linear states of polarization - electrical, computer &...
TRANSCRIPT
ECE 5616Curtis
Polarization of Light
• Polarization states• Stokes Vectors• Mueller Matrices• Jones Matrices• Components• Examples
ECE 5616Curtis
Polarization statesLinear states of polarization
x
y
x
y
0 ,0 =≠ yx EELinear x polarized
0 ,0 ≠= yx EELinear y polarized
πφφ =−= xyyx EE , Linear -45o polarized
πφφ ,0 , =−≠ xyyx EELinear θ polarized
O’Shea 1.6, Saleh & Teich 6
x
y
x
y
ECE 5616Curtis
Polarization statesElliptical states of polarization
Right-hand circular 2 , πφφ =−= xyyx EE
Left-hand circular 2 , πφφ −=−= xyyx EE
Elliptical
2
and/or
,
πφφ ±≠−
≠
xy
yx EE
x
y
x
y
x
y
x
y
ECE 5616Curtis
Stokes vectorsComplete description of polarization state
Perform 6 irradiance measurements with ideal polarizers:Ex Horizontal linearEy Vertical linearE45 45o linearE135 135o linearER Right circularEL Left circular
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−+
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
LR
VH
VH
EEEEEEEE
SSSS
S13545
3
2
1
0
r
ECE 5616Curtis
Stokes vectorsComplete description of polarization state
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
0011
H
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
=
00
11
V
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
0101
45
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−=
01
01
135
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
1001
R
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
=
1001
L
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
6.8.01
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
6.8.
01
ECE 5616Curtis
Poincarè sphereUseful visualization of Stokes parameters
• Surface is polarized, center is unpolarized• Equator is linear polarized• North hemisphere is right elliptical, south is left elliptical• Orthogonal polarizations are on opposite points of sphere
332211 jijijiji SSSSSSSS ++=⋅
S1
S2
S3
R
L
H
V
45
135
ECE 5616Curtis
Stokes vectorsDegree of polarization
0
23
22
21
SSSS
DoP++
= Degree of polarization
0
3
SSDoCP = Degree of circular polarization
Degree of linear polarization0
22
21
SSS
DoLP+
=
Stokes vectors are polychromatic on addition:
- 1 - 0 . 5 0 . 5 1
- 1
- 0 . 5
0 . 5
1
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
+
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
0002
00
11
0011 H + V at different
frequencies give Lissagous figure with no
average polarization state
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
+
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
0013
00
11
0022
- 2 - 1 1 2
- 1
- 0 . 5
0 . 5
1
2H + V = H polarized DoP = 1/3
ECE 5616Curtis
Mueller MatricsComplete polarization modeling
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
0011
0011
21
21
21
21
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
0000
00
11
21
21
21
21
E.g. Linear polarizer
4x4 matrix of real values describing transformation of polarizations.Can describe de-polarizing elements.Eigenvectors are eigenpolarization of system (unchanged).
Moral: We are going to use Jones vectors/matrices to do designs, but you should know their limitations. Jones vectors deal only with systems with perfect temporal and spatial coherence. Systems with finite coherence can be partially polarized and then you must use Mueller matrices and Stokes parameters.
ECE 5616Curtis
Jones vectorsSimplified for fully polarized systems
⎥⎦
⎤⎢⎣
⎡≡
y
x
AA
Jr where Ax and Ay are the complex amplitudes of the x and y
polarized electric fields.
