Diffraction

1
Diffraction
SOLO HERMELIN
Updated: 16.01.10
4.01.15
http://www.solohermelin.com

2
Table of Content
SOLO Optics - Diffraction
James Gregory Diffraction Grating
History of Diffraction
Francesco Grimaldi
Huygens Principle
Thomas Young Interference
Laplace, Biot, Poisson, Arago, Fresnel - Particle versus Wave Theory
Fresnel – Huygens’ Diffraction Theory
Fraunhofer Diffraction Theory
Fresnel-Kirchhoff Diffraction Theory
Complementary Apertures. Babinet Principle
Rayleigh-Sommerfeld Diffraction Formula
Extensions of Fresnel-Kirchhoff Diffraction Theory
Fresnel and Fraunhofer Diffraction Approximations

3
SOLO Diffraction
Fraunhofer Diffraction and the Fourier Transform
Phase Approximations – Fresnel (Near-Field) Approximation
Phase Approximations – Fraunhofer (Near-Field) Approximation
Fraunhofer Diffraction Approximations Examples
Resolution of Optical Systems
Fresnel Diffraction Approximations Examples
Kirchhoff’s Solution of the Scalar Helmholtz Nonhomogeneous Differential Equation
Two Dimensional Fourier Transform (FT)
Appendices

4
SOLO Diffraction
The term Diffraction has been defined by Sommerfield
as any deviation of light rays from rectilinear paths
which cannot be interpreted as reflection or refraction.
Sommerfeld, A., “Optics, Lectures on Theoretical Physics”, vol. IV, Academic Press Inc.,
New York, 1954, Chapter V, “The Theory of Diffraction”, pg. 179,
english translation of
Arnold Johannes
Wilhelm
Sommerfeld
1868 - 1951
Sommerfeld, A. , “Mathematische Theorie der Diffraction”, Math Ann., 1896
Table of Content

5
SOLO Diffraction - History
The Grimaldi’s description of diffraction was published in
1665 , two years after his death: “Physico-Mathesis de lumine,
Coloribus et iride”
Francesco M. Grimaldi, S.J. (1613 – 1663) professor of mathematics and physics at the
Jesuit college in Bolognia discovered the diffraction of light and gave it the name
diffractio, which means “breaking up”.
http://www.faculty.fairfield.edu/jmac/sj/scientists/grimaldi.htm
“When the light is incident on a smooth white surface it will
show an illuminated base IK notable greater than the rays
would make which are transmitted in straight lines through
the two holes. This is proved as often as the experiment is
trayed by observing how great the base IK is in fact and
deducing by calculation how great the base NO ought to be
which is formed by the direct rays. Further it should not be
omitted that the illuminated base IK appears in the middle
suffused with pure light, and either extremity its light is
colored.”
Single Slit
Diffraction
Double Slit
Diffraction
http://en.wikipedia.org/wiki/Francesco_Maria_Grimaldi
Table of Content

6
SOLO
Huygens Principle
Christiaan Huygens
1629-1695
Every point on a primary wavefront serves the source of spherical
secondary wavelets such that the primary wavefront at some later
time is the envelope o these wavelets. Moreover, the wavelets
advance with a speed and frequency equal to that of the primary
wave at each point in space.
“We have still to consider, in studying the
spreading of these waves, that each particle of
matter in which a wave proceeds not only
communicates its motion to the next particle to it,
which is on the straight line drawn from the
luminous point, but it also necessarily gives a motion
to all the other which touch it and which oppose its
motion. The result is that around each particle
there arises a wave of which this particle is a
center.”
Huygens visualized the propagation of light in
terms of mechanical vibration of an elastic
medium (ether).
Diffraction - History
Table of Content

7
SOLO
James Gregory (1638 – 1675) a Scottish mathematician and
astronomer professor at the University of St. Andrews and the
University of Edinburgh discovered the diffraction grating by
passing sunlight through a bird feather and observing the
diffraction produced.
http://en.wikipedia.org/wiki/James_Gregory_%28astronomer_and_mathematician%29
http://microscopy.fsu.edu/optics/timeline/people/gregory.html
Table of Content
1661

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SOLO
M.C. Hutley, “Diffraction Gratings”, Academic Press., 1982, p. 3
Diffraction Gratings
1786
The invention of Diffraction Gratings is ascribed to David Rittenhouse who in 1786
had been intriged by the effects produced when viewing a distant light source through
a fine handkerchief.
In order to repeat the phenomenon under controlled
conditions, he made up a square of parallel hairs laid across
two fine screws made by a watchmaker. When he looked
through this at a small opening in the window shutter of a
darkened room, he saw three images of approximately equal
brightness and several others on either side “fainter and
growing more faint, coloured and indistinct, the further they
were from the main line”. He noted that red light was bent
more than blue light and ascribed these effects to
diffraction.
http://experts.about.com/e/d/da/david_rittenhouse.htm
David Rittenhouse
1732 - 1796

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Optics - History
SOLO
History (continue)
In 1801 Thomas Young uses constructive and destructive interference
of waves to explain the Newton’s rings.
Thomas Young
1773-1829
1801-1803
In 1803 Thomas Young explains the fringes at the edges of shadows
using the wave theory of light. But, the fact that was belived that the
light waves are longitudinal, mad difficult the explanation of double
refraction in certain crystals.
Table of Content

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SOLO
“Between 1805 and 1815 Laplace, Biot and (in part) Malus created an elaborate
mathematical theory of light, based on the notion that light rays are streams of particles
that interact with the particles of matter by short range forces. By suitably modifying
Newton’s original emission theory of light and applying superior mathematical
methods, they were able to explain most of the known optical phenomena, including
the effect of double refraction which had been the focus of Huyghen’s work.
http://microscopy.fsu.edu/optics/timeline/people/gregory.html
http://www.schillerinstitute.org/fid_97-01/993poisson_jbt.html
Pierre-Simon Laplace
(1749-1827)
In 1817, expecting to soon celebrate the final triumph of their neo-Newtonian optics’
Laplace and Biot arranged for the physics prize of the French Academy of Science to
be proposed for the best work on theme of diffraction – the apparent bending of light
rays at the boundaries between different media.”

11
SOLO
Fraunhofer’s solar dark lines
In 1813 Joseph Fraunhofer rediscovered William Hyde Wollaston’s
dark lines in the solar system, which are known as Fraunhofer’s lines.
He began a systematic measurement of the wavelengths of the solar
Spectrum, by mapping 570 lines.
http://www.musoptin.com/spektro1.html
1813
Fraunhofer Telescope.
Fraunhofer placed a narrow slit in front of a prism and viewed the spectrum of light
passing through this combination with a small telescope eypiece. By this technique he
was able to investigate the spectrum bit by bit, color by color.

12
POLARIZATION - History
Arago and Fresnel investigated the interference of
polarized rays of light and found in 1816 that two
rays polarized at right angles to each other never
interface.
SOLO
History (continue)
Dominique François
Jean Arago
1786-1853
Augustin Jean
Fresnel
1788-1827
Arago relayed to Thomas Young in London the results
of the experiment he had performed with Fresnel. This
stimulate Young to propose in 1817 that the oscillations
in the optical wave where transverse, or perpendicular
to the direction of propagation, and not longitudinal as
every proponent of wave theory believed. Thomas Young
1773-1829
1816-1817
longitudinal
waves
transversal
waves

13
SOLO
Augustin Jean
Fresnel
1788-1827
In 1818 Fresnel, by using Huygens’ concept of secondary wavelets
and Young’s explanation of interface, developed the diffraction
theory of scalar waves.
1818Diffraction - History

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SOLO
In 1818 August Fresnel supported by his friend André-Marie Ampère submitted to the
French Academy a thesis in which he explained the diffraction by enriching the Huyghens’
conception of propagation of light by taking in account of the distinct phases within each
wavelength and the interaction (interference) between different phases at each locus of the
propagation process.
http://microscopy.fsu.edu/optics/timeline/people/gregory.html http://www.schillerinstitute.org/fid_97-01/993poisson_jbt.html
André-Marie Ampère
(1775-1836)
Dominique François
Jean Arago
1786-1853
Siméon Denis Poisson
1781-1840
Pierre-Simon Laplace
(1749-1827)
Joseph Louis
Guy-Lussac
1778-1850
Judging
Committee
of
French
Academy
1818

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SOLO
http://microscopy.fsu.edu/optics/timeline/people/gregory.html http://www.schillerinstitute.org/fid_97-01/993poisson_jbt.html
Dominique François
Jean Arago
1786-1853
Siméon Denis Poisson an Academy member rise the objection that if the Fresnel
construction is valid a bright spot would have to appear in the middle of the shadow cast by a
spherical or disc-shaped object, when illuminated, and this is absurd.
Soon after the meeting, Dominique Francois Arago, one of the judges for the Academy
competition, did the experiment and there was the bright spot in the middle of the shadow.
Fresnel was awarded the prize in the competition.
Siméon Denis Poisson
1781-1840
Poisson’s or Arago’s Spot

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SOLO
In 1821 Joseph Fraunhofer build the first diffraction grating,
made up of 260 close parallel wires. Latter he built a diffraction
grating using 10,000 parallel lines per inch.
Utzshneider, Fraunhfer, Reichenbach, Mertz
http://www.musoptin.com/fraunhofer.html
1821-1823
In 1823 Fraunhofer published his theory of diffraction.
Table of Content

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SOLO
Dffraction Grating
Diffraction - History 1835
By 1835 at the latest, the physicist F. M. Schwerd was able
to take exact measurements of the visible spectrum with
the aid of such a diffraction grating, and show that red
light has a longer wavelength than blue light, and that
yellow and blue light lie in the middle of the spectrum.
http://colorsystem.com/projekte/engl/16haye.htm
1835 - Schwerd developed a "wave" theory of the diffraction grating.
http://www.thespectroscopynet.com/Educational/Masson.htm
“Die Beugungserscheinungen aus den Fundamentalgesetzen der Undulationstheorie”
http://www.worldcatlibraries.org/wcpa/ow/3e723a9c5ac2a2b2.html
Friedrich Magnus Schwerd
1792 - 1871
http://193.174.156.247/FMSG/wir_ueber_uns/wer_war_Schwerd.php

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SOLO
H.A. Rowland at the John Hopkins University greatly improved diffraction gratings,
introducing curved grating.
Diffraction - History 1882
http://thespectroscopynet.com/educational/Kirchhoff.htm
American physicist who invented the concave diffraction
grating, which replaced prisms and plane gratings in many
applications, and revolutionized spectrum analysis—the
resolution of a beam of light into components that differ in
wavelength.
http://www.britannica.com/eb/article-9064251/Henry-Augustus-Rowland
Henry Augustus Rowland
1848 - 1901
http://chem.ch.huji.ac.il/~eugeniik/history/rowland.html
Rowland gratings
Rowland invented the ruling machine that can engrave as many
as 20,000 lines to inch for diffraction gratings

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SOLO Optics History
Debye-Sears Effect
1932 Nobel Prize in
Chemistry 1936
Peter Josephus Wilhelmus
Debye
1884 – 1966
Diffraction of light by ultrasonic waves.
Francis Weston Sears
1898 - 1975
P. Debye and F. W. Sears, ``On the Scattering of Light by
Supersonic Waves'', Proc. Natl. Acad. Sci. U.S.A. 18, 409
(1932).
Acousto-optic effect, also known in the scientific literature as acousto-
optic interaction or diffraction of light by acoustic waves, was first
predicted by Brillouin in 1921 and experimentally revealed by Lucas,
Biquard and Debye, Sears in 1932.
The basis of the acousto-optic interaction is a more general effect of
photoelasticity consisting in the change of the medium permittivity
under the action of a mechanical strain a. Phenomenologically, this
effect is described as variations of the optical indicatrix coefficients
caused by the strain
http://www.mt-berlin.com/frames_ao/descriptions/ao_effect.htm

