Matched Solitary Waves

Abstract

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 Ω_j = 2d_j E _j/ħ, j=1,2, where d_j 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 = (A_l ²+B₁ ²+A₂ ²+B₂ ²)/4, b = (A₂ ²+B₂ ²−A_l 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 μ_j=πN d_jω_j/ћc, N is the density of the atoms in the medium, ω_j is the laser frequency, and it is assumed that μ₁=μ₂. Notice that f₁ and f₂ satisfy f ²₁ + f ²₂ = 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.