18. Active polarization control - Brown University · 2017-03-10 · The Kerr effect: the polarization rotation is proportional to the Kerr constant and E2 ∆nis the induced birefringence,

18. Active polarization control

Ways to actively control polarization

Pockels' Effect – inducing birefringence

Kerr Effect

Optical Activity

Principal axes are circular, not linear

Faraday Effect – inducing optical activity

The Pockels Effect

Friedrich Carl Alwin Pockels(1865 - 1913)

Pockels discovered that, for certain materials, applying an electric field can cause them to become birefringent, or change the existing birefringence.

voltage on

voltage off

An example: BaTiO3 has a cubic lattice, but an applied voltage distorts the lattice into a tetragonal shape.

In most materials, the Pockels Effect

does not exist.

Consider a material that possesses “inversion symmetry”, which means that reflecting the position of every atom through a given central point doesn’t change the crystal.

Examples: many crystalline solids (e.g. silicon or diamond)any liquid or gas, amorphous solids with random atomic positions

If applying an electric field causes a change in the refractive index in proportion to the field:

∆ =n Eβthen applying the opposite electric field must cause the same change in the index: ( )∆ = −n Eβ

This can only be true if β = 0. Thus there is no Pockels effect in materials with inversion symmetry.

The Pockels Effect: Electro-optic constants3

0

0 / 2

2∆ = ≡ijn r V V

Vλ

π πϕ

λ

∆ϕ is the relative phase shift between the two polarization axes V is the applied voltage rij is the “electro-optic tensor” of the material. i,j = (x,y,z) are indices that depend on the crystal orientation

Vλ/2 is referred to as the “half-wave voltage”.

potassium di-hydrogen phosphate (KDP)

quartz

barium titanateBaTiO3

lithium niobateLiNbO3

r41 = 8.6r63 = 10.6

r41 = 0.2r63 = 0.9

r33 = 23r13 = 8.0r42 = 820

r33 = 30.8r13 = 8.6r42 = 28r22 = 3.4

Non-zero elements of the electro-optic tensor, for some materials that are often used. The units are 10-12 meters/volt.

A Pockels Cell

If we add polarizers, the Pockels' effect allows control over the amplitude of the wave.

a commercial Pockels cell

Applications of Pockels cells

• creating an amplitude-modulated or phase-modulated laser beam

• picking one pulse out of a train of pulses

Input pulse train

Electro-optic modulator

voltage pulse

Output single pulse

• switching energy out of a laser cavity - this is known as “Q-switching”. It is the way many high power lasers work.

Pockels effect: phase modulation

Suppose we start with an ordinary input wave from a laser:(polarized along one of the principle axes of the device)

0= j t

inE E e ω

( )

( )( )

sin

0

0 1 sin

Ω=

≈ + Ω

j tj t

out

j t

E E e e

E e j t

δω

ω δ

and the output wave is:

Suppose the voltage applied to the electro-optic material is a sinusoidally oscillating voltage, with small amplitude η and frequency Ω. Then the phase acquired by the light wave is:

( )sin∆ = Ωtϕ δ2

1= <<Vλ

πηδwhere

( ) ( )

0

0

12 2

2 2

Ω − Ω

+Ω −Ω

= + −

= + −

j t j t j t

out

j t j tj t

E E e e e

E e e e

ω

ω ωω

δ δ

δ δ

output light has new frequency components! This is known as ‘sideband generation’.

The Kerr effect: the polarization rotation

is proportional to the Kerr constant and E2

∆n is the induced birefringence, E is the electric field strength, K is the "Kerr constant“ of the material.

The Kerr effect exists in all materials, but is usually much weaker than the Pockels effect, so the practical applications are much more limited.

2

0n KEλ∆ =

The Kerr effect is sometimes called the “quadratic electro-optic effect.”

Optical Activity

Unlike birefringence, optical activity maintains a linear polarization

throughout. The rotation angle is proportional to the distance.

The effect of optical activity was first discovered in 1811 by Francois Arago, who was studying the optical properties of quartz crystals.

Francois Arago1786 - 1853

Principal Axes for Optical Activity

In media with optical activity, the principal axes correspond to circular polarizations.

We consider the component of light along each principal axis independently in the medium and recombine them afterward.

Just like birefringent media, the principal axes of an optically active medium are the medium's symmetry axes.

Unit vectors for circular polarization

( )( )

ˆ ˆ ˆ / 2

ˆ ˆ ˆ / 2

= −

= +

R x jy

L x jy

( )( )

ˆ ˆˆ / 2

ˆ ˆˆ / 2

= +

= −

x L R

y L R j

Of course we can invert these expressions to solve for the usual linear unit vectors x and y in terms of R and L:

What do we mean by ‘circular principle axes’?