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DiffractionSOLO
Augustin Jean
Fresnel
1788-1827
In 1818 Fresnel, by using Huygens’ concept of secondary wavelets
and Young’s explanation of interface, developed the diffraction
theory of scalar waves.
P
0P
Q 1x
0x
1y
0y
η
ξ
Fr

Sr

ρ
 r

O
'θ
θ
Screen
Image
plane
Source
plane
0O
1O
Sn1
Σ
Σ - Screen Aperture
Sn1 - normal to Screen
F
rPP

=0
SrQP

=0
rQP

=
From a source P0 at a distance from a aperture a spherical wavelet
propagates toward the aperture: ( ) ( )Srktj
S
source
Q e
r
A
tU −
= '
' ω
According to Huygens Principle second wavelets will start at the aperture and will add
at the image point P.
( ) ( ) ( )( )
( ) ( )( )
∫∫ Σ
++−
Σ
+−−
== dre
rr
A
Kdre
r
U
KtU rrktj
S
sourcerkttjQ
P
S 2/2/'
',', πωπω
θθθθ
where: ( )',θθK obliquity or inclination factor ( ) ( )SSS nrnr 11cos&11cos' 11
⋅=⋅= −−
θθ
( )
( )


===
===
0',0
max0',0
πθθ
θθ
K
K Obliquity factor and π/2 phase were introduced by Fresnel to explain
experiences results.
Fresnel Diffraction Formula
Fresnel took in consideration the phase of each wavelet to obtain:
Fresnel – Huygens’ Diffraction Theory
Table of Content
Fresnel –Kirchoff
Diffraction Formula

21
SOLO
In 1882 Gustav Kirchhoff, using mathematical foundation,
succeeded to show that the amplitude and phases ascribed to the
wavelets by Fresnel, by enhancing the Huyghen’s Principle, were a
consequence of the wave nature of light.
HBED

µε == &
For an Homogeneous, Linear and Isotropic Medium where
are constant scalars, we have
µε,
t
E
t
D
H
t
t
H
t
B
E
ED
HB
∂
∂
=
∂
∂
=×∇
∂
∂
∂
∂
−=
∂
∂
−=×∇×∇
=
=






εµ
µ
ε
µ
Since we have also
tt ∂
∂
×∇=∇×
∂
∂
t
D
H
∂
∂
=×∇


t
B
E
∂
∂
−=×∇


For Source less
Medium
( )
( ) ( )


















=⋅∇=
∇−⋅∇∇=×∇×∇
=
∂
∂
+×∇×∇
0&
0
2
2
2
DED
EEE
t
E
E




ε
µε
02
2
2
=
∂
∂
−∇
t
E
E


µε
Maxwell Equations are ( ) eJ
t
D
HA



+
∂
∂
=×∇
mBGM ρ=⋅∇

)(
( ) mJ
t
B
EF



−
∂
∂
−=×∇
( ) eDGE ρ=⋅∇

James C. Maxwell
(1831-1879)
Gustav Robert Kirchhoff
1824-1887
Diffraction

22
DiffractionSOLO
0
1
2
2
2
2
=





∂
∂
−∇ U
tv
Scalar Differential Wave Equation
For a monochromatic wave of frequency f ( ω = 2πf ) a solution is:
( ) ( ) ( )[ ] ( ) ( )[ ] ( ){ }tjPjPUPtPUtPU ωφφω expexpRecos, −=+=
Define the phasor ( ) ( ) ( )[ ]PjPUPU φ−= exp
U
v
U
tv 2
2
2
2
2
1 ω
−=
∂
∂
λ
π
π
ω 2
2 ===
v
f
v
k
( ) 022
=+∇ UkPhasor Scalar Differential Wave Equation
This is the Scalar Helmholtz Differential Equation
Hermann von Helmholtz
1821-1894
Boundary Conditions for the Helmholtz Differential Equation:
• Dirichlet (U given on the boundary)
• Neumann (dU/dn given on the boundary)
Johann Peter Gustav
Lejeune Dirichlet
1805-1859
Franz Neumann
1798-1895
εµ
1
0
11 2
2
2
2
2
2
2
2
2
==





∂
∂
−∇=
∂
∂
−∇ vE
tvt
E
v
E



Vector Differential Wave Equation

23
To find the solution of the Scalar Helmholtz Differential Equation we need to use the
following:
• Scalar Green’s Identity
( ) ( )∫∫
→
⋅∇−∇=∇−∇
SV
dSGUUGdVGUUG 22
• Green’s Function
( )
( )
SF
SF
FS
rr
rrkj
rrG 


−
−
=
exp
;
This Green’s Function is a particular solution of the following Helmholtz
Non-homogeneous Differential Equation:
( ) ( ) ( )SFFSFSS rrrrGkrrG

−=+∇ πδ4;; 22
SOLO
Diffraction
provided that and are
continuous in volume V
UUU 2
,, ∇∇ GGG 2
,, ∇∇
Free-Space Green’s Function

n
i
iSS
1=
=
iS
nS
dV
dSnS
→
1
V
Fr

Sr

F
0r SF rrr

−=
PositionSourcerS

PositionFieldrF

( ) 022
=+∇ Uk Scalar Helmholtz Differential Equation

24
SOLO
• Scalar Green’s Identities
( ) ( )∫∫
→
⋅∇−∇=∇−∇
SV
dSGUUGdVGUUG 22
Let start from the Gauss’ Divergence
Theorem
∫∫
→
⋅=⋅∇
SV
dSAdVA

Karl Friederich Gauss
1777-1855
where is any vector field (function of position and time)
continuous and differentiable in the volume V bounded by the
enclosed surface S. Let define .
A

UGA ∇=

( ) UGUGUGA 2
∇+∇⋅∇=∇⋅∇=⋅∇

Then
( ) ( ) ∫∫∫
→
⋅∇=∇+∇⋅∇=∇⋅∇
S
Gauss
VV
dSUGdVUGUGdVUG 2
( ) ( ) ∫∫∫
→
⋅∇=∇+∇⋅∇=∇⋅∇
S
Gauss
VV
dSGUdVGUUGdVGU 2
Subtracting the second equation from the first we obtain
First Green’s Identity
Second Green’s Identity
We have
GEORGE GREEN
1793-1841
Diffraction
To find a general solution of the Scalar Helmoltz Differential
Equation we need to use the
If we interchange with we obtainG U

25
Integral Theorem of Helmholtz and Kirchhoff
( ) ( )( ) ( )F
V
sF
V
SS rUdVUrrUGkUGkdVGUUG

πδπ 442222
−=−−+−=∇−∇ ∫∫
Using:
( ) ( ) ( )SFFSFSS rrrrGkrrG

−=+∇ πδ4;; 22

n
i
iSS
1=
=
iS
nS
dV
dSnS
→
1
V
Fr

Sr

F
0r SF rrr

−=
PositionSourcerS

PositionFieldrF

SOLO
Diffraction
( ) ( ) 0,22
=+∇ SFS rrUk

From the left side of the Second Scalar Green’s Identity we have:
( ) ∫∫ 





∂
∂
−
∂
∂
=⋅∇−∇
→
SS
SS dS
n
G
U
n
U
GdSGUUG
( )
( )
SF
SF
FS
rr
rrkj
rrG 


−
−
=
exp
;Using:
we obtain: ( )
( ) ( )
∫ 















−
−
∂
∂
−
∂
∂
−
−
−=
S SF
SF
SF
SF
F dS
rr
rrkj
n
U
n
U
rr
rrkj
rU 



 expexp
4
1
π
This is the Integral Theorem of Helmholtz and Kirchhoff that enables to calculate
as function of the values of and on the enclosed surface S.nU ∂∂ /UU
Note: This Theorem was developed first by H. von Helmholtz in acoustics.
1821-1894
Gustav Robert Kirchhoff
1824-1887
From the right side of the Second Scalar Green’s
Identity, using we have:dS
n
U
dSnUdSU SSS
∂
∂
=⋅∇=⋅∇
→→
1
Scalar Helmholtz Differential
Equation

26
Sommerfeld Radiation Conditions
SOLO
Diffraction
( )
∫∫
∫
∞+Σ+






∂
∂
−
∂
∂
−=






∂
∂
−
∂
∂
−=
SS
S
F
dS
n
G
U
n
U
G
dS
n
G
U
n
U
GrU
1
4
1
4
1
π
π

P
Fr

Sr

r

Σ
1S
∞S
R
Screen
Aperture
ωd
→
Sn1
→
Sn1
since the condition that the previous integral be finite is:
( ) ( )
R
Rkj
rrG
SFS
exp
; →
∞

Consider the surface of integration ∞+Σ+= SSS 1
1S - on the screen
∞S - hemisphere with radius ∞→R
( ) Gkj
R
Rkj
R
kj
n
G
≅





−=
∂
∂ exp1
∫∫∫ Ω






−
∂
∂
=





∂
∂
−
∂
∂
∞
ωdRUkj
n
U
GdS
n
G
U
n
U
G
S
2
( ) 1
exp
limlim ==
∞→∞→ R
Rkj
RGR
RR
0lim =





−
∂
∂
∞→
Ukj
n
U
R
R
This is Sommerfeld Radiation Conditions
Σ - on the aperture

27
is known as optical disturbance. Being a scalar quantity, it cannot accurately
represent an electromagnetic field. However, the square of this scalar quantity can
be regarded as a measure of the irradiance at a given point.
U
Sommerfeld Radiation Conditions (continue)
SOLO
Diffraction
( )
∫∫
∫
∞+Σ+






∂
∂
−
∂
∂
−=






∂
∂
−
∂
∂
−=
SS
S
F
dS
n
G
U
n
U
G
dS
n
G
U
n
U
GrU
1
4
1
4
1
π
π

0lim =





−
∂
∂
∞→
Ukj
n
U
R
R
This is Sommerfeld Radiation Conditions
This implies that: 0
4
1
=





∂
∂
−
∂
∂
∫∫
∞S
dS
n
G
U
n
U
G
π
and the Integral of Helmholtz and Kirchhoff becomes:
( ) ∫∫Σ+






∂
∂
−
∂
∂
−=
1
4
1
S
F dS
n
G
U
n
U
GrU
π

P
Fr

Sr

r

Σ
1S
∞S
R
Screen
Aperture
ωd→
Sn1
→
Sn1
0P
Q
Arnold Johannes
Wilhelm
Sommerfeld
1868 - 1951

28
The Kirchhoff Boundary Conditions
SOLO
Diffraction
Kirchhoff assumed the following boundary conditions:
( ) ∫∫Σ






∂
∂
−
∂
∂
−= dS
n
G
U
n
U
GrU F
π4
1
1. The field distribution and its derivative ,
across the aperture , are the same as in the
absence of the screen.
U nU ∂∂ /
Σ
2. On the shadowed part of the screen and0
1
=S
U
0/
1
=∂∂ S
nU
The Integral of Helmholtz and Kirchhoff becomes:
The field at point P is the superposition of the aperture values 0=Σ
U 0/ =∂∂ Σ
nU
Note:
Moreover, mathematically the condition implies0/&0
11
=∂∂= SS
nUU 0/&0 =∂∂= ΣΣ
nUU
However, if the dimensions of the aperture are large relative to the
wavelength λ, the integral agrees well with the experiment.
P
Fr

Sr

r

Σ
1S
∞S
R
Screen
Aperture
0P 0,0
1
1
=
∂
∂
=
S
S n
U
U
Σ
Σ ∂
∂
n
U
U ,
Q
→
Sn1
→
Sn1
Sθ
θ
Kirchhoff boundary conditions are not physical since the presence of the screen
changes field values on the aperture and on the screen.

