## Abstract

A practical method of slowing and stopping an incident ultra-short light pulse with a resonantly absorbing Bragg reflector is demonstrated numerically. It is shown that an incident laser pulse with suitable pulse area evolves from a given pulse waveform into a stable, spatially-localized oscillating or standing gap soliton. We show that multiple gap solitons can be simultaneously spatially localized, resulting in efficient optical energy conversion and storage in the resonantly absorbing Bragg structure as atomically coherent states.

©2003 Optical Society of America

Pulse propagation in photonic band gap structures and other Bragg reflectors (BR) has been of great research interest over the past decade [1]. Theoretical studies have found that BR can support a variety of gap solitons [2, 3, 4, 5, 6, 7], self-localized pulses in the BR that can be reflected, transmitted, slowed, or stored in the structure. Bragg grating solitons, pulses spectrally outside the Bragg reflection stopband but near the edge, were predicted to form in BR filled with a Kerr nonlinearity [3]. Experimentally, only Bragg grating solitons in fiber Bragg gratings have been observed [8], as well as gap solitons in Kerr nonlinear dielectric structures [9, 10]. Very intense light pulses (typically 10 *GW/cm*
^{2} or greater) were required to form the solitons. Gap solitons in resonantly absorbing Bragg-periodic reflectors (RABR), that is Bragg-periodic thin layers of resonantly absorbing two-level systems, have been predicted to form by a principally different mechanism at much lower input intensities (10 *MW/cm*
^{2} or less) [2, 4, 6, 11, 12], which is significant for practical applications. RABR have been successfully fabricated with Bragg-periodic semiconductor quantum well structures, and initial experimental studies of their linear and nonlinear properties have been performed [13, 14, 15].

An intriguing possibility for RABR is the storage of optical pulses as zero-velocity gap solitons followed by release from the structure at a controllable delay. Zero-velocity gap solitons were shown to exist in resonantly absorbing Bragg reflectors consisting of nonresonant dielectric layers doped periodically with thin layers of resonant two-level systems [5]. Optical pulse storage and access is also being explored via electromagnetically-induced transparency (EIT) [16, 17]. EIT media can capture optical pulses with at shortest microsecond pulse lengths. In contrast, RABR can store multiple optical pulses, as we show in this work, with temporal widths from 100’s of femtoseconds to picoseconds to nanoseconds depending on the density and *T*
_{1} time of the two-level systems. Methods for releasing the pulse from the structure are discussed.

It is well known that 2*π* pulses can transmit through an otherwise opaque medium without attenuation via self-induced transparency (SIT) [18, 19]. Optical pulses propagating in RABR can also form an effective 2*π* SIT two-wave soliton. The linear spectral response of RABR can be a reflection stopband not unlike a dielectric mirror. Weak optical pulses centered on the reflection stopband are efficiently reflected. However, very interesting nonlinear properties arise because the reflection stopband is formed from the two-level atoms themselves [14, 20, 21]. Optical pulses with peak intensity high enough to induce Rabi flopping of the two-level systems with period significantly shorter than the enhanced radiative response time of the collective structure can penetrate the reflection stopband [2].

Such a two-wave moving soliton solution in a RABR was first analytically derived in Ref. [2]. Later, the same author showed that such structures can support unstable gap solitons and stable oscillating gap solitons [22, 23], i.e, gap solitons that oscillate back and forth spatially in the structure but are essentially trapped by it. Numerically, parameters were found only for excitation of the unstable gap solitons in finite structures by an incident external pulse.

In this work, we demonstrate numerically that excitation of a stable oscillating gap soliton by a single external incident pulse is not only possible but robust with respect to details of the incident pulse shape and area. Moreover, such a trapped gap soliton is shown to be very stable against stochastic perturbation. We further demonstrate that the transverse and longitudinal relaxation times *T*
_{1} and *T*
_{2}, respectively, of the two-level systems do not qualitatively change the stability of the solitons, but do ultimately limit the time the pulse can be stored before dissipating. Gap solitons are also subject to pulse splitting for pulse areas greater than 4*π*. Additionally, we show that multiple optical pulses can be stored as trapped gap solitons in a RABR. Finally, we show that such stored pulses can be pushed out of the structure by collision with a weak excitation pulse at early times. A stored pulse can not be detected at later times by a moving gap soliton.

