the science of light - university of oxfordewart/optics 2011 part 2... · 2011-05-06 · fringes...

P. Ewart

The science of light

Oxford Physics: Second Year, Optics

• Geometrical optics: image formation

• Physical optics: interference → diffraction

• Phasor methods: physics of interference

• Fourier methods: Fraunhofer diffraction

= Fourier Transform

Convolution Theorem

• Diffraction theory of imaging

• Fringe localization: interferometers

The story so far


t

2t=x

sourceimages

2t=x

path difference cos x

circular fringe

constant

Parallel reflecting surfaces Extended source

Fringes localized at infinity

∞

Circular fringe constant

Fringes of equal inclination


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


• Dispersion: dq/dl, angular separation of wavelengths

• Resolving Power: lDl, dimensionless figure of merit

• Free Spectral Range: extent of spectrum covered by interference pattern before

overlap with fringes of same l and different order

• Instrument width: width of pattern formed by instrument with monochromatic light.

• Etendu: or throughput –

a measure of how efficiently the instrument uses available light.

Optical Instruments for SpectroscopySome definitions:


• Fringe formation – phasors

• Effects of groove size – Fourier methods

• Angular dispersion

• Resolving power

• Free Spectral Range

• Practical matters, blazing and slit widths

The Diffraction Grating Spectrometer


0

I

N = 2

4

N-slit grating

N-1 minima

I ~ N2

Peaks at = n2


0

I

N = 3

4

9

N-slit grating

Peaks at n2, N-1 minima, I ~ N2 1st minimum at 2/N


N

N m N

I

0

4

N2

N-slit grating


• Dispersion: dq/dl, angular separation of wavelengths

• Resolving Power: lDl, dimensionless figure of merit

• Free Spectral Range: extent of spectrum covered by interference pattern before

overlap with fringes of same l and different order

• Instrument width: width of pattern formed by instrument with monochromatic light.

• Etendu: or throughput –

a measure of how efficiently the instrument uses available light.

Optical Instruments for SpectroscopySome definitions:

N-slit diffraction grating

Order

Order

0 1 2 3 4

0

2

(a)

(b)

(c)

I(q) = I(0) sin2{N/2} sin2{/2}

sin2{/2} {/2}2

= (2l) d sinq (l) a sinq

. slit separation slit width

x 2 ,

N-slit diffraction grating

Order

Order

0 1 2 3 4

0

2

(a)

(b)

(c)

. slit separation slit width

x 2 ,


l ldl

D min2N

2n

(a)

l ldl

q

Dq min Dql

qp

(b) = (2l)dsinq


Diffractedlight

Reflectedlight

Diffractedlight

Reflectedlight

q

q

(a)

(b)

Diffracted lightReflected

light


Order

Order

0 1 2 3 4

0

2

(a)

(b)

(c)

Unblazed

Blazed


(a)

Dxs

(b)

Dxs q

grating

f1

f1

f2

f2


Instrument width =

Resolving Power = lDlInst

= nDnInst

= 2Wl

Maximum path difference in units of wavelength

Maximum path difference

1 .

Instrument function

DnInst =1

2W

nInstrument width (Grating spectrometer):

• Interference by division of wavefront:

The Diffraction Grating spectrograph

• Interference by division of amplitude:

2-beams -

The Michelson Interferometer


Optical Instruments for Spectroscopy


Michelson interferometer

• Fringe properties – interferogram

• Resolving power

• Instrument width

Albert Abraham Michelson

1852 –1931


t

Light source

Detector

M2

M/

2 M1

CP BS

Michelson

Interferometer


t

2t=x

sourceimages

2t=x

path difference cos x

circular fringe

constant

Fringes of equal inclinationLocalized at infinity


xn

Input spectrum Detector signal Interferogram

x = 2t

I(x) = ½ I0 [ 1 + cos 2nx ]

½ I0

I(x)


x

x

x

xmax

I( )n

I( )n

I( )n

1

2

(a)

(b)

(c)


DvInst = 1/xmax

Instrument width = 1 .


