Back Matter

doi:10.1117/3.626141.bm

Ebook Topic:
Back Matter

Book: Field Guide to Polarization

Author(s): Edward Collett

Published: 2005

https://doi.org/10.1117/3.626141.bm

Author Affiliations +

Abstract

This back matter contains the bibliography and index.

Equation Summary

General equations

Malus’ Law and Brewster’s Law:

I (θ) = I_{0} \cos^{2} θ \tan i = n_{2} / n_{1} = n

Field components propagating in the z direction:

\begin{array}{l} E_{x} (z, t) = E_{0 x} \cos (ω t - k z + δ_{x}) \\ E_{y} (z, t) = E_{0 y} \cos (ω t - k z + δ_{y}) \end{array}

Equation of the polarization ellipse:

\frac{E_{x} {(z, t)}^{2}}{E_{0 x}^{2}} + \frac{E_{y} {(z, t)}^{2}}{E_{0 y}^{2}} - \frac{2 E_{x} (z, t) E_{y} (z, t)}{E_{0 x} E_{0 y}} \cos δ = \sin^{2} δ

Parameters of the polarization ellipse:

\begin{array}{c} \tan 2 ψ = \frac{2 E_{0 x} E_{0 y}}{E_{0 x}^{2} - E_{0 y}^{2}} \cos δ & 0 \leq ψ \leq π \\ \sin 2 χ = \frac{2 E_{0 x} E_{0 y} \sin δ}{E_{0 x}^{2} + E_{0 y}^{2}} & - π / 4 < χ \leq π / 4 \end{array}

Auxilliary angle definition and polarization parameters:

\begin{array}{c} \tan α = \frac{E_{0 y}}{E_{0 x}}, 0 \leq α \leq π / 2 \\ \tan 2 ψ = (\tan 2 α) \cos δ \\ \sin 2 χ = (\sin 2 α) \sin δ 0 \leq δ < 2 π \end{array}

Stokes polarization parameters:

\begin{array}{l} S_{0} = E_{0 x}^{2} + E_{0 y}^{2} \\ S_{1} = E_{0 x}^{2} - E_{0 y}^{2} \\ S_{2} = 2 E_{0 x} E_{0 y} \cos δ \end{array}

\begin{array}{c} S_{3} = 2 E_{0 x} E_{0 y} \sin δ, δ = δ_{y} - δ_{x} \\ S_{0}^{2} = S_{1}^{2} + S_{2}^{2} + S_{3}^{2} \end{array}

Stokes vector:

S = (\begin{array}{c} S_{0} \\ S_{1} \\ S_{2} \\ S_{3} \end{array}) = (\begin{array}{c} E_{0 x}^{2} + E_{0 y}^{2} \\ E_{0 x}^{2} - E_{0 y}^{2} \\ 2 E_{0 x} E_{0 y} \cos δ \\ 2 E_{0 x} E_{0 y} \sin δ \end{array})

Stokes parameters in complex notation:

E_{x} (t) = E_{0 x} \exp (i δ_{x}), E_{y} (t) = E_{0 y} \exp (i δ_{y})

\begin{array}{l} S_{0} = E_{x} E_{x}^{*} + E_{y} E_{y}^{*} \\ S_{1} = E_{x} E_{x}^{*} - E_{y} E_{y}^{*} \\ S_{2} = E_{x} E_{y}^{*} + E_{y} E_{x}^{*} \\ S_{3} = i (E_{x} E_{y}^{*} + E_{y} E_{x}^{*}) \end{array}

Stokes vector on the Poincaré sphere:

\begin{array}{c} S = (\begin{array}{c} S_{0} \\ S_{1} \\ S_{2} \\ S_{3} \end{array}) = [\begin{array}{c} 1 \\ \cos (2 χ) \cos (2 ψ) \\ \cos (2 χ) \sin (2 ψ) \\ \sin (2 χ) \end{array}] \\ ψ = \frac{1}{2} \tan^{- 1} (\frac{S_{2}}{S_{1}}) 0 \leq ψ \leq π, χ = \frac{1}{2} \sin^{- 1} (\frac{S_{3}}{S_{0}}) - \frac{π}{4} \leq χ \leq \frac{π}{4} \end{array}

