Abstract

Analysis of photonic crystal coupled resonator optical waveguide (CROW) structures with a highly dispersive background medium is presented. A finite-difference time-domain algorithm was employed which contains an exact representation of the permittivity of a three-level atomic system which exhibits electromagnetically induced transparency (EIT). We find that the coupling strength between nearest-neighbor cavities in the CROW decreases with increasing steepness of the background dispersion, which is continuously tunable as it is directly related to the control field Rabi frequency. The weaker coupling decreases the speed of pulse propagation through the waveguide. In addition, due to the dispersive nature of the EIT background, the CROW band shape is tuned around a fixed k-point. Thus, the EIT background enables dynamic tunability of the CROW band shape and the group velocity in the structure at a fixed operating point in momentum space.

© 2007 Optical Society of America

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References

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  1. D. Marcuse, Theory of Dielectric Optical Waveguides,(Academic Press, New York, 1974).
  2. P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
    [CrossRef]
  3. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
    [CrossRef]
  4. E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  5. S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, S. Schultz, "Microwave propagation in two-dimensional dielectric lattices," Phys. Rev. Lett. 67, 2017-2020 (1991).
    [CrossRef] [PubMed]
  6. M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in threedimensional photonic crystal," Phys. Rev. Lett. 84, 2140-2143 (2000).
    [CrossRef] [PubMed]
  7. K.-J. Boller, A. Imamoglu, and S. E. Harris, "Observation of electromagnetically induced transparency," Phys. Rev. Lett. 66, 2593-2596 (1991).
    [CrossRef] [PubMed]
  8. L. V. Hau, Z. Dutton, C. H. Behroozi, and S. E. Harris, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature (London) 397, 594-598 (1999).
    [CrossRef]
  9. M. D. Lukin, M. Fleischhauer, M. O. Scully, and V. L. Velichansky, "Intracavity electromagnetically induced transparency," Opt. Lett. 23, 295-297 (1998).
    [CrossRef]
  10. G. Müller, M. Müller, A. Wicht, R.-H. Rinkleff, and K. Danzmann, "Optical resonator with steep internal dispersion," Phys. Rev. A 56, 2385-2389 (1997).
    [CrossRef]
  11. M. Soljačić, E. Lidorikis, L. V. Hau, and J. D. Joannopoulos, "Enhancement of microcavity lifetimes using highly dispersive materials," Phys. Rev. E 71, 026602 (2005).
    [CrossRef]
  12. C. W. Neff, L. M. Andersson, and M. Qiu, "Modelling electromagnetically induced transparency media using the finite-difference time-domain method," New J. Phys. 9, 48 (2007).
    [CrossRef]
  13. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, England, 1997).
  14. P. Jänes, J. Tidström, and L . Thyl’en, "Limits on optical pulse compression and delay bandwidth product in electromagnetically induced transparency media," J. Lightwave Technol. 23, 3893-3899 (2005).
    [CrossRef]
  15. M. Okoniewski, M. Mrozowski, and M. Stuchly, "Simple treatment of multi-term dispersion in FDTD," IEEE Micro. Guided Wave Lett. 7, 121-123 (1997).
    [CrossRef]
  16. A. Taflove and S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method, 2nd ed. (Artech House, Boston, 2000).
  17. M. Han, R. Dutton, and S. Fan, "Model dispersive media in finite-difference time-domain method with complexconjugate pole-residue pairs," Microwave andWireless Components Letters, IEEE [see also IEEE Micro. Guided Wave Lett.] 16, 119-121 (2006).
  18. J.-P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comp. Phys. 114, 185-200 (1994).
    [CrossRef]
  19. W.-H. Guo, W.-J. Li, and Y.-Z. Huang, "Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation," Microwave and Wireless Components Letters, IEEE [see also IEEE Micro. Guided Wave Lett.] 11, 223-225 (2001).
  20. G. A. Baker and J. L. Gammel, The Pad’e Approximant in Theoretical Physics, (Academic, New York, 1970).
  21. manuscript in preparation
  22. C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature (London) 409, 490-493 (2001).
    [CrossRef]
  23. J. Tidström, P. Jänes, and L. M. Andersson, "Delay bandwidth product of electromagnetically induced transparency media," Phys. Rev. A 75, 53803 (2007).
    [CrossRef]
  24. M. F. Yanik and S. Fan, "Stopping light all optically," Phys. Rev. Lett. 92, 083901 (2004).
    [CrossRef] [PubMed]