*21
*2121 yyxx AAAAJJ +≡⋅ Inner product
⎥⎦
⎤⎢⎣
⎡≡
01
H ⎥⎦
⎤⎢⎣
⎡ −≡
11
V
⎥⎦
⎤⎢⎣
⎡≡
11
2145 ⎥
⎦
⎤⎢⎣
⎡−
≡1
12
1135
⎥⎦
⎤⎢⎣
⎡≡
jR
12
1⎥⎦
⎤⎢⎣
⎡−
≡j
L1
21
ECE 5616Curtis
Rotation matricesComponents not aligned with x or y
( )JRJrr
θ=′
( ) ⎥⎦
⎤⎢⎣
⎡ −=
θθθθ
θcossin
sincosR
( ) ( )θθ −=′ RTRT
Coordinate Transform Matrix
Transform for Jones Vectors
Transform for Jones Matrices
ECE 5616Curtis
Three types of physics1. Diattenuation (polarization dependent loss)
– Transmission is polarization dependent– “Polarizers”– A.k.a.: polarization dependent loss (PDL), dichroism
2. Retardance– Optical path length is polarization dependent.– “Wave plates”, optical activity, electro-optic– A.k.a.: polarization mode dispersion (PMD)– Poincarè sphere geometry
3. Depolarization– The degree of polarization may decrease depending on input
polarization– A.k.a.: polarization scrambling
ECE 5616Curtis
Polarizers⎥⎦
⎤⎢⎣
⎡≡
0001
xP Jones matrix of linear polarizer passing H
( ) ⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡==
0cos
sincos
0001 θ
θθ
θLPJ xOut
Power transmission of analyzer and arbitrary linear polarization:
θ2cos=⋅= OutOut JJT Malus’ Law
Dichroism: polarization dependent absorption
⎥⎦
⎤⎢⎣
⎡≡
−
−
z
z
x ee
P
00
β
αE.g.: Sheet polarizer, polarizing sunglasses,
PolarcorTM, wire-grid
Crystal polarizersReflective polarizers: Brewster’s angle, thin-film coatings
E.g.: Wollaston, Rochon, Sénarmont E.g.: Glan-Focault
ECE 5616Curtis
Jones matrix exampleCascaded polarizers
Crossed polarizers:
0 0 1 0 0 00 1 0 0 0 0
⎡ ⎤ ⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦y xA A
x
y z
1 0y xE E= A A
so no light leaks through.
Rotation tolerance
( )0 0 1 0 00 1 0 0
εε
ε ε⎡ ⎤ ⎡ ⎤ ⎡ ⎤
= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
y xA A
( )00 0
0x x
y y x
E EE E E
εεε
⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥
⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦y xA A So Iout ≈ ε2 Iin,x
0E1E
x-pol
y-pol
0E 1E
rotatedx-pol
y-pol
ECE 5616Curtis
Retarders⎥⎦
⎤⎢⎣
⎡≡ Γ− je
R0
01 Jones matrix of retarder with fast axis in x
242 ππ
==Γ Quarter-wave plate. Converts linear into circular.
ππ==Γ
22 Half-wave plate.
Converts linear into linear.
( ) ⎭⎬⎫
⎩⎨⎧
+= 2
2
2
2
22
SinC,11
eoo nnos
nnθθ
θ
( )
n
oe
o
o
L
Lnn
Δ=
=−=Γ
2
2
λ
λπ π
E.g. Half-wave plate.
Quartz QWP is only 48 microns thick at 1550 nm.Crossing plates of differing dispersion can reduce λ, θ dependence
ECE 5616Curtis
Power walk-offConsider a Fourier propagation problem in which the k-surface is a tilted line:
Fourier propagator
Shift theorem
x
z
kr
- 2 - 1.5 - 1 - 0.5 0.5 1 1.5 2nt
- 0.5
0.5
1
1.5
2nn sr
kx
kz
θ
( ) ( ){ }
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
=−
−
0,
0,, 11
zdkdkxE
exEFFzxE
x
z
zkdkdkj
xx
xx
z
Poynting vector is normal to k surface
ECE 5616Curtis
Beam displacing polarizer
Separates polarizations with very high isolation (limited only by crystal scatter). Emerging polarizations are parallel to a very high degree since the crystal can be flat to an arc-second.
