Lecture 09: Interferometers

Transcription

1 1/31/16 PHYS 45 Fall semester 16 Lecture 9: Interferometers Ron Reifenberger Birck Nanotechnology Center Purue University Lecture 9 1 Two Types of Interference Division of Wavefront primary wavefront emits seconary waves which recombine an interfere with primary wave Young s two-slit Lloy s mirror Division of Amplitue primary wave is ivie into two or more waves which travel ifferent paths before recombining λ air to eye Thin film interference Michelson Fabry-Perot n film λ film t 1

2 1/31/16 Brief History Interferometry uses superposition to combine waves so that the result of their combination exhibits a property that provies information about the original state of the wave Historical Milestones 1665 Hooke s Micrographia - investigate color in mica plates, soap bubbles, an thin films of oil on water 174 Newton s Rings (observe by Boyle an Hooke) explaine in Newton s Opticks 183 Young s ouble slit 1856 Jamin s interferometer 186 Fizeau s fringes 188 Michelson s interferometer 1887 Michelson-Morely experiment 1891 Max & Zehner interferometer 1896 Rayleigh s interferometer 1899 Fabry & Perot etalon 196 Twyman & Green interferometer etc, etc 3 Coherence of Light If a light wave maintains a constant phase relation (a constant wave front) in space, we say the light is spatially coherent Spatial coherence can be classifie as either longituinal (temporal) or lateral (spatial) ANY monochromatic source will have a banwith natural sprea in wavelength which I A equals the full-with at half-maximum Source (FWHM) of the spectral emission about some central wavelength o f c c c f c f o 1 1 o coh f c 1 o o coh c coh c c o The lateral coherence between E A an E B epens on the spatial coherence length of the source The longituinal (temporal) coherence between E B an E B epens on how r B,B compare to the coherence length l coh of the EM wave emitte from S lateral B r AB r BB' 4 B'

3 1/31/16 Coherence time an Coherence length of Light (typical values) Coherence times an coherence lengths (typical values) Light Source (nm) (FWHM) f=c / coh =1/f l coh =c coh HeNe laser 638 nm 15x1 9 Hz 67x1-1 s m Low pressure Na ischarge tube nm 5x1 11 Hz x1-1 s 6 x1-4 m re LED ~6 3 nm 5x1 13 Hz 4x1-14 s 1 x1-6 m Incanescent light source, visible ~4-~7 ~ nm x1 14 Hz 5x1-15 s x1-6 m 5 Fabry-Perot Interferometer (1899) Two Parallel, Partially-Silvere Mirrors High resolution optical instrument (interferometer) OR an optical cavity OR a soli Etalon Two partially silvere mirrors an M Transparent plate with two partially silvere mirrors interferometer optical cavity soli Etalon (optical filter) M E o E o t r 3 E o tr 4 E o t r E o E o tr 3 E o tr E o tr E o t t r4 E o t r E o t E o t r + E o t r 3 + Excellent surface flatness an plate parallelism are require 6 Etalon is from the French étalon, meaning "measuring gauge" or "stanar" E o E o t +E o t r + E o t r 4 + M E o t r + E o t r 3 + M E o E o t +E o t r + E o t r 4 + 3

4 1/31/16 ONE partially silvere mirror Fabry-Perot Interferometer (Basic Ieas) I o I o r =I o R I o t =I o T I o =I o R+ I o T 1=T+R E o t r 3 M E o t r 4 M I o I trans =? E o t r E o tr 3 E o tr E o tr 4 E o tr E o t r E o t move E o E o t 7 What is OPL? E o y y =/- e a h f M E o t r b c ❷ ❶ E o t OPL ab be ef ( ab bc) be ef bc Let h = be ef bc ysin( ) y hsin( ) bc hsin ( ) be ef bc h h sin ( ) h cos ( ) but cos( ) / h h / cos( ) OPL cos( ) Phase ifference : OPL 4 cos cos 8 4

