Reference Optics by Hecht

Transcription

1 Reference Optics by Hecht

2 EquationOfEllipse.nb Optics 55 - James C. Wyant Equation of Ellipse e x = a x Cos@k -wtd e y = a y Cos@k -wt +fd FullSimplifyA i k e x y þþþþþþþ a x { Sin@fD Therefore, i e x y þþþþþþþ k a x { + e y i y þþþþþþþ k a y { + e y e y i y þþþþþþþ - i k a y { k e x y i y þþþþþþþ þþþþþþþ a x { k a Cos@fDE y { e y - i k e x y i y þþþþþþþ þþþþþþþ Cos@fD = Sin@fD a x { k a y {

3 JonesCalculus.nb Optics 55 - James C. Wyant Jones Calculus Jones Vectors Linear Horiontal lhjones = J N Linear Vertical lvjones = J N Linear at +45 degrees lp45jones = þþþþþþþþþþ!!!! J N Linear at -45 degrees lm45jones = þþþþþþþþþþ!!!! J - N Right Circular rcjones = þþþþþþþþþþ!!!! J N -É Left Circular lcjones = þþþþþþþþþþ!!!! J ÉN Right Elliptical rejones@ax_, ay_d :=!!!!!!!!!!!!!!!!!!!!! þþþþþþþþþþþþþþþþþþþþþþþþþþþ ax + ay J ax -É ay N

4 JonesCalculus.nb Optics 55 - James C. Wyant Left Elliptical lejones@ax_, ay_d :=!!!!!!!!!!!!!!!!!!!!! þþþþþþþþþþþþþþþþþþþþþþþþþþþ ax + ay J ax É ay N Jones Matrices Horiontal linear polarier hlpjones = J N Vertical linear polarier vlpjones = J N Linear polarier at + 45 degrees lpp45jones = þþþþ J N Linear polarier at - 45 degrees lpm45jones = þþþþ J - - N Quarter-wave plate with fast axis vertical qfavjones = È Épƒ4 J N -É Quarter-wave plate with fast axis horiontal qfahjones = È -É pƒ4 J N É Retarder with fast axis vertical rfavjones@f_d := È Éfƒ i k È -É f y { Retarder with fast axis horiontal rfahjones@f_d := È -É fƒ i k È Éf y {

5 y y Ex =Ex cos θ + Ey sin θ Ey =-Ex sin θ +Eycosθ Ey cos θ θ Ey Ey sin θ Ex cos θ x -Ex sin θ θ Ex Ex Ey = x cosθ sinθ sinθ cosθ Ex Ey Optics 55 - James C. Wyant

6 JonesCalculus.nb Optics 55 - James C. Wyant 3 Rotation Matrix rotjones@q_d := J -Sin@qD Cos@qD Cos@qD Sin@qD N Rotated Matrix = rotjones@-qd R@ D rotjones@qd OutputPolariation = rotjones@-qd R@ D rotjones@qd InputPolariation Calculation of matrix of a retarder of retardation f having a fast axis at an angle q from the horiontal rrot@f_, q_d := FullSimplify@rotJones@-qD.rfahJones@fD.rotJones@qDD MatrixForm@rrot@f, qdd i Éf È - þþþþþþþ HCos@qD +È Éf Sin@qD L -É Sin@ qd Sin@ þþþþ f D y k -É Sin@ qd Sin@ þþþþ f Éf þþþþþþþ D È- HÈ Éf Cos@qD + Sin@qD L {

7 þþþ þþþþ þþþ þþþþ þþþ þþþþ þþþ þþþþ ClassExamples.nb Optics 55 - James C. Wyant Class Examples Convert vertical linear polarier to linear polarier at 45 degrees MatrixForm@rotJones@45 DegreeD. vlpjones. rotjones@-45 DegreeDD i þþþþ þþþþ y þþþþ þþþþ k { Convert quarter-wave plate with fast axis vertical to quarter wave plate with fast axis horiontal MatrixForm@ rotjones@9 DegreeD.qfavJones.rotJones@-9 DegreeDD Ép i -É È 4 y Ép k È 4 { or MatrixForm@rotJones@-9 DegreeD.qfavJones.rotJones@9 DegreeDD Ép i -É È 4 y Ép k È 4 { Proof that a half-wave plate at angle q rotates plane of polariation q MatrixForm@TrigReduce@rotJones@-qD.rfavJones@pD.rotJones@qD.lhJonesDD É Cos@ qd J É Sin@ qd N Two half-wave plates at angle q between them are equivalent to rotator through angle q MatrixForm@TrigReduce@rotJones@-qD.rfavJones@pD.rotJones@qD.rfavJones@pDDD -Cos@ qd Sin@ qd J -Sin@ qd -Cos@ qd N

