Abstract

In this paper we discuss the possibility of producing Bragg solitons in an electromagnetically induced transparency medium. We show that a coherent medium can be engineered to be a Bragg grating with a large Kerr nonlinearity through proper arrangements of light fields. The parameters of the medium can be easily controlled through adjusting the intensities and detunings of lasers. This scheme may provide an opportunity to study the dynamics of Bragg solitons with low power lights.

© 2007 Optical Society of America

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References

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  1. C. M. de Sterke and J. E. Sipe, "Gap solitons," in Progress in Optics, Vol. 33, E. Wolf, eds. (North-Holland, Amsterdam, 1994), Vol. 33 pp. 203-260.
  2. W. Chen and D. L. Mills, "Gap solitons and the nonlinear optical response of superlattices," Phys. Rev. Lett. 58, 160-163 (1987).
    [CrossRef] [PubMed]
  3. D. L. Mills and S. E. Trullinger, "Gap solitons in nonlinear periodic structures," Phys. Rev. B 36, 947-952 (1987).
    [CrossRef]
  4. D. N. Christodoulides and R. I. Joseph, "Slow Bragg solitons in nonlinear periodic structures," Phys. Rev. Lett. 62, 1746-1749 (1989).
    [CrossRef] [PubMed]
  5. A. B. Aceves and S. Wabnitz, "Self-induced transparency solitons in nonlinear refractive periodic media, " Phys. Lett. A 141, 37-42 (1989).
    [CrossRef]
  6. J. Feng and F. K. Kneubuhl, "Solitons in a periodic structure with kerr nonlinearity," IEEE J. Quantum Electr. 29, 590-597 (1993).
    [CrossRef]
  7. B. J. Eggleton, C. M. de Sterke and R. E. Slusher, "Nonlinear pulse propagation in Bragg gratings," J. Opt. Soc. Am. B 14, 2980-2993 (1997).
    [CrossRef]
  8. C. Conti and S Trillo, "Bifurcation of gap solitons through catastrophe theory, " Phys. Rev. E 64, 036617 (2001).
    [CrossRef]
  9. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
    [CrossRef] [PubMed]
  10. G. Van Simaeys, S. Coen, M. Haelterman, and S. Trillo, "Observation of Resonance Soliton Trapping due to a Photoinduced Gap in Wave Number," Phys. Rev. Lett. 92, 223902 (2004).
    [CrossRef] [PubMed]
  11. S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (1997).
    [CrossRef]
  12. E. Arimondo, "Coherent population trapping in laser spectroscopy," in Progress in Optics, Vol. 35, E.Wolf, eds. (North-Holland, Amsterdam, 1996), pp. 257-353.
  13. M. Fleischhauer, A. Imamoglu and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Rev. Mod. Phys. 77, 633-673 (2005).
    [CrossRef]
  14. H. Schmidt and A. Imamoğlu, "Giant Kerr nonlinearities obtained by electromagnetically induced transparency," Opt. Lett. 21, 1936-1938 (1996).
    [CrossRef] [PubMed]
  15. Yuri V. Rostovtsev, Andrey B. Matsko, and Marlan O. Scully, "Electromagnetic-induced transparency and amplification of electromagnetic waves in photonic band-gap materials," Phys. Rev. A 57, 4919-4924 (1998).
    [CrossRef]
  16. YuriV. Rostovtsev, Andrey B. Matsko, and Marlan O. Scully, "Electromagnetically induced photonic band gap," Phys. Rev. A 60, 712-714 (1999).
    [CrossRef]
  17. A. Andre and M. D. Lukin, "Manipulating Light Pulses via Dynamically Controlled Photonic Band gap," Phys. Rev. Lett. 89, 143602 (2002).
    [CrossRef] [PubMed]
  18. A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Nonlinear Optics with Stationary Pulses of Light," Phys. Rev. Lett. 94, 063902 (2005).
    [CrossRef] [PubMed]
  19. M. Bajcsy, A. S. Zibrov, and M. Lukin, "Stationary pulses of light in an atomic medium," Nature (London) 426, 638-640 (2003).
    [CrossRef]
  20. M. Artoni, G. La Rocca, and F. Bassani, "Resonantly absorbing one-dimensional photonic crystals," Phys. Rev. E 72, 046604 (2005).
    [CrossRef]
  21. M. Artoni and G. La Rocca, "Optically Tunable Photonic Stop Bands in Homogeneous Absorbing Media," Phys. Rev. Lett. 96, 073905 (2006).
    [CrossRef] [PubMed]
  22. S. E. Harris, J. E. Field and A. Imamo˘glu, "Nonlinear optical processes using electromagnetically induced transparency," Phys. Rev. Lett. 64, 1107-1110 (1990).
    [CrossRef] [PubMed]
  23. A. A. Krokhin and P. Halevi, "Influence of weak dissipation on the photonic band structure of periodic composites," Phys. Rev. B 53, 1205-1214 (1996).
    [CrossRef]
  24. A. Tip, A. Moroz and J. M. Combes, "Band structure of absorptive photonic crystals," J. Phys. A 33, 6223-6252 (2000).
    [CrossRef]
  25. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 2003).
  26. Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic Press, London, 2003).

