## Collective emission and absorption in a linear resonator chain

Optics Express, Vol. 17, Issue 17, pp. 15210-15215 (2009)

http://dx.doi.org/10.1364/OE.17.015210

Acrobat PDF (115 KB)

### Abstract

We show that a linear chain of circular macroscopic resonators coupled in parallel demonstrates the phenomena of “superabsorption” and superradiance. Both the frequency spectrum of the transmitted light through the resonator chain and the decay rate of the resonator chain being prepared in a proper initial state are proportional to the number of resonators in the chain, *N*, and the intensity of the emitted radiation grows as *N*^{2}. The spectral bandwidth, the growth of the decay rate, and the intensity are restricted by the dispersion of the waveguides connecting the resonators.

© 2009 Optical Society of America

## 1. Introduction

*N*), while the intensity of the decay is proportional to the square of the number of resonators (

*N*

^{2}), when all the resonators are prepared in the same excited state. We also show that the spectral linewidth of the resonator structure is proportional to the number of resonators (

*N*).

*N*coherent oscillators, or the far-field diffraction of plane waves from a periodic aperture of

*N*slits, or free induction decay. In those cases the intensity is also proportional to

*N*

^{2}. The coherence of the decay is enforced from the beginning. This is different from the case of superfluorescence [28], which does not simply arise from the the coherence of the spontaneous emission, but from the interaction among atoms or molecules, which necessarily involves nonlinear couplings via the Maxwell-Bloch equations. Therefore, the resonator chain is one more entirely classical system that demonstrates superradiance.

29. J. K. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett. **31**, 456–458 (2006). [PubMed]

30. A. M. Kapitonov and V. N. Astratov, “Observation of nanojet-induced modes with small propagation losses in chains of coupled spherical cavities,” Opt. Lett. **32**, 409–411 (2007). [PubMed]

37. S. F. Mingaleev, A. E. Miroshnichenko, and Y. S. Kivshar, “Coupled-resonator-induced reflection in photonic-crystal waveguide structures,” Opt. Express **16**, 11647–11659 (2008). [PubMed]

## 2. Superabsorption

16. Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. **96**, 123901 (2006). [PubMed]

*T*is the coupling constant,

*k*(ω) is the wave vector in the material,

*a*is the radius of the resonator. The spectral properties of the

*j*elementary section of the resonator chain is, hence, described by

^{th}*L*. It is convenient to rewrite Eq. (2) in the compact form v

_{j+}1=

*U*v

_{j}*. The transfer matrix*

_{j}*U*is characterized with

_{j}*det*(

*U*)=1 because we consider a lossless system.

_{j}*N*resonators is given by the product Π

^{j=N}^{j=1}

*U*. For a small number of resonators the spectrum of the add/drop ports is Lorentzian:

_{j}*t*and

*r*are the power transmission of the add/drop port of the chain, 2γ is the FWHM of the standing alone resonator, ω

_{0}is the resonant frequency of the selected mode. We have assumed that

*k*(ω)

*a*=

*m*

_{1}and

*k*(ω)

*L*=2

*πm*

_{2}, where

*m*

_{1}and

*m*

_{2}are integer numbers. Equations (3) and (4) show that adding a resonator to the chain broadens the spectrum of the system by 2γ. The constructive interference of the decays of the resonators into the waveguides results in the increased loading of the whole chain.

*L*=0, and

*N*formally can go to infinity. Using Bloch condition [1]

_{max}*L*=2

*πa*. The maximum spectral bandwidth in this case is less than the FSR (see Fig. 2(b)). The maximum number of the constructively interfering resonators is restricted by

*F*is the finesse of the individual resonator. Equation (6) was derived using numerical simulations defining

*N*as the ratio of the width of the spectral passband of the infinite resonator chain and the FWHM of a single resonator. Similar expression can be obtained from simple physical reasoning. The decay time of the set of constructively interacting resonators τ

_{max}_{d1}=(2γ

*N*)

^{-1}should exceed the phase delay resulting from light traveling through the off-resonant part of the structure τ

_{d2}=

*nNL/c*. Comparing those equations we get an expression like Eq. (6) with a slightly different numerical coefficient. Therefore, the higher is the finesse, the larger is the number of the constructively interfering resonators for a given distance between the resonators. On the other hand, the absolute spectral bandwidth of the passband decreases with increasing finesse.

*N*→∞ if

_{max}*L*/

*a*→0, so the decay time of the resonator structure goes to zero and the spectral width goes to infinity. This condition can be fulfilled if the resonators are located in different planes, however its practical implementation is hardly possible. Fundamentally, though, only the spectral width less than the free spectral range (FSR) of the resonators, or the spectral width of the first Brillouin zone, makes sense since the travel time of light from the input to the drop port is always larger that the round trip time within a resonator, which is necessary for the interference process to be established.

