## Coupled-mode theory and propagation losses in photonic crystal waveguides

Optics Express, Vol. 11, Issue 13, pp. 1490-1496 (2003)

http://dx.doi.org/10.1364/OE.11.001490

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### Abstract

Mode coupling phenomena, manifested by transmission “mini-stopbands”, occur in two-dimensional photonic crystal channel waveguides. The huge difference in the group velocities of the coupled modes is a new feature with respect to the classical Bragg reflection occurring, e.g., in distributed feedback lasers. We show that an adequate ansatz of the classical coupled-mode theory remarkably well accounts for this new phenomenon. The fit of experimental transmission data from GaAs-based photonic crystal waveguides then leads to an accurate determination of the propagation losses of both fundamental and higher, low-group-velocity modes.

© 2003 Optical Society of America

## 1. Introduction

1. S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Oesterle, and R. Houdré, “Mini stopbands of a one dimensional system: the channel waveguide in a two-dimensional photonic crystal,” Phys. Rev. B **63**, 113311 (2001). [CrossRef]

2. S. Olivier, H. Benisty, C. J. M. Smith, M. Rattier, C. Weisbuch, T. F. Krauss, R. Houdré, and U. Oesterle, “Transmission properties of two-dimensional photonic crystal channel waveguides,” Opt. Quantum Electron. **34**, 171–181 (2002). [CrossRef]

3. C. J. M. Smith, R. M. De La Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdré, and U. Oesterle, “Coupled guide and cavity in a two-dimensional photonic crystal,” App. Phys. Lett. **78**, 1487–1489 (2001). [CrossRef]

## 2. Theory

### 2.1. Loss-less case

1. S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Oesterle, and R. Houdré, “Mini stopbands of a one dimensional system: the channel waveguide in a two-dimensional photonic crystal,” Phys. Rev. B **63**, 113311 (2001). [CrossRef]

2. S. Olivier, H. Benisty, C. J. M. Smith, M. Rattier, C. Weisbuch, T. F. Krauss, R. Houdré, and U. Oesterle, “Transmission properties of two-dimensional photonic crystal channel waveguides,” Opt. Quantum Electron. **34**, 171–181 (2002). [CrossRef]

4. H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. **79**, 7483–7492 (1996). [CrossRef]

5. M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. **81**, 1163–1165 (2002). [CrossRef]

_{eff}=11 and the air-filling factor is

*f*=0.37, which leads to a photonic band gap in TE polarization covering the reduced frequency range

*u*=

*a*/λ=0.22-0.3.

_{z}(x,y) is represented in Fig. 2(c), basically resembles a traditional index-guided mode. Its dispersion relation is folded at the Brillouin zone edge due to the periodicity. The high-order modes follow at higher frequencies. The highest modes #4, #5 and #6, which have their cut-off frequency inside the photonic bandgap of the crystal, are purely Bragg-guided modes. They have a very low group velocity (or high group index

*n*

_{g}=

*c/ν*

_{g}) when they cross the lower-order modes. If symmetry allows, anti-crossings instead of crossings occur between modes of different order, labeled “a” and “b”, when they obey the first-order Bragg condition:

*π*/

*a*[1

1. S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Oesterle, and R. Houdré, “Mini stopbands of a one dimensional system: the channel waveguide in a two-dimensional photonic crystal,” Phys. Rev. B **63**, 113311 (2001). [CrossRef]

*u*

_{0}of the anticrossing, by the apparently classical coupled-mode formulation:

_{ab}is their coupling constant. δ

_{a}(u) and δ

_{b}(u), which are the detuning of the uncoupled modes with respect to the Bragg condition, can be expressed as a function of the reduced frequency

*u*, assuming that the dispersion relations of the uncoupled modes are linear around the central frequency

*u*

_{0}(i.e., their group velocities or group indices

*n*

_{ga}and

*n*

_{gb}are constant):

4. H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. **79**, 7483–7492 (1996). [CrossRef]

*n*

_{ga}and

*n*

_{gb}of both modes are extracted from the slope of their dispersion relation, taken far from the coupling region. The value of the coupling constant

*κ*

_{ab}can then be derived from the knowledge of these three parameters, using Eq. (3). For the W3 waveguide considered above, the parameters extracted from the dispersion diagram of Fig. 2 are

*n*

^{ga}=3.3,

*n*

_{gb}=33 and

*Δu*

_{ab}=2.5 10

^{-3}, leading to

*κ*

_{ab}=0.14

*a*

^{-1}.

