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Polarization and Optics

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Polarization and Optics

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Polarization and Polarizing Optics

WE804 Notes Module 16

Materials from BEA Saleh, MC Teich Fundamentals of Photonics John Wiley & Sons, NY (1991)

Polarization of Light

Refers to time variation of the direction of the Efield vector ? Our specialized case: monochromatic optical beams (paraxial waves)

? ? z ?? ? E (z, t ) = Re ?A exp ? j 2?Ð?Í ? t ? ?? ? ? c ?? ? ? ? ? ? A = Ax x + Ay y where Ax Ay are complex components Trace endpoint of E ( z, t ) at each z with time

Polarization Ellipse

Express complex Ax Ay as magnitude/phase

Ax = a x e

j? x

Ay = a y e

j? y

? E (z, t ) =Ex x +Ey y

where

Important! The phase of the complex coefficient adds to the phase of the propagating sinusoid!

? ? z? Ex = a x cos ? j 2?Ð?Í ? t ? ? + ? x ? ? c? ? ?

? ? z? Ey = a y cos ? j 2?Ð?Í ? t ? ? + ? y ? ? c? ? ? are the x and y components of the field and also are parametric equations of an ellipse ExEy Ex + 2 ? 2 cos ? = sin 2 ? 2 ax a y ax a y

2

Ey 2

= ? y ??x

Elliptical Trajectories

At fixed z, the tip of the E-field vector rotates periodically in X-Y plane, tracing an ellipse ? At fixed t, the tip of the E-field vector traces a helical path on the surface of an elliptical cylinder and ?Ë=c/?Í ? State of polarization depends on shape of ellipse, which depends on ay/ax and ?Õ= ?Õy? ?Õx ? Size of ellipse depends on I = (ax2+ay2)/?Ç where ?Ç is impedance

Polarization Ellipse

Linear Polarization

Examples : ax = 0 ? E (z, t) =Ey y

= 0, ?Ð (+ / ? below)

Ey = ?À (a y a x ) x E

Circular Polarization

If ? = ? y ? ? x = ?À?Ð / 2 and a x = a y = a0 ? ? ? z? Ex = a0 cos ?2?Ð?Í ? t ? ? + ? x ? ? c? ? ? ? ? ? z? Ey = m a0 sin ?2?Ð?Í ? t ? ? + ? x ? ? c? ? ?

2 Ex 2 +Ey 2 = a0 Circle

= +?Ð / 2 right polarized (clockwise from front) ? = ??Ð / 2 left polarized (counterclockwise from front)

Circular Polarization

A x ? J=? ? ?A y ? Given J, can find I = Ax + Ay

2

Jones Matrix

(

2

) 2?Ç ,

a y ax = A y A x and ? = ? y ? ? x = Arg {Ay }? Arg {Ax }

Notes for figure Ax + Ay = 1

2 2

x = 0

Orthogonal Polarizations and Decomposition

(J

J 1 , J 2 are orthogonal J 1 ?Í J 2 if inner product is zero

* * , J 2 ) = A1x A2 x + A1 y A2 y = 0 1

where A1x , A1 y are components of J 1 , * is complex conjugate, etc

Examples ?Linearly polarized waves in x and y directions ?Right- and left-circularly polarized waves ?Orthogonal Jones vectors J1, J2 can be used as a basis set, arbitrary Jones vectors can be expressed as the sum of their projections onto this set

?Á1 = (J , J 1 ), ?Á 2 = (J, J 2 )

J = ?Á 1J 1 + ?Á 2 J 2

Matrix Representation of Polarization Optics

Using Jones vectors, the affect of polarizing optics is naturally expressed in matrix form ? x,y axes are defined by the device

?C input vector J1 should be expressed in terms of these axes

J 2 = TJ 1 ? A2 x ? ?T11 T12 ? ? A1x ? ?A ? = ? ??A ? ? 2 y ? ?T21 T22 ? ? 1 y ?

Linear Polarizer

Allows only 1 axis (e.g. x) to pass

J 2 = TJ 1 ? A1x ? ?1 0? ? A1x ? ? 0 ? = ?0 0 ? ? A ? ? ? ? ? ? 1y ?

Wave Retarder (Wave Plate)

Passes one axis (e.g. x) axis faster than other (e.g. y) ? Amount of phase retardation ?? on y axis depends on thickness

A2 x ? ? A1x ? ?1 0 ? ? A1x ? ? A ? = ? A e ? j?? ? = ? ? ? j?? ? ? ? 2 y ? ? 1y ? ?0 e ? ? A1 y ?

