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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 10 — May. 9, 2011
  • pp: 9690–9698
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Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs

Vincent Laude, Jean-Charles Beugnot, Sarah Benchabane, Yan Pennec, Bahram Djafari-Rouhani, Nikos Papanikolaou, Jose M. Escalante, and Alejandro Martinez  »View Author Affiliations


Optics Express, Vol. 19, Issue 10, pp. 9690-9698 (2011)
http://dx.doi.org/10.1364/OE.19.009690


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Abstract

We demonstrate theoretically that photons and acoustic phonons can be simultaneously guided and slowed down in specially designed nanostructures. Phoxonic crystal waveguides presenting simultaneous phononic and photonic band gaps were designed in perforated silicon membranes that can be conveniently obtained using silicon-on-insulator technology. Geometrical parameters for simultaneous photonic and phononic band gaps were first chosen for optical wavelengths around 1550 nm, based on the finite element analysis of a perfect phoxonic crystal of circular holes. A plain core waveguide was then defined, and simultaneous slow light and elastic guided modes were identified for some waveguide width. Joint guidance of light and elastic waves is predicted with group velocities as low as c/25 and 180 m/s, respectively.

© 2011 OSA

1. Introduction

The propagation of photons and acoustic phonons can be controlled separately by nanostructuration of the supporting materials. In particular, the introduction of periodicity leads to the photonic [1

1. J. Joannopoulos and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 2008).

] and phononic [2

2. M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, “Acoustic band structure of periodic elastic composites,” Phys. Rev. Lett. 71, 2022–2025 (1993). [CrossRef] [PubMed]

] crystal concepts, respectively. Strong spatial confinement of the wave energy and reduction of group velocities by several orders of magnitude are made possible. In addition, periodic structures also allow one for tailoring the spatial dispersion of both photons and phonons, leading to non trivial effects such as negative refraction. It has to be noted that though the velocity of acoustic waves are five orders of magnitude lower than the velocity of light in a vacuum, it is the wavelength that governs the frequency ranges where waves strongly feel periodicity. Photonic crystals operating in the visible and near-IR spectrum, and phononic crystals operating in the GHz range have in common that the relevant wavelengths are in the sub-micron range.

In this paper, we aim at the theoretical demonstration that photons and acoustic phonons can be simultaneously guided and slowed down in specially designed nanostructures, via a phoxonic band gap effect. The silicon phoxonic crystal slab geometry has been selected for its simplicity and potential ease of fabrication. Starting from known optimal geometrical parameters to obtain a complete phoxonic band gap [9

9. Y. Pennec, B. Rouhani, E. El Boudouti, C. Li, Y. El Hassouani, J. Vasseur, N. Papanikolaou, S. Benchabane, V. Laude, and A. Martinez, “Simultaneous existence of phononic and photonic band gaps in periodic crystal slabs,” Opt. Express 18, 14301–14310 (2010). [CrossRef] [PubMed]

], we investigate the dispersion of phoxonic crystal waveguides created by managing a line of defects in the periodic structure. The very strong spatial confinement created by the phoxonic band gap is found to be favorable for obtaining low group velocities for both sound and light waves.

2. Phoxonic crystal slab modes

A phoxonic crystal slab can be made by perforating periodically a thin membrane of a solid material (silicon in this work). Figure 1(a) depicts the unit-cell for computation of the dispersion relation and the modes of a phoxonic crystal slab. The basic geometrical parameters of such a structure are the thickness of the slab, h, the pitch of the array, a, and the diameter of the air holes, d. Figure 1(c) shows the first irreducible Brillouin zone associated with the square lattice. The highest symmetry points Γ, X and M have wavevector coordinates (0, 0), (π/a,0) and (π/a,π/a), respectively.