29
Fresnel-Kirchhoff Diffraction Formula
SOLO
Diffraction
( ) ∫∫Σ






∂
∂
−
∂
∂
−= dS
n
G
U
n
U
GrU F
π4
1
The Integral of Helmholtz and Kirchhoff:
ΣAssume that the aperture is illuminated by a single
spherical wave:
( ) ( )
S
Ssource
S
r
rkjA
rU
exp
=

( ) ( ) ( )
( )






⋅





−=
⋅





∇=⋅∇=
∂
∂
→→
→→
SS
S
Ssource
S
S
S
Ssource
SSSS
S
nr
r
rkjA
r
kj
n
r
rkjA
nrU
n
rU
11
exp1
1
exp
1


( )
( )
SF
SF
FS
rr
rrkj
rrG 


−
−
=
exp
;
( ) ( )
( )
( )






⋅





−−=
⋅








−
−
∇=⋅∇=
∂
∂
→→
→=−→
S
S
SF
SF
S
rrr
SFSS
FS
nr
r
rkj
r
kj
n
rr
rrkj
nrrG
n
rrG SF
11
exp1
1
exp
1,
,



 
P
Fr

Sr

r

Σ
1S
∞S
R
Screen
Aperture
0P 0,0
1
1
=
∂
∂
=
S
S
n
U
U
Σ
Σ ∂
∂
n
U
U ,
Q
→
Sn1
→
Sn1
Sθ
θ

30
SOLO
Diffraction
( ) ∫∫Σ






∂
∂
−
∂
∂
−= dS
n
G
U
n
U
GrU F
π4
1
The Integral of Helmholtz and Kirchhoff:
ΣAssume that the aperture is illuminated by a single
spherical wave, and:
Srr,<<λ
( ) ( )






⋅≅
∂
∂ →→
SS
S
SsourceS
nr
r
rkjA
j
n
rU
11
exp2
λ
π

( ) ( )
r
rkj
rrG FS
exp
; =

( ) ( )






⋅−≅
∂
∂ →→
S
FS
nr
r
rkj
j
n
rrG
11
exp2,
λ
π

Srr
k
1
,
12
>>=
λ
π
( ) ( )( )
∫∫Σ
→→→→


















⋅−





⋅−





 +
= dS
nrnr
rr
rrkjA
jrU
SSS
s
ssource
F
2
1111
exp
λ

( ) ( )
S
Ssource
S
r
rkjA
rU
exp
=

P
Fr

Sr

r

Σ
1S
∞S
R
Screen
Aperture
0P 0,0
1
1
=
∂
∂
=
S
S
n
U
U
Σ
Σ ∂
∂
n
U
U ,
Q
→
Sn1
→
Sn1
Sθ
θ

31
SOLO
Diffraction
( ) ( )( )
( )
( )∫∫
∫∫
Σ
Σ
→→→→


















++
=


















⋅−





⋅−





 +
=
dSK
rr
rrkj
A
dS
nrnr
rr
rrkjA
jrU
S
s
s
source
SSS
s
ssource
F
θθ
π
λ
λ
,
2
exp
2
1111
exp
( ) 





⋅−=





⋅−=
+
=






⋅−





⋅−
=
→→→→
→→→→
SSSS
S
SSS
S nrnr
nrnr
K 11cos&11cos
2
coscos
2
1111
, θθ
θθ
θθ
1. Obliquity or Inclination Factor:
( ) ( ) 0,0&10,0 ====== πθθθθ SS KK
2. Additional phase π/2
3. The amplitude is scaled by the factor 1/λ (not found in Fresnel derivation)
P
Fr

Sr

r

Σ
1S
∞S
R
Screen
Aperture
0P 0,0
1
1
=
∂
∂
=
S
S
n
U
U
Σ
Σ ∂
∂
n
U
U ,
Q
→
Sn1
→
Sn1
Sθ
θ
We recovered Fresnel Diffraction Formula with:

32
Reciprocity Theorem of Helmholtz
SOLO
Diffraction
( ) ( )( )
( )
( )∫∫
∫∫
Σ
Σ
→→→→


















++
=


















⋅−





⋅−





 +
=
dSK
rr
rrkj
A
dS
nrnr
rr
rrkjA
jrU
S
s
s
source
SSS
s
ssource
F
θθ
π
λ
λ
,
2
exp
2
1111
exp
We can see that the Fresnel-Kirchhoff Diffraction Formula is symmetrical with respect
to r and rS, i.e. point source and observation point. Therefore we can interchange them
and obtain the same relation. This result is called Reciprocity Theorem of Helmholtz.
P
Fr

Sr

r

Σ
1S
∞S
R
Screen
Aperture
0P 0,0
1
1
=
∂
∂
=
S
S
n
U
U
Σ
Σ ∂
∂
n
U
U ,
Q
→
Sn1
→
Sn1
Sθ
θ
1821-1894
Note:
This is similar to Lorentz’s Reciprocity Theorem in Electromagnetism.

33
Huygens-Fresnel Principle
SOLO
Diffraction
( )
( )
( )∫∫Σ


















++
= dSK
rr
rrkj
A
rU S
s
s
source
F θθ
π
λ
,
2
exp

The Fresnel Diffraction Formula can be rewritten as:
( ) ( ) ( )
∫∫Σ
= dS
r
rkj
QVrU F
exp
where:
( ) ( )
s
s
S
source
r
rkj
K
A
QV






+
=
2
exp
,
π
θθ
λ
The interpretation of this formula is that each point
of a wavefront can be considered as the center of a
secondary spherical wave, and those secondary spherical
waves interfere to result in the total field, is known as the
Huygens-Fresnel Principle.
P
Fr

Sr

r

Σ
1S
∞S
R
Screen
Aperture
0P 0,0
1
1
=
∂
∂
=
S
S
n
U
U
Σ
Σ ∂
∂
n
U
U ,
Q
→
Sn1
→
Sn1
Sθ
θ
Table of Content

34
SOLO Diffraction
Consider a diffracting aperture Σ. Suppose that the aperture
is divided into two portions Σ 1 and Σ 2 such that Σ = Σ1 + Σ2.
The two aperture Σ1 and Σ2 are said to be complementary.
Complementary Apertures. Babinet Principle
From the Fresnel Diffraction Formula:
( ) ( ) ( )
( ) ( ) ( ) ( )
∫∫∫∫
∫∫
ΣΣ
Σ
+=
=
21
dS
r
rkj
QVdS
r
rkj
QV
dS
r
rkj
QVrU F
expexp
exp
P
Fr

1S
r

1r

2
Σ
1S
Screen
Apertures
0P
1
Q→
Sn1
2S
r

2
r
1
Σ
2
Q
We can see that the result is the added effect of all complimentary
apertures. This is known as Babinet Principle.
Table of Content
The result can be very helpful when Σ is a very complicated
aperture, that can be decomposed in a few simple apertures.

35
SOLO Diffraction
The Kirchhoff Diffraction Formula is an approximation since for zero field and
normal derivative on any finite surface the field is zero everywhere. This was pointed
out by Poincare in 1892 and by Sommerfeld in 1894.
The first rigorous solution of a diffraction problem was given by Sommerfeld in
1896 for a two-dimensional case of a planar wave incident on an infinitesimally thin,
perfectly conducting half plane. This solution is not given here.
Arnold Johannes
Wilhelm
Sommerfeld
1868 - 1951
Jules Henri Poincaré
1854-1912
Sommerfeld, A. : “Mathematische Theorie der Diffraction”,
Math. Ann., 47:317, 1896 translated in english as
“Optics, Lectures on Theoretical Physics”, vol. IV,
Academic Press Inc., New York, 1954

36
SOLO
Diffraction
Let start from the Helmholtz and Kirchhoff Integral:
P
Fr

Sr

r

Σ
1S
∞S
R
Screen
Aperture
0P
0,0
1
1
=
∂
∂
=
S
S n
U
U
Σ
Σ
∂
∂
n
U
U ,
Q
→
Sn1
→
Sn1
Sθ
θ
0'PFr'

→→






⋅−= SSSSS nnrrr 112'

( ) ∫∫+Σ






∂
∂
−
∂
∂
−=
1
4
1
S
F dS
n
G
U
n
U
GrU
π

Suppose that the Scalar Green Function is generated not only by P0 located at ,
but also by a point P’0 located symmetric relative to the screen at
→→







Sr

G
( )
( ) ( )
SF
SF
SF
SF
SF
rr
rrkj
rr
rrkj
rrG
'
'expexp
,_ 




−
−
−
−
−
=
( )
( ) ( )
SF
SF
SF
SF
SF
rr
rrkj
rr
rrkj
rrG
'
'expexp
, 




−
−
+
−
−
=+
or:
We have 11 ,,
' SSFSSF rrrr ΣΣ
−=−

( ) ( )
→
Σ
→
Σ
⋅−−=⋅− SSSFSSSF nrrnrr 1'1 11 ,,

0
1,
=
Σ− S
G ( ) 011
exp1
2
11
,,
_
≠











⋅





−−=
∂
∂
Σ
→→
Σ S
S
S
nr
r
rkj
r
kj
n
G
0
1,
=
∂
∂
Σ
+
S
n
G( ) 0
exp
2
1
1
,
,
≠



=
Σ
Σ+
S
S r
rkj
G

37
SOLO
Diffraction
1. Start from the Helmholtz and Kirchhoff Integral:
P
Fr

Sr

r

Σ
1S
∞S
R
Screen
Aperture
0P
0,0
1
1
=
∂
∂
=
S
S n
U
U
Σ
Σ ∂
∂
n
U
U ,
Q
→
Sn1
→
Sn1
Sθ
θ
0'PFr'

→→







( ) ∫∫+Σ






∂
∂
−
∂
∂
−=
1
4
1
S
F dS
n
G
U
n
U
GrU
π

( )
( ) ( )
SF
SF
SF
SF
SF
rr
rrkj
rr
rrkj
rrG
'
'expexp
,_ 




−
−
−
−
−
=Choose
0
1,
=
Σ− S
G ( ) 011
exp1
2
11
,,
_
≠











⋅





−−=
∂
∂
Σ
→→
Σ S
S
S
nr
r
rkj
r
kj
n
G
On the shadowed part of the screen and0
1
=S
U 0/
1
≠∂∂ S
nU
( ) ( ) ( )
∫∫∫∫
Σ
→→=
>>Σ





 ⋅−≅








∂
∂
= dSnr
r
rkj
rU
j
dS
n
G
UrU SS
k
r
kj
F 11
exp
4
1 /2
1
_ 
λπ
λπ
This is Rayleigh-Sommerfeld Diffraction Formula of the first kind
SF rrr

−=
Arnold Johannes
Wilhelm
Sommerfeld
1868 - 1951
( ) ( )
S
Ssource
S
r
rkjA
UrU
exp
== Σ

John William
Strutt
Lord Rayleigh
(1842-1919)
( ) ( ) ( )
∫∫Σ
→→






⋅−= dSnr
r
rkj
r
rkjAj
rU S
S
Ssource
F 11
expexp
λ

we obtain:

38
SOLO
Diffraction
2. Start from the Helmholtz and Kirchhoff Integral:
P
Fr

Sr

r

Σ
1S
∞S
R
Screen
Aperture
0P
0,0
1
1
=
∂
∂
=
S
S n
U
U
Σ
Σ ∂
∂
n
U
U ,
Q
→
Sn1
→
Sn1
Sθ
θ
0'PFr'

→→







( ) ∫∫+Σ






∂
∂
−
∂
∂
−=
1
4
1
S
F dS
n
G
U
n
U
GrU
π

( )
( ) ( )
SF
SF
SF
SF
SF
rr
rrkj
rr
rrkj
rrG
'
'expexp
, 




−
−
+
−
−
=+
Choose
On the shadowed part of the screen and0
1
≠S
U 0/
1
=∂∂ S
nU
SF rrr

−=0
1,
=
∂
∂
Σ
+
S
n
G( ) 0
exp
2
1
1
,
,
≠





=
Σ
Σ+
S
S r
rkj
G
( ) ( )
∫∫∫∫ ΣΣ
+
∂
∂
−=





∂
∂
−= dS
n
U
r
rkj
dS
n
U
GrU F
exp
2
1
4
1
ππ

( ) ( )
S
Ssource
S
r
rkjA
UrU
exp
== Σ
 ( ) ( )






⋅≅
∂
∂ →→
SS
S
SsourceS
nr
r
rkjA
j
n
rU
11
exp2
λ
π

( ) ( ) ( )
∫∫Σ
→→






⋅−= dSnr
r
rkj
r
rkjAj
rU SS
S
Ssource
F 11
expexp
λ

For
we obtain:
Table of Content
This is Rayleigh-Sommerfeld Diffraction Formula of the second kind

39
P
0P
Q 1x
0x
1y
0y
η
ξ
Sr'

Sr

ρ
 r

O
Sθ θ
Screen
Image
plane
Source
plane
0O
1O
Sn1
Σ
1r
0r

SSS rn θcos11 =⋅
θcos11 =⋅ rnS
z
Sn1
'r
 Fr

F
rPP

=0
S
rQP

=0
rQP

=
SrOP '0

=
'1 rOO

=
SOLO Diffraction
( ) ( ) ( )
∫∫Σ
+
= dS
r
rkj
r
rkjAj
rU S
S
Ssource
F
2
coscosexpexp θθ
λ

Start with Fresnel-Kirchhoff (or Rayleigh-Sommerfeld) Diffraction Formula
1. If the inclination factor is nearly constant over the aperture ( ) constKK S ==θθ ,
Extensions of Fresnel-Kirchhoff Diffraction Theory
( ) ( ) ( )
∫∫Σ
≈ dS
r
rkj
r
rkjAKj
rU
S
Ssource
F
expexp
λ

( ) ( ) ( )
∫∫Σ
= dS
r
rkj
rU
Kj
rU SF
exp
λ
2. Replace the incident point source wavefront
with a general waveform
( )
S
S
r
rkjexp
( )Sinc rU

3. Characterize the aperture by a
transfer function τ to model amplitude
or phase changes due to optic system
( ) ( ) ( ) ( )
∫∫Σ
= dS
r
rkj
rrU
j
rU SSF
exp
τ
λ Table of Content

40
SOLO Diffraction
Phase Approximations – Fresnel (Near-Field) Approximation
Fresnel Approximation or Near Field Approximation
can be used when aperture dimensions are
comparable to distance to source rS or image r.
( ) ( ) ( )
∫∫Σ
+
= dS
r
rkj
r
rkjAj
rU S
S
Ssource
F
2
coscosexpexp θθ
λ

Start with Fresnel-Kirchhoff Diffraction Formula
If the inclination factor is nearly constant over the aperture
( ) constKK S ==θθ ,
( ) ( ) ( ) ( ) ( )
∫∫∫∫ ΣΣ
== dS
r
rkj
rU
Kj
dS
r
rkj
r
rkjAKj
rU S
S
Ssource
F
expexpexp 
λλ
P
Q 1x
1y
η
ξ
ρ
 r

O
θ
Screen
Image
plane
1O
Sn1
Σ
1r

z
Sn1
'r

rQP

=
'1 rOO

=
P
0P
Q 1x
0x
1y
0y
η
ξ
Sr'

Sr

ρ
 r

O
Sθ θ
Screen
Image
plane
Source
plane
0O
1O
Sn1
Σ
1r
0r

SSS rn θcos11 =⋅
θcos11 =⋅ rnS
z
Sn1
'r
 Fr

( ) ( ) ( )
1
''2
'
'
1''2'
'
2
2
1
2
1
2
11
2/1
2
2
1
2/1
11
0
1
2
1
<<
−
+
−
+≈







 −
+=








−⋅−+−⋅+=
−+=
++=+
r
r
k
r
r
r
r
r
rrrrrrr
rrr
x
x
ρρ
ρ
ρρρ
ρ









( ) ( )















 −
≈
'2
exp
'
'expexp
2
1
r
r
kj
r
rkj
r
rkj ρ
 ( ) ( ) ( )
2
max
2
1
2
1
'
'2
exp
'
'exp
rrk
dS
r
r
kjrU
r
rkjKj
rU SF
<<−







 −
≈ ∫∫Σ
ρ
ρ
λ



Augustin Jean Fresnel
1788-1827

41
SOLO Diffraction
Phase Approximations – Fraunhofer (Near-Field) Approximation
Fraunhofer Approximation or Far Field Approximation
can be used when aperture dimensions are very small
comparable to distance to source rS or image r.
( ) ( ) ( )
∫∫Σ
+
= dS
r
rkj
r
rkjAj
rU S
S
Ssource
F
2
coscosexpexp θθ
λ

Start with Fresnel-Kirchhoff Diffraction Formula
If the inclination factor is nearly constant over the aperture
( ) constKK S ==θθ ,
( ) ( ) ( ) ( ) ( )
∫∫∫∫ ΣΣ
=≈ dS
r
rkj
rU
Kj
dS
r
rkj
r
rkjAKj
rU S
S
Ssource
F
expexpexp 
λλ
P
Q 1x
1y
η
ξ
ρ
 r

O
θ
Screen
Image
plane
1O
Sn1
Σ
1r

z
Sn1
'r

rQP

=
'1 rOO

=
P
0P
Q 1x
0x
1y
0y
η
ξ
Sr'

Sr

ρ
 r

O
Sθ θ
Screen
Image
plane
Source
plane
0O
1O
Sn1
Σ
1r
0r

SSS rn θcos11 =⋅
θcos11 =⋅ rnS
z
Sn1
'r
 Fr

( ) ( )





 ⋅
−≈
'
exp
'
'expexp 1
r
r
kj
r
rkj
r
rkj ρ
 ( ) ( ) ( )
( ) 2
max
22
1
1
'
2
'
exp
'
'exp
rr
k
dS
r
r
kjrU
r
rkjKj
rU SF
<<+





 ⋅
−≈ ∫∫Σ
ρ
ρ
λ


( ) ( ) ( )
1
'2'
'
'2'
'
'
2
1''2'
'
2
22
11
2
22
11
2
11
2/1
2
2
1
2
1
2/1
11
0
1
2
1
<<
+⋅
−≈+
+
+
⋅
−≈





 +⋅−
+=








−⋅−+−⋅+=
−+=
++=+
r
rk
r
r
r
r
r
r
r
r
r
rr
rrrrrrr
rrr
x
x
ρρρρ
ρρ
ρρρ
ρ









Table of Content

42
0P
Q
0x
0y
η
Sr'

Sr

ρ

O
Sθ
ScreenSource
plane
0O
Σ
Sn10r

S
rQP

=0
SrOP '0

=
SOLO Diffraction
Fresnel Approximations at the Source
[ ]
( ) 








+
⋅
−+⋅+≈














+⋅+=
+⋅+=
+=
+−+=+
S
S
SS
S
S
xx
x
SS
S
S
SSS
SS
r
r
rr
r
r
rr
r
r
rrr
rr
'2
'1
'2'
'
'
''
'
21'
'2'
'
2
282
11
2/12
2
2/122
2
ρρ
ρ
ρ
ρ
ρρ
ρ
( ) ( ) ( ) ( )







 ⋅−
⋅=
S
S
S
S
S
S
S
r
r
kjrkj
r
rkj
r
rkj
'2
'1
exp'1exp
'
'expexp
2
2
ρρ
ρ


( )
S
S
r
rkj
'
'exp
( )ρ

⋅Srkj '1exp
( )







 ⋅−
S
S
r
r
kj
'2
'1
exp
2
2
ρρ

Spherical wave centered at P0.
Lowest order approximation to the phase of
a spherical wavefront
Planar wave propagating in directionSr'1
P
0P
Q 1x
0x
1y
0y
η
ξ
Sr'

Sr

ρ
 r

O
Sθ θ
Screen
Image
plane
Source
plane
0O
1O
Sn1
Σ
1r
0r

SSS rn θcos11 =⋅
θcos11 =⋅ rnS
z
Sn1
'r
 Fr


43
SOLO Diffraction
P
Q 1x
1y
η
ξ
ρ
 r

O
θ
Screen
Image
plane
1O
Sn1
Σ
1r

z
Sn1
'r

rQP

=
'1 rOO

=
( ) ( ) ( )








+
⋅
−
+
+≈





 +⋅−
+=








−⋅−+−⋅+=
−+=
++=+
''2
'
'
2
1'
'2'
'
1
22
1
2
11
2/1
2
2
1
2
1
2/1
11
0
1
2
1
r
r
r
r
r
r
rr
r
rrrrrr
rrr
x
x
ρρ
ρρ
ρρρ
ρ
( ) ( )













 ⋅
−






























≈
'
exp
'2
exp
'2
exp
'
'expexp 1
22
1
r
r
kj
r
kj
r
r
kj
r
rkj
r
rkj ρρ

Fresnel Approximations at the Image plane
P
0P
Q 1x
0x
1y
0y
η
ξ
Sr'

Sr

ρ
 r

O
Sθ θ
Screen
Image
plane
Source
plane
0O
1O
Sn1
Σ
1r
0r

SSS rn θcos11 =⋅
θcos11 =⋅ rnS
z
Sn1
'r
 Fr

( )
'
'exp
r
rkj
( )ρ

⋅− '1exp rkj
( )







 ⋅−
'2
'1
exp
2
2
r
r
kj
ρρ

Spherical wave centered at O.
Lowest order approximation to the phase of
a spherical wavefront
Planar wave propagating in direction'1r

44
SOLO Diffraction
Fresnel and Fraunhofer Diffraction Approximations (1st
way)
( ) ( ) ( )
( )
∫∫Σ
+
= dS
r
rkj
r
rkjAj
rU
SK
S
S
Ssource
F
  

θθ
θθ
λ
,
2
coscosexpexp
Fresnel Approximation
( ) ( )[ ] [ ] ( )[ ] ( ) ( )
∫∫Σ






















−⋅−−
+
⋅−
⋅−⋅
+
≈ dS
r
rrr
r
r
kjrrkjrrkj
rr
rrkjKAj
rU
S
S
S
S
Ssource
F
'2
'1
'2
'1
exp'1'1exp'1exp
''
''exp
2
1
2
1
2
2
1
ρρρρ
ρ
λ


Fraunhofer Approximation

( ) ( )
1
'2
'1
'2
'12
2
1
2
1
2
2
<<
−⋅−−
+
⋅−
S
S
k
r
rrr
r
r ρρρρ
λ
π

or S
MAX
rr ','
2
<<
λ
ρ
( ) ( )[ ] [ ] ( )[ ]∫∫Σ
⋅−⋅
+
≈ dSrrkjrrkj
rr
rrkjKAj
rU S
S
Ssource
F ρ
λ

