Ebook Topic:
Back Matter
Abstract
This back matter contains the bibliography and index.

Equation Summary

General equations

Malus’ Law and Brewster’s Law:

I(θ)=I0cos2θtani=n2/n1=n

Field components propagating in the z direction:

Ex(z,t)=E0xcos(ωtkz+δx)Ey(z,t)=E0ycos(ωtkz+δy)

Equation of the polarization ellipse:

Ex(z,t)2E0x2+Ey(z,t)2E0y22Ex(z,t)Ey(z,t)E0xE0ycosδ=sin2δ

Parameters of the polarization ellipse:

tan2ψ=2E0xE0yE0x2E0y2cosδ0ψπsin2χ=2E0xE0ysinδE0x2+E0y2π/4<χπ/4

Auxilliary angle definition and polarization parameters:

tanα=E0yE0x, 0απ/2tan2ψ=(tan2α)cosδsin2χ=(sin2α)sinδ0δ<2π

Stokes polarization parameters:

S0=E0x2+E0y2S1=E0x2E0y2S2=2E0xE0ycosδS3=2E0xE0ysinδ, δ=δyδxS02=S12+S22+S32

Stokes vector:

S=(S0S1S2S3)=(E0x2+E0y2E0x2E0y22E0xE0ycosδ2E0xE0ysinδ)

Stokes parameters in complex notation:

Ex(t)=E0xexp(iδx), Ey(t)=E0yexp(iδy)S0=ExEx*+EyEy*S1=ExEx*EyEy*S2=ExEy*+EyEx*S3=i(ExEy*+EyEx*)

Stokes vector on the Poincaré sphere:

S=(S0S1S2S3)=[1cos(2χ)cos(2ψ)cos(2χ)sin(2ψ)sin(2χ)]ψ=12tan1(S2S1)0ψπ, χ=12sin1(S3S0)π4χπ4

Stokes vector on the observable polarization sphere:

S=(S0S1S2S3)=(1cos2αsin2αcosδsin2αsinδ)

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

2α=cos1(S1S0)02α<πδ=tan1(S3S2)0δ<2π

Ellipticity and orientation angles relations:

tan(2ψ)=tan(2α)cosδsin(2χ)=sin(2α)sinδcos(2α)=cos(2χ)cos(2ψ)cotδ=cot(2χ)sin(2ψ)

Stokes vectors for unpolarized and partially polarized light:

SUNP=S0(1000)S=(S0S1S2S3)=(1P)(S0000)+P(S0S1S2S3)0P1

Definition of the degree of polarization P:

P=IpolItot=S12+S22+S32S00P1S02S12+S22+S32

Mueller matrices

Mathematical form of the Mueller matrix:

M=(m00m01m02m03m10m11m12m13m20m21m22m23m30m31m32m33)

Mueller matrix for a polarizer:

MPOL(px,py)=12(px2+py2px2py200px2py2px2+py200002pxpy00002pxpy)0px10py1

Mueller matrices for a linear horizontal and vertical polarizer:

MPOL=12(1100110000000000)MPOL=12(1100110000000000)

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

MPOL=12(1010000010100000)

Mueller matrix for a wave plate:

MWP(ϕ)=(1000010000cosϕsinϕ00sinϕcosϕ)ϕ=2πλ(none)d

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

MQWP=(1000010000010010)MHWP=(1000010000100001)

Mueller matrix for a rotator:

MROT(θ)=(10000cos2θsin2θ00sin2θcos2θ00001)

Mueller matrix for a rotated polarizing element:

M(θ)=MROT(2θ)MMROT(2θ)

Mueller matrix for a rotated linear polarizer:

MPOL(θ)=12(1cos2θsin2θ0cos2θcos22θsin2θcos2θ0sin2θsin2θcos2θsin22θ00000)

Mueller matrix of a circular polarizer:

MCP=(1010000000001010)

Reflection and transmission

Fresnel’s Equations for reflection and refraction:

Rp=tan(ir)tan(i+r)EpRs=sin(ir)sin(i+r)EsTp=2sinrcosisin(i+r)cos(ir)EpTS=2sinrcosisin(i+r)ES

Stokes parameters for reflection:

S0R=fR[(cos2α+cos2α+)S0+(cos2αcos2α+)S1]S1R=fR[(cos2αcos2α+)S0+(cos2α+cos2α+)S1]S2R=fR(2cosαcosα+)S2S3R=fR(2cosαcosα+)S3fR=12(tanαtanα+)2

