The science of light P. Ewart Oxford Physics: Second Year, Optics - - PowerPoint PPT Presentation

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The science of light P. Ewart Oxford Physics: Second Year, Optics - - PowerPoint PPT Presentation

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The science of light P. Ewart Oxford Physics: Second Year, Optics The story so far Geometrical optics: image formation Physical optics: interference diffraction Phasor methods: physics of interference Fourier methods:

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  • P. Ewart

The science of light

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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

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Oxford Physics: Second Year, Optics

t 2t=x source images 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

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Oxford Physics: Second Year, Optics

Wedged Parallel Point Source Non-localised Equal thickness Non-localised Equal inclination Extended Source Localised in plane

  • f Wedge

Equal thickness Localised at infinity Equal inclination

Summary: fringe type and localisation

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Oxford Physics: Second Year, Optics

  • 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

  • verlap 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:

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Oxford Physics: Second Year, Optics

  • 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

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Oxford Physics: Second Year, Optics

   I N = 2         

4

N-slit grating N-1 minima I ~ N2 Peaks at  = n2

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Oxford Physics: Second Year, Optics

      I             N = 3

4 9

N-slit grating

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

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Oxford Physics: Second Year, Optics

      N    N     m N

I

  4 N

2

N-slit grating

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Oxford Physics: Second Year, Optics

  • 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

  • verlap 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:

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SLIDE 11

N-slit diffraction grating

Order Order 1 2 3 4 2

(a) (b) (c)

I(q) = I(0) sin2{N/2} sin2{/2} sin2{/2} {/2}2  = (2l) d sinq   (l) a sinq

. slit separation

slit width x 2  , 

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N-slit diffraction grating

Order Order 1 2 3 4 2

(a) (b) (c)

. slit separation

slit width x 2  , 

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Oxford Physics: Second Year, Optics

l l d l  D 

min

2 N

2n

(a)

l l d l q Dq 

min Dql

qp

(b)

 = (2l)dsinq

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Oxford Physics: Second Year, Optics Diffracted light Reflected light Diffracted light Reflected light

   q q  (a) (b)

Diffracted light Reflected light

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Oxford Physics: Second Year, Optics

Order Order 1 2 3 4 2

(a) (b) (c)

Unblazed Blazed

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Oxford Physics: Second Year, Optics

(a) Dxs (b) Dxs q grating f1 f1 f2 f2

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Oxford Physics: Second Year, Optics

Instrument width = Resolving Power = lDlInst = nDnInst = 2Wl

Maximum path difference in units of wavelength

Maximum path difference 1 . Instrument function

DnInst = 1 2W n

Instrument width

(Grating spectrometer):

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SLIDE 18
  • Interference by division of wavefront:

The Diffraction Grating spectrograph

  • Interference by division of amplitude:

2-beams - The Michelson Interferometer

Oxford Physics: Second Year, Optics

Optical Instruments for Spectroscopy

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Oxford Physics: Second Year, Optics

Michelson interferometer

  • Fringe properties – interferogram
  • Resolving power
  • Instrument width

Albert Abraham Michelson 1852 –1931

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Oxford Physics: Second Year, Optics

t

Light source Detector

M2 M

/

2

M1

CP BS

Michelson Interferometer

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Oxford Physics: Second Year, Optics

t 2t=x source images 2t=x  path difference cos x  circular fringe constant 

Fringes of equal inclination Localized at infinity

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Oxford Physics: Second Year, Optics

x n

Input spectrum Detector signal Interferogram x = 2t

I(x) = ½ I0 [ 1 + cos 2nx ]

½ I0

I(x)

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Oxford Physics: Second Year, Optics

x x x xmax I( ) n I( ) n I( ) n

1

2

(a) (b) (c)

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Oxford Physics: Second Year, Optics

DvInst = 1/xmax

Instrument width = 1 . Maximum path difference

Resolving Power = Maximum path difference in units of wavelength

  • Michelson Interferometer

Size of instrument

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Oxford Physics: Second Year, Optics

LIGO, Laser Interferometric Gravitational-Wave Observatory 4 Km

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LIGO, Laser Interferometric Gravitational-Wave Observatory Vacuum ~ 10-12 atmosphere Precision ~ 10-18 m

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Oxford Physics: Second Year, Optics

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

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Oxford Physics: Second Year, Optics

x x x xmax I( ) n I( ) n I( ) n

1

2

(a) (b) (c)

