Möbius transformation and coupled-wave theory: Complete identification of the transfer matrix

A M\"obius transformation which conformally maps the unit circle onto itself is applied to the scalar coupled-wave equations, describing electromagnetic wave propagation in Bragg gratings, and reduces them to a first-order nonlinear differential equation of a single real variable. This equation is analytically integrated for linear detuning and numerically for more complicated refractive index modulation scenarios, e.g., chirped and apodized Bragg gratings, offering a platform for identifying both the amplitude and phase of all elements of the transfer matrix of arbitrarily complex cases. A link between coupled-wave theory and coupled oscillators is established, and exploring the transformation's geometrical properties leads to alternative definitions of the photonic band gap.