In this paper, a rigorous leaky mode propagation method has been used to investigate the influence of the grating period and grating index profile on the design of grating-assisted directional couplers (GADC's). A detailed explanation of resonance condition and radiation loss in terms of electromagnetic field contribution in the grating region as a function of the grating period and profile is given. Optimal design parameters have been found for well-defined structures in order to achieve either minimum coupling length or maximum coupling efficiency. A very fast method to extract the resonance condition in any grating-assisted structure by using a sinusoidal profile is proposed. Numerical results are presented for both moderately and strongly asymmetric structures in terms of normalized propagation constant, mode radiation loss, coupling length and coupling efficiency. Comparisons with grating period and coupling length predictions obtained by other methods are also shown. The rectangular profile with optimized duty cycle has been demonstrated to be the best choice in order to minimize the GADC coupling length.
Vittorio M. N. Passaro, "Optimal Design of Grating-Assisted Directional Couplers," J. Lightwave Technol. 18, 973- (2000)