The residual amplitude modulation ($mathrm{RAM}$) is the undesired, non-zero amplitude modulation that usually occurs when a phase modulation based on the electro-optic effect is imprinted on a laser beam. In this work, we show that electro-optic modulators (EOMs) that are used to generate the sidebands on the laser beam also generate a $mathrm{RAM}$ in the optical setup. This result contradicts standard textbooks, which assume the amplitude remains unchanged in the process and should be considered as a fundamental $mathrm{RAM}$ ($mathrm{RAM_{F}}$) for these devices. We present a classical model for the propagation of an infrared laser with frequency $omega_{0}$ in a wedge-shaped crystal and an EOM with an RF modulating signal of frequency $Omega$. Since ${Omega}ll omega_{0}$, we solve Maxwells equations in a time-varying media via a WKB approximation and we write the electromagnetic fields in terms of quasi-plane waves. From the emerging fields of the setup, we compute the associated $mathrm{RAM_{F}}$ and show that it depends on the phase-modulation depth $m$ and the quotient $left(frac{Omega}{omega_{0}}right)$. The $mathrm{RAM_{F}}$ values obtained for the EOMs used in gravitational wave detectors are presented. Finally, the cancellation of $mathrm{RAM_{F}}$ is analyzed.
تحميل البحث