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Oxford Physics: Second Year, Optics

Coherence Longitudinal coherence Coherence length: lc ~ 1/Dn

_

Transverse coherence Coherence area: size of source or wavefront with fixed relative phase

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Oxford Physics: Second Year, Optics

Transverse coherence r ws d   r >> d d sin  < l

Interference Fringes

 < ld d defines coherence area

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Michelson Stellar Interferometer

Measures angular size of stars

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Oxford Physics: Second Year, Optics

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

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Oxford Physics: Second Year, Optics

d Eo t r

1 2

t1 Eot t

1 2

t r r

1 2 1

2

t r r

1 2 1

3 2

t r r

1 2 1

Eot t r r

1 2 1 2e

  • i

t r r

1 2 1

2

2

Eot t r r

1 2 1 2

2 2

  • i2

e

Eot t r r

1 2 1 2

3 3

  • i3

e

q t2 t1

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Oxford Physics: Second Year, Optics

d Eo tr t Eot

2

tr

3

tr

5

tr

2

Eot r

2 2

  • i

e

tr

4

Eot r

2 4

  • i2

e

Eot r

2 6

  • i3

e

q

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Oxford Physics: Second Year, Optics

d Eo tr t Eot

2

tr

3

tr

5

tr

2

Eot r

2 2

  • i

e

tr

4

Eot r

2 4

  • i2

e

Eot r

2 6

  • i3

e

q

Airy Function

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Oxford Physics: Second Year, Optics

m2 (m+1)2  I( ) 

The Airy function: Fabry-Perot fringes   (l)d.cosq

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Oxford Physics: Second Year, Optics

Extended Source Lens Lens Fabry-Perot interferometer d Screen

  (l)2d.cosq Fringes of equal inclination Localized at infinity

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Oxford Physics: Second Year, Optics

m2 (m+1)2  I( ) 

The Airy function: Fabry-Perot fringes DFWHM  Finesse = DFWHM

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Finesse = 10 Finesse = 100

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Oxford Physics: Second Year, Optics

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

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Oxford Physics: Second Year, Optics

m2 (m+1)2  I( ) 

The Airy function: Fabry-Perot fringes DFHWM 

Finesse, F = DFHWM DInst = F

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Oxford Physics: Second Year, Optics

DvInst = 1 . 2d.F = 1/xmax

Instrument width = 1 . Maximum path difference

  • Fabry-Perot Interferometer: Instrument width

effective

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Oxford Physics: Second Year, Optics

m2 (m+1)2  I( ) 

The Airy function: Fabry-Perot fringes DInst  Finesse = D

Free Spectral Range

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Oxford Physics: Second Year, Optics

FSR

(m+1) d n nDn mth

th

I( ) d

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Oxford Physics: Second Year, Optics

I( ) d

d n nDnR DnInst

Resolution criterion:

DnR  DnInst

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Oxford Physics: Second Year, Optics

qm-1 mth fringe (on axis) rm-1

Aperture size to admit only mth fringe

Typically aperture ~

10

rm-1 Centre spot scanning

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Oxford Physics: Second Year, Optics

Maxwell’s equations

2 2 2

t E E

r

  • r

      

2 2 2

t H H

r

  • r

      

) . ( r k t i

  • e

E E

H n H E

  • r
  • r

     1  

E n H    constant

E.M. Wave equations

index refractive n 

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Oxford Physics: Second Year, Optics

n2

E1 E2 E1 n1 .n2

Boundary conditions: Tangential components of E and H are continuous

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Oxford Physics: Second Year, Optics

n2

Eo ET E1 Eo E1 .no .n1 ..nT ko .k1 .k1 ko .kT  (a) (b)

l

Layer

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T T O O

E k n n i k E E                    

1 1 1

sin cos

 

T T O O O

E k n k i E E n  

1 1

cos sin ) (     

(9.7) (9.8)

; cos

1

k A  ; sin 1

1 1

 k n i B          

; sin

1 1

 k in C   

1

cosk D 

T T O O T T O O O O

Dn C n Bn An Dn C n Bn An r E E         

T T O O O O T

Dn C n Bn An n t E E      2

(9.10) (9.11)

l l l l l l l l

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SLIDE 51

Quarter Wave Layer: A=0, B=-i/n1, C=-in1, D=0

, 4 l  

T T O O T T O O O O

Dn C n Bn An Dn C n Bn An r E E         

R =

2 2 1 2 1 2

n n n n n n r

T O T O

  

