## Relevance of the light line in planar photonic crystal waveguides with weak vertical confinement |

Optics Express, Vol. 19, Issue 24, pp. 24344-24353 (2011)

http://dx.doi.org/10.1364/OE.19.024344

Acrobat PDF (1621 KB)

### Abstract

The concept of the so-called light line is a useful tool to distinguish between guided and non-guided modes in dielectric slab waveguides. Also for more complicated structures with 2D mode confinement, the light lines can often be used to divide a dispersion diagram into a region of a non-guided continuum of modes, a region of discrete guided modes and a forbidden region, where no propagating modes can exist. However, whether or not the light line is a concept of practical relevance depends on the geometry of the structure. This fact is sometimes ignored. For instance, in the literature on photonic crystal waveguides, it is often argued that substrate-type photonic crystal waveguides with a weak vertical confinement are inherently lossy, since the entire bandgap including the line defect modes is typically located above the light line of the substrate. The purpose of this article is to illustrate that this argument is inaccurate and to provide guidelines on how an improved light line concept can be constructed.

© 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, 594–598 (1999). [CrossRef]

2. S. Mahnkopf, M. Kamp, A. Forchel, and R. März, “Tunable distributed feedback laser with photonic crystal mirrors,” Appl. Phys. Lett. **82**, 2942–2944 (2003). [CrossRef]

3. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science **284**, 1819–1821 (1999). [CrossRef] [PubMed]

4. B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics **3**, 206–210 (2009). [CrossRef]

5. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**, 944–947 (2003). [CrossRef] [PubMed]

6. T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics **1**, 49–52 (2007). [CrossRef]

7. H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. **94**, 073903 (2005). [CrossRef] [PubMed]

8. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature **438**, 65–69 (2005). [CrossRef] [PubMed]

9. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: Role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. **94**, 033903 (2005). [CrossRef] [PubMed]

10. L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, “Loss engineered slow light waveguides,” Opt. Express **18**, 27627–27638 (2010). [CrossRef]

## 2. The background line concept

### 2.1. The buried rectangular waveguide

11. Let *σ _{y}* denote a reflection in the

*x*–

*z*plane, i.e.,

*σ*

_{y}**x̂**=

**x̂**,

*σ*

_{y}**ŷ**= −

**ŷ**,

*σ*

_{y}**ẑ**=

**ẑ**for the unit vectors

**x̂**,

**ŷ**, and

**ẑ**, respectively.

**E**transforms like a vector, whereas

**H**transforms like a pseudovector under an orientation-reversing map. A mode of even parity is characterized by

**E**(

*σ*

_{y}**r**) =

*σ*

_{y}**E**(

**r**) and

**H**(

*σ*

_{y}**r**) = −

*σ*

_{y}**H**(

**r**), a mode of odd parity by

**E**(

*σ*

_{y}**r**) = −

*σ*

_{y}**E**(

**r**) and

**H**(

*σ*

_{y}**r**) =

*σ*

_{y}**H**(

**r**). Note that different conventions of parity might be used in other contexts.

*x*–

*z*mirror plane (

*E*=

_{x}*H*=

_{y}*E*= 0 in the symmetry plane), computed by Lumerical (bullets), a commercially available 2D finite difference frequency domain (FDFD) eigenmode solver [12

_{z}12. Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express **10**, 853–864 (2002). [PubMed]

*k*→ ∞), the waveguide mode crosses the dispersion line of the substrate (

_{x}*n*

_{2}) and converges (not shown here) to the dispersion line of the core (

*n*

_{1}). For small propagation constants, a cutoff is found near an effective refractive index of 2.6. This observation requires explanation, since it contradicts a widespread understanding of how the waveguiding mechanism works. Conventional wisdom has it that the effective refractive index lies between the core index and the largest index of the materials involved in the guiding action. Formally, this translates into the statement that all guided modes of a longitudinally invariant dielectric waveguide must satisfy the condition [13] where ∂ℝ

^{2}is the “boundary at infinity” of the cross-section ℝ

^{2}of the waveguide. The condition is based on the definition of guided modes, i.e., on the prerequisite that all fields decay exponentially and no oscillatory fields shall be supported at infinite distances from the core. For our BRWG, Eq. (1) writes as

*n*

_{1}= 3.33 >

*n*

_{eff}>

*n*

_{2}= 3.15, which is in clear disagreement with those results of Fig. 1 which are located above the dispersion line of the substrate.