Fig. 1 Examples of unit-cells used in the analysis of phoxonic crystal slab structures. Two-dimensional periodic boundary conditions are applied at the lateral sides of the unit-cells. (a) For elastic waves, only the solid part (silicon) needs to be meshed. Traction-free boundary conditions are applied at the top and the bottom surfaces. (b) For photonic modes, the vacuum or air inside the holes and surrounding the slab needs to be taken into account. Free boundary conditions are applied at the top and the bottom surfaces of the air region, resulting in an artificial truncation of the computation domain. (c) First irreducible Brillouin zone showing the highest symmetry points.

Various numerical methods can be employed to compute the photonic and the phononic properties of periodic structures. We have in this work opted for the finite element method (FEM). In the phononic crystal slab case, FEM was recently shown to be highly efficient for obtaining band structures [12

12. A. Khelif, B. Aoubiza, S. Mohammadi, A. Adibi, and V. Laude, “Complete band gaps in two-dimensional phononic crystal slabs,” Phys. Rev. E 74, 046610 (2006). [CrossRef]

]. The FEM variational formulation for solid-air phononic crystals uses the three displacements as the dependent variables and 3D Lagrange finite elements. The practical advantage of FEM in the case of solids resides in the small size of the domain that has to be meshed. Indeed, only the solid part of the perforated silicon slab supports elastic waves and the free boundary conditions on the top and bottom faces yield perfect reflection of all elastic waves. There are at least two ways to obtain Bloch waves with FEM. One possibility is to introduce the Bloch-Floquet wavevector directly in the equations of propagation and thus in the variational formulation of the problem, together with symmetry boundary conditions for the lateral sides [13

13. M. I. Hussein, “Reduced bloch mode expansion for periodic media band structure calculations,” Proc. R. Soc. London, Ser. A 465, 2825–2848 (2009). [CrossRef]

]. A second possibility is to use periodic boundary conditions for the lateral sides [12

12. A. Khelif, B. Aoubiza, S. Mohammadi, A. Adibi, and V. Laude, “Complete band gaps in two-dimensional phononic crystal slabs,” Phys. Rev. E 74, 046610 (2006). [CrossRef]

]. Both methods give exactly the same band structures but for numerical errors.

In the photonic case, there is an additional difficulty. Vacuum or air (with unit refractive index) surrounds the silicon slab (with refractive index 3.6 at a wavelength in a vacuum of 1550 nm) and extends in principle to infinity. However, the computation domain has to be restricted to a finite region of space in practice. Since we are looking for eigenmodes of the system, the use of artificial boundary conditions such as perfectly matched layers (PML) is not adequate. We instead consider simple free boundary conditions a certain distance away from the slab surfaces. Consequently, waves radiated away from the slab can be reflected at the limits of the computation domain, and care has to be taken in the analysis of the obtained dispersion diagrams, in connection with the concept of the light line [14

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

]. The light line is defined by the dispersion relation for light waves in a vacuum, ck/ω = 1, with k the wavevector, ω the angular frequency, and c the speed of light in a vacuum. The light cone or radiative region is defined by ck/ω < 1. Outside the light cone, any plane wave is necessarily evanescent and thus cannot radiate energy outside of the system. The air thickness is in practice adjusted so that evanescent waves have attenuated enough, so that the band structure for guided waves (lying outside the light cone) has converged. Inside the light cone, the artificial truncation of the air domain induces spurious bands, the number of which increases with the thickness of the air layer. No attempt was made at eliminating such spurious solutions from the photonic band structures reported in this paper, but this can be accomplished by examination of the energy distribution of each eigenmode, as illustrated later. Because of the symmetry of the slab structure with respect to the middle plane, it is further possible to separate photonic modes between even and odd. Even (respectively, odd) modes are also termed quasi-TE (resp., quasi-TM) modes. TE and TM here refer to the electric field vector being mostly transverse electric or magnetic, respectively. As a result of this symmetry property, only one half of the structure needs to be meshed, as shown in Fig. 1(b). The variational formulation for dielectric photonic crystals uses the magnetic field vector H as the dependent variable and 3D vector finite elements with vanishing divergence [15

15. J. Jin, The finite element method in electromagnetics, 2nd ed. (Wiley, 2002).

].

