraction integral . . . . . . . . . . . . . . . . 2
1.3 Babinet’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Poisson’s spot 7
2.1 Field behind a circular aperture . . . . . . . . . . . . . . . . . . . . . 7
2.2 Field and intensity behind a circular block . . . . . . . . . . . . . . . 9

3 The Fresnel and Fraunhofer approximations 11

3.1 The Fresnel approximation . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 The Fraunhofer approximation . . . . . . . . . . . . . . . . . . . . . . 12

ii
Chapter 1

The Huygens-Fresnel principle and

diffraction integral

1.1 The principle

Diffraction is intimately connected to the notion of interference. Actually, as Feyn-
man wrote in his famous lectures:
“no one has ever been able to define the difference
between interference and diffraction satisfactorily. It is
just a question of usage, and there is no specific,
important physical difference between them”

Diffraction can nonetheless be viewed as follows: it accounts for the phenomena in

which light deviates from following straight-line paths in the absence of any reflec-
tion and/or refraction effects. For instance, diffraction typically accounts for the
possibility of observing some light (i.e. a nonzero intensity) even in the geometrical
shadow of an obstacle (such as the famous Poisson spot, which we’ll discuss later in
chapter 2).
Just like interference, diffraction is a distinct wave phenomenon. As such, it
hence applies of course to light waves, but also to any other kind of waves: water
waves, sound waves, etc. . . In order to provide some physical intuition, some picture
of why waves can somehow deviate from straight-line paths in order to “go around”
obstacles, Huygens proposed the following principle, which was later completed by
Fresnel, hence making it what is now known as the Huygens-Fresnel principle:

Each point of a wavefront may be viewed as the source of

a secondary, spherical wavelet. These secondary wavelets,
emerging from each point of a wavefront, may interfere
constructively and destructively so as to form interference
(or diffraction) fringes.

1
CHAPTER 1. THE HUYGENS-FRESNEL PRINCIPLE. . . 2

While Huygens did not say why such secondary wavelets are emitted by a point
of a wavefront, his principle has the merit of providing a picture so as to why we
should, for instance, expect seeing light even in the geometrical shadow behind
an obstacle: it’s because of such secondary spherical wavelets that propagate in
all forward directions. This can indeed be viewed for a plane wave (i.e. plane
wavefronts) impinging on an obstacle: the resulting spherical wavelets spread also
behind the obstacle, as is illustrated in figure 1.

obstacle
secondary
spherical
wavelets

MY

tea
spherical
waves can

spread behind
plane the obstacle
wavefronts

Figure 1: Schematic illustration of the Huygens-Fresnel principle for a plane wave

impinging on an opaque obstacle.

While Huygens proposed this physical picture in the 17th century, it took almost
two centuries until Fresnel gave some mathematical ground to this principle: this
resulted in the so-called Huygens-Fresnel diffraction integral (or formula).

1.2 The Huygens-Fresnel diffraction integral

Here we consider the following problem, which is schematized in figure 2: mono-

chromatic light, of wavelength λ = 2π/k, is incident on a planar screen, located at
position z = 0, so that the screen hence spans the (x, y) plane. This screen is pierced
with an aperture, but is otherwise perfectly opaque, i.e. no light can go through the
screen except through the aperture. The aperture hence corresponds to some region
CHAPTER 1. THE HUYGENS-FRESNEL PRINCIPLE. . . 3

Incident light Screen

Wavelength I ECK y z
I

lit R x n'la

y y y
zig
I

Figure 2: Schematic representation of the problem of diffraction of light through an

aperture. Such a physical situation is described by the Huygens-Fresnel diffraction
integral (1.1).

of the (x, y) plane. Light can thus go through this aperture, and keeps propagating
further on the right of this screen, i.e. to positions z > 0. For simplicity, from now
on we neglect the vectorial nature of the electric field that is associated with light,
and merely consider a scalar 1 electric field E(x, y, z).

We then ask the question:

Can we relate the electric field E(x, y, z) at an observation point

(x, y, z) on the right of the aperture to the electric field E(x′ , y ′ , 0)
inside the aperture?

