Download Oxford Physics: Second Year, Optics

The science of light
P. Ewart
Oxford Physics: Second Year,
Optics
• Lecture notes: On web site
NB outline notes!
• Textbooks:
Hecht, Optics
Lipson, Lipson and Lipson, Optical Physics
Further reading:
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
Astronomical observatory, Hawaii, 4200m above sea level.
Oxford Physics: Second Year,
Optics
Multi-segment
Objective mirror,
Keck Obsevatory
Oxford Physics: Second Year,
Optics
Hubble Space Telescope, HST,
In orbit
Oxford Physics: Second Year,
HST Deep Field
Oldest objects
in the Universe:
13 billion years
Optics
Oxford Physics: Second Year,
Optics
HST Image: Gravitational lensing
Oxford Physics: Second Year,
SEM Image:
Insect head
Optics
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
Oxford Physics: Second Year,
Optics
Optics in Physics
•
•
•
•
•
•
Astronomy and Cosmology
Microscopy
Spectroscopy and Atomic Theory
Quantum Theory
Relativity Theory
Lasers
Oxford Physics: Second Year,
Optics
Lecture 1: Waves and Diffraction
• Interference
• Analytical method
• Phasor method
Oxford Physics: Second Year,
Optics
T

u
u
t, x
Time
or distance
axis
t,z
Phase change of 2p
Oxford Physics: Second Year,
Optics
P
r1
r2

d
dsin
D
Phasor diagram
Optics
Imaginary
Oxford Physics: Second Year,
u

Real
Oxford Physics: Second Year,
Optics
Phasor diagram for 2-slit interference
up

uuo/r
o
r

uuoo/r
r
Oxford Physics: Second Year,
Optics
Lecture 2: Diffraction theory
• Diffraction at a finite slit
- Analytical method
- Phasor method
• 2-D diffraction at apertures
• Fraunhofer diffraction
Oxford Physics: Second Year,
Optics
Diffraction from a single slit
+a/2
y
dy

r+ ysin
r
ysin
-a/2
D
P
Oxford Physics: Second Year,
Optics
Intensity pattern from
diffraction at single slit
1.0
0.8
0.6
2
sinc ()
0.4
0.2
0.0
-10
p
p
-5
p
0

p
5
p
10
p
Oxford Physics: Second Year,
Optics
P
+a/2

-a/2
r
asin
D
Oxford Physics: Second Year,
Phasors and resultant
at different angles 
Optics

RP= RO
/

RP

Oxford Physics: Second Year,
Optics

R
R sin /2
RP
R

Oxford Physics: Second Year,
Optics
Phasor arc to
first minimum
Phasor arc to
second minimum
Oxford Physics: Second Year,
Optics
y
x

z

Diffraction from a rectangular aperture
Oxford Physics: Second Year,
Optics
Diffraction pattern from circular aperture
Intensity
y
x
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,
Lecture 3:
Optics
Diffraction theory
and wave propagation
• Fraunhofer diffraction
• Fresnel’s theory of wave propagation
• Fresnel-Kirchoff diffraction integral
Oxford Physics: Second Year,
Optics
Fraunhofer Diffraction
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
How linear is linear?
Oxford Physics: Second Year,
Optics

R
source
 < 

a
R
a
R
R
diffracting
aperture
observing
point
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
P
A
O
B
f
C
Oxford Physics: Second Year,
Optics
u
v
Equivalent lens
system
Diffracted
waves
imaged
Fraunhofer diffraction: in image plane of system
Oxford Physics: Second Year,
Optics
(a)
O
P
(b)
O
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
-z
P
0
z
Plane wave
surface
unobstructed
Huygens secondary sources on wavefront at -z
radiate to point P on new wavefront at z
=0
Oxford Physics: Second Year,
Optics
n
n
rn
rn
P
qq
Construction of elements of equal area on wavefront
Oxford Physics: Second Year,
Optics
p
p
Rp
Rp
(q+/2)
(q+/2)
q
q
First Half Period Zone
Resultant, Rp, represents amplitude from 1st HPZ
Oxford Physics: Second Year,
/2
p
Optics
Phase difference of /2
at edge of 1st HPZ
q
P
O
q
Oxford Physics: Second Year,
Optics
Rp
As n a infinity resultant a ½ diameter of 1st HPZ
Oxford Physics: Second Year,
Optics
Fresnel-Kirchoff diffraction integral
i

uo dS
ikr
up   
 (n, r ) e

r
Fresnel-Kirchoff diffraction integral
Oxford Physics: Second Year,
Babinet’s Principle
Optics
Oxford Physics: Second Year,
Optics
Lectures 1 - 3: The story so far
• Scalar diffraction theory:
Analytical methods
Phasor methods
• Fresnel-Kirchoff diffraction integral:
propagation of plane waves
Oxford Physics: Second Year,
Joseph Fraunhofer
(1787 - 1826)
Phase at observation is
linear function of position
in aperture:
 = k sin y
Optics
Augustin Fresnel
(1788 - 1827)
up  
i
Gustav Robert Kirchhoff
(1824 –1887)

uo dS
ikr

(
n,
r
)
e
 r
Fresnel-Kirchoff Diffraction Integral
Oxford Physics: Second Year,
Optics
Lecture 4: Fourier methods
• Fraunhofer diffraction as a
Fourier transform
• Convolution theorem –
solving difficult diffraction problems
• Useful Fourier transforms
and convolutions
Oxford Physics: Second Year,
Optics
Fresnel-Kirchoff diffraction integral:
i
uo dS
up   
 (n.r )eikr

r
Simplifies to:

u p  A(  )    u ( x)eix dx

where
 = ksin
Note: A() is the Fourier transform of u(x)
The Fraunhofer diffraction pattern is proportional to the
Fourier transform of the transmission function
(amplitude function) of the diffracting aperture
Oxford Physics: Second Year,
Optics
The Convolution function:
h( x )  f ( x )  g ( x ) 

 f ( x' ).g ( x  x' ).dx'

