Abstract

In this paper, a periodic chain composed of two-dimensional dielectric cylindrical inclusions was studied based on the Fourier series expansion method with perfectly matched layers. Phase and attenuation constants associated with guided modes, forward propagation leaky modes, and backward propagation leaky modes, were conceptually proposed and numerically examined. In particular, the relationships between the backward propagation mode, leaky mode, and propagation constant were explained in the second-order Bragg reflection region. This simple structure was investigated with the goal of realizing an efficient guiding device. Phase and attenuation constant results were compared with the results obtained using the Lattice Sums technique with the T-matrix approach and FDTD method; very good agreement was observed between these methods.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals (CRC Press, 2005).
  2. H. Nemec, P. Kuzel, J.-L. Coutaz, and J. Ctyroky, “Transmission properties and band structure of a segmented dielectric waveguide for the terahertz range,” Opt. Commun. 273(1), 99–104 (2007).
    [Crossref]
  3. B. Vial, F. Zolla, A. Nicolet, and M. Commandre, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89(2), 023829 (2014).
    [Crossref]
  4. A. Nicolet, G. Demésy, F. Zolla, and B. Vial, “Quasi-modal analysis of segmented waveguides,” in Proc. IEEE Conf. Antenna Meas. Appl. (CAMA, 2014), pp. 1–4.
  5. B. N. Behnken, G. Karunasiri, D. R. Chamberlin, P. R. Robrish, and J. Faist, “Real-time imaging using a 2.8 THz quantum cascade laser and uncooled infrared microbolometer camera,” Opt. Lett. 33(5), 440–442 (2008).
    [Crossref] [PubMed]
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    [Crossref]
  8. H. Guo, Y. Zhang, and B. Li, “Periodic dielectric waveguide-based cross- and T-connections with a resonant cavityat the junctions,” Opt. Commun. 284(9), 2292–2297 (2011).
    [Crossref]
  9. N. Talebi and M. Shahabdi, “Analysis of the Propagation of Light Along an Array of Nanorods Using the Generalized Multipole Techniques,” J. Comput. Theor. Nanosci. 5(4), 711–716 (2008).
    [Crossref]
  10. Y. H. Cho and D.-H. Kwon, “Efficient Mode-Matching Analysis of 2-D Scattering by Periodic Array of Circular Cylinders,” IEEE Trans. Antenn. Propag. 61(3), 1327–1333 (2013).
    [Crossref]
  11. J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
    [Crossref]
  12. S. K. Gray and T. Kupka, “Propagation of light in metallic nanowire arrays: Finite-difference time-domain studies of silver cylinders,” Phys. Rev. B 68(4), 045415 (2003).
    [Crossref]
  13. S. Li and Y. Y. Lu, “Efficient method for computing leaky modes in two-dimensional photonic crystal waveguides,” J. Lightwave Technol. 28(6), 978–983 (2010).
    [Crossref]
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    [Crossref]
  15. D. Zhang and H. Jia, “Numerical analysis of leaky modes in two-dimensional photonic crystal waveguides using Fourier series expansion method with perfectly matched layer,” IEICE Trans. Electron. E90-C(3), 613–622 (2007).
    [Crossref]
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    [Crossref]
  18. K. Yasumoto, H. Toyama, and R. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antenn. Propag. 52(10), 2603–2611 (2004).
    [Crossref]

2014 (1)

B. Vial, F. Zolla, A. Nicolet, and M. Commandre, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89(2), 023829 (2014).
[Crossref]

2013 (1)

Y. H. Cho and D.-H. Kwon, “Efficient Mode-Matching Analysis of 2-D Scattering by Periodic Array of Circular Cylinders,” IEEE Trans. Antenn. Propag. 61(3), 1327–1333 (2013).
[Crossref]

2011 (2)

H. Guo, Y. Zhang, and B. Li, “Periodic dielectric waveguide-based cross- and T-connections with a resonant cavityat the junctions,” Opt. Commun. 284(9), 2292–2297 (2011).
[Crossref]

