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Chapter 6: Polarization of light

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First, review the chapter First, review the chapter onon

Atomic StructureAtomic Structure

The The elementselements

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Preliminaries and definitionsPreliminaries and definitions

Plane-wave approximationPlane-wave approximation: : EE((rr,,tt) and ) and BB((rr,,tt) are uniform in the plane ) are uniform in the plane kk

We will say that light We will say that light polarization vectorpolarization vector is along is along EE((rr,,tt) (although it was along ) (although it was along BB((rr,,tt) in classic optics literature)) in classic optics literature)

Similarly, Similarly, polarization planepolarization plane contains contains EE((rr,,tt) ) andand kk

kk

BB EE

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Simple polarization statesSimple polarization states Linear Linear or or plane polarizationplane polarization Circular polarizationCircular polarization

Which one isWhich one is LCP LCP, 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 states Which one isWhich one is LCP LCP, and which is , and which is

RCPRCP?? Warning: optics definition is Warning: optics definition is

opposite to that in high-energy opposite to that in high-energy physics; physics; helicityhelicity

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

Go to Polarization

Tutorial

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More definitionsMore definitions LCP and RCP are defined w/o LCP and RCP are defined w/o

reference to a particular quantization reference to a particular quantization axisaxis

Suppose we define a z-axisSuppose we define a z-axis -polarization-polarization : linear along z : linear along z

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

-- : : 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 – a – semi-major axissemi-major axis; ; b – b – semi-major axissemi-major axis

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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 exist Need to talk about non-monochromatic lightNeed to talk about non-monochromatic light The three-independent light-source model (all The three-independent light-source model (all

three sources have equal average intensity, and three sources have equal average intensity, and emit three orthogonal polarizationsemit three orthogonal polarizations

Anisotropic light (a light beam) cannot be Anisotropic light (a light beam) cannot be unpolarized !unpolarized !

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Angular momentum carried by Angular momentum carried by lightlight

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 momentum Orbital angular momentum and Laguerre-Orbital angular momentum and Laguerre-

Gaussian Modes (theory and experiment)Gaussian Modes (theory and experiment)

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Helical Light: WavefrontsHelical Light: Wavefronts

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Formal description of light Formal description of light polarizationpolarization

The The spherical basis spherical basis ::

ee+1 +1 LCP for light propagating along + LCP for light propagating along +zz ::

Lagging by /2zz

yy xx

LCP

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

vectorsvectors

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:

*| |q q

q

Tr E E 2E

• 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 0

0 1/ 3 0

0 0 1/ 3

•… 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 1

0 1 0 ; 0 1 0 0 0 0 ; 0 0 03 9

0 0 1 0 0 1 0 0 0 0 0 0

1

3

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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

Angular-momentum probability surface

Examples

• z-polarized light

2sin

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Visualization of polarizationVisualization of polarizationExamples

• circularly polarized light propagating along z

21 cos 2

1 cos

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Visualization of polarizationVisualization of polarizationExamples

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

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

Result :

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Visualization of polarizationVisualization of polarizationExamples

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

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Description of polarization withDescription of polarization with Stokes parametersStokes parameters

• P0 = I = Ix + Iy Total intensity

• P1 = Ix – Iy Lin. pol. x-y

• P2 = I/4 – I- /4 Lin. pol. /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 with Stokes parameters and Poincaré Stokes parameters and Poincaré

SphereSphere• P0 = I = Ix + Iy Total intensity

• P1 = Ix – Iy Lin. pol. x-y

• P2 = I/4 – I- /4 Lin. pol. /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 22 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

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