PHYS 450 Spring semester Lecture 10: Interferometers. Ron Reifenberger Birck Nanotechnology Center Purdue University.
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1 3/10/017 PHYS 450 Spring semester 017 Lecture 10: Interferometers Ron Reifenberger Birck Nanotechnology Center Purue University Lecture 10 1 Brief History Interferometry uses superposition of coherent waves to eluciate a physical property of the original state of the wave or the material through which the wave propagates Historical milestones along the way 1665 Hooke s Micrographia - investigate color in mica plates, soap bubbles, an thin films of oil on water 1704 Newton s Rings (observe by Boyle an Hooke) explaine in Newton s Opticks 1803 Young s ouble slit 1834 Lloy s mirror 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 1
2 3/10/017 Key Iea: Aing in-phase an out-of-phase signals The phase of the sine wave is measure in raians a) x x Asin o π Asin o π Δφ λ λ In Phase SUM E (N/m) Δφ =0,π, 4π, superposition of two waves: two extremes b) position (m) Out of Phase SUM=0 Superposition only works because the E-fiels of the two waves lie in the SAME plane E (N/m) position (m) Principle of Superposition Δφ = π, 3π, 3 3 Two Types of Interference Division of Wavefront primary wavefront emits seconary waves which recombine an interfere with each other or with the 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 an interfering λ air to eye Thin film interference Michelson Fabry-Perot n film λ film t 4
3 3/10/017 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 C epens on how r B,C compares to the coherence length l coh of the EM wave emitte from the Source lateral B r AB r BC 5 C Coherence time an Longituinal 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 nm 15x10 9 Hz 67x10-10 s 0 m spectral line (low pressure Na nm 5x10 11 Hz x10-1 s 6 x10-4 m ischarge tube) re LED nm 5x10 13 Hz 4x10-14 s 1 x10-6 m sunlight, typical ~545 ~800 nm 8x10 14 Hz x10-15 s 06 x10-6 m 6 3
4 3/10/017 Interferometers Exploit Coherence Two Important Interferometers Fabry-Perot Michelson 7 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 θ 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 8 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 + 4
5 3/10/017 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 θ E o E o t 9 What is OPL between ❶ an ❷? E o θ y y =/-θ e a θ θ h f M E o t r e iδ θ b c ❷ ❶ E o t OPL abbeef ( abbc) 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 10 5
6 3/10/017 E o t r 3 E o t r θ E o trans E o tr 3 E o tr E o tr 4 E o tr E o t M What is 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 11 Transmitte Irraiance Io Itrans I o A 1 F sin 4R where F 1 R 3 max when,,, =/ R=00 The Airy Function A(θ) E o t r 3 E o t r 04 R=050 0 R= m (m+1) =δ+δ r (in raians) θ E o E o tr 4 E o tr 3 E o tr E o tr E o t M E o t r 4 e iδ E o t r e iδ E o t Typical values for 4 cos( ) r Typically 0 01 m, 600 nm, 0, r m 35,
7 3/10/017 Fabry-Perot Alignment froste glass M ❹ move ❸ 1 Both mirrors are partially silvere, mirror surfaces face each other Remove mirror 3 Ajust laser height to hit center of mirror M 4 Using auto-reflection, ajust the tilt on laser to make M perpenicular to laser beam 5 Insert mirror an move by han to make perpenicular to beam Use auto-reflection 6 Lock position of 7 Chase ots until they all collapse 8 Insert +5 mm converging lens to expan laser beam 9 Fringes shoul be observe after minor tweaking ❷ ❶ Top view, looking own insert f~ +5 mm Laser 3,1,3? 4, 3,,4? 4,1 3,1 3, Transmitte beams Four sets of parallel planes: ❸❷ strongest ❸❶ ❹❷ ❹❶ weakest Try to ientify the origin of ALL the reflections! 13 Experiment Extene Source Fabry- Perot Lens Screen E o t r 3 M E o t r 4 e iδ E o t r E o tr 3 E o tr E o tr 4 E o tr E o t r e iδ E o t θ E o E o t Screen 14 7
8 3/10/017 Fabry-Perot Setup Photoioe Alignment is critical Froste Glass Fabry Perot Orange HeNe Laser Mirror M 5 cm converging lens 1 Position 10% filter mask over photoioe Na Lamp Center interference pattern on filter mask Piezoelectric Displacement Fringes on froste glass 15 Fabry-Perot Data Acquisition Orange HeNe Laser Piezoelectric Effect L Piezo bar few m s from experiment Fabry Perot Photoioe Screen V L Reflecting Mirror Piezo bar L+L 0 0 ~1000 V V 16 8
