## Coupled resonator optical waveguide structures with highly dispersive media

Optics Express, Vol. 15, Issue 16, pp. 10362-10369 (2007)

http://dx.doi.org/10.1364/OE.15.010362

Acrobat PDF (185 KB)

### 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

## 1. Introduction

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]

*Q*micro-resonators are aligned within proximity to one another such that photons can tunnel from one resonator to its nearest-neighbor and thus propagate as if in a waveguide. Figure 1 shows a schematic of such a coupled-resonator optical waveguide (CROW) formed by an infinite straight line of defect cavities in a two-dimensional (2D) photonic crystal (PhC) [4

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, and 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 three-dimensional photonic crystal,” Phys. Rev. Lett. **84**, 2140–2143 (2000). [CrossRef] [PubMed]

*κ*:

*ω*is the mode frequency of a single, isolated resonator of the waveguide,

_{res}*R*is the spacing between resonators, and

*k*is the wave vector. Therefore, the group velocity, Eq. (2), of the pulse can be decreased by decreasing

*κ*. However,

*κ*is typically a fixed quantity based on the design of the structure, e.g. the intrinsic

*Q*of a the individual resonators and/or the spacing

*R*. Yet, for realizing active components in an optical circuit, a dynamic control over

*v*

*is desirable. The problem then becomes how one can manipulate κ in a seemingly fixed system. We propose that a tunable, highly dispersive process such as electromagnetically induced transparency (EIT) [7*

_{g}7. K.-J. Boller, A. Imamoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. **66**, 2593–2596 (1991). [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]

*b*and

*c*to the upper level

*a*show destructive quantum interference under the two-photon resonance condition. Phenomenologically, a transparency window opens between two absorption peaks in the susceptibility, which allows a probe pulse to propagate through the otherwise optically opaque medium. At the same time, this transparency window occurs at a highly dispersive region of the susceptibility. It has been reported that such a lossless, highly dispersive EIT medium within an optical resonator cavity has the effect of dramatically narrowing the cavity linewidth and increasing the cavity

*Q*-factor [9

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, J. D. Hau, and Joannopoulos, “Enhancement of microcavity lifetimes using highly dispersive materials,” Phys. Rev. E **71**, 026602 (2005). [CrossRef]

*v*

*in the waveguide.*

_{g}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]

14. P. Jänes, J. Tidström, and L. Thylén, “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]

*ε*and the electric flux density,

*D*⃗, via the polarization,

*P*⃗

*. This link introduces a phasor polarization current,*

_{l}*J*⃗

*(*

_{l}*t*)=

*∂P*⃗

*/*

_{l}*∂t*, from which time-step update-equations can be derived. Since it adds only one extra step to the FDTD update algorithm, the ADE method does not increase the computation load significantly.

### 1.1. Electromagnetically induced transparency

14. P. Jänes, J. Tidström, and L. Thylén, “Limits on optical pulse compression and delay bandwidth product in electromagnetically induced transparency media,” J. Lightwave Technol. **23**, 3893–3899 (2005). [CrossRef]

*ε*is the background dielectric constant and

_{b}*A*

_{1,2}and

*B*

_{1,2}are the coefficients:

### 1.2. Coupled resonator optical waveguide structure

*Q*, point defect PhC cavities separated by a distance of

*R*=3

*a*(see Fig. 1). The cavities are formed in 2D square lattice of high-dielectric rods,

*ε*=11.56, embedded in an EIT medium. The EIT medium has the same physical parameters as those given in section 1.1 except that

*ω*is chosen so that the EIT resonance frequency matches the mid-band frequency of the vacuum CROW,

_{ab}*ω*

_{0}=0.3796, the subscript denoting the vacuum background structure. The size of the computational domain is 3

*a*× 14

*a*and contains and array of circular rods with diameter

*d*=0.4

*a*arranged in a square lattice. The point defect cavities are formed by removing a rod in the center of the array. Periodic boundaries are used along the long sides of the domain to form an infinitely long series of defect cavities separated by the distance

*R*=3

*a*. Perfectly matched layer [18

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

*x*=Δ

*y*=0.025

*a*is used in all of the calculations. A transverse-magnetic polarized field (magnetic field components normal to the rod axis) is used in the calculations since a photonic band gap exists for this polarization, and the fields are randomized throughout the domain at the beginning of the calculation. Data collected from a point detector positioned near the center of the point defect is used to calculate the resonant frequency of the CROW using the Padé approximation and Baker’s algorithm [19, 20].

## 2. Results and discussion

*. For the vacuum background CROW, the frequency width Δ*

_{c}*ω*of the waveguide band is 0.0109, and the maximum group velocity

*v*

_{g,0}=0.1025

*v*/

_{g}*c*

_{0}of the band occurs near the zone center. We can calculate the coupling factor from

*κ*=Δ

*ω*/(2

*ω*) if the resonant frequency of the isolated resonator

_{res}*ω*is known [6

_{res}6. M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystal,” Phys. Rev. Lett. **84**, 2140–2143 (2000). [CrossRef] [PubMed]

*ω*=0.3793, thus giving a coupling factor of

_{res}*κ*

_{0}=0.0144. For the EIT background isolated resonator, the value of

*ω*changes for each value of Ω

_{res}*because of a frequency pulling effect [21].*

_{c}*ω*of the CROW and case 2) when the opposite is true.