The phononic band structure is displayed in Fig. 2(a) and shows a complete elastic band gap appearing around 5 GHz, between the 6-th and the 7-th bands (8% relative band gap width). The first six Bloch modes at the M point of the first Brillouin zone are also shown for illustration in Figs. 2(b)–2(g). It can be observed that none of these Bloch waves has a pure polarization, instead, their polarization has components along all three directions in space.

Fig. 2 (a) Phononic band structure for a periodic array of circular holes in a silicon slab with parameters a = 651 nm, h = 390 nm, and r = 280 nm. A complete band gap appears around 5 GHz. (b)–(g) Modal distribution of the displacements for the first six modes at the M point of the first Brillouin zone. The color bar is for the modulus of the total displacement while the deformation of the mesh is proportional to the algebraic displacements.

The photonic band structure is presented in Fig. 3(a). The dark gray region above the light line is the light cone for air; all waves having their dispersion represented by a point below the light line are guided by the phoxonic crystal slab. Two photonic band gaps for guided waves are found (appearing in white in Fig. 3(a)), around the reduced frequencies 0.35 and 0.42. The lower band gap is valid for even modes only, while the upper one is valid for both even and odd modes (i.e., is a complete band gap for guided waves). The first six even photonic modes are shown in Figs. 3(b)–3(g) for the reduced wavevector value k x a/2π = k y a/2π = 0.3, somewhat midway along the ΓM direction. Only the first two even modes, Figs. 3(b) and 3(c), are outside the light cone and are thus certainly guided by the slab. However, even though it is apparent that the third, fourth and sixth modes are leaking in the surrounding air, the fifth mode is well guided inside the slab (Fig. 3(f)). From this observation, we can infer that a Bloch wave exhibing a dispersion lying within the light cone is only likely to be leaky; radiation to the surrounding air does not necessarily happens, as discussed by Xu et al. [16

16. T. Xu, M. S. Wheeler, S. V. Nair, H. E. Ruda, M. Mojahedi, and J. S. Aitchison, “Highly confined mode above the light line in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. 93, 241105 (2008). [CrossRef]

].

Fig. 3 (a) Photonic band structure for a periodic array of circular holes in a silicon slab with parameters a = 651 nm, h = 390 nm, and r = 280 nm. The left vertical axis is in units of the reduced frequency, ωa/2πc = a/λ. The two photonic band gaps for guided waves appear in white. (b)–(g) Modal distribution of the modulus of the magnetic field vector H (a.u.) for the reduced wavevector value k x a/2π = k y a/2π = 0.3, somewhat midway along the ΓM direction, for the first six even modes.

3. Guided slow phoxonic crystal slab modes

Fig. 4 Phononic band structure of a solid core phoxonic crystal waveguide defined along the ΓX direction in a square-lattice phoxonic crystal slab. In the supercell computation, three rows of holes surround a central solid core of width (a) w = a and (b) w = 0.9a. The complete phononic band gap appears in white. The band supporting guided waves with the lowest group velocity is shown as a solid line. (c)–(e) Modal distributions of the displacements for three particular eigenmodes with k x a/2π = 0.3. The color bar is for the modulus of the total displacement while the deformation of the mesh is proportional to the algebraic displacements.

Fig. 5 Photonic band structure for waves in a solid core phoxonic crystal waveguide defined along the ΓX direction in a square-lattice phoxonic crystal slab. In the supercell computation, three rows of holes surround a central solid core of width (a) w = a and (b) w = 0.9a. Two photonic band gaps for guided waves appear in white. The bands supporting waves latterally guided by the photonic band gap effect are shown with solid lines. (c),(d) Modal distribution for two of these eigenmodes. The modulus of the magnetic field vector H is presented for the reduced wavevector value kxa/2π = 0.45.