## 3. Superradiance

*N*<

*N*. The free decay of the resonators can be described by a set of linear differential equations in this case:

_{max}*e*is the amplitude of the field in

*j**j*resonator. The set has one collective mode with a decay rate equal to γ

^{th}*N*. This is the mode hat is coupled with the external environment. The other modes are not decaying and are the “dark state” modes. If all the modes are initially prepared in the same state

*e*=

_{j}*e*, the chain of resonators decays with time τ

_{0}_{2}. The intensity of the emitted light is proportional to

*N*

^{2}, as in conventional superradiative systems. To observe superradiant decay experimentally one ultimately needs to excite all resonators with a single phase-coherent source, which should be uncoupled after the resonators are pumped. The coupling-uncoupling of resonators can be realized, e.g., either with MEMS technique [42] or electro-optically [14

14. M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. **93**, 233903 (2004). [PubMed]

## 4. Conclusion

## References and links

1. | J. Heebner, R. Grover, and T. A. Ibrahim, |

2. | R. J. C. Spreeuw, N. J. van Druten, M. W. Beijersbergen, E. R. Eliel, and J. P. Woerdman, “Classical realization of a strongly driven two-level system,” Phys. Rev. Lett. |

3. | C. Pare, L. Gagnon, and P. A. Belanger, “Aspherical laser resonators: An analogy with quantum mechanics,” Phys. Rev. A |

4. | R. J. C. Spreeuw, M.W. Beijersbergen, and J. P. Woerdman, “Optical ring cavities as tailored four-level systems: An application of the group U(2,2),” Phys. Rev. A |

5. | D. Bouwmeester, N.H. Dekker, F.E.v. Dorsselaer, C.A. Schrama, P.M. Visser, and J.P. Woerdman, “Observation of Landau-Zener dynamics in classical optical systems,” Phys. Rev. A |

6. | A. E. Siegman, “Laser beams and resonators: Beyond the 1960s ,” IEEE J. Sel. Top. Quantum Electron. |

7. | L. Maleki, A. B. Matsko, D. Strekalov, and A. A. Savchenkov “Photonic media with whispering-gallery modes,” Proc. SPIE |

8. | T. Opatrny and D. G. Welsch, “Coupled cavities for enhancing the cross-phase-modulation in electromagnetically induced transparency,” Phys. Rev. A |

9. | D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A |

10. | D. D. Smith and H. Chang, “Coherence phenomena in coupled optical resonators,” J. Mod. Opt. |

11. | L. Maleki, A. B. Matsko, A. A. Savchenkov, and V. S. Ilchenko, “Tunable delay line with interacting whispering gallery-mode resonators,” Opt. Lett. |

12. | A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Interference effects in lossy resonator chains,” J. Mod. Opt. |

13. | W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. |

14. | M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. |

15. | A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, “Induced transparency and absorption in coupled whispering-gallery microresonators,” Phys. Rev. A |

16. | Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. |

17. | Q. Xu, J. Shakya, and M. Lipson, “Direct measurement of tunable optical delays on chip analogue to electromagnetically induced transparency, Opt. Express |

18. | A. Imamoglu, “Interference of radiatively broadened resonances,” Phys. Rev. A |

19. | S. E. Harris, “Electromagnetically induced transparency,” Phys. Today , |

20. | A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonance in nanoscale structures,” arXiv.org>condmat>arXiv:0902.3014 |

21. | S. T. Chu, B. E. Little, W. Pan, T. Kaneko, and Y. Kukubun, “Second-order filter response from parallel coupled glass microring resonators,” IEEE Phot. Tech. Lett. |

22. | D. Dragoman and M. Dragoman, |

23. | D. Dragoman, “Classical versus complex fractional Fourier transformation,” J. Opt. Soc. Am. A |

24. | S. Chavez-Cerda, H. M. Moya Cessa, and J. R. Moya Cessa, “Quantum-like entanglement in classical optics,” Optics and Photonics News |

25. | Y.-F. Xiao, X.-B. Zou, W. Jiang, Y.-L. Chen, and G.-C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A |

26. | R. H. Dicke, “Coherence in spontaneous radiation process,” Phys. Rev. |

27. | A. E. Siegman, |

28. | R. Bonifacio and L. A. Lugiato, “Cooperative radiation process in two-level systems: Superfluorescence,” Phys. Rev. A |

29. | J. K. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett. |

30. | A. M. Kapitonov and V. N. Astratov, “Observation of nanojet-induced modes with small propagation losses in chains of coupled spherical cavities,” Opt. Lett. |