*h*

_{bot}are considered, while

*h*

_{top}is kept constant at 1

*μ*m. The four curves correspond to

*h*

_{bot}values of 2

*μ*m, 4

*μ*m, 6

*μ*m, and 8

*μ*m. The size of the simulation domain (16 × 16

*μ*m) and the mesh resolution (grid spacing between 40 nm, in the core, and 80 nm, far from the core) are the same for all curves. The PML loss near cutoff is substantial for

*h*

_{bot}= 2

*μ*m, but is drastically reduced for the larger values of

*h*

_{bot}. The limit of numerical accuracy is reached around 10

^{−10}dB/cm. It is interesting to note that if the position of the PML is moved further away from the waveguide core, the change in PML loss is negligible. For instance, with

*h*

_{bot}= 4

*μ*m and a frequency of 140 THz (

*n*

_{eff}= 2.8), the PML loss fluctuates by ± 0.0041 % when we vary the position of the lower PML boundary in a range that spans 10

*μ*m (95 % confidence interval from a set of 32 data points). This indicates that the main contribution to the PML loss is given by an oscillatory field rather than an exponentially decaying one. In this respect, we cannot classify the modes as guided modes. Nevertheless, the results of Fig. 2 suggest that arbitrarily low PML losses can be obtained by increasing

*h*

_{bot}to infinity. We conclude that, for a BRWG with

*h*

_{bot}→ ∞, guided modes exist with effective refractive indices outside the range given in Eq. (1).

*h*

_{bot}→ ∞, we have to go back to the prerequisites from which the condition follows. As indicated above, Eq. (1) is based on the requirement that the dielectric “background” at infinity must not support any oscillating solutions of Maxwell’s equations. For simplicity, we will let

*h*

_{top}→ ∞, as well. The background at infinity of this system is composed of the materials of refractive indices

*n*

_{2}(thin slab) and

*n*

_{3}(two half-spaces), i.e., it is essentially a slab waveguide structure [14

14. It might be interesting to note that the condition of Eq. (1) for guided modes can be proven by a rigorous mathematical analysis, as long as there is a radius *R* around the core, for which *n*(**r**) = *n*_{∞} holds for all **r** ∈ ℝ^{2} with |**r**| > *R* [A.-S. Bonnet-Ben Dhia and P. Joly, “Mathematical analysis and numerical approximation of optical waveguides,” in *Mathematical Modeling in Optical Science*, G. Bao, L. Cowsar, and W. Masters, eds. (Siam, Philadelphia, 2001), pp. 273–324, Frontiers in Applied Mathematics]. In other words, Eq. (1) holds for all guided modes, if outside of a circle of radius *R* the background is made up of a homogeneous medium of refractive index *n*_{∞}. This condition is not fulfilled by the structure of Fig. 1 if we let *h*_{bot} → ∞. [CrossRef]

*n*

_{eff}=

*n*

_{2}. Therefore, the dispersion line of the substrate (

*n*

_{2}) cannot be relevant in the determination of the BRWG cutoff. Instead, the relevant separatrix in the dispersion diagram, which separates guided from non-guided modes, is the dispersion curve of the fundamental mode of the background system (the

*n*

_{3}/

*n*

_{2}/

*n*

_{3}slab waveguide). We will refer to this separatrix as the “background line”. Propagating modes of the full BRWG system, which are located below the background line are guided modes and theoretically loss-free.

*x*–

*z*symmetry plane) in the core of the BRWG fundamental mode (bullets), the TE or TM modes of the background slab waveguide have to be considered (solid lines). The simulation of the BRWG is again performed with the Lumerical mode solver, and the results of odd parity correspond to those of Fig. 1(b). The background lines are calculated from analytical equations [15]. Cutoff for both parities occurs as predicted. This confirms that the background line bears more practical relevance than the condition of Eq. (1) (dotted vertical line in Fig. 3) if

*h*

_{bot}is large.

