Matched Solitary Waves
In a recent papers1, we presented exact analytic solutions for a pair of solitary waves which can propagate through a medium of three-level and a medium of five-level atoms without loss and with their shapes invariant. The solitary-wave pair or matched pulses for the three-level A system is of particular interest. Denote the Rabi frequencies of the two pulses by Ωj = 2dj E j/ħ, j=1,2, where dj is the dipole matrix element between levels j and j+1, and Ej is the slowly varying electromagnetic field amplitude. The Rabi frequencies are functions of both space z and time t. For the solitary pulses, the Rabi frequencies depend on t and z through ξ= (t - z/v)/τ, where r is the pulse length and v is the velocity of the pulse pair. Our analytic solutions give Ω1(ξ) = A1f1(ξ) + A2f2(ξ) and Ω2(ξ) = B1f1(ξ) + B2f2(ξ), where A1, A2, B1 and B2 are real constants, and where f1 and f2 are dimensionless functions of ξ. There are two independent sets of solutions for the pair of elliptic functions f1 and f2. The canonical solutions to which other forms of solutions can be transformed are given in Table 1 where a = (Al 2+B1 2+A2 2+B2 2)/4, b = (A2 2+B2 2−Al 2−B1 2)/4 ≥ 0 and A1A2+B1B2=0 The solitary-wave pair has a constant velocity v given by 1/v=1/c+4μ1τ2/R2, where μj=πN djωj/ћc, N is the density of the atoms in the medium, ωj is the laser frequency, and it is assumed that μ1=μ2. Notice that f1 and f2 satisfy f 1 2 + f 2 2 = 1. The constant R, its relation to the squared modulus 0 ≤ k2 ≤ 1 (or the squared complementary modulus k′2=1−k2) of f1 and f2, and one other relationship which must be satisfied by the amplitudes and length of the pulses, are given in the last two rows of Table 1.