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P. Ewart
The science of light
• Lecture notes: On web site
NB outline notes!
• Textbooks:
Hecht, Optics
Klein and Furtak, Optics
Lipson, Lipson and Lipson, Optical Physics
Brooker, Modern Classical Optics
• Problems: Material for four tutorials plus past Finals papers A2
• Practical Course: Manuscripts and Experience
Oxford Physics: Second Year, Optics
Structure of the Course
1. Physical Optics (Interference)
Diffraction Theory (Scalar)
Fourier Theory
2. Analysis of light (Interferometers)
Diffraction Gratings
Michelson (Fourier Transform)
Fabry-Perot
3. Polarization of light (Vector)
Oxford Physics: Second Year, Optics
Electronics
Optics
Quantum
Electronics
Quantum
Optics
Photonics
10-7 < T < 107 K; e- > 109 eV; superconductor
Electromagnetism
Oxford Physics: Second Year, Optics
Astronomical observatory, Hawaii, 4200m above sea level.
Oxford Physics: Second Year, Optics
Hubble Space Telescope, HST,
In orbit
Oxford Physics: Second Year, Optics
HST Deep Field
Oldest objects
in the Universe:
13 billion years
Oxford Physics: Second Year, Optics
HST Image: Gravitational lensing
Oxford Physics: Second Year, Optics
SEM Image:
Insect head
Oxford Physics: Second Year, Optics
Coherent Light:
Laser physics:
Holography,
Telecommunications
Quantum optics
Quantum computing
Ultra-cold atoms
Laser nuclear ignition
Medical applications
Engineering
Chemistry
Environmental sensing
Metrology ……etc.!
Oxford Physics: Second Year, Optics
CD/DVD Player: optical tracking assembly
• Astronomy and Cosmology
• Microscopy
• Spectroscopy and Atomic Theory
• Quantum Theory
• Relativity Theory
• Lasers
Oxford Physics: Second Year, Optics
Optics in Physics
Oxford Physics: Second Year, Optics
Lecture 1: Waves and Diffraction
• Interference
• Analytical method
• Phasor method
• Diffraction at 2-D apertures
Oxford Physics: Second Year, Optics
u
t, x
T
Time
or distance
axis
t,z
u
Phase change of 2p
Oxford Physics: Second Year, Optics
dsinq
d q
r1r2
P
D
Oxford Physics: Second Year, Optics
Real
Imag
inar
y
u
Phasor diagram
Oxford Physics: Second Year, Optics
u /roup
u /ro
Phasor diagram for 2-slit interference
uor
uor
Oxford Physics: Second Year, Optics
ysinq
+a/2
-a/2
qr
r y+ sinq
P
D
y dy
Diffraction from a single slit
Oxford Physics: Second Year, Optics
-10 -5 0 5 10
0.0
0.2
0.4
0.6
0.8
1.0
pp p ppp
sinc2()
Intensity pattern from
diffraction at single slit
Oxford Physics: Second Year, Optics
asinq
+a/2
-a/2
qr
P
D
Oxford Physics: Second Year, Optics
q
q/
RO
R
R =
P
P
Phasors and resultant
at different angles q
Oxford Physics: Second Year, Optics
R RP
R sin /2
R
Oxford Physics: Second Year, Optics
Phasor arc tofirst minimum
Phasor arc tosecond minimum
Oxford Physics: Second Year, Optics
y
x
z
q
Diffraction from a rectangular aperture
Oxford Physics: Second Year, Optics
Intensity
y
x
Diffraction pattern from circular aperture
Point Spread Function
Diffraction from a circular aperture
Diffraction from circular apertures
Oxford Physics: Second Year, Optics
Dust
pattern
Diffraction
pattern
Basis of particle sizing instruments
Oxford Physics: Second Year, Optics
Lecture 2: Diffraction theory
and wave propagation
• Fraunhofer diffraction
• Huygens-Fresnel theory of
wave propagation
• Fresnel-Kirchoff diffraction
integral
Oxford Physics: Second Year, Optics
y
x
z
q
Diffraction from a circular aperture
Oxford Physics: Second Year, Optics
Fraunhofer Diffraction
How linear is linear?
