18. Active polarization control

Transcription

1 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

2 The Pockels Effect Pockels discovered that, for certain materials, applying an electric field can cause them to become birefringent, or change the existing birefringence. An example: BaTiO 3 has a cubic lattice, but an applied voltage distorts the lattice into a tetragonal shape. voltage on Friedrich Carl Alwin Pockels ( ) voltage off

3 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= β E then applying the opposite electric field must cause the same change in the index: n= β ( E ) This can only be true if β = 0. Thus there is no Pockels effect in materials with inversion symmetry.

4 The Pockels Effect: Electro-optic constants ϕ = 2π n r V 3 0 ij πv 0 V λ / 2 λ V λ/2 is referred to as the half-wave voltage. ϕ is the relative phase shift between the two polarization axes V is the applied voltage r ij is the electro-optic tensor of the material. i,j = (x,y,z) are indices that depend on the crystal orientation potassium di-hydrogen phosphate (KDP) quartz barium titanate BaTiO 3 lithium niobate LiNbO 3 r 41 = 8.6 r 63 = 10.6 r 41 = 0.2 r 63 = 0.9 r 33 = 23 r 13 = 8.0 r 42 = 820 r 33 = 30.8 r 13 = 8.6 r 42 = 28 r 22 = 3.4 Non-zero elements of the electro-optic tensor, for some materials that are often used. The units are meters/volt.

5 A Pockels Cell If we add polarizers, the Pockels' effect allows control over the amplitude of the wave. a commercial Pockels cell

6 Applications of Pockels cells creating an amplitudemodulated or phasemodulated laser beam picking one pulse out of a train of pulses voltage pulse Input pulse train Output single pulse Electro-optic modulator switching energy out of a laser cavity - this is known as Q-switching. It is the way many high power lasers work.

7 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) Ein = j t E e ω 0 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: and the output wave is: ϕ = δ sin( Ωt) jωt E = E e e out ω δ Ω δ Eout = E0e 1+ e e 2 2 j t j t jωt jωt δ δ = E0 e + e e where jδ sin πη δ = <<1 ( Ωt) V λ 2 ( δ ( )) jωt E0e 1+ j sin Ωt ( ω+ω) ( ω Ω) j t j t output light has new frequency components! This is known as sideband generation.

8 The Kerr effect: the polarization rotation is proportional to the Kerr constant and E 2 The Kerr effect is sometimes called the quadratic electro-optic effect. n=λ KE 0 2 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.

9 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 Arago

10 Principal Axes for Optical Activity In media with optical activity, the principal axes correspond to circular polarizations. Just like birefringent media, the principal axes of an optically active medium are the medium's symmetry axes. We consider the component of light along each principal axis independently in the medium and recombine them afterward.

11 Unit vectors for circular polarization What do we mean by circular principle axes? We can define unit vectors which point along the right-handed and left-handed circular directions: ( ) ( ) Rˆ = xˆ jyˆ / 2 Lˆ = xˆ + jyˆ / 2 Of course we can invert these expressions to solve for the usual linear unit vectors x and y in terms of R and L: ( ˆ ˆ) xˆ = L+ R / 2 ( ˆ ˆ) yˆ = L R / 2 j 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.

12 Math of Optical Activity Circular Principal Axes Suppose that, at the input of an optically active medium, we have linear polarization oriented along the x axis. This x-polarized beam can be written as R + L (neglecting the 2 in all terms), where: [ ω ] [ ω ] { } Ex ( z, t) = Re E0 exp j( kz t) R : % Ey ( z, t) = Re je0 exp j( kz t) % { } [ ω ] [ ω ] { } Ex ( z, t) = Re E0 exp j( kz t) L : % Ey ( z, t) = Re je0 exp j( kz t) % { } Note that this is just a complicated way of writing x-polarized light!

13 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. After propagating through an optically active medium of length d, the R and L components acquire different phases: [ + knrd ω ] [ j k + k ωt ] { } Ex ( z, t) = Re E0 exp j( kz t) R : % Ey ( z, t) = Re j 0 exp ( R ) % { E z n d } [ + knld ω ] [ j k + kn ωt ] { } Ex ( z, t) = Re E0 exp j( kz t) L : % Ey ( z, t) = Re j 0 exp ( L ) % { E z d } where n R and n L are the refractive indices for the R and L components:

14 Math of Optical Activity Circular Principal Axes (continued) Polarization State : [ kz+ kn d ωt ] E [ j kz+ d ωt ] { } 0 R 0 L % % { 0 [ + knrd ω ] 0 [ + L ω ]} E ( z, t) = Re E exp j( ) + exp ( kn ) x Adding up the field components, we have: Ey ( z, t) = Re je exp j( kz t) + je exp j( kz kn d t) % % or: Define: ϕ = kn d and: ϕ = kn d L ( kn d) ( kn d) ( jωt) { } 0 R L Ex ( z, t) = Re E exp j + exp j exp jkz ( kn d) ( kn d) ( z jωt) { } 0 R L Ey ( z, t) = Re je exp j + exp j exp jk L R 1 1 exp( ) exp( ) / = jϕl jϕr Ey Ex j exp( jϕ ) + exp( jϕ ) L R R

15 Math of Optical Activity Circular Principal Axes (continued) ( ϕ ) ϕ ( ϕ ) Now, define: ϕave = R+ ϕl 2 and: = ϕr L exp( jϕ ) exp( j ) = exp( jϕave) exp( j ϕ) exp( j ϕ) L ϕr j j exp( ) + exp( ) exp( ) exp( ) + exp( ) jϕl jϕr jϕave j ϕ j ϕ 1 = tan ϕ ϕ= k( n n ) d / 2 R L 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 x when ϕ = mπ and y when ϕ = (m + 1/2)π. (m = any integer)

16 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 mirrorsymmetric, then it is chiral. It is different from its own mirror image. Example: glucose D-glucose L-glucose Such molecular pairs are called enantiomers.

17 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.

18 The Faraday Effect 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. Magneto-optic medium Polarizer Analyzer Magnetic field 0 +V The Faraday effect allows active control over the polarization rotation.

19 The Faraday effect: the polarization rotation is proportional to the magnetic field strength β = V B d where: βis the polarization rotation angle, B is the DC magnetic field strength, d is the distance, V is the "Verdet constant" of the material.

20 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 backreflection from an optical amplifier can travel back to the laser and destroy it!