5 1/31/16 E o t r 3 E o t r E o trans E o tr 3 E o tr E o tr 4 E o tr E o t M What is the Transmitte Irraiance? E o t r 4 e iδ E o t r e iδ E o t i i e e E E t E t r E t r I trans E E 4 o o o t o see Appenix i 1 r e 4 o t i 1 r e ( ) OPL 4 cos ir Let r r e ( r is possible phase change uner reflection) 4 4 t t Itrans Eo I o i 1 re 1 r ir i e e 1 r ir i e e 4 t T I 4 i o r i r 1R Rcos r Io 1 r r e e a1sin a Using cos an r T T Itrans Io I 1R 1 T 1 Io Io 1 R 4R 1 sin 1 sin 1 R 4R T where F an 1 no losses o R sin 1R 4Rsin 1 R 1 R F 9 Transmitte Irraiance The Airy Function A() I M o E Itrans I o A o t r 4 e iδ 1 F sin E o t r 3 E E o t r e iδ o tr 4 4R E o t where F r E o tr 3 E o t 1 R E o tr E o tr 3 E max when,,, o t 1 E o Typical values for 8 4 cos( ) r 6 R= Typically 1 m, 4 6 nm, R=5, r R= m 35, 9-9 m (m+1) =δ+δ r (in raians) 5

6 1/31/16 Fabry-Perot Setup Lens Photoioe (1% filter mask, Gain=1) Mirror Lens Orange HeNe Laser Mirror M Mirror 5 cm converging lens Na Lamp To avoi chasing ots, learn how to ientify the origin of ALL the reflections! 11 Experiment Screen Extene Source Fabry- Perot Lens E o t r 3 M E o t r 4 e iδ E o tr 4 E o t r e iδ E o t r E o tr 3 E o tr E o t E o tr E o E o t 1 6

7 1/31/16 Experimental Data Laser M Lens Fabry-Perot Interferometer Photoioe w 1% mask, Gain=1 1 8 HeNe laser, =638 nm, mirror spee assume constant at 13 nm/s, R=6 Fabry_Perot - HeNe laser / M3 6 4 Piezo- Translator Micrometer Moveable Carriage HV c amplifier, Gain Data Stuio, -5 V Mirror Displacement (nm) positive ramp 5 V ~1 s time 13 Piezo nonlinearity - Hysteresis Piezoelectric Effect L Piezo bar few m s L from experiment V Piezo bar L+L ~1 V V 14 7

8 1/31/16 Estimate Mirror Spee vstime (use peak positions in F-P pattern as fiucial marks at every m/) Least-squares fit to quaratic function: 15 Velocity vs Time Velocity (nm/s) y = 13x x R² = Time (s) 15 Ajuste Data Variable mirror spee (from previous slie) 1 Fabry_Perot - HeNe laser Mirror Displacement (nm) Zoome-in; mirror reflectance R=6 1 8 Fabry_Perot - HeNe laser /=3164 nm Mirror Displacement (nm) 16 8

9 1/31/ Application: Resolving two spectral lines Fabry_Perot: a =65 nm, b =65 nm Mirror Displacement (nm) Maximum fringe Fabry_Perot: a =65 SUM Maximum fringe nm, b =65 nm visibility visibility washout Mirror Displacement (nm) 17 Measuring the splitting 1 ~675 nm N fringes ~88 nm N=5 16 washout Mirror Displacement (nm) Measure - 1 an a b a a b a 1 N 1 N N a a N 1 N N a a N 1 N 1 N 1 NN 1 a a a 1 N

10 1/31/16 Putting in some numbers 1 5 a 65 nm nm nm 815 a for the Na oublet a nm nm mm 1 31 nm 3mm nm Use the micrometer! Beware of Backlash! 19 Resolving the Na oublet washout Maximum 1 fringe 3 Minimum 4 fringe visibility visibility Mirror Displacement (nm) Photos from astrgsueu/hbase/phyopt/fabry3html#c1 1