Stokes vector on the observable polarization sphere:

S = (\begin{array}{c} S_{0} \\ S_{1} \\ S_{2} \\ S_{3} \end{array}) = (\begin{array}{c} 1 \\ \cos 2 α \\ \sin 2 α \cos δ \\ \sin 2 α \sin δ \end{array})

Angle relation of 2a and δ to the Stokes parameters:

\begin{array}{c} 2 α = \cos^{- 1} (\frac{S_{1}}{S_{0}}) 0 \leq 2 α < π \\ δ = \tan^{- 1} (\frac{S_{3}}{S_{2}}) 0 \leq δ < 2 π \end{array}

Ellipticity and orientation angles relations:

\begin{array}{l} \tan (2 ψ) = \tan (2 α) \cos δ & \sin (2 χ) = \sin (2 α) \sin δ \\ \cos (2 α) = \cos (2 χ) \cos (2 ψ) & \cot δ = \cot (2 χ) \sin (2 ψ) \end{array}

Stokes vectors for unpolarized and partially polarized light:

\begin{array}{c} S_{UNP} = S_{0} (\begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \end{array}) \\ S = (\begin{array}{c} S_{0} \\ S_{1} \\ S_{2} \\ S_{3} \end{array}) = (1 - P) (\begin{array}{c} S_{0} \\ 0 \\ 0 \\ 0 \end{array}) + P (\begin{array}{c} S_{0} \\ S_{1} \\ S_{2} \\ S_{3} \end{array}) 0 \leq P \leq 1 \end{array}

Definition of the degree of polarization P:

\begin{array}{c} P = \frac{I_{pol}}{I_{tot}} = \frac{\sqrt{S_{1}^{2} + S_{2}^{2} + S_{3}^{2}}}{S_{0}} 0 \leq P \leq 1 \\ S_{0}^{2} \geq S_{1}^{2} + S_{2}^{2} + S_{3}^{2} \end{array}

Mueller matrices

Mathematical form of the Mueller matrix:

M = (\begin{array}{l} m_{00} & m_{01} & m_{02} & m_{03} \\ m_{10} & m_{11} & m_{12} & m_{13} \\ m_{20} & m_{21} & m_{22} & m_{23} \\ m_{30} & m_{31} & m_{32} & m_{33} \end{array})

Mueller matrix for a polarizer:

\begin{array}{l} M_{P O L} (p_{x}, p_{y}) & = & \frac{1}{2} (\begin{array}{c} p_{x}^{2} + p_{y}^{2} & p_{x}^{2} - p_{y}^{2} & 0 & 0 \\ p_{x}^{2} - p_{y}^{2} & p_{x}^{2} + p_{y}^{2} & 0 & 0 \\ 0 & 0 & 2 p_{x} p_{y} & 0 \\ 0 & 0 & 0 & 2 p_{x} p_{y} \end{array}) \\ 0 \leq p_{x} \leq 1 0 \leq p_{y} \leq 1 \end{array}

Mueller matrices for a linear horizontal and vertical polarizer:

\begin{array}{l} M_{POL} = \frac{1}{2} (\begin{array}{c} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}) & M_{POL} = \frac{1}{2} (\begin{array}{c} 1 & - 1 & 0 & 0 \\ - 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}) \end{array}

Mueller matrix for a linear polarizer with its transmission axis at +45°:

M_{POL} = \frac{1}{2} (\begin{array}{c} 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array})

Mueller matrix for a wave plate:

M_{WP} (ϕ) = (\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos ϕ & - \sin ϕ \\ 0 & 0 & \sin ϕ & \cos ϕ \end{array}) ϕ = \frac{2 π}{λ} (n_{o} - n_{e}) d

Mueller matrices for a quarter- and half-wave plate:

\begin{array}{l} M_{QWP} = (\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & 1 & 0 \end{array}) & M_{HWP} = (\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & - 1 \end{array}) \end{array}