9 3/10/017 Laser Data Acquisition Piezo- Translator Lens M M3 Moveable Carriage Micrometer Fabry-Perot Interferometer Photoioe w 10% mask, Gain=1 075 m HV c amplifier, Gain 00 Data Stuio, 0-5 V Intensity (au) positive ramp 5 V 0 0 ~100 s time 5 Fabry Perot Data March 8, Time (s) Difference between Run 1 an Run? About 1/8 of a turn of mirror alignment! Don t give up Run 1 Run s of ata HeNe laser, =638 nm, mirror spee assume constant at 13 nm/s ~6,000 ata points Fabry_Perot - HeNe laser / Mirror Displacement (nm) 18 9
10 3/10/017 Estimate Mirror Spee vstime (use peak positions in F-P pattern as fiucial marks at every m/) Least-squares fit to quaratic function: Velocity (nm/s) Velocity vs Time constant velocity y = 00103x x R² = Time (s) 19 Correcte Data Variable mirror spee (from previous slie) 100 Fabry_Perot - HeNe laser Mirror Displacement (nm) Zoome-in: mirror reflectance R=06 Fabry_Perot - HeNe laser trans /=3164 nm 1 F sin Mirror Displacement (nm) Fit to: Io I OPL 4 cos 0 10
11 3/10/017 Michaelson Interferometer -Historical Michelson Interferometer Source filter M Perform Three Experiments a) Calibrate wavelength of laser b) Determine inex of refraction of transparent winow c) Determine inex of refraction of air V B C 11
12 3/10/017 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 3 OPL=cosθ P' P' I θ Conition for Constructive Interference P'' P'' I M ' M Constructive interference when cosθ=m; m=0,1,,3 1 If M for fixe, m, an, θ must equal a constant circular fringes Other values of θ coul satisfy above equation (for same an ) provie m changes many concentric circular fringes S' P Viewing Screen Image Planes, I an I M moveable (istance between mirrors) Source Lens P''' P maps to P Each ring forms at a ifferent 4 1
13 3/10/017 A few consequences 1 Constructive interference when cosθ m =m; m=0,1,,3 cosθ m+ = (m+) cosθ m+1 = (m+1) cosθ m = m cosθ m-1 = (m-1) cosθ m- = (m-) Ring location specifie by angle θ Say = 5 mm, =500 nm If Θ m =0 = m m=/~0,000 3 For Θ m =0, change in to + causes collapse of N fringes = m (+) = (m +N) =/N 5 Beware of backlash! Mirror M (moveable) Michelson Setup Mirror (fixe) Lens Cube Beamsplitter Laser Cube Lens Beamsplitter Laser Two right angle views 6 13
14 3/10/017 Mirror Drive A Calibrate Laser Wavelength Motor Calibration 80E-05 Michelson mirror isplacement vs time Micrometer Distance move (m) 60E-05 40E-05 0E-05 00E+00 velocity Elapse time (s) 05 mm=50 units on spinle of micrometer=500 m 001 mm = 1 unit on spinle of micrometer=10 m 000 mm = 0 units on spinle of micrometer (smallest ivision) = 0 m 7 Counting Fringes Do not use center fringe too easy to miss a fringe Mark fringe counting location with hea of pin Requires a steay chin rest =/N Hea of a pin 8 14
15 3/10/017 B Inex of Refraction for Transparent Winow Suppose we coul change length of a block of material M OPL OPL OPL final n LL nl initial nl Suppose change in OPL isplaces N fringes nl N N n L Source B ajust L L V 9 Experimental Setup M Source B Rotate Transparent Plate Beamsplitter, B Transparent plate, thickness t V Lock knob Precision rotation 30 15
16 3/10/017 Change in OPL by rotation C D A B F G t K t Must set ϕ=0 accurately use auto reflection to efine ϕ=0 E H N, Fringe Number Rotating Plate Insie Michelson Interferometer Count N 10 fringes ϕ, rotation angle (egrees) 31 C Inex of Refraction for Air Suppose we coul change inex of refraction of a block of material Source B M OPL OPL OPL nn LnL nl Suppose change in OPL isplaces N fringes nl N N n ajust n L L final initial V 3 16
17 3/10/017 Experimental Setup Source B M Pressure Gauge Isolation Valve X To vacuum pump L N P L Pfinal P init atm 1 n P Count N fringes as pressure changes from P init to P final atm V ~0 mins for cell to leak up to atmosphere 33 L Convenient Chin Rest Stainless steel vacuum cell with orange vacuum pump in backgroun 34 17
18 3/10/017 Comparing Michelson 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 compatible with 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 calibration Laser cavities/optical resonators Inter-cavity frequency iscrimination Astronomical observations 35 Up Next Fresnel Diffraction 36 18
19 3/10/017 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 37 A 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 φ A A φ A n φ n Real Axis 38 19
20 3/10/017 Appenix A: Distance between two images (P an P ) of an object 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 P an P are images of object P prouce by two mirrors ( an M) which are offset one from the other by a istance +s o =(+s o ) = 39 Appenix B: Analytical summation of a Geometric Series n k n Sn r 1 rr r k 0 rs r r r n k 0 n1 S rs 1r S 1r n n n n k 1 r Sn r 1 r n1 n1 40 0
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