Introduction

We have previously described a method for the analytic solution of the Maxwell-Bloch equations that govern the shape-preserving (self-similar) propagation of coupled pairs of short optical pulses in both lambda and V media, including the influence of inho-mogeneous (e.g., Doppler) broadening [1]. Our self-similar solutions were found to have the typical soliton-like sech shape. A restriction to matched pulses may be considered severe, and the sech shape is very specific, so the generality of the results was not easy to judge.

In this paper we present results under more general conditions. We relax both restrictions and show results for the propagation of non-sech pulses and pulses that are not matched, while restricting attention to V media. We present numerical results in five short movies. The movies demonstrate the anticipated result that unmatched pulses have the ability to match themselves to each other. They also show that the sech solution shape is not mathematically singular but the natural steady-state form for the V-medium pulses. Finally, and perhaps most interesting, we demonstrate the quantitative validity of the two-pulse Area Theorem presented recently [2], and show that it is obeyed even under conditions of pulse breakup.

 

Figure 1. Schematic diagram of a V-type three-level atom and a two-level atom. The pulses Ω_a and Ω_b (driving transitions 1–2 and 2–3 respectively) are in two-photon resonance with intermediate detuning Δ in the V-type medium. The pulse Ω drives the 1–2 transition in the two-level medium.

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A two-level medium and a V-type three-level medium are illustrated in Fig. 1. The electric field vector for the two optical pulses in a V-type medium can be written as:

where k_ac = ω_a and k_bc = ω_b ; and ε_a and ε_b , are the amplitudes of the electric fields of the two pulses. The Rabi frequencies that correspond to these fields are given by: Ω_a (≡ 2d ₁₂ ε_a /ħ) and Ω_b (≡ 2d ₂₃ ε_b /ħ), where d_ij is the dipole moment between levels i and j. In the Rotating Wave Approximation, the evolution equations for the atomic levels with complex amplitudes C ₁, C ₂, and C ₃ can be obtained for V-type media from Schrodinger’s equation as:

where Δ is the detuning shown in Fig. 1. The evolution equations for the fields can be written from the Maxwell equations as:

where g(Δ) is the distribution of detunings resulting from inhomogeneous (e.g. Doppler) broadening. Let $T_2^{*}$ stand for the inhomogeneous broadening lifetime. The standard propagation coefficient is assumed to be equal for all transitions and is given by: μ = 4πd ² Nω/ħc, where N is the density of atoms in the medium. The equations (2–6) are written in local-time coordinates ζ and τ in the frame propagating with velocity c in the medium: cτ ≡ ct - z and ζ ≡ z. The medium is in the ground state so, for all depths ζ of the medium, at the initial time τ = 0 we have,

The propagation equation of an optical pulse and the atomic response equations in a two level medium can be deduced from the V-type medium equations by ignoring level 3 and pulse ‘b’. Namely, the propagation equation of a pulse with Rabi frequency Ω in a two-level medium can be found from equations (2–6) by putting C ₃ = 0 and Ω_b = 0 and assuming Ω ≡ Ω_a. In the following section we quickly review previous studies in two-level media.

Two level review

The propagation problem in a two-level medium has been studied extensively in the past [3]. McCall and Hahn [4] found that a pulse in an inhomogeneously broadened two-level medium obeys an “area theorem” [5]. If Ω is the Rabi frequency of the optical pulse then the area θ is defined by:

The area θ of evolves as function of the propagation distance ζ according to:

where α = μπg(0). Equation (9) is the area theorem. It predicts that pulses whose areas are multiples of 2π (i.e., 0, 2π, 4π, ⋯) are stable. Input pulses of other areas will approach one of these values during propagation. Moreover, we know that in an inhomogeneously broadened two-level medium, a 2π pulse gives rise to a sech-type soliton during propagation [4–7]:

where K is a propagation constant and τ_p is the pulse width. The area theorem and the formation of sech-type solitons in two-level media are illustrated in the following movie linked from Fig. 2.

The movie in Fig. 2 shows the evolution of a pulse as it propagates inside the medium. The pulse at the entry surface of the medium ζ = 0 has a super-Gaussian (“smooth square”) shape and an area of 2.5π. As it propagates inside, the shape changes and it becomes a sech-shaped pulse. Simultaneously the pulse area becomes 2π.

 

Figure 2. (312KB) Snapshot of a frame of a movie showing the changing shape of an initially nearly square pulse as it propagates in a two-level medium. The pulse shape (≡ Ω$T_2^{*}$) is plotted as a function of time T (≡ τ/$T_2^{*}$) and space Z (≡ ${\mu \zeta T}_2^{*}$)-The inset shows the pulse area as a function of space Z.

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Matched pulses in V-type media

Now consider a V-type three-level system and send two pulses (Ω_a and Ω_b,) inside the medium. We look at the propagation of a special kind of input pair called ‘matched’ pulses [8]. Two input pulses are called matched if they have proportional time envelopes when they enter the medium. So at the entry surface ζ = 0 the matched pulses are of the form:

where r is a proportionality constant. By direct substitution, we can show that the following relations are satisfied by Eqns. (2–6) and are consistent with the initial and boundary conditions in Eqns. (7–11):

Using Eqn. (13) we can reduce Eqns. (2–6) and show that they are equivalent to the Maxwell-Schrödinger equations for two-level media [2]. This extends in a significant way an earlier result [9] showing the correspondence of the two-level and matched three-level Schrödinger equations. Thus a V-type medium with two matched pulses becomes equivalent to a two-level medium with a single pulse. This extended correspondence lets us translate the results for propagation from the two-level medium to the V-type medium. For example, the area theorem in Eqn. (9) can be translated to [2]:

where θ_a and θ_b are the areas of pulse a and b respectively. Eqn. (14) shows that, for two matched pulses in V-type media, the quantity $\sqrt{{\theta }_a^2+{\theta }_b^2}$ will become a multiple of 2π during propagation. Also by extending Eqn. (10) we find that pulses in a V-type medium will approach a soliton pair of the following type during propagation:

The movies linked from Fig. 3 and Fig. 4 illustrate the propagation of two matched pulses inside V-type media.

