document: We describe three ways to solve the reflection problem. The first way is very simple (Section 4). We exploit the consequences of shifting the semi-infinite row by one period (to the right or left). In effect, we regard the semi-infinite row as two scatterers, one of which is another semi-infinite row. This idea goes back to a series of papers by Millar in the 1960s, starting with [2]. He used it for several two-dimensional grating problems. A similar approach was used for layered media by Shenderov [3]. In our one-dimensional context, we obtain a quadratic equation for R; we show how to select the correct solution. We remark that there has been much recent interest in related two-dimensional waveguide problems; see, for example, [4–6], where the shifting-by-one-period idea is again employed, leading to a quadratic equation for a certain operator.
doc_bio_tags: O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O
extractive_keyphrases: layered media obtain a quadratic equation quadratic equation reflection problem two scatterers waveguide problems
abstractive_keyphrases: semi-infinite row shifting-by-one-period idea shifting the semi-infinite row two-dimensional grating problems two-dimensional waveguide problems