2007

C. W. Neff, L. M. Andersson, and M. Qiu, "Modelling electromagnetically induced transparency media using the finite-difference time-domain method," New J. Phys. 9, 48 (2007).
[CrossRef]

J. Tidström, P. Jänes, and L. M. Andersson, "Delay bandwidth product of electromagnetically induced transparency media," Phys. Rev. A 75, 53803 (2007).
[CrossRef]

2005

M. Soljačić, E. Lidorikis, L. V. Hau, and J. D. Joannopoulos, "Enhancement of microcavity lifetimes using highly dispersive materials," Phys. Rev. E 71, 026602 (2005).
[CrossRef]

P. Jänes, J. Tidström, and L . Thyl’en, "Limits on optical pulse compression and delay bandwidth product in electromagnetically induced transparency media," J. Lightwave Technol. 23, 3893-3899 (2005).
[CrossRef]

2004

M. F. Yanik and S. Fan, "Stopping light all optically," Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

2001

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature (London) 409, 490-493 (2001).
[CrossRef]

2000

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in threedimensional photonic crystal," Phys. Rev. Lett. 84, 2140-2143 (2000).
[CrossRef] [PubMed]

1999

L. V. Hau, Z. Dutton, C. H. Behroozi, and S. E. Harris, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature (London) 397, 594-598 (1999).
[CrossRef]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
[CrossRef]

1998

1997

G. Müller, M. Müller, A. Wicht, R.-H. Rinkleff, and K. Danzmann, "Optical resonator with steep internal dispersion," Phys. Rev. A 56, 2385-2389 (1997).
[CrossRef]

M. Okoniewski, M. Mrozowski, and M. Stuchly, "Simple treatment of multi-term dispersion in FDTD," IEEE Micro. Guided Wave Lett. 7, 121-123 (1997).
[CrossRef]

1994

J.-P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comp. Phys. 114, 185-200 (1994).
[CrossRef]

1991

K.-J. Boller, A. Imamoglu, and S. E. Harris, "Observation of electromagnetically induced transparency," Phys. Rev. Lett. 66, 2593-2596 (1991).
[CrossRef] [PubMed]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, S. Schultz, "Microwave propagation in two-dimensional dielectric lattices," Phys. Rev. Lett. 67, 2017-2020 (1991).
[CrossRef] [PubMed]

1987

E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

1976

P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
[CrossRef]

IEEE Micro. Guided Wave Lett.

M. Okoniewski, M. Mrozowski, and M. Stuchly, "Simple treatment of multi-term dispersion in FDTD," IEEE Micro. Guided Wave Lett. 7, 121-123 (1997).
[CrossRef]

J. Comp. Phys.

J.-P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comp. Phys. 114, 185-200 (1994).
[CrossRef]

J. Lightwave Technol.

Nature (London)

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature (London) 409, 490-493 (2001).
[CrossRef]

L. V. Hau, Z. Dutton, C. H. Behroozi, and S. E. Harris, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature (London) 397, 594-598 (1999).
[CrossRef]

New J. Phys.

C. W. Neff, L. M. Andersson, and M. Qiu, "Modelling electromagnetically induced transparency media using the finite-difference time-domain method," New J. Phys. 9, 48 (2007).
[CrossRef]

Opt. Commun.

P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
[CrossRef]

Opt. Lett.

