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 505 - 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 2 1 + E 2 2 + 2E 1 E 2 ( a ˆ 1 a ˆ 2 )cos[(ω 1 ω 2 )t + α 1 α 2 ] [ ] I = I 1 + I 2 + 2 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 505 - James C. Wyant Page 2 of 26 Page 1
Intefeence Finges I = I 1 + I 2 + 2 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 2 + 2 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 + 1 2 Optics 505 - 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 + 1 2 Optics 505 - James C. Wyant Page 4 of 26 Page 2
Finge Spacing I = I 1 + I 2 + 2 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 505 - 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 505 - James C. Wyant Page 6 of 26 Page 3
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 505 - James C. Wyant Page 7 of 26 Moié Patten - Two Plane Waves Optics 505 - James C. Wyant Page 8 of 26 Page 4
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 505 - 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 505 - James C. Wyant Page 10 of 26 Page 5
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 505 - James C. Wyant Page 11 of 26 Moié Patten - Spheical Waves Optics 505 - James C. Wyant Page 12 of 26 Page 6
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 505 - James C. Wyant Page 13 of 26 Moié Patten - Staight Line Finges Optics 505 - James C. Wyant Page 14 of 26 Page 7
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 505 - James C. Wyant Page 15 of 26 Concentic Cicula Finges Optics 505 - James C. Wyant Page 16 of 26 Page 8
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 505 - James C. Wyant Page 17 of 26 Two Basic Classes of Intefeometes Division of Wavefont Division of Amplitude Optics 505 - James C. Wyant Page 18 of 26 Page 9
Division of Wavefont (Young s Two Pinholes) Souce Two Pinholes Intefeence of two spheical waves Optics 505 - James C. Wyant Page 19 of 26 Division of Wavefont (Lloyd s Mio) S 1 S 2 Mio Intefeence of two spheical waves Optics 505 - James C. Wyant Page 20 of 26 Page 10
Division of Wavefont (Fesnel smios) S M 1 S 1 M 2 S 2 Optics 505 - James C. Wyant Page 21 of 26 Division of Wavefont (Fesnel s Bipism) S 1 S S 2 Optics 505 - James C. Wyant Page 22 of 26 Page 11
Division of Amplitude (Beamsplitte) Beamsplitte Optics 505 - James C. Wyant Page 23 of 26 Division of Amplitude (Diffaction) θ d sin θ = mλ Diffaction Gating Optics 505 - James C. Wyant Page 24 of 26 Page 12
Division of Amplitude and Division of Wavefont Polaization Lateal Displacement Angula Displacement E O O E OA OA Savat Plate Wollaston Pism Optics 505 - James C. Wyant Page 25 of 26 Division of Amplitude and Division of Wavefont Plane Paallel Plate Optics 505 - James C. Wyant Page 26 of 26 Page 13