Photonic quantum computing is a strong contender in the race to fault tolerance. Recent proposals using qubits encoded in frequency modes promise a large reduction in hardware footprint and have garnered much attention. In this encoding, linear optics, i.e., beam splitters and phase shifters, is necessarily not energy conserving and is costly to implement. In this work, we present designs of frequency-mode beam splitters based on modulated arrays of coupled resonators. We develop a methodology to construct their effective transfer matrices based on the SLH formalism for quantum input-output networks. Our methodology is flexible and highly composable, allowing us to define $N$ mode beam splitters either natively based on arrays of $N$-resonators of arbitrary connectivity or as networks of interconnected $l$ mode beam splitters with $l<N$. We apply our methodology to analyze a two-resonator device, a frequency-domain phase shifter, and a Mach-Zehnder interferometer obtained from composing these devices, a four-resonator device, and derive a no-go theorem forbidding the native implementation of a subclass of $N$ mode frequency-domain beam splitters with arrays of $N$-resonators.

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• Electro-optic effects
• Frequency conversion
• Integrated optics
• Optical quantum information processing
• Quantum optics
• Devices
• Cavity resonators