2006

M. Artoni and G. La Rocca, "Optically Tunable Photonic Stop Bands in Homogeneous Absorbing Media," Phys. Rev. Lett. 96, 073905 (2006).
[CrossRef] [PubMed]

2005

M. Fleischhauer, A. Imamoglu and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Rev. Mod. Phys. 77, 633-673 (2005).
[CrossRef]

A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Nonlinear Optics with Stationary Pulses of Light," Phys. Rev. Lett. 94, 063902 (2005).
[CrossRef] [PubMed]

M. Artoni, G. La Rocca, and F. Bassani, "Resonantly absorbing one-dimensional photonic crystals," Phys. Rev. E 72, 046604 (2005).
[CrossRef]

2004

G. Van Simaeys, S. Coen, M. Haelterman, and S. Trillo, "Observation of Resonance Soliton Trapping due to a Photoinduced Gap in Wave Number," Phys. Rev. Lett. 92, 223902 (2004).
[CrossRef] [PubMed]

2003

M. Bajcsy, A. S. Zibrov, and M. Lukin, "Stationary pulses of light in an atomic medium," Nature (London) 426, 638-640 (2003).
[CrossRef]

2002

A. Andre and M. D. Lukin, "Manipulating Light Pulses via Dynamically Controlled Photonic Band gap," Phys. Rev. Lett. 89, 143602 (2002).
[CrossRef] [PubMed]

2001

C. Conti and S Trillo, "Bifurcation of gap solitons through catastrophe theory, " Phys. Rev. E 64, 036617 (2001).
[CrossRef]

2000

A. Tip, A. Moroz and J. M. Combes, "Band structure of absorptive photonic crystals," J. Phys. A 33, 6223-6252 (2000).
[CrossRef]

1999

YuriV. Rostovtsev, Andrey B. Matsko, and Marlan O. Scully, "Electromagnetically induced photonic band gap," Phys. Rev. A 60, 712-714 (1999).
[CrossRef]

1998

Yuri V. Rostovtsev, Andrey B. Matsko, and Marlan O. Scully, "Electromagnetic-induced transparency and amplification of electromagnetic waves in photonic band-gap materials," Phys. Rev. A 57, 4919-4924 (1998).
[CrossRef]

1997

1996

H. Schmidt and A. Imamoğlu, "Giant Kerr nonlinearities obtained by electromagnetically induced transparency," Opt. Lett. 21, 1936-1938 (1996).
[CrossRef] [PubMed]

A. A. Krokhin and P. Halevi, "Influence of weak dissipation on the photonic band structure of periodic composites," Phys. Rev. B 53, 1205-1214 (1996).
[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, 1627-1630 (1996).
[CrossRef] [PubMed]

1993

J. Feng and F. K. Kneubuhl, "Solitons in a periodic structure with kerr nonlinearity," IEEE J. Quantum Electr. 29, 590-597 (1993).
[CrossRef]

