  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.