L
Real space
ks
θ
Fourier space
α
Wave in crystal is going “sideways”
S
ECE 5616Curtis
Beam displacing polarizer
( )2
sincos
arctan
2221
2
2
2
2eo
eomaxBDe
o
emaxBD
nnnn
n
nn
+=⎥
⎦
⎤⎢⎣
⎡+=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
−θθθ
θ Crystal orientation for maximum walk-off
Phase velocity at this propagation angle
Differential optical path length
1045.09447.12
1486.29447.1
2
22
22
=⎟⎟⎠
⎞⎜⎜⎝
⎛−
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛−
+=Δ o
eo nnnn
YVO4 at 1.55 μm
θθα −⎥⎦
⎤⎢⎣
⎡= tanarctan 2
2
e
o
nn Walk-off angle vs. crystal cut angle
ECE 5616Curtis
Poincarè and retarders• Retarders rotate the polarization state on the Poincarè sphere.• Axis of rotation connects eigenpolarizations of the retarder.
• Example1: QWP with horizontal axis converts 45o linear into RHC• Example2: Optically active (or Faraday) rotator converts H to 45o
S1
S2
S3
R
L
H
V
45
135
ECE 5616Curtis
Useful combinationsQWP
@-45o
QWP@45o
Pol@0o
Circular polarizer
HWP@θ
Pol@0o
Adjustable linear polarizer
Pol@0o
Adjustable transmission
HWP@θ
Babinet-Soleil adjustable retarder
ECE 5616Curtis
Optical activityRetarders with circular eigenstates
⎥⎦
⎤⎢⎣
⎡−
≡22
22
cossinsincos
δδ
δδ
OA
ECE 5616Curtis
Wollaston• Wollaston polarizer is made of two birefringent material prisms that are
cemented together. The deviations of the ordinary and extraordinary beams are nearly symmetrical about the input beam axis. The separation angle exhibits chromatic dispersion, as shown in the blow. Any separation angle can be designed upon the requirement. Typical separation angle vs wavelength is shown in the plot below for Calcite.
Extinsion Ratio: 5x10 -6Beam Separation Angle:16.7-22.5 at 980nm
ECE 5616Curtis
Faraday Rotator• A Faraday rotator is an optical device that rotates the polarization of light due to
the Faraday effect, which in turn is based on a magneto-optic effect.• The Faraday rotator works because one polarization of the input light is in
ferromagnetic resonance with the material which causes its phase velocity to be higher than the other.
• The plane of linearly polarized light is rotated when a magnetic field is applied parallel to the propagation direction (permanent magnets). The empirical angle of rotation is given by:
• Where β is the angle of rotation (in radians).B is the magnetic flux density in the direction of propagation (in teslas). d is the length of the path (in metres) where the light and magnetic field interact.Then V is the Verdet constant for the material. This empirical proportionality constant (in units of radians per tesla per metre, rad/(T·m)) varies with wavelength and temperature and is tabulated for various materials.
• Note that direction of rotation given by sign of B Rotation is independent of propagation direction (NOT like waveplate)
• Faraday rotators are used in optical isolators to prevent undesired back propagation of light from disrupting or damaging an optical system.
VBd=β
ECE 5616Curtis
Polarization Dependent Isolator• The polarization dependent isolator, or Faraday isolator, is made of
three parts, an input polarizer (polarized vertically), a Faraday rotator, and an output polarizer, called an analyzer (polarized at 45 degrees)
• Light traveling in the forward direction becomes polarized vertically by the input polarizer. The Faraday rotator will rotate the polarization by 45 degrees. The analyser then enables the light to be transmitted through the isolator.