1990

S. E. Harris, J. E. Field and A. Imamo˘glu, "Nonlinear optical processes using electromagnetically induced transparency," Phys. Rev. Lett. 64, 1107-1110 (1990).
[CrossRef] [PubMed]

1989

D. N. Christodoulides and R. I. Joseph, "Slow Bragg solitons in nonlinear periodic structures," Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

A. B. Aceves and S. Wabnitz, "Self-induced transparency solitons in nonlinear refractive periodic media, " Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

1987

W. Chen and D. L. Mills, "Gap solitons and the nonlinear optical response of superlattices," Phys. Rev. Lett. 58, 160-163 (1987).
[CrossRef] [PubMed]

D. L. Mills and S. E. Trullinger, "Gap solitons in nonlinear periodic structures," Phys. Rev. B 36, 947-952 (1987).
[CrossRef]

IEEE J. Quantum Electr.

J. Feng and F. K. Kneubuhl, "Solitons in a periodic structure with kerr nonlinearity," IEEE J. Quantum Electr. 29, 590-597 (1993).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. A

A. Tip, A. Moroz and J. M. Combes, "Band structure of absorptive photonic crystals," J. Phys. A 33, 6223-6252 (2000).
[CrossRef]

Nature (London)

M. Bajcsy, A. S. Zibrov, and M. Lukin, "Stationary pulses of light in an atomic medium," Nature (London) 426, 638-640 (2003).
[CrossRef]

Opt. Lett.

Phys. Lett. A

A. B. Aceves and S. Wabnitz, "Self-induced transparency solitons in nonlinear refractive periodic media, " Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

Phys. Rev. A

Yuri V. Rostovtsev, Andrey B. Matsko, and Marlan O. Scully, "Electromagnetic-induced transparency and amplification of electromagnetic waves in photonic band-gap materials," Phys. Rev. A 57, 4919-4924 (1998).
[CrossRef]

YuriV. Rostovtsev, Andrey B. Matsko, and Marlan O. Scully, "Electromagnetically induced photonic band gap," Phys. Rev. A 60, 712-714 (1999).
[CrossRef]

Phys. Rev. B

D. L. Mills and S. E. Trullinger, "Gap solitons in nonlinear periodic structures," Phys. Rev. B 36, 947-952 (1987).
[CrossRef]

A. A. Krokhin and P. Halevi, "Influence of weak dissipation on the photonic band structure of periodic composites," Phys. Rev. B 53, 1205-1214 (1996).
[CrossRef]

Phys. Rev. E

C. Conti and S Trillo, "Bifurcation of gap solitons through catastrophe theory, " Phys. Rev. E 64, 036617 (2001).
[CrossRef]

M. Artoni, G. La Rocca, and F. Bassani, "Resonantly absorbing one-dimensional photonic crystals," Phys. Rev. E 72, 046604 (2005).
[CrossRef]

Phys. Rev. Lett.

M. Artoni and G. La Rocca, "Optically Tunable Photonic Stop Bands in Homogeneous Absorbing Media," Phys. Rev. Lett. 96, 073905 (2006).
[CrossRef] [PubMed]

S. E. Harris, J. E. Field and A. Imamo˘glu, "Nonlinear optical processes using electromagnetically induced transparency," Phys. Rev. Lett. 64, 1107-1110 (1990).
[CrossRef] [PubMed]

A. Andre and M. D. Lukin, "Manipulating Light Pulses via Dynamically Controlled Photonic Band gap," Phys. Rev. Lett. 89, 143602 (2002).
[CrossRef] [PubMed]

A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Nonlinear Optics with Stationary Pulses of Light," Phys. Rev. Lett. 94, 063902 (2005).
[CrossRef] [PubMed]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

G. Van Simaeys, S. Coen, M. Haelterman, and S. Trillo, "Observation of Resonance Soliton Trapping due to a Photoinduced Gap in Wave Number," Phys. Rev. Lett. 92, 223902 (2004).
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, "Slow Bragg solitons in nonlinear periodic structures," Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

W. Chen and D. L. Mills, "Gap solitons and the nonlinear optical response of superlattices," Phys. Rev. Lett. 58, 160-163 (1987).
[CrossRef] [PubMed]

Phys. Today

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (1997).
[CrossRef]

Rev. Mod. Phys.

