Principles of calculating alignment signals in complex resonant optical interferometers

In the long-baseline laser interferometers that are now under construction to measure gravitational waves, the alignment of the optical components with respect to the incoming laser beam is crucial for maintaining maximum phase sensitivity. We present a basic formalism to calculate the effects of misalignment and beam distortions analytically in an arbitrarily complex optical system, including coupled cavities and Michelson interferometer configurations coupled with cavities. The electromagnetic field is decomposed into a superposition of higher-order Gaussian modes, while misaligned and distorting optical components along with free-space propagators are represented by matrix operators that act on the state vectors in this basis. We show how to deduce useful alignment signals generally, in order to design angular control systems.