Polarized Light : linear, circular, elliptical Jones Vectors for Polarized Light

Polarized Light : linear, circular, elliptical Jones Vectors for Polarized Light

Jones Matrices for Polarizers, Phase Retarders, Rotators

14. Matrix treatment of Polarization

14. Matrix treatment of

Polarization

Electromagnetic Radiation

• Combined Faraday’s Law and Ampere’s Law

– time varying B-field induces E-field – time varying E-field induces B-field

x

z y

• E-field and B-field are perpendicular

2 2

2 2

x x

o o

E E

z t

∂ μ ε ∂

∂ ⁼ ∂

E

B

S

s b

e ˆ × ˆ = ˆ

• B

_y

is in phase with E

_x

• B

_max

= E

_max

/ c

max

sin( )

E

x

= E kz − ω t

(plane wave solution)

(Linear) Polarization

)

0

sin( kz t E

E

_x

= − ω E

_y

= 0 E

_z

= 0

0

sin( )

y

B E kz t

c ω

= − B

_z

= 0

= 0 B

x

in terms of components:

eˆ

x y

0

cos sin( )

E

x

= E θ kz − ω φ t +

0

sin sin( )

E

y

= E θ kz − ω t + φ

= 0 E

z

) ˆ

₀

sin( − ω + φ

= e E kz t

E r _φ is constant

phase

• This wave is an example of a linearly polarized wave.

– We always define the direction of polarization as the direction of the oscillation of the Electric field vector (x-direction in this case)

– In general, a linearly polarized wave traveling in the +z direction can be written as:

x

z y

θ

a) E

_x

= E

_o

sin (kz + ω t) b) E

_y

= E

_o

sin (kz - ω t)

c) B

_y

= B

_o

sin (kz - ω t)

6) Which equation correctly describes this electromagnetic wave?

7) In which direction is this wave polarized?

a) x b) y c) z

질 문

질 문

• At t = 0, z = 0, the electric field of an electro- magnetic wave is oriented at an angle θ with respect to the x-axis, as shown.

– Which arrow indicates the direction of the

magnetic field at the same location and instant of time?

(a) A (b) B

• This question cannot be answered unless the direction of propagation is specified:

• If the wave propagates in the +z direction, then B-field is along A

• If the wave propagates in the –z direction, then B-field is along B

θ eˆ

x y

A

B

Time Dependence of Linear Polarization Time Dependence of Linear Polarization

Linear Polarization, θ to x-axis, z = 0

y

x

$

0

( ) ^$

0

( )

( , )

_x

cos

_y

cos

E z t r = x E ω t + y E ω t

t = 0

y

x

t = 2π/ω * (1/4)=π/2ω

y

x

t = π/ω

y

x

t = 3π/2ω

θ θ sin cos

0 0

o y

o x

E E

E E

=

=

θ

Polarization of Laser Radiation Polarization of Laser Radiation

Linear Polarization, +θ or -θ to x-axis

( )

( )

0

0

( , ) ˆ sin ˆ sin

x y

E z t x E k z t

y E k z t

ω ω

= −

+ −

r

( )

( )

( )

( )

0

0

0

0

( , ) ˆ sin

ˆ sin

ˆ sin

ˆ sin

x

y x

y

E z t x E k z t

y E k z t

x E k z t

y E k z t

ω

ω π ω

ω

= −

+ − +

= −

− −

r Z=0

y

x

+θ

E_x leads E_y by π phase.

E_x and E_y are in same phase.

−θ

More Polarizations More Polarizations

General linear polarization state: $ $

(cos sin )

0

sin( ) E r = θ x + θ y E kz − ω t

E

0

≡ r

Polarized at –45°.

Another way to write these:

0_x

sin( )

0_y

sin( )

E E x r = ) kz − ω t + E y ) kz − ω t + Δ ϕ

In general, this phase shift can take other values!

θ = 45 ° $ $

0

2

E x y + r =

E₀

What if instead we had ? $ $

0

2

E r = x y −

E₀

pol.

linear for

- or or

0 π π

ϕ =

Δ

Other Polarization States?

• Are there polarizations other than linear?

– Sure!!