M. Fleischhauer, A. Imamoglu and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Rev. Mod. Phys. 77, 633-673 (2005).
[CrossRef]

Other

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 2003).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic Press, London, 2003).

E. Arimondo, "Coherent population trapping in laser spectroscopy," in Progress in Optics, Vol. 35, E.Wolf, eds. (North-Holland, Amsterdam, 1996), pp. 257-353.

C. M. de Sterke and J. E. Sipe, "Gap solitons," in Progress in Optics, Vol. 33, E. Wolf, eds. (North-Holland, Amsterdam, 1994), Vol. 33 pp. 203-260.

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Figures (4)

Fig. 1.
Fig. 1.

(a) Energy diagram of the atom. (b) Geometric configuration of the lights. Coupling beam and probe beam are co-propagating. A standing wave is formed by a forward and a backward controlling fields E→sf and E→sb . A small angle ϕ between the standing wave and probe beam is chosen so that kscosϕ = kp is fulfilled.

Fig. 2.
Fig. 2.

Real and imaginary parts of χ a and χ(3) versus ∆1 a (see the text for parameters). Near the two-photon resonance the absorption is small while the Kerr nonlinearity is large.

Fig. 3.
Fig. 3.

(a) Reflectivity of the medium(see text for parameters). A band gap appears near the two-photon resonance. Solid lines are results when absorption is included while dash lines are results when there is no absorption. (b) Dispersion relation (with absorption). d is the period of the Bragg grating. (c) Dispersion relation (without absorption). The number of periodic structure is 4000 corresponding to a propagation distance (size of the cold atomic cloud) of 3mm.

Fig. 4.
Fig. 4.

Pin (v) and T 0(v) with different T 0.

Equations (14)

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χ ( ω p ) = K 0 2 4 δ ˜ Δ ˜ 42 Δ ˜ 52 + Δ ˜ 42 Ω s 2 + Δ ˜ 52 Ω p ' 2 4 δ ˜ Δ ˜ Δ ˜ 42 Δ ˜ 52 Δ ˜ Δ ˜ 52 Ω p ' 2 Δ ˜ Δ ˜ 42 Ω s 2 Δ ˜ 42 Δ ˜ 52 Ω c 2 ,
χ a ( ω p ) = K 0 2 4 δ ˜ Δ ˜ 52 + Ω s 2 4 δ ˜ Δ ˜ Δ ˜ 52 Δ ˜ 52 Ω c 2 ,
χ ( 3 ) ( ω p ; ω p , ω p , ω p ) = K 1 2 1 4 δ ˜ Δ ˜ Δ ˜ 42 Δ ˜ 42 Ω c 2 ,
n 2 ( z ) = 1 + χ a ( ω ) + χ ( 3 ) ( ω p ; ω p , ω p , ω p ) E p 2 ,
= 1 + χ ̅ a + δχ cos ( 2 k B z ) + χ ( 3 ) E p 2 ,
2 E p z 2 n 2 ( z ) c 2 2 E p t 2 = 0 .
E p = A + z t e i ( k p z ω p t ) + A z t e i ( k p z ω p t ) ,
z A + + v g 1 i A + e 2 iΔkz A ( A + 2 + 2 A 2 ) A + = 0 ,
z A + v g 1 t A e 2 iΔkz A + ( A 2 + 2 A + 2 ) A = 0 ,
A + z t = a + sech ( ζ 2 ) e ,
A z t = a sech ( ζ + 2 ) e ,
a ± = ± ( 1 ± v 1 ± v ) 1 4 κ ( 1 v 2 ) γ ( 2 v 2 ) sin ψ ,
ζ = z V G t 1 v 2 κ sin ψ ,
θ = v ( z V G t ) 1 v 2 κ cos ψ 4 v 3 v 2 tan 1 [ cot ψ 2 coth ζ ] .

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