The science of light P. Ewart
Oxford Physics: Second Year, Optics Parallel reflecting surfaces t images source Extended source path difference xcos 2t=x Fringes localized at infinity Circular fringe constant 2t=x circular fringe constant Fringes of equal inclination
Oxford Physics: Second Year, Optics Summary: fringe type and localisation 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
Structure of the Course 1. Physical Optics (Interference) Diffraction Theory (Scalar) Fourier Theory Properties of interference fringes 2. Analysis of light (Interferometers) Diffraction Gratings Michelson (Fourier Transform) Fabry-Perot 3. Polarization of light (Vector)
Optical Instruments for Spectroscopy Some definitions: Dispersion: dq/dl, angular separation of wavelengths Resolving Power: ldl, dimensionless figure of merit Free Spectral Range: Instrument width: Instrument function: extent of spectrum covered by interference pattern before overlap with fringes of same l and different order width of pattern formed with monochromatic light. shape of the interference fringe Étendu: or throughput measure of how efficiently the instrument uses available light.
The Diffraction Grating Spectrometer Calculate the shape of fringe pattern Fringe formation phasors Effects of groove size Fourier methods Angular dispersion Resolving power Free Spectral Range Practical matters - brightness
Oxford Physics: Second Year, N-slit grating Optics 4 N = 2 I ~ N 2 I 0 N-1 minima Peaks at = n2
N-slit grating I 9 4 N = 3 0 Peaks at n2, N-1 minima, I ~ N 2 1 st minimum at 2/N
N-slit grating N 2 I 4 0 N N mn
N-slit diffraction grating (a) Order 0 1 2 3 4 x 2, (b) I(q) = I(0) sin 2 {N/2} sin 2 {/2} sin 2 0 {/2} {/2} 2 (c) = (2l) d sinq (l) a sinq. slit separation slit width Order 2
l ldl l ldl 2n q p q D min 2 N Dq min Dq l (a) = (2l)dsinq (b)
Reflected light Reflected light Diffracted light Diffracted light (a) q Reflected light Diffracted light q (b)
(a) Order 0 1 2 3 4 (b) Unblazed 0 (c) Blazed Order 2
Diffraction Grating Spectrographs Transmission gratings not used Lenses have chromatic aberration
Diffraction Grating Spectrographs Reflection grating used Mirrors have no chromatic aberration
f 1 f 2 Dx s (a) f 1 grating Dx s q f 2 (b)
Instrument function Instrument width (Grating spectrometer): Instrument width = Resolving Power Dn Inst = 1. Maximum path difference = ldl Inst = ndn Inst = 2Wl 1 2W Maximum path difference in units of wavelength n
Optical Instruments for Spectroscopy Interference by division of wavefront: The Diffraction Grating spectrograph Interference by division of amplitude: 2-beams - The Michelson Interferometer
Albert Abraham Michelson 1852 1931 Michelson interferometer Fringe properties interferogram Resolving power Instrument width
M 2 Michelson Interferometer M / 2 M 1 t CP BS Detector Light source
Oxford Physics: Second Year, Optics images t source path difference xcos 2t=x Fringes of equal inclination Localized at infinity 2t=x circular fringe constant
I(x) ½ I 0 n Input spectrum x x = 2t Detector signal Interferogram I(x) = ½ I 0 [ 1 + cos 2nx ]
(a) I( n ) 1 x (b) I( n ) 2 x x max (c) I( n) x
