## Low-loss guided modes in photonic crystal waveguides

Optics Express, Vol. 13, Issue 13, pp. 4939-4951 (2005)

http://dx.doi.org/10.1364/OPEX.13.004939

Acrobat PDF (561 KB)

### Abstract

We study disorder-induced propagation losses of guided modes in photonic crystal slabs with line-defects. These losses are treated within a theoretical model of size disorder for the etched holes in the otherwise periodic photonic lattice. Comparisons are provided with state-of-the-art experimental data, both in membrane and Silicon-on-Insulator (SOI) structures, in which propagation losses are mainly attributed to fabrication imperfections. The dependence of the losses on the photon group velocity and the useful bandwidth for low-loss propagation are analyzed and discussed for membrane and asymmetric as well as symmetric SOI systems. New designs for further improving device performances are proposed, which employ waveguides with varying channel widths. It is shown that losses in photonic crystal waveguides could be reduced by almost an order of magnitude with respect to latest experimental results. Propagation losses lower than 0.1 dB/mm are predicted for suitably designed structures, by assuming state-of-the-art fabrication accuracy.

© 2005 Optical Society of America

## 1. Introduction

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**, 2059–2062 (1987). [CrossRef] [PubMed]

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**, 2486–2489 (1987). [CrossRef] [PubMed]

6.
See papers in *IEEE J. Quantum Electron.*38, Feature section on *Photonic Crystal Structures and Applications*, edited by
T. F. Krauss and T. Baba, pp.724–963 (2002). [CrossRef]

7. A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B **62**, 4488–4492 (2000). [CrossRef]

8. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic crystal slabs,” Phys. Rev. B **62**, 8212–8222 (2000). [CrossRef]

9. M. Qiu, “Band gap effects in asymmetric photonic crystal slabs,” Phys. Rev. B **66**, 033103 (2002). [CrossRef]

10. H. Benisty, D. Labilloy, C. Weisbuch, C. J. .M. Smith, T. F. Krauss, D. Cassagne, A. Béraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate,” Appl. Phys. Lett. **76**, 532 (2000). [CrossRef]

11. W. Bogaerts, P. Bienstman, D. Taillaert, R. Baets, and D. De Zutter, “Out-of-plane scattering in photonic crystal slabs,” IEEE Photon. Techn. Lett. **13**, 565–567 (2001). [CrossRef]

12. M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, “Structural tuning of guiding modes of line-defect waveguides of Silicon-on-Insulator photonic crystal slabs,” IEEE J. Quantum Electron. **38**, 736–742 (2002). [CrossRef]

13. S. J. McNab, N. Moll, and Yu. A. Vlasov, “Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,” Opt. Express **11**, 2927–2939 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2927. [CrossRef] [PubMed]

14. M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H.-Y. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express **12**, 1551–1561 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1551. [CrossRef] [PubMed]

15. W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, and R. Baets, “Basic structures for photonic integrated circuits in Silicon-on-insulator,” Opt. Express **12**, 1583–1591 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1583. [CrossRef] [PubMed]

16. Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, K. Asakawa, and K. Inoue, “Low propagation loss of 0.76 dB/mm in GaAs-based single-line-defect two-dimensional photonic crystal slab waveguides up to 1 cm in length,” Opt. Express **12**, 1090–1096 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1090. [CrossRef] [PubMed]

17. K. K. Lee, D. R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO_{2} waveguide: experiments and model,” Appl. Phys. Lett. **77**, 1617–1619 (2000). [CrossRef]

18. Yu. A. Vlasov and S. J. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express **12**, 1622–1631 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1622. [CrossRef] [PubMed]

19. W. Bogaerts, P. Bienstman, and R. Baets, “Scattering at sidewall roughness in photonic crystal slabs,” Opt. Lett. **28**, 689–691 (2003). [CrossRef] [PubMed]

20. M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, and M. Soljačić, “Effect of a photonic band gap on scattering from waveguide disorder,” Appl. Phys. Lett. **84**, 3639–3641 (2004). [CrossRef]

21. Y. Tanaka, T. Asano, Y. Akahane, B.-S. Song, and S. Noda, “Theoretical investigation of a two-dimensional photonic crystal slab with truncated cone air holes,” Appl. Phys. Lett. **82**, 1661–1663 (2003). [CrossRef]

