Lecture 39: Mueller Calculus and Optical Activity Physical Optics II (Optical Sciences 330) (Updated: Friday, April 29, 2005, 8:29 PM) W.J.

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1 C:\Dallas\1_Courses\1_OpSci_33\1 Lecture Notes\39 MuellerCalculus.doc: Page 1 of 5 Lecture 39: Mueller Calculus and Optical Activity Physical Optics II (Optical Sciences 33) (Updated: Friday, April 29, 25, 8:29 PM) W.J. Dallas Introduction The Mueller calculus is useful for coherent, partially coherent and incoherent light. In this calculus, the Mueller matrices represent optical elements that operate on Stokes vectors that represent lights with various states of polarization. Stokes parameters The Stokes parameters are defined in terms of the total irradiance (= flux-density) of a beam and changes in the irradiance when various polarizers, linear and circular, are inserted in the optical path of the beam. The changes are the difference in irradiance between having the first polarizer inserted and having the second polarizer inserted. Zero degrees is vertical and, for us, is in the x-direction. The polarizers are numbered from zero to three,. Isotropic (neutral density filter with irradiance transmittance of.5, i.e., 5% transmission) 1. Linear at (x-axis) 2. Linear Right-circular Let s define a set of irradiances that are transmitted by the polarizers, I = the irradiance in the presence of a 5% neutral density filter I 1 = the irradiance in the presence of a linear polarizer at I 2 = the irradiance in the presence of a linear polarizer at 45 I 3 = the irradiance in the presence of a right-circular polarizer We now look at some examples to see how these numbers behave when various polarizers are inserted. We let I = the transmitted irradiance. The incident light is unpolarized with irradiance 2I. No filter I = 2I Linear polarizer at I = I 1 Linear polarizer at 9 I = 2I - I 1 The Stokes parameters are defined as S = Incident irradiance 2I S 1 = irradiance for P and P 9 I 1 -( 2I - I 1 )= 2I 1-2I S 2 = irradiance for P 45 and P -45 2I 2-2I S 3 = irradiance for P RC and P LC 2I 3-2I Stokes vectors The Stokes parameters are inserted as components of a vector to compose the Stokes vector,

2 S S1 S = S2 S 3 C:\Dallas\1_Courses\1_OpSci_33\1 Lecture Notes\39 MuellerCalculus.doc: Page 2 of 5 The vector represents the polarization state of the light. Contrary to the Jones matrices, the light need not be coherent. Some Stokes Vectors Left circularly polarized: 1 1 Right circularly polarized: 1 Horizontal (x) linear polarization: 1 Vertical (y) linear polarization: º linear polarization: 1-45º linear polarization: 1 1 Mueller Matrix M The Mueller matrices represent the action of an optical on the polarization state of light that is described a Stokes vector. S out = MS in 4x4 matrix, 16 elements, 7 independent elements Some Mueller Matrices Polarizers Linear horizontal (x): Linear vertical (y):

3 Linear +45º: Linear -45º: C:\Dallas\1_Courses\1_OpSci_33\1 Lecture Notes\39 MuellerCalculus.doc: Page 3 of Circular left: Circular right: Retarders Quarter-wave plate horizontal (x): Quarter-wave plate vertical (y): Quarter-wave plate +45º : Quarter-wave plate -45º : Half-wave plate º or 9º : Optical Activity Half-wave plate ± 45 : Materials that rotate the plane of linearly polarized light are termed optically active. This is natural rotation as it is a property of the material. Consider a source on one side of the material and the observer on the other side. If the rotation, from the observer s perspective is clockwise, then the material is said to be dextrorotatory or d-rotatory. In the figure, θ > for a d-rotatory material. If the rotation of the plane of polarization is counter-clockwise, θ <, the material is said to be levorotatory or l-rotatory. In order to describe the action of optically active elements on polarized light, we will be using Jones vectors and matrices.

4 C:\Dallas\1_Courses\1_OpSci_33\1 Lecture Notes\39 MuellerCalculus.doc: Page 4 of 5 d θ Figure 1: Rotating the plane of linear polarization Specific rotation or rotating power The specific rotation is a property of the material and is defined as, rotation angle degrees β = e.g., distance millimeter For Figure 1, θ = βd. Examples Material β (degrees/mm) Hg S Lead Hyposulphate +5.5 Quartz What is the origin of optical activity? The origin is that the material has different indices of refraction for right- and left- polarized light. A material can be optically active and not birefringent. The properties are associated with different types of polarization: Optically active (circular birefringence): circular polarization Birefringence: linear polarization Why does an optically active material rotate linear polarization? Let s look at the Jones vectors, right circular 1 2 i left circular i We now compose a beam of the two circular polarizations, but with a phase-shift introduced between the two.

5 iψ 1 1 e iψ + + e = i i iψ i( 1 e ) C:\Dallas\1_Courses\1_OpSci_33\1 Lecture Notes\39 MuellerCalculus.doc: Page 5 of 5 Let s look at this expression for ψ = π. It becomes 1 1 2i i ( 1 1 = =, horizontally polarized.. + ) 2i 1 Linearly polarized light can be decomposed into a sum of two circularly-polarized components. For example, =. 4i 2 i 2 i Because we are dealing with plane waves directed parallel to the z-axis, the propagation formulas are ikz straight forward; we just multiply by e. The new aspect is the fact that the index of refractions differ for right and left polarized light. This has the consequence that there are two wave numbers, 2 2 k = π π r nrk and kl nk l λ n = = r λ n =. The propagation equation for x-direction polarized linear light moving l a distance z then becomes, 1 1 in 1 1 rkz inlkz e e 4i i 4i i It is convenient to define the indices in term of an average index and deviation from that average n n nr = n+ nl = n i nkz i nkz inkz 1 We then have, e e 2 e 2 4i i i Now we define ψ = nkz so that the expression becomes 2 1 inkz cosψ + isinψ cosψ isinψ e 4i sinψ + icosψ sinψ icosψ Finally we define and get π π nkz θ = ψ = e 2 inkz cosθ sinθ The direction of the linear polarization has been rotated by an angle θ.