'1'1exp'1exp
''
''exp
1
If
( ) ( )
1
'2
'1
'2
'1
exp
2
1
2
1
2
2
≈



















 −⋅−−
+
⋅−
S
S
r
rrr
r
r
kj
ρρρρ

we obtain
1788-1827
( ) constKK S =≈θθ ,
Start with
'1'1 rrq S
−=

P
0P
Q 1x
0x
1y
0y
η
ξ
Sr'

Sr

ρ
 r

O
Sθ θ
Screen
Image
plane
Source
plane
0O
1O
Sn1
Σ
1r
0r

SSS rn θcos11 =⋅
θcos11 =⋅ rnS
z
Sn1
'r
 Fr

F
rPP

=0
SrQP

=0
rQP

=
S
rOP '0

=
'1
rOO

=

45
SOLO Diffraction
Fresnel and Fraunhofer Diffraction Approximations (2nd
way)
Fresnel Approximation
( ) ( ) ( ) ( ) ( ) ( )
∫∫Σ





 −⋅−
= dS
r
rr
kjrrU
r
rkjj
rU SSF
'2
exp
'
'exp 11
ρρ
τ
λ



( ) '1
'2
2 max
2
1
22
1
2
r
r
r
r
k
<<
+
⇒<<
+
λ
ρρ
λ
πIf
we obtain
1788-1827
Start with ( ) ( ) ( ) ( )
∫∫Σ
= dS
r
rkj
rrU
j
rU SSF
exp
τ
λ
- aperture optical transfer function( )Sr

τ
- disturbance at the aperture( )SrU

( ) ( ) ( ) ( )∫∫Σ











 ⋅
−= dS
r
r
kjrrU
r
rkjj
rU SSF
'
exp
'
'exp 1 ρ
τ
λ


( ) ( )











 ⋅
−















 +
≈
'
exp
'2
exp
'
'expexp 1
2
1
2
r
r
kj
r
r
kj
r
rkj
r
rkj ρρ

P
Q 1x
1y
η
ξ
ρ
 r

O
θ
Screen
Image
plane
1O
Sn1
Σ
1r

z
Sn1
'r

rQP

=
'1
rOO

=
ρ

−+= 1
' rrr ( ) ( ) ( ) ( ) ( ) ( ) ( )





 −⋅−
+≅




 −⋅−
+=








−⋅−+−⋅+= 2
11
2/1
2
11
2/1
11
0
1
2
'2
1'
'
1''2'
r
rr
r
r
rr
rrrrrrr
ρρρρ
ρρρ




( ) ( ) ( ) ( )∫∫Σ











 ⋅
−




 +
= dS
r
r
kj
r
r
kjrrU
r
rkjj
rU SSF
'
exp
'2
exp
'
'exp 1
2
1
2
ρρ
τ
λ


( ) ( ) ( ) 





+
+
+
⋅
−≈




 +⋅−
+=








−⋅−+−⋅+=
++=+
'2'
'
'
2
1''2'
22
11
2
112/1
2
2
1
2
1
2/1
11
0
1
2
r
r
r
r
r
r
rr
rrrrrrr
x
x
ρρρρ
ρρρ

46
SOLO Diffraction
1788-1827
1x
1yη
ξ
max
ρ=D
Screen
1O
1r

z
λ
2
D
R <
Fresnel Region Fraunhofer Region
λ
2
D
R >
R
Σ
O

( ) '1
'2
2 max
2
1
22
1
2
r
r
r
r
k
<<
+
⇒<<
+
λ
ρρ
λ
π
If
Table of Content

47
SOLO Diffraction
( ) ( ) ( ) ( )∫∫Σ











 ⋅
−= dS
r
r
kjrrU
r
rkjj
rU SSF
'
exp
'
'exp 1 ρ
τ
λ


( )ηξ
λ
πρ
11
1
'
2
'
yx
rr
r
k +=




 ⋅

( ) ( ) ( ) ( ) ( )∫∫Σ






+−= ηξηξ
λ
π
τ
λ
ddyx
r
jrrU
r
rkjj
rU SSF 11
'
2
exp
'
'exp 
The integral is the two dimensional Fourier Transform
of the field within the aperture ( ) ( )SS rrU

τ
( )
( )
( ) ( )[ ] { }Σ
Σ
=+−= ∫∫ fFTddkkjfkkF yxyx ηξηξηξ
π
exp,
2
1
:, 2
( ) ( ) ( ) ( ) ( ){ }SSF rrUFT
r
rkjj
rU

τπ
λ
2
2
'
'exp
=Therefore
P
0P
Q 1x
0x
1y
0y
η
ξ
Sr'

Sr

ρ
 r

O
Sθ θ
Screen
Image
plane
Source
plane
0O
1O
Sn1
Σ
1r
0r

SSS rn θcos11 =⋅
θcos11 =⋅ rnS
z
Sn1
'r
 Fr

F
rPP

=0
SrQP

=0
rQP

=
SrOP '0

=
'1 rOO

=
Two Dimensional
Fourier Transform
Table of Content

48
SOLO Diffraction
Rectangular Aperture
P
Q 1x
1y
η
ξ
ρ
 r

O
θ
Screen
Image
plane
1O
Sn1Σ
1r

z
Sn1
'r
1ξ
2ξ
1η
2η
( ) ( ) ( ) ( )
( ) ( )∫∫
∫∫
−−
=
Σ






⋅−





⋅−=





 ⋅
−




 ⋅
−





=
1
1
1
1
11
0
2
11
2
10
'
2
exp
'
2
exp
'2
'exp
'
exp
'
exp
'2
exp
'
'exp
η
η
ξ
ξ
λ
π
ηη
λ
π
ξξ
λ
π
π
ηξ
ηξ
λ
dy
r
jdx
r
j
r
rkjUkj
dd
r
y
kj
r
x
kj
r
r
kj
r
rkjUj
rU
k
F

( ) ( )


 ≤≤≤≤
=
elsevere
U
rrU SS
0
& 21110 ηηηξξξ
τ

For a Rectangular Aperture
Therefore
( ) ( ) ( ) ( )∫∫Σ











 ⋅
−





= dS
r
r
kjrrU
r
r
kj
r
rkjj
rU SSF
'
exp
'2
exp
'
'exp 1
2
1 ρ
τ
λ








⋅






⋅
=






−






⋅+−





⋅−
=





⋅−∫−
11
11
1
1
1111
1
'
2
'
2
sin
2
'
2
'
2
exp
'
2
exp
'
2
exp
1
1
ξ
λ
π
ξ
λ
π
ξ
λ
π
ξ
λ
π
ξ
λ
π
ξξ
λ
π
ξ
ξ
x
r
x
r
x
r
j
x
r
jx
r
j
dx
r
j






⋅






⋅
=






−






⋅+−





⋅−
=





⋅−∫−
11
11
1
1
1111
1
'
2
'
2
sin
2
'
2
'
2
exp
'
2
exp
'
2
exp
1
1
η
λ
π
η
λ
π
η
λ
π
η
λ
π
η
λ
π
ξη
λ
π
ξ
ξ
y
r
y
r
y
r
j
y
r
jy
r
j
dy
r
j
( ) ( )







⋅






⋅






⋅






⋅
=
11
11
11
11
4/
11
0
'
2
'
2
sin
'
2
'
2
sin
'
'exp2
η
λ
π
η
λ
π
ξ
λ
π
ξ
λ
π
ηξπ
y
r
y
r
x
r
x
r
r
rkjUkj
rU
A
F


49
SOLO Diffraction
Rectangular Aperture (continue – 1)
Since U stands for scalar field intensity
(E or H), the irradiance I is given by
where < > is the time average and * is
the complex conjugate.
( ) ( )






⋅






⋅






⋅






⋅
=
11
11
11
11
0
'
2
'
2
sin
'
2
'
2
sin
'
'exp8
η
λ
π
η
λ
π
ξ
λ
π
ξ
λ
π
π
y
r
y
r
x
r
x
r
Ar
rkjUkj
rU F

( ) ( ) ( ) >⋅< ∗
FFF rUrUrI

~
Therefore
( ) ( ) 2
11
11
2
2
11
11
2
'
2
'
2
sin
'
2
'
2
sin
0






⋅






⋅






⋅






⋅
=
η
λ
π
η
λ
π
ξ
λ
π
ξ
λ
π
y
r
y
r
x
r
x
r
IrI F

I (0) is the irradiance at O1 (x1 = y1 = 0).
Hecht pg.466

50
SOLO Diffraction
Single Slit Aperture
Let substitute in the
rectangular aperture ξ1 → 0
where < > is the time average and * is
the complex conjugate.
( ) ( )






⋅






⋅






⋅






⋅
=
11
11
11
11
0
'
2
'
2
sin
'
2
'
2
sin
'
'exp8
η
λ
π
η
λ
π
ξ
λ
π
ξ
λ
π
π
y
r
y
r
x
r
x
r
Ar
rkjUkj
rU F

( ) ( ) ( ) >⋅< ∗
FFF rUrUrI

~
Therefore
( ) ( ) 2
11
11
2
'
2
'
2
sin
0






⋅






⋅
=
η
λ
π
η
λ
π
y
r
y
r
IrI F

to obtain the single (vertical) slit diffraction
( ) ( )






⋅






⋅
=
11
11
0
'
2
'
2
sin
'
'exp2
η
λ
π
η
λ
π
π
y
r
y
r
Ar
rkjUkj
rU FSLITSINGLE

Since U stands for scalar field intensity
(E or H), the irradiance I is given by
Hecht, pg. 453
Hecht, pg. 456

51
SOLO Diffraction
Single Slit Aperture (continue)
( ) ( ) 2
11
11
2
'
2
'
2
sin
0






⋅






⋅
=
η
λ
π
η
λ
π
y
r
y
r
IrI F

Hecht, pg. 456
Hecht 455
Define: 11
'
2
: η
λ
π
β ⋅= y
r
( ) ( ) 2
2
sin
0
β
β
β II =
The extremum of I (β) is obtained from:
( ) ( ) ( ) 0
sincossin2
0 3
=
−
=
β
ββββ
β
β
I
d
Id
The results are given by:
minimum,3,2,0sin πππββ ±±±=⇒=
maximumββ =tan
The solutions can be obtained
graphically as shown in the figure and
are: ,4707.3,4590.2,4303.1 πππβ ±±±=

52
SOLO Diffraction
Double Slit Aperture
( ) ( ) ( )
( )
( )
( )
( )
( )













 ⋅
−+




 ⋅
−





= ∫∫
+
−
+−
−−
ξ
ξ
ξ
ξ
λ
d
r
x
kjd
r
x
kj
r
r
kj
r
rkjUj
rU
ba
ba
ba
ba
F
2/
2/
1
2/
2/
1
2
10
'
exp
'
exp
'2
exp
'
'exp
P
Q
1x
1y
η
ξ
ρ
 r

O
θ
Screen
Image
plane
1O
Sn1Σ
1r

z
Sn1
'r

1η
b
b
a( ) ( ) ( ) ( )∫∫Σ











 ⋅
−





= dS
r
r
kjrrU
r
r
kj
r
rkjj
rU SSF
'
exp
'2
exp
'
'exp 1
2
1 ρ
τ
λ


( )
( ) ( ) ( )