Quarter-wave Plate

?? = ?Ð/2 ? Convert linear ? circular

linear to left circularly polarized 0 ? ?1? ? 1 ? ?1 ?? j ? = ?0 e ? j (?Ð / 2 ) ? ?1? ? ? ? ?? ? right circularly polarized to linear 0 ? ?1 ? ?1? ?1 ?1? = ?0 e ? j (?Ð / 2 ) ? ? j ? ?? ? ?? ?

Half-wave Plate

? ??=?Ð Convert:

?C linear ? linear, angle depends on rotation of wave plate! ?C right ? left circular

2X

linear to linear @ - ?Ð ? 1 ? ?1 0 ? ?1? ?? 1? = ?0 e ? j?Ð ? ?1? ? ? ? ?? ? right circular to left circular ? 1 ? ?1 0 ? ? 1 ? ? ? j ? = ? 0 e ? j?Ð ? ? j ? ? ? ? ?? ?

Rotation of Jones Matrices

Previous wave plate examples had slow axis aligned with coordinate system y axis ? If the plate is rotated to another angle, the Jones matrix needs to be rotated ? Coordinate transformation or Jones vectors and matrices for rotation from old x-y system to new x??-y?? system

y y??

X??

?È x

cos(?È ) sin (?È ) ? R(?È ) = ? ? sin (?È ) cos(?È )? ? ? ? x? ? x'? ? y '? = R(?È )? y ? ? ? ? ? ? Ax ' ? ? Ax ? ? A ? = R(?È )? A ? ? y' ? ? y? T ' = R(?È )TR(? ?È )

Reflection and Refraction

Polarization devices are based on reflection, refraction properties

?C Reflectivity depends on polarization wrt to plane of incidence

x = TE = transverse electric = ??s?? = ger. senkrecht - perpendicular ? y = TM = transverse magnetic = ??p?? = parallel

?C Index of refraction depends on crystal axes

Snells Law of Refraction

A1x ? ? A2 x ? ? A3 x ? J1 = ? ? J 2 = ? ? J 3 = ? ? ? A1 y ? ? A2 y ? ? A3 y ? J 2 = tJ 1 J 3 = rJ 1 ?t x 0 ? ? rx 0 ? t=? ? r = ?0 r ? y? ? 0 ty ? ? notes t, r are generally complex ? A1x ? ? E1x ? ? A ? = ? E ? etc, in this analysis ? 1y ? ? 1y ? ? E2 x ? ?t x 0 ? ? E1x ? ? E3 x ? ? rx 0 ? ? E1x ? ?E ? = ? 0 t ? ?E ? ; ?E ? = ? 0 r ? ?E ? y ? ? 1y ? y ? ? 1y ? ? 2y ? ? ? 3y ? ?

Boundary Conditions

?È1 = ?È 3 n1 sin ?È1 = n2 sin ?È 2

Snell's Law Applying BC' s to TE and TM components separately yields Fresnel Equations (derivation follows) rx = n1 cos ?È1 ? n2 cos ?È 2 n1 cos ?È1 + n2 cos ?È 2 n2 cos ?È1 ? n1 cos ?È 2 n2 cos ?È1 + n1 cos ?È 2 n1 1 + ry n2 TE polarization

t x = 1 + rx ry = ty = TM polarization

(

)

1 2

In the above equations, n1 , n2 , ?È1 are known, need cos ?È 2 = (1 ? sin 2 ?È 2 ) ? ? n ?2 ? 2 1 = ?1 ? ? ? sin ?È1 ? ? ? ? ? n2 ? ? ? ?

1 2

Complex values possible!

Derivation of Fresnel Equations

At the boundary, incident, reflected and transmitted waves all exist

?C Also true for both TM and TE polarizations

Say that the relationship between amplitudes cannot depend on the position of the boundary or time ? This implies that the phases of all waves are equal at the reflection point

Incident TE wave E1 = E1x e j (k1 ?r ??Ø1t ) Reflected TE wave E3 = E3 x e j (k 3 ?r ??Ø3t ) Transmitted TE wave E 2 = E 2 x e j ( k 2 ?r ? ?Ø 2 t )

Note on Propagation ??k?? Vectors

In previous, have assume k is scalar that is related to wavelength as

k= 2?Ð

In general it is a vector quantity ? ? ? k = kx x + k y y + kz z ? And the projection onto a direction of interest k ? r = k x rx + k y ry + k z rz ? gives the propagation of each component, and

2 2 k = k x + k y + k z2 =

2?Ð

where ?Ë is the wavelength in the propagation direction

Derivation (cont.)

Equal phases : k 1 ? r ? ?Ø1t = k 3 ? r ? ?Ø3t = k 2 ? r ? ?Ø2t In particular, if r = 0 ?Ø1 = ?Ø3 = ?Ø2 (not surprisingly!) But, if t = 0 k1 ? r = k 3 ? r = k 2 ? r From which several conclusions follow

1. Any two components subtract to 0

(k 1 ? k 3 ) ? r = (k 1 ? k 2 ) ? r = (k 2 ? k 3 ) ? r = 0

which implies that k2,k3 lie in k1-r (y-z or incident) plane

Derivation (cont.)