The theory used here for the interaction of light pulses with periodic thin layers of resonantly absorbing two-level systems is based on the two-wave Maxwell-Bloch equations (TWMB) in the slowly varying envelope approximation of the forward and backward propagating electric fields *E*
^{±}(*x, t*). Note thin layers of two-level atoms means that the thickness of the two-level layer is much smaller than the emission wavelength of the two-level system. With the additional assumption of Bragg spacing of the two-level systems, the TWMB equations can be expressed in terms of real valued functions [2]:

where Ω^{±}(*x, t*)=(2*τ*_{c}*µ/h̅*)*E*
^{±}(*x, t*); *E*
^{±}(*x, t*) are the smooth field-amplitude envelopes of the forward and backward Bloch waves; ${\tau}_{c}^{2}$=8*πT*
_{1}/3*cρλ*
^{2} is the cooperative time; *ρ* is the density of two-level systems; *µ* is the matrix element of the dipole transition moment; *λ* is the wavelength; *P*(*x, t*) and *n*(*x, t*) are the polarization and density of inverse population, respectively; *c* is the speed of light; *t*′ and *x*′ are, respectively, the time and spatial coordinates along the normal to the resonance planes in the structure; and the subscripts *x* and *t* imply partial derivatives. Note that relaxation of the Bloch vector due to transverse and longitudinal relaxation times, *T*
_{1} and *T*
_{2}, is neglected at this point in the equations because *T*
_{1} and *T*
_{2} are assumed to be much larger than either *τ*_{c}
or the pulse duration *τ*
_{0} as required for a SIT condition. The coupled equations are solved with a finite-difference time-domain method. Dimensionless space and time variables *x*=*x*′/*cτ*_{c}
and *t*=*t*′/*τ*_{c}
are used.

Note the extremely compact notation of the theory. The work presented here is in scaled units of the cooperative time *τ*_{c}
, i.e., *τ*_{c}
=1. Specific material parameters such as density and oscillator strength need not be specified, because they are contained in *τ*_{c}
. *τ*_{c}
is the absorption time of the photon in the structure, and is thus closely related to the inverse Einstein B coefficient. The structure length is in scaled units of *cτ*_{c}
also. Specific material realizations that specify parameters *τ*_{c}
and structure length are planned as the subject of future work.

To search for conditions that lead to the formation of a stable, oscillating, stored gap soliton, we began with initial conditions for the material and the external, incident optical pulse of the form:

$${\Omega}^{\pm}(x,t=0)=0,\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}P(x,t=0)=0,\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}n(x,t=0)=-1.$$

We initially assumed a sech shape of the electric field, and varied its temporal duration *τ*
_{0} and area *θ*=${\int}_{-\infty}^{+\infty}$${\mathrm{\Omega}}_{0}^{+}$(*t*)*dt*.

The general characteristics for the nonlinear interaction dynamics for varying pulse amplitude ${\mathrm{\Omega}}_{0}^{+}$ are shown in Figs.1(a–d) with standard sech incident pulses ${\mathrm{\Omega}}_{0}^{+}$(*t*)=${\mathrm{\Omega}}_{0}^{+}$
*sech*((*t*-*t*
_{0})/*τ*
_{0}). Parameters kept constant for all simulations in Fig. 1 were the total length *l*=40 (in units of *cτ*_{c}
) and pulse duration *τ*
_{0}=0.5 (in units of *τ*_{c}
). Figure 1 shows that the response to the incident pulse ranges from non-delayed full linear Bragg reflections to nonlinear soliton splitting when the pulse amplitude ${\mathrm{\Omega}}_{0}^{+}$, and hence the pulse area, increases gradually.

