## The Serpentine Optical Waveguide: engineering the dispersion relations and the stopped light points |

Optics Express, Vol. 19, Issue 12, pp. 11517-11528 (2011)

http://dx.doi.org/10.1364/OE.19.011517

Acrobat PDF (1329 KB)

### Abstract

We present a study a new type of optical slow-light structure comprising a serpentine shaped waveguide were the loops are coupled. The dispersion relation, group velocity and GVD are studied analytically using a transfer matrix method and numerically using finite difference time domain simulations. The structure exhibits zero group velocity points at the ends of the Brillouin zone, but also within the zone. The position of mid-zone zero group velocity point can be tuned by modifying the coupling coefficient between adjacent loops. Closed-form analytic expressions for the dispersion relations, group velocity and the mid-zone zero v_{g} points are found and presented.

© 2011 OSA

## 1. Introduction

1. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature **397**(6720), 594–598 (1999). [CrossRef]

10. B. Z. Steinberg, J. Scheuer, and A. Boag, “Rotation-induced superstructure in slow-light waveguides with mode-degeneracy: optical gyroscopes with exponential sensitivity,” J. Opt. Soc. Am. B **24**(5), 1216–1224 (2007). [CrossRef]

11. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. **15**(6), 998–1005 (1997). [CrossRef]

14. R. C. Polson, G. Levina, and Z. V. Vardeny, “Spectral analysis of polymer microring lasers,” Appl. Phys. Lett. **76**(26), 3858–3860 (2000). [CrossRef]

2. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**(11), 711–713 (1999). [CrossRef]

5. J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, “Coupled resonator optical waveguides: towards slowing and storing of light,” Opt. Photon. News **16**(2), 36–40 (2005). [CrossRef]

6. J. Heebner, P. Chak, S. Pereira, J. Sipe, and R. Boyd, “Distributed and localized feedback in microresonator sequences for linear and nonlinear optics,” J. Opt. Soc. Am. B **21**(10), 1818–1832 (2004). [CrossRef]

16. J. Heebner, R. Boyd, and Q. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B **19**(4), 722–731 (2002). [CrossRef]

17. S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Y. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express **16**(2), 1104–1114 (2008). [CrossRef] [PubMed]

20. O. Weiss and J. Scheuer, “Side coupled adjacent resonators CROW--formation of mid-band zero group velocity,” Opt. Express **17**(17), 14817–14824 (2009). [CrossRef] [PubMed]

_{g}= 0) at the band-edge. In most of the more extensively studied slow-light structures (SLSs) such as the CROW and the SCISSOR, the zero group-velocity points are formed at the edges of the Brillouin zone (i.e. at

*K*= 0 and

*K*= π where

*K*is the Bloch wavenumber) due to the Bragg resonances and the cavity resonances of the structure. Consequently, the dispersion relations of these structure exhibit cosine-like shapes and the ability to engineer and modify these relations is quite limited. Formation of zero group velocity points inside the Brillouin zone has also been shown to exist in structures allowing for two distinct, counter-propagating, light-paths such as the side-coupled adjacent resonators CROW (SC-CROW) [19

19. P. Chak, J. K. Poon, and A. Yariv, “Optical bright and dark states in side-coupled resonator structures,” Opt. Lett. **32**(13), 1785–1787 (2007). [CrossRef] [PubMed]

20. O. Weiss and J. Scheuer, “Side coupled adjacent resonators CROW--formation of mid-band zero group velocity,” Opt. Express **17**(17), 14817–14824 (2009). [CrossRef] [PubMed]

18. A. A. Sukhorukov, A. V. Lavrinenko, D. N. Chigrin, D. E. Pelinovsky, and Y. S. Kivshar, “Slow-light dispersion in coupled periodic waveguides,” J. Opt. Soc. Am. B **25**(12), C65–C74 (2008). [CrossRef]

19. P. Chak, J. K. Poon, and A. Yariv, “Optical bright and dark states in side-coupled resonator structures,” Opt. Lett. **32**(13), 1785–1787 (2007). [CrossRef] [PubMed]

20. O. Weiss and J. Scheuer, “Side coupled adjacent resonators CROW--formation of mid-band zero group velocity,” Opt. Express **17**(17), 14817–14824 (2009). [CrossRef] [PubMed]