22. Y. Tanaka, T. Asano, R. Hatsuta, and S. Noda, “Analysis of a line-defect waveguide on a Silicon-on-Insulator two-dimensional photonic-crystal slab,” J. Lightwave Technol. **22**, 2787–2792 (2004). [CrossRef]

23. B. Cluzel, D. Gérard, E. Picard, T. Charvolin, V. Calvo, E. Hadji, and F. de Fornel, “Experimental demonstration of Bloch mode parity change in photonic crystal waveguide,” Appl. Phys. Lett. **85**, 2682–2684 (2004). [CrossRef]

24. D. Gerace and L. C. Andreani, “Disorder-induced losses in photonic crystal waveguides with line defects,” Opt. Lett. **29**, 1897–1899 (2004). [CrossRef] [PubMed]

25. L. C. Andreani, D. Gerace, and M. Agio, “Gap maps, diffraction losses and exciton-polaritons in photonic crystal slabs,” Phot. Nanostruct. **2**, 103–110 (2004). [CrossRef]

26. 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]

24. D. Gerace and L. C. Andreani, “Disorder-induced losses in photonic crystal waveguides with line defects,” Opt. Lett. **29**, 1897–1899 (2004). [CrossRef] [PubMed]

*v*

_{g}) of the defect mode, by using a more fundamental theoretical approach. The importance of the group velocity for low-loss propagation was previously stressed in Refs. [27

27. K. Yamada, H. Morita, A. Shinya, and M. Notomi, “Improved line-defect structures for photonic-crystal waveguides with high group velocity,” Opt. Commun. **198**, 395–402 (2001). [CrossRef]

28. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. **87**, 253902 (2001). [CrossRef] [PubMed]

## 2. A theoretical model of disorder

24. D. Gerace and L. C. Andreani, “Disorder-induced losses in photonic crystal waveguides with line defects,” Opt. Lett. **29**, 1897–1899 (2004). [CrossRef] [PubMed]

25. L. C. Andreani, D. Gerace, and M. Agio, “Gap maps, diffraction losses and exciton-polaritons in photonic crystal slabs,” Phot. Nanostruct. **2**, 103–110 (2004). [CrossRef]

*a*triangular lattice of holes with lattice constant a, in which a line-defect consisting of a missing row of holes along the Γ-K direction is introduced (see Fig. 1 for a schematic picture of the structure and a definition of the symmetry directions of the triangular lattice). The system is realized on a PhC slab of thickness

*d*patterned in a high-index material. A waveguide with channel width

*W*=

*W*

_{0}=√3

*a*is called a W1.0 waveguide. Increased- or reduced-width waveguides can be defined by modifying the channel width as

*W*=

*X*·

*W*

_{0}, where

*X*is a real multiplication factor. A schematic picture of the in-plane supercell employed in the calculations of disorder-induced losses is shown in Fig. 1(a), where the r.m.s. deviation has been exaggerated for clarity. The disordered lattice is constructed after supercell repetition of the fundamental cell (shaded holes in figure) along the Γ-K direction. The dielectric perturbation leading to extrinsic losses is given by Δ

*ε*(

**r**)=

*ε*

_{dis}(

**r**)-

*ε*(

**r**), where

*ε*(

**r**) is the spatially-dependent dielectric function of the ideal lattice and

*ε*

_{dis}(

**r**) is the dielectric function of the disordered lattice. Only size variation along the Γ-K direction is relevant and is taken into account.

_{2}-clad structures in which the Silicon layer is grown on top of an oxide cladding, and symmetric SiO

_{2}-clad PhC waveguides where an upper SiO

_{2}cladding is deposited after patterning of the core. The dielectric constants used throughout this work are

*ε*

_{diel}=12.11 and

*ε*

_{oxide}=2.08 for Silicon and SiO

_{2}materials, respectively. These values are appropriate for such materials when considering the wavelength range around 1.55

*µ*m. In the present work, a supercell of size 8

*W*

_{0}+

*W*is used in the direction perpendicular to the line defect for the calculations of photonic mode dispersion; disorder-induced losses are calculated by using a supercell of size 49

*a*along the Γ-K direction in the perturbative treatment, and an average over six different Gaussian distributions of hole radii with the same r.m.s. deviation is performed to get final results. The number of plane waves and guided modes taken in the basis set has been chosen, according to convergence tests, in order to get an accuracy of the order of a percent for the losses.

