Frozen and broadband slow light in coupled periodic nanowire waveguides

We develop novel designs enabling slow-light propagation with vanishing group-velocity dispersion (“frozen light”) and slow-light with large delay-bandwidth product, in periodic nanowires. Our design is based on symmetry-breaking of periodic nanowire waveguides and we demonstrate its vailidy through twoand three-dimensional simulations. The slow-light is associated with a stationary inflection point which appears through coupling between forward and backward waveguide modes. The mode coupling also leads to evanescent modes, which enable efficient light coupling to the slow mode. © 2012 Optical Society of America OCIS codes: (230.7370) Waveguides; (250.5300) Photonic integrated waveguides. References and links 1. M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express 16, 1300–1320 (2008). 2. P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94, 121106–3 (2009). 3. E. Kuramochi, H. Taniyama, T. Tanabe, K. Kawasaki, Y. G. Roh, and M. Notomi, “Ultrahigh-q one-dimensional photonic crystal nanocavities with modulated mode-gap barriers on SiO2 claddings and on air claddings,” Opt. Express 18, 15859–15869 (2010). 4. M. D. Settle, R. J. P. Engelen, M. Salib, A. Michaeli, L. Kuipers, and T. F. Krauss, “Flatband slow light in photonic crystals featuring spatial pulse compression and terahertz bandwidth,” Opt. Express 15, 219–226 (2007). 5. T. F. Krauss, “Why do we need slow light?” Nature Photon. 2, 448–450 (2008). 6. T. Baba, “Slow light in photonic crystals,” Nature Photon. 2, 465–473 (2008). 7. J. B. Khurgin and R. S. Tucker, eds., Slow Light: Science and Applications (Tailor and Francis, New York, 2009). 8. R. J. P. Engelen, Y. Sugimoto, Y. Watanabe, J. P. Korterik, N. Ikeda, N. F. van Hulst, K. Asakawa, and L. Kuipers, “The effect of higher-order dispersion on slow light propagation in photonic crystal waveguides,” Opt. Express 14, 1658–1672 (2006). 9. J. Ma and M. L. Povinelli, “Effect of periodicity on optical forces between a one-dimensional periodic photonic crystal waveguide and an underlying substrate,” Appl. Phys. Lett. 97, 151102 (2010). 10. A. Y. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. 85, 4866–4868 (2004). 11. 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). 12. 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). 13. T. P. White, L. C. Botten, C. M. de Sterke, K. B. Dossou, and R. C. McPhedran, “Efficient slow-light coupling in a photonic crystal waveguide without transition region,” Opt. Lett. 33, 2644–2646 (2008). 14. 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). #159068 $15.00 USD Received 1 Dec 2011; revised 18 Jan 2012; accepted 23 Jan 2012; published 30 Jan 2012 (C) 2012 OSA 13 February 2012 / Vol. 20, No. 4 / OPTICS EXPRESS 3519 15. S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010). 16. 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,” Nature Photon. 3, 206–210 (2009). 17. C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron. 16, 344–356 (2010). 18. L. O’Faolain, D. M. Beggs, T. P. White, T. Kampfrath, K. Kuipers, and T. F. Krauss, “Compact optical switches and modulators based on dispersion engineered photonic crystals,” IEEE Photon. J. 2, 404–414 (2010). 19. A. Sukhorukov, A. Lavrinenko, D. Chigrin, D. Pelinovsky, and Y. Kivshar, “Slow-light dispersion in coupled periodic waveguides,” J. Opt. Soc. Am. B 25, C65–C74 (2008). 20. C. Bao, J. Hou, H. Wu, X. Zhou, E. Cassan, and X. Gao, and D. Zhang, “Low dispersion slow light in slot waveguide grating,” IEEE Photon. Technol. Lett. 23, 1700–1702 (2011). 21. S. J. McNab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,” Opt. Express 11, 2927–2939 (2003). 22. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large groupvelocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001). 23. A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” Waves Random Complex Media 16, 293–382 (2006). 24. A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B 67, 165210 (2003). 25. A. Figotin and I. Vitebskiy, “Oblique frozen modes in periodic layered media,” Phys. Rev. E 68, 036609 (2003). 26. J. Ballato, A. Ballato, A. Figotin, and I. Vitebskiy, “Frozen light in periodic stacks of anisotropic layers,” Phys. Rev. E 71, 036612–12 (2005). 27. M. Spasenovic, T. P. White, S. Ha, A. A. Sukhorukov, T. Kampfrath, Y. S. Kivshar, C. M. de Sterke, T. F. Krauss, and L. Kuipers, “Experimental observation of evanescent modes at the interface to slow-light photonic crystal waveguides,” Opt. Lett. 36, 1170–1172 (2011). 28. S. Ha, M. Spasenovic, A. A. Sukhorukov, T. P. White, C. M. de Sterke, L. K. Kuipers, T. F. Krauss, and Y. S. Kivshar, “Slow-light and evanescent modes at interfaces in photonic crystal waveguides: optimal extraction from experimental near-field measurements,” J. Opt. Soc. Am. B 28, 955–963 (2011). 29. S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002). 30. A. Hosseini, X. C. Xu, D. N. Kwong, H. Subbaraman, W. Jiang, and R. T. Chen, “On the role of evanescent modes and group index tapering in slow light photonic crystal waveguide coupling efficiency,” Appl. Phys. Lett. 98, 031107 (2011). 31. C. M. de Sterke, K. B. Dossou, T. P. White, L. C. Botten, and R. C. McPhedran, “Efficient coupling into slow light photonic crystal waveguide without transition region: role of evanescent modes,” Opt. Express 17, 17338–17343 (2009). 32. B. Desiatov, I. Goykhman, and U. Levy, “Nanoscale mode selector in silicon waveguide for on chip nanofocusing applications,” Nano Lett. 9, 3381–3386 (2009). 33. A. Figotin and I. Vitebsky, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E 63, 066609 (2001). 34. 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). 35. A. A. Sukhorukov, S. Ha, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Dispersion extraction with near-field measurements in periodic waveguides,” Opt. Express 17, 3716–3721 (2009). 36. S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, C. M. de Sterke, and Y. S. Kivshar, “Bloch-mode extraction from near-field data in periodic waveguides,” Opt. Lett. 34, 3776–3778 (2009). 37. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible freesoftware package for electromagnetic simulations by the fdtd method,” Comput. Phys. Commun. 181, 687–702 (2010). 38. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nature Photon. 1, 65–71 (2007). 39. D. Tan, K. Ikeda, P. Sun, and Y. Fainman, “Group velocity dispersion and self phase modulation in silicon nitride waveguides,” Appl. Phys. Lett. 96, 061101 (2010). 40. N. Gutman, L. C. Botten, A. A. Sukhorukov, and C. M. de Sterke, “Slow and frozen light in optical waveguides with multiple gratings: Degenerate band edges and stationary inflection points,” submitted to Phys. Rev. A. 41. N. Gutman, L. C. Botten, A. A. Sukhorukov, and C. M. de Sterke, “Degenerate band edges in optical fiber with multiple grating: efficient coupling to slow light,” Opt. Lett. 36, 3257–3259 (2011). #159068 $15.00 USD Received 1 Dec 2011; revised 18 Jan 2012; accepted 23 Jan 2012; published 30 Jan 2012 (C) 2012 OSA 13 February 2012 / Vol. 20, No. 4 / OPTICS EXPRESS 3520