We can define unit vectors which ‘point’ along the right-handed and left-handed circular directions:

Just as any vector in the x-y plane can be written as a sum of two linear polarizations x and y, it can also be written as a sum of two circular polarizations R and L.

Math of Optical Activity–Circular

Principal Axes

This x-polarized beam can be written as R + L(neglecting the √2 in all terms), where:

Note that this is just a complicated way of writing x-polarized light!

[ ] [ ]

0

0

( , ) Re exp ( ):

( , ) Re exp ( )

x

y

E z t E j kz t

E z t jE j kz tR

= −

= − −

ω

ω%

%

[ ] [ ]

0

0

( , ) Re exp ( ):

( , ) Re exp ( )

x

y

E z t E j kz t

E z t jE j kz tL

= −

= −

ω

ω%

%

Suppose that, at the input of an optically active medium, we have linear polarization oriented along the x axis.

Math of Optical Activity–Circular

Principal Axes (cont’d)

In optical activity, each circular polarization can be regarded as having a different refractive index, as in birefringence.

[ ] [ ]

0

0

( , ) Re exp ( ):

( , ) Re exp ( )R

x R

y

E z t E j kz t

E z t j

kn dR

kE z n dj k t

+

+

= −

= − −

ω

ω%

%

[ ] [ ]

0

0

( , ) Re exp ( ):

( , ) Re exp ( )L

x L

y

E z t E j kz t

E z t j

kn d

knL

E z dj k t

+

+

= −

= −

ω

ω%

%

where nR and nL are the refractive indices for the R and L components:

After propagating through an optically active medium of length d, the R and L components acquire different phases:

Math of Optical Activity–Circular

Principal Axes (continued)

( )( )

11

exp( ) exp/

exp( ) exp

Polarization State :

= − +

y x

j jE E j

j j

ϕϕ

ϕϕ

L

RL

R

[ ] [ ] 0 0( , ) Re exp ( ) exp ( )+= − + −+% %

R Lx kn dE z t E j knkz dt E j kz tω ω

[ ] [ ] 0 0( , ) Re exp ( ) exp ( )+= − − + −+% %

R Ly kn dE z t jE j kz t jE j kz kn d tω ω

Adding up the field components, we have:

Define: and: = =L Rkn d kn dϕϕRL

or:( ) ( ) ( ) 0( , ) Re exp exp exp= + − R Lx knE z t E j j jkzkn d d j tω

( ) ( ) ( ) 0( , ) Re exp exp exp= − + − LRy kn dE z t jE j j jkn d zk j tω

Math of Optical Activity–Circular

Principal Axes (continued)

R L( ) / 2∆ = −k n n dϕ

( ) ( )Now, define: 2 and: 2= + ∆ = −L LR Rave ϕ ϕϕ ϕϕϕ

( ) ( )( ) ( )

( ) ( ) ( )( ) ( ) ( )

11

exp exp expexp exp

exp exp exp exp exp

− ∆ − ∆ =− + − ∆ + ∆

ave

ave

j j jj jj j

j j j j j

ϕ ϕϕ ϕ

ϕ ϕϕϕ

ϕϕR

R

L

L

1

tan

= ∆ ϕ

Remarkably, the polarization state of the output is simply linear, for any value of the relative phase delay!

For an x-polarized input beam, the output polarization is xwhen ∆ϕ = mπ and y when ∆ϕ = (m + 1/2)π.

(m = any integer)

Why are some materials optically active?

Optical activity arises when a molecule or a crystal interact with right-handed circular polarization differently from left-handed circular polarization.

If the structure of a molecule or a crystal is not mirror-symmetric, then it is “chiral.” It is different from its own mirror image.

Example: glucose

D-glucose L-glucose

Such molecular pairs are called “enantiomers”.

Measuring optical activity

Optical activity is used in the sugar industry to measure syrup concentration, in optics to manipulate polarization, in chemistry to characterize substances in solution, and is being developed as a method to measure blood sugar concentration in diabetics.

Magnetic field

Magneto-optic medium

Polarizer Analyzer

0 +V

The Faraday Effect

The Faraday effect allows active control over the polarization rotation.

The Faraday effect is analogous to the Pockel’s effect, except with an applied magnetic field instead of an applied electric field.The magnetic field can induce optical activity in certain materials.

The Faraday effect: the polarization rotation is

proportional to the magnetic field strength

β = V B d

where: β is the polarization rotation angle,

B is the DC magnetic field strength,

d is the distance,

V is the "Verdet constant" of the material.

The Faraday isolator: non-reciprocal optics

Unlike almost any other passive optical component, the Faraday effect is not reciprocal. Beams passing one way through the system don’t necessarily do the same thing as beams passing the other way. This can therefore be used for optical isolation.

Polarizer

Permanent magnet

A Faraday isolator

This can be extremely useful in amplified laser systems, where a back-reflection from an optical amplifier can travel back to the laser and destroy it!