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
T
Monochromatic
Wave
.o  p/T
Fourier
Transform

Oxford Physics: Second Year,
-function
Optics
V
V(x)
xo
Fourier transform
Power spectrum
V()V()* = V2
= constant
x
V()


Oxford Physics: Second Year,
Optics
Comb of -functions
V(x)
xS
Fourier transform
x
V()

Oxford Physics: Second Year,
Optics
Constructing a double slit function by convolution
g(x-x’ )
h(x)
f(x’ )
Oxford Physics: Second Year,
Optics
Triangle as a convolution of two “top-hat” functions
g(x-x’ )
f(x’ )
h(x)
This is a self-convolution or Autocorrelation function
Oxford Physics: Second Year,
Optics
Lecture 5: Theory of imaging
• Fourier methods in optics
• Abbé theory of imaging
• Resolution of microscopes
• Optical image processing
• Diffraction limited imaging
Oxford Physics: Second Year,
Optics
• Heat transfer theory:
- greenhouse effect
• Fourier series
• Fourier synthesis and analysis
• Fourier transform as analysis
Joseph Fourier (1768 –1830)
Ernst Abbé (1840 -1905)
Oxford Physics: Second Year,
Optics
Abbé theory of imaging (coherent light)
Fourier plane
d’
d
a

v(x)
u(x)
u
f
v
D
Fourier plane
Oxford Physics: Second Year,
Optics
The compound microscope
Objective magnification = v/u
Eyepiece magnifies real image of object
Diffracted orders from high spatial frequencies miss
the objective lens
max defines the numerical aperture… and resolution
So high spatial frequencies are
missing from the image.
Oxford Physics: Second Year,
Fourier
plane
Image
plane
Optics
Image processing
Oxford Physics: Second Year,
Optics
Optical simulation of
“X-Ray diffraction”
(a) and (b) show objects:
double helix
at different angle of view
a
b
a’
b’
Diffraction patterns of
(a) and (b) observed in
Fourier plane
Computer performs
Inverse Fourier transform
To find object “shape”
Oxford Physics: Second Year,
Optics
amplitude or phase object
Fourier plane
d’
d
a

v(x)
u(x)
u
f
v
D
Oxford Physics: Second Year,
Optics
Schlieren photography
Refractive
index
variation
Fourier
Plane
Source
Collimating
Lens
Imaging
Lens
Knife
Edge
Image
Plane
Schlieren photography of combustion
Schlieren film of ignition
Courtesy of Prof C R Stone
Eng. Science, Oxford University
Oxford Physics: Second Year,
Optics
Diffraction pattern from circular aperture
Intensity
y
x
Point Spread Function, PSF
Oxford Physics: Second Year,
Optics
Lecture 6: Optical instruments and
Fringe localisation
• Interference fringes
• What types of fringe?
• Where are fringes located?
• Interferometers
Oxford Physics: Second Year,
Optics
• Interference by:
• Division of wavefront
- Young’s slits:
- N-slit grating:
2 beams
multiple beams
• Division of amplitude
- Michelson:
- Fabry-Perot:
2 beams
multiple beams
What kind of interference fringes
and where are they?
Oxford Physics: Second Year,
Optics
What kind of interference fringes
and where are they?
Depends on:
Type of light source – Point source
- Extended source
Division of wavefront – Point sources
Division of amplitude by – Reflection from:
Wedged surfaces
Parallel surfaces
Oxford Physics: Second Year,
Optics
Division of wavefront
Young’s slits
non-localised
Non-localized
fringes
fringes
Plane
waves
Usually observed under
Fraunhofer conditions –
large distance from slits
Oxford Physics: Second Year,
Division of wavefront
Optics
Diffraction grating
to
8
Fringes localized at infinity
Z
Plane
waves


f
Fraunhofer condition
Oxford Physics: Second Year,
P’
Optics
Division of Amplitude
Point source
Wedged reflecting surfaces
Non-localized
P
Fringes of
Equal thickness
O
Oxford Physics: Second Year,
Optics
Division of Amplitude
Point source
Parallel reflecting surfaces
P’
P
O

’
Non-localized
Fringes of
Equal inclination
Oxford Physics: Second Year,
Optics
Extended source
Wedged Reflecting surfaces
P’
R’
Division of Amplitude
Localized in
plane of wedge,
near apex
P
Equal thickness
R
S
O
Oxford Physics: Second Year,
Optics
Division of Amplitude
Extended source
Parallel reflecting surfaces
t
images
source
Localized at infinity
Fringes of
equal inclination
2t=x
path difference xcos

2t=x
circular fringe
constant 
Oxford Physics: Second Year,
Optics
Summary: fringe type and localisation
Point
Source
Extended
Source
Wedged
Parallel
Non-localised
Equal thickness
Non-localised
Equal inclination
Localised in plane Localised at infinity
of Wedge
Equal thickness
Equal inclination