Quarter Wave Plate Uses Birefringence To Create Circularly Polarized Light

•  Since they have different phase velocities, the component of E || to optic axis and the component of E ⊥ to axis can be phase-shifted relative to each other to rotate plane of polarization.

•  For normal incidence and plate thickness to produce 90° phase difference between 2 components of E, circularly-polarized light will exit the plate.

•  For other thicknesses, the phase differences lead to the following polarizations of the emerging light èè

1

Polarized monochromatic light incident normally (k || x) on birefringent crystal. E in y-z plane at angle to optic axis (|| z)�

•  Polarized white light with E at 45° to optic axis enters crystal normally (k || x)•  E components: Ez and Ey || and ⊥ to

optic axis•  Indices of refraction: ny and nz for Ey, Ez

•  ky = (ω/c)ny and kz = (ω/c)nz

•  Ey and Ez waves have same amplitude. •  Ey, Ez waves exit crystal with phase

difference, Δψ = dΔk = ωdΔn/c •  Different polarization for different Δψ

•  Quarter-wave plate: Δψ = ±π/2 +2πj gives circular polarization

•  Half-wave plate: Δψ = ±π +2πj rotates plane of polarization of entering wave

•  Elliptic polarization for other Δψ

Back (x=d)

z, optic axis

yFront k, x

Front (x=0)

E

E d

Ez

Ey

E

Front: E z = Acos 0−ωt +φ( ), E y = Acos 0−ωt +φ( )

Inside: E z = Acos ωcnzx −ωt +φ

⎛

⎝⎜

⎞

⎠⎟, E y = Acos ω

cnyx −ωt +φ

⎛

⎝⎜

⎞

⎠⎟

Back: E z = Acos ωcnzd −ωt +φ

⎛

⎝⎜

⎞

⎠⎟, E y = Acos ω

cnyd −ωt +φ

⎛

⎝⎜

⎞

⎠⎟

Phase diff. betw. E z , E y at back: ωnzd / c−ωnyd / c =ωdΔn / c

After leaving: x > d : E z,y = Acos ωcx −d( )−ωt +φ +ω

cnz,yd

⎛

⎝⎜

⎞

⎠⎟

2

Birefringent material between crossed polarizers

Inside the birefringent plate the two components of the wave E-field have different wavenumbers:k1 = (ω/c)n1 and �k2 = (ω/c)n2 so the phasedifference between the two components upon exiting the plate = (ω/c)Δn·(plate thickness)

3

Analyzing stress in plastics between crossed linear polaroid filter

•  Stress causes plastics to become birefringent which causes the plane of polarization to rotate

•  E-fields corresponding to different wavelength rays are rotated by different amounts depending on stress and thickness, leading to pattern of colors after second crossed linear polarizer.

4

Why do we see different colors from incident white light when a birefringent material is put between

crossed linear polarizers?•  White light has spectrum of wavelengths

(colors) which do not interfere with each other.

•  After 1st polarizer each wavelength is linearly polarized

•  After 2nd polarizer each wavelength is again linearly polarized with contributions from ordinary and extraordinary waves.

•  Birefringent crystals have different indices of refraction for different ω in white light spectrum because Δψ = dΔk = ωdΔn/c

•  Ordinary and extraordinary waves at each wavelength (color) interfere constructively, destructively or in between.

–  Different colors experience different degrees of constructive or destructive interference making them brighter or dimmer

5

π/2 phase difference but could be any phase difference

6

•  Waves interact with each other to gives rise to light and dark and color "fringes" –  Interference cannot be

understood by ray-tracing–  It occurs because E-fields from

different rays have different phases and can add or subtract

•  Adding together E-fields of waves of same wavelength, frequency and polarization–  Phase of one wave in relation to

other(s)•  Examples of interference

–  Double slit experiment and diffraction gratings

–  Thin films (color in oil slicks, and soap bubbles; photographic filters)

–  Iridescence: butterfly wings and peacock's feathers

Optics Physics 4150 �Interference and diffraction

Review of waves and interference

Review: Waveform of monochromatic (single wavelength) wave along a ray?

Speed of light in vacuumc = 3 x 108 meters/sec

RayWaveform

Amplitude�(maximum height)Intensity of light ∝ A2 Large A means bright light. Small A means dim light

Wav

elen

gth

Review:

• Light consists of E and B fields moving along ray but pointing ⊥ to ray

• Would exert force on charge if it were placed anywhere on ray

– Direction of that force is perpendicular to ray

– Strength and direction (up or down) given by waveform

electric force on that charge�(this is a bright spot)

positive charge

Waveform can be replaced by a set of electric field vectors

perpendicular to ray

How do two waves of same wavelength interact?

•  They do not interact unless they are in the same place at the same time with same polarization

•  If they are in same place at same time net amplitude will be greater or less than amplitude of each, depending on phase

•  Net E-field force on hypothetical charge located where rays of two waves overlap depends on phase.

•  This is called interference–  Waves moving in the same direction

can interfere

These two waves DON'T INTERACT because their rays DON'T intersect or overlap

anywhere

The two waves below DO INTERACT because their rays overlap. They ADD TO

MAKE A BRIGHTER WAVE

How do two waves of the same wavelengths add

•  Adding waves means adding E- fields wherever they intersect or overlap

Electric force fields of each wave separately at

this point on rayElectric force field resulting from constructive interference of the two rays at this point on the ray

Electric force fields cancel here because equal and opposite

forces on same object cancel. Hence this light is extinquished

(destructive interference)

Two waves in phase�(slightly displaced to see them)

We don't always show the two waves overlapping. It

is understood!