*v*vs. Ω

_{g}*relationship of the EIT background [23*

_{c}23. J. Tidström, P. Jönes, and L. M. Andersson, “Delay bandwidth product of electromagnetically induced transparency media,” Phys. Rev. A **75**, 53803 (2007). [CrossRef]

*ab*-transition frequency we can derive an expression for

*v*(Ω

_{g}*):*

_{c}*α*is (2C

*ω*)

_{ab}^{-1}. From the values given earlier, we calculate

*α*=1.317×10

^{2}for the EIT background medium and plot Eq. (7) in Figure 5. We find that the relationship of the EIT CROW

*κ*vs. Ω

*follows that of the EIT background. We used a minimization algorithm to fit the FDTD data points to an equation of the form of Eq. (7) to find*

_{c}*α*=1.940×10

^{2}. The shape of

*κ*(Ω

*) can be explained as follows: Since the coupling is dependent upon the overlap of the field between adjacent resonators,*

_{c}*κ*∝1/

*Q*. At the same time, as Soljačić et al. have shown,

*Q*increases as the steepness of the dispersion increases, thus

*v*∝1/

_{g}*Q*. It follows that

*κ*∝

*v*, meaning that in the EIT CROW system, manipulation of the coupling coefficient is directly coupled to a controllable parameter, namely Ω

_{g}*.*

_{c}## 3. Conclusion

*ε*of the highly dispersive EIT background into the time-stepping algorithm. We found that by adjusting the control field strength, dynamic control the shape of the CROW band was enabled, whereas the shape of the CROW band is typically fixed since it is controlled by the coupling strength of the cavities, eg. distance

*R*between the defects. We classified two regimes of the dynamic system by the relative frequency band width of the vacuum CROW structure verses the band width of the EIT transparency window. In the two regimes, the rate of change in the coupling factor (or the group velocity) as a function of the control field Rabi frequency differed, being larger in the case where Ω

*≥ Δ*

_{c}*ω*. We also found that the relationship of the EIT-CROW coupling factor,

*κ*(Ω

*) follows that of the EIT group velocity,*

_{c}*v*(Ω

_{g}*). In the spirit of the TB analysis, one can draw a parallel between the atomic separation distance and the resonator cavity separation in the CROW. Therefore, one can also think of the dispersion of the media or Ω*

_{c}*in the cavity as the ‘atomic potential’. As Ω*

_{c}*decreases, the atomic potential increases, reducing the coupling between modes. To achieve a usable slow light effect, a large bandwidth is required to accept the input pulse into the system before the band is flattened and the pulse stopped [24*

_{c}24. M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. **92**, 083901 (2004). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | D. Marcuse, |

2. | P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. |

3. | A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. |

4. | E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. |

5. | S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. |

6. | M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystal,” Phys. Rev. Lett. |

7. | K.-J. Boller, A. Imamoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. |

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) |

9. | M. D. Lukin, M. Fleischhauer, M. O. Scully, and V. L. Velichansky, “Intracavity electromagnetically induced transparency,” Opt. Lett. |

10. | G. Müller, M. Müller, A. Wicht, R.-H. Rinkleff, and K. Danzmann, “Optical resonator with steep internal dispersion,” Phys. Rev. A |

11. | M. Soljačić, E. Lidorikis, J. D. Hau, and Joannopoulos, “Enhancement of microcavity lifetimes using highly dispersive materials,” Phys. Rev. E |

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. |

13. | M. O. Scully and M. S. Zubairy, |

14. | P. Jänes, J. Tidström, and L. Thylén, “Limits on optical pulse compression and delay bandwidth product in electromagnetically induced transparency media,” J. Lightwave Technol. |

15. | M. Okoniewski, M. Mrozowski, and M. Stuchly, “Simple treatment of multi-term dispersion in FDTD,” IEEE Micro. Guided Wave Lett. |

16. | A. Taflove and S. C. Hagness, |

17. | M. Han, R. Dutton, and S. Fan, “Model dispersive media in finite-difference time-domain method with complex-conjugate pole-residue pairs,” Microwave andWireless Components Letters, IEEE [see also IEEE Micro. Guided Wave Lett.] |

18. | J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. |

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.] |

20. | G. A. Baker and J. L. Gammel, |

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) |

23. | J. Tidström, P. Jönes, and L. M. Andersson, “Delay bandwidth product of electromagnetically induced transparency media,” Phys. Rev. A |

24. | M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. |

**OCIS Codes**

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(230.4555) Optical devices : Coupled resonators

(130.5296) Integrated optics : Photonic crystal waveguides

**ToC Category:**

Optical Devices

**History**

Original Manuscript: June 28, 2007

Revised Manuscript: July 24, 2007

Manuscript Accepted: July 27, 2007

Published: August 1, 2007

**Citation**

Curtis W. Neff, L. Mauritz Andersson, and Min Qiu, "Coupled resonator optical waveguide structures with highly dispersive media," Opt. Express **15**, 10362-10369 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-16-10362

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### References

- D. Marcuse, Theory of Dielectric Optical Waveguides,(Academic Press, New York, 1974).
- P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976). [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]
- 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]
- 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]
- M. D. Lukin, M. Fleischhauer, M. O. Scully, and V. L. Velichansky, "Intracavity electromagnetically induced transparency," Opt. Lett. 23, 295-297 (1998). [CrossRef]
- 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. 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]
- 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]
- M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, England, 1997).
- 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]
- 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]
- 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).
- J.-P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comp. Phys. 114, 185-200 (1994). [CrossRef]
- 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
- 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]
- J. Tidström, P. Jänes, and L. M. Andersson, "Delay bandwidth product of electromagnetically induced transparency media," Phys. Rev. A 75, 53803 (2007). [CrossRef]
- M. F. Yanik and S. Fan, "Stopping light all optically," Phys. Rev. Lett. 92, 083901 (2004). [CrossRef] [PubMed]

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