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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 12 — Jun. 4, 2012
  • pp: 13108–13114
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Control of dispersion in photonic crystal waveguides using group symmetry theory

Pierre Colman, Sylvain Combrié, Gaëlle Lehoucq, and Alfredo De Rossi  »View Author Affiliations


Optics Express, Vol. 20, Issue 12, pp. 13108-13114 (2012)
http://dx.doi.org/10.1364/OE.20.013108


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Abstract

We demonstrate dispersion tailoring by coupling the even and the odd modes in a line-defect photonic crystal waveguide. Coupling is determined ab-initio using group theory analysis, rather than by trial-error optimisation of the design parameters. A family of dispersion curves is generated by controlling a single geometrical parameter. This concept is demonstrated experimentally with very good agreement with theory.

© 2012 OSA

The linear and nonlinear propagation of waves depends crucially on the dispersion. The ability to control the dispersion is therefore a highly desired property of the physical medium supporting the propagation. For instance, the minimization of chromatic distortion in optical fibers [1

1. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

] is the main concern in optical communications. Moreover, some nonlinear interactions e.g. parametric conversion/generation, involving different frequencies, are completely determined by the dispersion, through the phase matching condition [2

2. R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

]. In optical fibers and waveguides, dispersion control is possible because the waveguide dispersion, which can be controlled through the geometry, can compensate for the material dispersion, which is material-dependent [3

3. L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. 31, 1295–1297 (2006). [CrossRef] [PubMed]

]. In an optical waveguide, e.g. the so-called photonic wires, the dispersion can be controlled by changing the height and the width of the high index silicon wire on top of the silica layer [3

3. L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. 31, 1295–1297 (2006). [CrossRef] [PubMed]

]. The ability to generate a prescribed dispersion function, also referred as dispersion engineering [4

4. J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16, 6227–6232 (2008). [CrossRef] [PubMed]

, 5

5. O. Khayam and H. Benisty, “General recipe for flatbands in photonic crystalwaveguides,” Opt. Express 17, 14634–14648 (2009). [CrossRef] [PubMed]

], is a more demanding goal. By offering many degrees of freedom in the design, photonic crystals (PhC) fibers [6

6. J. C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003). [CrossRef] [PubMed]

] and waveguides [4

4. J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16, 6227–6232 (2008). [CrossRef] [PubMed]

, 7

7. D. Mori and T. Baba, “Dispersion-controlled optical group delay device by chirped photonic crystal waveguides,” Appl. Phys. Lett. 85, 1101–1103 (2004). [CrossRef]

10

10. Y. Hamachi, S. Kubo, and T. Baba, “Slow light with low dispersion and nonlinear enhancement in a lattice-shifted photonic crystal waveguide,” Opt. Lett. 34, 1072–1074 (2009). [CrossRef] [PubMed]

] have proven to enable dispersion engineering.

The particular dispersion of line-defect photonic crystal waveguides is often described as the result of an anti-crossing between index-guided (e.g. guided by total internal reflection) and gap-guided modes (which only exist in PhCs) [11

11. A. Y. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. 85, 4866–4868 (2004). [CrossRef]

]. However the inherent complexity of photonic crystals, as opposed to simpler optical structures, limits the understanding of the mechanisms underlying the control of the dispersion. As a result, in order to tune a PhC waveguide, the design is optimized regardless of the nature of the underlying mechanism, e.g. the changes in the mode field distribution or in the coupling between modes. This said, dispersion engineering has been demonstrated in PhC waveguides by optimizing the coupling of two waveguides [7

7. D. Mori and T. Baba, “Dispersion-controlled optical group delay device by chirped photonic crystal waveguides,” Appl. Phys. Lett. 85, 1101–1103 (2004). [CrossRef]

,8

8. A. Petrov and M. Eich, “Dispersion compensation with photonic crystal line-defect waveguides,” IEEE J. Sel. Area. Commun. 23, 1396–1401 (2005). [CrossRef]

] or by coupling modes with different polarizations [12

12. M. Patterson, S. Hughes, D. Dalacu, and R. L. Williams, “Broadband purcell factor enhancements in photonic-crystal ridge waveguides,” Phys. Rev. B 80, 125307 (2009). [CrossRef]

].

