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PHYS 450Fall semester 2016

Lecture 12: Polarized Light

Ron ReifenbergerBirck Nanotechnology Center

Purdue University

Lecture 121

For light (electromagnetic radiation), polarization refers to the orientation of the electric (E-field) vector E.

Polarization direction is by convention defined by the direction of the E-field.

Polarization and the Plane of the E-field

ˆ, cos cos

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oz t kz t E kz t x

k f

oE E

Eo

Bo

No fields at all!No fields at all!E=Ey(z,t)

B=Bz(z,t)

Snap shot at fixed time

z

Wave traveling in +z direction(cos could be replaced by sin).

Phase between E, B radiative wave2

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Historical Timeline1669 Bartholinus – describes image doubling properties of calcite

Start of a 150 year odyssey: To explain light, you must explain birefringent properties of calcite!1808 Malus discovers that calcite can modulate brightness of light passing through it.

• Measures light reflected from various surfaces• First to describes light as being “polarized”• Malus’ Law for crossed polarizers: I=Iocos2()

1817 Young realizes light must have a perpendicular component1821 Fresnel claims light must be 100% transverse1828 Nicol cuts thin slab of calcite – makes first transmissive polarizer, produces plane polarized light1928 Land invents a sheet-type dichroic linear polarizer (while an undergrad at Harvard University)

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Calcite Crystal StructureThe understanding of polarization is closely related to study of crystalline

CaCO3, the most abundant mineral on earth after SiO2

Crystal structure: https://www.youtube.com/watch?v=t_w00mgJzd0

Unit Cell:Green=CaBlack=CRed=O

two CO3 molecular ions twisted by 60o

+

++

O

OC

Eapp

Eind

‐‐

‐

b) Eapp // CO3 plane

Eind reinforces Eapp

a) Reduced polarizability → lower ε → lower n → higher v → fast

b) Enhanced polarizability → higher ε → higher n → lower v → slow

v

v//

b) Slow directiona) Fast direction

+

++ OO

C

Eapp

Eind

‐‐

‐

a) Eapp CO3 plane

Eind opposes Eapp

Birefringence means the refractive index depends on the polarization direction

of an EM wave

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5https://www.microscopyu.com/tutorials/birefringence-in-calcite-crystals

A102o

102o102o

Optical Axis

AA102o

102o102o

Optical Axis

Calcite Rhomb

Ordinary ray

Extra-ordinary ray

Birefringence in Calcite

Slide over

What you see when viewed from the top

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“O ray” passes through crystal undeviated, suffering no refraction as

it should. Obeys Snell’s law for ALL angles of incidence.

Unpolarizedlight, angle of incidence i=0o

Optical Axis

102o

71o

109o

Principal section (including all

parallel planes)

n1=1.00

n2

“E ray” does not obey Snell’s law. The ratio

depends on i → velocity of E-ray depends on direction of incident ray.

sin isin r = n2

Birefringence in Calcite

Snell’s lawn1 sin i = n2 sin r r

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Action of an IDEAL Polarizer

Not all polarizers are ideal!

Polarizing direction

INPUT: Unpolarized

light

OUTPUT: Polarized

light

All planes of E-field are possible

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A102o

102o102o

Optical Axis

Polarizer

Ordinary ray

102o

102o102o

Optical Axis

A

PolarizerExtra-

ordinary ray

Birefringence when viewed through a polarizer

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-Eosin(θ)

θx

x'

y'

y

Eo

Eocos(θ)

ˆ, cos

ˆ ˆ, cos cos ' sin cos '

LP o

LP o o

z t E kz t

or

z t E kz t E kz t

x

x y

E

E

Essential Mathematics of Polarization

or

Short hand notation:Linear polarized:

Unpolarized:z

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Reflection from an interface depends on incident polarization

Incoming Ray

Reflected Ray

θi

p-polarization (E-field lies in plane

of incidence)

n1

n2

s-polarization(E-field perpendicular to

plane of incidence)

