Optical filters I

Mar. 8, 2018

• Principles of optical filters: Interference, diffraction, absorption. Operate on fixed freq or tunable freq.

Fabry-Perot (FP) resonator (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`

Transfer function of FP resonator (ref. 2 pp. 70-72 1st Ed, pp. 64-66 2nd 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., pp. 368-377 2nd 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.

`ccF -= (Delta f_(FSR)) /{Delta nu}` and `Q -= nu_o / (Delta nu)`, ` nu_o` is the resonant freq.
High loss `->` low `ccF` or `Q` factor, i.e. `ccF = (pi sqrt{R}) /(1-R)`,
must be larger than the number of channels for a system
with certain FSR.

Tunable by incident angle `theta_g` and `d`: Oblique incident angle `phi = 2nd k_o cos theta_g`

• Bragg grating (pp. 123-130 2nd Ed, pp. 129-136 3rd Ed;
ref2. pp. 68-70 1st Ed, pp. 62-64 2nd Ed.): Consider N reflective surfaces with very low field reflectivity, i.e. `r` << 1. We only consider 1st reflection from each surface. Each surface differs by the factor `M = t^2 e ^{-j phi}` where `t` is the field transmission coef. and `phi = k 2 Lambda`

`E_("Rtot") = E_("in") (0) r ( 1+ M + M^2 + ... )`

(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. AdjustingΛ thermally or mechanically to tune filter.
FBGs are fabricated with interferometer method and phase mask method .

Last Modified: March 4, 2018
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