Closed-Form Approximations to the Threshold Quantities of Distributed-Feedback Lasers With Varying Phase Shifts and Positions

We derive analytical approximations to the threshold gain, detuning, and power distribution ratios of distributed-feedback lasers with a variable phase shift at a variable longitudinal position. These closed-form approximations exhibit the direct influence of the grating parameters, and are in excellent agreement with the exact solution of the coupled mode equations for a wide range of phase shifts, positions, grating strengths and lengths. It is shown that the product of grating strength and phase shift offset distance from the grating center has a dominant influence on most threshold parameters. The threshold gain and output power splitting ratio grow exponentially with this product, whereas the intracavity peak power and the Q-factor of the cavity decay exponentially with it. It is also shown that a nonvanishing sine of the phase shift leads to a linear change of the detuning, a quadratic increase of the threshold gain, and a quadratic decrease of the output power splitting ratio, intracavity peak power, and Q-factor. Our results are independent of the source of the gain, such as semiconductor, rare-earth, and Raman fiber lasers, provided that the gain is approximately constant along the grating length and that reflections from facets can be neglected. As an example, we simulate a Raman DFB laser and demonstrate that our closed-form approximations are in perfect agreement with steady state results from involved time-domain simulations.