Mueller matrix for a rotator:

M_{ROT} (θ) = (\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & \cos 2 θ & \sin 2 θ & 0 \\ 0 & - \sin 2 θ & \cos 2 θ & 0 \\ 0 & 0 & 0 & 1 \end{array})

Mueller matrix for a rotated polarizing element:

M (θ) = M_{ROT} (- 2 θ) \cdot M \cdot M_{ROT} (2 θ)

Mueller matrix for a rotated linear polarizer:

M_{P O L} (θ) = \frac{1}{2} (\begin{array}{c} 1 & \cos 2 θ & \sin 2 θ & 0 \\ \cos 2 θ & \cos^{2} 2 θ & \sin 2 θ \cos 2 θ & 0 \\ \sin 2 θ & \sin 2 θ \cos 2 θ & \sin^{2} 2 θ & 0 \\ 0 & 0 & 0 & 0 \end{array})

Mueller matrix of a circular polarizer:

M_{CP} = (\begin{array}{c} 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \end{array})

Reflection and transmission

Fresnel’s Equations for reflection and refraction:

\begin{array}{c} R_{p} = \frac{\tan (i - r)}{\tan (i + r)} E_{p} & R_{s} = - \frac{\sin (i - r)}{\sin (i + r)} E_{s} \\ T_{p} = \frac{2 \sin r \cos i}{\sin (i + r) \cos (i - r)} E_{p} & T_{S} = \frac{2 \sin r \cos i}{\sin (i + r)} E_{S} \end{array}

Stokes parameters for reflection:

\begin{array}{c} S_{0 R} = f_{R} [(\cos^{2} α_{-} + \cos^{2} α_{+}) S_{0} + (\cos^{2} α_{-} - \cos^{2} α_{+}) S_{1}] \\ S_{1 R} = f_{R} [(\cos^{2} α_{-} - \cos^{2} α_{+}) S_{0} + (\cos^{2} α_{-} + \cos^{2} α_{+}) S_{1}] \\ S_{2 R} = - f_{R} (2 \cos α_{-} \cos α_{+}) S_{2} \\ S_{3 R} = - f_{R} (2 \cos α_{-} \cos α_{+}) S_{3} f_{R} = \frac{1}{2} {(\frac{\tan α_{-}}{\tan α_{+}})}^{2} \end{array}

Stokes parameters for transmission (refraction):

\begin{array}{c} S_{0 T} = f_{T} [(\cos^{2} α_{-} + 1) S_{0} + (\cos^{2} α_{-} - 1) S_{1}] \\ S_{1 T} = f_{T} [(\cos^{2} α_{-} - 1) S_{0} + (\cos^{2} α_{-} + 1) S_{1}] \\ S_{2 T} = - f_{T} (2 \cos α_{-}) S_{2} \\ S_{3 T} = - f_{T} (2 \cos α_{-}) S_{3} f_{T} = \frac{1}{2} \frac{\sin 2 i \sin 2 r}{{(\sin α_{+} \cos α_{-})}^{2}} \end{array}

Mueller matrices for reflection and transmission:

\begin{array}{l} M_{R} = {(\frac{n - 1}{n + 1})}^{2} (\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & - 1 \end{array}) & M_{T} = \frac{4 n}{{(n + 1)}^{2}} (\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}) \end{array}

Mueller matrices for Brewster angle reflection and transmission:

\begin{array}{c} M_{R, B} = \frac{1}{2} \cos^{2} 2 i_{B} (\begin{array}{c} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}) \\ M_{T, B} = \frac{1}{2} (\begin{array}{c} \sin^{2} 2 i_{B} + 1 & \sin^{2} 2 i_{B} - 1 & 0 & 0 \\ \sin^{2} 2 i_{B} - 1 & \sin^{2} 2 i_{B} + 1 & 0 & 0 \\ 0 & 0 & 2 \sin 2 i_{B} & 0 \\ 0 & 0 & 0 & 2 \sin 2 i_{B} \end{array}) \end{array}