θ

= tan

ox oy

E E

π φ

φ

φ ≡

_x

−

_y

= 0 , ±

0

s in ( )

x x x

E = E k z − ω t + φ )

0 y

sin(

y

y

E kz t

E = − ω + φ

θ

E r

x y

Linear Polarization

– The general harmonic solution for a plane wave traveling in the +z-direction is:

2 φ π

φ

φ ≡ − = ±

Δ

_{x y}

ox

oy

E

E =

E r

x y

Circular Polarization

(

E_0x

and

E_0y

are ±90° out of phase.)

Polarization of Laser Radiation Polarization of Laser Radiation

Circular Polarization: electric field vector moves in a circle

Z=0 y

x

( )

0

0

( , ) ˆ cos

ˆ cos

2

x

y

E z t x E k z t

y E k z t

ω ω π

= −

⎛ ⎞

+ ⎜ − + ⎟

⎝ ⎠

r

t = 0

t 2 π

= ω

Visualization

Circular polarization stems from the intrinsic

angular momentum (“spin”) of the photons that make up the beam.

Note: If you shine circularly polarized light onto an absorber, it will in principle start to rotate

→ conservation of angular momentum!

Application: Light-Driven Micro Machines Application: Light-Driven Micro Machines

Fact 1: Small particles are

attracted to regions of high E field gradient (induced dipole force) → laser “tweezers”

Fact 2: Because birefringent crystals convert

linear ↔ circular polarization,

they acquire angular momentum

→ rotation

Uses: Biophysics: Manipulating DNA, proteins, etc.

Microscopic fluid pumps

Parts are only 10μm across!

Mathematical Representation of Polarized Light: Jones Vectors Mathematical Representation of

Polarized Light: Jones Vectors

The electric field shown in the figure is propagating out of the page and is polarized at a certain angle to the x - axis.

The electric field vector is given by :

y

x

E_x E

E_y

Two-element matrix

“Jones vector”

y

x

⎥ ⎦

⎢ ⎤

⎣

= ⎡ b E ~

_o

a

form.

normalized is

it , 1 b

a

²

+

²

= If

⎥ ⎦

⎢ ⎤

⎣

= ⎡ 1 0 E

o

⎥ ⎦

⎢ ⎤

⎣

= ⎡ 0 1 E

o

⎥ ⎦

⎢ ⎤

⎣

= ⎡ 1 1 E

o

y

x

y

x

y

x

α α α

α α α

− −

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞

= ⎢⎣ ⎥⎦ = ⎢⎣ ⎥⎦ = ⎢⎣ ⎥⎦ = ⎜⎝ ⎟⎠ = ⎜ ⎟⎝ ⎠

0 1 1

0

0

cos cos

, tan tan

sin sin

x oy

oy x

E A E b

E A

E A E a

⎥ ⎦

⎢ ⎤

⎣

⎡

= −

1 1 E

o

y

x

Mathematical Representation of Polarized Light: Jones Vectors Mathematical Representation of

Polarized Light: Jones Vectors

x

y

ϕ

ϕ

φ = −

Δ

Circularly Polarized Light Circularly Polarized Light

( ) ( ) ^{( )}

π

ϕ ϕ π

− − °

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= ⎢ ⎥ = ⎢ ⎥ = ⎢ ⎥ = ⎢ ⎥

⎣ ⎦ ⎣ ⎦

⎢ ⎥ ⎣ ⎦

⎣ ⎦

% %

r

0

0 0

0

/ 2 (90 ),

exp 1 1 1

exp / 2 :

exp 2

x x

y y

When the phase difference between the x and y components is the light will be circularly polarized when the amplitudes are equal

E i A

E A normalized E

A i i i

E i

(

^ω

)

⎡ ⎤ ⎡ ⎤

= ⎢ ⎥ ⎣ − ⎦

⎣ ⎦

−

− + 1 exp

. ,

( ).

