Spatial switching Mar. 24, 2014

+ Directional coupler (Sect 7.4B in 1st Ed, Sect 8.5B in 2nd Ed): Two planar waveguides are brought very close to each other [equation] wave in one waveguide can completely escape to another if the propagation constant of the wave in both waveguides are phase matched.
Waves -- [equation] & [equation]
Coupled mode eqn -- [equation], [equation] where [equation] & [equation] are coupling coef. and [equation] is the phase mismatch.
Variation of waves with propagation distance -- [equation] where [equation], [equation].
[equation], phase match [equation] complete exchange after a distance of [equation], i.e. an optical coupler
If distance = [equation], we have a 3dB splitter.

Derivation of solution for coupled mode equations.

+ Spatial switching by phase matching: Build an optical coupler with an electro-optical material as substrate. Without voltage [equation] complete transfer, otherwise reduced transmission.
[equation] when [equation] and [equation] when [equation]
[equation] where [equation] and d is the spacing of the electrodes.

+ Spatial light modulator: 2D device makes of photoconductive and electrooptic materials sandwiched by two electrodes
Writing image from the photoconductive side and causing drop in electric field [equation] drop in [equation] and creation of a phase plate according to the image
Read image by input light to the electrooptic side and the reflected light is phase modulation by the phase plate.
The image retained for a short time.
E.g. Pockel readout optical modulator -- BSO which is electrooptic for red and photoconductive for blue
Setting up image -- apply voltage, write image with intense blue light, read with uniform red light, erase with uniform blue light and voltage

Anisotropic EO materials Mar. 24, 2014

+ Modified index ellipsoid: [equation] where impermeability [equation] modified to include Kerr and Pockel effects.
Expect the orientation and size of the index ellipsoid affected by orientation (polarization) and strength of the field
Notation on indices -- symmetry [equation] [equation]
Remap [equation] 1,1 [equation] 1; 2,2 [equation] 2; 3,3 [equation] 3; 2,3 [equation] 4; 3,1 [equation] 5; 1,2 [equation] 6 and 18 [equation] instead of 27
symmetry [equation] [equation]
After remapping, 36 [equation] instead of 81

+ Determine n when [equation]:
1. At [equation], find [equation], [equation], [equation] and principal axes.
2. Find [equation] and [equation] from table.
3. [equation].
4. Write eqn for modified index ellipsoid and see if the principal axes rotated.
5. Find [equation], [equation] and [equation] and/or new axes by rewrite the index ellipsoid eqn into [equation].
6. Use method in Ch. 6 to find n for certain direction of propagation.
Expect a) rotation of ellipsoid depending on wave polarization and crystal structure
b) shape of ellipsoid changed with E

Example for calculation of index ellipsoid rotation

EO material tables

+ Matrix notation for isotropic Kerr media:
[equation] where [equation]
Notice the cross terms have factor of 2 owing to 2 permutations

+ Modulator design:
Recognize the direction of E-field and propagation direction of light and choose the appropriate n & r.
Use the correct formula for [equation].

+ Liquid crystals:
The ellipsoids in LC align with external E-field.
[equation] twisted nematic LC can be unwinded by E-field, i.e. a switch.
[equation] nematic LC modulator -- the extension of unwinding angle [equation] depends on strength of E-field where [equation] for [equation] and [equation] is a critical voltage.
A voltage dependent retarder formed.
Applications -- phase modulator, wave retarder and becomes intensity modulator when put between cross polarizers.
HW #7 due 4/1/14
1. Problem 18.1-1 (page 735 1st Ed.; problem 20.1-2 page 871 2nd Ed.)
2. Exercise 18.2-1 (page 721 1st Ed.; exercise 20.2-1 page 856 2nd Ed.)
3. Problem 18.1-2 (page 735 1st Ed.; problem 20.1-3 page 871 2nd Ed.)
4. Problem 18.1-3 (page 735 1st Ed.; problem 20.1-4 page 871 2nd Ed.)

Last Modified: Mar 24, 2014
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