the science of light - university of oxford...2011/10/19 · optical instruments for spectroscopy...

P. Ewart

The science of light

Oxford Physics: Second Year, Optics

• 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


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


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


• Dispersion: dq/dl, angular separation of wavelengths

• Resolving Power: l/Dl, 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 Spectroscopy Some 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

Interference pattern for N-slit diffraction grating

Infinitesimal slit width

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

sin2{/2}

= (2/l) d sinq


finite slit width

Order

Order

0 1 2 3 4

0

2

(a)

(b)

(c)

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

sin2{/2} {g/2}2

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

. slit separation slit width

x 2 , g


Elements of Diffraction Grating Spectrograph


• Dispersion: dq/dl, angular separation of wavelengths

• Resolving Power: l/Dl, 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 Spectroscopy Some definitions:


finite slit width

Order

Order

0 1 2 3 4

0

2

(a)

(b)

(c)

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

sin2{/2} {g/2}2

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

. slit separation slit width

x 2 , g


Section 5.3.2 The angular separation dq between spectral components

differing in wavelength by dq dl

Notes error


l ldl

D min2N

2n

(a)

l ldl

q

Dq min Dql

qp

(b) = (2/l)dsinq

Strictly speaking this is incorrect.

Principal maxima for all wavelengths

occur at = n2

So the maxima at both l and l dl

would lie at the same positions on

the -axis.

Plotted vs angle q the maxima of

these two wavelengths will have

an angular separation Dq

Notes error Fig 5.4(a)


l ldl

D min2N

2n

(a)

l ldl

q

Dq min Dql

qp

(b) = (2/l)dsinq


Diffractedlight

Reflectedlight

Diffractedlight

Reflectedlight

q

q

(a)

(b)

Diffracted light Reflected

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


Resolving Power = Maximum path difference

in units of wavelength

DvInst = 1 ...

2W

-

Resolving Power: l

l

l maxx

D

D

1

Maximum path difference Instrument width =

max

1

x


• Michelson interferometer

• Fringe properties – interferogram

• Resolving power

• Instrument width: Michelson Interferometer

Lecture 9

Instrument width and Resolving power depend only on the

size of the interferometer !

• Interference by division of wavefront:

The Diffraction Grating spectrograph

• Interference by division of amplitude:

2-beams -

The Michelson Interferometer


Optical Instruments for Spectroscopy


t

Light source

Detector

M2

M/

2 M1

CP BS

Michelson

Interferometer


t

2t=x

sourceimages

2t=x


circular fringe

constant

Fringes of equal inclination Localized at infinity


t

2t=x

sourceimages

2t=x


circular fringe

constant Detector

x

Input spectrum Detector signal Interferogram

Detector

signal

Measure on-axis fringe intensity as function of x


x


x = 2t


21 ll qpx 21 ll

p

q

Unknown wavelength

Known wavelength


Michelson (Fourier transform) interferometer

• Measuring wavelength differences

- Resolving power and instrument width.

• Fourier transform interferometer

• Fringe visibility and relative intensities

• Coherence and correlation of light

Lecture 10


t

Light source

Detector

M2

M/

2 M1

CP BS

On axis: xx l

2

2

iaeau Amplitude:

)( 2/2/2/ iii eeaeu

)2/cos(22/ iaeu

Intensity: )2/(cos4 22 aI

)2/(cos2 OII

)]cos(1)[2/1( OII

)]2cos(1[)2/1( xII O Interferogram


x

x

x

xmax

I( )

I( )

I( )

1

2

(a)

(b)

(c)

)]2cos(1[)2/1( 11 xII O

)]2cos(1[)2/1( 22 xII O

Source with two wavelengths

Cycle distance depends on

wavelength or wavenumber

difference


)]2cos(1[)2/1( xII O

)]2cos(1)[()2/1()]2cos(1)[()2/1( 2211 xIxII OO

)]2cos()2cos(2)[()2/1( 21 xxII O

For equal intensities

For two sources:

)()()( 21 OOO III

]2

2cos.2

2cos22)[()2/1( 2121 xxII O

]2

2cos.2

2cos1)[( 2121 xxII O

Oscillation at ~ mean Envelope at ~ D


]2

2cos21

x

The envelope function is slowly varying in x

The cycle from max to min to max occurs over a range If this occurs in the maximum range of movement xmax

then is the minimum wavenumber difference resolvable

Dmax

min

22 x

max

min

1

xInst DD Instrument width

Resolving Power l

maxx

Inst

D

minD

(6.5)

This minimum resolvable wavenumber difference is the instrument width as it

represents the width of the spectrum produced by the instrument for a monochromatic

wave.