Incoming Ray

Reflected Ray

θi

n2

n1

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n1

n2

ik

rk

tk

θi θr

θt

EiBi Er

Br

EtBt

Interface x

y

z

p-polarization

n1

n2

ik

rk

tk

θi θr

θt

EiBi

Er

Br

Et

Bt

Interfacex

y

z

s-polarization

plan

e of

inci

denc

e

plan

e of

inci

denc

e

1 2

1 2

cos cos

cos cosi tor

soi i t

n nEr

E n n

1 2

1 2

cos cos

cos cost ior

poi t i

n nEr

E n n

2s sR r2

p pR r

Fresnel Equations

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0.00

0.25

0.50

0.75

1.00

0 15 30 45 60 75 90

Refl

ecti

vit

y

Incident Angle, θi (in degrees)

s-polarized

p-polarized

Grazing incidence

Normal incidence

Reflectivity vs. Incidence Angle

≃

2

1

tan tani B

n

n

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polarizer

p-polarization

E

Eprism

slit

Na lamp

plane of incidence

i

Measure Brewster’s Angle for Prism

polarizer

s-polarization

E

E

plane of incidence

slit

Na lamp

prism

i

PLUS

Top view Top view

1. Continuously adjust polarizer to block s and pass p2. Continuously adjust prism angle to minimize p3. When intensity is minimum, measure 4. Infer i=B

collimator collimator

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Phase Shifting Light

ˆ ˆ, cos cosLP ox oyz t E kz t E kz t x + yE

Linearly polarized wave, standard notation:

Phase shifted wave:

ˆ ˆ, cos cosLP ox oyz t E kz t E kz t x + yE'

If =0 or =π, wave is still linearly polarized

ˆ, cosLP oz t E kz t xE

air

glass, n

d 2

ˆ, cosLP oair

z t E d t

xE

2 2ˆ ˆ, cos cos

2ˆcos

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a

LP o oairg

io

air

las

r

s

n d

z t E d t E d t

E d t

n

x = x

= x

E

BASIC IDEA:

More Generally:

Example: for d=1 m, n=1.5, air=545 nm, then =1.83π

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Extension to Birefringent Material

ˆ, cosLP oz t E kz t xE

air

d

2 2ˆ ˆ, cos

2

cos

2ˆc 1os slow

a

Lair

P o o

ir

glass

slo

o

w

air

z t E d t E d t

E d

n

nt d

x = x

= x

E

Birefringent slab, nfast & nslowBASIC IDEA:

nfast

nslow

ˆ, cosLP oz t E kz t E y

2 2ˆ ˆ, cos

2

cos

2ˆc 1os fast

a

Lair

P o o

ir

glass

fas

o

t

air

z t E d t E d t

E d

n

nt d

y = y

= y

E

In calcite, nslow=1.658, nfast=1.486

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The ¼-wave plate (converts linearly polarized light into circular polarized light)

Vertical polarization

goes through plate faster

Horizontal polarization

goes through plate slower

1 2

1 2

2 21 1

21 1

2 2

1441

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fast slowair air

fast slowair

fast slow OPLair air

airOPL

air

fast slow

n d n d

dn n

n n d

If wave plate

then dn n

d

nslow

nfast

What is typical time delay for ¼-plate? • Let’s say the wave plate is designed to operate

at 560 nm• plate=air/n ≈ 560 nm/1.5 = 375 nm • One-quarter of 375 nm is d = 95 nm. • How long does it take light to travel 95 nm? • Time delay Δt= d/c = 0.32 fs

Tracing the tip of the total electric vector reveals a helix, with

a period of just one wavelength.

Arrows show E-fields

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Na Lamp

Polarizer #2 ¼-wave

platePolarizer

#1Dial for azimuthal angle of ¼-wave plate and polarizers

¼-wave plate VERY FRAGILE

Equipment

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In principle, waveplates are wavelength specific.

Zero Order Waveplate: the total retardation is the desired value without excess.True zero order waveplates for visible light are made from a single crystallinebirefringent material that has been processed into a fragile ultra-thin plate only afew microns thick.

Multiple Order Waveplates: total retardation is the desired value plus an integernumber of wavelengths. In principle, the excess integer portion has no effect onperformance.

Both zero order and multiple order waveplates require precise control of thethickness of the plate.

A ¼-wave plate converts linearly polarized light to circularly polarized light.

