Hydrodynamic and Optical Waves: A Common Approach for Unidimensional Propagation

Abstract

The aim of this chapter is to build a bridge between water and optical waves. After a brief introduction on the role played by the so-called normal variable in the D’Alembert equation and a short description of the Hamiltonian formulation of water waves, we introduce a similar formalism for describing optical waves. We restrict our analysis to one-dimensional propagation. Under a number of assumptions, we rewrite the Maxwell equations in a very general form that account for three- and four-wave interactions. Those equations are very similar to the one describing water waves. Analogies and differences between hydrodynamic and optical waves are also discussed.

Appendix

Coupling coefficient of the Zakharov equation (70) in optical waves in the presence of χ ² and χ ³ media:

$$\displaystyle{ \tilde{T}_{1,2,3,4} = T_{1,2,3,4} + W_{1,2,3,4}, }$$

(77)

with

$$\displaystyle{ \begin{array}{rl} W_{1,2,3,4} & = -V _{1,3,1-3}V _{4,2,4-2}\left [ \frac{1} {k_{3}+k_{1-3}-k_{1}} + \frac{1} {k_{2}+k_{4-2}-k_{4}} \right ] \\ &\quad - V _{2,3,2-3}V _{4,1,4-1}\left [ \frac{1} {k_{3}+k_{2-3}-k_{2}} + \frac{1} {k_{1}+k_{4-1}-k_{4}} \right ] \\ &\quad - V _{1,4,1-4}V _{3,2,3-2}\left [ \frac{1} {k_{4}+k_{1-4}-k_{1}} + \frac{1} {k_{2}+k_{3-2}-k_{3}} \right ] \\ &\quad - V _{2,4,2-4}V _{3,1,3-1}\left [ \frac{1} {k_{4}+k_{2-4}-k_{2}} + \frac{1} {k_{1}+k_{3-1}-k_{3}} \right ] \\ &\quad - V _{1+2,1,2}V _{3+4,3,4}\left [ \frac{1} {k_{1+2}-k_{1}-k_{2}} + \frac{1} {k_{3+4}-k_{3}-k_{4}} \right ] \\ &\quad - V _{-1-2,1,2}V _{-3-4,3,4}\left [ \frac{1} {k_{1+2}+k_{1}+k_{2}} + \frac{1} {k_{3+4}+k_{3}+k_{4}} \right ]. \end{array} }$$

(78)