Modulators Tuesday, 11/14/2006 Physics 158 Peter Beyersdorf Document info 17. 1
Class Outline Birefringence Optical Activity Faraday Rotation Optical Modulators Electrooptic Modulators Accoustooptic Modulators 17. 2
Birefringence Crystals have regular arrangement of atoms The resonant frequency of charge vibration in the crystal depends on n 2 1 + Nq 2 e ɛ 0 m e (ω 2 0 ω2 ) So the index of refraction depends on polarization 17. 3
Index Ellipsoid The directional dependance of the index of refraction is shown by the index ellipsoid The index for a wave propagating along z, polarized an an angle θ with respect to the x-axis is is n(θ) = n 2 x cos 2 θ + n 2 y sin 2 θ Along the optical axis the index of refraction is independent of the angle of polarization 17. 4
Crystal Types biaxial positive uniaxial negative uniaxial What direction are the optical axes for each crystal? 17. 5
Double Refraction Calcite exhibits double refraction as shown The ordinary ray propagates through the crystal obeying normal refraction The extraordinary ray propagates at an angle through the crystal Consider Fermat s principle 17. 6
Double Refraction The index seen by the extraordinary ray depends on the angle of propagation of the ray n(θ) = -α n 2 x cos 2 (θ t α) + n 2 y sin 2 (θ t α) θ d the optical path length is therefore Λ(θ) = n(θ)d cos θ the minimum of this function is not at θ=0 17. 7
Optical Activity Material with non-zero chirality (i.e. helical shaped molecules)exhibit optical activity that causes the direction of polarization to be rotated These materials can be thought of as birefringent crystal with a circularly polarized basis Optically active materials include quartz, sugar water, organic materials β = πd λ 0 (n l n r ) β is the angle through which the polarization is rotated when traveling through a distance d Specific rotary power is β/d 17. 8
Optical Activity Quartz has a specific rotary power of 21.7 /mm. Calculate the thickness of quartz necessary to rotate the polarization by 45 d=45 /(21.7 /mm)=2.07mm What is the polarization state of light that passes through the quartz and then is retroreflected back through the quartz a second time? β 1 =45, β 2 =-45 so the polarization is unchanged after returning through the quartz. 17. 9
Faraday Effect Some materials exhibit induced optical activity in the presence of a magnetic field The Verdet constant V gives the rotation, per unit magnetic field per unit distance β = VBd If B points in the direction of propagation the rotation of the polarization is counterclockwise as seen from an observer facing the light (left handed rotation) 17. 10
Faraday Effect Terbium Gallium Garnet (TGG) is a crystal with a verdet constant of 40 rad/tm at λ=1064 nm What magnetic field is necessary to produce a 45 rotation in a 3cm length of TGG? B = π/4 rad 40 rad/tm 0.03 m = 0.65 T What is the polarization state of light that passes through the TGG and then is retroreflected back through the crystal a second time? β 1 =45, β 2 =45 so the polarization is rotated by 90 17. 11
Faraday Isolator Polarization rotation is a function of propagation direction because retro-reflection isn t equivalent to time reversal unless the magnetic field direction also changes. What properties does the following device have? 17. 12
Phase Modulators John Kerr found in 1875 that an applied electric field can make an isotropic substance behave like a birefringent crystal with birefringence Δn=λ 0 KE 2 with an optical axis along the direction of the applied field. The phase shift is therefore Δφ=2πKlV 2 /d 2 where the electrodes have a length of l and a separation of d 17. 13
Phase Modulators The half wave voltage of such a modulator is typically several tens of kilovolts. The device acts like a variable waveplate that can operate with a speed of about 10 GHz that is limited by the slew rate of the electronics (the modulator itself behaves as a capacitor limiting the slew rate in most systems) 17. 14
Phase Modulators The Pockels effect is a linear electrooptic effect in non-centrosymmetric crystals such as Potassium Dihydrogen Phosphate (KDP) and Lithium Niobate (LiNbO 3 ). The phase shift is proportional to the electrooptic constant r 63 Δφ=2πn 03 r 63 V/λ 0 and has typical halfwave voltages of a few kilovolts Speeds of 30 GHz are possible, limited by transit time of the crystal 17. 15
Amplitude Modulators Show that polarizers placed around a phase modulator so that the optical axis is at 45 with respect to the transmission axis of the modulator will convert the phase modulator into an amplitude modulator [ ] Ex,out E y,out = [ ] Ex,out E y,out [ 0 0 0 1 ] [ cos 45 o sin 45 o sin 45 o cos 45 o ] [ 1 0 0 e i φ = ie i φ/2 sin φ/2 [ 0 1 ] ] [ cos 45 o sin 45 o ] [ 1 0 sin 45 o cos 45 o 0 0 ] [ ] Ex,in E y,in I out = sin 2 ( φ/2)i x,in 17. 16
Amplitude Modulators How could the amplitude modulator shown be modified to have a linear (rather than quadratic) response? A static phase shift of Δφ s =π/4 could be added by a quarter wave plate or one polarizer could be rotated by 45 17. 17
Summary Crystals can have different indices of refraction for different polarization states because of the anisotropy of the material The birefringence of materials can be modified by applying an electric field Isolators, phase modulators and amplitude modulators are common optical devices that make use of induced birefringence 17. 18