The most striking result is shown in Fig. 1(b), where a stable gap soliton decelerates gradually from a moving to an oscillating then to an almost standing one. This is similar to what occurs in the case of Kerr and quadratic nonlinear photonic bandgap structures [7], where the transition can be either abrupt or asymptotical. As described in Ref. [22], the nonlinear interaction of a pulse with the RABR can be understood as motion of a quasiparticle in a potential. The incident pulse corresponds to the quasiparticle with its initial velocity proportional to the pulse intensity, and the potential is created by the pulse itself and the matter excitation, set-up also by the light pulse. The potential can either be repulsive or attractive. For a repulsive potential [23], the particle enters into a nonnegative potential and gradually decelerates. If the particle enters with exactly the right velocity, it asymtotically approaches an *unstable* equilibrium (standstill), resulting in a delayed nonlinear reflection or transmission of the pulse. Alternatively, the particle can enter into an attractive potential. For suitable initial velocity, the particle will be trapped by the potential, resulting in *stable* backward and forward oscillation of the particle in the potential as in Fig. 1(b). If the particle’s initial velocity is too low, the particle rebounds, as in Fig. 1(a). If the particle’s initial velocity is too high, the particle moves through the structures as in Fig. 1(c). Finally, in Fig. 1(d), a particle with an area of multiple *π* (4*π* in Fig. 1(d)) splits in a way analogous to the splitting that occurs in the SIT of a uniform medium. Localization as in Fig. 1(b) only occurs for particles with an intermediate initial velocity. For the above parameters of length and pulse width, the steady localization results for ${\mathrm{\Omega}}_{0}^{+}$=3.57 to 3.63, each value resulting in standing solitons at different positions in the structure.

Further insight can be gained into the trapped gap soliton in Fig. 1(b) through comparison of the distributions Ω^{±}(x), *P*(*x*), and *n*(*x*) at the initial trapping time. Figure 2 shows these distributions for the same initial conditions as in Fig. 1 at the time of *t*=90, when the oscillating soliton begins to form. Note the intensity of the backward wave already has increased almost to that of the forward one, which is induced from the Bragg reflection by the lattice. The sum of forward and backward waves, in fact, nearly vanishes during the localization process, and the total energy is stored in the structure almost completely in coherent two-level states. This means that the storage time is ultimately limited by the dephasing time *T*
_{2} of the two level systems.

The decelerating soliton is found to be fairly stable against stochastic perturbation. Random intensities added to the incident pulse profile of Fig. 1(b) ${\Omega \prime}_{0}^{+}$(*t*)=${\mathrm{\Omega}}_{0}^{+}$(*t*)(1+0.3(rand(*t*)-0.5)) do not qualitatively change the formation of the decelerating soliton, as is shown in Fig. 3(a). Moreover, the self-localizing regime remains almost unchanged, 3.57≤${\mathrm{\Omega}}_{0}^{+}$≤3.63.

Figure 3(b) demonstrates that the effects of relaxations on the decelerating gap solitons also do not qualitatively change the formation of the trapped gap soliton. To include the effects of material relaxation parameters, *T*
_{1} and *T*
_{2}, on the decelerating gap solitons, we follow the model of Ref.[2], i.e., the phenomenological relaxation terms, -${T}_{2}^{-1}$
*P*(*x, t*) and -${T}_{1}^{-1}$(*n*(*x, t*)+1), are added on to the Bloch equations, Eqs. (2) and (3), respectively. Finite relaxation times *T*
_{1} and *T*
_{2} have two main effects. First, the trapped gap soliton forms only with a greater intial incident intensity, which is necessary to overcome the energy lost to the relaxation process. Second, for times longer than *T*
_{2}, the trapped gap soliton begins to dissipate. We further note that introducing finite *T*
_{1} and *T*
_{2} does not affect the stability of the trapped gap soliton. The stable regime is still as wide as Δ${\mathrm{\Omega}}_{0}^{+}$=0.06 with its peak intensity increased to ${\mathrm{\Omega}}_{0}^{+}$=4.00.

We also point out that the existence of two-wave solitons does not depend on the condition of *τ*
_{0}≤*τ*_{c}
[2] in our simulations for the finite structure (*l*=40). In fact, even when *τ*
_{0}=1.5*τ*_{c}
, we still can obtain results analogous to those in Fig. 1 by increasing the peak intensity by a corresponding amount. However, for a pulse with significantly longer temporal width such as *τ*
_{0}=10*τ*_{c}
but the same pulse area, the pulse is efficiently reflected. Reflection occurs because the period of the Rabi oscillation becomes correspondingly slower for temporally long pulses than the enhanced radiative response time of collective structure; hence the pulse can not penetrate the structure. Thus for propagation into the RABR, sufficient pulse area is a necessary but not sufficient condition; the pulse must additionally have high enough peak intensity.