16. T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. **31**, 1941–1943 (1995). [CrossRef]

17. J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science **282**, 1476–1478 (1998). [CrossRef] [PubMed]

*effective index method*, which is a popular analytical approximation to find the modes of integrated optical waveguides (such as ridge waveguides), makes use of the background modes as one of the steps in the simplification procedure for an analytical mode analysis. It is important to note that, while the effective index method makes considerable errors in determining the dispersion near cutoff of certain waveguides, the cutoff itself is always predicted on the correct separatrix in the dispersion diagram (but not at the correct location on the separatrix).

### 2.2. The role of total internal reflection

## 3. Substrate-type photonic crystal waveguides

18. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B **60**, 5751–5758 (1999). [CrossRef]

### 3.1. The background line of PhC waveguides

*h*

_{top}and

*h*

_{bot}(as defined in Fig. 1(a)), the background is formed (if

*h*

_{top},

*h*

_{bot}→ ∞) by an infinitely extended 2D line defect PhC waveguide. The set of background modes is a lot more complicated than it was for the BRWG system studied above. To numerically determine the background line, the modes of the line defect waveguide have to be simulated for all propagation directions. All oscillating modes of the background system have to be included into a band diagram which is projected onto the core axis of the full system. In Fig. 5 we consider a single line defect (W1) PhC waveguide based on an InP/InGaAsP/InP layer structure, which resembles the BRWG of Fig. 1(a) in terms of geometry of the cross-section. The lateral mode confinement is no longer given by a homogeneous material but by a PhC structure of triangular geometry and an

*r/a*ratio of 0.34. The simulations are performed by the plane wave expansion method (MPB package [19

19. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001). [CrossRef] [PubMed]

*instead*of the substrate light line.

- Is it possible to design a theoretically loss-free substrate-type PhC waveguide based on realistic materials? The above considerations show that it is not hopeless, but Fig. 5 also shows that for our particular material system it will be very challenging.
- How much loss do we have for a given mode above the background line? The background line marks the separatrix for guided modes. However, in a typical PhC waveguide, such as the one of Fig. 5, the main component of the Bloch mode is located in the second Brillouin zone [20] and below the background line. Only the Bloch component of the first Brillouin zone lies above the background line, and only that component will contribute to the waveguide loss [21
20. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, “Fourier analysis of Bloch wave propagation in photonic crystals,” J. Opt. Soc. Am. B

**22**, 1179–1190 (2005). [CrossRef]]. The quantitative connection between the strength of this component and the theoretical propagation loss has not been established to our knowledge.21. W. Kuang and J. D. O’Brien, “Reducing the out-of-plane radiation loss of photonic crystal waveguides on high-index substrates,” Opt. Lett.

**29**, 860–862 (2004). [CrossRef] [PubMed]

## 4. Conclusion

## 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,” |

2. | S. Mahnkopf, M. Kamp, A. Forchel, and R. März, “Tunable distributed feedback laser with photonic crystal mirrors,” Appl. Phys. Lett. |

3. | O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science |

4. | B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics |

5. | Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature |

6. | T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics |

7. | H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. |

8. | Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature |

9. | S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: Role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. |

10. | L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, “Loss engineered slow light waveguides,” Opt. Express |

11. | Let x–z plane, i.e., σ_{y}x̂ = x̂, σ_{y}ŷ = −ŷ, σ_{y}ẑ = ẑ for the unit vectors x̂, ŷ, and ẑ, respectively. E transforms like a vector, whereas H transforms like a pseudovector under an orientation-reversing map. A mode of even parity is characterized by E(σ_{y}r) = σ_{y}E(r) and H(σ_{y}r) = −σ_{y}H(r), a mode of odd parity by E(σ_{y}r) = −σ_{y}E(r) and H(σ_{y}r) = σ_{y}H(r). Note that different conventions of parity might be used in other contexts. |

12. | Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express |

13. | R. März, |

14. | It might be interesting to note that the condition of Eq. (1) for guided modes can be proven by a rigorous mathematical analysis, as long as there is a radius |