31. | F. Xia, L. Sekaric, and Y. Vlasov, “Resonantly enhanced all optical buffers on a silicon chip,” IEEE Proceedings of Photonics in Switching Symposium, pp. 7–8 (2007). |

32. | F. L. Kien and K. Hakuta, “Cooperative enhancement of channeling of emission from atoms into a nanofiber,” Phys. Rev. A |

33. | P. Urquhart, “Compound optical-fiber-based resonators,” J. Opt. Soc. Am. A |

34. | K. Oda, N. Takato, and H. Toba, “A wide-FSR waveguide double-ring resonator for optical FDM transmission systems”, J. Lightwave Technol. |

35. | B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. |

36. | O. Schwelb and I. Frigyes, “A design for a high finesse parallel-coupled microring resonator filter,” Microwave Opt. Technol. Lett. |

37. | S. F. Mingaleev, A. E. Miroshnichenko, and Y. S. Kivshar, “Coupled-resonator-induced reflection in photonic-crystal waveguide structures,” Opt. Express |

38. | E. L. Ivchenko, A. I. Nesvizhskii, and S. Jorda, “Bragg reflection of light from quantum-well structures,” Phys. Solid State |

39. | M. Hubner, J. P. 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. |

40. | L. Pilozzi, A. DAndrea, and K. Cho, “Optical response in multi-quantum wells under Bragg conditions,” Phys. Stat. Sol. (c) |

41. | L. I. Deych, M. V. Erementchouk, A. A. Lisyansky, E. L. Ivchenko, and M. M. Voronov, “Exciton luminescence in one-dimensional resonant photonic crystals: A phenomenological approach,” Phys. Rev. B |

42. | J. Yao, D. Leuenberger, M.-C. M. Lee, and M. C. Wu, “Silicon microtoroidal resonators with integrated MEMS tunable coupler,” IEEE J. Sel. Top. Quantum. Electron. |

**OCIS Codes**

(140.4780) Lasers and laser optics : Optical resonators

(230.5750) Optical devices : Resonators

(230.4555) Optical devices : Coupled resonators

**ToC Category:**

Optical Devices

**History**

Original Manuscript: June 22, 2009

Revised Manuscript: August 3, 2009

Manuscript Accepted: August 4, 2009

Published: August 12, 2009

**Citation**

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, "Collective emission and absorption in a linear resonator chain," Opt. Express **17**, 15210-15215 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-15210

