(c) Dr Paul Kinsler. [Acknowledgements & Feedback]
This work was done in conjunction with Prof. GHC New along with JCA. Tyrrell and SB. Radnor in the Department of Physics at Imperial College.
Optical pulses are useful because they travel in a specific direction, either in free space or in a waveguide or optical fibre. Since Maxwell's equations give the electromagnetic field field evolution in all directions simultaneously, it is usually inefficient to solve them when we are only interested in how a pulse travels in its chosen direction from its starting point. Because of this, we look to simplify the description, and save effort (either theoretical or computational) by rewriting or approximating Maxwell's equations.
I started working on this in 2001/2, after Geoff New brought the 1970 paper by Fleck to my attention. The (Fleck) wave equations had features similar to those present in few-cycle corrections to envelope propagation, which is the topic we were then most interested in. After some random theoretical experimentation, the basic ideas came together clearly, resulting in the 2005 conference presentations and the PRA paper below.
In progress/ under review
The earliest rewriting of Maxwell's equations in a in a directional form was by Fleck PRB1, 84 (1970), who treated a dispersionless medium and plane polarized wave. Unfortunately, the idea was not used (at that time) beyond its appearance in that paper, and played only a minor role even in that. Fleck constructed his directional fields by combining the sum and difference of the E and H fields, weighted by the square roots of the permittivity and permability respectively. The result is a simple first-order wave equation for the forward (or backward) propagation of these ``Fleck'' field variables.
Most work on modelling optical pulses tend to use some kind of directionality assumption, with the exception of those using FDTD in a full solution of Maxwell's equations. For the ubiquitous envelope methods based on the standard second-order wave equation, the direction is imposed by the form for the carrier function, which is usually a plane wave travelling in the chosen direction.
Most quantization schemes also impose a directionality assumption, at least in the case of free fields. They do this by chosing mode functions for the vector potential which are directional, and are often the same planes waves as used by envelope methods. An interesting case for directional waves in unstable resonators was quantized by Brown and Dalton in 2002.
Approaches like that of Fleck, where the full field is explicitly decomposed into forward and backward travelling parts, was not revisited until the work of Kolesik et.al. in PRL89, 283902 (2002), followed up by a more comprehensive PRE70, 036604 (2004). The 2002 letter introduces the idea of a projection operator, which was not fully developed (in the letter), and did not appear in their later paper. Their idea is to chose an appropriate direction, then separate out the forward-like and backward-like parts of the propagating fields, resulting in first order wave equations for the propagation of the forward and backward field components. The long 2004 paper applies their theory to a variety of different examples.
Prior to Kolesik et.al, Casperson, PRA44, 3291 (1991) had constructed a forward-backward split up of the field components. He used envelopes and carrier exponentials, but in his sum-and-difference of normalised E and H fields, you can see some similarity with the Fleck style approach later comprehensively generalised by Kinsler et.al.
Appearing in early 2005, Ferrando et.al. PRA71, 016601 (2005) contained another idea for constructing directional wave equations, this time by factoring the standard second order wave equation into two first order equations. Because of its starting point, it is slightly more restricted than those starting from Maxwell's equations, although the differences in practise will likely be small.
In December 2005 Mizuta et.al. PRA72, 063802 (2005) was published. They had rediscovered the idea of Fleck (they cite neither Fleck, Casperson, or Kolesik et.al.). We can see that the mechanism for generating directional fields is the same, however, since inspection of their equation (11a,b) shows that they combine the sum (or differences) of E and H using the square root of the refractive index. The new feature they add is an averaging procedure for the transverse properties of the field, allowing its application in e.g. optical fibres etc. Like the long Kolesik et.al. paper, they include a number of very comprehensive example applications.
Appearing at the same time as Mizuta et.al., is a paper by Kinsler et.al. PRA72, 063807 (2005). This work follows directly on from the simple construction of Fleck, but generalised it to include the full vectorial and dispersive properties in the construction of the field variables. The result is a set of first-order equations for the directional field variables, which, athough non-trivial, simplify greatly in the usual case(s) of transverse and/or paraxial propagation regimes. They also give an ideal way of constructing the directional fields from a set of E and H fields that have been obtained by other methods -- such as a direct FDTD simulation of Maxwell's equations.
All of these methods amount to the same basic concept -- combining the E and H fields in the right way so as to create a pair of forward and backward-like fields. Fleck introduced the simplest and clearest implementation, but it was too simple to be useful. To my mind, Kolesik et.al. and Mizuta et.al. have less clear formulations, but clearly both approaches work very well as evidenced by the examples in their paper. The Kinsler et.al. construction is the most transparent, although its generality tends to obscure how it might be applied, although the paper does contain simple examples explaining the basics in a way missing from the other two.
Mizuta et.al. use a method they call FDM (Fourier direct method) to propagate pulses which are functions of time forward in space using Maxwell's equations. In this they use the same approach as Tyrrell et.al. in a paper appearing in J. Mod. Opt. earlier in 2005 which was called PSSD (pseudo-spectral spatial domain). However, Tyrrell et.al. made no attempt to separate forward and backward field components.
Date=05052006 1213 20050313 Author=P.Kinsler Created=20050313