The group velocity can be estimated from the band structures for guided waves from the definition vg=ωk. In the phononic case, it is customary to consider the group index defined by n g = c/v g. Figure 6(a) displays the dependence of the group index with reduced wave vector for the guided photonic mode labeled “c” in Fig. 5. It can be seen that this quantity can be either positive or negative, depending on the wave vector. However, from an experimental point of view, the most useful part of the spectrum is arguably found around the reduced frequency ωa/2πc = 0.343, where the group index remains approximately constant (n g ≈ 25). Figure 6(b) shows the dependence of the group velocity with reduced wave vector, for the guided phononic mode labeled “c” in Fig. 4. Around the 5 GHz phonon frequency, the group velocity varies around a mean value of 180 m/s, more than 30 times less than the speed of any bulk wave in silicon and less than the speed of sound in air.

Fig. 6 (a) Dependence of the group index and of the reduced frequency with reduced wave vector, for the guided photonic mode labeled “c” in Fig. 5. (b) Dependence of the group velocity and of the frequency with reduced wave vector, for the guided phononic mode labeled “c” in Fig. 4.

4. Conclusion

We have presented the principle of a phoxonic crystal slab waveguide managed by inserting a linear solid core defect in a nanostructured silicon slab presenting simultaneously a phononic and a photonic band gap. Such a nanostructure can be conveniently obtained using silicon-on-insulator technology. Geometrical parameters for simultaneous photonic and phononic band gaps were first chosen, based on the finite element analysis of a perfect phoxonic crystal of circular holes. A plain core waveguide was then introduced, and simultaneous guidance of slow photons and slow acoustic phonons was identified. Optical and acoustic group velocities as small as c/25 and 180 m/s over a usable frequency range were predicted, respectively. The structure of the proposed phoxonic crystal slab waveguide is markedly different from the opto-mechanical waveguide structures proposed by Safavi et al. [10

10. A. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express 18, 14926–14943 (2010). [CrossRef] [PubMed]

]. The former structures were designed to optimize the optomechanical interaction, where coupling of photons and phonons would be provided by the periodic motion of the boundaries. In contrast, the phoxonic crystal waveguides we have presented guide photons and phonons in a small central solid core, and are thus better suited to situations where acousto-optical coupling is provided by the elasto-optical effect. As such, they bear an analogy to photonic crystal fibers which were shown to support the joint guidance of photons and phonons [18

18. V. Laude, A. Khelif, S. Benchabane, M. Wilm, T. Sylvestre, B. Kibler, A. Mussot, J. M. Dudley, and H. Maillotte, “Phononic band-gap guidance of acoustic modes in photonic crystal fibers,” Phys. Rev. B 71, 045107 (2005). [CrossRef]

, 19

19. P. Dainese, P. Russell, N. Joly, J. Knight, G. Wiederhecker, H. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2, 388–392 (2006). [CrossRef]

]. A difference is that the periodic nanostructuration in the plane of the slab makes it possible to tailor slow wave propagation, which is less easy when the 2D structuration runs along the core of a photonic crystal fiber.

Acknowledgments

This research has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number 233883 (TAILPHOX).

References and links

1.

J. Joannopoulos and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 2008).

2.

M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, “Acoustic band structure of periodic elastic composites,” Phys. Rev. Lett. 71, 2022–2025 (1993). [CrossRef] [PubMed]

3.

M. Maldovan and E. Thomas, “Simultaneous localization of photons and phonons in two-dimensional periodic structures,” Appl. Phys. Lett. 88, 251907 (2006). [CrossRef]

4.

M. Maldovan and E. Thomas, “Simultaneous complete elastic and electromagnetic band gaps in periodic structures,” Appl. Phys. B: Lasers and Optics 83, 595–600 (2006). [CrossRef]

5.

A. Akimov, Y. Tanaka, A. Pevtsov, S. Kaplan, V. Golubev, S. Tamura, D. Yakovlev, and M. Bayer, “Hypersonic modulation of light in three-dimensional photonic and phononic band-gap materials,” Phys. Rev. Lett. 101, 033902 (2008). [CrossRef] [PubMed]

6.

