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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 4 — Feb. 13, 2012
  • pp: 3519–3528
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Frozen and broadband slow light in coupled periodic nanowire waveguides

Nadav Gutman, W. Hugo Dupree, Yue Sun, Andrey A. Sukhorukov, and C. Martijn de Sterke  »View Author Affiliations


Optics Express, Vol. 20, Issue 4, pp. 3519-3528 (2012)
http://dx.doi.org/10.1364/OE.20.003519


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Abstract

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 two- and 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 OSA

1. Introduction

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

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). [CrossRef] [PubMed]

7

7. J. B. Khurgin and R. S. Tucker, Slow Light: Science and ApplicationsTailor and FrancisNew York2009

]. For such scalability, it is necessary to lower the group-velocity dispersion (GVD) which may lead to undesirable pulse broadening [8

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). [CrossRef] [PubMed]

]. For a single periodic nanowire waveguide slow-light occurs close to a photonic band-edge [9

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). [CrossRef]

], 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

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

15

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). [CrossRef]

], and importantly in such PCWs strong enhancement of nonlinear processes was demonstrated [16

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). [CrossRef]

18

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). [CrossRef]

].

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 zA. (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 wz. In (c) the structure is supported by a low refractive index material such as silica.

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.

2. 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 vg = ω/∂k (with real k) have been successfully engineered in periodic structures, such as PCWs [6

6. T. Baba, “Slow light in photonic crystals,” Nature Photon. 2, 465–473 (2008). [CrossRef]

].

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 features a quadratic stationary point, where ΔωD2Δk2 [21

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). [CrossRef] [PubMed]

], where Δω and Δk are the frequency and wavenumber detunings from the band-edge, and Dj = (jω/∂kj)/j! is proportional to the j’th order dispersion coefficient; we write vg for D1. Then the group velocity can be expressed as vg ∝ |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

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). [CrossRef] [PubMed]

, 22

22. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. 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]

].

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 D2 = 0 and thus Δωvg0Δk+D3Δk3. 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
ΔωD3Δk3+D4Δk4.
(1)

The realization of a SIP was first suggested for one-dimensional (1D) periodic photonic crystals [23

23. A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” Waves Random Complex Media 16, 293–382 (2006). [CrossRef]

], composed of magnetic [24

24. A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B 67, 165210 (2003). [CrossRef]

] and anisotropic [25

25. A. Figotin and I. Vitebskiy, “Oblique frozen modes in periodic layered media,” Phys. Rev. E 68, 036609 (2003). [CrossRef]

, 26

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). [CrossRef]

] 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

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). [CrossRef] [PubMed]

,27

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). [CrossRef] [PubMed]

,28

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). [CrossRef]

], 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 D2 = 0 and D3 = 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 Δk1,2,3. We can choose the first solutions to be real-valued, corresponding to the propagating slow-light mode, i.e.,
Δk1Δk.
(2)

The two other, complex solutions have the simple form
Δk2=Δk(3/2i/2),
(3)
Δk3=Δk(3/2+i/2).
(4)
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

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). [CrossRef]

] and intermediate PCW [30

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). [CrossRef]

] 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

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). [CrossRef]

]. As shown above, evanescent modes are always present around an SIP, enabling efficient in-coupling [13

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). [CrossRef] [PubMed]

, 23

23. A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” Waves Random Complex Media 16, 293–382 (2006). [CrossRef]

], which is a very desirable feature.

3. 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

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). [CrossRef] [PubMed]

]. 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

23. A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” Waves Random Complex Media 16, 293–382 (2006). [CrossRef]

]. Unlike the 1D PCs considered by Figotin et al. [25

25. A. Figotin and I. Vitebskiy, “Oblique frozen modes in periodic layered media,” Phys. Rev. E 68, 036609 (2003). [CrossRef]

,26

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). [CrossRef]

,33

33. A. Figotin and I. Vitebsky, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E 63, 066609 (2001). [CrossRef]

], 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 ±ko, as shown below.