 

Figure 3. (398KB) Snapshot of a frame of a movie showing changing pulse shape during propagation in a V-type medium. The pulse shapes (≡ Ω_a $T_2^{*}$, Ω_b $T_2^{*}$) are plotted as a function of time T (≡ τ/$T_2^{*}$) and space Z (≡ ${\mu \zeta T}_2^{*}$). The inset shows the pulse areas as a function of space Z.

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The movie in Fig. 3 shows two pulses that have super-Gaussian temporal shapes at the entry surface of the medium ζ = 0. As the pulses travel together, they change their shape and become a pair of sech pulses (checked by curve fitting). Also it is interesting to observe the pulse areas as they propagate. At the entry surface the pulses have θ_a = 2.24π and θ_b = 1.12π so that $\sqrt{{\theta }_a^2+{\theta }_b^2}=2.5\pi .$ As the pulses travel inside the medium, we observe that $\sqrt{{\theta }_a^2+{\theta }_b^2}$ approaches 2π.

 

Figure 4. (400KB) Snapshot of a frame of a movie showing changing pulse shape during propagation in a V-type medium. The pulse shapes (≡ Ω_a $T_2^{*}$, Ω_b $T_2^{*}$) are plotted as a function of time T (≡ τ/$T_2^{*}$) and space Z (≡ ${\mu \zeta T}_2^{*}$). The inset shows the pulse areas as a function of space Z.

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The movie in Fig. 4 shows another example of matched pulses in a V-type medium. In this case, the pulses start as super-Gaussians with input areas of θ_a = 3.12π and θ_b = 1.56π so that $\sqrt{{\theta }_a^2+{\theta }_b^2}=3.5\pi .$ As the pulses travel together, they change their shape and break up into two pairs of sech pulses (checked by curve fitting). We also observe that $\sqrt{{\theta }_a^2+{\theta }_b^2}$ approaches 4π during propagation, which is in accordance with Eqn. (14).

In both of the simulations of Fig. 3 and Fig. 4, if we measure the amplitude of each output sech pulse and denote the amplitudes of pulse a and b by A and B respectively then we observe that they satisfy an amplitude relation given by:

where τ_p is the pulse width. This is consistent with the predictions in Eqn. (15–16), and reproduces a result first found for simulton propagation [10].

Arbitrary pulses in V-type media

So far we have discussed the special case of matched pulses in V-type media and showed the correspondence with two-level media. Now we will show the propagation of arbitrary pulses in V-type media. Fig. 5 is linked with a movie that shows the propagation of two unmatched pulses. The input pulses have Gaussian temporal shape. They start at different times and overlap for some time but they have their peaks at different times. As they propagate inside the medium, the pulses break into three secft-shaped pulse pairs. The inset of the movie shows that the pulse areas θ_a , θ_b do not follow the three-level area theorem as in Fig. 3 or Fig. 4 because the quantity $\sqrt{{\theta }_a^2+{\theta }_b^2}$ does not approach a multiple of 2π. Nevertheless, if we measure the amplitudes of each sech pulse pair we find that Eqn. (17) is satisfied by all the pairs separately.

 

Figure 5. (522KB) Snapshot of a frame of a movie showing changing pulse shape during propagation in a V-type medium. The pulse shapes (≡ Ω_a $T_2^{*}$, Ω_b $T_2^{*}$) are plotted as a function of time T (≡ τ/$T_2^{*}$) and space Z (≡ ${\mu \zeta T}_2^{*}$). The inset shows the pulse areas as a function of space Z.

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A last example of two-pulse propagation in a V-type medium is shown in the movie linked from Fig. 6. We start with two sech-shaped pulses that do not overlap in time. These pulses have 2π area but different widths. As the pulses propagate in the medium, the faster pulse overtakes the slower pulse and the slower pulse gets a 180° phase shift.

 

Figure 6. (462KB) Snapshot of a frame of a movie showing changing pulse shape during a pulse collision in a V-type medium. The pulse shapes (≡ Ω_a $T_2^{*}$, Ω_b $T_2^{*}$) are plotted as a function of time T (≡ τ/$T_2^{*}$) and space Z (≡ ${\mu \zeta T}_2^{*}$). The inset shows the pulse areas as a function of space Z.

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Summary

We have shown the results of numerical experiments on propagation of different types of input pulse inside inhomogeneously broadened V-type media. When the input pulses are matched, the propagation dynamics of V-type three-level media is equivalent to that of two-level media. We have also shown that the input pulses become sech shaped pulse pairs during propagation, and obey a new area theorem for pulses in three-level media. When the pulses are not matched, the area theorem is not obeyed globally, but is still obeyed by the individual pairs of output pulses, which are still sech shaped and still obey the amplitude relation A ² + B ² = 4/${\tau }_p^2$. Finally, we have shown a collision of two sech shaped pulses in a V-type medium.

This research was partially supported by NSF through grants PHY94-15583 and PHY97-22024.