Stokes parameters for transmission (refraction):

S0T=fT[(cos2α+1)S0+(cos2α1)S1]S1T=fT[(cos2α1)S0+(cos2α+1)S1]S2T=fT(2cosα)S2S3T=fT(2cosα)S3fT=12sin2isin2r(sinα+cosα)2

Mueller matrices for reflection and transmission:

MR=(n1n+1)2(1000010000100001)MT=4n(n+1)2(1000010000100001)

Mueller matrices for Brewster angle reflection and transmission:

MR,B=12cos22iB(1100110000000000)MT,B=12(sin22iB+1sin22iB100sin22iB1sin22iB+100002sin2iB00002sin2iB)

Mueller matrix for total internal reflection:

MR=(1000010000cosδsinδ00sinδcosδ)tanδ2=cosin2sin2i1nsin2i

Mueller matrix of the Fresnel rhomb:

M=(1000010000cos(δU+δL)sin(δU+δL)00sin(δU+δL)cos(δU+δL))

Jones matrix calculus

Jones vector:

E=(ExEy)=(E0xeiδxE0yeiδy)

Jones matrix formulation of optical intensity:

I=EEI=(Ex*Ey*)(ExEy)=ExEx*+EyEy*

Jones matrix definition:

J=(jxxjxyjyxjyy)

Jones matrices for linear polarizers:

JPOL=(px00py)0px,py1JLHP=(1000)JLVP=(0001)

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

JWP=(ei(ϕ/2)00ei(ϕ/2))(100eiϕ)JQWP=(100i)JHWP=(1001)

Jones matrix for a rotator:

JROT(θ)=(cosθsinθsinθcosθ)

Jones matrix for a rotated polarizing element:

J(θ)=JROT(θ)JJROT(θ)

Polarizer characterization

Transmittance of a single linear polarizer:

TUNP=k1+k22TLHP=k12TLVP=k22

Transmittance of a pair of linear polarizers:

TUNP=H0=12(k12+k22)TLHP=k122TLVP=k222

Transmittance of a pair of crossed polarizers:

TUNP=H90=k1k2

Transmittance for a single rotated polarizer at an angle θ:

T(θ)=k1sin2θ+k2cos2θ

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

T(θ)=k1k2sin2θ+12(k12+k22)cos2θ=H90+(H0H90)cos2θ

Logarithmic definition of transmittance (density):

D0=log10H0D90=log10H90

Contrast ratio of a polarizer:

C=TmaxTminTmax+Tmin=k1k2k1+k2

Kerr and Pockel cell characterization

Birefringence of the Kerr effect:

npns=λBε2

Pockel cell phase shift:

ϕ=ϕyϕx=ωn03r63cV

Stokes vectors for radiating charges

Stokes vector for a linearly oscillating charge:

S=(ez0c2R)2sin2θω04(1100)

Stokes vector for a randomly oriented oscillating charge:

S=8π3(eAc2R)2ω04(1000)

Stokes vector for a charge moving in a circle:

S=(eAc2R)2ω04(1+cos2θsin2θ02cosθ)

Stokes vector for the Zeeman effect:

S=23(eA2c2R)2[ω4(1+cos2θsin2θ02cosθ)+2ω04(sin2θsin2θ00)+ω+4(1+cos2θsin2θ02cosθ)]

Mueller matrix for Thomson scattering:

M=12(e2mc2R)2(1+cos2θsin2θ00sin2θ1+cos2θ00002cosθ00002cosθ)

Stokes vector for Rayleigh scattering:

S=12[e2mc2(ω2ω02)]2ω4(S0(1+cos2θ)S1sin2θS0sin2θ+S1(1+cos2θ)2S2cosθ2S3cosθ)

Absorbing media—semiconductors and metals

Complex dielectric constant and refractive index for metals:

ε=εi(4πσω)n=n(1iκ)

Reflectivity at normal incidence for absorbing media:

Rs=Rp=[(n1)2+(nκ)2(n+1)2+(nκ)2]

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

Rs=|RsEs|2=[(ncosi)2+(nκ)2(n+cosi)2+(nκ)2]Rp=|RpEp|2=[(n1/cosi)2+(nκ)2(n+1/cosi)2+(nκ)2]
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