Fabry-Perot fiber etalons (FPE) built from three or more reflectors are attractive for a variety of applications including communications and sensing. For accelerating a research and development work, one often desires to use off-the-shelf components to build an FPE with a required transmission profile for a particular application. Usually, multistage FPEs are designed with equal lengths of cavities followed by determination of the required reflectivities for realizing a desired transmission profile. As seen in previous works, fabricated reflectors are usually slightly different from the designed ones leading to departure from the desired transmission profile of the FPE. Here, we show a novel digital synthesis of multistage etalons with off-the-shelf reflectors and unequal lengths of involved cavities. We find that, in contrast to equal cavity lengths, unequal lengths of cavities provide more number of poles in the $z$-domain to achieve a desired multicavity FPE transmission response. For given reflectivities and by determining correct unequal lengths of cavities with our synthesis technique, we demonstrate a design example of increasing the FSR followed by its experimental validation. This work is generalizable to ring resonators, mirrored, and fiber Bragg grating based cavities; enabling the design and optimization of cavity systems for a wide range of applications including lasers, sensors, and filters.
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