### 2.2. Influence of propagation losses

_{a}and α

_{b}of both modes must be taken into account, in order to obtain an accurate model of the shape and intensity level of the transmission spectrum. The high-order mode, which has a much lower group velocity and a much larger penetration in the guide boundaries, experiences much larger propagation losses than the fundamental mode, with a realistic ratio of α

_{b}/α

_{a}=10. For short waveguides (typically 30 rows), the losses of the fundamental mode will have a negligible effect, whereas they will become more visible for longer waveguides. The impact of the propagation losses is therefore studied separately in the following, where simulations are performed for a W3 waveguide of length 60

*a*.

_{b}=0.1 µm

^{-1}, one obtains the spectra represented in Figs. 3(a) and (b), superimposed on the lossless spectra. The transmission and reflection spectra are essentially the bottom envelope of the spectrum calculated in the lossless case, washing out secondary oscillations. The transmission level of the fundamental mode still drops to zero at the central frequency, while the reflection of the high-order mode at the input is now well below 100%.

_{a}=0.01 µm

^{-1}, we obtain the spectra displayed in Figs. 3(c) and (d). The transmission of the fundamental mode at the output is 70%, while the reflection level of the high-order mode at the input is essentially dominated by its propagation losses and suffers negligibly from the losses of the fundamental mode that excited it.

## 3. Experimental results and comparison with theory

### 3.1 Experimental procedure

*a*, 120

*a*and 240

*a*(with

*a*=260 nm), carved into a GaAs-based heterostructure have been measured using the internal source technique described in earlier work [10

10. D. Labilloy, H. Benisty, C. Weisbuch, T.F. Krauss, R. Houdré, and U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensionam photonic crystals,” Appl. Phys. Lett. **71**, 738–740 (1997). [CrossRef]

*selectively*measured, whatever the other excited guided modes.

### 3.2 Determination of the propagation losses

*u*

_{0}=0.265, in good agreement with the frequency of the anticrossing in the dispersion diagram of Fig. 2. Note that the shift of the mini-stopband frequency towards the higher frequencies when the guide length increases is mainly an artefact due to the proximity effects during e-beam patterning (slightly larger holes for larger overall exposure area). From the fit of the three experimental spectra with the coupling constant

*κ*

_{ab}and the losses α

_{a}and α

_{b}as the adjustable parameters, we obtain an accurate measurement of the losses of the fundamental mode and of the 5

^{th}order mode. We find α

_{a}=35 cm

^{-1}=1.5 dB/100 µm for mode #1 and α

_{b}=400 cm

^{-1}=17 dB/100 µm for mode #5, with an accuracy of 10%. The value of the coupling constant is

*κ*

_{ab}=0.09

*a*

^{-1}, a little below the calculated value 0.14

*a*

^{-1}, maybe on account of a minute width variation along the guide. It is remarkable that the spectral width of the transmission dip, simultaneously with the transmission level of the fundamental mode on both sides of the dip, are very well reproduced by the model. The propagation losses of the high-order mode are responsible for the apparently surprising fact that the MSB takes a broader and broader appearance when the length of the waveguide increases, whereas the coupling coefficient is unchanged.

### 3.3 Comparison with other methods

11. E. Schwoob, H. Benisty, S. Olivier, C. Weisbuch, C.J.M Smith, T.F. Krauss, R. Houdre, and U. Oesterle, “Two-mode fringes in planar photonic crystal waveguides with constrictions: a sensitive probe to propagation losses,” J. Opt. Soc. Am. B **19**, 2403–2412 (2002). [CrossRef]

^{-1}for the fundamental mode of a W3 waveguide, similar to the one studied above (same hole diameter, period and, being fabricated by RIE, same depth as well), has been found using this alternative method.

## 4. Conclusion

## References and Links

1. | S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Oesterle, and R. Houdré, “Mini stopbands of a one dimensional system: the channel waveguide in a two-dimensional photonic crystal,” Phys. Rev. B |

2. | S. Olivier, H. Benisty, C. J. M. Smith, M. Rattier, C. Weisbuch, T. F. Krauss, R. Houdré, and U. Oesterle, “Transmission properties of two-dimensional photonic crystal channel waveguides,” Opt. Quantum Electron. |

3. | C. J. M. Smith, R. M. De La Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdré, and U. Oesterle, “Coupled guide and cavity in a two-dimensional photonic crystal,” App. Phys. Lett. |

4. | H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. |

5. | M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. |

6. | H. Kogelnick and C.V. Shank, “Coupled wave theory of distributed feedback lasers,” J. Appl. Phys. |