The main idea of this paper is to define a suitable perturbation of PhC waveguide such that the even mode, which is commonly considered for PhC waveguide devices, anti-crosses with the odd mode. The parity here is defined relative to the plane parallel to the propagation direction and perpendicular to the PhC slab. Those two modes are typically found in line-defect PhC waveguides and are uncoupled because they have different symmetries; however, the odd mode is rather regarded as a nuisance limiting the spectral extent of single mode operation. Here, we show that, owing to even-odd mode coupling, a variety of useful dispersion profiles can be controlled by a single parameter, which is a quite desirable feature. More generally, the idea of coupling different modes in a PhC waveguide, as a mean to to control the dispersion in PhCs has been considered [13

13. S. Lü, J. Zhao, and D. Zhang, “Flat band slow light in asymmetric photonic crystal waveguide based on microfluidic infiltration,” Appl. Opt. 49, 3930–3934 (2010). [CrossRef] [PubMed]

16

16. N. Gutman, W. Dupree, Y. Sun, A. Sukhorukov, and C. de Sterke, “Frozen and broadband slow light in coupled periodic nanowire waveguides,” Opt. Express 20, 3519–3528 (2012). [CrossRef] [PubMed]

]; here we show how group symmetry theory can provide the right guidance to the design. We have fabricated a waveguide according to these design rules and found out that, not only the achievement of the desired dispersion profile was straightforward, owing to the very simple design rule, but the experimental evidence also allowed us to conclude that the propagation loss of such a waveguide were very low, compared to our initial design.

To fix the ideas, let us consider first the case of two coupled PhC waveguides [7

7. D. Mori and T. Baba, “Dispersion-controlled optical group delay device by chirped photonic crystal waveguides,” Appl. Phys. Lett. 85, 1101–1103 (2004). [CrossRef]

] within the formalism of coupled mode theory (CMT). The dispersion of the super-mode resulting from the coupling γ(k) of the modes of the individual waveguides, with dispersion ωwg{1,2} (k), results from the diagonalisation of:
|ωwg{1}(k)γ(k)γ*(k)ωwg{2}(k)|
(1)
Introducing the coupling results into a modification of the dispersion of the original modes, which can be controlled within some extent by playing with some parameters, such as the radius of the holes or the separation between the two waveguides. We point out that γ(k) could be modified, in principle, almost independently from ωwg{i} i = {1, 2}.

Here, we explore the possibility of coupling the two modes of a single waveguide in order to control the dispersion. Operating with a single waveguide, rather than a set of coupled waveguides, has many practical advantages: the excitation (i.e. the coupling) of the mode is easier and the linear and nonlinear effective areas are smaller, resulting into stronger light-matter interaction, a desirable feature in nanophotonics. Specifically, we consider the odd mode, defined relative to the plane of symmetry xz, where x is the direction of the propagation and z is perpendicular to the 2D PhC lattice.

Let us consider a line-defect waveguide with dispersion represented in Fig. 1(c), calculated using the 3D plane-wave expansion method implemented in the MPB code [17

17. 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]

]. The corresponding parameters, normalized with the lattice period, are: radius r/a = 0.26, slab thickness h/a = 0.41, width of the line defect W/a = 0.95 √3, outward displacement of the first row of holes s/a = 0.14, with radius reduced to r1/a = 0.23. The refractive index of the material is assumed to be 3.17. Here, the parameters have been chosen in order to lower the edge of the odd mode (at k = 0.5K, K being the reciprocal lattice vector); indeed, here the frequency spacing is only δfa/c = 610−3. If a suitable coupling between these two modes, which are now closer in frequency, is introduced, then the anti-crossing will result into a change in the dispersion. As a consequence, controlling the coupling strength would enable, for example, the flattening of the lower frequency branch.

Fig. 1 PhC waveguide without (a) and with (b) perturbation. (c) Corresponding band diagram of the initial (perturbed) structure in black (red) line. The cyan markers indicate the modes for which the field distribution is shown in Figs. 2(a) and 2(b). The thin blue line corresponds to the dispersion obtained for a different perturbation as discussed in Fig. 2(d).
Fig. 2 Spatial distribution of the Hz-field, calculated at K=0.445, (a) Even mode and Odd mode. (b) even-like and odd-like mode for a dielectric perturbatation as presented in (c). (c) Calculated |γee|, |γoo| and |γeo| coefficients for T=0.15a. Inset : corresponding Δε and its dominant symmetry (colors refers to opposite signs). (d) Idem as (c) but holes are moved in the same direction so the symmetry of Δε is now B1 instead of A2.

Hereafter we give a procedure to generate a perturbation Δε (r⃗) such that γeo (k) is non-zero, but also dominates also substantially over the other terms. Let us first look at the integrand Eodd*(r)Δε(r)Eeven(r). The integral can be non-zero only if this is an even function. Considering that the planar PhC waveguide belongs to the group of symmetry C2v, that statement is equivalent to requiring the integrand to have at least the element of symmetry A1 (namely, the fully symmetric function). Conversely, if that condition is not respected, then integrand will have both positive and negative parts which would mutually cancel out.

