Polarized and unpolarised transverse waves, with applications to optical systems

Size: px

Start display at page:

Download "Polarized and unpolarised transverse waves, with applications to optical systems"

Phebe Horton
5 years ago
Views:

1 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 6 1 Polarized and unpolarised transverse waves, with applications to optical systems T. Johnson

2 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 6 2 Outline Previous lecture: The quarter wave plate Set up coordinate system suitable for transverse waves Jones calculus; matrix formulation of how wave polarization changes when passing through polarizing component Examples: linear polarizer, quarter wave plate, Faraday rotation This lecture Statistical representation of incoherent/unpolarized waves Polarization tensors and Stokes vectors The Poincaré sphere Muller calculus; matrix formulation for the transmission of partially polarized waves

3 Incoherent/unpolarised waves Many sources of electromagnetic radiation are not coherent they do not radiate perfect harmonic oscillations (not sinusoidal wave) over a few wave lengths the oscillations may look harmonic over longer periods the wave appear unpredictable, incoherent, or even stochastic such waves are often referred to as unpolarised To model such waves, consider the electric field to be a stochastic process, with statistical properties: an average: < E α (t,x) > a variance: < [ E α (t,x) ] * E β (t,x) > a covariance: < [ E α (t,x) ] * E β (t+s,x+y) > In this chapter we will focus on the variance, here called the intensity tensor I αβ = < [E α (t,x) ] * E β (t,x) > and the polarization tensor (where e M =E / E is the polarization vector) p αβ = < [ e Mα (t,x) ] * e Mβ (t,x) > 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 6 3

4 Representations for the polarization tensor 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 6 4

5 Examples 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 6 5

6 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 2 - T. Johnson 6 Table of ideal polarisations By Dan Moulton - CC BY-SA 3.0,

Since e M1 and e M2 are uncorrelated the

7 The polarization tensor for unpolarized waves (1) What are the Stokes parameters for unpolarised waves? Let the e M1 and e M2 be stochastic variable Since e M1 and e M2 are uncorrelated the offdiagonal term vanish The vector e M is normalised: By symmetry (no statistical difference between e M1 and e M2 ) the polarization tensor then reads i.e. unpolarised have {q,u,v}={0,0,0}! 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 6 7

The polarization tensor for unpolarized waves (2) Alternative derivation: Note first that the polarization vector is normalised the polarization is complex and stochastic: where θ, φ 1 and φ 2 are

8 The polarization tensor for unpolarized waves (2) Alternative derivation: Note first that the polarization vector is normalised the polarization is complex and stochastic: where θ, φ 1 and φ 2 are uniformly distributed in [0,2π] The corresponding polarization tensor here the average is over the three random variables θ, φ 1 and φ 2 i.e. unpolarised have {q,u,v}={0,0,0}! 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 6 8

$vector { q/r, u/r, v/r }; the polarised fraction since this vector is real and normalised it will represent points on a sphere, the so called Poincaré sphere u$ Thus, a transverse wave field is described by a point on the Poincaré sphere Polar coordinates χ and ψ are useful!

Thus, a transverse wave field is described by a point on the Poincaré sphere Polar coordinates χ and ψ are useful!

9 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 6 9 Poincaré sphere Define the degree of polarisation: r = q $ + u $ + v $ Consider the normalised vector { q/r, u/r, v/r }; the polarised fraction since this vector is real and normalised it will represent points on a sphere, the so called Poincaré sphere u Thus, a transverse wave field is described by a point on the Poincaré sphere Polar coordinates χ and ψ are useful! v 2ψ 2χ q A polarizing element induces a motion on the sphere Example: passing though a birefringent crystal traces a circle! Birefringence induces a rotation in χ Faraday rotation is a rotation in ψ Poincare sphere

10 The Stokes vector The intensity tensor, I +, = E + E,, is also Hermitian: I +, = 1 2 I Q U V 0 i i 0 = I + Q U iv U + iv I Q The four real parameters I, Q, U, V are the Stokes parameter The Stokes vector is defined as S 9 = I, Q, U, V Using unit- and Pauli-matrixes, we define τ A αβ as: Using index notation the intensity tensor and the Stokes vector are related by: The matrixes τ A αβ, defines a transformation between Hermitian 2x2 matrixes and real 4-vectors 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 6 10