The answer to this question is given by the Huygens-Fresnel diffraction integral, or

Huygens-Fresnel diffraction formula, which reads

i eikR
E(x, y, z) = − dx′ dy ′ E(x′ , y ′ , 0) , (1.1)
λ R
aperture

1
We can for instance view this as having a linearly polarized monochromatic plane wave incident
on the aperture, which then remains linearly polarized: we hence merely focus on the amplitude
(which is indeed by definition a scalar quantity) of this vector field.
CHAPTER 1. THE HUYGENS-FRESNEL PRINCIPLE. . . 4

where
p
R≡ (x − x′ )2 + (y − y ′ )2 + z 2 (1.2)

merely denotes the distance that separates the observation point (x, y, z) and the
point (x′ , y ′ , 0) inside the aperture.
The equation (1.1) is the mathematical formulation of the Huygens-Fresnel prin-
ciple. Indeed:

i) each point (x′ , y ′ , 0) inside the aperture can be viewed as the source of a sec-
ondary, spherical wavelet eikR /R, with the same wavelength λ = 2π/k as the
incident light;

ii) each of these secondary wavelets is weighted by the electric field E(x′ , y ′ , 0) at
the point of the aperture which is the source of this wavelet;

iii) R can also be seen as the radius of the spherical wavelet (originating from the
point (x′ , y ′ , 0) inside the aperture) that intersects the plane, located at position
z along the z axis, at the point (x, y, z);

iv) by integrating over the aperture, we take into account the contribution of all
these secondary spherical wavelets to the field at the point (x, y, z).

We emphasize that the diffraction integral (1.1) must be seen as an approxima-

tion: it is actually valid when the vector

R ≡ (x − x′ )x̂ + (y − y ′ )ŷ + z ẑ , (1.3)

which joins a point (x′ , y ′ , 0) inside the aperture to the observation point (x, y, z),
and the vector ẑ make an angle θR,ẑ that is sufficiently small. This means that (1.1)
is accurate for observation points (x, y, z) such that

x − x′ ≪ z and y − y′ ≪ z , ∀x′ , y ′ ∈ aperture . (1.4)

This can be shown from a more general version of (1.1), which was obtained
by Kirchhoff. This more general version is hence known as the Fresnel-Kirchhoff
diffraction integral, which reads2

i ′ ′ ′ ′ eikR 1 + cos θR,ẑ
E(x, y, z) = − dx dy E(x , y , 0) . (1.5)
λ R 2
aperture

2
The proof of (1.5) actually naturally shows why we must have the −i/λ factor.
CHAPTER 1. THE HUYGENS-FRESNEL PRINCIPLE. . . 5

The additional factor (1 + cos θR,ẑ )/2 in (1.5) is known as the obliquity factor : to
neglect it, i.e. to say that it is basically 1 (i.e. assuming that (1.4) is satisfied),
indeed gives back the Huygens-Fresnel integral (1.1).
Since (1.5) is more general than (1.1), it is also mandatorily more difficult to
handle. Therefore, here we’ll stick to the Huygens-Fresnel diffraction integral (1.1).
Actually, even (1.1) is unfortunately still difficult to handle: this led Fresnel (and
others) to use additional approximations to try to simplify it. We will discuss two
such approximations in chapter 3 below.

1.3 Babinet’s principle

Depending on the geometrical shape of the aperture that we consider, it may be dif-
ficult to parametrize the double integral involved in the Huygens-Fresnel diffraction
integral (1.1). It may however be the case that a given aperture can be divided into
simpler apertures.
For instance, a circular annulus can be viewed as a “difference” of two disks.
Indeed, let’s call R the annulus that is delimited by two circles of radii r1 and r2 ,
with r1 < r2 (both circles have the same center). Let’s then call D1 and D2 the disks
of radii r1 and r2 . We can then formally write:

R = D2 − D1 . (1.6)

We can then write the electric field, call it ER (x, y, z), that results from light going
through the aperture R in terms of the fields ED1 and ED2 that result from light
separately going through the aperture D1 and D2 , and we have

ER (x, y, z) = ED2 (x, y, z) − ED1 (x, y, z) . (1.7)

The same logic then applies in order to describe diffraction from obstacles rather
than diffraction through apertures. Therefore, suppose for concreteness that a
monochromatic plane wave E0 eik·r impinges onto an opaque disk D of radius r.
Let’s call Eblock (x, y, z) the resulting electric field at a point (x, y, z) behind the
block. This field is then merely the difference between the free, unblocked plane
wave E0 eik·r and the field, let’s call it Eap , that results from light going through the
corresponding disk aperture (and not obstacle!), that is

Eblock (x, y, z) = E0 eik·r − Eap (x, y, z) , (1.8)

where we emphasize that Eap is thus obtained from the Huygens-Fresnel diffraction
integral (1.1) for a disk aperture. This remains valid for any kind of obstacles, and
CHAPTER 1. THE HUYGENS-FRESNEL PRINCIPLE. . . 6

the resulting relation of the form (1.8) is known as the Babinet principle.
This allows us in particular to treat an important historical example for which
the diffraction integral (1.1) can be computed, and is then seen to give rise to a
surprising phenomenon: the so-called Poisson spot.
Chapter 2

