## Coupled mode theory for photonic crystal cavity-waveguide interaction

Optics Express, Vol. 13, Issue 13, pp. 5064-5073 (2005)

http://dx.doi.org/10.1364/OPEX.13.005064

Acrobat PDF (210 KB)

### Abstract

We derive a coupled mode theory for the interaction of an optical cavity with a waveguide that includes waveguide dispersion. The theory can be applied to photonic crystal cavity waveguide structures. We derive an analytical solution to the add and drop spectra arising from such interactions in the limit of linear dispersion. In this limit, the spectra can accurately predict the cold cavity quality factor (Q) when the interaction is weak. We numerically solve the coupled mode equations for the case of a cavity interacting with the band edge of a periodic waveguide, where linear dispersion is no longer a good approximation. In this regime, the density of states can distort the add and drop spectra. This distortion can lead to more than an order of magnitude overestimation of the cavity Q.

© 2005 Optical Society of America

## 1. Introduction

1. M. Loncar et al. “Low-threshold photonic crystal laser,” App. Phy. Lett. **81**, 2680–2682 (2002). [CrossRef]

2. Y. Akahane et al. “Design of a channel drop filter by using a donor-type cavity with high-quality factor in a two-dimensional photonic crystal slab” App. Phys. Lett. **82**, 1341–1343 (2003). [CrossRef]

3. J. Vuckovic and Y. Yamamoto. “Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot,” App. Phys. Lett. **82**, 2374–2376 (2003). [CrossRef]

4. T. Asano et al. “Investigation of channel-add/drop-filtering device using acceptor-type point defects in a two-dimensional photonic-crystal slab,” App. Phys. Lett. **83**, 407–409 (2003) [CrossRef]

5. T. Asano et al. “A channel drop filter using a single defect in a 2d photonic crystal slab - defect engineering with respect to polarization mode and ratio of emissions from upper and lower sides,” J. Lightwave Technol. **21**, 1370–1376 (2003) [CrossRef]

6. C. Seassal et al. “Optical coupling between a two-dimensional photonic crystal-based microcavity and single-line defect waveguide on inp membranes,” IEEE J. Quantum Electron. **38**811–815 (2002) [CrossRef]

7. B.K. Min, J.E. Kim, and H.Y. Park. “Channel drop filters using resonant tunneling processes in two-dimensional triangular lattice photonic crystal slabs,” Opt. Commun. **237**59–63 (2004) [CrossRef]

8. M.F. Yanik and S. Fan. “High-contrast all-optical bistable switching in photonic crystal microcavities,” App. Phy. Lett. **83**, 2739 (2003) [CrossRef]

11. S. Olivier et al. “Cascaded photonic crystal guidesand cavities: spectral studies and their impact on intergrated optics design,” IEEE J. Quantum Electron. **38**, 816–824 (2002) [CrossRef]

12. G.H. Kim et al. “Coupling of small, low-loss hexapole mode with photonic crystal slab waveguide mode,” Opt. Express **12**, 6624–6631 (2004) [CrossRef] [PubMed]

13. M. Okano, S. Kako, and S. Noda. “Coupling between a point-defect cavity and a line-defect waveguide in three-dimensional photonic crystal,” Phys. Rev. B **68**, 235110 (2003) [CrossRef]

14. Ziyang Zhang Min Qiu. “Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2d photonic crystal slabs,” Opt. Express **13**, 2596–2604 (2005) [CrossRef] [PubMed]

9. C. Manolatou et al. “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. **35**, 1322 (1999) [CrossRef]

10. Y. Xu et al. “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E **62**, 7389–7404 (2000) [CrossRef]

9. C. Manolatou et al. “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. **35**, 1322 (1999) [CrossRef]

## 2. Coupled mode theory

*ε*(

**r**) is the relative dielectric constant, and

*c*is the speed of light in vacuum. We define εc as the relative dielectric constant for the cavity,

*εw*as the dielectric constant for the waveguide, and

*ε*for the coupled system. We assume the waveguide dielectric constant to be periodic. Thus, the solutions to Eq. (1) with

_{t}*ε*=

*ε*, denoted

_{w}*ε*is the crystal momentum, and

_{w}, k*z*the direction of propagation in the waveguide. The cavity mode, which is the solution to Eq. (1) with

*ε*=

*ε*as the index, is defined as

_{c}**r**).