Proceeding similarly as in [18

18. O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,” Phys. Rev. B 68, 035110 (2003). [CrossRef]

], the Bloch modes in a PhC with a 2D hexagonal lattice can be labelled according to Table 1. Based on the mode field distribution of the unperturbed structure in Fig. 2(a), we conclude that the even and the odd modes belong to the symmetry subgroups B1 and B2 respectively. The symmetry condition on the integrand in γoe (k) is then formulated as:
A1=B2XB1
(5)
where X is the symmetry of Δε (r⃗), still to be determined. It is found that X must have the symmetry A2. This implies that the dielectric perturbation is anti-symmetric with respect both to the x = 0 and the y = 0 symmetry planes. This also implies that γoo (k) = γee (k) = γss (k) = γse (k) = 0, i.e., the only non-zero elements are γeo (k) and γso (k). Indeed, since the even and the substrate modes have the same symmetry, they are not coupled together by a such perturbation. It can be observed that the coupling γso (k) between the odd and the substrate modes is minimized if the initial frequency spacing between the two modes is large and if the changes of the dielectric function Δε (r⃗) is localized where the field of the even and odd modes is much stronger than that of the substrate mode. This is in general the case in the region close to the line defect (see Fig. 2(a)).

Table 1. C2v point group symmetry table

table-icon
View This Table

We point out that this analysis applies to other design, for instance that based on the B1 perturbation, as used in ref. [10

10. Y. Hamachi, S. Kubo, and T. Baba, “Slow light with low dispersion and nonlinear enhancement in a lattice-shifted photonic crystal waveguide,” Opt. Lett. 34, 1072–1074 (2009). [CrossRef] [PubMed]

]. There, the even mode and the substrate modes of a simple line-defect waveguide are very close in frequency. Thereby the introduction of the B1 readily produces a strong change in the dispersion.

Let us consider now the dispersion, namely the group index as a function of the wavelength, as the parameter T is changed (Fig. 3(c)). Indeed, while a flat-band over more then 10nm is obtained with T=0.05 (delay-bandwidth product of 0.15), the other values generate a dispersion where the group dispersion changes sign twice and a local maximum of the group index increasing sharply as T is further increased.

Fig. 3 (a) Group index dependence as a function of the wavelength calculated for T= 0.05a, 0.1a, 0.2a. (b) Corresponding measured dispersion. The lattice period is a = 465nm.

We have fabricated a sample based on GaInP membrane [19

19. S. Combrié, Q. V. Tran, A. D. Rossi, C. Husko, and P. Colman, “High quality gainp nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. 95, 221108 (2009). [CrossRef]

] with the above specified design and we measured the dispersion using an interferometric technique [20

20. A. Parini, P. Hamel, A. De Rossi, S. Combrie, N.-V.-Q. Tran, Y. Gottesman, R. Gabet, A. Talneau, Y. Jaouen, and G. Vadala, “Time-wavelength reflectance maps of photonic crystal waveguides: a new view on disorder-induced scattering,” J. Lightwave Technol . 26, 3794–3802 (2008). [CrossRef]

]. The use of mode adapters [21

21. Q. V. Tran, S. Combrié, P. Colman, and A. D. Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett. 95, 061105 (2009). [CrossRef]

] prevents any coupling issue, although the mode is slightly asymmetric. The result is shown is Fig. 3(b). The agreement with the predicted dispersion is very good. Residual differences (e.g. for T/a = 0.05) can be attributed to the fabrication tolerances (e.g. the radius of the holes). The important point is that the transition from one of these dispersion profiles to another follows continuously the change in the parameter T; which can be further adjusted in order to reach the desired dispersion profile.

The same concept can also be extended to the coupling between two arbitrary modes (not only odd and even ones). This result is relevant to a broad range of applications, particularly parametric effects (e.g. Four-Wave Mixing) and it enables the optimization of the phase matching. We believe that it could be applied to other systems as well (e.g. PCF) and enable an even more accurate control of the dispersion.

Acknowledgments

This work has been supported by the 7th framework of the European commission through the GOSPEL project (www.gospel-project.eu) and COPERNICUS (www.copernicusproject.eu) projects. We thank M. Santagiustina, G. Eisenstein and S. Trillo for enlightening discussions.

References and links

1.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

2.

R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

3.

