By A. Schuster, J. Nicholson
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Additional info for An Introduction to the Theory of Optics
In principle, a single beam can be represented by a superposition of several plane waves (spatial frequencies), with wave vectors k of equal magnitude but slightly diﬀerent directions with respect to the optical axis. Since each plane-wave component can be characterized by a diﬀerent projection of its wave vector k on the optical axis, each component propagates at a diﬀerent phase velocity with respect to that axis. In this way, the component that propagates “on” the optical axis propagates faster than a component i that propagates at some angle Θi , whose propagation constant is proportional to cos (Θi ).
Since we deal with self-focused light structures in saturable photorefractive nonlinear media in Chap. 4, we also make use of the term “soliton” in this broader sense. However, the reader should keep in mind the essential diﬀerence between solitons in integrable and nonintegrable systems. Even though a saturable medium is not integrable, it remains conservative in many respects. 30)) can still be deﬁned and exploited for the purpose of ﬁnding soliton solutions and their subsequent characterization.
Since each plane-wave component can be characterized by a diﬀerent projection of its wave vector k on the optical axis, each component propagates at a diﬀerent phase velocity with respect to that axis. In this way, the component that propagates “on” the optical axis propagates faster than a component i that propagates at some angle Θi , whose propagation constant is proportional to cos (Θi ). The narrowest width of the spatial beam is obtained at a particular plane in space at which all components are in phase.