oxford physics: second year, optics - university of oxford diffraction pattern formed in the image...

P. Ewart

The science of light

• 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

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, 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

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

dsin

d

r1

r2

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

uo r

uo 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

ysin

+a/2

-a/2

r

r y+ sin

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

asin

+a/2

-a/2

r

P

D

Oxford Physics: Second Year, Optics

/

RO

R

R =

P

P

Phasors and resultant

at different angles

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

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 3: Diffraction theory

and wave propagation

• Fraunhofer diffraction

• Fresnel’s theory of wave propagation

• Fresnel-Kirchoff diffraction integral

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

diffracting aperture

source observingpoint

<

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

n

P

Construction of elements of equal area on wavefront

rn

q

n

Oxford Physics: Second Year, Optics

(q+ /2)

q

p

Rp

First Half Period Zone

(q+/2)

q

p

Rp

Resultant, Rp, represents amplitude from 1st HPZ

Oxford Physics: Second Year, Optics

p

/2

q

q

PO

Phase difference of /2

at edge of 1st HPZ

Oxford Physics: Second Year, Optics

As n a infinity resultant a ½ diameter of 1st HPZ

Rp

Oxford Physics: Second Year, Optics

Fresnel-Kirchoff diffraction integral

ikrop e

r

Suiu )rn,(

d

Fresnel-Kirchoff diffraction integral

Oxford Physics: Second Year, Optics

Babinet’s Principle

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, 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 sin y

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

ikrop ern

r

uiu ).(

dS

xexuAu xi

p d)()(

Fresnel-Kirchoff diffraction integral:

Simplifies to:

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

'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

Lecture 5: Theory of imaging

• Fourier methods in optics

• Abbé theory of imaging

• Resolution of microscopes

• Optical image processing

• Diffraction limited imaging

Joseph Fourier (1768 –1830)

Oxford Physics: Second Year, Optics

• Heat transfer theory:

- greenhouse effect

• Fourier series

• Fourier synthesis and analysis

• Fourier transform as analysis

Ernst Abbé (1840 -1905)

Oxford Physics: Second Year, Optics

a

dd’

f Du v

u(x) v(x)

Fourier plane

Abbé theory of imaging (coherent light)

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

So high spatial frequencies are missing from the image.

max 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 performs Inverse Fourier transform To find object “shape”

Oxford Physics: Second Year, Optics

a

dd’

f Du v

u(x) v(x)

Fourier plane

amplitude or phase object

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 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: 2 beams

- N-slit grating: multiple beams

• Division of amplitude - Michelson: 2 beams

- Fabry-Perot: 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

non-localised fringes

Plane

waves

Division of wavefront Young’s slits

Usually observed under Fraunhofer conditions – large distance from slits

Non-localized fringes

Z

Oxford Physics: Second Year, Optics

to 8

f

Plane

waves

Fraunhofer condition

Division of wavefront Diffraction grating

Fringes localized at infinity

Oxford Physics: Second Year, Optics

P’

P

O

Wedged reflecting surfaces Point source

Division of Amplitude

Fringes of Equal thickness

Non-localized

Oxford Physics: Second Year, Optics

OPP’

’

Parallel reflecting surfaces Point source

Division of Amplitude

Fringes of Equal inclination

Non-localized

Oxford Physics: Second Year, Optics

P’

P

R

OS

R’

Wedged Reflecting surfaces

Extended source

Division of Amplitude

Localized in plane of wedge, near apex

Equal thickness

Oxford Physics: Second Year, Optics

t

2t=x

sourceimages

2t=x

path difference cos x

circular fringe

constant

Parallel reflecting surfaces Extended source

Division of Amplitude

Localized at infinity

Fringes of equal inclination

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