Optical filters II Mar. 13, 2018

• Acousto-optics tunable filters (pp. 143-146 2nd Ed, pp. 149-152 3rd Ed): Use RF signal to induce acoustic oscillation -> Use RF signal to induce generate a Bragg grating that diffracts light.
Acoustic wave parameters - freq f_a, velocity v_a and wavelength Lambda = v_a /f_a
RF signal induces strain s(x,t) = S_o cos ( Omega t - qx) where  q = (2pi)/Lambda and Omega = 2 pi f_a
acousto-optical effect causes refractive index change Delta n = -(p n^3)/2 s(x,t) where p is the photoelastic constant.
Select wavelength lambda = Delta n Lambda
Response time D / v_a where D is the optical beam diameter.
Operation - Use birefringent material to make a polarizing beam splitter. The resulting TE wave will be converted into TM wave if grating condition lambda = |n_(TE) - n_(TM) | Lambda = Delta n Lambda is satisfied. The polarizing beam splitter selects only TM waves.
Freq response T(lambda) = (sin^2 (pi/2 sqrt{1+X^2}))/(1+X^2) where X = (2 Delta lambda)/(Delta_(AO)), Delta lambda = lambda - lambda_B and Delta_(AO) = lambda_B^2/(D Delta n)
• Mach-Zehnder (MZ) filter (pp. 135-139 2nd Ed, pp. 141-145 3rd Ed): Similar to Mach-Zehnder interferometer; replace beam splitter with directional coupler.
Directional coupler is a 2 ports device (pp. 108-112 2nd Ed, pp. 114-118 3rd Ed).
Its transfer function for straight thru (e.g. input 1 -> output 1) is T_{11} = cos^2 ( kappa l) and for cross over (e.g. input 1 -> output 2) is T_{12} = sin^2 (kappa l) where kappa is the coupling coefficient and l is the coupling length..
By choosing appropriate l , we can construct a 3dB coupler (i.e. 50%/50% splitter).
For 3dB coupler, signal from each input port is split across 2 output ports. Phase shift of pi/2 occurs when signal crosses ports, e.g. input port 1 to output port 2 or input port 2 to output port 1.
Two input ports separate freq depending on the phase difference between the two paths for input signals, e.g. signal at output port 1 satisfies Delta phi_1 = (2 m -1 ) pi for lambda_1 ;
signal at output port 2 satisfies Delta phi_2 = 2 m pi for lambda_2
Channel frequency spacing f_2 - f_1 = Delta f = c / (2n Delta L)
Tunable output freq by RF, thermal or mechanical controls.
Cascading MZ to get more than 2 outputs.