Introduction
Optical nanowire waveguides are emerging as a versatile building block for manipulation of optical pulses in compact photonic chips.Nanowires facilitate strong light confinement combined with flexible engineering of the modal dispersion, enabling efficient pulse manipulation through the enhancement of nonlinear optical effects [1].Perforated waveguides containing arrays of holes can be designed to realize optical cavities [2,3], further enhancing light-matter interactions with applications ranging from all-optical switching to environmental sensing.Nanowire cavities can have very high Q-factors, comparable to those in two-dimensional photonic crystal (2D PC) cavities [2,3].
Whereas high-Q cavities can be used to trap pulses and strongly enhance nonlinear effects, the tradeoff is the narrow-band operation.On the other hand, in periodic waveguides it is possible to realize broadband slow-light propagation, where the total pulse delay and nonlinearity enhancement can be potentially increased by scaling the device length [4][5][6][7].For such scalability, it is necessary to lower the group-velocity dispersion (GVD) which may lead to undesirable pulse broadening [8].For a single periodic nanowire waveguide slow-light occurs close to a photonic band-edge [9], and in this regime strong GVD is unavoidable.So far, slow-light propagation with suppressed GVD has been only demonstrated in specially designed 2D PC waveguides (PCW), where the geometry of otherwise periodic PCs is distorted in the waveguiding region by shifting rows of holes or changing the hole sizes [10][11][12][13][14][15], and importantly in such PCWs strong enhancement of nonlinear processes was demonstrated [16][17][18].
In this paper we present a new general approach to the design of perforated nanowire waveguides, which can have slow-light characteristics similar to those achievable in 2D PCWs.We demonstrate that by introducing a longitudinal shift between parallel rows of holes in the nanowires (see Fig. 1), it is possible to realize broadband slow-light with large delay-bandwidth product and "frozen light" with extremely small group velocity, which, based on previous work [19], shows efficient in-coupling to slow-light modes.These results suggest new opportunities for application of compact nanowire waveguides, which have smaller footprint compared to 2D photonic crsytal-based waveguides, for the realization of optical delays and slow-light enhanced light-matter interactions.Recent work of Bao et.al. [20], suggested similar structures but these structures have four modes and have a symmetric period.For these reason these structures have a slightly larger footprint." The paper is organized as follows.In Sec. 2 we review the key properties of dispersion relations which are suited for slow-light propagation.In Sec. 3 we present the design of nanowire waveguides featuring such dispersion, and show explicitly the presence of evanescent modes which are known to enable efficient coupling.Finally in Sec. 4 we discuss our findings and conclude.