• Light traveling in the backward direction becomes polarized at 45 degrees by the analyzer. The Faraday rotator will again rotate the polarization by 45 degrees. This means the light is polarized horizontally (the rotation is insensitive to direction of propagation). Since the polarizer is vertically aligned, the light will be extinguished
ECE 5616Curtis
Nonreciprocal opticsaka Faraday rotators
HWP @ 22.5o 45o FRWollaston Wollaston
Example: Polarization independent isolator
Similar optics can form a pseudo-circulator:
in
out
FBG
ECE 5616Curtis
Design notation for polarization manipulation
HWP @ 22.5o 45o FRWollaston Wollaston
View of polarization as if looking into laser. This is the EE convention and yields RHC with thumb pointing into viewers eyes, along with laser. To use the Physics convention, consider view as with laser.
Angle of propagation
Position in system
FORWARD
REVERSE
In Out
OutIn
Collimator Collimator
ECE 5616Curtis
Another exampleSpatial phase-shifting interferometer
PBS HWP @ 22.5O Vertical beam displacer
Combine Rotate Interfere
0
π
Hei
ght o
f pro
paga
tion
Many thanks to Bill Chang for teaching me this notation.
ECE 5616Curtis
PBS Example
Newport Catalogue
2 kW/cm2 CW, 1 J/cm2 with 10 nsec pulses, typicalDamage Threshold
Non-abrasive method, acetone or isopropyl alcoholon lens tissue recommended (see Care and Cleaning of
Optics)Cemented optic, do not immerse in a solvent
Cleaning
MIL-C-675CDurability
-50 °C–90 °CTemperature Range
Broadband, multilayer coating, Ravg <1.0% per surfaceAntireflection Coating
±0.25 mmDimensions Tolerance
0° ±5°Angle of Incidence
90° ±5 arc minReflected Beam Deviation
≤5 arc minTransmitted Beam Deviation
Tp/Ts >500:1, 1000:1 averageExtinction Ratio
Tp >80%, >90% average, Rs >99.5% averageEfficiency
20-10 scratch-digSurface Quality
Central diameter, >80% of dimensionClear Aperture
≤λ/4 at 632.8 nm over the clear apertureWavefront Distortion
SF 2, NSSK grade, precision annealed optical glassMaterial
ECE 5616Curtis
Zero Order Waveplates
Newport CatalogueBlack anodized aluminumHousing
500 W/cm2 CW, 0.3 J/cm2 with 10 nsec pulses, visibleDamage Threshold
Non-abrasive method, acetone or isopropyl alcoholCleaning
Broadband, multilayer coating, Ravg <0.5%Antireflection Coating
-20 °C to 50 °CTemperature Range
6.2 mmHousing Thickness
25.4 ±0.13 mmHousing Diameter
3.56 mmThickness
± 7°Acceptance Angle
≤1 arc minTransmitted Beam Deviation
40-20 scratch-digSurface Quality
10.2 mmClear Aperture
≤λ/4 at 632.8 nm over the full apertureWavefront Distortion
±λ/100Retardation Accuracy
λ/4 or λ/2Retardation
BK 7, grade A, fine annealed optical glassSubstrate Material
Birefringent polymer film stackRetarder Material
ECE 5616Curtis
Multiple order Waveplates
Newport Catalogue
2 MW/cm2 CW, 2 J/cm2 with 10 nsec pulsesDamage
Threshold
Non-abrasive method, acetone or isopropyl alcohol
on lens tissue recommended, caution: fragile, thin optic
Cleaning
Single wavelength: 1 mm, nominalDual wavelength: 0.5–4 mm, nominalThickness
Single wavelength: R <0.5% total, or 0.25% per surface
Dual wavelength: Ravg <1.5%
Reflectivity per Surface
+0/-0.25 mmDiameter Tolerance
<0.5 arc secWedge
10-5 scratch-digSurface Quality
≥central 90% of diameterClear Aperture
≤λ/10 at 632.8 nm over the full apertureWavefront Distortion
±λ/300 at 20°C ±1°CRetardation Accuracy
λ/4 or λ/2Retardation
Single plateConstruction
Quartz, schlieren gradeMaterial
Must order for laser lineIncident angles must be close to normalMultiple nulls with polarizer