The generation of 0π pulses in coherent pulse propagation is studied for a simple initial pulse profile consisting of two separated square pulses with one inverted to give a total pulse area of zero. When the two pulses are close together, it is found that the threshold for lossless propagation occurs at an absolute area of 21/2π. But as the pulses become farther separated, one finds two thresholds. The first one occurs for an absolute area of about π and the second one at about 2π. Although one can have lossless propagation occurring for the absolute area between π and 2π, it is accomplished only by generating very low amplitude 0π pulses. It is only after the second threshold, 2π, is achieved that one can expect to find moderate to large amplitude 0π pulses, and these may be accompanied by several low amplitude 0π pulses.
In coherent optics the pulse area θ is defined as the total angle the atomic state vector rotates around the electric field of the resonant driving pulse. The atomic excitation remaining after passage of the pulse is determined by this angle. For example, a π/2 pulse will put the atoms in a coherent superposition of the ground and excited states, a π pulse will put the atoms in the excited state, and a 2π pulse will take the atoms to the excited state and then back to the ground state. Another type of pulse which has generated a considerable amount of theoretical interest1 but for which there has been only a few experimental investigations2 is the 0π pulse. For this pulse, the initial part of the pulse excites the atoms and then due to a phase change of π in the electric field, the latter part of the pulse takes the atoms back to the ground state. Thus, just as in self-induced transparency, the atoms are completely in the ground state after passage of the pulse. There are two ways to produce such a 0π pulse. The first is to switch the phase of the second half of the pulse by electro-optic techniques. For this method it is difficult to get the 0π pulse area to the desired accuracy. The second method is to rely on the automatic reshaping to the 0π pulse which occurs when a weak (θ(t) ≪ 1) pulse is propagated through a resonant vapor. This method will produce the 0π pulse area to any desired degree of accuracy. This fact is due to an area theorem for these weak pulses which states that the pulse area decays exponentially with the on-resonance absorption coefficient.1
The inverse scattering technique is used to show that the phase term in the Maxwell-Bloch equations should be set to zero when the slowly varying envelope approximation is assumed for self-induced transparency. A solvable model of two square pulses shifted and inverted with respect to each other is considered, and the threshold for "breather" generation for this profile is shown to correspond to an initial absolute area of $\ensuremath{\surd}2\ensuremath{\pi}$ when the separation between the two pulses is zero, and approaches $\ensuremath{\pi}$ as the separation is increased. When the two pulses are identical, only breathers are found. Upper bounds on the energy of the $0\ensuremath{\pi}$ pulses are derived, along with threshold conditions for soliton generation. For an absolute area less than $\ensuremath{\pi}$ there are no bound solutions.
The dynamics of coherent propagation of a unipolar subcycle pulse of a large electric-field area (the integral of the electric-field strength with respect to time) in a nonlinear two-level resonant medium is studied theoretically. The propagating pulse is shown to be capable of splitting into a pair of individual components, each of which behaves like a pulse of self-induced transparency. This phenomenon is similar to the well-known phenomenon of splitting of a 4π or 6π pulse into a pair of 2π pulses of self-induced transparency, which occurs in the case of propagation of long pulses, when the notion of the pulse area is applicable and the area theorem holds.
For pt.I see ibid., vol.8, no.3, 339 (1975). The transient behaviour of two short optical pulses in two-photon coherent propagation is obtained by calculating numerically the coherent propagation equations which couple the two electromagnetic pulses and the atomic system in self-consistent fashion. The coherent interaction of the two different frequency pulses gives rise to lossless propagation in the absorptive medium. This description is characterized by a two-photon area theorem. Novel coherent propagation effects of peak amplification and multiple pulse breakup are found. These effects are ascribed to two-photon self-induced transparency analogous to that of a one-photon transition in a two-level system. A difference of input intensity between the two different frequency pulses yields curious modulation effects involving sharp peak amplification and marked pulse breakup for the low-intensity input. In contrast, the high-intensity input pulse displays a smooth variation with distance and time. Low-loss pulse transmission above threshold, and pulse advance and delay are found for the present case in contrast to that for one-photon self-induced transparency.
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
The spectrum evolution of a few-cycle optical pulse in a resonant two-level atom medium is studied theoretically by using the full Maxwell-Bloch equations. On the propagating pulse, significantly much faster oscillation components separated with the main pulse appear due to strong self-phase modulation and pulse reshaping. In this case, ideal self-induced transparency cannot occur for a 2π pulse. The spectrum of the 4π pulse shows an evident oscillatory feature because of the continuum interference of the separate pulses. For larger pulse areas, continuum generation from near ultraviolet to infrared occurs.