For this column vector the electric field vector will rotate in the

direction when viewed head on EM wave propagating in

E A i kz t

i

counterclockwise

This is left circularly polarized

the z direc light

tion

Right −

= ⎢ ⎥⎡ ⎤⎣ ⎦−

%0

1

circularly polarized light is represented by

E A

i

Elliptically Polarized Light Elliptically Polarized Light

( ) ( ) ^{( )}

π

ϕ ϕ π

− − °

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= ⎢ ⎥ = ⎢ ⎥ = ⎢ ⎥ =

+

⎣ ⎦

⎢ ⎥ ⎣ ⎦

⎣ ⎦

% ⁰ %

0 0 2 2

0

/ 2 (90 ),

exp 1

exp / 2 : exp

x x

y y

When the phase difference between the x and y components is the light will be elliptically polarized when the amplitudes are not equal

E i A A

E normalized E

B i Bi

E i A B

⎡ ⎤⎢ ⎥

⎣ ⎦

− +

⎡ ⎤

= ⎢⎣− ⎥⎦

%0

1

( ).

i

The electric field vector will

direction when viewed head on EM wave propagating in the z direction

Elliptically polarized light that rotates clockwis is represente rotate in the counterclockwise

d

Bi

e by

E A

General Case General Case

( )

( ) ( )

ε

ε

ε ε ε

ε ⁻

− −

⎡ ⎤ ⎡ ⎤

= ⎢⎣ ⎥⎦ = ⎢⎣ + ⎥⎦

= + = +

= ⎛ ⎞⎜ ⎟⎝ ⎠

=

%₀

1

0

( ),

exp

exp cos sin

tan

x

elliptically polarized w

When the phase difference between the x and y componen hen the ampli

ts is some angle the light will be

A A

E b

tudes ar

i B iC

b i b i B iC

C B

e not e l

E

qua

A α ε

= +

= −

2 2

0

0 0

2 2

0 0

2 cos

tan 2

y

x y

x y

E B C

E E

E E

Example

Example

Summary of Jones Vectors

Summary of Jones Vectors

Summary of Jones Vectors

Summary of Jones Vectors

Summary of Jones Vectors

Summary of Jones Vectors

Mathematical Representation of Polarizers Mathematical Representation of Polarizers

Polarizer

Phase retarder

Rotator

Birefringence (복굴절) 현상 이용

⎥ ⎦

⎢ ⎤

⎣

⎡

d c

b

a

• How can we make polarizations other than linear, e.g., circular?

– Birefringence!

» Birefringent materials (e.g., crystals or stressed plastics) have the property that the speed of light is different for light polarized in the two transverse dimensions (polarization-dependent speed), i.e.,

• light polarized along the “fast axis” propagates at speed v_fast

• light polarized along the “slow axis” propagates at speed v_slow

» Thus, if the “fast” and “slow” polarizations start out in phase, inside the birefringent material the “fast” polarization will ‘pull ahead’:

For a given birefringent

material, the relative phase is determined by the thickness

d, and frequency

ω .

slow slow

slow

t d v

ϕ ω

ω

=

=

1 1

fa s t s lo w

fa s t s lo w

d v v

ϕ ϕ

ω

−

⎛ ⎞

= ⎜ ⎜ ⎝ − ⎟ ⎟ ⎠

thickness, d

“fast” wave

“slow” wave y

x

Birefringence (복 굴절률)

Birefringence (복 굴절률)

fast fast

fast

t

d v

ϕ ω

ω

=

=

• light polarized along the “fast axis” propagates at speed v

_fast

• light polarized along the “slow axis” propagates at speed v

_slow

/

fa st fa st

fa st

fa st

fa st

fa st o

t d v

d c n

d n

c n k d

ϕ ω

ω

ω ω

=

=

=

⎛ ⎞

= ⎜ ⎟

⎝ ⎠

=

( )

1 1

fa s t s lo w

fa s t s lo w

fa s t s lo w o

d v v

n n k d

ϕ − ϕ = ω ^⎛ ⎜ ⎜ ⎝ − ^⎞ ⎟ ⎟ ⎠

= −

Birefringence (복 굴절률)

Birefringence (복 굴절률)

slo w slo w

slo w o

t

n k d

ϕ

=

ω

=

fast slow

n

n >

What Causes Birefringence?

What Causes Birefringence?

Birefringence can occur in any material that possesses some asymmetry in its structure, so that the material is more “springy”

in one direction than another.