L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. 31, 1295–1297 (2006). [CrossRef] [PubMed]

4.

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16, 6227–6232 (2008). [CrossRef] [PubMed]

5.

O. Khayam and H. Benisty, “General recipe for flatbands in photonic crystalwaveguides,” Opt. Express 17, 14634–14648 (2009). [CrossRef] [PubMed]

6.

J. C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003). [CrossRef] [PubMed]

7.

D. Mori and T. Baba, “Dispersion-controlled optical group delay device by chirped photonic crystal waveguides,” Appl. Phys. Lett. 85, 1101–1103 (2004). [CrossRef]

8.

A. Petrov and M. Eich, “Dispersion compensation with photonic crystal line-defect waveguides,” IEEE J. Sel. Area. Commun. 23, 1396–1401 (2005). [CrossRef]

9.

L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express 14, 9444–9450 (2006). [CrossRef] [PubMed]

10.

Y. Hamachi, S. Kubo, and T. Baba, “Slow light with low dispersion and nonlinear enhancement in a lattice-shifted photonic crystal waveguide,” Opt. Lett. 34, 1072–1074 (2009). [CrossRef] [PubMed]

11.

A. Y. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. 85, 4866–4868 (2004). [CrossRef]

12.

M. Patterson, S. Hughes, D. Dalacu, and R. L. Williams, “Broadband purcell factor enhancements in photonic-crystal ridge waveguides,” Phys. Rev. B 80, 125307 (2009). [CrossRef]

13.

S. Lü, J. Zhao, and D. Zhang, “Flat band slow light in asymmetric photonic crystal waveguide based on microfluidic infiltration,” Appl. Opt. 49, 3930–3934 (2010). [CrossRef] [PubMed]

14.

X. Mao, Y. Huang, W. Zhang, and J. Peng, “Coupling between even- and oddlike modes in a single asymmetric photonic crystal waveguide,” Appl. Phys. Lett. 95, 183106 (2009). [CrossRef]

15.

J. Ma and C. Jiang, “Demonstration of ultraslow modes in asymmetric line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 20, 1237–1239 (2008). [CrossRef]

16.

N. Gutman, W. Dupree, Y. Sun, A. Sukhorukov, and C. de Sterke, “Frozen and broadband slow light in coupled periodic nanowire waveguides,” Opt. Express 20, 3519–3528 (2012). [CrossRef] [PubMed]

17.

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]

18.

O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,” Phys. Rev. B 68, 035110 (2003). [CrossRef]

19.

S. Combrié, Q. V. Tran, A. D. Rossi, C. Husko, and P. Colman, “High quality gainp nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. 95, 221108 (2009). [CrossRef]

20.

A. Parini, P. Hamel, A. De Rossi, S. Combrie, N.-V.-Q. Tran, Y. Gottesman, R. Gabet, A. Talneau, Y. Jaouen, and G. Vadala, “Time-wavelength reflectance maps of photonic crystal waveguides: a new view on disorder-induced scattering,” J. Lightwave Technol . 26, 3794–3802 (2008). [CrossRef]

21.

Q. V. Tran, S. Combrié, P. Colman, and A. D. Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett. 95, 061105 (2009). [CrossRef]

22.

Y. Vlasov, W. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2, 242–246 (2008). [CrossRef]

23.

A. Shinya, S. Matsuo, Yosia, T. Tanabe, E. Kuramochi, T. Sato, T. Kakitsuka, and M. Notomi, “All-optical on-chip bit memory based on ultra high Q InGaAsP photonic crystal,” Opt. Express 16, 19382–19387 (2008). [CrossRef]

24.

A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. Ó Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “On-chip tunable delay lines in silicon photonics,” IEE Photon. J. 2, 181–194 (2010). [CrossRef]

25.

S. Schultz, L. O’Faolain, D. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010). [CrossRef]

26.

P. Colman, S. Combrié, I. Cestier, A. Willinger, G. Eisenstein, A. de Rossi, and G. Lehoucq, “Observation of gain due to four-wave-mixing in dispersion engineered GaInP photonic crystal waveguides,” Opt. Lett. 36, 2629–2631 (2011). [CrossRef] [PubMed]

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(260.2030) Physical optics : Dispersion
(130.5296) Integrated optics : Photonic crystal waveguides

ToC Category:
Photonic Crystals

History
Original Manuscript: March 7, 2012
Revised Manuscript: April 12, 2012
Manuscript Accepted: April 12, 2012
Published: May 25, 2012