Poisson’s spot

Here our objective is to find the intensity behind a circular block (i.e. an opaque disk)
of diameter D. A monochromatic plane wave of wavelength λ = 2π/k, propagating
along z, is assumed to impinge on this block from the left. We assume that the
origin of our Cartesian system of coordinates is taken at the center of this block.
The z axis is then chosen perpendicular to the plane of the block (the latter hence
being in the (x, y) plane), and oriented to the right of the block.
We now want to determine the intensity at a point (0, 0, z), that is the intensity
on the z axis behind the block. To do this, we make use of the Babinet principle: the
latter hence first requires to compute the field at a point (0, 0, z) behind a circular
aperture.

2.1 Field behind a circular aperture

The field Eap at a point (0, 0, z) behind a circular aperture is given by the Huygens-
Fresnel diffraction integral (1.1). Therefore, rewriting (1.1) and (1.2) here yields

i eikR
Eap (0, 0, z) = − dx′ dy ′ E(x′ , y ′ , 0) , (2.1)
λ R
circular
aperture

with
p
R= x′2 + y ′2 + z 2 . (2.2)

First of all, the incident light is a monochromatic plane wave of wavelength

λ = 2π/k that propagates along z: it is thus of the form E0 eikz . Therefore, inside
the aperture, which is by construction located at z = 0, the amplitude of this

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CHAPTER 2. POISSON’S SPOT 8

incident plane wave is thus merely E0 , that is

E(x′ , y ′ , 0) = E0 . (2.3)

Then, because the aperture is circular, it seems reasonable to work with polar in-
tegration variables (ρ, θ) rather than the Cartesian ones (x′ , y ′ ). These coordinates
are connected through

x′ = ρ cos θ and y ′ = ρ sin θ . (2.4)

Then, the infinitesimal surface element dx′ dy ′ reads, in polar coordinates,

dx′ dy ′ = ρ dρ dθ . (2.5)

Finally, the integration limits in polar coordinates are given by

ρ ∈ [0, D/2] and θ ∈ [0, 2π] . (2.6)

Since in view of (2.2) and (2.4) we have

q p
R= ρ2 cos2 θ + ρ2 sin2 θ + z 2 = ρ2 + z 2 , (2.7)

we get upon combining (2.1) with (2.3)-(2.7)

  √
D/2 2π 2 2
i eik ρ +z
Eap (0, 0, z) = − dρ dθ ρE0 p , (2.8)
λ 0 0 ρ2 + z 2

where the integral with respect to the angle θ is straightforward to compute, so that
we get
 √
D/2 2 2
2iπ eik ρ +z
Eap (0, 0, z) = − E0 dρ ρ p . (2.9)
λ 0 ρ2 + z 2

Now, note that

√
λ ∂ ik√ρ2 +z2
2 2
eik ρ +z
ρp = e . (2.10)
ρ2 + z 2 2iπ ∂ρ

Substituting (2.10) into (2.9) hence yields

 √ 
2 2
Eap (0, 0, z) = −E0 eik z +D /4 − eikz . (2.11)

We can now use (2.11) to compute the field Eblock (0, 0, z) behind a circular block.
CHAPTER 2. POISSON’S SPOT 9

2.2 Field and intensity behind a circular block

Let’s now apply the Babinet principle (1.8) in order to compute the electric field
Eblock (0, 0, z) (and then the corresponding intensity) behind a circular block. We
have from (1.8), because the incident plane wave propagates along z here,

Eblock (0, 0, z) = E0 eikz − Eap (0, 0, z) , (2.12)

that is in view of the expression (2.11) of Eap

√
z 2 +D2 /4
Eblock (0, 0, z) = E0 eik . (2.13)

We can now easily compute the corresponding intensity I(0, 0, z) of the light
behind the circular block. Since we have by definition

I(0, 0, z) ≡ |Eblock (0, 0, z)|2 , (2.14)

we immediately obtain from (2.13)

I(0, 0, z) = |E0 |2 , ∀z > 0 . (2.15)