*ε*=

*ε*in Eq. (1). Using the standard arguments of coupled mode theory [16], we assume the solution of the coupled system to take on the form

_{t}*a*(

*t*) is the slowly varying component of the cavity, and

*b*(

*k, t*) and

*c*(

*k, t*) are slowly varying components of the forward and backward propagating Bloch states respectively. Plugging the above solution back into Eq. (1), we derive the coupled mode equations

*ν*is a phenomenological decay constant which is added to account for the finite lifetime of the cavity resulting from mechanisms other than cavity-waveguide coupling.

*P*and

_{c}*P*(

_{w}*k*) are external driving terms that can potentially drive the cavity or waveguide at a frequency

*ω*. The damping term

_{p}*η*is also included to give the waveguide modes a finite lifetime. In the analytical calculations we take the limit

*η*→0. In the numerical simulations, however, we set this damping term to a very small value in order to have a well defined steady state solution. The coupling constants are given by

*ε*=

_{c,w}*ε*-

_{t}*ε*.

_{c,w}## 3. Linear dispersion

*ω*(

*k*) to

*k*. For some systems, we can assume that this relation is linear, taking on the form

*V*is the group velocity. When this linearized approximation is valid, an analytical solution can be derived for Eq. (4)–(6). This solution is most easily obtained using the method of Laplace transforms. We take the Lapace transform in time of Eq. (5) and Eq. (6) and plug into Eq. (4). We make the additional approximation

_{g}*P*represents the Cauchy principal value of the expression. This leads to

*a*0 is the initial cavity field, and the other constants are given by

*A*(

*r*) is a real function, so that

*γ*(

*k*)=

*γ*(-

*k*).

*P*=

_{w}*P*=0. The cavity is assumed to contain an initial field

_{c}*a*(0) at time 0. The solution of the cavity field is obtained from the equations of motion to be

*ν*is the rate at which the cavity field escapes into leaky modes, while Γ is the rate at which the cavity field escapes into the waveguide. The total decay rate of the cavity field is simply the sum of these two rates. It is important to note that the coupling rate into the waveguide is inversely proportional to the group velocity. This dependence is simply a reflection of the increased interaction time between the cavity and waveguide at slower group velocities.

*P*=0. One can show that the cavity source term will drive the waveguide field to a steady state value given by

_{w}*P*=0. In this case the waveguide spectrum is

_{c}## 4. Weakly periodic waveguides

9. C. Manolatou et al. “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. **35**, 1322 (1999) [CrossRef]

*c*=1, and the effective index of refraction is

*n*=1. The bandgap constant is set to

_{eff}*D*=0.1. We set

*κ*=

_{ab}*κ*=10

_{ba}^{-2}, and assume that these coupling constants are independent of

*k*. This is a good approximation for small cavities which are highly delocalized in momentum. The cavity decay constant is set to 0.01, which corresponds to a cavity Q of 35 for

*ω*=0.35. This value was selected because it corresponds to a sufficiently narrow linewidth for the simulation, but is not exceedingly narrow that it requires very long simulation times. To simulate drop filtering we set both waveguide and cavity to be initially empty, and pump the waveguide modes with a pump source whose resonant frequency

_{0}*ω*is swept across the cavity resonance. We set the waveguide modes to have a decay constant

_{p}*η*=0.0005, which is much smaller than the decay of the cavity, and pump the system until a steady-state value is reached. We then calculate the transmitted power which is defined as

*t*is a large enough time for all transients to decay so that the system is in steady state. The transmitted power is normalized by the transmitted power of the waveguide without a cavity. This normalization constant is calculated by evolving the system with

_{f}*κ*=

_{ab}*κ*=0.

_{ba}*ω*=0.45, the cavity spectrum significantly narrows. This linewidth distortion is caused by the divergence of the density of states near the band edge. The linewidth when

_{0}*ω*=0.45 corresponds to a quality factor of 180, which is significantly larger than the cold cavity Q of 45. The effect is even more dramatic when

_{0}*ω*=0.46, at which point the cavity resonance is completely inside the bandgap. Despite the fact that the cavity does not resonate with any of the waveguide modes, the extremely high density of states near the band edge still allows the cavity to efficiently scatter light. This results in an extremely sharp resonance right at the band edge frequency, whose linewidth corresponds to a Q of 1800.