⋅






⋅






⋅
=






−






−−⋅−−





+−⋅−
=





⋅−∫
+−
−−
ax
r
j
bx
r
bx
r
b
x
r
j
bax
r
jbax
r
j
dx
r
j
ba
ba
1
1
1
1
112/
2/
1
'
exp
'
'
sin
1
'
2
'
exp
'
exp
'
2
exp
λ
π
λ
π
λ
π
λ
π
λ
π
λ
π
ξξ
λ
π
( )
( ) ( ) ( )






⋅−






⋅






⋅
=






−






−⋅−−





+⋅−
=





⋅−∫
+
−
ax
r
j
bx
r
bx
r
b
x
r
j
bax
r
jbax
r
j
dx
r
j
ba
ba
1
1
1
1
112/
2/
1
'
exp
'
'
sin
1
'
2
'
exp
'
exp
'
2
exp
λ
π
λ
π
λ
π
λ
π
λ
π
λ
π
ξξ
λ
π
( ) ( )






⋅






⋅






⋅






−= ax
r
bx
r
bx
r
br
r
kj
r
rkjUj
rU F 1
1
12
10
'
cos
'
'
sin
2
'2
exp
'
'exp
λ
π
λ
π
λ
π
λ

( ) ( ) 





⋅






⋅






⋅
= ax
r
bx
r
bx
r
IrI F 1
2
2
1
1
2
'
cos
'
'
sin
0
λ
π
λ
π
λ
π

( ) ( ) ( ) >⋅< ∗
FFF rUrUrI

~
( ) ( )
( ) ( ) ( ) ( )


 +≤≤−+−≤≤−−
=
elsevere
babababaU
rrU SS
0
2/2/&2/2/0 ξξ
τ


53
SOLO Diffraction
Double Slit Aperture (continue -= 1)
Hecht p.458
( ) ( ) ( ) γ
β
β
λ
π
λ
π
λ
π
2
2
2
12
2
1
12
cos
sin
0
'
cos
'
'
sin
0 I
a
r
x
b
r
x
b
r
x
IrI F =





⋅






⋅






⋅
=

P
Q
1x
1y
η
ξ
ρ
 r

O
θ
Screen
Image
plane
1O
Sn1Σ
1r

z
Sn1
'r

1η
b
b
a
The factor (sin β/ β)2
that
was previously found as the
distribution function for a
single slit is here the envelope
for the interference fringes
given by the term cos2
γ.
Bright fringes occur for
γ = 0,±π ,±2π,…
The angular separation
between fringes is Δγ = π.

54
Hecht 459
SOLO Diffraction
Double Slit Aperture (continue – 2)

55
SOLO Diffraction
Multiple Slit Aperture
P
1y
r

Image
plane
1O
1r

Q
η
ξ
ρ

O
θ
Screen
Σ
Sn1
'r

b
b
a
a
b
b
a
b
a
The Aperture consists of a large number N of identical
parallel slits of width b and separation a.
( ) ( ) ( )
∑ ∫
−
=
+
− 












 ⋅
−





=
1
0
2/
2/
1
2
10
'
exp
'2
exp
'
'exp N
k
bak
bak
F d
r
x
kj
r
r
kj
r
rkjUj
rU ξ
ξ
λ

( ) ( ) ( ) ( )∫∫Σ











 ⋅
−





= dS
r
r
kjrrU
r
r
kj
r
rkjj
rU SSF
'
exp
'2
exp
'
'exp 1
2
1 ρ
τ
λ








⋅−






⋅






⋅
=






−












−⋅−−











+⋅−
=





⋅−∫
+
−
akx
r
j
bx
r
bx
r
b
x
r
j
b
kax
r
j
b
kax
r
j
dx
r
j
bka
bka
1
1
1
1
112/
2/
1
'
2
exp
'
'
sin
1
'
2
2'
2
exp
2'
2
exp
'
2
exp
λ
π
λ
π
λ
π
λ
π
λ
π
λ
π
ξξ
λ
π
( ) ( )
( ) ( )






⋅






⋅






⋅






⋅






=






⋅−−






⋅−−






⋅






⋅






=




















⋅−






⋅






⋅






= ∑
−
=
ax
r
aNx
r
bx
r
bx
r
br
r
kj
r
rkjUj
ax
r
j
aNx
r
j
bx
r
bx
r
br
r
kj
r
rkjUj
akx
r
j
bx
r
bx
r
br
r
kj
r
rkjUj
rU
N
k
F
1
1
1
12
10
1
1
1
12
10
1
0
1
1
12
10
'
sin
'
sin
'
'
sin
1
'2
exp
'
'exp
'
2
exp1
'
2
exp1
'
'
sin
1
'2
exp
'
'exp
'
2
exp
'
'
sin
1
'2
exp
'
'exp
λ
π
λ
π
λ
π
λ
π
λ
λ
π
λ
π
λ
π
λ
π
λ
λ
π
λ
π
λ
π
λ

( ) ( )


 −=+≤≤+
=
elsevere
NkbkabkaU
rrU SS
0
1,,1,02/2/0  ξ
τ

56
SOLO Diffraction
Multiple Slit Aperture (continue – 1)
P
1y
r

Image
plane
1O
1r

Q
η
ξ
ρ

O
θ
Screen
Σ
Sn1
'r

b
b
a
a
b
b
a
b
a
The Aperture consists of a large number N of identical
parallel slits of width b and separation a.
( ) ( )






⋅






⋅






⋅






⋅






=
ax
r
aNx
r
bx
r
bx
r
br
r
kj
r
rkjUj
rU F
1
1
1
12
10
'
sin
'
sin
'
'
sin
1
'2
exp
'
'exp
λ
π
λ
π
λ
π
λ
π
λ

( ) ( ) ( )
α
α
β
β
λ
π
λ
π
λ
π
λ
π
22
2
2
2
1
22
1
2
2
1
1
2
sin
sinsin
0
'
sin
'
sin
'
'
sin
0
N
N
I
ax
r
N
aNx
r
bx
r
bx
r
IrI F =






⋅






⋅






⋅






⋅
=

( ) ( ) ( ) >⋅< ∗
FFF rUrUrI

~

57
SOLO Diffraction
Hecht p.462
Hecht p.463
( ) ( ) ( )
α
α
β
β
λ
π
λ
π
λ
π
λ
π
22
2
2
2
1
22
1
2
2
1
1
2
sin
sinsin
0
'
sin
'
sin
'
'
sin
0
N
N
I
ax
r
N
aNx
r
bx
r
bx
r
IrI F =






⋅






⋅






⋅






⋅
=


58
SOLO Diffraction
( ) ( ) ( )
α
α
β
β
λ
π
λ
π
λ
π
λ
π
22
2
2
2
1
22
1
2
2
1
1
2
sin
sinsin
0
'
sin
'
sin
'
'
sin
0
N
N
I
ax
r
N
aNx
r
bx
r
bx
r
IrI F =






⋅






⋅






⋅






⋅
=

Sears p.222
Hecht p. 462
Sears p.236
Interference Irradiation
for 1, 2, 3 and 4 slits as
function of observation
angle.
Diffraction Pattern for
1, 2, 3, 4 and 5 slits.

59
SOLO Resolution of Optical Systems
According to Huygens-Fresnel Principle, a differential area dS, within an optical
Aperture, may be envisioned as being covered with coherent secondary point sources.
φρφρ sincos == yz
Φ=Φ= sincos qYqZ
Differential area dS, coordinates
Image , coordinates
φρρ dddS =
( )
dSe
r
E
dE rktiA −






= ω
The spherical wave that
propagates from dS to Image is
where
( ) ( )[ ] ( )[ ] ( )[ ]22/122/1222
/1/21 RZzYyRRZzYyRzZyYXr +−≈+−≈−+−+= [ ] 2/1222
ZYXR ++=
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )RkaqJkaqRe
R
E
ddee
R
E
dSee
R
E
dEE RktiA
a
RkpqiRktiA
Aperture
RzZyYkiRktiA
Aperture
// 1
0
2
0
cos// −
= =
Φ−−+−






=





=





== ∫ ∫∫∫∫
ω
ρ
π
φ
φωω
φρρ
The spherical wave at Image, for a Circular Aperture, is

60
( )
( ) ( )RkaqJkaqRe
R
E
E RktiA
// 1
−






= ω
where
( ) ( )
∫
+
−
=
π
π
2
0
cos
2
dve
i
uJ vuvmi
m
m
Bessel Functions (of the first kind)
E. Hecht, “Optics”
The spherical wave at Image, for a Circular Aperture, is

61
Irradiance ∗
=
∗
==×== EEHEHESI
EH
µ
εµ
ε
2
1
2
1
2
1
( ) ( )
( )
( ) ( )
( )
2
1
2
1
2
22
/
/2
0
/
/22
2
1






=





== ∗
Rkaq
RkaqJ
I
Rkaq
RkaqJ
R
aE
EEI A π
µ
ε
µ
ε
D
nnn
λ
θθ 44.22 ==
Circular Aperture
( )
Daaak
RkaquuJ n
Rq
λλ
λ
π
θ
θ
22.1
2
22.1
2
83.383.3
sin83.3/0
sin/
1 ====⇒==⇒=
=

62
Distribution of Energy in the Diffraction Pattern
at the Focus of a Perfect Circular Lens
Ring f/(λf#) Peak Energy in ring
Illumination (%)
Central max 0 1 83.9
1st
dark ring 1.22 0
1st
bright ring 1.64 0.017 7.1
2nd
dark 2.24 0
2nd
bright 2.66 0.0041 2.8
3rd
dark 3.24 0
3rd
bright 3.70 0.0016 1.5
4th
dark 4.24 0
4th
bright 4.74 0.00078 1.0
5th
dark 5.24 0

63
SOLO Diffraction
Circular Aperture
Hecht p.469

64
Airy Rings
In 1835, Sir George Biddell Airy, developed the formula for diffraction pattern,
of an image of a point source in an aberration-free optical system, using the wave
theory.

65

66
Rayleigh’s Criterion (1902)
The images are said to be just resolved when the
center of one Airy Disk falls on the first minimum
of the Airy pattern of the other image.
The minimum resolvable angular separation or
angular limit is:
D
nnn
λ
θθ 44.22 ==
Sparrow’s Criterion
At the Rayleigh’s limit there is a central minimum
Or saddle point between adjacent peaks.
Decreasing the distance between the two point
sources cause the central dip to grow shallower and
ultimately to disappear. The angular separation
corresponding to that configuration is the Sparrow’s
Limit.

