The initial value problem for the propagation of a pulse through a resonant two-level optical medium is solved by the inverse scattering method. In general, an incident pulse decomposes not only into a special class of pulses to which the medium is transparent but also yields radiation which is absorbed by the medium. In this respect ``this problem'' has properties markedly different from other dispersive and reversible wave phenomena some of which are tractable by the inverse scattering method. Indeed, it is remarkable that in the present case the method still applies. In particular, we show that, while there are an infinite number of local conservation laws, the integrated densities, and in particular the energy, are only conserved for a very special class of initial conditions. The theoretical results obtained are in close agreement with all the qualitative features observed in the experiments on coherent pulse propagation. Finally, we also show that causality is preserved. Two new and novel features are introduced and briefly discussed. First, we show that if the homogeneous broadening effect is a function of position in the medium, the pulses may speed up and slow down accordingly, without losing their permanent identities. Second, we have found a new kind of solution mode corresponding to a proper eigenvalue of the scattering problem which is not a bound state.
The inverse scattering method is used to investigate the resonant propagation of an optical pulse assumed to be initially unchirped and to have a hyperbolic-secant shape. An exact solution is found for the two-component initial eigenvalue problem. The discrete spectrum consists of a finite set of purely imaginary, equidistant values. The number of discrete eigenvalues (number of solitons) is determined by the initial pulse area
A. The scattering amplitudes are expressed in terms of Euler gamma functions and hyperbolic functions. In addition to poles in the upper half-plane, the function
ab has a pole on the real axis for $A=(2n+1)\ensuremath{\pi}$.
We shall consider two resonant optical models. One is the reflection and transmission of ultrafast pulses on the boundary of a two-level medium. The other model is an ensemble of Rydberg atoms in a resonant cavity, where an external pulse is also coupled into the cavity to initiate the radiation. Both problems can be modelled by a nonlinear density matrix-type equation, where the effective Hamiltonian itself depends on the state of the atomic system. We present a new version of the inverse scattering method, which allows one to find soliton-like analytical solutions for the problems considered. Our work enlarges the class of optical problems having soliton type solutions.
A simple model for optical pulse propagation in a nondegenerate two-level amplifying medium is considered, under the assumption of extreme Doppler broadening. Starting with a "Pade approximant" left reflection coefficient with N simple poles and N$\sb{\rm b}\leq$ N bound states, and employing Lamb's one-component inverse scattering method*, pulses whose initial area can be
>π are obtained. Then, employing the asymptotic behavior of the eigenvalues of the kernel of the right Marchenko equation, the asymptotic behavior of the pulses far into the medium is analyzed in detail when N
≤ 3. In addition to the expected portion of area
π near the light cone which undergoes amplification and compression, pulses with a continuous leading edge develop forward-moving oscillations, and some pulses trail behind one or more 2
π (or 0
π) solitons. Pulses whose final area is
π, 3
π, and 5
π are obtained. Interestingly, the number of trailed solitons is in general not equal to the number of bound states. These solitons are associated with a subset of the zeroes of the transmission coefficient rather than of its poles. Conditions for the appearance of a soliton are given in terms of the poles and residues of the left reflection coefficient. A connection is established between the values of the pulse profiles and their first 2N
− 1 derivatives at the light cone, and the residue and pole parameters of the left reflection coefficient. For N = 2 and the case in N = 3 where the pulse has a continuous leading edge, simple conditions for the appearance of a soliton are obtained in terms of the values of the pulse profiles and their first N
− 1 derivatives at the light cone, and the poles of the left reflection coefficient. It is established for N
≤ 3 that for each 2
π soliton there is a purely imaginary zero of the left reflection coefficient