S. Sadat-Saleh, S. Benchabane, F. Baida, M. Bernal, and V. Laude, “Tailoring simultaneous photonic and phononic band gaps,” J. Appl. Phys. 106, 074912 (2009). [CrossRef]

7.

N. Papanikolaou, I. Psarobas, and N. Stefanou, “Absolute spectral gaps for infrared light and hypersound in three-dimensional metallodielectric phoxonic crystals,” Appl. Phys. Lett. 96, 231917 (2010). [CrossRef]

8.

S. Mohammadi, A. Eftekhar, A. Khelif, and A. Adibi, “Simultaneous two-dimensional phononic and photonic band gaps in opto-mechanical crystal slabs,” Opt. Express 18, 9164–9172 (2010). [CrossRef] [PubMed]

9.

Y. Pennec, B. Rouhani, E. El Boudouti, C. Li, Y. El Hassouani, J. Vasseur, N. Papanikolaou, S. Benchabane, V. Laude, and A. Martinez, “Simultaneous existence of phononic and photonic band gaps in periodic crystal slabs,” Opt. Express 18, 14301–14310 (2010). [CrossRef] [PubMed]

10.

A. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express 18, 14926–14943 (2010). [CrossRef] [PubMed]

11.

Y. El Hassouani, C. Li, Y. Pennec, E. H. El Boudouti, H. Larabi, A. Akjouj, O. Bou Matar, V. Laude, N. Papanikolaou, A. Martinez, and B. Djafari Rouhani, “Dual phononic and photonic band gaps in a periodic array of pillars deposited on a thin plate,” Phys. Rev. B 82, 155405 (2010). [CrossRef]

12.

A. Khelif, B. Aoubiza, S. Mohammadi, A. Adibi, and V. Laude, “Complete band gaps in two-dimensional phononic crystal slabs,” Phys. Rev. E 74, 046610 (2006). [CrossRef]

13.

M. I. Hussein, “Reduced bloch mode expansion for periodic media band structure calculations,” Proc. R. Soc. London, Ser. A 465, 2825–2848 (2009). [CrossRef]

14.

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

15.

J. Jin, The finite element method in electromagnetics, 2nd ed. (Wiley, 2002).

16.

T. Xu, M. S. Wheeler, S. V. Nair, H. E. Ruda, M. Mojahedi, and J. S. Aitchison, “Highly confined mode above the light line in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. 93, 241105 (2008). [CrossRef]

17.

V. Laude, Y. Achaoui, S. Benchabane, and A. Khelif, “Evanescent Bloch waves and the complex band structure of phononic crystals,” Phys. Rev. B 80, 092301 (2009). [CrossRef]

18.

V. Laude, A. Khelif, S. Benchabane, M. Wilm, T. Sylvestre, B. Kibler, A. Mussot, J. M. Dudley, and H. Maillotte, “Phononic band-gap guidance of acoustic modes in photonic crystal fibers,” Phys. Rev. B 71, 045107 (2005). [CrossRef]

19.

P. Dainese, P. Russell, N. Joly, J. Knight, G. Wiederhecker, H. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2, 388–392 (2006). [CrossRef]

OCIS Codes
(160.1050) Materials : Acousto-optical materials
(350.7420) Other areas of optics : Waves
(130.5296) Integrated optics : Photonic crystal waveguides
(160.5298) Materials : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: March 18, 2011
Manuscript Accepted: April 16, 2011
Published: May 3, 2011

Citation
Vincent Laude, Jean-Charles Beugnot, Sarah Benchabane, Yan Pennec, Bahram Djafari-Rouhani, Nikos Papanikolaou, Jose M. Escalante, and Alejandro Martinez, "Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs," Opt. Express 19, 9690-9698 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9690