Mueller matrix for total internal reflection:

\begin{array}{l} M_{R} = (\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos δ & - \sin δ \\ 0 & 0 & \sin δ & \cos δ \end{array}) & \tan \frac{δ}{2} = \frac{\cos i \sqrt{n^{2} \sin^{2} i - 1}}{n \sin^{2} i} \end{array}

Mueller matrix of the Fresnel rhomb:

M = (\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos (δ_{U} + δ_{L}) & - \sin (δ_{U} + δ_{L}) \\ 0 & 0 & \sin (δ_{U} + δ_{L}) & \cos (δ_{U} + δ_{L}) \end{array})

Jones matrix calculus

Jones vector:

E = (\begin{array}{l} E_{x} \\ E_{y} \end{array}) = (\begin{array}{l} E_{0 x} e^{i δ_{x}} \\ E_{0 y} e^{i δ_{y}} \end{array})

Jones matrix formulation of optical intensity:

I = E^{†} \cdot E I = (E_{x}^{*} E_{y}^{*}) (\begin{array}{l} E_{x} \\ E_{y} \end{array}) = E_{x} E_{x}^{*} + E_{y} E_{y}^{*}

Jones matrix definition:

J = (\begin{array}{l} j_{x x} & j_{x y} \\ j_{y x} & j_{y y} \end{array})

Jones matrices for linear polarizers:

\begin{array}{l} J_{POL} = (\begin{array}{c} p_{x} & 0 \\ 0 & p_{y} \end{array}) & 0 \leq p_{x}, p_{y} \leq 1 \\ J_{LHP} = (\begin{array}{c} 1 & 0 \\ 0 & 0 \end{array}) & J_{LVP} = (\begin{array}{c} 0 & 0 \\ 0 & 1 \end{array}) \end{array}

Jones matrices for a wave plate, quarter-wave plate, and a half-wave plate:

\begin{array}{l} J_{WP} = (\begin{array}{c} e^{i (ϕ / 2)} & 0 \\ 0 & e^{- i (ϕ / 2)} \end{array}) \to (\begin{array}{c} 1 & 0 \\ 0 & e^{- i ϕ} \end{array}) \\ J_{QWP} = (\begin{array}{c} 1 & 0 \\ 0 & - i \end{array}) J_{HWP} = (\begin{array}{c} 1 & 0 \\ 0 & - 1 \end{array}) \end{array}

Jones matrix for a rotator:

J_{ROT} (θ) = (\begin{array}{c} \cos θ & \sin θ \\ - \sin θ & \cos θ \end{array})

Jones matrix for a rotated polarizing element:

J (θ) = J_{ROT} (- θ) \cdot J \cdot J_{ROT} (θ)

Polarizer characterization

Transmittance of a single linear polarizer:

T_{UNP} = \frac{k_{1} + k_{2}}{2} T_{LHP} = \frac{k_{1}}{2} T_{LVP} = \frac{k_{2}}{2}

Transmittance of a pair of linear polarizers:

T_{UNP} = H_{0} = \frac{1}{2} (k_{1}^{2} + k_{2}^{2}) T_{LHP} = \frac{k_{1}^{2}}{2} T_{LVP} = \frac{k_{2}^{2}}{2}

Transmittance of a pair of crossed polarizers:

T_{UNP} = H_{90} = k_{1} k_{2}

Transmittance for a single rotated polarizer at an angle θ:

T (θ) = k_{1} \sin^{2} θ + k_{2} \cos^{2} θ

Transmittance for a fixed and rotated polarizer at an angle θ:

\begin{array}{c} T (θ) = k_{1} k_{2} \sin^{2} θ + \frac{1}{2} (k_{1}^{2} + k_{2}^{2}) \cos^{2} θ \\ = H_{90} + (H_{0} - H_{90}) \cos^{2} θ \end{array}

Logarithmic definition of transmittance (density):

D_{0} = - \log_{10} H_{0} D_{90} = - \log_{10} H_{90}

Contrast ratio of a polarizer:

C = \frac{T_{\max} - T_{\min}}{T_{\max} + T_{\min}} = \frac{k_{1} - k_{2}}{k_{1} + k_{2}}

Kerr and Pockel cell characterization

Birefringence of the Kerr effect:

n_{p} - n_{s} = λ B ε^{2}

Pockel cell phase shift:

ϕ = ϕ_{y} - ϕ_{x} = \frac{ω n_{0}^{3} r_{63}}{c} V

Stokes vectors for radiating charges

Stokes vector for a linearly oscillating charge:

S = {(\frac{e z_{0}}{c^{2} R})}^{2} \sin^{2} θ ω_{0}^{4} (\begin{array}{l} 1 \\ 1 \\ 0 \\ 0 \end{array})

Stokes vector for a randomly oriented oscillating charge:

S = \frac{8 π}{3} {(\frac{e A}{c^{2} R})}^{2} ω_{0}^{4} (\begin{array}{l} 1 \\ 0 \\ 0 \\ 0 \end{array})

Stokes vector for a charge moving in a circle:

S = {(\frac{e A}{c^{2} R})}^{2} ω_{0}^{4} (\begin{array}{c} 1 + \cos^{2} θ \\ \sin^{2} θ \\ 0 \\ 2 \cos θ \end{array})

Stokes vector for the Zeeman effect:

S = \frac{2}{3} {(\frac{e A}{2 c^{2} R})}^{2} [ω_{-}^{4} (\begin{array}{c} 1 + \cos^{2} θ \\ - \sin^{2} θ \\ 0 \\ - 2 \cos θ \end{array}) + 2 ω_{0}^{4} (\begin{array}{c} \sin^{2} θ \\ \sin^{2} θ \\ 0 \\ 0 \end{array}) + ω_{+}^{4} (\begin{array}{c} 1 + \cos^{2} θ \\ - \sin^{2} θ \\ 0 \\ 2 \cos θ \end{array})]

Mueller matrix for Thomson scattering:

M = \frac{1}{2} {(\frac{e^{2}}{m c^{2} R})}^{2} (\begin{array}{c} 1 + \cos^{2} θ & - \sin^{2} θ & 0 & 0 \\ - \sin^{2} θ & 1 + \cos^{2} θ & 0 & 0 \\ 0 & 0 & 2 \cos θ & 0 \\ 0 & 0 & 0 & 2 \cos θ \end{array})

Stokes vector for Rayleigh scattering:

S^{'} = \frac{1}{2} {[\frac{e^{2}}{m c^{2} (ω^{2} - ω_{0}^{2})}]}^{2} ω^{4} (\begin{array}{c} S_{0} (1 + \cos^{2} θ) - S_{1} \sin^{2} θ \\ - S_{0} \sin^{2} θ + S_{1} (1 + \cos^{2} θ) \\ 2 S_{2} \cos θ \\ 2 S_{3} \cos θ \end{array})

Absorbing media—semiconductors and metals

Complex dielectric constant and refractive index for metals:

ε = ε - i (\frac{4 π σ}{ω}) n = n (1 - i κ)

Reflectivity at normal incidence for absorbing media:

R_{s} = R_{p} = [\frac{{(n - 1)}^{2} + {(n κ)}^{2}}{{(n + 1)}^{2} + {(n κ)}^{2}}]

s- and p- reflectivity at non-normal incidence:

\begin{array}{l} R_{s} = {| \frac{R_{s}}{E_{s}} |}^{2} = [\frac{{(n - \cos i)}^{2} + {(n κ)}^{2}}{{(n + \cos i)}^{2} + {(n κ)}^{2}}] \\ R_{p} = {| \frac{R_{p}}{E_{p}} |}^{2} = [\frac{{(n - 1 / \cos i)}^{2} + {(n κ)}^{2}}{{(n + 1 / \cos i)}^{2} + {(n κ)}^{2}}] \end{array}

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KEYWORDS

Wave plates

Linear polarizers

Polarizers

Polarization

Dielectric polarization

Geometrical optics

Crystals

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