Unifying planar and cylindrical Kerr waveguides: exact amplitude-phase invariants and a practical analysis roadmap

<jats:p>We present a systematic amplitude-phase formalism for transverse electric (TE) and transverse magnetic (TM) modes in nonlinear Kerr waveguides, covering both planar (slab) and cylindrical (fiber) geometries. Starting from the normalized Maxwell equations, we derive two fundamental invariants for each polarization: a flow invariant related to the phase difference between field components, and an energy invariant derived from the Kerr potential function $U(I) = \varepsilon_l I + \frac{\alpha}{2} I^2$. In planar geometry, these invariants yield a fully separable quadrature for the dielectric function $\varepsilon(z)$: $\int d\varepsilon/\sqrt{\mathcal{F}(\varepsilon)} = z - z_0$. Extending the formalism to cylindrical symmetry, we demonstrate that a simple rescaling of the transverse fields, $\Psi \to r\Psi$, absorbs the radial metric term $1/r$ and preserves the Hamiltonian structure. This rescaling yields the cylindrical analogues of the invariants: \[ r \cdot \frac{1}{2} |E_r| |H_\phi| \sin\Delta\Phi = C_1, \] \[ \left|\dot{E}_r + \frac{E_r}{r}\right|^2 = \beta^2 |E_z|^2 - \varepsilon_l I_{\mathrm{TM}} - \frac{\alpha}{2} r^2 I_{\mathrm{TM}}^2 + \frac{C_2}{r^2}, \] and their TE counterparts. These reduce the problem to a single first-order ODE for $\varepsilon(r)$. For guided modes with zero radial power flow ($C_1 = 0$), the cylindrical ODE reduces to the pure quadrature $\int d\varepsilon/\mathcal{F}(\varepsilon) = \ln r + C$, providing the exact cylindrical analogue of the planar solution. For leaky or radiation modes ($C_1 \neq 0$), the ODE retains explicit $r$-dependence, yielding a semi-analytical reduction that eliminates the need for multi-parameter shooting. Our results provide a unified framework for analyzing nonlinear wave propagation in Kerr media, explicitly demonstrating how the geometry (slab vs. fiber) manifests in the structure of the invariants and the separability of the resulting ODEs.</jats:p>