Slow light and dispersion stationary inflection points
Insight into the physical properties of optical waveguides can be obtained by examining the dispersion relation, the dependence of the guided mode frequency (ω) on the wavenumber (k).In periodic waveguides, Photonic Stop Bands (PSB) can emerge.For all frequencies ω inside a PSB the associated modes are evanescently growing or decaying, which means that their wavenumbers are complex.By contrast, in allowed bands at least one mode exists with a purely real wavenumber, representing a propagating mode.In the last decade, waveguides with low group velocity v g = ∂ ω/∂ k (with real k) have been successfully engineered in periodic structures, such as PCWs [6].
A traditional approach for realizing slow light is based on the generic feature of the dispersion relation close to a photonic band-edge.Here the dispersion curves have a maximum or a minimum and the group velocity has to vanish.Most commonly, the band-edge dispersion fea- tures a quadratic stationary point, where Δω ∝ D 2 Δk 2 [21], where Δω and Δk are the frequency and wavenumber detunings from the band-edge, and D j = (∂ j ω/∂ k j )/ j! is proportional to the j'th order dispersion coefficient; we write v g for D 1 .Then the group velocity can be expressed as v g ∝ |Δω| 1/2 .Thus the group velocity changes rapidly versus the frequency detuning close to the band-edge, which is why GVD is unavoidable in this case [8,22].
To realize slow-light with small or vanishing GVD, the region of small group velocity should appear inside a transmission band, away from the band-edges.This occurs when the dispersion features an inflection point (IP) for finite group velocity for which D 2 = 0 and thus Δω ∝ v g0 Δk +D 3 Δk 3 .However, for the geometries we consider here we can take two of the dispersion coefficients to vanish and so we consider a stationary inflection points (SIP) where ( The realization of a SIP was first suggested for one-dimensional (1D) periodic photonic crystals [23], composed of magnetic [24] and anisotropic [25,26] layers.However in such 1D structures light would exhibit strong refraction and diffraction due to a strong dependence of the group velocity on the angle of incidence, which so far prevented experimental demonstration of these concepts.A robust realization of dispersion with an SIP or IP was achieved in specially designed 2D PCW [13,27,28], where the mode can only propagate along the waveguide.In the following sections we show how to generate a SIP in periodic nanowires and show how light can be coupled to the SIP frequency were the group velocity is zero, hence "frozen light".We also show that for small but finite group velocity, we can have D 2 = 0 and D 3 = 0, creating a robust IP with low and constant group velocity.
The dispersion relation near a SIP ω(k) can be formally inverted to find the values of wavenumber k for each frequency ω.Since near a SIP the dispersion is cubic to lowest order (see Eq. ( 1)), there are three such solutions Δk 1,2,3 .We can choose the first solutions to be real-valued, corresponding to the propagating slow-light mode, i.e., Δk 1 ≡ Δk. (2) The two other, complex solutions have the simple form describing evanescent modes which are exponentially decaying or growing.
As we show in Sec 3.2, such modes are excited at waveguide boundaries.An important consideration for practical applications of slow-light waveguides is the in-coupling efficiency.In order to overcome the mismatch between the impedance of the incoming fast mode and the slow modes, specially designed tapers [29] and intermediate PCW [30] have been explored.Recently, it was shown that efficient coupling can be achieved without tapering by harnessing the properties evanescent modes of the slow light waveguide, which can strongly enhance the coupling efficiency [31].As shown above, evanescent modes are always present around an SIP, enabling efficient in-coupling [13,23], which is a very desirable feature.

