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Frequency-modulated (FM) combs with a linearly-chirped frequency and nearly constant intensity occur naturally in certain laser systems; they can be most succinctly described by a nonlinear Schrödinger equation with a phase potential. In this work, we perform a comprehensive analytical study of FM combs in order to calculate their salient properties. We develop a general procedure that allows mean-field theories to be constructed for arbitrary sets of master equations, and as an example consider the case of reflective defects. We derive an expression for the FM chirp of arbitrary Fabry-Perot cavities—important for most realistic lasers—and use perturbation theory to show how they are affected by finite gain bandwidth and linewidth enhancment in fast gain media. Lastly, we show that an eigenvalue formulation of the laser’s dynamics can be useful for characterizing all of the stable states of the laser: the fundamental comb, the continuous-wave solution, and the harmonic states. 

We report the magnetoresistance of Co/Pt superlattices having thickness gradients at different orientations relative to an applied current. We measure the magnetoresistance at a fixed field as a function of the out-of-plane field angle, and find a unidirectional magnetoresistance (UMR) in addition to the expected anisotropic magnetoresistance (AMR) and spin Hall magnetoresistance (SMR). Specifically, the UMR signal is the difference in resistance between the (out-of-plane) +z and −z field orientation. The magnitude of UMR is minimized when the thickness gradient is parallel to the applied current and maximized when the gradient is nearly perpendicular to the current. The results imply the possibility of an alternative source of UMR in Co/Pt multilayers in addition to the previously considered anomalous Hall effect.