Light propagation in a planar dielectric slab waveguide with step discontinuities

Extremely asymmetrical scattering in gratings with varying mean structural parameters

“…This means that boundary conditions (8) (except for the first one) have to be modified in order to include these bulk and guided waves. This can again be done in the general case in the same way as in the mode matching theory [22][23][24][25][26]. However, manufacturing gratings with small amplitude usually results in small variations of the mean thickness of the slab.…”

“…Using speculations similar to those in section 2, one comes to the conclusion that equations (4) also describe EAS of guided modes in gratings with varying mean thickness at the grating boundaries. However, the coupling coefficients Γ 0 and Γ 1 are obviously different from those for bulk TE waves and are determined in the approximate theories of conventional Bragg scattering in corrugation gratings with grooves parallel to the grating boundaries [22][23][24][25]. The wave numbers for the incident and scattered modes (e.g.…”

“…For example, interaction of the incident mode with a step-like variation of h at the front boundary results in generation of other guided modes in the slab and bulk electromagnetic waves outside the slab [22][23][24][25][26]. In the approximate theory, however, all these waves do not contribute to scattering in the grating, except for one guided mode for which the Bragg condition is satisfied.…”

“…Therefore, the situation is similar to the approximate theory of EAS of bulk electromagnetic waves in section 2. For example, from consideration of the interaction of the incident mode with the front boundary (by means of the mode matching theory [22][23][24][25][26]), one can determine the amplitudes and the directions of propagation of the resulting bulk and guided waves, and then regard the guided mode with the amplitude S 200 , the angle of propagation θ 20 , and the wave vector k 20 that satisfies the Bragg condition as the incident wave at x = 0 in the grating (inside the grating, the amplitude of this wave is S 20 (x)).…”

“…Similarly to bulk electromagnetic waves experiencing reflection and refraction at the grating boundaries with non-zero ∆ǫ 1,3 , guided incident and scattered modes also interact with the step-like variations of the mean thickness of the slab. For example, interaction of the incident mode with a step-like variation of h at the front boundary results in generation of other guided modes in the slab and bulk electromagnetic waves outside the slab [22][23][24][25][26]. In the approximate theory, however, all these waves do not contribute to scattering in the grating, except for one guided mode for which the Bragg condition is satisfied.…”