(6.6)

max

min

1

xD

max

1

xInst D

Notes error: section 6.2 should be


The Michelson interferometer as a

Fourier transform spectrometer


x


Input spectrum Interferogram

wo time, t

Fourier transform

x


t1~

l1~


( ) )]2cos(1[)2/1( xIxI O

( ) )]2cos(1)[()2/1()]2cos(1)[()2/1( 2211 xIxIxI OO

For two sources:

)]2cos(1)[()2/1(..... source third 33 xIO

For multiple discrete sources:

( ) ( ) )]2cos(1[ )2/1( xIxI iiO

i

( ) ( ) ( ) )2cos( )2/1( )2/1( xIIxI iiO

i

iO

i

For continuous spectral distribution sources:

( ) ( ) dxSIxI iO

0

)2cos( )2/1(

( ) ( ) dxSIxI iO

0

)2cos( 2/

( ) ( ) dxSxF i

)2cos(

( )S Cosine Fourier transform of

( ) ( ) dxxIxIS O

)2cos(2

Spectrum = F.T. (Interferogram - constant background)


Coherence

Longitudinal coherence

Coherence length: lc ~ 1/D _

Spectral width of light

The maximum path difference over which fringes are visible is a measure of the coherence length and the spectral bandwidth


Coherence

Longitudinal coherence

Coherence length: lc ~ 1/D _

Transverse coherence

Coherence area: size of source or wavefront

with fixed relative phase


Transverse coherence

r

ws d

r >> d

d sin < l

Interference

Fringes < l/d

d defines coherence area

Michelson Stellar Interferometer

Measures angular size of stars


LIGO, Laser Interferometric Gravitational-Wave Observatory

4 Km

Two instruments: Washington and Louisiana, USA

LIGO, Laser Interferometric Gravitational-Wave Observatory

Vacuum ~ 10-12 atmosphere

Precision ~ 10-18 m


The Fabry-Perot interferometer

• Multiple beam interference (division of amplitude)

• The Airy function

• Fringe sharpness - Finesse

Lecture 11


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 inclination Localized at infinity


m2 (m+1)2

I( )


D

Finesse = /D

Finesse = 10 Finesse = 100


Scanning Fabry-Perot interferometer

Sequence of fringe patterns as plate separation, d, varies

d


The Fabry-Perot interferometer

Fringe sharpness set by Finesse

F = √R

(1-R)

• Multiple beam interference (division of amplitude)

• Instrument width

• Free spectral range, FSR

• Resolving power

• Practical matters: designing a Fabry-Perot

Lecture 12

InstD


m2 (m+1)2

I( )


D

Finesse = /D


m2 (m+1)2

I( )


D

Finesse = /D

D

on axis d22 For monochromatic light

Instrument width

( )I


DvInst = 1 .

2d.F

= 1/xmax

Instrument width = 1 .

Maximum path difference

-

Fabry-Perot Interferometer


m2 (m+1)2

I( )


DInst

Finesse = /D

Free Spectral Range


FSR

(m+1) d

D

mth th

I( )d


FSR

d

I(d)

D

m m+1 m m+1

FSR D


I( )d

d

DR

DInst

Resolution criterion:

DR DInst


qm-1

mth fringe (on axis)

rm-1

Aperture size to admit only mth fringe

Typically aperture ~ 10

rm-1

Centre spot scanning


Fine structure of Balmer -line {32S½ – 22P1/2 , 3/2 }

in cooled deuterium:

Doppler width m

T

nm 016.0D fsl

1cm 365.0 D fs

x

frr pp

l22

1

2 2

Notes chapter 4... radii of fringes in focal plane of lens with focal length f:

(4.3)


Reflection at dielectric boundaries

• Reflection at air/dielectric surface

• Effects of a single dielectric layer on a surface

• Anti-reflection coatings

• High reflection coatings

• Interference filters

Lecture 13


Maxwell’s equations

2

22

t

EE roro

2

22

t

HH roro

).( rkti

oeEE w

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)(

(8.7)

(8.8)

;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

(8.10)

(8.11)

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


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)(

(8.7)

(8.8)

;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


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


.nT.nH.nHnL .nH

nL nL.nH

.nH.nHnL nL

l/

Interference filter; composed of multi-layer stacks


Reflection at dielectric boundaries (2)

• Reflection at oblique incidence

• Effects of polarization – direction of E-field

• Fresnel’s equations

• Total internal reflection

• Evanescent waves

Lecture 14


Oblique incidence

senkrecht: perpendicular

x

z


Reflection of p-polarized light

H-field

out of plane

( ) ( )(8.23)

cossin

sincos2

cossincossin

sincos2

1

2

qq

q

qq

q

P

P

E

E

Error in notes

( ) ( )(8.23)