Periodic wire waveguides
We now present a simple but systematic and powerful approach for achieving a SIP in coupled periodic wire waveguides.Such waveguides, which are illustrated in Fig. 1, have been extensively investigated in recent years, and we choose parameters based on previously fabricated structures [32].Since a SIP is associated with three modes (one propagating and two evanescent-see Eqs ( 2)-( 4)), the unperturbed waveguide, i.e., the waveguide without the holes shown in Fig. 1, needs to have three modes.As shown below, these modes are coupled by the holes, leading to an SIP [23].Unlike the 1D PCs considered by Figotin et al. [25,26,33], where at any frequency there are four modes, in the optical waveguides considered here the number of modes may be chosen by varying the waveguide width.In 1D PCs, non-reciprocal materials or tilted incident beam are needed to couple three of the modes (two modes propagating in one direction and one in the opposite direction).In our three mode waveguide (with three forward and three backward modes), two sets of three modes can be used to create SIPs at ±k o , as shown below.

Dispersion engineering in wire waveguides
We calculate the dispersion relations of the nanowire waveguides using 2D simulations with effective refractive index 2.798, which corresponds to a waveguide thickness of w z = 0.5a (we validate our results with full 3D simulations in Sec.3.4 below).The calculations were performed using the MIT Photonic Bands (MPB) software package utilizing the plane-wave expansion method [34].
As a reference, we calculate the band structure of a single mode waveguide without holes (see Fig. 2(a)).Due to the formal periodicity of the structure, it is sufficient to consider only the 1 st Brillouin zone (BZ), since at the BZ edges (ka/2π = ±0.5) the dispersion relations are folded back.We consider TE modes (magnetic field pointing in the z-axis) of silicon waveguide.
A three mode waveguide can be achieved by placing identical single mode wires on both sides of a central waveguide.Figure 2(b) shows the splitting of the single mode by the coupling between the waveguides.The mode spacing can be adjusted by changing the distance g between the waveguides centers (see Fig. 1).The three modes are either symmetric (blue) or anti-symmetric (red) about the center.In a symmetric periodic structure [Fig.2(c)] the modes with the same symmetry are coupled, creating an avoided crossing.The anti-asymmetric mode does not couple to the symmetric modes and its dispersion relation intersects that of the other modes.An asymmetric periodic structure is required to couple all three modes.It is well known in optical and electrical lithography that fabricating holes with high accuracy in diameter is more difficult than positioning them.Hence, we choose to make the structure asymmetric by shifting one line of holes laterally with respect to the other, so that as illustrated in Fig. 2(d), all modes become coupled.In this structure, none of the modes have a simple odd or even transverse symmetry.The modes with positive dispersion slope are dielectric modes, for which the optical field is concentrated in the high dielectric medium, whereas the modes with a negative slope are air modes, for which the field is concentrated in the low index medium (here air).To create a SIP, we tune the parameters to adjoin the two stationary points, marked in red circles in Fig. 2(d).This can be achieved by adjusting the lateral shift z A .The dispersion with SIP is illustrated in Fig. 2(e).If the shift is further increased, then the dispersion transforms to an IP with v g0 = 0, see Fig. 2(i).Away from the SIP the modes preserve their symmetry.Figures 2(f)-(h) show the mode profile along the band with a SIP.As k increases the mode evolves from being approximately symmetric (f) to being approximately antisymmetric (h).At the SIP, the mode has no distinct symmetry (g).
They show how the mode evolve from a symmetric to an anti-symmetric.At the SIP, Fig. 2(h), there is no distinct symmetry.This very general procedure can be applied to structures with any value of the waveguide separation g.The results of such calculations are summarized in Fig. 3(a) in which the blue curve gives the pairs of waveguide separation g and longitudinal shift z A for which a SIP results, for fixed radius r = 0.22a and width of w = 0.8a.On this curve the dispersion relation transitions between a band with two local stationary points [Fig.2(d)] and a monotonically increasing band [Fig.2(i)], providing a practical and rigorous condition for the presence of a SIP.