7. | See for example Tamir, Guided Wave Optoelectronics, Springer Verlag, Berlin, chaps 2,6 (1988). |

8. | P. Ferrand, R. Romestain, and J.C. Vial, “Photonic band-gap properties of a porous silicon periodic planar waveguide,” Phys. Rev. B 63, 115106 (2001). [CrossRef] |

9. | E. Peral and A. Yariv, “Supermodes of grating-coupled multimode waveguides and application to mode conversion between copropagating modes mediated by backward Bragg scattering,” J. Lightwave Tech. |

10. | D. Labilloy, H. Benisty, C. Weisbuch, T.F. Krauss, R. Houdré, and U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensionam photonic crystals,” Appl. Phys. Lett. |

11. | E. Schwoob, H. Benisty, S. Olivier, C. Weisbuch, C.J.M Smith, T.F. Krauss, R. Houdre, and U. Oesterle, “Two-mode fringes in planar photonic crystal waveguides with constrictions: a sensitive probe to propagation losses,” J. Opt. Soc. Am. B |

12. | M. Qiu, B. Jaskorzynska, M. Swillo, and H. Benisty, “Time-domain 2D modeling of slab-waveguide-based photonic-crystal devices in the presence of radiation losses,” Microwave and Opt. Tech. Lett. |

**OCIS Codes**

(130.0130) Integrated optics : Integrated optics

(230.7380) Optical devices : Waveguides, channeled

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 13, 2003

Revised Manuscript: June 11, 2003

Published: June 30, 2003

**Citation**

Segolene Olivier, H. Benisty, C. Weisbuch, C. Smith, T. Krauss, and R. Houdre, "Coupled-mode theory and propagation losses in photonic crystal waveguides," Opt. Express **11**, 1490-1496 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-13-1490

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### References

- S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Oesterle and R. Houdré, �??Mini stopbands of a one dimensional system: the channel waveguide in a two-dimensional photonic crystal,�?? Phys. Rev. B 63, 113311 (2001). [CrossRef]
- S. Olivier, H. Benisty, C. J. M. Smith, M. Rattier, C. Weisbuch, T. F. Krauss, R. Houdré and U. Oesterle, �??Transmission properties of two-dimensional photonic crystal channel waveguides,�?? Opt. Quantum Electron. 34, 171-181 (2002). [CrossRef]
- C. J. M. Smith, R. M. De La Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdré and U. Oesterle, �??Coupled guide and cavity in a two-dimensional photonic crystal,�?? App. Phys. Lett. 78, 1487-1489 (2001). [CrossRef]
- H. Benisty, �??Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,�?? J. Appl. Phys. 79, 7483-7492 (1996). [CrossRef]
- M. Qiu, �??Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,�?? Appl. Phys. Lett. 81, 1163-1165 (2002). [CrossRef]
- H. Kogelnick and C.V. Shank, �??Coupled wave theory of distributed feedback lasers,�?? J. Appl. Phys. 43, 2328 (1972).
- See for example Tamir, Guided Wave Optoelectronics, Springer Verlag, Berlin, chaps 2,6 (1988).
- P. Ferrand, R. Romestain and J.C. Vial, �??Photonic band-gap properties of a porous silicon periodic planar waveguide,�?? Phys. Rev. B 63, 115106 (2001). [CrossRef]
- E. Peral and A. Yariv, �??Supermodes of grating-coupled multimode waveguides and application to mode conversion between copropagating modes mediated by backward Bragg scattering,�?? J. Lightwave Tech. 17, 942-947 (1999). [CrossRef]
- D. Labilloy, H. Benisty, C. Weisbuch, T.F. Krauss, R. Houdré and U. Oesterle, �??Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensionam photonic crystals,�?? Appl. Phys. Lett. 71, 738-740 (1997). [CrossRef]
- E. Schwoob, H. Benisty, S. Olivier, C. Weisbuch, C.J.M Smith, T.F. Krauss, R. Houdre and U. Oesterle, �??Two-mode fringes in planar photonic crystal waveguides with constrictions: a sensitive probe to propagation losses,�?? J. Opt. Soc. Am. B 19, 2403-2412 (2002). [CrossRef]
- M. Qiu, B. Jaskorzynska, M. Swillo and H. Benisty, �??Time-domain 2D modeling of slab-waveguide-based photonic-crystal devices in the presence of radiation losses,�?? Microwave and Opt. Tech. Lett. 34, 387-393 (2002). [CrossRef]

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