Michelson Interferometer Dv- Inst = 1/x max Instrument width = 1. Maximum path difference Size of instrument Resolving Power = Maximum path difference in units of wavelength
LIGO, Laser Interferometric Gravitational-Wave Observatory 4 Km
LIGO, Laser Interferometric Gravitational-Wave Observatory Vacuum ~ 10-12 atmosphere Precision ~ 10-18 m
Michelson interferometer x = pl Path difference: measure l by reference to known l calibration Instrument width: Dv Inst = 1/x max WHY? Fourier transform interferometer Fringe visibility and relative intensities Fringe visibility and coherence Albert Abraham Michelson 1852 1931
(a) I( n ) 1 x (b) I( n ) 2 x x max (c) I( n) x
Coherence Longitudinal coherence Coherence length: l c ~ 1/Dn _ Transverse coherence Coherence area: size of source or wavefront with fixed relative phase
Transverse coherence d sin < l w s d r >> d r < ld Interference Fringes d defines coherence area
Michelson Stellar Interferometer Measures angular size of stars
Division of wavefront Young s slits (2-beam) Diffraction grating (N-beam) Sharper fringes Division of amplitude Michelson (2-beam) Fabry-Perot (N-beam)? fringes cos 2( /2) sin 2 (N/2) sin 2 (/2) fringes cos 2 (2)
d t r r 1 2 1 t 1 t r r 1 2 1 2 2 t r r 1 2 1 3 2 2 t1r2 r1 tr 1 2 q 3 3 -i3 E o t1t2r1 r2 e 2 2 -i2 E o t1t2r1 r2 e E o t t r r 1 2 1 2 e-i E o tt 12 E o t 1 t 2
d E o tr 2 6 e -i3 tr 4 tr 2 tr 5 tr 3 tr q E o tr 2 4 e -i2 E o tr 2 2 e -i E o t 2 E o t
d E o tr 2 6 e -i3 tr 4 tr 2 tr 5 tr 3 tr q E o tr 2 4 e -i2 E o tr 2 2 e -i E o t 2 E o t Airy Function
The Airy function: Fabry-Perot fringes I( ) m2 (m+1)2 (l)d.cosq
Fabry-Perot interferometer Lens Screen d (l)2d.cosq Extended Source Lens Fringes of equal inclination Localized at infinity
I( ) The Airy function: Fabry-Perot fringes D FWHM m2 (m+1)2 Finesse = D FWHM
Finesse = 10 Finesse = 100
Fabry-Perot Interferometer Multiple beam interference: Fringe sharpness set by Finesse F = R (1-R) Instrument width Dv Inst Free Spectral Range FSR Resolving power Designing a Fabry-Perot
I( ) The Airy function: Fabry-Perot fringes D FHWM m2 (m+1)2 Finesse, F = D FHWM D Inst = F
Fabry-Perot Interferometer: Instrument width Dv- Inst = 1. 2d.F = 1/x max effective Instrument width = 1. Maximum path difference
I( ) The Airy function: Fabry-Perot fringes D Inst Free Spectral Range m2 (m+1)2 Finesse = D
n ndn I( d) m th (m+1) th d FSR
n ndn R I( d) Dn Inst Resolution criterion: Dn R Dn Inst d
Centre spot scanning q m-1 r m-1 m th fringe (on axis) Aperture size to admit only m th fringe Typically aperture ~ r m-1 10
Maxwell s equations 2 2 2 t E E r o r o 2 2 2 t H H r o r o ). ( r k t i E o e E H n H E o o r o r o 1 E n H constant E.M. Wave equations index refractive n
E 1 E 2 E 1 n 2 n 1.n 2 Boundary conditions: Tangential components of E and H are continuous
(a) Layer (b) E o E 1 k o.k 1 E T E o E 1.k T k o.k 1 n 2.n o.n 1..n T l
n O E O E O n cos k n 1 l sin 1 T i k1 ( E O E O ) isin k1 cos l E l n T k1 l T E T (9.7) (9.8) 1 A cos k 1 ; B i sin k1; n l l C in 1 sin k 1 l ; D cos k 1 l 1 EO AnO BnOnT C DnT r (9.10) E An Bn n C Dn O O O T T E E T O t An O Bn 2n O n O T C Dn T (9.11)
