Abstract

A numerical method combining complex-k band calculations and absorbing boundary conditions for Bloch waves is presented. We use this method to study photonic crystals with Dirac cones. We demonstrate that the photonic crystal behaves as a zero-index medium when excited at normal incidence, but that the zero-index behavior is lost at oblique incidence due to excitation of modes on the flat band. We also investigate the formation of monomodal and multimodal cavity resonances inside the photonic crystals, and the physical origins of their different line-shape features.

© 2015 Optical Society of America

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References

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd Edition (Princeton University Press, 2008).
  2. P. Markos and C. M. Soukoulis, Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University Press, 2008).
    [Crossref]
  3. C. T. Chan, S. Datta, K. M. Ho, and C. M. Soukoulis, “A7 structure: A family of photonic crystals,” Phys. Rev. B 50, 1988–1991 (1994).
    [Crossref]
  4. R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, and C. M. Soukoulis, “Negative refraction and superlens behavior in a two-dimensional photonic crystal,” Phys. Rev. B 71, 085106 (2005).
    [Crossref]
  5. S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
    [Crossref] [PubMed]
  6. M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010).
    [Crossref]
  7. X. Huang, Y. Lai, Z. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011).
    [Crossref]
  8. F. D. M. Haldane and S. Raghu, “Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
    [Crossref] [PubMed]
  9. T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
    [Crossref]
  10. X. Zhang, “Observing Zitterbewegung for Photons near the Dirac point of a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 100, 113903 (2008).
    [Crossref] [PubMed]
  11. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20, 3898–3917 (2012).
    [Crossref] [PubMed]
  12. K. Sakoda, Optical Properties of Photonic Crystals, 2nd Edition (Springer, 2004).
  13. M. Davanco, Y. Urzhumov, and G. Shvets, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express 15, 9681–9691 (2007).
    [Crossref] [PubMed]
  14. C. Fietz, Y. Urzhumov, and G. Shvets, “Complex k band diagrams of 3D metamaterial/photonic crystals,” Opt. Express 19, 19027–19041 (2011).
    [Crossref] [PubMed]
  15. M. Jin, The Finite Element Method in Electromagnetics, 2nd Edition (Wiley, 2002).
  16. 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]
  17. M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weisberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, “Analysis of general geometric scaling perturbations in a transmitting waveguide: fundamental connection between polarization-mode dispersion and group-velocity dispersion,” J. Opt. Soc. Am. B 19, 2867–2875 (2002).
    [Crossref]
  18. C. Fietz, “Absorbing boundary condition for Bloch-Floquet eigenmodes,” J. Opt. Soc. Am. B 30, 2615–2620 (2013).
    [Crossref]
  19. U. Fano, “Effects of configuration interation on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
    [Crossref]

2013 (1)

2012 (1)

2011 (2)

C. Fietz, Y. Urzhumov, and G. Shvets, “Complex k band diagrams of 3D metamaterial/photonic crystals,” Opt. Express 19, 19027–19041 (2011).
[Crossref] [PubMed]

X. Huang, Y. Lai, Z. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011).
[Crossref]

2010 (1)

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010).
[Crossref]

2009 (1)

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
[Crossref]

2008 (2)

X. Zhang, “Observing Zitterbewegung for Photons near the Dirac point of a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 100, 113903 (2008).
[Crossref] [PubMed]

F. D. M. Haldane and S. Raghu, “Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

2007 (1)

2005 (1)

R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, and C. M. Soukoulis, “Negative refraction and superlens behavior in a two-dimensional photonic crystal,” Phys. Rev. B 71, 085106 (2005).
[Crossref]

2003 (1)

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[Crossref] [PubMed]

2002 (2)

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]

M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weisberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, “Analysis of general geometric scaling perturbations in a transmitting waveguide: fundamental connection between polarization-mode dispersion and group-velocity dispersion,” J. Opt. Soc. Am. B 19, 2867–2875 (2002).
[Crossref]

1994 (1)

C. T. Chan, S. Datta, K. M. Ho, and C. M. Soukoulis, “A7 structure: A family of photonic crystals,” Phys. Rev. B 50, 1988–1991 (1994).
[Crossref]

1961 (1)

U. Fano, “Effects of configuration interation on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
[Crossref]

Bienstman, P.

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]

Chan, C. T.

X. Huang, Y. Lai, Z. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011).
[Crossref]

C. T. Chan, S. Datta, K. M. Ho, and C. M. Soukoulis, “A7 structure: A family of photonic crystals,” Phys. Rev. B 50, 1988–1991 (1994).
[Crossref]

Datta, S.

C. T. Chan, S. Datta, K. M. Ho, and C. M. Soukoulis, “A7 structure: A family of photonic crystals,” Phys. Rev. B 50, 1988–1991 (1994).
[Crossref]

Davanco, M.

Diem, M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010).
[Crossref]

Economou, E. N.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[Crossref] [PubMed]

Engeness, T. D.

Fano, U.

U. Fano, “Effects of configuration interation on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
[Crossref]

Fietz, C.

Fink, Y.

Foteinopoulou, S.