A diffraction pattern for which the
phase of the light at the
observation point is a
linear function of the position
for all points in the diffracting
aperture is Fraunhofer diffraction
Oxford Physics: Second Year, Optics
R
R R
R
a a
r r
diffracting aperture
source observingpoint
r <
Oxford Physics: Second Year, Optics
Fraunhofer Diffraction
A diffraction pattern formed in the image
plane of an optical system is
Fraunhofer diffraction
Oxford Physics: Second Year, Optics
O
P
A
BC
f
Oxford Physics: Second Year, Optics
u v
Equivalent lenssystem
Fraunhofer diffraction: in image plane of system
Diffracted
waves
imaged
Oxford Physics: Second Year, Optics
(a)
(b)
O
O
P
P
Equivalent lens system:
Fraunhofer diffraction is independent of aperture position
Oxford Physics: Second Year, Optics
Fresnel’s Theory of wave propagation
Oxford Physics: Second Year, Optics
dS
n
r
Pz-z 0
Plane wave surface
Huygens secondary sources on wavefront at -z
radiate to point P on new wavefront at z = 0
unobstructed
Oxford Physics: Second Year, Optics
rn
q
rn
P
Construction of elements of equal area on wavefront
rn
q
rn
Oxford Physics: Second Year, Optics
(q+ /2)
q
rp
Rp
First Half Period Zone
(q+/2)
q
rp
Rp
Resultant, Rp, represents amplitude from 1st HPZ
Oxford Physics: Second Year, Optics
rp
/2
q
q
PO
Phase difference of /2
at edge of 1st HPZ
Oxford Physics: Second Year, Optics
As n a infinity resultanta ½ diameter of 1st HPZ
Rp
Oxford Physics: Second Year, Optics
Fresnel-Kirchoff diffraction integral
ikrop e
r
Suiu )rn,(
d
Oxford Physics: Second Year, Optics
Babinet’s Principle
Oxford Physics: Second Year, Optics
Lectures 1 - 2: The story so far
• Scalar diffraction theory:
Analytical methods
Phasor methods
• Fresnel-Kirchoff diffraction integral:
propagation of plane waves
Oxford Physics: Second Year, Optics
Gustav Robert Kirchhoff
(1824 –1887)
Joseph Fraunhofer
(1787 - 1826)
Augustin Fresnel
(1788 - 1827)
Fresnel-Kirchoff Diffraction Integral
ikrop e
r
Suiu )rn,(
d
Phase at observation is
linear function of position
in aperture:
= k sinq y
Oxford Physics: Second Year, Optics
Lecture 3: Fourier methods
• Fraunhofer diffraction as a
Fourier transform
• Convolution theorem –solving difficult diffraction problems
• Useful Fourier transforms and convolutions
Oxford Physics: Second Year, Optics
ikrop ern
r
uiu ).(
dS
xexuAu xip d)()(
Fresnel-Kirchoff diffraction integral:
Simplifies to:
where = ksinq
Note: A() is the Fourier transform of u(x)
The Fraunhofer diffraction pattern is the Fourier transformof the amplitude function in the diffracting aperture
Oxford Physics: Second Year, Optics
'd).'().'()()()( xxxgxfxgxfxh
The Convolution function:
The Convolution Theorem:
The Fourier transform, F.T., of f(x) is F()
F.T., of g(x) is G()
F.T., of h(x) is H()
H() = F().G()
The Fourier transform of a convolution of f and g
is the product of the Fourier transforms of f and g
Oxford Physics: Second Year, Optics
Monochromatic
Wave
Fourier
Transform
T
.o p/T
Oxford Physics: Second Year, Optics
xo x
V(x)
V( )
-function
Fourier transform
Power spectrum
V()V()* = V2
= constant
V
Oxford Physics: Second Year, Optics
xS x
V(x)
V( )
Comb of -functions
Fourier transform
g(x-x’) f(x)
h(x)
Constructing a double slit function by convolution
Oxford Physics: Second Year, Optics
g(x-x’) f(x)
h(x)
Triangle as a convolution of two “top-hat” functions
This is a self-convolution or Autocorrelation function
Oxford Physics: Second Year, Optics
Oxford Physics: Second Year, Optics
ikrop ern
r
uiu ).(
dS
xexuAu xip d)()(
