Abstract
<jats:p>Delayed optical feedback can narrow semiconductor-laser linewidth while enhancing the relaxation-oscillation contribution to relative intensity noise (RIN). We treat both effects with one stochastic Lang--Kobayashi delay operator. The neutral phase root gives phase diffusion, whereas the conjugate relaxation pair controls intensity and carrier noise through delay-corrected residues. A weak-feedback expansion yields an analytic phase criterion for simultaneous linewidth and RIN suppression. Full-resolvent maps and nonlinear stochastic delay simulations confirm stable trade-off and simultaneous-suppression states. The result persists under mixed field--carrier noise, finite detection bandwidth, gain compression, pump and carrier-time variations, and fixed-output-power comparison, providing a phase-resolved design rule for weak-feedback semiconductor lasers.</jats:p>