Theory of phase and amplitude bunching in traveling wave interactions

Abstract

A solution is given for the parametric interaction of two traveling waves ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\upsilon_{k, k}</tex> ) and ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega, k</tex> ) where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(\upsilon_{k, k}) \ll (\omega, K)</tex> . The strong pump wave at frequency υ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> is not affected but modulates the index of refraction for the weak wave at ω. The solution is for an arbitrary initial time dependence of the weak wave and can describe a large change in the fractional bandwidth of the modulated wave. If the phase velocities υ and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega/K</tex> are equal, the cycles of the high-frequency wave bunch together in every other half-cycle of the modulation wave and draw apart in the alternate half-cycles. This strong phase bunching is accompanied by strong amplitude modulation that is found to have the same <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> dependence as the instantaneous frequency.