18. A. A. Sukhorukov, A. V. Lavrinenko, D. N. Chigrin, D. E. Pelinovsky, and Y. S. Kivshar, “Slow-light dispersion in coupled periodic waveguides,” J. Opt. Soc. Am. B **25**(12), C65–C74 (2008). [CrossRef]

_{g}points were also found theoretically in the dispersion relations of optical microcoil resonator structures [21

21. M. Sumetsky, “Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation,” Opt. Express **13**(11), 4331–4340 (2005). [CrossRef] [PubMed]

*k*

_{0}⋅

*n*

_{eff}⋅(2π

*R*+ 2

*L*). The couplers in the structure serve as partial reflectors due to the geometry of the SOW, similar to the role of the interfaces between adjacent layers in a Bragg stack.

_{B}21. M. Sumetsky, “Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation,” Opt. Express **13**(11), 4331–4340 (2005). [CrossRef] [PubMed]

22. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. **75**(4), 1896–1899 (1994). [CrossRef]

26. S. Mookherjea, “Semiconductor coupled-resonator optical waveguide laser,” Appl. Phys. Lett. **84**(17), 3265–3267 (2004). [CrossRef]

22. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. **75**(4), 1896–1899 (1994). [CrossRef]

26. S. Mookherjea, “Semiconductor coupled-resonator optical waveguide laser,” Appl. Phys. Lett. **84**(17), 3265–3267 (2004). [CrossRef]

## 2. Structure parameters and transfer matrix

*A*); 2) The phase accumulation in the curved waveguide connecting the upper and lower loops (section

*B*); 3) the coupling coefficient,

*κ*, between the loops. Regarding the third parameter, we note that the structure can be generalized by allowing the coupling coefficients between the upper and lower loops to be different. However, such generalization does not modify substantially the properties of the structure and in order to simplify the analysis we strict ourselves to identical coupling.

*E*

_{1}..

*E*

_{6}as defined in Fig. 1. The electric field amplitudes at adjacent unit cells can be connected by a transfer matrix

*M*:where M is a 6 × 6 unitary matrix given by:

*ϕ*and

_{A}*ϕ*represent the phase accumulation through sections

_{B}*A*and

*B*respectively. Invoking the Bloch theorem, Eq. (1) can be used to derive the spectral properties of the structure as well as the field distribution for any set of the parameters.

## 3. Band structure, dispersion relations and group velocity

*ϕ*+ 2

_{A}*ϕ*=

_{B}*k*

_{0}⋅

*n*

_{eff}⋅(2π

*R*+ 2

*L*), where

_{B}*k*

_{0}is the wavenumber,

*n*

_{eff}is the effective index of the waveguide mode,

*R*is the radius of section

*A*and

*L*is the overall length of section

_{B}*B*(see Fig. 1) given by

*L*= 2

_{B}*αR*.

*A*is

*R*= 5μm and section

*B*consists of two circular sections with radius

*R*= 5μm and an angle of

*α*= 58.5°. Note, that the choice of

*α*not only affects the overall length of section

*B*but also the width of the gap between the waveguide loops and, hence, the coupling coefficient.

*λ*= 1.55mm the refractive indices of the Si and the silica are respectively

*n*

_{Si}= 3.477 and

*n*

_{silica}= 1.4455. The resulting effective index is

*n*

_{eff}= 2.362 for the lowest TE mode. Figure 3(a) depicts the dispersion relation of an SOW structure with the above-mentioned cross-section and coupling coefficient of

*κ*= 0.24. Figure 3(b) depicts the corresponding normalized group velocity (v

_{g}/Λ) for the band which is marked in Fig. 3(a). For simplicity, the dispersion of the waveguide cross-section and the material was neglected. Nevertheless, this dispersion can be readily introduced by modifying

*n*

_{eff}according to the wavelength.

*f*

_{FSR}=

*c/n*

_{eff}

*R*(2π + 4

*α*), is 2.45THz. Each passband consists of a pair of mirror image dispersion curves where the symmetry lines are located at frequencies which are “anti-resonant” with the optical length of the waveguide structure in each unit cell i.e. frequencies satisfying

*k*

_{0}⋅

*n*

_{eff}⋅(2π

*R*+ 2

*L*) = 2π⋅(

_{B}*m*+ 1/2) where

*m*is an integer. This can be easily obtained from Eq. (4) by substituting

*K*Λ = ± π which yields cos(4

*ϕ*+ 2

_{A}*ϕ*) = −1. The term “anti-resonant” is in quotation marks because there is no actual resonance or anti-resonance associated with this optical length.

