Optical filters II |
Mar. 13, 2018 |
• Acousto-optics tunable filters (pp.
143-146 2nd Ed, pp. 149-152 3rd Ed):
`->`
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.