Phys. Rev. A

G. Müller, M. Müller, A. Wicht, R.-H. Rinkleff, and K. Danzmann, "Optical resonator with steep internal dispersion," Phys. Rev. A 56, 2385-2389 (1997).
[CrossRef]

J. Tidström, P. Jänes, and L. M. Andersson, "Delay bandwidth product of electromagnetically induced transparency media," Phys. Rev. A 75, 53803 (2007).
[CrossRef]

Phys. Rev. E

M. Soljačić, E. Lidorikis, L. V. Hau, and J. D. Joannopoulos, "Enhancement of microcavity lifetimes using highly dispersive materials," Phys. Rev. E 71, 026602 (2005).
[CrossRef]

Phys. Rev. Lett.

E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, S. Schultz, "Microwave propagation in two-dimensional dielectric lattices," Phys. Rev. Lett. 67, 2017-2020 (1991).
[CrossRef] [PubMed]

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in threedimensional photonic crystal," Phys. Rev. Lett. 84, 2140-2143 (2000).
[CrossRef] [PubMed]

K.-J. Boller, A. Imamoglu, and S. E. Harris, "Observation of electromagnetically induced transparency," Phys. Rev. Lett. 66, 2593-2596 (1991).
[CrossRef] [PubMed]

M. F. Yanik and S. Fan, "Stopping light all optically," Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

Other

D. Marcuse, Theory of Dielectric Optical Waveguides,(Academic Press, New York, 1974).

W.-H. Guo, W.-J. Li, and Y.-Z. Huang, "Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation," Microwave and Wireless Components Letters, IEEE [see also IEEE Micro. Guided Wave Lett.] 11, 223-225 (2001).

G. A. Baker and J. L. Gammel, The Pad’e Approximant in Theoretical Physics, (Academic, New York, 1970).

manuscript in preparation

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, England, 1997).

A. Taflove and S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method, 2nd ed. (Artech House, Boston, 2000).

M. Han, R. Dutton, and S. Fan, "Model dispersive media in finite-difference time-domain method with complexconjugate pole-residue pairs," Microwave andWireless Components Letters, IEEE [see also IEEE Micro. Guided Wave Lett.] 16, 119-121 (2006).

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Figures (5)

Fig. 1.
Fig. 1.

Cross-section schematic of a CROW structure formed in a 2D PhC square lattice of high index rods where single rods were removed to create the resonant cavities.

Fig. 2.
Fig. 2.

(a) Schematic of the energy levels in a three level Λ system. Representative plot of the real and imaginary parts of the permittivity for an EIT medium at two Ω c values of the control field: 0.08 (solid) and 0.03 (dashed)

Fig. 3.
Fig. 3.

Evolution of EIT CROW bands (a) and group velocities (b) for case 1: Ω c ≥Δω.

Fig. 4.
Fig. 4.

Evolution of EIT CROW bands (a) and group velocities (b) for case 2: Ω c ≤Δω.

Fig. 5.
Fig. 5.

Coupling coefficient (normalized by the vacuum CROWκ 0) as a function of control field Rabi frequency. The points and crosses are taken from the FDTD and TB analysis data, respectively, and the solid line is a plot of the EIT group velocity relationship (right axis) with a fitted α parameter.

Equations (7)

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ω ( k ) = ω res [ 1 + κ cos ( k R ) ]
ν g ( k ) = ω res R κ sin ( k R ) ,
χ ( Δ ) = C Δ + i γ bc ( Δ + i γ bc ) ( Δ + i γ ab ) Ω c 2 4 ,
ε = ε b + A 1 B 1 + i ω + A 2 B 2 + i ω ,
A 1 , 2 = i C 2 { 1 γ bc γ ab [ ( γ bc γ ab ) 2 Ω c 2 ] 1 2 } ,
B 1 , 2 = i ω ab + 1 2 { γ bc + γ ab [ ( γ bc γ ab ) 2 Ω c 2 ] 1 2 } .
ν g = c 0 1 + ω ab 2 χ Δ = c 0 1 + ω ab 2 4 C Ω c 2 = c 0 α Ω c 2 1 + α Ω c 2 ,

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