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Appendix D
Bullets with two principal channels

In the analysis of central bullets in section 5.1, we have in fact been considering a special case, in which there is a single principal channel which is brighter than all the others. A plane of symmetry exists along this channel, with the channels on one side having the same field profile as the corresponding channel on the other side. However, it is also possible to have this plane of symmetry in the gap between channels. Therefore, there will be two principal channels, having equal brightness.

The type with a single principal channel is known as an odd bullet, and the type with two principal channels is known as an even bullet. By analogy with a similar phenomenon in condensed matter physics, the two types can also be refered to as site centred and bond centred [17, 28].

Even bullet solutions are readily found by using the same Newton-Raphson method described in appendix A.3, but by initialising the calculation with pulses in two neighboring channels. A typical example is given in figure D.1. The energy profiles of even and odd bullets are given in figure D.2. In both cases, 14 channels are used, providing a quasi-infinite regime.


Figure D.1: A bullet solution in a 14 channel array, calculated with q = 5 and c0 = 1.546. The bullet is of the "even" (or "bond centred") type, with two central channels in the middle.


Figure D.2: Total energy content of odd and even bullets (for a quasi-infinite number of channels), as a function of spatial wavenumber q divided by coupling coefficient c0. Plots are given for both odd and even bullet configurations. The unstable regions are denoted by dotted lines, whilst the total cutoff is denoted by the vertical dashed line.

The higher energy of even bullets makes them unstable (as a lower energy state exists). This makes them a much poorer candidate for the experimental observation of optical bullets. Therefore, having noted their existance, we will not consider this type of bullet any further.

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