Basic Interference and. Classes of of Interferometers

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1 Basic Intefeence and Classes of Intefeometes Basic Intefeence Two plane waves Two spheical waves Plane wave and and spheical wave Classes of of Intefeometes Division of of wavefont Division of of amplitude Optics James C. Wyant Page 1 of 26 Optical Detectos Respond to Squae of Electic Field E = E 1 + E 2 = E 1ˆ a 1 e i ( ω 1t +α 1 ) + E 2 a ˆ 2 e i ( ω 2t+α 2 ) I = Constant E 1 + E 2 2 = Constant E E E 1 E 2 ( a ˆ 1 a ˆ 2 )cos[(ω 1 ω 2 )t + α 1 α 2 ] [ ] I = I 1 + I I 1 I 2 ( a ˆ 1 a ˆ 2 )cos[(ω 1 ω 2 )t + α 1 α 2 ] Iadiance at each point vaies cosinusoidally with time at the diffeence fequency Optics James C. Wyant Page 2 of 26 Page 1

2 Intefeence Finges I = I 1 + I I 1 I 2 ( a ˆ 1 a ˆ 2 )cos[(ω 1 ω 2 )t + α 1 α 2 ] Let ω 1 = ω 2 I = I 1 + I I 1 I 2 ( a ˆ 1 a ˆ 2 )cos( α 1 α 2 ) Bight intefeence finge α 1 α 2 = 2πm Dak intefeence finge α 1 α 2 = 2π m Optics James C. Wyant Page 3 of 26 Intefeence of Two Plane Waves k1 k 2 θ 1 θ2 y α = α 1 α 2 = k 1 k 2 + φ 1 φ 2 x E 1 = E 1 e i ( k 1 ωt+φ 1 )ˆ a 1 E 2 = E 2 e i ( k 2 ωt+φ 2 )ˆ a 2 k 1 = k( cosθ 1 i ˆ + sin θ 1 ˆ j ) k 2 = k cosθ 2 i ˆ + sin θ ˆ 2 j = x i ˆ + y ˆ j, k = 2π λ ( ) Let φ 1 = φ 2 Bight finge α = 2π m = kxcosθ ( 1 cosθ 2 )+ y( sinθ 1 sinθ 2 ) [ ] Dak finge ( ) α = 2π m Optics James C. Wyant Page 4 of 26 Page 2

3 Finge Spacing I = I 1 + I I 1 I 2 ( a ˆ 1 a ˆ 2 )cos( α 1 α 2 ) Bight finge α = α 1 α 2 = 2πm = kxcosθ ( 1 cosθ 2 )+ y( sinθ 1 sinθ 2 ) Staight equi-spaced finges Look in x=0 plane Finge spacing y = [ ] λ sinθ 1 sinθ 2 Optics James C. Wyant Page 5 of 26 Finge Visibility I I 1 + I 2 + 2( a ˆ 1 a ˆ 2 ) I 1 I 2 AC I 1 + I 2 I 1 + I 2 2( a ˆ 1 a ˆ 2 ) I 1 I 2 DC y Finge Visibility = V = I max I min I max + I min if a ˆ 1 a ˆ 2 = 1 V = 2 I 1I 2 = I 1 + I AC 2 DC Optics James C. Wyant Page 6 of 26 Page 3

4 Finge Spatial Fequency (1) φ φ (2) /2 /2 φ λ=633 nm (1) ν s (l / mm) 2sin ( φ /2) ν s = λ ν s = sin φ λ (2) φ (Degees ) Optics James C. Wyant Page 7 of 26 Moié Patten - Two Plane Waves Optics James C. Wyant Page 8 of 26 Page 4

5 Effect of Polaization Diection E p y E s θ 2 E p θ 1 x E s Dependence of a ˆ 1 a ˆ 2 on angle fo s and p polaization s polaization: a ˆ 1 a ˆ 2 = 1 fo all angles p polaization: a ˆ 1 a ˆ 2 depends upon angle Optics James C. Wyant Page 9 of 26 Intefeence of Two Spheical Waves S 1 1 P( ) S 2 2 E 1 = a ˆ B 1 1 e ik [ 1 ωt+φ 1 ] 1 E 2 = a ˆ B 2 2 e ik [ 2 ωt+φ 2 ] 2 Optics James C. Wyant Page 10 of 26 Page 5

6 Two Spheical Waves - Finge Shape E 1 = a ˆ B 1 1 e ik [ 1 ωt+φ 1 ] 1 E 2 = a ˆ B 2 2 e ik [ 2 ωt+φ 2 ] 2 I 1 = I 2 = B 1 Constant 1 B 2 Constant 2 α = 2π { λ 1 2 }+ φ 1 φ 2 = Constant fo given finge Hypebolic Finges Optics James C. Wyant Page 11 of 26 Moié Patten - Spheical Waves Optics James C. Wyant Page 12 of 26 Page 6

7 Spheical Waves - Special Case #1 1 2c 2 x o if x o >> 2c then mλ = 2cy x o Same esult as fo two plane waves Optics James C. Wyant Page 13 of 26 Moié Patten - Staight Line Finges Optics James C. Wyant Page 14 of 26 Page 7

8 Spheical Waves - Special Case #2 y 2c Y o x Fo bight finge mλ = 1 2 ( = 2c x2 + z 2 ) c Y o 2 Finges ae concentic cicles = 2c mλ c Y o spatial fequency = 1 = 2c y o2 λ Optics James C. Wyant Page 15 of 26 Concentic Cicula Finges Optics James C. Wyant Page 16 of 26 Page 8

9 Intefeence of Plane Wave and Spheical Wave x o y x If x o >> y and z then θ α = 2π ysin θ x λ o 1 + y2 + z 2 2x2 o If θ = 0 bight finge when y2 + z 2 2x o = (mλ + x o ) = m λ Cicula finges of adius = 2x o m λ Optics James C. Wyant Page 17 of 26 Two Basic Classes of Intefeometes Division of Wavefont Division of Amplitude Optics James C. Wyant Page 18 of 26 Page 9

10 Division of Wavefont (Young s Two Pinholes) Souce Two Pinholes Intefeence of two spheical waves Optics James C. Wyant Page 19 of 26 Division of Wavefont (Lloyd s Mio) S 1 S 2 Mio Intefeence of two spheical waves Optics James C. Wyant Page 20 of 26 Page 10

11 Division of Wavefont (Fesnel smios) S M 1 S 1 M 2 S 2 Optics James C. Wyant Page 21 of 26 Division of Wavefont (Fesnel s Bipism) S 1 S S 2 Optics James C. Wyant Page 22 of 26 Page 11

12 Division of Amplitude (Beamsplitte) Beamsplitte Optics James C. Wyant Page 23 of 26 Division of Amplitude (Diffaction) θ d sin θ = mλ Diffaction Gating Optics James C. Wyant Page 24 of 26 Page 12

13 Division of Amplitude and Division of Wavefont Polaization Lateal Displacement Angula Displacement E O O E OA OA Savat Plate Wollaston Pism Optics James C. Wyant Page 25 of 26 Division of Amplitude and Division of Wavefont Plane Paallel Plate Optics James C. Wyant Page 26 of 26 Page 13

Chapter 16. Fraunhofer Diffraction

$Chapter 16. Fraunhofer Diffraction$ Chapte 6. Faunhofe Diffaction Faunhofe Appoimation Faunhofe Appoimation ( ) ( ) ( ) ( ) ( ) λ d d jk U j U ep,, Hugens-Fesnel Pinciple Faunhofe Appoimation : ( ) ( ) ( ) λ π λ d d j U j e e U k j jk ep,,