Loxodromes in open multisection lasers

We introduce a formalism to efficiently calculate lasing modes and optical power flow in multisection lasers with open boundaries. The formalism is underpinned by a projection of the complex-valued electric field and its spatial derivative onto a suitably extended complex Z plane to reduce the order of the problem and simplify analysis. In a single-section laser, we show that a laser mode is a loxodrome on the extended complex Z plane. In a multisection laser, we obtain loxodromes for individual sections of the laser. Then, a multisection mode is constructed by continuously concatenating individual loxodromes from each section using the open boundary conditions. A natural visualization of this construction is given by stereographic projection of the extended complex Z plane onto the Riemann sphere. Our formalism simplifies the analysis of lasing modes in open multisection lasers and provides insight into the mode geometry and degeneracy.