Two waves a half wavelength out of phase

What is difference between coherent and incoherent wave?

•  The phase of an incoherent mono-chromatic wave jumps randomly after a random number of wavelengths–  A neon light is an example of an

incoherent monochromatic light source

–  An incandescent light or a fluorescent light are incoherent white light sources (all of the spectral components jump around randomly)

•  The phase of a coherent monochromatic wave does not change (does not jump around–  Light from a laser is coherent

phase jump phase jump

INCOHERENT MONOCHROMATIC LIGHT

COHERENT MONOCHROMATIC LIGHT

Interference requires that the two waves interfering be coherent or if they are

incoherent that they are incoherent in exactly the same way. In the previous example the two reflected waves are coherent with each

other because they come from the same wave

B

Two coherent waves can be in phase at 2 different locations where they do not overlap. The waves will

interfere at other spatial locations where they do overlap•  Here is a wave going to the right from

point A at an instant when it has a trough (bottom) at point A

•  Here is another wave of the same wavelength also going to the right, but starting from point B, where it also has a trough (bottom)

–  The (yellow) wave at B is in phase with the (red) wave at A at all times if the waves are coherent

•  How will the two waves interfere to the right of point B if the distance from A to B is 2 1/2 wavelengths?

a)  Destructively b)  Constructivelyc)  They will not interfere

•  What if distance from A to B is two wavelengths?

A

The waves interfere destructively because when wave from A arrives at B it is a half-wavelength out of phase with the emerging wave from B

Now waves interfere constructively at B because waveform of wave starting at A is in phase at B with the wave starting at B LESSON: Just because two waveforms are in phase at two separate locations does NOT mean that they will be in phase where their rays cross

BA

B

Two coherent waves can be a half-wavelength out of phase at 2 different locations where they do not overlap. The waves

will interfere at other spatial locations where they do overlap

•  Here is wave going to the right from point A at an instant when it has a trough (valley) at point A

•  Here is another wave of the same wavelength also going to the right, but starting from point B, where it has a peak instead of a valley

–  The (yellow) wave at B is a half-wavelength out of phase with the (red) wave at A at all times if the waves are coherent

–  If distance from A to B is 2 1/2 wavelengths, the two waves interfere constructively to right of point B

•  What if distance from A to B is 2 wavelengths?

A

The waves interfere constructively because when wave from A arrives at B it is in phase with the emerging wave from B

Now the waves interfere destructively because when the wave from A arrives at B it is a half-wavelength out of phase with the wave from B

BA

Even if the rays of the two waves make a small angle to each other, the waves will interfere wherever they

overlap

•  How do the white and red waveforms interfere at point X?

a)  Constructivelyb)  Destructivelyc)  Not at all

•  How do the white and red waveforms interfere at point Y?

a)  Constructivelyb)  Destructivelyc)  Not at all

•  How about for this second case?

X

YX

Y

Questions

In order for two waves to interfere destructively and cancel each other completely at all times they must

a)  Have the same wavelengthb) Be going almost in the same

directionc)  Have their rays cross where

they interfered) Be completely out of phase

(e.g., crest on trough)e)  All of the above

A light wave is coherent if its a)  Speed doesn't changeb)  Amplitude doesn't changec)  Phase doesn't change

Along a wavefront of a wave what never changes?

a)  The wave's direction of travel (direction of a ray)

b)  The wave's phasec)  The wave's amplitude

Interference from slits �and gratings

Rays of a water or light wave coming out of a slit go out radially. The wavefronts are circles or half-circles

•  Problem from HW1: –  interference of two

spherically-symmetric waves emerging in-phase from two pinholes (rather than two slits)

–  note dark and bright "fringes" everywhere in space (not just on a screen)

Dark Dark

DarkDark

BrightBrightBright

Interference of monochromatic light passing through two slits, S1 and S2

•  Light of one wavelength�passsing through �slit S0 comes out with�cylindrical wavefronts

•  Waves passing through �slits S1 and S2 come out�as two cylindrical waves �in phase with each other�at S1 and S2

•  Those waves interfere�everywhere in space and�thus on a distant flat �screen where the inter-�ference pattern is visible

eiφ0

eiφ0

Locations of bright and �dark fringes on a screen

In limit of large L, rays are parallel

•  Ray from each slit to screen defines optical �path of ray. ( In this case straight lines)

•  Optical path length = r2 or r1 •  Optical path length difference is δ = r2-r1•  Rays in phase at slits S1 and S2 so �

kr2−kr1 =kδ = phase difference which �determines whether there is constructive or�destructive interference at point P on screen

Q.eiφ0

eiφ0

ei kr1+φ0( )

ei kr2+φ0( )

Bright fringe: kδ j = 2π j, j = ± integer or 0, δ j = d sinθ j

kd sinθ j = 2π j, or, δ j = jλ . For small θ j,

sinθ ≈ tanθ j =yjL

, θ j <<1⇒ sinθ ≈ 2π jkd

= j λd

Center is bright. 1st bright fringe (j = 1) at: y1bright

L=λd

Dark fringes: kδ = π + 2π j, j = ± integer or 0

sinθi ≈y jdark

L=π + 2π j[ ]kd

=λd

12+ j

⎡

⎣⎢⎤

⎦⎥