E E O O r An An O O Bn Bn O O n n T T C C Dn Dn T T l l 4, Quarter Wave Layer: A=0, B=-i/n 1, C=-in 1, D=0 R = r 2 n n O O n n T T n n 2 1 2 1 2 Anti-reflection coating, n 1 2 ~ n O n T, R 0 e.g. blooming on lenses
0.04 3 layer Uncoated glass Reflectance 2 layer 1 layer 0.01 400 500 600 700 wavelength nm
E o k o k o E o n 2.n o.n H n L.n n 2 H n L.n T Multi-layer stack
n O n EO EO cos k n 1l sin 1 ( E O E O ) isin k1l cos T i k1 n T k1 l E l T E T (9.7) (9.8) 1 A cos k 1 ; B i sin k1; n l l C in 1 sin k 1 l ; D cos k 1 l 1 1 + r = ( A + Bn T )t n O ( 1 r ) = (C + Dn T )t 1 1 A B 1 + r = t n O -n O C D n T
.n H n L.n H n L.n H n L.n H.n H.n H n L n L.n T l Interference filter; composed of multi-layer stacks
Polarized Light Production Analysis Manipulation
z E oz E x E oy y
z Source y x
Circularly polarized light: E z = E o sin(kx o t) ; E y = E o cos(kx o t) z z E z = Eosin( o) E kx Source Source E = 0 z x x o E y = Eocos( kxo) y x x o E E = y y E o (a) x = x o, t = 0 (b) x = x o, t = kx / o
z Source y x Looking towards source: Clockwise rotation of E Right circular polarization
E L E L E P E R E R Superposition of equal R & L Circular polarization = Plane polarized light Superposition of unequal R & L Circular polarization = Elliptically polarized light
z z E oz E E oy y E q y (a) (b)
Polarization states defined by: E y = E oy cos(kx - t) E z = E oz cos(kx - t - ) = 0 0 Linearly Polarized Elliptically Polarized
Polarization states defined by: E y = E oy cos(kx - t) E z = E oz cos(kx - t - ) = 0 Linear = + /2 E oy = E oz Left/Right Circular /2 E oy E oz Left/Right Elliptical axes along y,z /2 E oy E oz Left/Right Elliptical axes at angle to y,z
Unpolarized light defined by: E y = E oy cos(kx - t) E z = E oz cos(kx - t (t) = (t) Phase varies randomly in time unpolarized light
Plane Right circular Left circular Looking towards source Right elliptical Left elliptical
= 0, 5 Note: if E oy = E oz these ellipses become circles
Polarization states defined by: E y = E oy cos(kx - t) E z = E oz cos(kx - t - ) = 0 0 Linearly Polarized Elliptically Polarized = 0, 5
Uniaxial Crystals z n ex z n o n o n ex x,y x,y (a) Optic Axis (b) Positive Negative
Uniaxial Crystals E z n ex n o propagation z n o n ex e-ray x,y Phase shift: x,y = (2l)[n o -n e ]l o-ray (a) (b) Positive Negative
Linear Input change Output q= 45 O, E oy =E oz, l Left/Right Circular q 45 O, E oy E oz, l L/R Elliptical q 45 O, E oy E oz,l Elliptical tilted axis q 45 O, E oy E oz, l Linear at q to input
z z E z q E P E P q E z E y x E y y x y Half-wave plate: introduces -phase shift
e-ray e-ray e-ray o-ray Optic axis o-ray Optic axis o-ray Prism Polarizers
A d 1 d 1 B d 2 d 2 (a) Babinet (b) Babinet-Soleil
Analysis of polarized light z z y z z y E E q y y Elliptically polarized light Linearly polarized light after passage through the l/4 plate (a) (b)
Analysis of polarized light z z E y q y Find E oz :E oy and z z E y q y (a) Elliptically polarized light Find ratio of major:minor axes and angle q - approximately by rotating linear polarizer E oz /E oy = tan (b) Linearly polarized light after passage through the l/4 plate Make linear polarization with l/4 plate Find angle of linear polarization using linear polarizer - precisely (q) q, tanq cos
A B A B C (a) (b) A B C (c). D