Instrumental

lecture 3

13 March 2020 Lecture 2 – summary

1. Optical materials (UV, VIS, IR) 2. Anisotropic media (ctd) • Double refraction

2. Anisotropic media • : no, ne() • Permittivity tensor • Birefringent materials – calcite, • Optical axis YVO4,

• Ordinary wave  perpendicular to o.a.-푘 plane • Extraordinary wave  Polarization in o.a.-푘 plane

• Birefringent walk-off 3. Polarization of light • Wave vector ellipsoid • states of polarization

Polarization of light

linear circular elliptical 휋 휋 Δ휙 = 0, sin Θ = 퐸 /퐸 Δ휙 = , 퐸 = 퐸 Δ휙 ∈ (0, ) 푦 푥 2 푥 푦 2 Plane wave, 푘||푧, 푧 = const:

• Fully polarized light 푖(휔푡+휙푥) 퐸푥(푡) 퐸0푥푒 퐸0푥 • Partially polarized light 푖(휔푡+휙 ) i(휔푡+휙푥) iΔ휙 퐸 푡 = 퐸푦(푡) = 퐸 푒 푦 =e 퐸 e • Fully depolarized light 0푦 0푦 0 0 0

Δ휙 = 휙푦 − 휙푥 Note: electric polarization 푃(퐸) vs. polarization of a wave! Linear

Key parameter: Other parameters: 퐼 extinction coefficient, 푦 • Spectral range, 퐼푥 • damage threshold, (assumption: incident light is • angle of incidence, unpolarized!) • beam geometry For polarizing beamsplitters: 2 extinction coefficients, reflection typically worse • clear aperture

Polarizer types: • Dichroic • Crystalline • Thin-film Dichroic polarizers

Dichroic : • Made from material with anisotropic absorption (or reflection) coefficient

long, „conductive” molecules/structures

• Polaroid: polyvinyl alcohol + iodine – one polarization component is absorbed. • Cheap, large aperture, broadband, poor extinction, low damage threshold

• Wire grid / nanowire: sub-wavelength spaced metallic wires reflect/absorb one polarization • Good extinction (> 103), large aperture, broadband, better damage threshold • Variation: elongated Ag nanoparticles Crystalline (birefringent) polarizers

High extinction (104 ÷ 105) Some designs have very high damage threshold Small apertures Complicated geometry Restricted spectral range

Poor transmission (Fresnel reflection), Wollaston Glan-Foucault Beam deflection reflected beam not glue polarized

Good transmission, Good transmission reflected beam (~Brewster angle), Glan-Thompson polarized , but low Glan-Taylor/ reflected beam damage threshold Glan laser (air gap) polarized Dielectric (thin-film) polarizers

Thin dielectric layers. Polarization-dependent reflectivity based on polarization-dependent interference effects in thin (sub-휆) layers of varying refractive index .

Polarizing beam splitter Extinction: 102 ÷ 103 Very high damage threshold possible Spectral range – UV-VIS / VIS-NIR / NIR-IR Acceptance angle: ~10 degrees

Wave plates o.a.

→

Half-wave plate: 횪 = 흅 + 2푚휋 Phase shift: Quarter-wave plate: 흅 휞 = + 2푚휋 ퟐ Wave plate order: 푚 Phase shift is wavelength-dependent! fast axis Half-wave plate acting on linear polarization

퐸in

퐸out

Quarter-wave plate action

퐸 e푖휔푡 0 1 Linear polarization at 45 deg. to optical axis of wave plate 퐸in = 2 1

푖휔푡 1 푖휔푡 퐸0e 퐸0e 1 퐸0 푖휋 cos 휔푡 퐸out = = Re 퐸out = 2 1 ⋅ 푒 2 2 푖 2 −sin 휔푡 ! Construction of a wave plate

Quartz 2휋 푛푒 휆 −푛표 휆 푑 Phase shift: 훤 휆 = (: 0.009): 휆 푑 = 28 μm for zero-order HWP at 500 nm Quarter waveplate for 휆 = 550 nm In practice two thicker quartz plates with perpendicular SiO2 5th order + MgF2 5th order optical axes

True-zero-order wavplate: single thin quartz plate. SiO2, 0-order achromatic

SiO2, 5th order Jones calculus

2 푉푥 ∗ ∗ Jones vector: 푉 = , normalization: 푉 = 푉푥푉푥 + 푉푦푉푦 = 1. 푉푦

Examples: linear polarization

Circular polarization

Eliptical polarization, with ellipse axes aligned to reference frame:

cos휗 General eliptical polarization = sin휗ei휙

Jones vector is not unique: 푉 & 푒푖휑푉 correspond to the same state of polarization! (but remember about the global phase in interferometers etc.!) Partially polarized light – coherence matrix

Example: circularily polarized light: 휋 퐸0 퐸0 i(휔푡+ ) 퐸 = ei휔푡, 퐸 = e 2 푥 2 푦 2 퐸2 퐸 퐸∗ = 퐸 퐸∗ = 0 Unpolarized light 푥 푥 푦 푦 2 2 휋 2 퐸0 −i 퐸0 퐸 퐸∗ = e 2 = − 푖 푥 푦 2 2 2 휋 2 ∗ 퐸0 i 퐸0 퐸푦퐸푥 = e 2 = 푖 Linearly polarized light 2 2 (for fully polarized light in complex notation: 퐸푖퐸푗 = 퐸푖퐸푗) 2 퐸0 1 −푖 퐽 = 2 푖 1 Intensity Example 2: unpolarized light: i휔푡 i휔푡+휙random 퐸푥 = 퐸0e , 퐸푦 = 퐸0e Degree of polarization

∗ ∗ 2 퐸푥퐸푥 = 퐸푦퐸푦 = 퐸0

∗ ∗ 퐸푥퐸푦 = 퐸푦퐸푥 = 0 Jones matrices

Change of Jones vector due to polarization elements is described by the Jones matrix 푊

푉out = 푊푉in

Constant intensity -> Jones matrix is unitary: 푊†푊 = ퟏ

푉in 푉out Jones matrices for basic polarization elements

Free space propagation (distance 푑):

1 0 푊 = (same for isotropic dielectric) empty−space 0 1

Wave plate with optical axis aligned to the reference frame

1 0 푊 = waveplate 0 eiΓ

Polarizer along 푥 axis 1 0 푊 = polarizer 0 0

Jones matrices can also be used to transfrom coherence matices 퐽: 퐽′ = 푊†퐽푊 Change of basis Linear (퐿) – circular (퐾)

Reference frame rotation

Example: wave plate

푊plate Half-wave plate, /2

Γ =  푊plate

푊plate

Fast axis of waveplate

퐸ou푡 퐸in Quarter-wave plate, /4

Γ =  /2 푊plate

For α =  /4

Optically active material Circular birefringence: 푛푃 ≠ 푛퐿 (푃 – right, 퐿 – left )

Rotation angle independent of input! 푇 Stokes vector 푆0, 푆1, 푆2, 푆3 Measuring the state of polarization Natężenie

NPBS1 (2:1) NPBS2 (1:1) 푇 Stokes vector 푆0, 푆1, 푆2, 푆3 Poincaré sphere

Normalized Stokes vector

For fully polarized light:

In general ≤

Fully polarized light:

→ angle 2휗 around 푠3, angle 휑 wrt. : rotation by angle 훤 wrt. axis equatorial plane defined by eigenpolarizations