Complex band structure of a stationary inflection point
The dispersion relation of a waveguide, which strictly speaking refers to an infinite structure, usually only includes propagating modes.The plane wave method, for example, does not calculate the evanescent modes.Nevertheless, waveguides support evanescent modes with complex wavenumbers (see Eqs. ( 3) and ( 4)), which play a significant role at interfaces.To confirm explicitly that our structures support these evanescent modes, and that they can thus be expected to exhibit effiient slow light coupling, we find the complex wavenumbers from the electric field distribution using the dispersion extraction method.Briefly, in this method the field is written as a superposition of Bloch modes, propagating and evanescent, with the paramters extracted using a least-square approach [28,35,36].We simulated light propagation through the structure using the finite-difference time-domain (FDTD) method [37].The simulation was run for different frequencies around the SIP.The complex field in the propagation direction in one of the waveguides was taken from these sim- ulation, and the transmission coefficient was calculated.Because all modes of the periodic waveguides, both propagating and evanescent, satisfy Bloch's theorem, the complex amplitude of each mode can be expressed as where k m are the complex wavenumber; x is the direction of the periodicity, and the Φ m are the periodic Bloch-wave envelope function which satisfy Φ m (y, x) = Φ m (y, x + a).The total field inside the waveguide is a linear superposition of six guided and evanescent modes (three pairs of forward and backward modes): where the a m are the mode amplitudes and w(x, y; ω) is the radiation field due to scattering or the excitation of non-guided waves.In Fig. 4(c) the best fitting forward Bloch functions are shown, at frequency ω = 0.281, to the calculated field in Fig. 4(b), indicating one propagating mode and two evanescent modes, one decaying and one growing, around a center wavenumber.This process is carried out for a range of frequencies around the SIP frequency ω o = 0.28.The real and imaginary parts of the wavenumbers are plotted against frequency in Fig. 4(d) and (e) respectively.The real wavenumber (blue), follow Eq. ( 2); the complex conjugate pair in red and green follow Eqs.( 3) and (4).These three modes become degenerate at the SIP consistent with Eqs (2)-( 4), where the group index diverges to infinity (f).In Fig. 4(g) the transmission coefficient around the SIP is shown versus frequency.The light is either transmitted or reflected back.Light is transmitted thanks to the presence of the evanescent modes, leading to non-zero coupling into the mode with zero velocity, thus demonstrating the characteristic of frozen light

Slow light bandwidth product
Though we have concentrated on band structures with an SIP, we show that the same procedure can be used to obtain a slow light mode with a high, constant group index over a large frequency interval.The two main structures that are known to support such modes are PCWs [6] and coupled ring resonators [38].The relative merits of these and other structures may be assessed using the Group index/Bandwidth Product (GBP) [15] figure of merit.In particular, GBP = n g Δω/ω o , where n g = c/v g , and the bandwidth (Δω/ω o ) is the frequency range where n g does not vary by more than 10%.Consequently, we search for structures for which D 2 = 0 and D 3 = 0.In Fig. 3(a) the values of z A and g for which these equalities hold are indicated by the red curve; an example of a flat band is given in Fig. 3(e).The structure, with the parameters given in the caption of Fig. 3, has a 5 nm bandwidth for a wavelength of 1550 nm for n g = 25, resulting in a GBP of For PCWs the typical values is 0.16 [15], which is similar to that for the coupled nanowires we present.We have only changed the distance and the radius of the holes to optimize GBP.Further optimization can be carried out by changing additional parameters such as the hole position inside the waveguide, radius and the waveguides width; or, using the same methodology shown here, a quintic SIP (Δk 5 ) can be generated in a waveguide with five modes and with at least four adjustable parameters.The quintic SIP can then be turned into a slow light band.