R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, and C. M. Soukoulis, “Negative refraction and superlens behavior in a two-dimensional photonic crystal,” Phys. Rev. B 71, 085106 (2005).
[Crossref]

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[Crossref] [PubMed]

Guven, K.

R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, and C. M. Soukoulis, “Negative refraction and superlens behavior in a two-dimensional photonic crystal,” Phys. Rev. B 71, 085106 (2005).
[Crossref]

Haldane, F. D. M.

F. D. M. Haldane and S. Raghu, “Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

Hang, Z.

X. Huang, Y. Lai, Z. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011).
[Crossref]

Ho, K. M.

C. T. Chan, S. Datta, K. M. Ho, and C. M. Soukoulis, “A7 structure: A family of photonic crystals,” Phys. Rev. B 50, 1988–1991 (1994).
[Crossref]

Huang, X.

X. Huang, Y. Lai, Z. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011).
[Crossref]

Ibanescu, M.

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]

M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weisberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, “Analysis of general geometric scaling perturbations in a transmitting waveguide: fundamental connection between polarization-mode dispersion and group-velocity dispersion,” J. Opt. Soc. Am. B 19, 2867–2875 (2002).
[Crossref]

Jacobs, S. A.

Jin, M.

M. Jin, The Finite Element Method in Electromagnetics, 2nd Edition (Wiley, 2002).

Joannopoulos, J. D.

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]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd Edition (Princeton University Press, 2008).

Johnson, S. G.

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]

M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weisberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, “Analysis of general geometric scaling perturbations in a transmitting waveguide: fundamental connection between polarization-mode dispersion and group-velocity dispersion,” J. Opt. Soc. Am. B 19, 2867–2875 (2002).
[Crossref]

Koschny, T.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010).
[Crossref]

Lai, Y.

X. Huang, Y. Lai, Z. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011).
[Crossref]

Lidorikis, E.

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]

Markos, P.

P. Markos and C. M. Soukoulis, Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University Press, 2008).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd Edition (Princeton University Press, 2008).

Moussa, R.

R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, and C. M. Soukoulis, “Negative refraction and superlens behavior in a two-dimensional photonic crystal,” Phys. Rev. B 71, 085106 (2005).
[Crossref]

Ochiai, T.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
[Crossref]

Onoda, M.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
[Crossref]

Ozbay, E.

R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, and C. M. Soukoulis, “Negative refraction and superlens behavior in a two-dimensional photonic crystal,” Phys. Rev. B 71, 085106 (2005).
[Crossref]

Raghu, S.

F. D. M. Haldane and S. Raghu, “Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

Sakoda, K.

Shvets, G.

Skorobogatiy, M.

Skorobogatiy, M. A.

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]

Soljacic, M.

Soukoulis, C. M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010).
[Crossref]

R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, and C. M. Soukoulis, “Negative refraction and superlens behavior in a two-dimensional photonic crystal,” Phys. Rev. B 71, 085106 (2005).
[Crossref]

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[Crossref] [PubMed]

C. T. Chan, S. Datta, K. M. Ho, and C. M. Soukoulis, “A7 structure: A family of photonic crystals,” Phys. Rev. B 50, 1988–1991 (1994).
[Crossref]

P. Markos and C. M. Soukoulis, Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University Press, 2008).
[Crossref]

Tuttle, G.

R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, and C. M. Soukoulis, “Negative refraction and superlens behavior in a two-dimensional photonic crystal,” Phys. Rev. B 71, 085106 (2005).
[Crossref]

Urzhumov, Y.

Weisberg, O.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd Edition (Princeton University Press, 2008).

Zhang, L.

R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, and C. M. Soukoulis, “Negative refraction and superlens behavior in a two-dimensional photonic crystal,” Phys. Rev. B 71, 085106 (2005).
[Crossref]

Zhang, X.

X. Zhang, “Observing Zitterbewegung for Photons near the Dirac point of a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 100, 113903 (2008).
[Crossref] [PubMed]

Zheng, H.

X. Huang, Y. Lai, Z. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011).
[Crossref]

J. Opt. Soc. Am. B (2)

Nature Mater. (1)

X. Huang, Y. Lai, Z. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nature Mater. 10, 582–586 (2011).
[Crossref]

Opt. Express (3)

Phys. Rev. (1)

U. Fano, “Effects of configuration interation on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
[Crossref]

Phys. Rev. B (3)

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
[Crossref]

C. T. Chan, S. Datta, K. M. Ho, and C. M. Soukoulis, “A7 structure: A family of photonic crystals,” Phys. Rev. B 50, 1988–1991 (1994).
[Crossref]

R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, and C. M. Soukoulis, “Negative refraction and superlens behavior in a two-dimensional photonic crystal,” Phys. Rev. B 71, 085106 (2005).
[Crossref]

Phys. Rev. E (1)

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]

Phys. Rev. Lett. (3)

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[Crossref] [PubMed]

F. D. M. Haldane and S. Raghu, “Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

X. Zhang, “Observing Zitterbewegung for Photons near the Dirac point of a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 100, 113903 (2008).
[Crossref] [PubMed]

Physica B (1)

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405, 2990–2995 (2010).
[Crossref]

Other (4)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd Edition (Princeton University Press, 2008).

