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. Adjusting$\text{ }\Lambda$ thermally or mechanically to tune filter.
FBGs are fabricated with interferometer method and phase mask method .