*z*axis to be at the middle of the slab, specular reflection with respect to the (

*x*,

*y*) plane is a symmetry operation for air-clad or symmetric SiO

_{2}-clad PhC waveguides, which we denote by

_{xy}. When dealing with symmetric PhC waveguides, we consider only modes with σ

_{xy}=+1 (sometimes called TE-like modes) for which the triangular lattice has a band gap in all directions. For asymmetric structures, like SOI, modes with both parities must be included in the basis set. Furthermore, for any kind of vertical structure, defect modes can be classified according to reflection symmetry with respect to the vertical plane bisecting the waveguide channel, denoted as

_{kz}≡

_{xz}operation: odd (even) modes are classified as σ

_{kz}=-1 (σ

_{kz}=+1).

26. 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]

26. 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]

## 3. Determination of disorder parameters

14. M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H.-Y. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express **12**, 1551–1561 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1551. [CrossRef] [PubMed]

*r*by reproducing the experimentally measured values of propagation losses.

### 3.1. Losses in Silicon membranes

*a*=400 nm and a Silicon core thickness

*d*=220 nm (

*d*/

*a*=0.55). The average hole radius is fixed at

*r*/

*a*=0.275. Experimental data reported in Ref. [14

14. M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H.-Y. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express **12**, 1551–1561 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1551. [CrossRef] [PubMed]

_{kz}=-1 mode. Such modes are usually considered the most interesting for applications, as they have an even spatial profile of the field intensity (no nodes at the middle of the waveguide channel). In an ideal system without imperfections, modes with different

_{kz}should be completely uncoupled. In realistic structures, asymmetries between the two sides of the waveguide channel are responsible for parity mixing between σ

_{kz}=+1 and σ

_{kz}=-1 modes, as it was recently demonstrated [23

23. B. Cluzel, D. Gérard, E. Picard, T. Charvolin, V. Calvo, E. Hadji, and F. de Fornel, “Experimental demonstration of Bloch mode parity change in photonic crystal waveguide,” Appl. Phys. Lett. **85**, 2682–2684 (2004). [CrossRef]

*ka*/

*π*for the wave vector, on both axes. For a lattice constant

*a*=400 nm the σ

_{kz}=-1 defect mode dispersion is centered around

*λ*=1.5

*µ*m (

*a*/

*λ*⋍0.267) below the light line. The shaded region represents the continuum of TE-like slab modes of the triangular lattice folded in the line-defect Brillouin zone. On the right hand side, propagation losses are plotted only for what concerns the single-mode frequency region of the σ

_{kz}=-1 defect mode, on a logarithmic scale on the abscissas. The different curves correspond to different values of the disorder parameter. According to these calculations, experimental values of 0.5–0.6 dB/mm are reproduced with a disorder parameter Δ

*r*/

*a*=0.008, which means that the average uncertainty on the hole radius should be of the order of Δ

*r*=3.2 nm. As it can be noticed from Fig. 2, the actual low-loss frequency range for the mode considered here (which is highlighted in the figure) is much narrower than the whole single-mode propagation region. This is due to the strong dependence of propagation losses on the group velocity, which implies that the low group velocity region may be detrimental for future all-optical integration [26

**94**, 033903 (2005). [CrossRef] [PubMed]

27. K. Yamada, H. Morita, A. Shinya, and M. Notomi, “Improved line-defect structures for photonic-crystal waveguides with high group velocity,” Opt. Commun. **198**, 395–402 (2001). [CrossRef]

*a*/

*λ*⋍0.267 (

*λ*~1.5

*µ*m for

*a*=400 nm) and

*a*/

*λ*⋍0.279 (

*λ*~1.43

*µ*m) is highlighted, corresponding to a low-loss bandwidth of ~70 nm.

### 3.2. Losses in SOI waveguides

_{2}-clad PhC waveguides. The difference with respect to the symmetric air-clad structure is that the Si layer is grown on top of a semi-infinite SiO

_{2}cladding, as schematically depicted in the central picture of Fig. 1(b). In such kind of structures, it is well established from the literature that the low-loss propagation region below the light line is sensitively reduced by the presence of the SiO

_{2}light line. The problem of having a defect mode dispersion with a high group velocity below the light line is solved by using reduced-width PhC waveguides [12

12. M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, “Structural tuning of guiding modes of line-defect waveguides of Silicon-on-Insulator photonic crystal slabs,” IEEE J. Quantum Electron. **38**, 736–742 (2002). [CrossRef]

27. K. Yamada, H. Morita, A. Shinya, and M. Notomi, “Improved line-defect structures for photonic-crystal waveguides with high group velocity,” Opt. Commun. **198**, 395–402 (2001). [CrossRef]

*W*=0.7·√3

*a*.