Fresnel-Kirchoff diffraction integral:
Simplifies to:
where = ksinq
Note: A() is the Fourier transform of u(x)
The Fraunhofer diffraction pattern is the Fourier transformof the amplitude function in the diffracting aperture
Joseph Fourier (1768 –1830)
Oxford Physics: Second Year, Optics
• Heat transfer theory:
- greenhouse effect
• Fourier series
• Fourier synthesis and analysis
• Fourier transform as analysis
Oxford Physics: Second Year, Optics
Lecture 4: Theory of imaging
• Fourier methods in optics
• Abbé theory of imaging
• Resolution of microscopes
• Optical image processing
• Diffraction limited imaging
Ernst Abbé (1840 -1905)
Oxford Physics: Second Year, Optics
a
dd’
f Du v
u(x) v(x)
Fourier plane
q
Abbé theory of imaging (coherent light)
Oxford Physics: Second Year, Optics
The compound microscope
Objective magnification = v/uEyepiece magnifies real image of object
Diffracted orders from high spatial frequencies miss the objective lens
So high spatial frequencies are missing from the image.
qmax defines the numerical aperture… and resolution
Oxford Physics: Second Year, Optics
Fourier
plane
Image
plane
Image processing
Oxford Physics: Second Year, Optics
Optical simulation of
“X-Ray diffraction”
a b
a’ b’
(a) and (b) show objects:
double helix
at different angle of view
Diffraction patterns of
(a) and (b) observed in
Fourier plane
Computer performsInverse Fourier transformTo find object “shape”
PIV particle image velocimetry
Oxford Physics: Second Year, Optics
• Two images recorded
in short time interval
• Each moving particle gives
two point images
• Coherent illumination
of small area produces
“Young’s” fringes in
Fourier plane of a lens
• CCD camera records
fringe system –
input to computer
to calculate velocity vector
PIV particle image velocimetry
Oxford Physics: Second Year, Optics
Oxford Physics: Second Year, Optics
a
dd’
f Du v
u(x) v(x)
Fourier plane
q
phase object
amplitude object
or
Oxford Physics: Second Year, Optics Schlieren photography
Source
CollimatingLens
ImagingLens
KnifeEdge
ImagePlane
Refractiveindex variation
Fourier
Plane
Schlieren photography of combustion
Schlieren film of ignition
Courtesy of Prof C R Stone
Eng. Science, Oxford University
Oxford Physics: Second Year, Optics
Intensity
y
x
Diffraction pattern from circular aperture
Point Spread Function, PSF
Oxford Physics: Second Year, Optics
Lecture 5: Optical instruments and
Fringe localisation
• Interference fringes
• What types of fringe?
• Where are fringes located?
• Interferometers
Oxford Physics: Second Year, Optics
non-localised fringes
Plane
waves
Young’s slits
Z
Oxford Physics: Second Year, Optics Diffraction grating
q q
to 8
f
Plane
waves
Fringes localised at infinity: Fraunhofer
Oxford Physics: Second Year, Optics
P’
P
O
Wedged
reflecting surfaces
Point
source
Oxford Physics: Second Year, Optics
OPP’
q
q’Parallel
reflecting surfaces
Point
source
Oxford Physics: Second Year, Optics
P’
P
R
OS
R’
Wedged
Reflecting surfaces
Extended source
Oxford Physics: Second Year, Optics
t
2t=x
sourceimages
2t=x
path difference cos x
circular fringe
constant
Parallel
reflecting
surfaces
Extended
source
Oxford Physics: Second Year, Optics
Wedged Parallel
Point
Source
Non-localised
Equal thickness
Non-localised
Equal inclination
Extended
Source
Localised in plane
of Wedge
Equal thickness
Localised at infinity
Equal inclination
Summary: fringe type and localisation
Oxford Physics: Second Year, Optics
Lecture 6: Fourier methods
• Useful Fourier transforms
• Useful convolutions
• Fourier synthesis and analysis