Quantum coherence and interference in atomic systems can lead to interesting optical phenomena such as nonabsorbing resonances, lasing without inversion, generating an ultrahigh index of refraction without absorption, propagation of light pulses without losses. In the last-named is given the title transparency. Up to now it is known some types of transparency in multi-level resonant media. One of them is electromagnetically induced transparency (EIT). The phenomenon of EIT in the propagation of laser radiation through an absorbing medium has been recently investigated and explained in terms of quantum coherence and interference for multi-level atomic system. For instance in the (Lambda) -system, if the atomic medium is prepared in a quantum superposition of states, matched pulses, i.e., a pair of optical pulses whose amplitude and phase have a well defined relation, propagate without absorption. The mechanism of coherent population trapping is responsible for EIT in a (Lambda) -system. There is another, well known mechanism for creating transparency in a two-state atom: self-induced transparency, or SIT. Here, a single pulse whose area (the integral of its Rabi frequency over time) is 2(pi) is applied to the medium. Such a pulse causes the atoms to cycle smoothly from the ground to the excited state and then back again. In principle, such a pulse can propagate over long distances. SIT can be generalized to multi-level systems and multiple quasi-monochromatic fields. Generalization of SIT in a (Lambda) -system was obtained in reference 2. In this report we present a new type of transparency (combined transparency). This type describes propagation of laser pulse through a (Lambda) -type medium where the initial atomic state is prepared in the form of spatially varying coherent superposition of low-energetic states (initial coherency gradient). The common envelope behaves as usual 2(pi) hyperbolic secant pulse, but the ratio between the components of frequencies transitions 1> yields 3> and 2> yields 3> changes due to coherent Raman scattering so (phi) equals atan(E1/E2) equals (pi) /2-Gz, where z is the coordinate, G is the constant determined by both the incident pulse parameters and the initial coherence between the atomic states. Our solution demonstrates a direct connection of EIT in adiabatic regime to the SIT effects. This type of a transparency is given the title screw-type transparency, because for the given form of incident pulse (pitch of screw) medium is transparent, if medium's the initial coherence gradient is correspondingly prepared (thread on a engine nut). Besides a transparency, this phenomenon contains a transformation of the incident pulse frequency during the process of the propagation. This property of the phenomena can be result in a complete conversion of the frequency.
The conservation laws for coherent resonant propagation are studied, and an interpretation of these laws as conservation of pulse spectral moments is proposed. The nonlinear spectral density is expressed in terms of the scattering data for an initially hyperbolic secant pulse. In this way spectral moments are exactly determined at any distance and are used for a partial description of the propagating pulse. The conservation laws for the three lowest moments are investigated in detail. The evolution of the zero-order moment leads to an explicit formula for the attenuation of the electromagnetic energy. The rate at which the energy is absorbed depends critically on the pulse area. There is an essential difference between propagation of pulse with area smaller than $\ensuremath{\pi}$ and those with area larger than $\ensuremath{\pi}$. The evolution of the first moment corresponds to change of the pulse average frequency. The calculated shift of the average frequency due to detuning is compared with the experimental result of Slusher and Gibbs. The second-order moment is a measure of the pulse spectral width. The evolution of the Fourier second-order moment is used to explain the observed spectral narrowing of a $\ensuremath{\pi}$ pulse and broadening of a $3\ensuremath{\pi}$ pulse.
The propagation behavior of nontrivial $0\ensuremath{\pi}$ pulses [$\ensuremath{\int}{\ensuremath{-}\ensuremath{\infty}}^{\ensuremath{\infty}}\mathcal{E}(t,z)dt=0$, $\ensuremath{\int}{\ensuremath{-}\ensuremath{\infty}}^{\ensuremath{\infty}}{\mathcal{E}}^{2}(t,z)\mathrm{dt}>0$] in attenuating media is examined. It is found that under appropriate conditions $0\ensuremath{\pi}$ pulses exhibit anomalously low absorption which is a direct consequency of atomic coherence. Specific analytical results valid for an unbroadened medium are presented, while the corresponding solutions for an inhomogeneously broadened resonance are calculated numerically. These two results are consistent in the limit of ${T}_{2}^{*}\ensuremath{\rightarrow}\ensuremath{\infty}$. Furthermore, the influences of
T2,
T2∗, and level degeneracy on the characteristics of propagation are determined. Finally, the behavior of more general $0\ensuremath{\pi}$ pulses is studied. The criteria for the production of $0\ensuremath{\pi}$ pulses are qualitatively established by numerical methods, and the influence of the relative phase of pulse pairs on propagation is examined. The latter indicates a method for the measurement of relative pulse phase.