_{B}*K*Λ≈ ± 0.491π for the parameters chosen in Fig. 3). It should be noted that in the vicinity of the mid-zone zero group velocity point there are two possible (positive) group velocities for every frequency. For example, at

*f*= 190.4THz there are two possible group velocities –

*v*

_{g}/π = 1.06ps

^{−1}and

*v*

_{g}/π = 2.38ps

^{−1}. The reason for the two possible group velocities stem for the two paths through which the light can propagate along the structure – along the waveguide or by tunneling through the directional couplers. As a result, for a given frequency there are two possible Bloch modes (with different direct/tunneled propagation combination) which possesses different Bloch wavenumbers and, correspondingly, group velocities.

*κ*, exceeds a certain value, a zero group velocity point is formed within the Brillouin zone; 2) As the coupling coefficient is increased, the zero group velocity point shifts to larger Bloch wavenumbers,

*K*, in the dispersion diagram, approaching

*K*Λ = 2π/3 for

*κ*→1; 3) At

*κ =*0.5, the two curves forming each transmission band collide at

*K*= 0. Increasing the coupling coefficient further results in a formation of a crossing of the two curves.

*ω*and substituting v

_{g}= (d

*K*/d

*ω*)

^{−1}yields the group velocity at each point:

_{g}= 0 to Eq. (5) yields a quadratic equation for cos(

*K*Λ) which solutions provide the zero v

_{g}point:

*κ*= 1 in Eq. (6) yields the maximal Bloch number for which such point is formed –

*K*Λ = cos

^{−1}(1/2).

_{g}= d

*ω*/d

*K*for

*κ*= 0.1, i.e. at the coupling level where the mid-zone zero group velocity is formed. Figure 6 depicts the group velocity of the structure defined in Section 3 for

*κ*= 0.1. As shown in the figure, an inflection point accompanied by a flat region is formed around

*K*= 0.

## 4. Finite difference time domain simulations

*x*(corresponding to the horizontal coordinate in Fig. 1) and perfectly matched layers in all other boundaries. The dimensions and parameters of the structure were as detailed in Section 3.

*n*

_{eff}= 2.362 and

*κ*= 0.245 respectively. The excellent agreement between the simulations and the analytic model is clear, thus validating the theoretical analysis of Section 3.

## 5. Practical considerations – finite SOW structure and propagation losses

*N*unit cells, one can relate the fields at the output to the fields at the input by a product of three matrices: 1) an input matrix, connecting the fields at the input to the fields at the first unit cell; 2)

*N*-unit-cell transfer matrix which is given by

*M*=

_{N}*M*

^{N}where

*M*is given by Eq. (2); and 3) an output matrix, connecting the fields at the last unit cell to the fields at the output. Note that assuming the SOW is excited from one of its sides, the outputs of the structures are the transmitted field and the reflected field (see Fig. 1).

*t*|

^{2}in Fig. 1) of a lossless SOW structure consisting of five unit cells for coupling coefficients ranging from 0.01 to 0.99. The dependence of the transfer function on the coupling coefficient is quite complex. It seems that the SOW response can be divided into several sub-sections depending on the coupling level. At low coupling levels (

*κ<*0.3) the transmission function consists of several transmission peaks (see inset 1). Although similar property is exhibited by several, finite, slow-light structures (e.g. CROW) it should be noted that the number of the transmission peaks decreases as the coupling coefficient is increased. This is in contrast to CROWs, for example, in which the number of transmission peaks is determined solely by the number of unit cells.

*κ*~0.35 and then broadening again as the coupling is further increased. The reason for this behavior is that for low coupling levels the SOW approaches a simple curved waveguide (which does not exhibit structure dependent spectral properties) while for

*κ*→1 the strong coupling provide a shorter route for the light, with basically no back reflections, thus yielding an almost unity and wavelength independent spectral transmission. This is in contrast to CROW where the transmission band monotonically increases when the coupling in increased.