_{0}*ω*

_{0}=0.46 and vary the cavity Q. The calculated drop spectra for different Q values are shown in Fig. 3. As can be seen, the linewidth remains almost completely unchanged as we sweep the Q from 46 to 460. We expect this trend to continue as the Q goes up to values of 10,000 and beyond, which are more realistic Q values for photonic crystal micro-cavities.

## 5. Photonic crystal cavity-waveguide system

*a*, slab thickness

*d*=0.65

*a*, hole radius

*r*=0.3

*a*, and refractive index

*n*=3.6. The waveguide is evanescently coupled to a cavity formed by a three hole defect. Fig. 5 shows three dimensional (3D) FDTD simulations of the cavity mode, which has a normalized resonant frequency of 0.251 in units of

*a*/

*λ*, where

*λ*is the free space wavelength. Figure 6 shows the dispersion relation of the waveguide modes, which are calculated by the same 3D FDTD method. The grey area represents the top of the air band and bottom of the dielectric band for the photonic crystal mirrors. The red line is represents the light-line of the slab waveguide. Any modes above this line will be extremely lossy, as they are not confined by total internal reflection. The white area, which lies approximately between the frequencies 0.23 and 0.33, represents the bandgap region of the mirrors. Waveguiding can only happen in this bandgap region. We see that inside the bandgap there are two waveguide bands, represented by light and dark cicles. These two bands cross, meaning that the are of opposite parity and hence do not couple. The insets show the

*z*component of the magnetic field of these two bands at the band edge, taken at the center of the slab. One of the modes has even parity across the center of the waveguide, while the other mode has odd parity. Looking at Fig. 5, one can see that the cavity mode has even parity, and will therefore couple only to the even parity Bloch state. Thus, the odd parity mode can be neglected in the simulations. It is important to note that both the even and odd modes feature a nearly flat dispersion near the band edge.

*κ*and

_{ab}*κ*. This is done by first using FDTD simulations to determine the field at the center of the slab waveguide for different in plane momenta. The overlap integrals in Eq. (7) and Eq. (8) are then evaluated numerically for different in plane momenta. The results are shown in Fig. 7. The cavity is most strongly coupled to waveguide modes near

_{ba}*k*=

*π/a*, which is the flattest region of the dispersion. The calculated coupling constants are used to simulate the waveguide transmission using the same technique as the weak periodicity waveguide. A three hole defect cavity of the type shown in Fig. 4 has a typical Q of about 2000. Such a high quality factor would require extremely long calculation times to properly simulate. Instead, we set the cavity

*Q*=350. The drop spectrum of the cavity is plotted in Fig. 8. From the full-width half-max bandwidth of the cavity one finds a Q of 1300, which is much larger than the cold cavity Q. The width of the transmission spectrum in Fig. 8 is limited by the spectral resolution of the simulation.

## References and links

1. | M. Loncar et al. “Low-threshold photonic crystal laser,” App. Phy. Lett. |

2. | Y. Akahane et al. “Design of a channel drop filter by using a donor-type cavity with high-quality factor in a two-dimensional photonic crystal slab” App. Phys. Lett. |

3. | J. Vuckovic and Y. Yamamoto. “Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot,” App. Phys. Lett. |

4. | T. Asano et al. “Investigation of channel-add/drop-filtering device using acceptor-type point defects in a two-dimensional photonic-crystal slab,” App. Phys. Lett. |

5. | T. Asano et al. “A channel drop filter using a single defect in a 2d photonic crystal slab - defect engineering with respect to polarization mode and ratio of emissions from upper and lower sides,” J. Lightwave Technol. |

6. | C. Seassal et al. “Optical coupling between a two-dimensional photonic crystal-based microcavity and single-line defect waveguide on inp membranes,” IEEE J. Quantum Electron. |

7. | B.K. Min, J.E. Kim, and H.Y. Park. “Channel drop filters using resonant tunneling processes in two-dimensional triangular lattice photonic crystal slabs,” Opt. Commun. |

8. | M.F. Yanik and S. Fan. “High-contrast all-optical bistable switching in photonic crystal microcavities,” App. Phy. Lett. |

9. | C. Manolatou et al. “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. |

10. | Y. Xu et al. “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E |

11. | S. Olivier et al. “Cascaded photonic crystal guidesand cavities: spectral studies and their impact on intergrated optics design,” IEEE J. Quantum Electron. |