Characteristics of Waveplate (aka Retarders)

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The ¼-wave plate experiment

1. Adjust Polarizer 2 slit

discharge lamp

Polarizer 1

Collimator

Telescope

2. Extinction, no light

Polarizer 2slit

discharge lamp

Polarizer 1

3. Adjust ¼-wave plate

Collimator

Telescope

4. Extinction, no light

Polarizer 2slit

discharge lamp

6. Rotate Polarizer 1 thru 360o

5. Rotate 45o

Collimator

Telescope

7. Always bright!

¼ wave plate

a) b) c)

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Rotate calcite crystal

Calcite crystal

paper

hole

1) Unpolarized light through hole

Calcite crystal

paper

hole

2) Polarized light through hole

Birefringent Properties of Calcite

Two bright spots always observed as

crystal is rotated

Intensity of 2nd bright spot is modulated as

crystal is rotated

spot disappears!

Rotate calcite crystal

b)

b) a)

a)

1

2

1

2

1

1

Spot 1 remains stationary (ordinary)

Spot 1 remains stationary (ordinary)

Insert polarizer

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Scattered light is polarized perpendicular to the plane containing the sky point, the

sun, and the Viking observer

Light is polarized by scattering

Horizon

Zenith

Viking Observer

Sun

Sky Point

Maximum polarization occurs in a direction 90o from the sun in a plane containing the sun, the zenith, and the Viking observer

Horizon

Sun

90o

Zenith max polarization

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Sun Stone (Icelandic Spar)

Viking Observer

Viking Sun Stone

Sun(hidden)

Search for maximum

polarization

For demo, see: https://www.youtube.com/watch?v=mp9yvV4vH6g

90o

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Altering the Polarization of Light Produces a Myriad of Optical Effects

1. Rotation of Plane Polarized Light by a Sugar Solution2. Birefringent polymer filters3. Photoelasticity4. . . . . etc.

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• Glucose (aka dextrose) (C6H12O6) and Fructose(C6H12O6) have identical chemical formulas

• Glucose and Fructose are chiral molecules (cannotbe superposed onto a mirror image).

• Molecules with a chiral structure exhibit circularbirefringence.

• Circular birefringence: different refractive indexfor left-hand circular (LHC) and right-hand circular(RHC) polarized light, i.e. nRHC (λ)≠nLHC (λ).

• No matter where the glucose comes from, it isalways d-rotary, i.e. it rotates light clockwise(defined from the viewpoint when light approachesan observer).

• Fructose on the other hand always rotates polarizedlight in a counter clockwise direction and is known asl-rotary.

from the Latin: d=dextera – right, l=laevus – left.

Some Facts about Sugar

Chromatographic separation

Regular corn syrup contains only glucose.

High fructose corn syrup (HFCS) contains mainly fructose and is allegedly obesigenic.

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Rotation of Plane Polarized Light by a Sugar Solution• Linearly polarized light is equivalent to the

superposition of LHC and RHC polarized light.• The interaction of linearly polarized light with

glucose (aka dextrose) molecules causes arotation in the plane of polarization.

• If naturally produced sugar has equal amounts ofleft-handed and right-handed glucose molecules,the plane of polarization of linearly polarized lightwill not rotate while in transit.

• If naturally produced sugar is comprised solely ofright-handed glucose molecules, then the plane ofpolarization of linearly polarized light will rotatewhile in transit.

• Experiment shows a linearly polarized beampassing through a solution rich in glucose issystematically rotated in only one direction.

• Thus nature preferentially makes one chiralityglucose molecule over the other.

transmitted light

scattered light

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Glass tube filled with sugar solution

Polarized laser

Experiment

What does it mean when the scattered

light disappears?

Polarized laser

Glass tube E-field

d

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Color by subtraction in thin polymer films

Unstretched polymer strands Stretched polymer strands

Optically Isotropic →             Optically Anisotropic cellophane acts as a birefringent filter

for white light

Linearly polarized

white light

(,s)

film thickness

s

2s

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Polarascope: Two Crossed Polarizers

cellophane wrapper

Cellophane acquires color by subtraction

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Photoelasticity – Stress-induced birefringence(Coloration Indicates Stress Distribution)

Polarizer #1

Light Box

Molded Plastic Cuvette

Polarizer #1

Crossed Polarizer #2

Light Box

Cuvette

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Up Next – Wrap up

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Appendix: Phase shifts between E and B

When the E and B-fields are phase shifted, the fields become reactive, not radiative. The energy oscillates back and forth instead of

actually propagating as it would in radiation. Such EM-fields are not produced by a dipole antenna.

Example:

Most of this lecture was about shifting the phase of E-fields in an EM wave. Don’t mix this up with phase shifts between the E and B-fields.