P. Markos and C. M. Soukoulis, Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University Press, 2008).
[Crossref]

K. Sakoda, Optical Properties of Photonic Crystals, 2nd Edition (Springer, 2004).

M. Jin, The Finite Element Method in Electromagnetics, 2nd Edition (Wiley, 2002).

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Figures (6)

Fig. 1
Fig. 1

(a) The band structure for TM modes (electric field parallel to the rod axis). There is a triple degeneracy at the Γ point and the Dirac frequency is 23.05161 GHz. (b) The full band structure in the 1st Brillouin zone. (c) The real lattice and its reciprocal lattice of the PhC. The embedded rods are made of alumina, with ε = 9.8, μ = 1. The radius of the cylinder is r = 1.59 mm, and the lattice constant is a = 4.66153r.

Fig. 2
Fig. 2

(a) The transmission spectrum versus the incident angle. The PhC is finite in the propagation direction (x direction), in which 11 rows are present (refer to Fig. 4 for a visual sketch). (b) A cut-plane of the Dirac frequency over the full band. There are intersections with the middle band. (c) A top view of the cut-plane and the intersections. Each blue horizontal straight dash line indicates a certain ky (component of the wave vector parallel to the interface between the air and PhC), or the incident angle. Numbers of the horizontal straight lines correspond to peaks in (a). It is shown that peak 1 corresponds to the zero-index mode with kx = ky = 0, due to the coupling with the linear branch. Peak 2 to 10 indicate there are 2 propagating modes with their reflective counterparts in the PhC, while peak 11 corresponds to only 1 propagating mode and its reflective counterpart, both due to the coupling with the flat band.

Fig. 3
Fig. 3

Complex-k band diagram. x-axis: the eigenvalue of the propagating constant. y-axis: frequency of the EM wave. The black dots denote the real part of the eigen-kx , and the red dots denote the imaginary part of the eigen-kx . The blue dash shows where the Dirac frequency lies. (a)&(b): Complex-k band structure for normal incidence (ky = 0) at different ranges of frequencies. (c)&(d): Complex-k band structure for the incident angle of 0.4 rad (ky = 188 m−1) at different ranges of frequencies. From (a) and (c), we plot the extended Brillouin zone, from which we can see the periodicity for the real part of kx .

Fig. 4
Fig. 4

(a) A sketch for multiple reflections in the PhC. The incoming and outgoing waves are plane waves. The field inside the PhC is the superposition of a set of Bloch waves. In the propagating direction (x direction), 11 rows of rods are present. In (b) and (c), |E| is plotted, with f = 23.05161 GHz and ky = 188 m−1. (b) Plane waves are excited and absorbed at the leftmost boundary, and the relevant Bloch modes are absorbed at the rightmost boundary. (c) One Bloch mode is excited and its reflective counterpart is absorbed at the leftmost boundary; plane waves are absorbed at the right.

Fig. 5
Fig. 5

Energy transmission (|M|2) vs. incident angle. The black lines denote the transfer function M which comes from the direct measurement of a full simulation in the air/PhC/air system. The red dots denote the transfer function M determined from the multiple reflection assumption. (a) and (b) correspond to a lossless system, where the alumina rods have ε = 9.8. (c) and (d) correspond to a lossy system, where the alumina rods have ε = 9.8 + 6.8 × 10−4 i.

Fig. 6
Fig. 6

(a) Energy transmission of both Bloch modes in the small incident angle regime, ignoring the coupling between them. The blue curve corresponds to the steep-slope mode in Fig. 2(c), while the red curve corresponds to the small-slope mode. (b) Dispersion relations of two excited modes with different incident angles. The black dashed-dot line marks the excitation frequency of 23.05161 GHz. Its intersection with the blue curve shows the propagating and counterpropagating Bloch waves excited at peak 11 in Fig. 2(a). The intersection with the red curve shows the two Bloch modes and their counterpropagating parts at small incident angle with respect to peak 7 in Fig. 2(a).

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

E ( r ) = u ( r ) exp ( i k r ) E ^ , u ( r ) = u ( r + a )
0 = d Ω [ k 2 μ v u i v μ k u + i u μ ( v ) k ( v ) u μ + ε ω 2 c 2 v u ] = d Ω W
M FP = t pa A t ap + t pa ( A r pa A r pa ) A t ap + t pa ( A r pa A r pa ) 2 A t ap + = t pa ( I A r pa A r pa ) 1 A t ap
t ap = [ t ap 1 t ap 2 ] r pa = [ r pa 11 r pa 12 r pa 21 r pa 22 ] A = [ exp ( i k x 1 x ) 0 0 exp ( i k x 2 x ) ] t pa = ( t pa 1 t pa 2 )
W b = v [ n ^ × ( 1 μ × E ) ] = ± i ω c v H y
H y = n | n > < n | H y > = n h yn d y ( e zn * H y + E z h yn * ) d y ( e zn * h yn + e zn h yn * ) ,

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