8 E A x x E + A y y E E x y cosφ = sin Ax Ay φ A x φ = < φ < π/ φ = π/ π/ < φ < π A y φ = π π < φ < 3π/ φ = 3π/ 3π/ < φ < π Optics 55 - James C. Wyant

9 StokesParameters.nb Stokes Parameters Ref: Optics by Hecht and Polaried Light by Shurcliff Let us have 4 filters, each of which transmits 5% of natural radiation # isotropic filter, passing all states equally, transmits I o # linear polarier, transmission axes are horiontal, transmits I #3 linear polarier, at 45, transmits I #4 circular polarier, opaque to L-state, I 3 Define Stokes parameters S o = I o S = I - I o S = I - I o S 3 = I 3 - I o S o is the incident irradiance. The other 3 Stokes parameters specify the state of polariation é S If S >, resembles horiontal polariation If S <, resembles vertical polariation If S =, beam shows no preferential orientation Hi.e. elliptical at 45, circular, unpolariedl

10 StokesParameters.nb é S If S >, resembles polariationat + 45 If S <, resembles polariationat - 45 If S =, neither é S 3 If S 3 >, tendency toward right handed If S 3 <, tendency toward left handed Calculating Stokes parameters é Electric field Ȩ x = A x Cos@k -wtd ì Ȩ y = A y Cos@k -wt +fd ` Ȩ = é Stokes Parameters S o = XA x \ + XA y \ S = X A x \ - XA x + A y \ = XA x \ - XA y \ S = [ þþþþþþþþþ!!! HA x + A y È Éf L _ - XA x + A y \ = X A x A y Cos@fD\ S 3 = We must calculate using Jones Calculus Multiplying the incident radiation times a right circular polarier yields þþþþ J É -É N.i XA x\ y kxa y È Éf i þþþ HXA x\ +ÉXÈ Éf A y \L y = \ { k þþþ H-É XA x\ + XÈ Éf A y \L { intensity = Z þþþþ 4 IA x + A y + A x A y CosAf + þþþþ p EM^ = Z þþþþ HA x + A y - A x A y Sin@fDL^ S 3 = X- A x A y Sin@fD\ Note : If f=-þþþþ p we have right handed circular. Often normalie Stokes parameters by dividing by S o.

11 StokesParameters.nb 3 Degree of Polariation If beam is unpolaried S o = XA x \ + XA y \ and S = S = S 3 = For completely polaried light S o = S + S + S3 It can be shown that the degree of polariation, V, is given by V = I p þþþþþþþþþþþþþþþþ I p + I u = "######################## S + S + S 3 þþþþþþþþþþþþþþþþ þþþþþþþþþþþþþþþþ S o Addition of Stokes parameters For incoherent beams the Stokes parameters add S o = S o + S o S = S + S S = S + S S 3 = S 3 + S 3

12 StokesVectorsandMuellerMatrices.nb Optics 55 - James C. Wyant Stokes Vectors and Mueller Matrices Stokes Vectors Unpolaried Light iy upstokes = k{ Linear Horiontal iy lhstokes = k{ Linear Vertical lvstokes = i y - k { Linear at +45 degrees iy lp45stokes = k{ Linear at -45 degrees i lm45stokes = k - y {

13 StokesVectorsandMuellerMatrices.nb Optics 55 - James C. Wyant Right Circular iy rcstokes = k{ Left Circular i y lcstokes = k -{ Mueller Matrices Horiontal linear polarier hlpmueller = þþþþ i y k { Vertical linear polarier vlpmueller = i þþþþþ k - y - { Linear polarier at + 45 degrees lpp45mueller = þþþþ i y k { Linear polarier at - 45 degrees lpm45mueller = i þþþþ k - y - {

14 StokesVectorsandMuellerMatrices.nb Optics 55 - James C. Wyant 3 Quarter-wave plate with fast axis vertical i y qfavmueller = - k { Quarter-wave plate with fast axis horiontal i y qfahmueller = k - { Rotation Matrix i y Cos@ qd Sin@ qd rotmueller@q_d := -Sin@ qd Cos@ qd k { Rotated Matrix = rotmueller@-qd R@ D rotmueller@qd OutputPolariation = rotmueller@-qd R@ D rotmueller@qd InputPolariation