Three dimension structures
Until now we have considered an ideal 2D structure which is uniform in the direction orthogonal to the periodicity.As our 2D structure is suspended in air, this is not realistic for the long lengths required for sufficient nonlinear interactions.Here we confirm that our analysis holds for more realistic 3D structures, and in particular we confirm the existence of an SIP for two different architectures.The standard platform for asymmetric wire waveguides is a semiconductor film with Buried Oxide (BOX-see Fig. 1(c)), typically consisting of a membrane bound to a silica substrate.
Figures 5(a) and (b) show the band structures for high-and low-index semiconductor BOX structures, with the waveguide made of, respectively, silicon (n Si = 3.5) and silicon nitride (n SiN = 2 [39]).These figures confirm that, using the procedure described in Sect.3, both were designed to exhibit a SIP.
A practical 3D symmetric structure is shown in Fig. 1(d), in which the three wire waveguides are connected to create a rigid structure of width w.The spacing of the two side rows of holes is g.Since a single wide waveguide is more rigid than the three thin waveguides [Fig.1(a)] a longer suspended wire waveguides is possible [32].An example of its band structure, again obtained by the design procedure introduce in Sec. 3, and the required parameters for a SIP are given in Fig. 5(c).This demonstrates that SIPs can be created in any realistic nanowire material platform.We have shown that the same is true for the slow light designs presented in Section 3.3.

Discussion and
We have shown that a periodic three mode waveguide with an asymmetric unit cell can be engineered to have a SIP.The method was verified for three coupled nanowires and for a single wide nanowire, for both 2D and 3D structures with different material platforms.Our procedure can similarly be used to achieve a flat band with small group velocity for a large frequency range.Our work shows that the dispersion in nanowires can be manipulated and designed in a similar way as is possible in PCWs.However, the nanowire geometry has the advantage of having a smaller footprint than PCs.We surmise that in a waveguide supporting five modes, instead of three, a quintic SIP (Δω ∝ Δk 5 ) can be generated which can be engineered to have larger flat band.
Importantly, for of the low and even zero group velocity around a SIP the transmission through the waveguide is relatively high [40].The extracted complex band confirms the existence of evanescent modes at both interfaces of the structure which overcome the mismatch in impedance between the fast and slow modes, irrespective of the group index of the slow mode.We note that near degenerate band edges, which are of the form k 2m with m ≥ 2, evanescent modes also enhance the coupling to slow light [41].However near such a band edge the coupling vanishes as v g → 0.

Fig. 1 .
Fig. 1.Schematic of the nanowire waveguides considered.(a) Three identical wire waveguides of width w, separated by center-to-center distance g.The outer waveguides have holes of radius r and period a, which are shifted laterally by z A .(b) Rigid structure with similar geometry as in (a): total width w, and two sets of holes with g the distance from the center.(c),(d) 3D realizations of planar geometries shown in (a) and (b), respectively.The wire waveguides thickness is w z .In (c) the structure is supported by a low refractive index material such as silica.

Fig. 2 .
Fig. 2. Unit cell of a wire waveguide, surrounded by air, and the corresponding band structures.(a) Single mode wire waveguide with infinitesimal periodicity; waveguide width w = 0.8a.At the BZ edge, the band is folded.Dashed line: air light line.(b) Three coupled wire waveguides with infinitesimal periodicity and g = 1.16a showing symmetric (blue) and anti-symmetric (red) modes.(c) Symmetric periodicity with hole radius r = 0.28.(d-f) Asymmetric periodicity, through a lateral shift (d) of z A = 0.1a, (e) of z A = 0.24a, and (i) z A = 0.4.(f-h): The real part of the magnetic field, perpendicular to the waveguides, at different wavenumber along the band with SIP.(f) k = 0.91 k o , (g) k = k o and (h) k = 1.08 k o .

Fig. 3 .
Fig. 3. (a) Overview of the parameter space of z A and g, for fixed r = 0.22a and w = 0.8a.Blue curve: SIP as shown in (e).Red curve: flat band, as in (f).Other insets show the shape of the dispersion curve in the respective areas.(g) Group index of a slow light band (along the red curve, where D 2 = 0 and D 3 = 0) versus wavelength for parameters w = a, g = 1.28a, z A = 0.46a, and r = 0.28a.The black line indicates where the group index varies by less than 10%.

Fig. 4 .
Fig. 4. Field decomposition for frequencies around a SIP for the structure in (a), with r = 0.28, g = 1.16, w = 0.8 and Z A = 0.24 featuring a SIP at 0.28 in frequency.(b) The field at the SIP frequency.(c) The three modes with k ∼ k o .Blue: propagating mode; green and red: evanescently decaying and growing modes, respectively.Detail of the complex band structure around the SIP showing frequency versus the real and imaginary wavenumber in (d) and (e) respectively.(f) The frequency vs. the group index, calculated from the extracted dispersion.(g) Transmission for a structure with 60 periods.