Equivalence of Electro-optic Modulator Electrical and Optical Sidebands via Neumann's Addition Theorem for Bessel Functions

In photonic circuits and optical communications links, the electro-optic modulator (EOM) can be considered the heart of the entire system, allowing analog and digital data in the RF or microwave domain to be modulated onto an optical carrier for transmission over low-loss, high-bandwidth optical links. EOMs can be fabricated in a number of material systems and operated in various modes that optimize some measure of performance depending on the application. One common configuration is the push-pull configuration, where a phase modulator in each arm of a Mach-Zehnder interferometer is modulated with equal amplitude but opposite phase. The resulting optical fields of the EOM, and most importantly, the photodetected currents generated from those fields, are well-characterized in the literature. Here, we explicitly demonstrate how the complex optical fields, represented by a sinusoidal series with Bessel function amplitudes, are combined by the detector nonlinearity into RF and microwave currents, also represented by a sinusoidal series with Bessel function amplitudes, and extracted at the end of the optical link.