Note that I(0, 0, z) is thus exactly the same as the intensity of the incident plane
wave. In other words, the intensity of the light that we observe along the z axis
behind a circular obstacle is the same as the intensity of the incident light that we
send on this obstacle. In particular, this means that we have a nonzero intensity in
the geometrical shadow behind a circular obstacle, a region where we would naively
expect to have no light!
This fact is of historical importance. Indeed, in 1817, the French Academy of
Sciences organized a call for manuscripts in order for candidates to apply for the
biannual Great Prize. The subject of this call was: “the diffraction of light”. They
received two manuscripts, one of them being from Fresnel. In his manuscript, Fresnel
developed his theory of diffraction, which was based on viewing light as a wave.
Now, remember that in 1817, we were still far from having Maxwell’s equations
(which came like 50 years later). Furthermore, Young only performed his double-
slit experiment a few years before. Therefore, there were still many proponents of
Newton’s theory of light (hence viewing light as particles rather than as a wave).
And actually, there were proponents of Newton’s theory in the very panel of the
French Academy of Sciences in charge of evaluating Fresnel’s manuscript. Among
these proponents was Poisson: it is him who saw that Fresnel’s theory would lead
to the result (2.15) regarding the intensity of light behind a circular block. This
led Poisson to consider this prediction as a proof that Fresnel’s theory (and more
CHAPTER 2. POISSON’S SPOT 10

generally the wave theory of light itself) could not be true. But, another member
of the panel, named Arago, actually performed the experiment in order to test
the prediction (2.15): Arago indeed observed the occurrence of a bright spot (i.e.,
a nonzero intensity) in the geometrical shadow behind a circular block! At the
beginning of the 19th century, this provided yet another striking argument in favor
of the wave theory of light!
In recognition of the importance of the conclusion drawn from the result (2.15),
the bright spot observed behind a circular block is referred to as Poisson’s spot, or
else as Arago’s spot or Fresnel’s spot.
Chapter 3

The Fresnel and Fraunhofer

approximations

The Huygens-Fresnel diffraction integral (1.1) proves to be rather difficult to evalu-

ate analytically, except in some specific cases. Therefore, in order to make such an
evaluation more tractable (but also of course, mandatorily, less general), approxi-
mations are necessary. Two common such approximations are the so-called Fresnel
and Fraunhofer approximations.

3.1 The Fresnel approximation

The Fresnel approximation amounts to focus on small angles, that is we focus on
positions x, y, z that are such that

z 2 ≫ (x − x′ )2 + (y − y ′ )2 , ∀x′ , y ′ ∈ aperture . (3.1)

This allows, in view of (1.2), to rewrite R in the form

r
(x − x′ )2 + (y − y ′ )2
R=z 1+ ,
z2

which we can now Taylor expand around 0 because of the approximation (3.1), and
we get

(x − x′ )2 + (y − y ′ )2
R=z+ + ... (3.2)
2z

The expansion (3.2) now allows to simplify the diffraction integral (1.1). Since 1/R
varies more slowly than the exponential eikR , we can restrict to the zeroth order

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CHAPTER 3. THE FRESNEL AND FRAUNHOFER. . . 12

in (3.2) and write

1 1
≈ . (3.3)
R z

However, we must keep the first-order term in (3.2) for the exponential term, and
we hence write

x2 +y 2 x′2 +y ′2 xx′ +yy ′

eikR ≈ eikz eik 2z eik 2z e−ik z (3.4)

Substituting (3.3) and (3.4) into (1.1) yields the so-called Fresnel approximation of
the diffraction integral

i x2 +y 2 x′2 +y ′2 xx′ +yy ′
E(x, y, z) ≈ − eikz eik 2z dx′ dy ′ E(x′ , y ′ , 0) eik 2z e−ik z . (3.5)
λz
aperture

3.2 The Fraunhofer approximation

The Fraunhofer approximation amounts to focus on distances along the z axis that
are large as compared to the dimensions of the aperture: this is the so-called far
field. The Fraunhofer approximation is actually an approximation. . . of the Fresnel
approximation! This will yield yet a simpler expression of the diffraction integral.
The basic idea is to further simplify the Fresnel approximation (3.5) by writing

x′2 +y ′2
eik 2z ≈ 1. (3.6)

If we call D the typical dimension of the aperture (for instance the diameter for a
circular aperture), then (3.6) is justified in the so-called far-field regime described
by the condition

kD2
z≫ . (3.7)
2

We then get from (3.5) and (3.6) the so-called Fraunhofer diffraction integral

i x2 +y 2 xx′ +yy ′
E(x, y, z) ≈ − eikz eik 2z dx′ dy ′ E(x′ , y ′ , 0) e−ik z . (3.8)
λz
aperture