Citation
Pierre Colman, Sylvain Combrié, Gaëlle Lehoucq, and Alfredo De Rossi, "Control of dispersion in photonic crystal waveguides using group symmetry theory," Opt. Express 20, 13108-13114 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13108


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References

  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).
  2. R. W. Boyd, Nonlinear Optics (Academic Press, 2003).
  3. L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett.31, 1295–1297 (2006). [CrossRef] [PubMed]
  4. J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express16, 6227–6232 (2008). [CrossRef] [PubMed]
  5. O. Khayam and H. Benisty, “General recipe for flatbands in photonic crystalwaveguides,” Opt. Express17, 14634–14648 (2009). [CrossRef] [PubMed]
  6. J. C. Knight, “Photonic crystal fibres,” Nature424, 847–851 (2003). [CrossRef] [PubMed]
  7. D. Mori and T. Baba, “Dispersion-controlled optical group delay device by chirped photonic crystal waveguides,” Appl. Phys. Lett.85, 1101–1103 (2004). [CrossRef]
  8. A. Petrov and M. Eich, “Dispersion compensation with photonic crystal line-defect waveguides,” IEEE J. Sel. Area. Commun.23, 1396–1401 (2005). [CrossRef]
  9. L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express14, 9444–9450 (2006). [CrossRef] [PubMed]
  10. Y. Hamachi, S. Kubo, and T. Baba, “Slow light with low dispersion and nonlinear enhancement in a lattice-shifted photonic crystal waveguide,” Opt. Lett.34, 1072–1074 (2009). [CrossRef] [PubMed]
  11. A. Y. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett.85, 4866–4868 (2004). [CrossRef]
  12. M. Patterson, S. Hughes, D. Dalacu, and R. L. Williams, “Broadband purcell factor enhancements in photonic-crystal ridge waveguides,” Phys. Rev. B80, 125307 (2009). [CrossRef]
  13. S. Lü, J. Zhao, and D. Zhang, “Flat band slow light in asymmetric photonic crystal waveguide based on microfluidic infiltration,” Appl. Opt.49, 3930–3934 (2010). [CrossRef] [PubMed]
  14. X. Mao, Y. Huang, W. Zhang, and J. Peng, “Coupling between even- and oddlike modes in a single asymmetric photonic crystal waveguide,” Appl. Phys. Lett.95, 183106 (2009). [CrossRef]
  15. J. Ma and C. Jiang, “Demonstration of ultraslow modes in asymmetric line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett.20, 1237–1239 (2008). [CrossRef]
  16. N. Gutman, W. Dupree, Y. Sun, A. Sukhorukov, and C. de Sterke, “Frozen and broadband slow light in coupled periodic nanowire waveguides,” Opt. Express20, 3519–3528 (2012). [CrossRef] [PubMed]
  17. 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]
  18. O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,” Phys. Rev. B68, 035110 (2003). [CrossRef]
  19. S. Combrié, Q. V. Tran, A. D. Rossi, C. Husko, and P. Colman, “High quality gainp nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett.95, 221108 (2009). [CrossRef]
  20. A. Parini, P. Hamel, A. De Rossi, S. Combrie, N.-V.-Q. Tran, Y. Gottesman, R. Gabet, A. Talneau, Y. Jaouen, and G. Vadala, “Time-wavelength reflectance maps of photonic crystal waveguides: a new view on disorder-induced scattering,” J. Lightwave Technol. 26, 3794–3802 (2008). [CrossRef]
  21. Q. V. Tran, S. Combrié, P. Colman, and A. D. Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett.95, 061105 (2009). [CrossRef]
  22. Y. Vlasov, W. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics2, 242–246 (2008). [CrossRef]
  23. A. Shinya, S. Matsuo, Yosia, T. Tanabe, E. Kuramochi, T. Sato, T. Kakitsuka, and M. Notomi, “All-optical on-chip bit memory based on ultra high Q InGaAsP photonic crystal,” Opt. Express16, 19382–19387 (2008). [CrossRef]
  24. A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. Ó Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “On-chip tunable delay lines in silicon photonics,” IEE Photon. J.2, 181–194 (2010). [CrossRef]
  25. S. Schultz, L. O’Faolain, D. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt.12, 104004 (2010). [CrossRef]
  26. P. Colman, S. Combrié, I. Cestier, A. Willinger, G. Eisenstein, A. de Rossi, and G. Lehoucq, “Observation of gain due to four-wave-mixing in dispersion engineered GaInP photonic crystal waveguides,” Opt. Lett.36, 2629–2631 (2011). [CrossRef] [PubMed]

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