3.1. 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 wz = 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

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). [CrossRef] [PubMed]

].

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 1st 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.

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 zA = 0.1a, (e) of zA = 0.24a, and (i) zA = 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 ko, (g) k = ko and (h) k = 1.08 ko.

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 zA. The dispersion with SIP is illustrated in Fig. 2(e). If the shift is further increased, then the dispersion transforms to an IP with vg0 ≠ 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 zA 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.

Fig. 3 (a) Overview of the parameter space of zA 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 D2 = 0 and D3 = 0) versus wavelength for parameters w = a, g = 1.28a, zA = 0.46a, and r = 0.28a. The black line indicates where the group index varies by less than 10%.

3.2. 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

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). [CrossRef]

, 35

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). [CrossRef] [PubMed]

, 36

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). [CrossRef] [PubMed]

].

We simulated light propagation through the structure using the finite-difference time-domain (FDTD) method [37

37. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the fdtd method,” Comput. Phys. Commun. 181, 687–702 (2010). [CrossRef]

]. 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 simulation, 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
Φm(x,y;ω)exp(ikmx),
(5)
where km are the complex wavenumber; x is the direction of the periodicity, and the Φm are the periodic Bloch-wave envelope function which satisfy Φm(y, x) = Φm(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):
Hz(x,y;ω)=m=16amΦm(x,y;ω)+w(x,y;ω)
(6)
where the am 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 ωo = 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

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 ZA = 0.24 featuring a SIP at 0.28 in frequency. (b) The field at the SIP frequency. (c) The three modes with kko. 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.

3.3. Slow light bandwidth product

3.4. 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 (nSi = 3.5) and silicon nitride (nSiN = 2 [39

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). [CrossRef]

]). These figures confirm that, using the procedure described in Sect. 3, both were designed to exhibit a SIP.

Fig. 5 Band structures parameters showing a SIP in the 3D structure. (a)–(b) Three coupled nanowires on Silica, shown in Fig. 1(c). (a) Si on Silica: w = 1, wz = 0.5, g = 1.18, zA = 0.3 and r = 0.1. (b) SiN on Silica: w = 2, wz = 0.8, g = 2.25, zA = 0.28 and r = 0.2. (c) Multimode wire waveguide surrounded by air, Fig. 1(d): w = 2.8, wz = 0.5, g = 1.2, zA = 0.3 and r = 0.28.

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

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). [CrossRef] [PubMed]

]. 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.

4. Discussion and conclusions

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 ∝ Δk5) 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

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.

]. 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 k2m with m ≥ 2, evanescent modes also enhance the coupling to slow light [41

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). [CrossRef] [PubMed]

]. However near such a band edge the coupling vanishes as vg → 0.

Acknowledgments

The authors acknowledge useful discussions and assistance with numerical modeling by Dr. Tom White. This work was supported by the Australian Research Council. The computational work was done at the Australian National Computation Infrastructure (NCI).

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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). [CrossRef] [PubMed]

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). [CrossRef]

10.

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

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). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

20.

C. Bao, J. Hou, H. Wu, X. Zhou, E. Cassan, X. Gao, and D. Zhang, “Low dispersion slow light in slot waveguide grating,” IEEE Photon. Technol. Lett. 23, 1700–1702 (2011). [CrossRef]

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). [CrossRef] [PubMed]

22.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. 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]

23.

A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” Waves Random Complex Media 16, 293–382 (2006). [CrossRef]

24.

A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B 67, 165210 (2003). [CrossRef]

25.

A. Figotin and I. Vitebskiy, “Oblique frozen modes in periodic layered media,” Phys. Rev. E 68, 036609 (2003). [CrossRef]

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). [CrossRef]

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). [CrossRef] [PubMed]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef] [PubMed]

33.

A. Figotin and I. Vitebsky, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E 63, 066609 (2001). [CrossRef]

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). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

37.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the fdtd method,” Comput. Phys. Commun. 181, 687–702 (2010). [CrossRef]

38.