D. Zhang and A. Mase, “A Formula for Fourier Series Expansion Method with Complex Coordinate Stretching Layers,” J. Infrared Millim. Terahertz Waves 32(2), 196–203 (2011).
[Crossref]

2010 (1)

2009 (1)

J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
[Crossref]

2008 (3)

N. Talebi and M. Shahabdi, “Analysis of the Propagation of Light Along an Array of Nanorods Using the Generalized Multipole Techniques,” J. Comput. Theor. Nanosci. 5(4), 711–716 (2008).
[Crossref]

B. N. Behnken, G. Karunasiri, D. R. Chamberlin, P. R. Robrish, and J. Faist, “Real-time imaging using a 2.8 THz quantum cascade laser and uncooled infrared microbolometer camera,” Opt. Lett. 33(5), 440–442 (2008).
[Crossref] [PubMed]

M. Hammer, “Chains of coupled square dielectric optical microcavities,” Opt. Quantum Electron. 40(11-12), 821–835 (2008).
[Crossref]

2007 (2)

H. Nemec, P. Kuzel, J.-L. Coutaz, and J. Ctyroky, “Transmission properties and band structure of a segmented dielectric waveguide for the terahertz range,” Opt. Commun. 273(1), 99–104 (2007).
[Crossref]

D. Zhang and H. Jia, “Numerical analysis of leaky modes in two-dimensional photonic crystal waveguides using Fourier series expansion method with perfectly matched layer,” IEICE Trans. Electron. E90-C(3), 613–622 (2007).
[Crossref]

2005 (1)

H. Jia, D. Zhang, and K. Yasumoto, “Fast analysis of optical waveguides using an improved Fourier series method with perfectly matched layer,” Microw. Opt. Technol. Lett. 46(3), 263–268 (2005).
[Crossref]

2004 (1)

K. Yasumoto, H. Toyama, and R. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antenn. Propag. 52(10), 2603–2611 (2004).
[Crossref]

2003 (1)

S. K. Gray and T. Kupka, “Propagation of light in metallic nanowire arrays: Finite-difference time-domain studies of silver cylinders,” Phys. Rev. B 68(4), 045415 (2003).
[Crossref]

1998 (1)

1996 (1)

Aussenegg, F. R.

Behnken, B. N.

Chamberlin, D. R.

Cho, Y. H.

Y. H. Cho and D.-H. Kwon, “Efficient Mode-Matching Analysis of 2-D Scattering by Periodic Array of Circular Cylinders,” IEEE Trans. Antenn. Propag. 61(3), 1327–1333 (2013).
[Crossref]

Chui, S. T.

J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
[Crossref]

Commandre, M.

B. Vial, F. Zolla, A. Nicolet, and M. Commandre, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89(2), 023829 (2014).
[Crossref]

Coutaz, J.-L.

H. Nemec, P. Kuzel, J.-L. Coutaz, and J. Ctyroky, “Transmission properties and band structure of a segmented dielectric waveguide for the terahertz range,” Opt. Commun. 273(1), 99–104 (2007).
[Crossref]

Ctyroky, J.

H. Nemec, P. Kuzel, J.-L. Coutaz, and J. Ctyroky, “Transmission properties and band structure of a segmented dielectric waveguide for the terahertz range,” Opt. Commun. 273(1), 99–104 (2007).
[Crossref]

Demésy, G.

A. Nicolet, G. Demésy, F. Zolla, and B. Vial, “Quasi-modal analysis of segmented waveguides,” in Proc. IEEE Conf. Antenna Meas. Appl. (CAMA, 2014), pp. 1–4.

Du, J.

J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
[Crossref]

Faist, J.

Gray, S. K.

S. K. Gray and T. Kupka, “Propagation of light in metallic nanowire arrays: Finite-difference time-domain studies of silver cylinders,” Phys. Rev. B 68(4), 045415 (2003).
[Crossref]

Guo, H.

H. Guo, Y. Zhang, and B. Li, “Periodic dielectric waveguide-based cross- and T-connections with a resonant cavityat the junctions,” Opt. Commun. 284(9), 2292–2297 (2011).
[Crossref]

Hammer, M.

