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.

Last Modified: March 11, 2018
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