67
Resolution – Diffraction Limit
Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley

68
Diffraction Grating
Hecht 478

69
Diffraction Grating
Hecht 480

70
Diffraction Grating
Hecht 479a Hecht 479b
Hecht 485
Table of Content

71
SOLO Diffraction
Rectangular Aperture
( ) ( ) ( ) ( ) ( ) ( )
∫∫Σ





 −⋅−
= dS
r
rr
kjrrU
r
rkjj
rU SSF
'2
exp
'
'exp 11 ρρ
τ
λ


define
1788-1827
P
Q 1x
1y
η
ξ
ρ
 r

O
θ
Screen
Image
plane
1O
Sn1Σ
1r

z
Sn1
'r
1ξ
2ξ
1η
2η
( ) ( ) ( ) ( )
( ) ( ) ( )
∫∫
∫∫





 −





 −
=





 −





 −
=
=
Σ
2
1
2
1
'
2
2
exp
'
2
2
exp
'2
'exp
'2
exp
'2
exp
'
'exp
2
1
2
10
2
2
1
2
10
η
η
ξ
ξ
λ
π
η
λ
ηπ
ξ
λ
ξπ
π
ηξ
ηξ
λ
d
r
y
jd
r
x
j
r
rkjUkj
dd
r
y
kj
r
x
kj
r
rkjUj
rU
k
F

( ) ( )


 ≤≤≤≤
=
elsevere
U
rrU SS
0
& 21110 ηηηξξξ
τ

For a Rectangular Aperture
( ) ξ
λ
α
λ
ξ
α d
r
d
r
x
'
2
'
2
:
2
12
=
−
=
( )
∫∫ 





=




 − 2
1
2
1
2
2
1
2
exp
2
'
'
2
2
exp
α
α
ξ
ξ
αα
πλ
ξ
λ
ξπ
dj
r
d
r
x
j
( ) ( )212111
'
2
&
'
2
ξ
λ
αξ
λ
α −=−= x
r
x
r
Therefore
( ) η
λ
β
λ
η
β d
r
d
r
y
'
2
'
2
:
2
12
=
−
=
( ) ( )212111
'
2
&
'
2
η
λ
βη
λ
β −=−= y
r
y
r
( )
∫∫ 





=




 − 2
1
2
1
2
2
1
2
exp
2
'
'
2
2
exp
β
β
η
η
ββ
πλ
η
λ
ηπ
dj
r
d
r
y
j

72
SOLO Diffraction
1788-1827
P
Q 1x
1y
η
ξ
ρ
 r

O
θ
Screen
Image
plane
1O
Sn1Σ
1r

z
Sn1
'r
1ξ
2ξ
1η
2η
( ) ( ) ( ) ( )
( )
∫∫
∫∫












=





 −





 −
=
2
1
2
1
2
1
2
1
220
2
1
2
10
2
exp
2
exp
2
'exp
'
2
2
exp
'
2
2
exp
'
'exp
β
β
α
α
η
η
ξ
ξ
ββ
π
αα
π
η
λ
ηπ
ξ
λ
ξπ
λ
djdj
rkjUj
d
r
y
jd
r
x
j
r
rkjUj
rU F

Define Fresnel Integrals
( )
( ) ∫
∫






=






=
α
α
αα
π
α
αα
π
α
0
2
0
2
2
sin:
2
cos:
dS
dC
( ) ( )αααα
πα
SjCdj +=





∫0
2
2
exp
( ) ( ) ( ) ( )[ ] ( ) ( )[ ] 2
1
2
1
2
'exp0 β
β
α
α ββαα SjCSjC
rkjUj
rU F ++=

Using the Fresnel Integrals we can write
( ) ( ) 5.0±=∞±=∞± SC

73
SOLO Diffraction
1788-1827
Hecht p.499

74
SOLO Diffraction
1788-1827
P
Q 1x
1y
η
ξ
ρ
 r

O
θ
Screen
Image
plane
1O
Sn1Σ
1r

z
Sn1
'r
1ξ
2ξ
1η
2η

75
SOLO Diffraction
Cornu Spiral
Fresnel Integrals are defined as
( ) ( ) ∫∫ 





=





=
uu
duuuSduuuC
0
2
0
2
2
sin:&
2
cos:
ππ
( ) ( )uSjuCduuj
u
+=





∫0
2
2
exp
π
( ) ( ) 5.0±=∞±=∞± SC
Marie Alfred Cornu professor at the École Polytechnique in Paris
established a graphical approach, for calculating intensities in
Fresnel diffraction integrals.
The Cornu Spiral is defined as the
plot of S (u) versus C (u)
duuSd
duuCd






=






=
2
2
2
sin
2
cos
π
π
( ) ( ) duSdCd =+
22
Therefore u may be thought as measuring arc
length along the spiral.
“Méthode nouvelle pour la discussion des problèmes de diffraction dans le cas
d’une onde cylindrique”, J.Phys.3 (1874), 5-15,44-52

76
SOLO Diffraction
Cornu Spiral (continue – 1)
( ) ( ) ∫∫ 





=





=
uu
duuuSduuuC
0
2
0
2
2
sin:&
2
cos:
ππ
( ) ( )uSjuCduuj
u
+=





∫0
2
2
exp
π
( ) ( ) 5.0±=∞±=∞± SC
The Cornu Spiral is defined as the plot of S (u) versus C (u)
duuSdduuCd 





=





= 22
2
sin&
2
cos
ππ
( ) ( ) duSdCd =+
22






=












= 2
2
2
2
tan
2
cos
2
sin
u
u
u
Cd
Sd π
π
π
Therefore every point on the curve makes the angle
with the real ( C ) axis.
2
2
u
π
The radius of curvature of Cornu Spiral is
The tangent vector of Cornu Spiral is
SuCuT 1
2
sin1
2
cos 22






+





=
ππ
( ) ( ) u
SuCuu
udTdSdCdTd πππ
π
ρ
1
1
2
cos1
2
sin
1
/
1
/
1
22
22
=












+





−
==
+
= 
showing that the curve spirals toward the limit points.
∫ 





2
1
2
2
cos
u
u
duu
π
∫ 





2
1
2
2
sin
u
u
duu
π
∫ 





2
1
2
2
exp
u
u
duuj
π
Table of Content

77
ELECTROMAGNETICSSOLO
KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION
• GREEN’s FUNCTION
Define the Green’s Function as a particular solution of the following Helmholtz
Non-homogeneous Differential Equation:
( ) ( ) ( )SFFSFSS rrrrG
v
rrG

−−=+∇ πδ
ω
4;; 2
2
2
where δ (x) is the Dirac function
( )
( )









=
=∞
≠
=
∫
∞
∞−
1
0
00
dxx
x
x
x
δ
δ
Let use the Fourier Transformation to write
( ) ( ) ( ) ( )
( )[ ] ( )[ ] ( )[ ]
( )
( ) ( ) ( )[ ]{ }
( )
( )[ ]∫
∫ ∫ ∫
∫∫∫
−⋅−=
−+−+−−=








−−







−−







−−=
−−−=−
∞
∞−
∞
∞−
∞
∞−
∞
∞−
∞
∞−
∞
∞−
3
3
3
exp
2
1
exp
2
1
exp
2
1
exp
2
1
exp
2
1
dkrrkj
dkdkdkzzkyykxxkj
dkzzjkdkyyjkdkxxjk
zzyyxxrr
SF
zyxSFzSFySFx
zSFzySFyxSFx
SFSFSFSF


π
π
πππ
δδδδ
zyx
zyx
dkdkdkdk
zkykxkk
=
++=
→→→
3
111

where
Paul Dirac
1902-1984
Joseph Fourier
1768-1830

78
• GREEN’s FUNCTION (continue – 1)
Let use the Fourier Transformation to write
( ) ( ) ( )[ ]∫ −⋅−= SFFS rrkjkgdkrrG

exp,; 3
ω
Hence
( ) ( )[ ] ( )
( )[ ]∫∫ −⋅−−=−⋅−





+∇ SFSFS rrkjdkrrkjkgdk
v

exp
2
4
exp, 3
3
3
2
2
2
π
π
ω
ω
or
( ) ( )[ ] ( )[ ]{ } ( )
( )[ ]∫∫ −⋅−−=−⋅−+−⋅∇ SFSFSFS rrkjdkrrkjkrrkjkgdk

exp
2
4
expexp, 3
3
223
π
π
ω

n
i
iSS
1=
=
iS
nS
dV
dSn
→
1
V
Fr

Sr

F
0r SF rrr

−=
PositionSourcerS

PositionFieldrF


79
Let compute:
( )[ ] ( ) ( ) ( )[ ]{ }( )
( ) ( ) ( )[ ]{ }
( ) ( ) ( )[ ]{ }
( )[ ] ( ) ( ) ( )[ ] ( )[ ]SFSFSFS
SFzSFySFxSzyx
SFzSFySFxzyxS
SFzSFySFxSSSFS
rrkjkrrkjkjkjrrkjkj
zzkyykxxkjzjkyjkxjk
zzkyykxxkjzjkyjkxjk
zzkyykxxkjrrkj


−⋅−−=−⋅−−⋅−=−⋅−∇⋅−=
−+−+−−∇⋅





−−−=






−+−+−−





−−−⋅∇=
−+−+−−∇⋅∇=−⋅−∇
→→→
→→→
expexpexp
exp111
exp111
expexp
2
2
Therefore: ( ) ( )[ ] ( )
( )[ ]∫∫ −⋅−−=−⋅−





+− SFSF rrkjdkrrkj
v
kkgdk

exp
2
4
exp, 3
32
2
23
π
πω
ω
Because this is true for all k and ω, we obtain
( )












−
=
2
2
2
2
1
2
1
,
v
k
kg
ωπ
ω


80
( ) ( ) ( )[ ] ( )[ ]∫∫ −⋅−












−
=−⋅−= SFSFFS rrkj
v
k
d
dkrrkjkgdkrrG

exp
2
1
exp,;
2
2
2
3
2
3
ω
ω
π
ω
We can see that the integral in k has to singular points for
v
k
ω
±=
To find the integral let change ω by ω + jδ where δ is a small negative number
( ) [ ]
( )∫












+
−
⋅−
=
2
2
2
3
2
exp
2
1
;
v
j
k
rkj
dkrrG FS
δωπ


where .SF rrr

−=

81
In the plane ω we close the integration path by the semi-circle with
and the singular points on the upper side, for τ > 0 (for t > t’)
( )
( )
( )[ ]∫ ∫
∞
∞−
⋅−












+
−
= rkj
v
j
k
d
dktrtrG FS

ωτ
δω
ω
π
exp
4
1
',;,
2
2
2
3
3
∞→r
( ) ( ) ( )'00exp ttdjf
UPC
>>=∫ τωωτω
( ) ( ) ( )'00exp ttdjf
DOWNC
<<=−∫ τωωτω
( ) ( )
( ) ( )
( ) ( )







<−
>−
=−
∫
∫
∫
∞≤≤∞−
+
∞≤≤∞−
+∞
∞− 0exp
0exp
exp
τωωτω
τωωτω
ωωτω
ω
ω
DOWN
UP
C
C
djf
djf
djf
δjvk −− δjvk −
ωRe
ωIm

82
( )
( )[ ] ( )
( )[ ] ( )[ ]
( )[ ] ( )[ ]∫∫∫ 







−+−++
⋅−
=⋅−












+
−
=⋅−












+
−
=
∞
∞− CC
jvkjvk
rkjv
drkj
v
j
k
d
rkj
v
j
k
d
I
δωδω
ωτ
ωωτ
δω
ω
ωτ
δω
ω

 exp
expexp
2
2
2
2
2
2
2
Let use the Cauchy Integral for a complex function f (z) continuous on a
closed path C, in the complex z plane: ( ) ( ) ( )0
0
2lim2
0
zfjzfjdz
zz
zf
zz
C
ππ ==
− →∫
We have:
( )[ ]
( )[ ]
( )[ ]
( )[ ]
( ) ( ) ( ) ( ) ( )
k
kvrkj
v
vk
jkv
vk
jkv
rkjvj
jvk
rkjv
j
jvk
rkjv
jI
vkvk
τ
π
ττ
π
δω
ωτ
π
δω
ωτ
π
δωδω
sinexp
2
2
exp
2
exp
exp2
exp
2lim
exp
2lim
2
2
0,
2
0,



⋅−
=





+
−
−
⋅−−=
++
⋅−−
+
−+−
⋅−−
=
→→→−→
Therefore, we can write:
( ) ( )[ ] ( ) ( )
∫∫ ∫
⋅−
−=












−
⋅−
=
k
vkrkj
dk
v
v
k
rkj
ddktrtrG FS
τ
πω
ωτ
ω
π
sinexp
2
exp
4
1
',;, 3
2
2
2
2
3
3



