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 [equation] undergoes constructive interference (resonance) after multiple reflections between the two surfaces. [equation] Transmission and reflection peak at [equation].

+ 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 [equation] can be expressed as [equation]
Note - [equation] and [equation] are attenuation and gain coefficients for POWER.
At x=0, [equation] and at x=d, transmitted field after 1st pass [equation] where power transmittivity [equation] & power reflectivity [equation] are for the front surface, power transmittivity [equation] & power reflectivity [equation] are for the back surface. One more pass, i.e. after a round trip (traveling [equation]), field becomes [equation] where [equation] is change in amplitude and [equation] is change in phase. [equation] [equation].
Power transmittance [equation]

+ Resonance conditions (ref. 2: pp. 312-321 1st Ed., pp. 368-377 2nd): Similar to the transfer function of a comb filter or bandpass filter in DSP.
Freq condition - [equation] [equation] which gives freq spacing between resonances called free spectral range (FSR) [equation]
Amplitude condition - [equation] [equation] (gain and losses break even, !!lasing threshold!!)

+ Finesse: Another measurement of spectral width [equation] relates to Q factor of a filter.
[equation] and [equation], [equation] is the resonant freq.
High loss [equation] low F or Q factor, i.e. [equation],
must be larger than the number of channels for a system with certain FSR.

+ Tunable by incident angle [equation] and [equation]: Oblique incident angle [equation].

+ Bragg grating (pp. 123-127 2nd Ed, pp. 129-136 3rd Ed; ref. 2 pp. 68-70 1st Ed, pp. 62-64 2nd Ed.): Consider N reflective surfaces with very low field reflectivity, i.e. [equation]. We only consider 1st reflection from each surface. Each surface differs by the factor [equation] where [equation] is field transmission coef. and [equation].
[equation] (finite # layers)
[equation] [equation]
Power reflectivity [equation]
[equation] which resonates at Bragg condition [equation], [equation] - order of diffraction.

+ Fiber Bragg gratings (FBG): Wavelength selective reflection. Chirped FBG for dispersion compensate. Adjusting [equation] thermally or mechanically to tune filter.
FBGs are fabricated with interferometer method and phase mask method.

Last Modified: March 04, 2018
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