Computational Nanophotonics

$N_m\frac{d{B}_m}{dz}=+j{\displaystyle \sum _{n=1}{A}_n{\displaystyle \sum _{l=-\infty }^{+\infty }{C}_{mn}^{\left(l\right)}\exp \left[-j\left({\text{γ}}_n{\text{+γ}}_m-l\frac{2\text{π}}{\Lambda }\right)z\right]}}$

(8.56)

and for the co-directional propagation waves, we have

$N_m\frac{d{A}_m}{dz}=-j{\displaystyle \sum _{n=1}{A}_n{\displaystyle \sum _{l=-\infty }^{+\infty }{D}_{mn}^{\left(l\right)}\exp \left[-j\left({\text{γ}}_n-{\text{γ}}_m-l\frac{2\text{π}}{\Lambda }\right)z\right]}}$

(8.57)

$N_m\frac{d{B}_m}{dz}=+j{\displaystyle \sum _{n=1}{B}_n{\displaystyle \sum _{l=-\infty }^{+\infty }{D}_{mn}^{\left(l\right)}\exp \left[+j\left({\text{γ}}_n-{\text{γ}}_m+l\frac{2\text{π}}{\Lambda }\right)z\right]}}$

(8.58)

Further, for a given grating, the phase-matching condition is a function of wavelength through wavelength dependence of the mode propagation constants, that is, γ_n(λ) and γ_m(λ). It also depends on the mode index (m,n = 1,2,…) and the order of space harmonics in the Fourier expansion. Any combination of these parameters that lead to a phase matching condition will likely yield a distinct resonant signature in the mode coupling as illustrated later in the transmission and reflection spectra in this chapter. ...