## 6. Pulse propagation

*κ*= 0.15. The central frequency is set to be 191.72THz (1564.8nm) which is approximately at the center of the slow-light band (the shallow and linear slope regime of the red curve in Fig. 4) which is formed around

*K*Λ = 0.17π. The linked avi file (Media 1) shows the temporal evolution of the field in the structure. As seen in the figure the pulse maintains its shape as it propagates through the structure although some broadening is also observed. The distortion is formed by the GVD of the structure at the wavelength of the pulse.

*κ*= 0.15) the group velocity is found to be 0.35 Unit-cells/ps. Taking the length of the unit-cell to be the over-all length of the waveguide comprising it (i.e. its length for

*κ*= 0 – a simple curved waveguide), the corresponding group velocity is found to be 0.14 ×

*c*. Reducing the coupling coefficient towards 0.1 would yield slower group velocities.

## 7. Conclusions

27. J. B. Khurgin, “Dispersion and loss limitations on the performance of optical delay lines based on coupled resonant structures,” Opt. Lett. **32**(2), 133–135 (2007). [CrossRef]

28. D. Miller, “Fundamental limit for optical components,” J. Opt. Soc. Am. B **24**(10), A1–A18 (2007). [CrossRef]

## Acknowledgments

## References and links

1. | L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature |

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

3. | J. E. Heebner and R. W. Boyd, “'Slow’ and 'fast' light in resonator-coupled waveguides,” J. Mod. Opt. |

4. | A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. |

5. | J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, “Coupled resonator optical waveguides: towards slowing and storing of light,” Opt. Photon. News |

6. | J. Heebner, P. Chak, S. Pereira, J. Sipe, and R. Boyd, “Distributed and localized feedback in microresonator sequences for linear and nonlinear optics,” J. Opt. Soc. Am. B |

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

8. | B. Z. Steinberg, “Rotating photonic crystals: a medium for compact optical gyroscopes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

9. | J. Scheuer and A. Yariv, “Sagnac effect in coupled-resonator slow-light waveguide structures,” Phys. Rev. Lett. |

10. | B. Z. Steinberg, J. Scheuer, and A. Boag, “Rotation-induced superstructure in slow-light waveguides with mode-degeneracy: optical gyroscopes with exponential sensitivity,” J. Opt. Soc. Am. B |

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

12. | J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. |

13. | T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P.-T. Ho, and C. H. Lee, “All-optical switching in a laterally coupled microring resonator by carrier injection,” IEEE Photon. Technol. Lett. |

14. | R. C. Polson, G. Levina, and Z. V. Vardeny, “Spectral analysis of polymer microring lasers,” Appl. Phys. Lett. |

15. | X. Fengnian, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics |

16. | J. Heebner, R. Boyd, and Q. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B |

17. | S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Y. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express |

18. | A. A. Sukhorukov, A. V. Lavrinenko, D. N. Chigrin, D. E. Pelinovsky, and Y. S. Kivshar, “Slow-light dispersion in coupled periodic waveguides,” J. Opt. Soc. Am. B |

19. | P. Chak, J. K. Poon, and A. Yariv, “Optical bright and dark states in side-coupled resonator structures,” Opt. Lett. |

20. | O. Weiss and J. Scheuer, “Side coupled adjacent resonators CROW--formation of mid-band zero group velocity,” Opt. Express |

21. | M. Sumetsky, “Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation,” Opt. Express |

22. | J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. |

23. | A. G. Yamilov, M. R. Herrera, and M. F. Bertino, “Slow-light effect in dual-periodic photonic lattice,” J. Opt. Soc. Am. B |

24. | K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express |

25. | S. Nojima, “Enhancement of optical gain in two dimensional photonic crystal with active lattice points,” Jpn. J. Appl. Phys. |

26. | S. Mookherjea, “Semiconductor coupled-resonator optical waveguide laser,” Appl. Phys. Lett. |

27. | J. B. Khurgin, “Dispersion and loss limitations on the performance of optical delay lines based on coupled resonant structures,” Opt. Lett. |

28. | D. Miller, “Fundamental limit for optical components,” J. Opt. Soc. Am. B |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(260.2030) Physical optics : Dispersion

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: March 22, 2011

Revised Manuscript: May 2, 2011

Manuscript Accepted: May 24, 2011

Published: May 31, 2011

**Citation**

Jacob Scheuer and Ori Weiss, "The Serpentine Optical Waveguide: engineering the dispersion relations and the stopped light points," Opt. Express **19**, 11517-11528 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-12-11517