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References

  1. J. Joannopoulos and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 2008).
  2. M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, “Acoustic band structure of periodic elastic composites,” Phys. Rev. Lett. 71, 2022–2025 (1993). [CrossRef] [PubMed]
  3. M. Maldovan and E. Thomas, “Simultaneous localization of photons and phonons in two-dimensional periodic structures,” Appl. Phys. Lett. 88, 251907 (2006). [CrossRef]
  4. M. Maldovan and E. Thomas, “Simultaneous complete elastic and electromagnetic band gaps in periodic structures,” Appl. Phys. B: Lasers and Optics 83, 595–600 (2006). [CrossRef]
  5. A. Akimov, Y. Tanaka, A. Pevtsov, S. Kaplan, V. Golubev, S. Tamura, D. Yakovlev, and M. Bayer, “Hypersonic modulation of light in three-dimensional photonic and phononic band-gap materials,” Phys. Rev. Lett. 101, 033902 (2008). [CrossRef] [PubMed]
  6. S. Sadat-Saleh, S. Benchabane, F. Baida, M. Bernal, and V. Laude, “Tailoring simultaneous photonic and phononic band gaps,” J. Appl. Phys. 106, 074912 (2009). [CrossRef]
  7. N. Papanikolaou, I. Psarobas, and N. Stefanou, “Absolute spectral gaps for infrared light and hypersound in three-dimensional metallodielectric phoxonic crystals,” Appl. Phys. Lett. 96, 231917 (2010). [CrossRef]
  8. S. Mohammadi, A. Eftekhar, A. Khelif, and A. Adibi, “Simultaneous two-dimensional phononic and photonic band gaps in opto-mechanical crystal slabs,” Opt. Express 18, 9164–9172 (2010). [CrossRef] [PubMed]
  9. Y. Pennec, B. Rouhani, E. El Boudouti, C. Li, Y. El Hassouani, J. Vasseur, N. Papanikolaou, S. Benchabane, V. Laude, and A. Martinez, “Simultaneous existence of phononic and photonic band gaps in periodic crystal slabs,” Opt. Express 18, 14301–14310 (2010). [CrossRef] [PubMed]
  10. A. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express 18, 14926–14943 (2010). [CrossRef] [PubMed]
  11. Y. El Hassouani, C. Li, Y. Pennec, E. H. El Boudouti, H. Larabi, A. Akjouj, O. Bou Matar, V. Laude, N. Papanikolaou, A. Martinez, and B. Djafari Rouhani, “Dual phononic and photonic band gaps in a periodic array of pillars deposited on a thin plate,” Phys. Rev. B 82, 155405 (2010). [CrossRef]
  12. A. Khelif, B. Aoubiza, S. Mohammadi, A. Adibi, and V. Laude, “Complete band gaps in two-dimensional phononic crystal slabs,” Phys. Rev. E 74, 046610 (2006). [CrossRef]
  13. M. I. Hussein, “Reduced bloch mode expansion for periodic media band structure calculations,” Proc. R. Soc. London, Ser. A 465, 2825–2848 (2009). [CrossRef]
  14. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999). [CrossRef]
  15. J. Jin, The finite element method in electromagnetics , 2nd ed. (Wiley, 2002).
  16. T. Xu, M. S. Wheeler, S. V. Nair, H. E. Ruda, M. Mojahedi, and J. S. Aitchison, “Highly confined mode above the light line in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. 93, 241105 (2008). [CrossRef]
  17. V. Laude, Y. Achaoui, S. Benchabane, and A. Khelif, “Evanescent Bloch waves and the complex band structure of phononic crystals,” Phys. Rev. B 80, 092301 (2009). [CrossRef]
  18. V. Laude, A. Khelif, S. Benchabane, M. Wilm, T. Sylvestre, B. Kibler, A. Mussot, J. M. Dudley, and H. Maillotte, “Phononic band-gap guidance of acoustic modes in photonic crystal fibers,” Phys. Rev. B 71, 045107 (2005). [CrossRef]
  19. P. Dainese, P. Russell, N. Joly, J. Knight, G. Wiederhecker, H. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2, 388–392 (2006). [CrossRef]

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