12. | G.H. Kim et al. “Coupling of small, low-loss hexapole mode with photonic crystal slab waveguide mode,” Opt. Express |

13. | M. Okano, S. Kako, and S. Noda. “Coupling between a point-defect cavity and a line-defect waveguide in three-dimensional photonic crystal,” Phys. Rev. B |

14. | Ziyang Zhang Min Qiu. “Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2d photonic crystal slabs,” Opt. Express |

15. | Y. Akahane et al. “High-q photonic nanocavity in a two-dimensional photonic crystal,” Nature |

16. | A. Yariv. |

**OCIS Codes**

(230.5750) Optical devices : Resonators

(230.7400) Optical devices : Waveguides, slab

**ToC Category:**

Research Papers

**History**

Original Manuscript: April 13, 2005

Revised Manuscript: June 16, 2005

Published: June 27, 2005

**Citation**

Edo Waks and Jelena Vuckovic, "Coupled mode theory for photonic crystal cavity-waveguide interaction," Opt. Express **13**, 5064-5073 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-13-5064

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### References

- M. Loncar et al. �??Low-threshold photonic crystal laser,�?? App. Phy. Lett. 81, 2680�??2682 (2002). [CrossRef]
- Y. Akahane et al. �??Design of a channel drop filter by using a donor-type cavity with high-quality factor in a two-dimensional photonic crystal slab�?? App. Phys. Lett. 82, 1341�??1343 (2003). [CrossRef]
- J. Vuckovic and Y. Yamamoto. �??Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot,�?? App. Phys. Lett. 82, 2374�??2376 (2003). [CrossRef]
- T. Asano et al. �??Investigation of channel-add/drop-filtering device using acceptor-type point defects in a two-dimensional photonic-crystal slab,�?? App. Phys. Lett. 83, 407�??409 (2003) [CrossRef]
- T. Asano et al. �??A channel drop filter using a single defect in a 2d photonic crystal slab - defect engineering with respect to polarization mode and ratio of emissions from upper and lower sides,�?? J. Lightwave Technol. 21, 1370�??1376 (2003) [CrossRef]
- C. Seassal et al. �??Optical coupling between a two-dimensional photonic crystal-based microcavity and single-line defect waveguide on inp membranes,�?? IEEE J. Quantum Electron. 38 811�??815 (2002) [CrossRef]
- B.K. Min, J.E. Kim, and H.Y. Park. �??Channel drop filters using resonant tunneling processes in two-dimensional triangular lattice photonic crystal slabs,�?? Opt. Commun. 237 59�??63 (2004) [CrossRef]
- M.F. Yanik and S. Fan. �??High-contrast all-optical bistable switching in photonic crystal microcavities,�?? App. Phy. Lett. 83, 2739 (2003) [CrossRef]
- C. Manolatou et al. �??Coupling of modes analysis of resonant channel add-drop filters,�?? IEEE J. Quantum Electron. 35, 1322 (1999) [CrossRef]
- Y. Xu et al. �??Scattering-theory analysis of waveguide-resonator coupling,�?? Phys. Rev. E 62, 7389�??7404 (2000) [CrossRef]
- S. Olivier et al. �??Cascaded photonic crystal guidesand cavities: spectral studies and their impact on intergrated optics design,�?? IEEE J. Quantum Electron. 38, 816�??824 (2002) [CrossRef]
- G.H. Kim et al. �??Coupling of small, low-loss hexapole mode with photonic crystal slab waveguide mode,�?? Opt. Express 12, 6624�??6631 (2004) [CrossRef] [PubMed]
- M. Okano, S. Kako, and S. Noda. �??Coupling between a point-defect cavity and a line-defect waveguide in three-dimensional photonic crystal,�?? Phys. Rev. B 68, 235110 (2003) [CrossRef]
- Ziyang Zhang Min Qiu. �??Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2d photonic crystal slabs,�?? Opt. Express 13, 2596�??2604 (2005) [CrossRef] [PubMed]
- Y. Akahane et al. �??High-q photonic nanocavity in a two-dimensional photonic crystal,�?? Nature 425, 944�??947 (2003) [CrossRef] [PubMed]
- A. Yariv. Optical Electronics. Saunders College Publishing, Philadelphia, 1991.

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