Outline. Basics of interference Types of interferometers. Finite impulse response Infinite impulse response Conservation of energy in beam splitters

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1 ntefeometes lectue C 566 Adv. Optics Lab Outline Basics of intefeence Tpes of intefeometes Amplitude division Finite impulse esponse nfinite impulse esponse Consevation of eneg in beam splittes Wavefont division Fouie tansfom spectoscop ntefeence testing of abeations

2 Monochomatic intefeence C 566 Adv. Optics Lab Basics of intefeence + ntefeing plane waves ( ωt k + φ ) j ˆ ( ωt k + φ + e e ) j ˆ e e + + ( ) [( ) ( ) ( )] * eˆ eˆ cos ω ω t k k + φ φ, Real Summation of intensities ntefeence Tpicall we define * eˆ eˆ dentical polaization states ω K Φ ω k k π ˆ Λ G K G φ φ Yielding [ K + Φ] + + cos G Stationa finges Finges of peiod Λ and diection Kˆ G Phase shift Finge modulation depth, m Λ ma min m ma ma m + min min +

3 Monochomatic intefeence C 566 Adv. Optics Lab Basics of intefeence + ntefeing spheical waves j( ωt k + φ ) j( ωt k + φ ) ( ) eˆ e + ( ) eˆ e [ Φ] * ( R ) + ( R ) + ( eˆ eˆ ) ( R ) ( R ) cos k( R R ) Summation of intensities ntefeence Tpicall R we define i i ( R R ) Φ nπ k k ( R R ) Φ ( n )π + Distance fom focus Maima when path length diffeence n λ Minima Can sketch intefeence patten b connecting the dots : Hpebolas R R 3

4 Tpes of intefeometes Amplitude division C 566 Adv. Optics Lab Amplitude division Finite-impulse esponse intefeometes S S / Twman Geen Michelson S n ( )ds Mach-Zende Optical path length + + cos [ k S] + + πν cos S c ν c S Fee-spectal ange

5 Tpes of intefeometes Amplitude division C 566 Adv. Optics Lab Amplitude division nfinite-impulse esponse intefeometes Fab-Peot intefeomete, aka talon d S nd cosθ tans inc jk S jk S [ tt + tte + tte +...] This is a geometic seies. Witing R, T t T tans inc + F sin π ν ν F 4R ( R) ν c S c nd cosθ.8 F.6.4 F. F

6 Tpes of intefeometes Beam splittes C 566 Adv. Optics Lab Beam splitting Consevation of eneg t t Two identical fields incident on a beam splitte with amplitude eflectivit and amplitude tansmittance t: + t + t + + Consevation of eneg t + t + t + t φ φ φ φ + π * t * * t t * Divide though b + t t * * t φ φ π t Thus, thee must be a π/ phase shift on amplitude division. 6

7 Tpes of intefeometes Wavefont division C 566 Adv. Optics Lab ntefeometes Wavefont division intefeomet tot ( ) What is given N identical apetues which adiate an electic field? ap ( ) tot A N ( ) ( ) N n ap ap ( ) δ ( ) n n ( ) A( ) d n n A is the aa function We know that the adiation in the Faunhofe egime (angula spectum at infinite distance o in back focal plane of lens) is given b the Fouie tansfom of the electic field in the nea field: tot ( k ) F{ ( tot )} ( k ) A( k ) ap Convolution in space becomes multiplication in k-space ARRAY THORM: Diffaction fom aa is diffaction fom apetue multiplied b the FT of the aa distibution. Renolds, DeVelis, Paent, Thompson, Phsical Optics Notebook, SP, 989 7

8 Tpes of intefeometes Wavefont division C 566 Adv. Optics Lab ample Single and double slits L tot ( ) Sinc k L π L Sinc λ f ap / ( ) ect( L) f Sceen L L d tot ( ) Sinc k π L Sinc λ f L cos k d f π d cos λ f ap ( ) ect( / L) f A δ d + δ + d ( ) ( ) ( ) Sceen

9 Spectoscop and patial coheence C 566 Adv. Optics Lab Fouie tansfom spectoscop w/ Michelson intefeomete Powe spectum P ( k) / Path-length change () Detected ( ) ( + cos k ) P( k) P + P ( k) ( k) [ ( ) ] e jk dk dk jk e dk ntegate intefeence ove all wave-numbes () is Fouie tansfom of P(k) So P(k) is invese Fouie tansfom of () () P( k) F - 9

10 Spectoscop and patial coheence C 566 Adv. Optics Lab Coheence length mpotant app of FT spectoscop mpotant definitions and elations. t t ν t Coheence time: duation of single-fequenc opeation Bandwidth: single line width o multi-mode oscillation l c t c ν λ λ Coheence length: coheence time measued in [m] ample: ectangula spectum of bandwidth ν k + k / ( ) jk k + e dk + cc + sinc ( k ) cos k k / P( k) k π ν c () Visibilit k k o Finge modulation depth m (aka visibilit) goes to zeo at: π k c ν l c t Don t need to FT to get this impotant paamete

11 ntefeence testing of abeations ntoduction C 566 Adv. Optics Lab Wavefont abeations Geneal pictue Refeence sphee Wavefont Ra abeation Object Paaial focus ntance pupil it pupil A spheical wavefont in the eit pupil foms a pefect focus The diffeence between the actual wavefont and the efeence sphee is the wavefont abeation. Non-spheical wavefonts cause as to coss the optical ais NOT at the paaial focus a abeation

12 ntefeence testing of abeations ntoduction C 566 Adv. Optics Lab Coodinate sstem and appoach ρ θ Object Paaial image plane ntance pupil. Assume a otationall-smmetic optical sstem. pand the wavefont abeation in the pupil in a polnomial of the object a paametes (, ρ, θ). 3. Reject all tems which don t meet smmet assumption.

13 ntefeence testing of abeations C 566 Adv. Optics Lab Spheical abeation 4 W 4 ρ Wavefont eo in pupil W ntefeogam - 3

14 ntefeence testing of abeations C 566 Adv. Optics Lab Spheical abeation at CLC W 4 3 ( ρ ) 4 ρ Wavefont eo in pupil W ntefeogam

15 ntefeence testing of abeations C 566 Adv. Optics Lab Coma W 3 3 ρ cosθ W3 ρ Wavefont eo in pupil W ntefeogam

16 ntefeence testing of abeations C 566 Adv. Optics Lab Astigmatism W ρ cos θ W Wavefont eo in pupil W ntefeogam

17 ntefeence testing of abeations C 566 Adv. Optics Lab Astigmatism at CLC W ( ρ θ ρ ) W ( ) cos Wavefont eo in pupil W ntefeogam

18 ntefeence testing of abeations C 566 Adv. Optics Lab Field cuvatue W ρ Wavefont eo in pupil W ntefeogam

19 ntefeence testing of abeations C 566 Adv. Optics Lab Distotion 3 W3 ρ cos θ W 3 3 Wavefont eo in pupil W ntefeogam