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

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: Fraunhofer diffraction = Fourier Transform Convolution Theorem • Diffraction theory of imaging • Fringe localization: interferometers

Oxford Physics: Second Year, Optics Parallel reflecting surfaces Extended source t images source Fringes localized at infinity ∞  2t=x Circular fringe constant   path difference cos x  circular fringe Fringes of equal inclination constant  2t=x

Oxford Physics: Second Year, Optics Summary: fringe type and localisation Wedged Parallel Point Non-localised Non-localised Source Equal thickness Equal inclination Extended Localised in plane Localised at infinity of Wedge Source Equal inclination Equal thickness

Oxford Physics: Second Year, Optics Optical Instruments for Spectroscopy Some definitions: • Dispersion: d q / d l, 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.

Oxford Physics: Second Year, Optics The Diffraction Grating Spectrometer • Fringe formation – phasors • Effects of groove size – Fourier methods • Angular dispersion • Resolving power • Free Spectral Range • Practical matters, blazing and slit widths

Oxford Physics: Second Year, Optics N -slit grating N = 2 I ~ N 2 4    I          0 Peaks at  = n2  N-1 minima

Oxford Physics: Second Year, Optics N -slit grating    N = 3    9    I 4       0    Peaks at n2  , N-1 minima, I ~ N 2 1 st minimum at 2  /N

N -slit grating Oxford Physics: Second Year, Optics 2 N I 4    0         N       N m N

Oxford Physics: Second Year, Optics Optical Instruments for Spectroscopy Some definitions: • Dispersion: d q / d l, 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.

N -slit diffraction grating (a) x 2   ,  Order 0 1 2 3 4 (b) I (q) = I(0) sin 2 { N  /2} sin 2 {  /2} sin 2 {  /2} {  /2} 2 0  = (2 l ) d sin q   (l) a sin q (c) . slit separation slit width 2 Order

N -slit diffraction grating (a) x 2   ,  Order 0 1 2 3 4 (b) 0 (c) . slit separation slit width 2 Order

Oxford Physics: Second Year, Optics l l l l l l d d 2 n   q q p 2  Dq min Dq l  D  N min (a) (b)  = ( 2 l ) d sin q

Oxford Physics: Second Year, Optics Reflected Diffracted Reflected Diffracted light light light light  q  (a) Reflected Diffracted light light   q (b)

Oxford Physics: Second Year, Optics (a) Order 0 1 2 3 4 Unblazed (b) 0 Blazed (c) 2 Order

Oxford Physics: Second Year, Optics f 1 f 2 D x s (a) grating f 1 D x s q f 2 (b)

Oxford Physics: Second Year, Optics Instrument function n Instrument width 1 Dn Inst = (Grating spectrometer): 2W 1 . Instrument width = Maximum path difference = lDl Inst Resolving Power = nDn Inst = 2W l Maximum path difference in units of wavelength

Oxford Physics: Second Year, Optics Optical Instruments for Spectroscopy • Interference by division of wavefront: The Diffraction Grating spectrograph • Interference by division of amplitude: 2-beams - The Michelson Interferometer

Oxford Physics: Second Year, Optics Albert Abraham Michelson 1852 – 1931 Michelson interferometer • Fringe properties – interferogram • Resolving power • Instrument width

Oxford Physics: Second Year, Optics M 2 Michelson Interferometer / M M 1 2 CP BS t Detector Light source

Oxford Physics: Second Year, Optics t images source Fringes of equal inclination Localized at infinity 2t=x  path difference cos x  circular fringe constant  2t=x

Oxford Physics: Second Year, Optics I( x ) ½ I 0 n x x = 2t Input spectrum Detector signal Interferogram I( x ) = ½ I 0 [ 1 + cos 2 n x ]

Oxford Physics: Second Year, Optics n I( ) (a) 1 x n I( ) (b) 2 x x max n (c) I( ) x

Oxford Physics: Second Year, Optics Michelson Interferometer - D v Inst = 1/ x max Instrument width = 1 . Maximum path difference Size of instrument Resolving Power = Maximum path difference in units of wavelength