It is interesting that the above results are not restricted to sech-like incident pulses either. Our simulations (not shown here) show that the same behavior as in Fig. 1 results when incident pulse shapes are Gaussian or even rectangular. Only the initial pulse amplitude ${\mathrm{\Omega}}_{0}^{+}$ must be increased. Gap solitons evolve from the given input profile to sech-like. The pulses gradually dissipate their excess energy and transform into eigensolutions.

Finally, we investigated the possibility of releasing stored pulses from the structure via collision with another pulse at a controlled delay. Collision of decelerating solitons depends on two important factors: the delay time between the two input pulses and their intensities. For a short delay, even a weak linear pulse can push out the standing soliton by combining the two into a moving intensity. For sufficiently large time delay, the weak pulse is reflected back as if no standing soliton existed inside the material. On the other hand, two identical pulses with the same intensity excite two standing solitons. The one near the input end pushes the one already inside a little forward toward the output end, but both of them remain inside the structure. Interestingly, a moving soliton collides with the standing one and actually pushes it out of the structure. However, the pushed-out standing soliton emerges with the same characteristics as the triggering pulse, while the triggering soliton stops after the collision. Thus the input and output pulses can not be distinguished whether there is a stored pulse in the structure or not. Hence 1-d collisions are not an effective way to either detect or release stored pules. The latter two results are illustrated in Fig. 4.

In conclusion, we have numerically investigated the nonlinear pulse dynamics within a RABR. The results illustrate the general dynamics under different incident intensities and emphasize that a suitable resonant incident pulse can consistently evolve from a slow moving soliton into an oscillating then standing soliton without any initialized distributions within the structure. The optical pulse is stored in the structure as an atomically coherent state. The stability of the gap soliton strongly depends on the selected structure length. The effects of stochastic perturbation and of *T*
_{1},*T*
_{2} on the existence of decelerating solitons are discussed. Trapped gap solitons are also found to form with a variety of other incident pulse profiles. Interactions of multiple gap solitons are illustrated. Our numerical results show that the basic conclusion regarding the gap soliton does not change with the level shift induced when local field corrections are considered [27].

## Acknowledgments

Support for the project by the National Key Basic Research Special Foundation (NKBRSF) under Grant No. G1999075200 and Chinese National Natural Science Foundation (90201027) is acknowledged.

## References and links

**1. **R. E. Slusher and B. J. Eggleton (editors), *Nonlinear Photonic Crystals* (Springer-Verlag, Berlin, Heidelberg, 2003).

**2. **B. I. Mantsyzov and R. N. Kuz’min, “Coherent interaction of light with a discrete periodic resonant medium,” Sov. Phys. JETP **64**, 37–44 (1986).

**3. **W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. **58**, 160 (1987). [CrossRef] [PubMed]

**4. **A. Kozhekin and G. Kurizki,“Self-induced transparency in Bragg reflectors: gap solitons near absorption resonances,” Phys. Rev. Lett. **74**, 5020 (1995) [CrossRef] [PubMed]

**5. **A. E. Kozhekin and G. Kurizki,“Standing and moving gap solitons in resonantly absorbing gratings,” Phys. Rev. Lett. **81**, 3647 (1998). [CrossRef]

**6. **G. Kurizki, A. E. Kozhekin, T. Opatrny, and B. A. Malomed, “Optical solitons in periodic media with resonant and off-resonant nonlinearities,” Progress in Optics **42**, ed. E. Wolf, 93–140 (2001). [CrossRef]

**7. **C. Conti, G. Assanto, and S. Trillo, “Gap solitons and slow light,” J. Nonlinear Opt. Phys. & Mat. **11**, 239–259 (2002). [CrossRef]

