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Chapter 4: Polarization of light

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Preliminaries and definitionsPreliminaries and definitions

PlanePlane--wave approximationwave approximation: : EE((rr,,tt) and ) and BB((rr,,tt) ) are uniform in the plane are uniform in the plane ^̂ kkWe will say that light We will say that light polarization vectorpolarization vector is is along along EE((rr,,tt) (although it was along ) (although it was along BB((rr,,tt) in ) in classic optics literature)classic optics literature)Similarly, Similarly, polarization planepolarization plane contains contains EE((rr,,tt) ) andand kk

kkBB EE

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Simple polarization statesSimple polarization statesLinear Linear or or plane polarizationplane polarizationCircular polarizationCircular polarization

Which one isWhich one is LCPLCP, and which is , and which is RCP RCP ??

Electric-field vector is seen rotating counterclockwise by

an observer getting hit in their eye by the light (do not try

this with lasers !)

Electric-field vector is seen rotating clockwise by the said

observer

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Simple polarization statesSimple polarization statesWhich one isWhich one is LCPLCP, and which is , and which is RCPRCP??Warning: optics definition is opposite Warning: optics definition is opposite to that in highto that in high--energy physics; energy physics; helicityhelicity

There are many helpful resources There are many helpful resources available on the web, including available on the web, including spectacular animations of various spectacular animations of various polarization states, e.g., polarization states, e.g., http://www.enzim.hu/~szia/cddemo/http://www.enzim.hu/~szia/cddemo/edemo0.htmedemo0.htm

Go to Polarization

Tutorial

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More definitionsMore definitionsLCP and RCP are defined w/o reference to LCP and RCP are defined w/o reference to a particular quantization axisa particular quantization axisSuppose we define a zSuppose we define a z--axisaxis

pp--polarizationpolarization : linear along z: linear along z

ss++: : LCP (LCP (!!) light propagating along z ) light propagating along z

ss-- : : RCP (RCP (!!) light propagating along z) light propagating along z

If, instead of light, we had a right-handed wood screw, it would move opposite to the light propagation direction

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Elliptically polarized lightElliptically polarized light

a, b a, b –– semisemi--major axesmajor axes

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Unpolarized light ?Unpolarized light ?

Is similar to Is similar to free lunch free lunch in that such thing, in that such thing, strictly speaking, strictly speaking, does not existdoes not existNeed to talk about nonNeed to talk about non--monochromatic lightmonochromatic lightThe threeThe three--independent lightindependent light--source model (all source model (all three sources have equal average intensity, and three sources have equal average intensity, and emit three orthogonal polarizationsemit three orthogonal polarizationsAnisotropic light (a light beam) cannot be Anisotropic light (a light beam) cannot be unpolarized !unpolarized !

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Angular momentum carried by lightAngular momentum carried by light

The simplest description is in the The simplest description is in the photon picture photon picture ::A photon is a particle with intrinsic angular A photon is a particle with intrinsic angular momentum one ( )momentum one ( )Orbital angular momentumOrbital angular momentumOrbital angular momentum and Orbital angular momentum and LaguerreLaguerre--Gaussian Modes (theory and experiment)Gaussian Modes (theory and experiment)

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Helical Light: Helical Light: WavefrontsWavefronts

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Formal description of light polarizationFormal description of light polarization

The The spherical basis spherical basis ::

EE+1 +1 ¨̈ LCP for light propagating along +LCP for light propagating along +zz ::

Lagging by p/2zz

yy xx

ï LCP

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Decomposition of an arbitrary Decomposition of an arbitrary vector vector EE into spherical unit vectorsinto spherical unit vectors

Recipe for finding how much of a given basic

polarization is contained in the field E

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Polarization density matrixPolarization density matrix

• Diagonal elements – intensities of light with corresponding polarizations

• Off-diagonal elements – correlations

• Hermitian:

• “Unit” trace:

ρ ρ+ =

( )*| |q q

q

Tr E Eρ = =∑ 2E

• fl We will be mostly using normalized DM where this factor is divided out

For light propagating along z

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Polarization density matrixPolarization density matrix• DM is useful because it allows one to describe “unpolarized”

1/ 3 0 00 1/ 3 00 0 1/ 3

ρ⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

•… and “partially polarized” light

• Theorem: Pure polarization state ¨ ρ2=ρ

• Examples:

“Unpolarized” Pure circular polarization

2 2

2 2

1 0 0 1 0 0 1 0 0 1 0 01 10 1 0 ; 0 1 0 0 0 0 ; 0 0 03 9

0 0 1 0 0 1 0 0 0 0 0 013

ρ ρ ρ ρ

ρ ρ ρ ρ ρ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟= = = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= ≠ =

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Visualization of polarizationVisualization of polarization• Treat light as spin-one particles

• Choose a spatial direction (θ,φ)

• Plot the probability of measuring spin-projection =1 on this direction

fl

Angular-momentum probability surfaceExamples

• z-polarized light

2sin θ∝

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Visualization of polarizationVisualization of polarizationExamples

• circularly polarized light propagating along z

( )21 cosθ∝ +( )21 cosθ∝ −

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Visualization of polarizationVisualization of polarizationExamples

• LCP light propagating along θ=p/6; φ= p/3

• Need to rotate the DM; details are given, for example, in :

fl Result :

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Visualization of polarizationVisualization of polarizationExamples

• LCP light propagating along θ=p/6; φ= p/3

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Description of polarization withDescription of polarization withStokes parametersStokes parameters

• P0 = I = Ix + Iy Total intensity

• P1 = Ix – Iy Lin. pol. x-y

• P2 = Ip/4 – I- p/4 Lin. pol. ≤ p/4

• P3 = I+ – I- Circular pol.

Another closely related representation is the Poincaré Sphere

See http://www.ipr.res.in/~othdiag/zeeman/poincare2.htm

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Description of polarization withDescription of polarization withStokes parameters and Stokes parameters and PoincarPoincaréé

SphereSphere• P0 = I = Ix + Iy Total intensity

• P1 = Ix – Iy Lin. pol. x-y

• P2 = Ip/4 – I- p/4 Lin. pol. ≤ p/4

• P3 = I+ – I- Circular pol.• Cartesian coordinates on the Poincaré Sphere are normalized Stokes parameters:

P1/P0, P2/P0 , P3/P0

• With some trigonometry, one can see that a state of arbitrary polarization is represented by a point on the Poincaré Sphere of unit radius:

• Partially polarized light ⇒ R<1

• R ≡ degree of polarization

2 2 21 2 3

0

1P P P

RP

+ += =

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Jones CalculusJones Calculus• Consider polarized light propagating along z:

• This can be represented as a column (Jones) vector:

• Linear optical elements ⇒ 2×2 operators (Jones matrices), for example:

• If the axis of an element is rotated, apply

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Jones Calculus:Jones Calculus: an examplean example• x-polarized light passes through quarter-wave plate whose axis is at 45° to x

• Initial Jones vector:

• The Jones matrix for the rotated wave plate is:

• Ignore overall phase factor ⇒

• After the plate, we have:

• Or:

= expected circular polarization

10⎛ ⎞⎜ ⎟⎝ ⎠