15. | D. Marcuse, |

16. | T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. |

17. | J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science |

18. | S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B |

19. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

20. | B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, “Fourier analysis of Bloch wave propagation in photonic crystals,” J. Opt. Soc. Am. B |

21. | W. Kuang and J. D. O’Brien, “Reducing the out-of-plane radiation loss of photonic crystal waveguides on high-index substrates,” Opt. Lett. |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(230.7370) Optical devices : Waveguides

(130.5296) Integrated optics : Photonic crystal waveguides

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: August 18, 2011

Revised Manuscript: November 4, 2011

Manuscript Accepted: November 4, 2011

Published: November 14, 2011

**Citation**

P. Kaspar, R. Kappeler, D. Erni, and H. Jäckel, "Relevance of the light line in planar photonic crystal waveguides with weak vertical confinement," Opt. Express **19**, 24344-24353 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-24-24344

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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,” Nature397, 594–598 (1999). [CrossRef]
- S. Mahnkopf, M. Kamp, A. Forchel, and R. März, “Tunable distributed feedback laser with photonic crystal mirrors,” Appl. Phys. Lett.82, 2942–2944 (2003). [CrossRef]
- O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science284, 1819–1821 (1999). [CrossRef] [PubMed]
- B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics3, 206–210 (2009). [CrossRef]
- Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature425, 944–947 (2003). [CrossRef] [PubMed]
- T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics1, 49–52 (2007). [CrossRef]
- H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett.94, 073903 (2005). [CrossRef] [PubMed]
- Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature438, 65–69 (2005). [CrossRef] [PubMed]
- S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: Role of fabrication disorder and photon group velocity,” Phys. Rev. Lett.94, 033903 (2005). [CrossRef] [PubMed]
- L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, “Loss engineered slow light waveguides,” Opt. Express18, 27627–27638 (2010). [CrossRef]
- Let σy denote a reflection in the x–z plane, i.e., σyx̂ = x̂, σyŷ = −ŷ, σyẑ = ẑ for the unit vectors x̂, ŷ, and ẑ, respectively. E transforms like a vector, whereas H transforms like a pseudovector under an orientation-reversing map. A mode of even parity is characterized by E(σyr) = σyE(r) and H(σyr) = −σyH(r), a mode of odd parity by E(σyr) = −σyE(r) and H(σyr) = σyH(r). Note that different conventions of parity might be used in other contexts.
- Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express10, 853–864 (2002). [PubMed]
- R. März, Integrated Optics: Design and Modeling (Artech House, Norwood, 1995).
- It might be interesting to note that the condition of Eq. (1) for guided modes can be proven by a rigorous mathematical analysis, as long as there is a radius R around the core, for which n(r) = n∞ holds for all r ∈ ℝ2 with |r| > R [A.-S. Bonnet-Ben Dhia and P. Joly, “Mathematical analysis and numerical approximation of optical waveguides,” in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, and W. Masters, eds. (Siam, Philadelphia, 2001), pp. 273–324, Frontiers in Applied Mathematics]. In other words, Eq. (1) holds for all guided modes, if outside of a circle of radius R the background is made up of a homogeneous medium of refractive index n∞. This condition is not fulfilled by the structure of Fig. 1 if we let hbot → ∞. [CrossRef]
- D. Marcuse, Theory of Dielectric Optical Waveguides, Quantum Electronics: Principles and Applications (Academic Press, Boston, 1991), 2nd ed.
- T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett.31, 1941–1943 (1995). [CrossRef]
- J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science282, 1476–1478 (1998). [CrossRef] [PubMed]
- S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B60, 5751–5758 (1999). [CrossRef]
- S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express8, 173–190 (2001). [CrossRef] [PubMed]
- B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, “Fourier analysis of Bloch wave propagation in photonic crystals,” J. Opt. Soc. Am. B22, 1179–1190 (2005). [CrossRef]
- W. Kuang and J. D. O’Brien, “Reducing the out-of-plane radiation loss of photonic crystal waveguides on high-index substrates,” Opt. Lett.29, 860–862 (2004). [CrossRef] [PubMed]

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