M. Hammer, “Chains of coupled square dielectric optical microcavities,” Opt. Quantum Electron. 40(11-12), 821–835 (2008).
[Crossref]

Jia, H.

D. Zhang and H. Jia, “Numerical analysis of leaky modes in two-dimensional photonic crystal waveguides using Fourier series expansion method with perfectly matched layer,” IEICE Trans. Electron. E90-C(3), 613–622 (2007).
[Crossref]

H. Jia, D. Zhang, and K. Yasumoto, “Fast analysis of optical waveguides using an improved Fourier series method with perfectly matched layer,” Microw. Opt. Technol. Lett. 46(3), 263–268 (2005).
[Crossref]

Karunasiri, G.

Krenn, J. R.

Kupka, T.

S. K. Gray and T. Kupka, “Propagation of light in metallic nanowire arrays: Finite-difference time-domain studies of silver cylinders,” Phys. Rev. B 68(4), 045415 (2003).
[Crossref]

Kushta, R.

K. Yasumoto, H. Toyama, and R. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antenn. Propag. 52(10), 2603–2611 (2004).
[Crossref]

Kuzel, P.

H. Nemec, P. Kuzel, J.-L. Coutaz, and J. Ctyroky, “Transmission properties and band structure of a segmented dielectric waveguide for the terahertz range,” Opt. Commun. 273(1), 99–104 (2007).
[Crossref]

Kwon, D.-H.

Y. H. Cho and D.-H. Kwon, “Efficient Mode-Matching Analysis of 2-D Scattering by Periodic Array of Circular Cylinders,” IEEE Trans. Antenn. Propag. 61(3), 1327–1333 (2013).
[Crossref]

Leitner, A.

Li, B.

H. Guo, Y. Zhang, and B. Li, “Periodic dielectric waveguide-based cross- and T-connections with a resonant cavityat the junctions,” Opt. Commun. 284(9), 2292–2297 (2011).
[Crossref]

Li, L.

Li, S.

Lin, Z.

J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
[Crossref]

Liu, S.

J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
[Crossref]

Lu, Y. Y.

Mase, A.

D. Zhang and A. Mase, “A Formula for Fourier Series Expansion Method with Complex Coordinate Stretching Layers,” J. Infrared Millim. Terahertz Waves 32(2), 196–203 (2011).
[Crossref]

Nemec, H.

H. Nemec, P. Kuzel, J.-L. Coutaz, and J. Ctyroky, “Transmission properties and band structure of a segmented dielectric waveguide for the terahertz range,” Opt. Commun. 273(1), 99–104 (2007).
[Crossref]

Nicolet, A.

B. Vial, F. Zolla, A. Nicolet, and M. Commandre, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89(2), 023829 (2014).
[Crossref]

A. Nicolet, G. Demésy, F. Zolla, and B. Vial, “Quasi-modal analysis of segmented waveguides,” in Proc. IEEE Conf. Antenna Meas. Appl. (CAMA, 2014), pp. 1–4.

Quinten, M.

Robrish, P. R.

Shahabdi, M.

N. Talebi and M. Shahabdi, “Analysis of the Propagation of Light Along an Array of Nanorods Using the Generalized Multipole Techniques,” J. Comput. Theor. Nanosci. 5(4), 711–716 (2008).
[Crossref]

Talebi, N.

N. Talebi and M. Shahabdi, “Analysis of the Propagation of Light Along an Array of Nanorods Using the Generalized Multipole Techniques,” J. Comput. Theor. Nanosci. 5(4), 711–716 (2008).
[Crossref]

Toyama, H.

K. Yasumoto, H. Toyama, and R. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antenn. Propag. 52(10), 2603–2611 (2004).
[Crossref]

Vial, B.

B. Vial, F. Zolla, A. Nicolet, and M. Commandre, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89(2), 023829 (2014).
[Crossref]

A. Nicolet, G. Demésy, F. Zolla, and B. Vial, “Quasi-modal analysis of segmented waveguides,” in Proc. IEEE Conf. Antenna Meas. Appl. (CAMA, 2014), pp. 1–4.

