## Storage of ultrashort optical pulses in a resonantly absorbing Bragg reflector

Optics Express, Vol. 11, Issue 24, pp. 3277-3283 (2003)

http://dx.doi.org/10.1364/OE.11.003277

Acrobat PDF (222 KB)

### 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

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]

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

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]

*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

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]

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]

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]

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

*T*

_{1}time of the two-level systems. Methods for releasing the pulse from the structure are discussed.

*π*pulses can transmit through an otherwise opaque medium without attenuation via self-induced transparency (SIT) [18

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

*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

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]

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]

*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.

*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]:

^{±}(

*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;

*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.

*τ*

_{0}and area

*θ*=

*t*)

*dt*.

*t*)=

*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

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

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]

*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

^{±}(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.

*t*)=

*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≤

*T*

_{1}and

*T*

_{2}, on the decelerating gap solitons, we follow the model of Ref.[2], i.e., the phenomenological relaxation terms, -

*P*(

*x, t*) and -

*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

*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.

*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

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]

## Acknowledgments

## References and links

1. | R. E. Slusher and B. J. Eggleton (editors), |

2. | B. I. Mantsyzov and R. N. Kuz’min, “Coherent interaction of light with a discrete periodic resonant medium,” Sov. Phys. JETP |

3. | W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. |

4. | A. Kozhekin and G. Kurizki,“Self-induced transparency in Bragg reflectors: gap solitons near absorption resonances,” Phys. Rev. Lett. |

5. | A. E. Kozhekin and G. Kurizki,“Standing and moving gap solitons in resonantly absorbing gratings,” Phys. Rev. Lett. |

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 |

7. | C. Conti, G. Assanto, and S. Trillo, “Gap solitons and slow light,” J. Nonlinear Opt. Phys. & Mat. |

8. | B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. |

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. |

10. | N. G. R. Broderick, D. J. Richarson, and M. Ibsen, “Nonlinear switching in a 20-cm-long fiber Bragg grating,” Opt. Lett. |

11. | B. I. Mantsyzov, “Gap 2 |

12. | N. Akozbek and S. John, “Self-induced transparency solitary waves in a doped nonlinear photonic band gap material,” Phys. Rev. E |

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. |

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. |

15. | J. P. Prineas, C. Ell, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Exciton-polariton eigenmodes in light-coupled |

16. | D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. |

17. | V. G. Arkhipkin and I. V. Timofeev, “Electromagnetically induced transparency: writing, storing, and reading short optical pulses,” JETP Letters |

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

19. | P. Meystre and M. Sagent III, |

20. | E. L. Ivchenko, A. I. Nesvizhskii, and S. Jorda, “Bragg reflection of light from quantum wells” Fiz. Tverd. Tela (St. Petersburg) |

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. |

22. | B. I. Mantsyzov and R. A. Sil’nikov, “Oscillating gap 2 |

23. | B. I. Mantsyzov and R. A. Silnikov, “Unstable excited and stable oscillating gap 2 |

24. | P. Tran, “Optical switching with a nonlinear photonic crystal: a numerical study,” Opt. Lett. |

25. | A. Andre and M.D. Lukin, “Manupulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. |

26. | S. Chi, B. Luo, and H.Y. Tseng, “Ultrashort Bragg soliton in a fiber Bragg grating,” Opt. Comm. |

27. | J. Cheng and J. Y. Zhou, “Effects of the near-dipole-dipole interaction on gap solitons in resonantly absorbing gratings,” Phys. Rev. E |

**OCIS Codes**

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 29, 2003

Revised Manuscript: November 17, 2003

Published: December 1, 2003

**Citation**

W. Xiao, J. Zhou, and J. Prineas, "Storage of ultrashort optical pulses in a resonantly absorbing Bragg reflector," Opt. Express **11**, 3277-3283 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-24-3277

Sort: Journal | Reset

### References

- R. E. Slusher and B. J. Eggleton (editors), Nonlinear Photonic Crystals (Springer-Verlag, Berlin, Heidelberg, 2003).
- 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).
- W. Chen and D. L. Mills, �??Gap solitons and the nonlinear optical response of superlattices,�?? Phys. Rev. Lett. 58, 160 (1987). [CrossRef] [PubMed]
- A. Kozhekin and G. Kurizki, �??Self-induced transparency in Bragg reflectors: gap solitons near absorption resonances,�?? Phys. Rev. Lett. 74, 5020 (1995) [CrossRef] [PubMed]
- A. E. Kozhekin and G. Kurizki, �??Standing and moving gap solitons in resonantly absorbing gratings,�?? Phys. Rev. Lett. 81, 3647 (1998). [CrossRef]
- G. Kurizki, A. E. Kozhekin, T. Opatrny, 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]
- C. Conti, G. Assanto and S. Trillo, �??Gap solitons and slow light,�?? J. Nonlinear Opt. Phys. & Mat. 11, 239�??259 (2002). [CrossRef]
- 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]
- 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]
- 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]
- B. I. Mantsyzov, �??Gap 2�? pulse with an inhomogeneously broadened line and an oscillating solitary wave,�?? Phys. Rev. A 51, 4939 (1995). [CrossRef] [PubMed]
- 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]
- 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]
- J.P. Prineas, J.Y. Zhou, J. Kuhl, H. M. Gibbs, G. Khitrova, S. W. Koch, A. Knorr, �??Ultrafast ac Stark effect switching of active photonic bandgap from Bragg-periodic semiconductor quantum wells,�?? Appl. Phys. Lett. 81, 4332 (2002). [CrossRef]
- J. P. Prineas, C. Ell, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, �??Exciton-polariton eigenmodes in light-coupled In0.04Ga0.96As/GaAs semiconductor multiple quantum-well structures,�?? Phys. Rev. B, 61, 13863 (2000). [CrossRef]
- 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]
- V. G. Arkhipkin and I. V. Timofeev, �??Electromagnetically induced transparency: writing, storing, and reading short optical pulses,�?? JETP Letters 76, 66 (2002). [CrossRef]
- S.L. McCall and E.L. Hahn, Phys. Rev. 183, 457 (1969) [CrossRef]
- P. Meystre and M. Sagent III, Elements of Quantum Optics (Springer-Verlag, World Publishing Corp., 1992).
- 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)].
- 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]
- B. I. Mantsyzov and R. A. Sil�??nikov, �??Oscillating gap 2�? pulse in resonantly absorbing lattice,�?? JETP Letters 74, 456�??459 (2001). [CrossRef]
- B. I. Mantsyzov and R. A. Silnikov, �??Unstable excited and stable oscillating gap 2�? pulses,�?? J. Opt. Soc. Am B19, 2203-2207 (2002).
- P. Tran, �??Optical switching with a nonlinear photonic crystal: a numerical study,�?? Opt. Lett. 21, 1138�??1140 (1996). [CrossRef] [PubMed]
- A. Andre and M.D. Lukin, �??Manupulating light pulses via dynamically controlled photonic band gap,�?? Phys. Rev. Lett. 89, 143602 (2002). [CrossRef] [PubMed]
- S. Chi, B. Luo, H.Y. Tseng, �??Ultrashort Bragg soliton in a fiber Bragg grating,�?? Opt. Comm. 206, 115�??121 (2002) [CrossRef]
- J. Cheng, J. Y. Zhou, �??Effects of the near-dipole-dipole interaction on gap solitons in resonantly absorbing gratings,�?? Phys. Rev. E 66, 036606 (2002). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.