Resolving Power = Maximum path difference

in units of wavelength

-

Michelson Interferometer

Size of instrument


LIGO, Laser Interferometric Gravitational-Wave Observatory

4 Km

LIGO, Laser Interferometric Gravitational-Wave Observatory

Vacuum ~ 10-12 atmosphere

Precision ~ 10-18 m


Michelson interferometer

• Path difference: x = pl

measure l by reference to known lcalibration

• Instrument width: DvInst = 1/xmax

WHY?• Fourier transform interferometer

• Fringe visibility and relative intensities

• Fringe visibility and coherence

Lecture 10

Albert Abraham Michelson

1852 –1931


x

x

x

xmax

I( )n

I( )n

I( )n

1

2

(a)

(b)

(c)


Coherence

Longitudinal coherence

Coherence length: lc ~ 1/Dn_

Transverse coherence

Coherence area: size of source or wavefront

with fixed relative phase


Transverse coherence

r

wsd

r >> d

d sin < l

Interference

Fringes < ld

d defines coherence area

Michelson Stellar Interferometer

Measures angular size of stars


Division of wavefront

fringes

Young’s slits (2-beam) cos2(/2)

Diffraction grating (N-beam) sin2(N/2)

sin2(/2)

Division of amplitude

fringes

Michelson (2-beam) cos2(2)

Fabry-Perot (N-beam) ?

Sharper fringes


d

Eo

t r1 2

t1

Eot t1 2

t r r1 2 12

t r r1 2 13 2

t r r1 2 1

Eot t r r1 2 1 2e-i

t r r1 2 12 2

Eot t r r1 2 1 22 2 -i2

e

Eot t r r1 2 1 23 3 -i3

e

q

t2t1


d

Eo

tr

t

Eot2

tr3

tr5

tr2

Eot r2 2 -i

e

tr4

Eot r2 4 -i2

e

Eot r2 6 -i3

e

q

Oxford Physics: Second Year, Optics d

Eo

tr

t

Eot2

tr3

tr5

tr2

Eot r2 2 -i

e

tr4

Eot r2 4 -i2

e

Eot r2 6 -i3

e

q

Airy Function


m2 (m+1)2

I( )

The Airy function: Fabry-Perot fringes

(l)d.cosq


ExtendedSource

Lens

Lens

Fabry-Perotinterferometer

d

Screen

(l)2d.cosq

Fringes of equal inclinationLocalized at infinity


m2 (m+1)2

I( )


DFWHM

Finesse = DFWHM

Finesse = 10 Finesse = 100


Multiple beam interference:

Fringe sharpness set by Finesse

F = √R

(1-R)

• Instrument width DvInst

• Free Spectral Range FSR

• Resolving power

• Designing a Fabry-Perot

Fabry-Perot Interferometer


m2 (m+1)2

I( )


DFHWM

Finesse, F = DFHWM DInst = F


DvInst = 1 .

2d.F

= 1/xmax

Instrument width = 1 .


-

Fabry-Perot Interferometer: Instrument width

effective


m2 (m+1)2

I( )


DInst

Finesse = D

Free Spectral Range


FSR

(m+1) d

n nDn

mth th

I( )d


I( )d

d

n nDnR

DnInst

Resolution criterion:

DnR DnInst


qm-1

mth fringe (on axis)

rm-1

Aperture size to admit only mth fringe

Typically aperture ~10

rm-1

Centre spot scanning


Maxwell’s equations

2

22

t

EE roro

2

22

t

HH roro

).( rkti

oeEE

Hn

HEo

o

ro

ro

1

EnH constant

E.M. Wave equations

indexrefractiven


n2

E1

E2

E1

n1.n2

Boundary conditions:

Tangential components of E and H are continuous


n2

Eo

ET

E1

Eo E1

.no.n1

..nT

ko.k1

.k1ko

.kT

(a) (b)

l

Layer

TT

OO Ekn

nikEE

1

1

1 sincos

TTOOO EknkiEEn 11 cossin)(

(9.7)

(9.8)

;cos 1kA ;sin1

1

1

kn

iB

;sin 11 kinC 1cos kD

TTOO

TTOO

O

O

DnCnBnAn

DnCnBnAnr

E

E

TTOO

O

O

T

DnCnBnAn

nt

E

E

2

(9.10)

(9.11)

l l

l l

l l l l

Quarter Wave Layer:

A=0, B=-i/n1, C=-in1, D=0

,4l

TTOO

TTOO

O

O

DnCnBnAn

DnCnBnAnr

E

E

R =

2

2

1

2

12

nnn

nnnr

TO

TO

Anti-reflection coating, n12 ~ nO nT , R → 0

e.g. blooming on lenses

l


Ref

lect

ance

0.04

0.01

Uncoated glass

400 500 600 700 wavelength nm

3 layer

2 layer

1 layer


Eo

Eo

.no .nT

ko

ko

n2.nH nLn2.nH nL

Multi-layer stack

TT

OO Ekn

nikEE

1

1

1 sincos

TTOOO EknkiEEn 11 cossin)(

(9.7)

(9.8)

;cos 1kA ;sin1

1

1

kn

iB

;sin 11 kinC 1cos kD

1 + r = ( A + BnT )t

nO( 1 – r ) = (C + DnT )t

1 1 A B 1

+ r = t

nO -nO C D nT

l l

l l

l l l l


.nT.nH.nHnL .nH

nL nL.nH

.nH.nHnL nL

l

Interference filter; composed of multi-layer stacks


z

yx

EozE

Eoy


z

y

x

Source

Circularly polarized light: Ez = Eosin(kxo – t) ; Ey = Eocos(kxo – t)


z

y

x

E = z E sin( ) o okx Source Source

xo E = y E cos( ) o okx

E

z

y

x

E = 0z

xo E = y E o

E

(a) 0 x = x , t = o(b) x = x , t = kx /o o


z

y

x

Source

Looking towards source:

Clockwise rotation of E

Right circular polarization


EL

ER

EP

Superposition of equal

R & L Circular polarization

= Plane polarized light

EL

ER

Superposition of unequal

R & L Circular polarization

= Elliptically polarized light


E

qy

z

EEoz

Eoyy

z

(a) (b)


Polarization states defined by:

Ey = Eoy cos(kx - t) Ez = Eoz cos(kx - t - )

• = 0 Linearly Polarized

• ≠ 0 Elliptically Polarized


Polarization states defined by:

Ey = Eoy cos(kx - t) Ez = Eoz cos(kx - t - )

• = 0 Linear

• = + /2 Eoy = Eoz Left/Right Circular

• /2 Eoy ≠ Eoz Left/Right Elliptical

axes along y,z

• ≠ /2 Eoy ≠ Eoz Left/Right Elliptical

axes at angle to y,z


Unpolarized light defined by:

Ey = Eoy cos(kx - t) Ez = Eoz cos(kx - t – (t)

• = (t) Phase varies randomly in time

unpolarized light


= 0, 5

Note: if Eoy = Eoz

these ellipses become circles

Polarized Light

• Production

• Analysis

• Manipulation



z z

x,y x,y

nex

nexno

no

(a) (b)Positive Negative

Uniaxial Crystals

Optic Axis


z z

x,y x,y

nex

nexno

no


Uniaxial Crystals

o-ray

e-ray

Epropagation

Phase shift:

= (2l)[no-ne]l


z z

x,y x,y

nex

nexno

no


Uniaxial Crystals

o-ray

e-ray

E

o-ray


Linear Input change Output

q = 45O, Eoy=Eoz , l Left/Right Circular

q ≠ 45O, Eoy≠Eoz , l L/R Elliptical

q ≠ 45O, Eoy≠Eoz ≠ ,l Elliptical tilted axis

q ≠ 45O, Eoy≠Eoz , l Linear at q

to input


z

yx

EzEP

Ey

q

z

yx

Ez

EP

Ey

q

Half-wave plate: introduces -phase shift


e-ray e-raye-ray

o-ray

o-ray o-rayOptic axis Optic axis

Prism Polarizers


d1

d2

d1

d2

(a) (b)

A

B

Babinet-SoleilBabinet


Linearly polarized lightafter passage through the

/4 platel

Elliptically polarized light

E

y

z

yz

q

E

y

z

yz

(a) (b)

Analysis of polarized light


Linearly polarized lightafter passage through the

/4 platel

Elliptically polarized light

E

y

z

yz

q

E

y

z

yz

(a) (b)

Analysis of polarized light

Find ratio of

major:minor axes by

rotating linear polarizer

Find angle of

Linear polarization using

l/4 plate and linear polarizer

Eoz/Eoy = tan (q)→q, tanq→cos

Find Eoz:Eoy and


AB

(a)

AB

C

(b)

AB

C D

(c).

the science of light - university of oxfordewart/optics 2011 part 2... · 2011-05-06 · fringes...

Documents