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### References

- J. Heebner, R. Grover, and T. A. Ibrahim, Optical Microresonators: Theory, Fabrication, and Applications (Springer-Verlag, London, 2008).
- R. J. C. Spreeuw, N. J. van Druten, M. W. Beijersbergen, E. R. Eliel, and J. P. Woerdman, "Classical realization of a strongly driven two-level system," Phys. Rev. Lett. 65 2642-2645 (1990). [PubMed]
- C. Pare, L. Gagnon, and P. A. Belanger, "Aspherical laser resonators: An analogy with quantum mechanics," Phys. Rev. A 46, 4150-4160 (1992). [PubMed]
- R. J. C. Spreeuw, M. W. Beijersbergen, and J. P. Woerdman, "Optical ring cavities as tailored four-level systems: An application of the group U(2,2)," Phys. Rev. A 45, 1213-1229 (1992). [PubMed]
- D. Bouwmeester, N. H. Dekker, F. E. v. Dorsselaer, C. A. Schrama, P. M. Visser, and J. P. Woerdman, "Observation of Landau-Zener dynamics in classical optical systems," Phys. Rev. A 51 646-654 (1995). [PubMed]
- A. E. Siegman, "Laser beams and resonators: Beyond the 1960s," IEEE J. Sel. Top. Quantum Electron. 6, 1389-1399 (2000).
- L. Maleki, A. B. Matsko, D. Strekalov, and A. A. Savchenkov, "Photonic media with whispering-gallery modes," Proc. SPIE 5708 180-186 (2005).
- T. Opatrny and D. G. Welsch, "Coupled cavities for enhancing the cross-phase-modulation in electromagnetically induced transparency," Phys. Rev. A 64, 023805 (2001).
- D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, "Coupled-resonator-induced transparency," Phys. Rev. A 69, 063804 (2004).
- D. D. Smith and H. Chang, "Coherence phenomena in coupled optical resonators," J. Mod. Opt. 51, 25032513 (2004).
- L. Maleki, A. B. Matsko, A. A. Savchenkov, and V. S. Ilchenko, "Tunable delay line with interacting whispering gallery-mode resonators," Opt. Lett. 29, 626-628 (2004).
- A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, "Interference effects in lossy resonator chains," J. Mod. Opt. 51, 25152522 (2004).
- W. Suh, Z. Wang, and S. Fan, "Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities," IEEE J. Quantum Electron. 40, 15111518 (2004).
- M. F. Yanik, W. Suh, Z. Wang, and S. Fan, "Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency," Phys. Rev. Lett. 93, 233903 (2004). [PubMed]
- A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, "Induced transparency and absorption in coupled whispering-gallery microresonators," Phys. Rev. A 71, 043804 (2005).
- Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, "Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency," Phys. Rev. Lett. 96, 123901 (2006). [PubMed]
- Q. Xu, J. Shakya, and M. Lipson, "Direct measurement of tunable optical delays on chip analogue to electromagnetically induced transparency," Opt. Express 14, 63-68 (2006).
- A. Imamoglu, "Interference of radiatively broadened resonances," Phys. Rev. A 40, 2835 (1989). [PubMed]
- S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 3642 (1997).
- A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, "Fano resonance in nanoscale structures," arXiv.org > condmat > arXiv:0902.3014
- S. T. Chu, B. E. Little, W. Pan, T. Kaneko, and Y. Kukubun, "Second-order filter response from parallel coupled glass microring resonators," IEEE Phot. Tech. Lett. 11, 1426-1428 (1999).
- D. Dragoman and M. Dragoman, Quantum-Classical Analogies (Springer, 2004).
- D. Dragoman, "Classical versus complex fractional Fourier transformation," J. Opt. Soc. Am. A 26, 274-277 (2009).
- S. Chavez-Cerda, H. M. M. Cessa, and J. R. M. Cessa, "Quantum-like entanglement in classical optics," Opt. Photon. News 12, 38-38 (2007).
- Y.-F. Xiao, X.-B. Zou, W. Jiang, Y.-L. Chen, and G.-C. Guo, "Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems," Phys. Rev. A 75, 063833 (2007).
- R. H. Dicke, "Coherence in spontaneous radiation process," Phys. Rev. 93, 99-110 (1954).
- A. E. Siegman, Lasers (University Science Books, Mill Valley, California, 1986).
- R. Bonifacio and L. A. Lugiato, "Cooperative radiation process in two-level systems: Superfluorescence," Phys. Rev. A 11, 1507-1521 (1975).
- J. K. Poon, L. Zhu, G. A. DeRose, and A. Yariv, "Transmission and group delay of microring coupled-resonator optical waveguides," Opt. Lett. 31, 456-458 (2006). [PubMed]
- A. M. Kapitonov and V. N. Astratov, "Observation of nanojet-induced modes with small propagation losses in chains of coupled spherical cavities," Opt. Lett. 32, 409-411 (2007). [PubMed]
- F. Xia, L. Sekaric, and Y. Vlasov, "Resonantly enhanced all optical buffers on a silicon chip," IEEE Proceedings of Photonics in Switching Symposium, pp. 7-8 (2007).
- F. L. Kien and K. Hakuta, "Cooperative enhancement of channeling of emission from atoms into a nanofiber," Phys. Rev. A 77, 013801 (2008).
- P. Urquhart, "Compound optical-fiber-based resonators," J. Opt. Soc. Am. A 5, 803-812 (1988).
- K. Oda, N. Takato and H. Toba, "A wide-FSR waveguide double-ring resonator for optical FDM transmission systems," J. Lightwave Technol. 9, 728-736 (1991).
- B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997).
- O. Schwelb and I. Frigyes, "A design for a high finesse parallel-coupled microring resonator filter," Microwave Opt. Technol. Lett. 38, 125-129 (2003).
- S. F. Mingaleev, A. E. Miroshnichenko, and Y. S. Kivshar, "Coupled-resonator-induced reflection in photoniccrystal waveguide structures," Opt. Express 16, 11647-11659 (2008). [PubMed]
- E. L. Ivchenko, A. I. Nesvizhskii, and S. Jorda, "Bragg reflection of light from quantum-well structures," Phys. Solid State 36, 1156-1161 (1994).
- M. Hubner, J. P. 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, 2841-2844 (1999).
- L. Pilozzi, A. DAndrea, and K. Cho, "Optical response in multi-quantum wells under Bragg conditions," Phys. Stat. Sol. (C) 1, 14101419 (2004).
- L. I. Deych, M. V. Erementchouk, A. A. Lisyansky, E. L. Ivchenko, and M. M. Voronov, "Exciton luminescence in one-dimensional resonant photonic crystals: A phenomenological approach," Phys. Rev. B 76, 075350 (2007).
- J. Yao, D. Leuenberger, M.-C. M. Lee, and M. C. Wu, "Silicon microtoroidal resonators with integrated MEMS tunable coupler," IEEE J. Sel. Top. Quantum. Electron. 13, 202-208 (2007).

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