11 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 6 11 Previous lecture: The quarter wave plate Outline Set up coordinate system suitable for transverse waves Jones calculus; matrix formulation of how wave polarization changes when passing through polarizing component Examples: linear polarizer, quarter wave plate, Faraday rotation This lecture Statistical representation of incoherent/unpolarized waves Polarization tensors and Stokes vectors The Poincare sphere Müller (Mueller) calculus; matrix formulation for the transmission of partially polarized waves Example: Optical components General theory for dispersive media

12 Müller matrixes Müller matrixes maps incoming to outgoing Stokes vectors S 9 :;< = M 9> S >?@ Since S 9 is a four-vector, M 9> is four-by-four Müller matrixes generalises Jones matrixes by describing both coherent and incoherent waves How can we find the components of a Müller matrix, e.g. for a linear polariser? a) 4x4=16 unknown b) Consider four experiments with different polarisation, e.g. i. Horisontal polarisation, S = 1,1,0,0 ii. 45 o tilted polarisation, S = [1,0,1,0] iii. Circular polarisation, S = [1,0,0,1] iv. Incoherent polarisation, S = [1,0,0,0] c) Calculate the relation S 9 :;< (S >?@ ), for each of the four polarisations above. Note: This is four equations per polarisation, i.e. 16 equations! 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 2 - T. Johnson 12

13 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 6 13 Examples of Müller matrixes Examples: Linear polarizer (Horizontal Transmission) Linear polarizer (45 o transmission) Quarter wave plate (fast axis horizontal) Attenuating filter (30% Transmission) What is the Müller matrix for Faraday rotation?

=[1,0,0,0] Step 1: Linear polariser transmit linearly polarised light / 2 Step 2: Quarter wave

14 Examples of Müller matrixes In optics it is common to connect a series of optical elements Consider a system with: a linear polarizer and a quarter wave plate Insert unpolarised light, S A in =[1,0,0,0] Step 1: Linear polariser transmit linearly polarised light / 2 Step 2: Quarter wave plate transmit circularly polarised light 2 x 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 6 14

2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 6 15 Weakly

rewritten on a form suitable for studying the wave polarisation.

perturbation The wave equation when ΔK ij is a small, the 1 st order

15 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 6 15 Weakly anisotropic media In weakly anisotropic media the wave equation can be rewritten on a form suitable for studying the wave polarisation. Write the weakly anisotropic transverse response as where ΔK αβ is a small perturbation The wave equation when ΔK ij is a small, the 1 st order dispersion relation reads: n 2 n 0 2 the left hand side can then be expanded to give small, <<1

equation in Jones calculus Inverse Fourier

16 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 6 16 The wave equation in Jones calculus Inverse Fourier transform, when k 0 =ωn 0 /c: Factor our the eikonal with wave number k 0 : The wave equation can then be simplified The differential transfer equation in the Jones calculus! (We will use this relation in the next lecture)

The wave equation as an ODE Wave equation for the intensity

AB describes non-dissipative changes in polarization and the

17 The wave equation as an ODE Wave equation for the intensity tensor: from prev. page: Rewrite it in terms of the Stokes vector: we may call this the differential formulation of Müller calculus symmetric matrix ρ AB describes non-dissipative changes in polarization and the antisymmetric matrix µ AB describes dissipation (absorption) 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 6 17

18 The wave equation as an ODE The ODE for S A has the analytic solution (cmp to the ODE y =ky) cmp with Taylor series for exponential Here M AB is a Müller matrix M AB represents entire optical components / systems This is a component based Müller calculus 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 6 18

19 Summary Unpolarised waves is incoherent when studied on time-scales longer than the wave-period. Representation of unpolarised waves using Intensity tensor (Hermitian), p +, Polarisation tensor (Hermitian), I +, Stokes vector, S 9, components of I +, using the Pauli matrixes as basis Polarised part of a wave field may be represented on the Poincaré sphere Müller matrixes, M 9>, describe changes in the polarisation when passing an optical component. 2/16/17 Electromagnetic Processes In Dispersive Media, Lecture 2 - T. Johnson 19

Jones calculus for optical system

2/14/17 Electromagnetic Processes In Dispersive Media, Lecture 6 1 Jones calculus for optical system T. Johnson Key concepts in the course so far What is meant by an electro-magnetic response? What characterises