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### References

- L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 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(11), 711–713 (1999). [CrossRef]
- J. E. Heebner and R. W. Boyd, “'Slow’ and 'fast' light in resonator-coupled waveguides,” J. Mod. Opt. 49(14), 2629–2636 (2002). [CrossRef]
- A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35(4/5), 365–379 (2003). [CrossRef]
- J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, “Coupled resonator optical waveguides: towards slowing and storing of light,” Opt. Photon. News 16(2), 36–40 (2005). [CrossRef]
- J. Heebner, P. Chak, S. Pereira, J. Sipe, and R. Boyd, “Distributed and localized feedback in microresonator sequences for linear and nonlinear optics,” J. Opt. Soc. Am. B 21(10), 1818–1832 (2004). [CrossRef]
- M. F. Yanik and S. H. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92(8), 083901 (2004). [CrossRef] [PubMed]
- B. Z. Steinberg, “Rotating photonic crystals: a medium for compact optical gyroscopes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056621 (2005). [CrossRef] [PubMed]
- J. Scheuer and A. Yariv, “Sagnac effect in coupled-resonator slow-light waveguide structures,” Phys. Rev. Lett. 96(5), 053901 (2006). [CrossRef] [PubMed]
- B. Z. Steinberg, J. Scheuer, and A. Boag, “Rotation-induced superstructure in slow-light waveguides with mode-degeneracy: optical gyroscopes with exponential sensitivity,” J. Opt. Soc. Am. B 24(5), 1216–1224 (2007). [CrossRef]
- B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997). [CrossRef]
- J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12(3), 320–322 (2000). [CrossRef]
- T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P.-T. Ho, and C. H. Lee, “All-optical switching in a laterally coupled microring resonator by carrier injection,” IEEE Photon. Technol. Lett. 15(1), 36–38 (2003). [CrossRef]
- R. C. Polson, G. Levina, and Z. V. Vardeny, “Spectral analysis of polymer microring lasers,” Appl. Phys. Lett. 76(26), 3858–3860 (2000). [CrossRef]
- X. Fengnian, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2006).
- J. Heebner, R. Boyd, and Q. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 19(4), 722–731 (2002). [CrossRef]
- S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Y. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16(2), 1104–1114 (2008). [CrossRef] [PubMed]
- A. A. Sukhorukov, A. V. Lavrinenko, D. N. Chigrin, D. E. Pelinovsky, and Y. S. Kivshar, “Slow-light dispersion in coupled periodic waveguides,” J. Opt. Soc. Am. B 25(12), C65–C74 (2008). [CrossRef]
- P. Chak, J. K. Poon, and A. Yariv, “Optical bright and dark states in side-coupled resonator structures,” Opt. Lett. 32(13), 1785–1787 (2007). [CrossRef] [PubMed]
- O. Weiss and J. Scheuer, “Side coupled adjacent resonators CROW--formation of mid-band zero group velocity,” Opt. Express 17(17), 14817–14824 (2009). [CrossRef] [PubMed]
- M. Sumetsky, “Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation,” Opt. Express 13(11), 4331–4340 (2005). [CrossRef] [PubMed]
- J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994). [CrossRef]
- A. G. Yamilov, M. R. Herrera, and M. F. Bertino, “Slow-light effect in dual-periodic photonic lattice,” J. Opt. Soc. Am. B 25(4), 599–608 (2008). [CrossRef]
- K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express 4(5), 167–176 (1999). [CrossRef] [PubMed]
- S. Nojima, “Enhancement of optical gain in two dimensional photonic crystal with active lattice points,” Jpn. J. Appl. Phys. 37(Part 2, No. 5B), L565–L567 (1998). [CrossRef]
- S. Mookherjea, “Semiconductor coupled-resonator optical waveguide laser,” Appl. Phys. Lett. 84(17), 3265–3267 (2004). [CrossRef]
- J. B. Khurgin, “Dispersion and loss limitations on the performance of optical delay lines based on coupled resonant structures,” Opt. Lett. 32(2), 133–135 (2007). [CrossRef]
- D. Miller, “Fundamental limit for optical components,” J. Opt. Soc. Am. B 24(10), A1–A18 (2007). [CrossRef]

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