*d*=220 nm, with

*r*/

*a*=0.275 and

*a*=400 nm. The defect mode dispersion and the corresponding losses at different r.m.s. deviations are shown for the W0.7 SOI waveguide in Fig. 3. A disorder parameter Δ

*r*=2.2 nm is found to give a good quantitative agreement with the experimental value (1.5 dB/mm) reported for an analogous SOI structure [14

**12**, 1551–1561 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1551. [CrossRef] [PubMed]

*r*value with respect to membrane samples could be considered a theoretical confirmation of the partial smoothing and reduction of roughness in SOI structures with respect to membrane ones. It can be seen from the dispersion of Fig. 3 that the propagation region below the light line is also reduced by the presence of the projected TM-like slab modes, which form a continuum of states and allow defining a

*TM-slab light line*, as also discussed in Refs. [21

21. Y. Tanaka, T. Asano, Y. Akahane, B.-S. Song, and S. Noda, “Theoretical investigation of a two-dimensional photonic crystal slab with truncated cone air holes,” Appl. Phys. Lett. **82**, 1661–1663 (2003). [CrossRef]

22. Y. Tanaka, T. Asano, R. Hatsuta, and S. Noda, “Analysis of a line-defect waveguide on a Silicon-on-Insulator two-dimensional photonic-crystal slab,” J. Lightwave Technol. **22**, 2787–2792 (2004). [CrossRef]

_{xy}is no more a symmetry operation for the system. Such TM-slab light line may be very detrimental for achieving a large propagation bandwidth below the light line, thus considerably limiting the functionality of real devices fabricated on asymmetric SiO

_{2}-clad PhC waveguides. The low-loss frequency window highlighted in figure has a wavelength bandwidth of ~30 nm, i.e. from

*a*/

*λ*~0.251 to

*a*/

*λ*~0.256 in dimensionless units, and it is again limited by the strong dispersion as a function of the group velocity of the defect mode. On the low energy side, the highlighted region has been limited to the TE-slab modes band edge (dark gray area in figure). Indeed, propagation losses due to scattering between the guided mode and the extended slab modes is not supposed to occur in straight line-defect waveguides, because no overlap exists between such modes in the Brillouin zone. Yet, in realistic photonic integrated circuits the presence of point defects or bends could induce such unwanted scattering mechanism, which thus would limit the effective low-loss propagation bandwidth. For this reason, only the propagation region above the photonic band edge of TE-like slab modes is considered [22

22. Y. Tanaka, T. Asano, R. Hatsuta, and S. Noda, “Analysis of a line-defect waveguide on a Silicon-on-Insulator two-dimensional photonic-crystal slab,” J. Lightwave Technol. **22**, 2787–2792 (2004). [CrossRef]

**94**, 033903 (2005). [CrossRef] [PubMed]

*v*

_{g}) to propagation losses. In any case, the overall physical behavior and the design considerations discussed in the following should not be affected by this phenomenon.

## 4. Design concepts for large-bandwidth and low-loss guided modes

### 4.1. Increased-width waveguides in air-clad PhC slabs

**29**, 1897–1899 (2004). [CrossRef] [PubMed]

25. L. C. Andreani, D. Gerace, and M. Agio, “Gap maps, diffraction losses and exciton-polaritons in photonic crystal slabs,” Phot. Nanostruct. **2**, 103–110 (2004). [CrossRef]

21. Y. Tanaka, T. Asano, Y. Akahane, B.-S. Song, and S. Noda, “Theoretical investigation of a two-dimensional photonic crystal slab with truncated cone air holes,” Appl. Phys. Lett. **82**, 1661–1663 (2003). [CrossRef]

18. Yu. A. Vlasov and S. J. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express **12**, 1622–1631 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1622. [CrossRef] [PubMed]

*L*

_{c}of the size fluctuations [17

17. K. K. Lee, D. R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO_{2} waveguide: experiments and model,” Appl. Phys. Lett. **77**, 1617–1619 (2000). [CrossRef]