Oxford Physics: Second Year, Optics LIGO, Laser Interferometric Gravitational-Wave Observatory 4 Km

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

Oxford Physics: Second Year, Optics Lecture 10 Michelson interferometer • Path difference: x = p l  measure l by reference to known l calibration • Instrument width: D v Inst = 1/ x max WHY? • Fourier transform interferometer Albert Abraham Michelson • Fringe visibility and relative intensities 1852 – 1931 • Fringe visibility and coherence

Oxford Physics: Second Year, Optics n I( ) (a) 1 x n I( ) (b) 2 x x max n (c) I( ) x

Oxford Physics: Second Year, Optics Coherence Longitudinal coherence _ Coherence length: l c ~ 1 / Dn Transverse coherence Coherence area: size of source or wavefront with fixed relative phase

Oxford Physics: Second Year, Optics d sin  < l Transverse coherence  d w s  r r >> d Interference Fringes  < l d d defines coherence area

Michelson Stellar Interferometer Measures angular size of stars

Oxford Physics: Second Year, Optics Division of wavefront fringes cos 2(  /2) Young’s slits ( 2-beam ) sin 2 (N  /2) Diffraction grating ( N-beam ) Sharper fringes sin 2 (  /2) Division of amplitude fringes cos 2 ( 2) Michelson ( 2-beam ) Fabry-Perot ( N-beam ) ?

Oxford Physics: Second Year, Optics  3 3 -i3 E o t t r r e 1 2 1 2 d 3 2  t r r 2 2 -i2 E o t t r r e 1 2 1 1 2 1 2 2 2 t r r 1 2 1 -i  2 t r r E o t t r r 1 2 1 2 e 1 2 1 t r r 1 2 1 E o t t t r 1 2 1 2 q t 1 E o t 1 t 2

Oxford Physics: Second Year, Optics d  2 6 -i3 E o t r e  2 4 -i2 E o t r e 5 tr  2 2 -i 4 E o t r e tr 3 tr 2 2 tr E o t tr q E o t

Oxford Physics: Second Year, Optics d  2 6 -i3 E o t r e  2 4 -i2 E o t r e 5 tr  2 2 -i 4 E o t r e tr 3 tr 2 2 tr E o t tr q E o t Airy Function

Oxford Physics: Second Year, Optics The Airy function: Fabry-Perot fringes  I( ) m2   (m+1)2    (l) d.cos q

Oxford Physics: Second Year, Optics Fabry-Perot interferometer Screen Lens d   (l ) 2d.cos q Fringes of equal inclination Lens Extended Localized at infinity Source

Oxford Physics: Second Year, Optics The Airy function: Fabry-Perot fringes   I( ) D FWHM m2   (m+1)2  Finesse = D FWHM

Finesse = 10 Finesse = 100

Oxford Physics: Second Year, Optics Fabry-Perot Interferometer Multiple beam interference: Fringe sharpness set by Finesse F =  √R (1-R) • Instrument width D v Inst • Free Spectral Range FSR • Resolving power • Designing a Fabry-Perot

Oxford Physics: Second Year, Optics The Airy function: Fabry-Perot fringes   I( ) D FHWM m2   (m+1)2  Finesse, F = D FHWM D Inst =  F

Oxford Physics: Second Year, Optics Fabry-Perot Interferometer: Instrument width - D v Inst = 1 . 2d.F = 1/ x max effective Instrument width = 1 . Maximum path difference

Oxford Physics: Second Year, Optics The Airy function: Fabry-Perot fringes  Free Spectral Range  I( ) D Inst m2   (m+1)2  Finesse = D

Oxford Physics: Second Year, Optics nDn n I( ) d m th th d (m+1) FSR

Oxford Physics: Second Year, Optics nDn R n Resolution criterion: I( ) d Dn R  Dn Inst Dn Inst d

Centre spot scanning Oxford Physics: Second Year, Optics r m-1 q m-1 m th fringe (on axis) Aperture size to admit only m th fringe r m-1 Typically aperture ~ 10