11 1/31/16 You will nee to relocate the mirror, the small converging lens (f~5 mm), an the laser mount between Fabry-Perot an Michelson setups 1 Michaelson Interferometer Historical 11

12 1/31/16 Michelson Interferometer Source G M V B C 3 Michelson Interferometer (top view) moveable M l S Source C B Alignment is critical To avoi chasing ots, learn how to ientify the origin of ALL the reflections! V l 1 If you look through the viewing screen (DON T!), you woul see: Mirror M Virtual image of = Virtual image of Source=S 4 1

13 1/31/16 Conition for Constructive Interference OPL=cos P' P'' I I M P' P'' ' M Constructive interference when cos=m; m=,1,,3 1 If M for fixe, m, an, must equal a constant circular fringes Other values of coul satisfy above equation (for same an ) provie m increases many concentric circular fringes S' P Image Planes, I an I M moveable Lens V 5 A few consequences 1 Constructive interference when cos=m; m=,1,,3 cos m+ = (m+1) cos m+1 = (m+1) cos m = m cos m-1 = (m-1) cos m- = (m-) Equations efine all angles for constructive rings Say = 5 mm, =5 nm For Θ m = = m m=/~, 3 At Θ m =, change to + an count collapse of N fringes = m (+) = (m +N) =/N 6 13

14 1/31/16 Michelson Setup Mirror M Mirror Lens Cube Beamsplitter Laser Cube Lens Beamsplitter Laser Two right angle views 7 Comparing Michaelson to Fabry-Perot Attribute Michelson Fabry Perot Split-beam geometry Possibility of large separation between mirrors permits large path length ifference Small separation between mirrors ictates bench-top, high-resolution spectroscopic applications Fringe Visibility Relatively iffuse Controllable sharpness Typical uses: Calibration of stanar meter Measurement of coherence length Fourier transform spectroscopy Gravity wave etection Wavelength calibrations Laser cavities/optical resonators Inter-cavity frequency iscrimination Astronomical observations 8 14

15 1/31/16 Up Next Fresnel Diffraction 9 Review: Interference of TWO scalar monochromatic waves w arbitrary phases cos a t A t Re A e Re A cos 1 a t A t Re A e Re A where the phasors are efine as A t Imag Axis Ae 1 1 A t Ae A Phasors φ A 1 φ 1 t i t i 1 1 Real Axis it 1 it When the two waves interfere 1 1cos 1 cos ReA1 t A t Acost a t a t a t A t A t A A A AA cos A sin A sin tan A1cos1 Acos When one properly, Phasor summation is exactly the same as Vector summation A Imag Axis φ φ A 1 φ 1 Real Axis 3 A 15

16 1/31/16 Review: Interference of n scalar monochromatic waves A n φ n φ 1 φ A A 1 it1 1 1cos 1Re 1 Re 1 it cos Re Re a t A t A e A a t A t A e A Imag Axis Phasors Real Axis Aing n scalar monochromatic waves itn n ncos nre n Re n a t A t A e A The n waves interfere to give 1 n Re A1 t A t An t Acost a t a t a t a t When one properly, Phasor summation is exactly the same as Vector summation Imag Axis A 1 φ 1 A φ A φ A n φ n Real Axis 31 Appenix A: Difference in istance between two images P an P of a point P prouce by two mirrors ( an M) which are offset by a istance? a) I 1 I P s i =-s o M s o P b) P s i =-(+s o ) (+s o ) ' s o +s o P Image of point P prouce by two mirrors which are offset one from the other by a istance +s o =(+s o ) = 3 16

17 1/31/16 Appenix B: Analytical summation of a Geometric Series n k n Sn r 1 rr r k rs r r r n k n1 S rs 1r S 1r n n n n k 1 r Sn r 1 r n1 n