Yasumoto, K.

H. Jia, D. Zhang, and K. Yasumoto, “Fast analysis of optical waveguides using an improved Fourier series method with perfectly matched layer,” Microw. Opt. Technol. Lett. 46(3), 263–268 (2005).
[Crossref]

K. Yasumoto, H. Toyama, and R. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antenn. Propag. 52(10), 2603–2611 (2004).
[Crossref]

Zhang, D.

D. Zhang and A. Mase, “A Formula for Fourier Series Expansion Method with Complex Coordinate Stretching Layers,” J. Infrared Millim. Terahertz Waves 32(2), 196–203 (2011).
[Crossref]

D. Zhang and H. Jia, “Numerical analysis of leaky modes in two-dimensional photonic crystal waveguides using Fourier series expansion method with perfectly matched layer,” IEICE Trans. Electron. E90-C(3), 613–622 (2007).
[Crossref]

H. Jia, D. Zhang, and K. Yasumoto, “Fast analysis of optical waveguides using an improved Fourier series method with perfectly matched layer,” Microw. Opt. Technol. Lett. 46(3), 263–268 (2005).
[Crossref]

Zhang, Y.

H. Guo, Y. Zhang, and B. Li, “Periodic dielectric waveguide-based cross- and T-connections with a resonant cavityat the junctions,” Opt. Commun. 284(9), 2292–2297 (2011).
[Crossref]

Zi, J.

J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
[Crossref]

Zolla, F.

B. Vial, F. Zolla, A. Nicolet, and M. Commandre, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89(2), 023829 (2014).
[Crossref]

A. Nicolet, G. Demésy, F. Zolla, and B. Vial, “Quasi-modal analysis of segmented waveguides,” in Proc. IEEE Conf. Antenna Meas. Appl. (CAMA, 2014), pp. 1–4.

IEEE Trans. Antenn. Propag. (2)

Y. H. Cho and D.-H. Kwon, “Efficient Mode-Matching Analysis of 2-D Scattering by Periodic Array of Circular Cylinders,” IEEE Trans. Antenn. Propag. 61(3), 1327–1333 (2013).
[Crossref]

K. Yasumoto, H. Toyama, and R. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antenn. Propag. 52(10), 2603–2611 (2004).
[Crossref]

IEICE Trans. Electron. (1)

D. Zhang and H. Jia, “Numerical analysis of leaky modes in two-dimensional photonic crystal waveguides using Fourier series expansion method with perfectly matched layer,” IEICE Trans. Electron. E90-C(3), 613–622 (2007).
[Crossref]

J. Comput. Theor. Nanosci. (1)

N. Talebi and M. Shahabdi, “Analysis of the Propagation of Light Along an Array of Nanorods Using the Generalized Multipole Techniques,” J. Comput. Theor. Nanosci. 5(4), 711–716 (2008).
[Crossref]

J. Infrared Millim. Terahertz Waves (1)

D. Zhang and A. Mase, “A Formula for Fourier Series Expansion Method with Complex Coordinate Stretching Layers,” J. Infrared Millim. Terahertz Waves 32(2), 196–203 (2011).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (1)

Microw. Opt. Technol. Lett. (1)

H. Jia, D. Zhang, and K. Yasumoto, “Fast analysis of optical waveguides using an improved Fourier series method with perfectly matched layer,” Microw. Opt. Technol. Lett. 46(3), 263–268 (2005).
[Crossref]

Opt. Commun. (2)

H. Guo, Y. Zhang, and B. Li, “Periodic dielectric waveguide-based cross- and T-connections with a resonant cavityat the junctions,” Opt. Commun. 284(9), 2292–2297 (2011).
[Crossref]

H. Nemec, P. Kuzel, J.-L. Coutaz, and J. Ctyroky, “Transmission properties and band structure of a segmented dielectric waveguide for the terahertz range,” Opt. Commun. 273(1), 99–104 (2007).
[Crossref]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

M. Hammer, “Chains of coupled square dielectric optical microcavities,” Opt. Quantum Electron. 40(11-12), 821–835 (2008).
[Crossref]