36. F. P. Payne and J. P. R. Lacey, “A theoretical analysis of scattering loss from planar optical waveguides,” Opt. Quant. Electron. **26**, 977–986 (1994). [CrossRef]

37. F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size influence on the propagation loss induced by side-wall roughness in ultra-small SOI waveguides,” IEEE Photon. Techn. Lett. **16**, 1661–1663 (2004). [CrossRef]

*L*

_{c}is the hole circumference 2

*π r*, which is ~690 nm with the parameters of Fig. 4. If we compare with the calculated losses of single-mode Si/SiO

_{2}strip waveguides with σ=3.2 nm and

*L*

_{c}=690 nm, values of the order of 0.2-0.3 dB/mm are found [17

17. K. K. Lee, D. R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO_{2} waveguide: experiments and model,” Appl. Phys. Lett. **77**, 1617–1619 (2000). [CrossRef]

37. F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size influence on the propagation loss induced by side-wall roughness in ultra-small SOI waveguides,” IEEE Photon. Techn. Lett. **16**, 1661–1663 (2004). [CrossRef]

*L*

_{c}within a micro-roughness model for the hole sidewalls in PhC waveguides. Such an analysis is presently under study and will be reported elsewhere.

### 4.2. Reduced-width waveguides in asymmetric SiO_{2}-clad PhC slabs

**198**, 395–402 (2001). [CrossRef]

**22**, 2787–2792 (2004). [CrossRef]

**22**, 2787–2792 (2004). [CrossRef]

### 4.3. Waveguides in symmetric SiO_{2}-clad PhC slabs

_{2}claddings. In order to compare this structure to the corresponding SOI PhC waveguide of Fig. 5, the calculated photonic dispersion and propagation losses for the σ

_{kz}=-1 defect mode are shown in Fig. 6 for a symmetric SiO

_{2}-clad PhC waveguide. Both reduced- and increased-width waveguides have been systematically analyzed, and the choice of structure parameters for the calculations shown here follows from a detailed study aimed at optimizing both propagation bandwidth and losses.

*a*=400 nm, the useful propagation region of the defect mode is centered around 1.55

*µ*m. The symmetry operation

_{xy}is now allowed, and we consider only TE-like modes (σ

_{xy}=+1). As it can be seen by comparing Figs. 5 and 6(a), the low-loss propagation region for the W0.7 structure is slightly increased (~40 nm), owing to the existence of a complete band gap for TE-like modes (analogous to the air-clad structure), for which the TM-slab light line is not present. Results for propagation losses, shown on the right hand side of Fig. 6(a), show that they could be reduced by more than a factor of two with respect to the corresponding SOI structure, while having almost twice the propagation bandwidth of the W0.7 SOI waveguide. In Fig. 6(b), results are shown for increased-width waveguides in symmetric SiO

_{2}-clad structures. Although in this case the useful low-loss bandwidth is considerably smaller than in the optimized reduced-width waveguide, very low values for the propagation losses could be achieved in W1.1 waveguides, with an available bandwidth of ~20 nm. Such predicted values are comparable to measured values of state-of-the-art propagation losses in membrane-type W1.0 waveguides (see Fig. 2).

## 5. Conclusions

*W*=1.5

*W*

_{0}should allow achieving a large propagation bandwidth with predicted losses well below 0.1 dB/mm. For what concerns SOI-basedwaveguides, symmetrization of the vertical guiding structure, so-called symmetric SiO

_{2}-clad structures, should give larger propagation bandwidths and lower losses than in similar asymmetric structures.

*k*-

*ω*region available below the air light line. We believe that the proposed design concepts may be of importance for future research on photonic crystal-based optical circuits, not only for air-clad systems, but also for what concerns the use of large patterned areas on a single chip, in which robustness and integrability of a SOI structure would play a crucial role.