2. Incident and reflected waves are in same medium, so

k1 ? r = k 3 ? r k1r sin ?È1 = k3 r sin ?È 3 Since k1 = k3

?È1 = ?È 3 (angle of incidence = angle of reflection)

Derivation (cont.)

3. Transmitted and reflected wave propagation component equality implies??

k2 ? r = k3 ? r k 2 r sin ?È 2 = k3 r sin ?È 3 Since k 2 = n2

c c n2 sin ?È 2 = n1 sin ?È 3 = n1 sin ?È1 ( Snell' s Law)

; k3 = n1

Continuity of Fields Parallel to Boundary Across Boundary

Needs a sub-derivation from Maxwell??s equation analysis ? Consider normal incidence of plane wave traveling along z-axis in + direction, analyze for E fields only

Incident normal wave Ei = E 0i e j ( k1 ? z ??Ø1t ) Reflected normal wave E r = ? E 0 r e j ( k1 ? z ??Ø1t ) Transmitted normal wave Et = E ot e j ( k 2 ? z ??Ø2t ) Note that either E or B must change sign on reflection so that E ?Á B will point in the negative z direction. E is chosen negative here.

Gaussian Volume at Interface

Faraday??s & Stokes?? law and Maxwell eq. implies that tangential field is continuous across interface

Ei + E r ? Et = 0

Example: perfectly conducting metal implies that E field vanishes on both sides of interface, thus incident and reflected waves must subtract to zero, thus phase shift upon reflection is ?Ð r r 1 dB Smooth variation Maxwell's Eq. ??ÁE = ? Ei + E r c dt

?Ò?Ò ? ?Á E = ?

S S C

1 dB c ?Ò?Ò dt S

db dl ~ 0 db dl ~ 0

?Ò?Ò ? ?Á E = ?Ò E ? dr Stoke' s Law dl?ú0 0 as

(Ei + E r )db + ?Et db + ends = ? 1 dB dbdl c dt

assumed for differentiability of B ?Ì1,?Å1 Surface ?Ì2,?Å2

r Et (= 0 if perfect conductor) =

Implications of Field Continuity at Bounary

For TE

E1 + E3 = E2 B1 cos ?È1 ? B3 cos ?È 3 = B2 cos ?È 2

For TM

B1 + B3 = B2 ? E1 cos ?È1 + E3 cos ?È 3 = ? E2 cos ?È 2

Using these relations and E = (c/n)B, the Fresnel Equations may be derived

TE, External Reflection Coefficient

n2>n1: coefficient is real, negative

Mistake: arrow should be pointing down

n2 ? n1 rx = at ?È1 = 0 n1 + n2

TE, Internal Reflection Coefficient

n1>n2

rx =

n1 ? n2 at ?È1 = 0 n1 + n2 ?

1

?È c = sin ?1 ? n2 n ? ? ?

2 1 2

tan

x

2

(sin =

2

?È1 ? sin ?È c ) cos ?È1

TM, External Reflection Coefficient

n2>n1

n2 ? n1 rx = at ?È1 = 0 n1 + n2 ? n2 ? ?È B = tan ? n ? 1? ? Brewster'

s Angle

1

TM, Internal Reflection Coefficient

n1>n2

tan

y

2

(sin =

?È1 ? sin ?È c ) cos ?È1 sin 2 ?È c

2 2

1

2

?È B = tan ?1 ? n2 n ? ? ?

1? ? ?È C = sin ?1 ? n2 n ? ? ? 1? ?

Power Reflectance, Transmittance

R = r

2

T = 1?R Note, also ? n2 cos ?È 2 ? 2 2 T =? ? n cos ?È ? t ?Ù t in general ? 1 ? ? 1 ? n1 ? n2 ? R =? ? n + n ? at normal incidence ? ? 1 2? E.G. glass (n2 = 1.5) / air (n1 = 1) R = 0.04

2

?ÈB=tan-1(3.6)=74.5?ã

Power Control with a Half-Wave Plate

Linearly-polarized light is rotated by rotating wave plate ? Power not aligned with polarized axis is reflected ? In principle, fractions from 0 to 1 can be transmitted

Incoming laser beam has linear polarization

Beam is completely blocked at this angle.

Thin Film Polarizers

Optics used in femtosecond laser application ? Femtosecond pulses are stretched by thick optics, so thin film polarizer is used ? ?Ë= 700?C900 ? Tp >96% avg., ? Rs >80% avg. ? Tp/Ts >5:1

?Ë=775 nm

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