cossin

sincos2

cossincossin

sincos2

1

2

qq

q

qq

q

P

P

E

E


Reflection of s-polarized light

E-field

out of plane

(8.26) )sin(

)sin(

cossinsincos

cossinsincos

1

2

q

q

qq

qq

S

S

E

E

(8.26) )sin(

)sin(

cossinsincos

cossinsincos

1

1

q

q

qq

qq

S

S

E

E

Error in notes

(8.23) )cos()sin(

sincos2

cossincossin

sincos2

1

2

qq

q

qq

q

P

P

E

E

(8.24) )tan(

)tan(

cossincossin

cossincossin

1

1

q

q

qq

qq

P

P

E

E

)25.8( )sin(

sincos2

cossinsincos

sincos2

1

2

q

q

qq

q

S

S

E

E

(8.26) )sin(

)sin(

cossinsincos

cossinsincos

1

1

q

q

qq

qq

S

S

E

E


Fresnel Equations


Reflection vs angle of incidence

Unpolarized light

(8.23) )cos()sin(

sincos2

cossincossin

sincos2

1

2

qq

q

qq

q

P

P

E

E

(8.24) )tan(

)tan(

cossincossin

cossincossin

1

1

q

q

qq

qq

P

P

E

E

)25.8( )sin(

sincos2

cossinsincos

sincos2

1

2

q

q

qq

q

S

S

E

E

(8.26) )sin(

)sin(

cossinsincos

cossinsincos

1

1

q

q

qq

qq

S

S

E

E


Fresnel Equations

(8.24) )tan(

)tan(

cossincossin

cossincossin

1

1

q

q

qq

qq

P

P

E

E


Fresnel Equations

(8.24) )tan(

)tan(

1

1

q

q

r

E

EP

P

(8.24) )tan(

)tan(

1

1

q

q

r

E

EP

P

2)(

q )tan( q r tends to zero

P-polarized light suffers no reflection at this angle

(8.27) tan 1

2

n

nB q

qB is Brewster’s Angle



(8.23a) 2

12

1

1

2

nn

n

E

EP

P

(8.24a) 12

12

1

1

nn

nn

E

EP

P

)25.8( 2

12

1

1

2 ann

n

E

ES

S

(8.26a) 12

12

1

1

nn

nn

E

ES

S

Notes error

Fresnel equations (using Snell’s Law for q = 0 )

Should be


(8.23a) 2

12

1

1

2

nn

n

E

EP

P

(8.24a) 12

12

1

1

nn

nn

E

EP

P

)25.8( 2

12

1

1

2 ann

n

E

ES

S

(8.26a) 12

12

1

1

nn

nn

E

ES

S

Fresnel equations (using Snell’s Law for q = 0 )

SP EE

11


Total Internal Reflection, TIR

Evanescent wave

Note error


Total Internal Reflection, TIR

Evanescent wave • Reflected amplitude = incident amplitude

– total reflection.

•Transmitted beam has the following

characteristics:

(a) its wave velocity is v1/sinq (b) it travels parallel to the interface

(c) amplitude decays exponentially

perpendicular to the surface

(d) neither plane nor transverse wave

– has component parallel to surface

(e) its Poynting vector is zero.

Should be


Frustrated Total Internal Reflection, TIR

Evanescent wave

Wave “tunnels” through barrier: Quantum effect

Oxford Physics: Second Year, Optics Frustrated Total Internal Reflection, TIR

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)



Lecture 15

• Linear, circular and elliptical polarization:

- a mathematical description

Lecture 16

• Production of polarized light

• Manipulating polarized light

• Analysis of polarization state

• Interference of polarized light

Polarization of light


z

yx

EozE

Eoy


z

y

x

Source

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


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 w


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 - wt) Ez = Eoz cos(kx - wt - )

• = 0 Linearly Polarized

• ≠ 0 Elliptically Polarized


Lecture 16

• Production of polarized light

• Manipulating polarized light

• Analysis of polarization state

• Interference of polarized light

Polarization of light




• = 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 along y,z

Eoy ≠ Eoz axes at angle to y,z




• = 0 Linearly Polarized

• ≠ 0 Elliptically Polarized

/ / / / / / = 0, 5

2

2

2

2

2

sincos2 oz

z

oy

y

oz

z

oy

y

E

E

E

E

E

E

E

E


/ / / / / / = 0, 5

Note: if Eoy = Eoz these ellipses become circles


Unpolarized light defined by:

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

• = (t) Phase varies randomly in time

unpolarized light

Polarized Light

• Production

• Manipulation

• Analysis



qB 1. Fresnel reflection

at Brewster’s angle

s-polarized light

(partial reflection)

p-polarized and

remaining

s-polarized light

2. Polaroid sheet

(tourmaline crystals

in plastic)

Differential absorption

of orthogonally

polarized light


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

(a) (b)Positive Negative

Uniaxial Crystals

o-ray

e-ray

E propagation

Phase shift:

= (2/l)[no-ne]l

l

e- ray

o- ray


e-ray e-raye-ray

o-ray

o-ray o-rayOptic axis Optic axis

Prism Polarizers


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

*Quarter-wave plate Half-wave plate


z

yx

EzEP

Ey

q

z

yx

Ez

EP

Ey

q

Half-wave plate: introduces -phase shift


d1

d2

d1

d2

(a) (b)

A

B

Babinet-Soleil Babinet

Compensators


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,

(and angle q – approximately)

by rotating linear polarizer

• Make linear polarization with l/4 plate

• Find angle of linear polarization (q + )

using linear polarizer - precisely

Eoz/Eoy = tan (q)→q, tanq→cos

Find Eoz:Eoy and

q


AB

(a)

AB

C

(b)

AB

C D

(c).


1.


2.


3.


4.


5.

the science of light - university of oxford...2011/10/19 · optical instruments for spectroscopy...

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