## Acknowledgments

## References and links

1. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

3. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

4. | K. Sakoda, |

5. | S. G. Johnson and J. D. Joannopoulos, |

6. |
See papers in |

7. | A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B |

8. | S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic crystal slabs,” Phys. Rev. B |

9. | M. Qiu, “Band gap effects in asymmetric photonic crystal slabs,” Phys. Rev. B |

10. | H. Benisty, D. Labilloy, C. Weisbuch, C. J. .M. Smith, T. F. Krauss, D. Cassagne, A. Béraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate,” Appl. Phys. Lett. |

11. | W. Bogaerts, P. Bienstman, D. Taillaert, R. Baets, and D. De Zutter, “Out-of-plane scattering in photonic crystal slabs,” IEEE Photon. Techn. Lett. |

12. | M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, “Structural tuning of guiding modes of line-defect waveguides of Silicon-on-Insulator photonic crystal slabs,” IEEE J. Quantum Electron. |

13. | S. J. McNab, N. Moll, and Yu. A. Vlasov, “Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,” Opt. Express |

14. | M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H.-Y. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express |

15. | W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, and R. Baets, “Basic structures for photonic integrated circuits in Silicon-on-insulator,” Opt. Express |

16. | Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, K. Asakawa, and K. Inoue, “Low propagation loss of 0.76 dB/mm in GaAs-based single-line-defect two-dimensional photonic crystal slab waveguides up to 1 cm in length,” Opt. Express |

17. | K. K. Lee, D. R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO |

18. | Yu. A. Vlasov and S. J. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express |

19. | W. Bogaerts, P. Bienstman, and R. Baets, “Scattering at sidewall roughness in photonic crystal slabs,” Opt. Lett. |

20. | M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, and M. Soljačić, “Effect of a photonic band gap on scattering from waveguide disorder,” Appl. Phys. Lett. |

21. | Y. Tanaka, T. Asano, Y. Akahane, B.-S. Song, and S. Noda, “Theoretical investigation of a two-dimensional photonic crystal slab with truncated cone air holes,” Appl. Phys. Lett. |

22. | Y. Tanaka, T. Asano, R. Hatsuta, and S. Noda, “Analysis of a line-defect waveguide on a Silicon-on-Insulator two-dimensional photonic-crystal slab,” J. Lightwave Technol. |

23. | B. Cluzel, D. Gérard, E. Picard, T. Charvolin, V. Calvo, E. Hadji, and F. de Fornel, “Experimental demonstration of Bloch mode parity change in photonic crystal waveguide,” Appl. Phys. Lett. |

24. | D. Gerace and L. C. Andreani, “Disorder-induced losses in photonic crystal waveguides with line defects,” Opt. Lett. |

25. | L. C. Andreani, D. Gerace, and M. Agio, “Gap maps, diffraction losses and exciton-polaritons in photonic crystal slabs,” Phot. Nanostruct. |

26. | 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. |

27. | K. Yamada, H. Morita, A. Shinya, and M. Notomi, “Improved line-defect structures for photonic-crystal waveguides with high group velocity,” Opt. Commun. |

28. | M. Notomi, K. Yamada, A. Shinya, J. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. |

29. | L. C. Andreani and M. Agio, “Photonic bands and gap maps in a photonic crystal slab,” IEEE J. Quantum Electron. |

30. | T. Ochiai and K. Sakoda, “Nearly free-photon approximation for two-dimensional photonic crystal slabs” Phys. Rev. B |

31. | L. C. Andreani, “Photonic bands and radiation losses in photonic crystal waveguides,” Physica Status Solidi B |

32. | L. C. Andreani and M. Agio, “Intrinsic diffraction losses in photonic crystal waveguides with line defects,” Appl. Phys. Lett. |

33. | M. Skorobogatiy, G. Bégin, and A. Talneau, “Statistical analysis of geometrical imperfections from the images of 2D photonic crystals,” Opt. Express |

34. | M. Galli, D. Bajoni, M. Patrini, G. Guizzetti, D. Gerace, L. C. Andreani, M. Belotti, and Y. Chen, “Single-mode versus multi-mode behavior in Silicon photonic crystal waveguides measured by attenuated total reflectance,” submitted to Phys. Rev. B. |

35. | D. Marcuse, “Mode conversion caused by surface imperfections of a dielectric slab waveguide,” Bell. Syst. Tech. J. |

36. | F. P. Payne and J. P. R. Lacey, “A theoretical analysis of scattering loss from planar optical waveguides,” Opt. Quant. Electron. |

37. | F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size influence on the propagation loss induced by side-wall roughness in ultra-small SOI waveguides,” IEEE Photon. Techn. Lett. |

**OCIS Codes**

(230.7370) Optical devices : Waveguides

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 10, 2005

Revised Manuscript: June 10, 2005

Published: June 27, 2005

**Citation**

Dario Gerace and Lucio Andreani, "Low-loss guided modes in photonic crystal waveguides," Opt. Express **13**, 4939-4951 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-13-4939

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