F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nature Photon. 1, 65–71 (2007). [CrossRef]

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). [CrossRef]

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). [CrossRef] [PubMed]

OCIS Codes
(230.7370) Optical devices : Waveguides
(250.5300) Optoelectronics : Photonic integrated circuits

ToC Category:
Optoelectronics

History
Original Manuscript: December 1, 2011
Revised Manuscript: January 18, 2012
Manuscript Accepted: January 23, 2012
Published: January 30, 2012

Citation
Nadav Gutman, W. Hugo Dupree, Yue Sun, Andrey A. Sukhorukov, and C. Martijn de Sterke, "Frozen and broadband slow light in coupled periodic nanowire waveguides," Opt. Express 20, 3519-3528 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-3519


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References

  1. M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express16, 1300–1320 (2008). [CrossRef] [PubMed]
  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). [CrossRef]
  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. Express18, 15859–15869 (2010). [CrossRef] [PubMed]
  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. Express15, 219–226 (2007). [CrossRef] [PubMed]
  5. T. F. Krauss, “Why do we need slow light?” Nature Photon.2, 448–450 (2008). [CrossRef]
  6. T. Baba, “Slow light in photonic crystals,” Nature Photon.2, 465–473 (2008). [CrossRef]
  7. J. B. Khurgin and R. S. Tucker, Slow Light: Science and ApplicationsTailor and FrancisNew York2009
  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. Express14, 1658–1672 (2006). [CrossRef] [PubMed]
  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). [CrossRef]
  10. A. Y. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett.85, 4866–4868 (2004). [CrossRef]
  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. Express14, 9444–9450 (2006). [CrossRef] [PubMed]
  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. Express16, 6227–6232 (2008). [CrossRef] [PubMed]
  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). [CrossRef] [PubMed]
  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). [CrossRef] [PubMed]
  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). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  19. A. Sukhorukov, A. Lavrinenko, D. Chigrin, D. Pelinovsky, and Y. Kivshar, “Slow-light dispersion in coupled periodic waveguides,” J. Opt. Soc. Am. B25, C65–C74 (2008). [CrossRef]
  20. C. Bao, J. Hou, H. Wu, X. Zhou, E. Cassan, X. Gao, and D. Zhang, “Low dispersion slow light in slot waveguide grating,” IEEE Photon. Technol. Lett.23, 1700–1702 (2011). [CrossRef]
  21. S. J. McNab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,” Opt. Express11, 2927–2939 (2003). [CrossRef] [PubMed]
  22. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. 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]
  23. A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” Waves Random Complex Media16, 293–382 (2006). [CrossRef]
  24. A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B67, 165210 (2003). [CrossRef]
  25. A. Figotin and I. Vitebskiy, “Oblique frozen modes in periodic layered media,” Phys. Rev. E68, 036609 (2003). [CrossRef]
  26. J. Ballato, A. Ballato, A. Figotin, and I. Vitebskiy, “Frozen light in periodic stacks of anisotropic layers,” Phys. Rev. E71, 036612–12 (2005). [CrossRef]
  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). [CrossRef] [PubMed]
  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. B28, 955–963 (2011). [CrossRef]
  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. E66, 066608 (2002). [CrossRef]
  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). [CrossRef]
  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. Express17, 17338–17343 (2009). [CrossRef]
  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). [CrossRef] [PubMed]
  33. A. Figotin and I. Vitebsky, “Nonreciprocal magnetic photonic crystals,” Phys. Rev. E63, 066609 (2001). [CrossRef]
  34. 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]
  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. Express17, 3716–3721 (2009). [CrossRef] [PubMed]
  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). [CrossRef] [PubMed]
  37. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the fdtd method,” Comput. Phys. Commun.181, 687–702 (2010). [CrossRef]
  38. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nature Photon.1, 65–71 (2007). [CrossRef]
  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). [CrossRef]
  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). [CrossRef] [PubMed]

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