I am a retired Honeywell Principal Optical Engineer where I designed and fabricated fiber optic gyroscopes in Phoenix. Fiber optic gyroscopes are Sagnac interferometers, and their design requires an intimate understanding of the optical circuit to avoid polarization errors in the gyro (rotation) rate output and to avoid thermally induced non-reciprocities which also produce rate errors.
Jones matrices, which characterize components (lead lengths, birefringence between fiber axes, phase delay between axes, polarizer performance, phase modulation) , are used to ferret out polarization errors created by birefringent fiber axis misalignment at splices in the optical circuit brought me to Mathematica. After multiplying the matrices and obtaining the intensity terms, there are hundreds of thousands of error terms. Without Mathematica and its outstanding capabilites, this analysis would be impossible.