Phys. Rev. A (2)

B. Vial, F. Zolla, A. Nicolet, and M. Commandre, “Quasimodal expansion of electromagnetic fields in open two-dimensional structures,” Phys. Rev. A 89(2), 023829 (2014).
[Crossref]

J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
[Crossref]

Phys. Rev. B (1)

S. K. Gray and T. Kupka, “Propagation of light in metallic nanowire arrays: Finite-difference time-domain studies of silver cylinders,” Phys. Rev. B 68(4), 045415 (2003).
[Crossref]

Other (2)

A. Nicolet, G. Demésy, F. Zolla, and B. Vial, “Quasi-modal analysis of segmented waveguides,” in Proc. IEEE Conf. Antenna Meas. Appl. (CAMA, 2014), pp. 1–4.

K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals (CRC Press, 2005).

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Figures (8)

Fig. 1
Fig. 1 Infinite periodic chain of circular rods along x-axis with lattice constant h. Radius and dielectric permittivity of rods were r and ε, respectively. To apply FSEM, the periodic structure was bounded by PMLs at a distance Λ/2 from the global origin.
Fig. 2
Fig. 2 Phase constant βh/π and attenuation constant αh/π of the TE mode as a function of the normalized periodicity h/λ0 at r = 0.4167h and ε = 2.25, λ0 is a wavelength in a free space.
Fig. 3
Fig. 3 Phase constant βh/π and attenuation constant αh/π of the TM mode as a function of the normalized periodicity h/λ0. Other parameters are the same as those in Fig. 2.
Fig. 4
Fig. 4 Near field distributions for the periodic chain composed of 10 dielectric circular rods in the TE mode at h/λ0 = 0.3 (a) and h/λ0 = 0.5 (b).
Fig. 5
Fig. 5 Infinite periodic one-layer chain became a multilayered chain structure if there were no PMLs at the boundaries.
Fig. 6
Fig. 6 The convergence behavior compared for with and without PML at h/λ = 0.3 with M = 2.5Λ/h.
Fig. 7
Fig. 7 Convergence behavior compared with and without PML at h/λ = 0.4 with M = 2.5Λ/h.
Fig. 8
Fig. 8 The convergence behavior compared with and without PML at h/λ = 0.5 with M = 2.5Λ/h.

Tables (1)

Tables Icon

Table 1 Phase constant βh/π  and attenuation constant αh/π for the lowest TE and TM modes of the periodic chain of the dielectric circular rods calculated based on FSEM and Phase constant βh/π  calculated based on LST and FDTD.

Equations (13)

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

v(y) y E z =i k 0 H ˜ x , x E z =i k 0 H ˜ y , x H ˜ y v(y) y H ˜ x =i k 0 ε(y) E z
E z = m=M M e z,m (x) e i κ m y , H ˜ x(y) = m=M M h ˜ x(y),m (x) e i κ m y
2 x 2 e z (x)= k 0 2 C e z (x), h ˜ y (x)=i 1 k 0 x e z (x)
e z (x)= [ e z,M e z,M ] T , h ˜ y (x)= [ h ˜ y,M h ˜ y,M ] T
C=N (VA) 2 , [N] m m = 1 Λ 0 Λ ε(y) e i( κ m κ m )y dy
[V] m m = 1 Λ 0 Λ v(y) e i( κ m κ m )y dy, [A] m m = κ m k 0 δ m m
[ e z (x) h ˜ y (x) ]=FU(x x )a( x )
F=[ P P PB PB ],U(x)=[ U + (x) 0 0 U (x) ]
P=[ p 1 p 2 p 2M p 2M+1 ], U ± (x)=[ e ±i k 0 τ n x δ n n ], B=[ τ n δ n n ]
a(x)= [ a + (x) a (x)] T , a ± (x)=[ a 1 ± (x) a 2 ± (x) a 2M ± (x) a 2M+1 ± (x)]
[ a + (h) a (h) ]=K[ a + (0) a (0) ]
γ k =-ilog χ k /h
det[IT( λ 0 )L( λ 0 ,β)]=0

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