Optical filters I |
Mar. 8, 2018 |
• Principles of optical filters:
Interference, diffraction, absorption. Operate on fixed freq
or tunable freq.
•
(pp. 130-135 2nd Ed, pp.
134-139 3rd Ed): Also called etalon, FP interferometer and
FP cavity. It consists of two parallel reflective surfaces.
A wave with
appropriate `lambda_o`
undergoes constructive interference (resonance) after
multiple reflections between the two
surfaces.
`->` Transmission and reflection peak
at `lambda_o`
•
(ref. 2 pp. 70-72
1st Ed,
):
A wave inside a gain/lossy medium with
length `d`
can be expressed as ` E = A e^{-((alpha - g) d)/2} e^{-jkd}`
Note
- `alpha` and `g`
are attenuation and gain coefficients for POWER.
At
`x=0`, `E=A`
and at `x=d`, transmitted field after 1st
pass `E_0 = A sqrt{T_1 T_2} e^{-((alpha - g) d)/2} e^{-jkd}`
where power
transmittivity `T_1 = 1 -R_1`
& power
reflectivity `R_1`
are for the front surface, power
transmittivity `T_2 = 1 - R_2`
& power
reflectivity `R_2`
are for the back surface. One more pass, i.e. after a round
trip
(traveling `2d`), field
becomes `E_1 = E_0 h e^{-j phi}`
where ` h = sqrt{R_1 R_2} e^(-(alpha-g)d)`
is change in amplitude
and `phi = 2kd`
is change in
phase.
`E_("Ttot") = E_0 + E_1 +E_2 + ... -> (A sqrt{T_1 T_2} e^{-((alpha - g) d)/2} e^{-jkd}) /(1 - h e^{-j phi))`
Power
transmittance `=|E_("Ttot")|^2 /A^2 -> (T_1 T_2 e^(-(alpha-g)d))/((1-h)^2+4h sin^2 (phi/2)`
• Resonance conditions (ref. 2: pp. 312-321 1st Ed.,
):
Similar to the transfer
function of a comb filter or bandpass filter in DSP.
Freq condition
- `phi /2 = m pi -> lambda_o = (2dn)/m`
which gives freq spacing between resonances called free
spectral range
(FSR) `Delta f_{FSR} = c /(2nd)`
Amplitude condition
- `h = 1 -> sqrt{R_1 R_2} e^(-(alpha-g)d)=1`
(gain and losses break even, !!lasing threshold!!)
• Finesse: Another measurement of spectral
width `Delta nu`
relates to `Q` factor of a filter in circuit.
(finite # layers `N`)
`-> E_("in")(0) r ( 1 - M^N) / (1 - M)`Power reflectivity `= (|E_("Rtot")|^2)/(E_("in")^2(0))` ` -> R (sin^2 ( (N phi) / 2 )) /sin^2 (phi/2)`
which resonates at Bragg
condition `Lambda = m lambda_B /(2n_{"eff"})`, `m` - order of diffraction
and `lambda_B` - Bragg wavelength.
• Fiber Bragg gratings (FBG): Wavelength selective
reflection. Chirped FBG for dispersion compensate.
Adjusting$\text{\hspace{0.5em}}\Lambda $
thermally or mechanically to tune filter.
FBGs are fabricated with
and .