Anti-reflection coating, n1

2 ~ nO nT ,

R → 0

e.g. blooming on lenses l

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Oxford Physics: Second Year, Optics

R e f l e c t a n c e 0.04 0.01 Uncoated glass 400 500 600 700 wavelength nm 3 layer 2 layer 1 layer

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Oxford Physics: Second Year, Optics

Eo Eo .no .nT ko ko

n2

.nH nL

n2

.nH nL

Multi-layer stack

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SLIDE 54

T T O O

E k n n i k E E                    

1 1 1

sin cos

 

T T O O O

E k n k i E E n  

1 1

cos sin ) (     

(9.7) (9.8)

; cos

1

k A  ; sin 1

1 1

 k n i B          

; sin

1 1

 k in C   

1

cosk D 

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

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Oxford Physics: Second Year, Optics

.nT .nH .nH nL .nH nL nL .nH .nH .nH nL nL

l

Interference filter; composed of multi-layer stacks

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Oxford Physics: Second Year, Optics

z y x

Eoz E Eoy

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Oxford Physics: Second Year, Optics

z y x

Source

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

Oxford Physics: Second Year, Optics

z y x E =

z

E sin( )

  • kx

Source Source

xo E =

y

E cos( )

  • kx

E

z y x E = 0

z

xo E =

y

E

  • E

(a) x = x , t =

  • (b) x = x , t = kx /

  

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Oxford Physics: Second Year, Optics

z y x

Source

Looking towards source: Clockwise rotation of E

Right circular polarization

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Oxford Physics: Second Year, Optics

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

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SLIDE 61

Oxford Physics: Second Year, Optics

E

q

y z E Eoz Eoy y z

(a) (b)

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Oxford Physics: Second Year, Optics

Polarization states defined by: Ey = Eoy cos(kx - t) Ez = Eoz cos(kx - t - )

  •  = 0

Linearly Polarized

  •  ≠ 0

Elliptically Polarized

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Oxford Physics: Second Year, Optics

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

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Oxford Physics: Second Year, Optics

Unpolarized light defined by: Ey = Eoy cos(kx - t) Ez = Eoz cos(kx - t – (t)

  •  = (t)

Phase varies randomly in time unpolarized light

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Oxford Physics: Second Year, Optics

         = 0, 5

Note: if Eoy = Eoz these ellipses become circles

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SLIDE 66

Polarized Light

  • Production
  • Analysis
  • Manipulation

Oxford Physics: Second Year, Optics

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Oxford Physics: Second Year, Optics

z z x,y x,y nex nex no no

(a) (b)

Positive Negative Uniaxial Crystals

Optic Axis

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Oxford Physics: Second Year, Optics

z z x,y x,y nex nex no no

(a) (b)

Positive Negative Uniaxial Crystals

  • -ray

e-ray E

propagation

Phase shift:  = (2l)[no-ne]l

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SLIDE 69

Oxford Physics: Second Year, Optics

z z x,y x,y nex nex no no

(a) (b)

Positive Negative Uniaxial Crystals

  • -ray

e-ray E

  • -ray
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Oxford Physics: Second Year, Optics

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

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Oxford Physics: Second Year, Optics

z y x

Ez EP Ey

q

z y x

Ez EP Ey

q

Half-wave plate: introduces -phase shift

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SLIDE 72

Oxford Physics: Second Year, Optics

e-ray e-ray e-ray

  • -ray
  • -ray
  • -ray

Optic axis Optic axis

Prism Polarizers

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SLIDE 73

Oxford Physics: Second Year, Optics

d1 d2 d1 d2

(a) (b)

A B

Babinet-Soleil Babinet

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SLIDE 74

Oxford Physics: Second Year, Optics

Linearly polarized light after passage through the /4 plate l Elliptically polarized light E y z y z

 q

E y z y z

(a) (b)

Analysis of polarized light

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SLIDE 75

Oxford Physics: Second Year, Optics

Linearly polarized light after passage through the /4 plate l Elliptically polarized light E y z y z

 q

E y z y z

(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, tanq→cos

Find Eoz:Eoy and 

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SLIDE 76

Oxford Physics: Second Year, Optics

A B

(a)

A B C

(b)

A B C D

(c)

.
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