Complex modulation for Advanced LIGO*

Multistage optical phase modulation is used to control the interferometer length and alignment degrees of freedom in Advanced LIGO, VIRGO, and KAGRA. Phase modulation of the laser at RF frequency Ω produces sidebands at ω ± Ω, where ω is the laser frequency (the "carrier"). To achieve multiple modulation frequencies, multiple electro-optic modulators (EOMs) are used in series. The second and following EOMs impose sidebands not only on the carrier but on the sidebands produced by the upstream EOMs. This problem, called “sidebands on sidebands,” can affect the extraction of the interferometer control signals. A method of addressing it is called “complex modulation.” Complex modulation requires specific functions—produced by arbitrary-function generators—to be applied simultaneously to amplitude and phase modulators. The modulating functions are calculated to produce the desired modulation pattern on the laser. Two practical applications have been studied: single-sideband modulation and suppression of sidebands on sidebands. The results have been analyzed and compared to simulations. We also conducted characterization of practical modulators in an effort to understand the setup better and to provide accurate inputs for simulations.