Examples: Crystals: quartz, calcite

Different atom spacings in and .

x ˆ y ˆ

Long stretched molecular chains: wrap, cellophane tape

Birefringent Materials Birefringent Materials

•

Birefringent solids have a crystal structure such that the binding of the electron clouds about the nuclei in the lattice is anisotropic. Thus, the induced polarization in the medium will be anisotropic and the refractive index will depend on the

orientation of the electric field vector with respect to the crystal lattice.

Hecht, Optics, Chapter 8

Birefringent Materials Birefringent Materials

•

Amorphous solids and cubic crystals have isotropic refractive indices. On the other hand, solid crystals with asymmetric structure such as calcite will exhibit different refractive indices depending on the orientation of the electric field vector of the light field with respect to the crystal structure.

Birefringent Materials: Calcite Birefringent Materials: Calcite

•

Calcite is a natural, strongly birefringent material. It is commonly used for polarizers and exhibits the remarkable phenomenon of double refraction.

E

⊥

E

||

λ

⊥

=

=

=

||

1.4864 1.6584

589.3 n

n

for nm

Hecht, Optics, Chapter 8

Double Refraction in Birefringent Materials Double Refraction in Birefringent Materials

The phenomenon of double refraction was observed in calcite and in the early 1800’s was a key factor in the development of the theory of

polarization of light.

Hecht, Optics, Chapter 8

Wave Plates

• Birefringent crystals with precise thicknesses

The phase of the component along the fast axis is π/2 out of phase with the component along the slow axis. E.g.,

Before QWP

)

0

sin( kz t E

E

_x

= − ω

)

0

sin( kz t E

E

_y

= − ω

)

0

sin( kz t E

E

_x

= − ω

After

QWP )

sin( 2

0

ω − π

−

= E kz t

E

_y

RCP

Crystal which produces a phase change of π/2 → “quarter wave plate”

(a “full wave plate” produces a relative shift of → no effect).4× =^π₂ 2π

Light polarized along the fast or slow axis merely travels through at the appropriate speed → polarization is unchanged.

Light linearly polarized at 45° to the fast or slow axis will acquire a relative phase shift between these two components → alter the state of polarization.

Quarter Wave Plate summary:

•linear along fast axis Æ linear

•linear at 45

^°

to fast axis Æ circular

•circular Æ linear at 45

^°

to fast axis

Quarter Wave Plates

• Light linearly polarized at 45

^o

incident on a quarter wave plate produces the following wave after the quarter wave plate:

RCP )

cos(

2 )

sin( ₀

0 kz ωt E kz ωt

E

E_y = − −

π

= − −

Fast axis:

)

0 sin( kz ωt

E

E_x = −

Slow axis:

E r y

x

RCP = CW

Rotation at t = 0 :

(QWP: fast ahead of slow by λ

/4)

E

_f slow

45

^ο

E

TA

fast

z

y x

λ/4

Max vectors at t=0 :

E

_s

Half-Wave Plate

Hecht, Optics, 1987

Half-wave plates are used to rotate the polarization of linearly polarized light.

Quarter-wave plates are used to convert circular to linear polarization and vice versa.

Half-wave plates

Half-wave plates

Mathematical Representation of Polarizers

Mathematical Representation of Polarizers

Mathematical Representation of Polarizers Mathematical Representation of Polarizers

π

= ′

°

= −

% %

0 0

, ,

45 ,

exp

The operation of the polarizer or retarder on the electric field is described by

M E E

For example a QWP converts linearly polarized light when the light is at an angle of to the SA of the QWP to circularly polarized light

M

(

i

)

( )

( ) ( )

π

π π

⎡ ⎤ ⎡ ⎤

⎢ ⎥ = ⎢ ⎥

⎣ ⎦ ⎣ ⎦

− ⎡ ⎤

= ⎢ ⎥

⎣ ⎦

°

⎡ ⎤ ⎡ ⎤ − ⎡ ⎤

− ⎢⎣ ⎥⎦ ⎢ ⎥⎣ ⎦ = ⎢ ⎥⎣ ⎦−

%

%

0

0

1 0 1 1

/ 4 0 2 1

exp / 4 1 2

, 90

exp / 4

1 0 1 1 1

exp / 4

0 2 2 1

i E

M E i

i

A QWP will convert CP light back to LP light but now rotated by

i i

i i