83
Let use spherical coordinates relative to vector:
( ) ( )
∫
−
⋅−
=
2
2
2
3
2
exp
2
1
;
v
k
rkj
dkrrG FS
ωπ







=
=
=





=
=
=
rr
r
r
kk
kk
kk
z
y
x
z
y
x
0
0
cos
sinsin
cossin
θ
ϕθ
ϕθ
ϕθ dk sin
θdk
dk
( )( ) ϕθθ dddkkdk sin23
=
θ
ϕ
r
x
y
z

84
( )kvd
v
r
jkv
v
r
jkv
v
r
jkv
v
r
jkv
r ∫
∞
∞−












−−+











+−−











+−











−
=
4
expexpexpexp
1
ττττ
π
r
v
rr
tt
v
rr
tt SFSF







 −
+−−






 −
−−
=

'' δδ
r
v
r
v
r
dx
v
r
jx
v
r
jx
r






+−





−
=


















+−











−= ∫
∞
∞−
τδτδ
ττ
π
expexp
2
11
( ) ( )kvd
v
r
jkv
v
r
jkv
r
v
kvd
v
r
jkv
v
r
jkv
r ∫∫
∞
∞−
∞
∞−












+−−











−−
+












+−











−
=
4
expexp
4
expexp
1
ττ
π
ττ
π
( ) ( ) ( ) ( )
∫
∞
∞−





 −−





 −−
= dk
j
jvkjvk
j
jkrjkr
r
v
2
expexp
2
expexp ττ
π
( ) ( )∫∫
∞
∞−
∞
−
−=
−
−= dkkr
v
k
k
r
dkkr
v
k
k
r
sin
1
sin
2
2
2
2
2
0
2
2
2
2
ωπωπ
( ) ( ) ( )
∫∫
∞∞ =
=





 −−
−
−=
−
−
−
=
0
2
2
2
2
0 0
2
2
2
2
2
expexp2cosexp1
dk
j
jkrjkr
v
k
k
r
dk
jkr
jkr
v
k
k
ωπ
θ
ωπ
πθ
θ
( )
∫∫ ∫
∞
−
−
=
0 0
2
0
2
2
2
2
2
sin
cosexp
2
1
π π
θϕθ
ω
θ
π
dkddk
v
k
jkr
( ) ( )
∫
−
−
=
2
2
2
3
2
cosexp
2
1
;
v
k
jkr
dkrrG FS
ω
θ
π

( )
∫∫
∞
−
−
=
0 0
2
2
2
2
sin
cosexp1
π
θθ
ω
θ
π
dkdk
v
k
jkr

85
We can see that represents a progressive wave
and represents a regressive wave:







 −
+−=






 −
−−
v
rr
tt
v
rr
tt
SFSF

'' δδ







 −
−−=






 −
+−
v
rr
tt
v
rr
tt
SFSF

'' δδ
Hence ( )
SF
SFSF
FS
rr
v
rr
tt
v
rr
tt
trtrG 


−







 −
+−−






 −
−−
=
''
',;,
δδ
We shall consider only the progressive wave and use:
( )
SF
SF
FS
rr
v
rr
tt
trtrG 


−







 −
+−
=
'
',;,
δ
Retarded Green Function
The other solution is:
( )
SF
SF
FS
rr
v
rr
tt
trtrG 


−







 −
−−
=
'
',;,
δ
Advanced Green Function

n
i
iSS
1=
=
iS
nS
dV
dSn
→
1
V
Fr

Sr

F
0r SF rrr

−=
PositionSourcerS

PositionFieldrF

Table of Content

86
SOLO
( ) ( ) ( )[ ] { }gFTydxdyfxfjyxgffG yxyx =+−= ∫∫Σ
π2exp,:,
The two dimensional Fourier Transform F of the function f (x, y)
The Inverse Fourier Transform is
( ) ( ) ( )[ ] { }GFTfdfdyfxfjyxgyxg
F
yxyx
1
2exp,, −
=+= ∫∫ π
( )
( )
( ) ( )[ ] { }Σ
Σ
=+−= ∫∫ fFTddkkjfkkF yxyx ηξηξηξ
π
exp,
2
1
:, 2
Two Dimensional
Fourier Transform
In Fraunhofer Diffraction we arrived two dimensional
Fourier Transform of the field within the aperture
P
0P
Q 1x
0x
1y
0y
η
ξ
Sr'

Sr

ρ
 r

O
Sθ θ
Screen
Image
plane
Source
plane
0O
1O
Sn1
Σ
1r
0r

SSS rn θcos11 =⋅
θcos11 =⋅ rnS
z
Sn1
'r
 Fr

F
rPP

=0
SrQP

=0
rQP

=
SrOP '0

=
'1 rOO

=
Using kx = 2 π fx and ky = 2 π fy we obtain:
Diffractions

87
SOLO
( ) ( ){ } ( ){ } ( ){ }yxhFTyxgFTyxhyxgFT ,,,, βαβα +=+
1. Linearity Theorem
Fourier Transform Theorems
( ){ } ( )yx ffGyxgFT ,, =
2. Similarity Theorem
( ){ } 





=
b
f
a
f
G
ba
ybxagFT
yx
,
1
,If then
( ){ } ( )yx ffGyxgFT ,, =
3. Shift Theorem
( ){ } ( ) ( )[ ]bfafjffGbyaxgFT yxyx +−=−− π2exp,,
If
then
Diffractions

88
SOLO
Fourier Transform Theorems (continue – 1)
( ){ } ( )yx ffGyxgFT ,, =
4. Parseval’s Theorem
( ) ( )∫∫∫∫ = yxyx fdfdffGydxdyxg
22
,,
If
then
( ){ } ( )yx ffGyxgFT ,, =
5. Convolution Theorem
( ) ( ){ } ( ) ( )yxyx ffHffGddyxhgFT ,,,, =−−∫∫ ηξηξηξ
If
then
( ){ } ( )yx ffHyxhFT ,, =and
( ){ } ( )yx ffGyxgFT ,, =
6. Autocorrelation Theorem
( ) ( ){ } ( )2*
,,, yx ffGddyxggFT =−−∫∫ ηξηξηξ
If
then
similarly ( ){ } ( ) ( )∫∫ −−= ηξηξηξηξ ddffGGgFT yx ,,, *2
Diffractions

89
SOLO
Fourier Transform Theorems (continue – 2)
( ){ } ( ){ } ( )yxgyxgFTFTyxgFTFT ,,, 11
== −−
7. Fourier Integral Theorem
Diffractions

90
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Fourier Transform for a Circular Symmetric Optical Aperture
To exploit the circular symmetry of g (g (r,θ) = g (r) ) let
make the following transformation
( )
( ) φρφ
φρρ
θθ
θ
sin/tan
cos
sin/tan
cos
1
22
1
22
==
=+=
==
=+=
−
−
yxy
xyx
fff
fff
ryxy
rxyxr
{ } ( ) ( )[ ] ( ) ( )[ ]
( ) ( )
( ) ( )[ ]∫ ∫
∫ ∫∫∫
−−=
+−=+−=
=
=
Σ
a
o
rgrg
a
o
drdrydxd
yx
drjrdrrg
drjrgrdrydxdyfxfjyxggFT
πθ
πθ
θφθρπ
θθφθφρπθπ
2
0
,
2
0
cos2exp
sinsincoscos2exp,2exp,
Use Bessel Function Identity ( ) ( )[ ]∫ −−=
π
θφθ
2
0
0 cosexp dajaJ
( ) ( ){ } ( ) ( )∫==
a
o
rdrrgrJrgFTG ρπρ 2: 00
to obtain
J0 is a Bessel Function of the first kind, order zero.
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For a Circular Pupil of radius a we have
( )



>
≤
=
ar
ar
rg
0
1
Use Bessel Function Identity
J1 is a Bessel Function of the first kind, order one.
( ) ( ){ } ( )∫==
a
o
rdrrJrgFTG ρπρ 2: 00
( ) ( )xJxdJ
x
o
10 =∫ ςςς
( ) ( ){ }
( )
( ) ( )
( )
( ) ( )ρπ
ρπ
ςςς
ρπ
ρπρπρπ
ρπ
ρ
ρπ
aJ
a
dJ
rdrrJrgFTG
a
o
a
o
2
22
1
222
2
1
:
1
2
02
020
==
==
∫
∫
Bessel Functions of the first kind
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Circular Aperture
( ) ( ){ } ( )
( )ρπ
ρπ
ρ
a
aJ
argFTG
2
2
: 12
0 ==
Table of Content
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References
Diffractions
2. Goodman, J.,W., “Introduction to Fourier Optics”, McGraw-Hill, 1968
1. Sommerfeld, A., “Optics, Lectures on Theoretical Physics”, vol. IV,
Academic Press Inc., New York, 1954, Chapter V, “The Theory of Diffraction”,
7. M. Born, E. Wolf, “Principle of Optics – Electromagnetic Theory of Propagation,
Interference and Diffraction of Light”, 6th
Ed., Pergamon Press, 1980,
5. F.A. Jenkins, H.E. White, “Fundamentals of Optics”, 4th
Ed., McGraw-Hill, 1976
8. M.V.Klein, T.E. Furtak, “Optics”, 2nd
Ed., John Wiley & Sons, 1986
6. E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 1979
3. Elmore, W.C., Heald, M., A., “Physics of Waves”, Dover Publications, 1969,
4. Fowles, G., R., “Introduction to Modern Optics”, Dover Publications, 1968, 1975,
Ch. 5, Diffraction
9. J. Meyer-Arendt, “Introduction to Classical & Modern Optics”, 3th Ed.,
Prentince Hall, 1989

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References
[1] M. Born, E. Wolf, “Principle of Optics – Electromagnetic Theory of Propagation,
Interference and Diffraction of Light”, 6th
Ed., Pergamon Press, 1980,
[2] C.C. Davis, “Laser and Electro-Optics”, Cambridge University Press, 1996,
OPTICS

95
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References
[3] E.Hecht, A. Zajac, “Optics ”, 3th
Ed., Addison Wesley Publishing Company, 1997,
[4] M.V. Klein, T.E. Furtak, “Optics ”, 2nd
Ed., John Wiley & Sons, 1986
Table of Content
OPTICS

January 4, 2015 96
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Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 – 2013
Stanford University
1983 – 1986 PhD AA

97
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1788-1827
Circular Aperture
Hecht p.491
Hecht p.492

98
SOLO Diffraction
1788-1827
Circular Obstacles

99
SOLO Diffraction
1788-1827
Fresnel Zone Plate

100
SOLO Diffraction
1788-1827
Fresnel Diffraction by a Slit
Hecht p.504 a
Fresnel Diffraction
Hecht p.504 b

101
SOLO Diffraction
1788-1827
Semi-Infinite Opaque Screen
Hecht p.506 a
Hecht p.506

102
SOLO Diffraction
1788-1827
Semi-Infinite Opaque Screen
Hecht p.506 a
Hecht p.507

103
SOLO Diffraction
1788-1827
Diffraction by a Narrow Obstacle

104
Optics - DiffractionSOLO
Elmore p.409

105
Hecht p.487 a
Hecht p.487 b
Hecht p.489
Hecht p.490 a
Hecht p.490 b

106
Hecht p.500

107
Jenkins p.320 Jenkins p.323
Jenkins p.343
Klein Furata p.362

108
Reynolds 19
Reynolds 20 a
Reynolds 20 b

109
Reynolds 62 a
Reynolds 62 b
Reynolds 62 c

110
Reynolds 63 a
Reynolds 63 b

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