The propagation of wave packets in inhomogeneous anisotropic media with moderate absorption

Abstract

Hamilton's equations for geometric optics contain the quantity ∂ω/∂k which is complex in absorbing media. To investigate its physical meaning, the field of a wave packet (a pulsed beam) in a homogeneous medium with moderate absorption is calculated using the saddle-point method. The packet velocity v is approximately equal to the real part of ∂ω <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</inf> /∂k <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</inf> where k <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</inf> , ω <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</inf> are the saddle-point values of the propagation vector and the angular frequency. The imaginary part of ∂ω <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</inf> /∂k <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</inf> can physically be interpreted in terms of v and derivatives of Re k <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</inf> ċ σ, Re ω <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</inf> τ in the direction of v, where σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> and τ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> are beam-width and pulse duration, respectively.