**8. **B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. **76**, 1627 (1996). [CrossRef] [PubMed]

**9. **N. G. R. Broderick, P. Millar, D. J. Richardson, J. S. Aitchson, R. De La Rue, and T. Krauss, “Spectral features associated with nonlinear pulse compression in Bragg gratings,” Opt. Lett. **25**, 740 (2000). [CrossRef]

**10. **N. G. R. Broderick, D. J. Richarson, and M. Ibsen, “Nonlinear switching in a 20-cm-long fiber Bragg grating,” Opt. Lett. **25**, 536 (2000). [CrossRef]

**11. **B. I. Mantsyzov, “Gap 2*π* pulse with an inhomogeneously broadened line and an oscillating solitary wave,” Phys. Rev. A **51**, 4939 (1995). [CrossRef] [PubMed]

**12. **N. Akozbek and S. John, “Self-induced transparency solitary waves in a doped nonlinear photonic band gap material,” Phys. Rev. E **58**, 3876 (1998). [CrossRef]

**13. **M. Hübner, J. Prineas, C. Ell, P. Brick, E.S. Lee, G. Khitrova, H.M. Gibbs, and S.W. Koch, “Optical lattices achieved by excitons in periodic quantum well structures,” Phys. Rev. Lett. **83**, 2841 (1999). [CrossRef]

**14. **J.P. Prineas, J.Y. Zhou, J. Kuhl, H. M. Gibbs, G. Khitrova, S. W. Koch, and A. Knorr, “Ultrafast ac Stark effect switching of active photonic bandgap from Bragg-periodic semiconductor quantum wells,” Appl. Phys. Lett. **81**, 4332 (2002). [CrossRef]

**15. **J. P. Prineas, C. Ell, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Exciton-polariton eigenmodes in light-coupled *In*_{0.04}*Ga*_{0.96}*As*/*GaAs* semiconductor multiple quantum-well structures,” Phys. Rev. B , **61**, 13863 (2000). [CrossRef]

**16. **D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. **86**, 783 (2001). [CrossRef] [PubMed]

**17. **V. G. Arkhipkin and I. V. Timofeev, “Electromagnetically induced transparency: writing, storing, and reading short optical pulses,” JETP Letters **76**, 66 (2002). [CrossRef]

**18. **S.L. McCall and E.L. Hahn, Phys. Rev.183, 457 (1969). [CrossRef]

**19. **P. Meystre and M. Sagent III, *Elements of Quantum Optics* (Springer-Verlag, World Publishing Corp., 1992).

**20. **E. L. Ivchenko, A. I. Nesvizhskii, and S. Jorda, “Bragg reflection of light from quantum wells” Fiz. Tverd. Tela (St. Petersburg) **36**, 2118 (1994) [Phys. Solid State 36, 1156 (1994)].

**21. **M. Hübner, J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R. Hey, and K. Ploog, “Collective effects of excitons in multiple-quantum-well Bragg and anti-Bragg structures,” Phys. Rev. Lett. **76**, 4199 (1996). [CrossRef] [PubMed]

**22. **B. I. Mantsyzov and R. A. Sil’nikov, “Oscillating gap 2*π* pulse in resonantly absorbing lattice,” JETP Letters **74**, 456–459 (2001). [CrossRef]

**23. **B. I. Mantsyzov and R. A. Silnikov, “Unstable excited and stable oscillating gap 2*π* pulses,” J. Opt. Soc. Am. **B19**, 2203–2207 (2002).

**24. **P. Tran, “Optical switching with a nonlinear photonic crystal: a numerical study,” Opt. Lett. **21**, 1138–1140 (1996). [CrossRef] [PubMed]

**25. **A. Andre and M.D. Lukin, “Manupulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. **89**, 143602 (2002). [CrossRef] [PubMed]

**26. **S. Chi, B. Luo, and H.Y. Tseng, “Ultrashort Bragg soliton in a fiber Bragg grating,” Opt. Comm. **206**, 115–121 (2002). [CrossRef]

**27. **J. Cheng and J. Y. Zhou, “Effects of the near-dipole-dipole interaction on gap solitons in resonantly absorbing gratings,” Phys. Rev. E **66**, 036606 (2002). [CrossRef]