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PHYS 450Spring semester 2017

Lecture 13: Polarized Light

Ron ReifenbergerBirck Nanotechnology Center

Purdue University

Lecture 131

1669 Bartholinus – describes image doubling properties of calciteStart 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.

• Studied polarization of light reflected from various surfaces (1809)• 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% transverse1823 Fresnel derives equations for reflection of light (consistent with

Maxwell’s equations in 1860s)1828 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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Historical Timeline

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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 wave

3f remains constant, v and can change

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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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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

Polarizer

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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(deduced by Fresnel in 1823)

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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

Theory: Reflectivity vs. Incidence Angle

≃

~ 4%

2

1

tan tani B

n

n

x

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0.00

0.25

0.50

0.75

1.00

0 15 30 45 60 75 90

Ref

lect

ivit

y

Incident Angle, θi (in degrees)

s-polarized

p-polarized

p-pol data

s-pol data

0.001

0.010

0.100

1.000

0 15 30 45 60 75 90

Ref

lect

ivit

y

Incident Angle, θi (in degrees)

s-polarized

p-polarized

p-pol data

s-pol data

Typical data for =543.5 nm

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Measure Brewster’s Angle for Prism

1. Align face of prism with axis of prism table’s rotation

2. Continuously adjust polarizer to block s and pass p

3. Continuously adjust prism to minimize p

4. When intensity is minimum, measure

5. Infer i=(-)=B6. Repeat, take average value

collimator

fixed

Alignment!

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Equipment: Establishing =0

Beware: internal reflections occur inside prism

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Birefringent properties of calcite

The understanding of polarization is closely related to study of crystalline CaCO3, the most abundant mineral on earth after SiO2

Legend:Green=CaBlack=CRed=O

0.85

3 nm

Unit Cell

two CO3 molecular ions twisted by 60o

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Optical Properties of Calcite: Calcite Crystal Structure

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

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

+

++

O

OC

Eapp

Eind

‐‐

‐

Eind reinforces Eapp

b) Eapp // CO3 plane

14https://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 topa) b)

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What’s Going On?Cut three samples from a block of calcite

a) Sample A: Incident ray parallel to optical axis

b) Sample B: Incident ray perpendicular to optical axis

Incident ray is linearly polarized

Calcite

Calcite

Calcite

c) Sample C: Incident ray at oblique angle to optical axis (naturally occurring calcite)

Crystal Optical Axis

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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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Birefringence when viewed through a polarizer

A102o

102o102o

Optical Axis

Polarizer

Ordinary ray

102o

102o102o

Optical Axis

A

PolarizerExtra-

ordinary ray

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Wave Plates

A waveplate or retarder is an optical device that alters the polarization state of a light wave travelling through it.

Two common types of waveplates are • half-wave plate: rotates the polarization direction of linearly polarized

light

• quarter-wave plate: converts linearly polarized light into circularly polarized light and vice versa

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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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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'

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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2121

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=1.640-1.660, nfast=1.486

https://www.mindat.org/min‐859.html

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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

2 2

1441

4

fast slow

OPL fast slowair air

airOPL

air

fast slow

OPL n n d

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/n) ~ 0.5 fs

Tracing the tip of the total E-field reveals a helix, with a period of precisely one wavelength. This

represents circularly polarized light.

Arrows show E-fields

optical axis 

linearly polarized

phase shifted

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Operation Required ConfigurationChange linear polarizationto circular

Insert a 1/4 plate with axis at 45° to the input

polarization.

Change circularpolarization to linear

Insert a 1/4 plate with axis at 45° to the desired output

polarization.

¼-wave plate

45o

Fast axis

½-wave plate

2

Fast axis

Changes plane of linear polarization

Converts linear to circular polarization

Operation Required ConfigurationRotate linear

polarization todifferent orientation

Insert a 1/2 plate with axis at ½ the desired rotation

angle.Change the

handedness ofcircular polarization

Insert a 1/2 plate, orientation unimportant

The ¼-wave vs. the ½-wave plate

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Na Lamp

Polarizer #2 ¼-wave

platePolarizer

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

¼-wave mica plate VERY FRAGILE

Equipment

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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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The Viking Sun Stone

Sunstones could have helped the Vikings in their navigation from Norway to

America, well before a magnetic compass was introduced in Europe.

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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 polarizerbeneath calcite

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a) Scattered light is polarized perpendicular to the plane containing the sky point, the sun, and the Viking observer

Light is polarized by scattering from gas molecules

Horizon

Zenith

Viking Observer

Sun

Sky Point

b) 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

Application: Viking Sun Stone

Sun(hidden)

Search for direction of maximum polarization

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

90oMove screen around while

rotating calcite crystal until intensity transmitted is a

minimum

KEY IDEA: If the calcite is oriented to block out the

polarized component, the sky appears darker.

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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.

Light box

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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 dextro-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 aslevo-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

(,thickness)

film thickness

s

sOptic Axis

optic axis is parallel to plane of film

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Polarascope: Two Crossed Polarizers

cellophane wrapper

Cellophane acquires color by subtraction

Example: Cellophane wrappers

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Example: Cellophane wrappers viewed through crossed polarizers

Brightest image when optic axis is at 45o angle wrt two crossed polarizers

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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.