在我們的研究中，透過數值模擬與理論解析呈現表面電漿布洛赫震盪於同心圓金屬－介電質導波管陣列內，而且利用光學保角映射可將同心圓結構轉換成等效的矩形導波管結構，因同心圓結構半徑使該矩形導波管具有空間與材料特性的梯度；震盪週期與入射波長具正比關係，但震盪振幅並不隨入射波長而改變。此外我們也利用數值模擬與理論解析呈現表面電漿曾納穿隧效應在金屬－介電質導波管陣列，因為第一與第二能帶之間的能隙位置處於布里淵區中心位置，所以表面電漿曾納穿隧效應發生於入射位置，此特性與於介電質波導陣列內的光學曾納穿隧效應完全不同；而且借電常數的梯度與穿隧機率之間的關係與曾納穿隧理論吻合，因此可以確定此現象為金屬－介電質導波管陣列內的表面電漿曾納穿隧效應。最後，針對穿隧之後的波作探討，提出布洛赫－曾納震盪，並透過數值模擬與理論解析可得知布洛赫－曾納震盪具有圓形震盪軌跡，透過改變具有借電常數梯度的介電質層厚度，整個旋轉震盪就能被決定往後、往前甚至原地旋轉；介電質層厚度增加會使往右位移增加與往左位移減少，而整體的移動方向也能定義縱向的等效折射率。

This study investigates plasmonic Bloch oscillations (PBOs) in cylindrical metal–dielectric waveguide arrays (MDWAs) by performing numerical simulations and theoretical analyses. Optical conformal mapping is used to transform cylindrical MDWAs into equivalent chirped structures with permittivity and permeability gradients across the waveguide arrays, which is caused by the curvature of the cylindrical waveguide. The PBOs are attributed to the transformed structure. The period of oscillation increases with the wavelength of the incident Gaussian beam.However, the amplitude of oscillation is almost independent of wavelength.Besides, we elucidate plasmonicZener tunneling (PZT) in metal–dielectric waveguide arrays (MDWAs) by using numerical simulations and theoretical analyses. PZT in MDWAs occurs at the waveguide entrance and wherever the beam completes Bloch oscillations, because the bandgap between the first and second bands is minimal at the center of the first Brillouin zone. This feature significantly differs from that of optical Zener tunneling in dielectric waveguide arrays. The dependence of the simulated tunneling rate on the gradient of the relative permittivity of the dielectric layers correlates with the tunneling theory, thus confirming the occurrence of PZT in MDWAs. Finally, this work investigates plasmonic Bloch–Zener oscillation and beam curling in metal–dielectric waveguide arrays (MDWAs) using numerical simulations and theoretical analyses. The beam generated by plasmonicZener tunneling undergoes a plasmonic Bloch oscillation in the second band of MDWAs and becomes curled. Changing the width and the relative-permittivity gradient of the dielectric layers causes this curled beam to move backward, forward, or even unmoved. Increasing thewidth and the relative-permittivity gradient of the dielectric layers increases the rightward displacement and reduces the leftward displacement. The direction of motion of the curled beam is determined by the net longitudinal displacement.

[1.1] Hecht, Eugene. Optics 4th. (2002).
[1.2] E. N. Economou, Phys. Rev. 182(2), 539 (1969).
[1.3] C. C. Chao, S. H. Tu, C. M. Wang, H. I. Huang, C. C. Chen and J. Y. Chang, Plasmonics, DOI 10.1007/s11468-009-9114-2, (2009).
[1.4] J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, Phys. Rev. B 72(7), 075405 (2005).
[1.5] X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, Phys. Rev. Lett. 97, 073901 (2006)
[1.6] H. Shin and S. Fan, Appl. Phys. Lett. 89, 151102 (2006)
[1.7] C. Yan, D. H. Zhang, Y. Zhang, D. Li, and M. A. Fiddy, Optics Express, 18 (14), 14794-14801 (2010)
[1.8] X. Fan and G. P. Wang, Opt. Lett. 31, 1322 (2006).
[1.9] T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002)
[1.10] Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. Lett. 99, 153901 (2007)
[1.11] W. Lin, X. Zhou, and G. P. Wang, Appl. Phys. Lett. 91, 243113 (2007)
[1.12] A. R. Davoyan, A. A. Sukhorukov, I. V. Shadrivov, and Y. S. Kivshar, Phys. Rev. A 79, 013820-8 (2009).
[1.13] A. R. Davoyan, I. V. Shadrivov, A. A. Sukhorukov,and Y. S. Kivshar, Optics Express, 16 (5), 3299 (2008)
[1.14] A. R. Davoyan, I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, Appl. Phys. Lett. 91, 243113 (2007)
[1.15] W. Lin and L. Chen, J. Opt. Soc. Am. B 27, 112–117(2010).
[1.16] C. M. d. Sterke and J. E. Sipe, Opt. Lett. 16, 1141 (1991).
[1.17] G. Lenz, I. Talanina, and M. d. Sterke, Phys. Rev. Lett.83, 963-966 (1999).
[1.18] X. Kang and Z. Wang, Opt Commun282, 355 (2009)
[1.19] S. Longhi, Phys. Rev. Lett. 101, 193902 (2008).
[1.20] H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006)
[1.21] R. C. Shiu and Y. C. Lan, Opt. Lett. 36, 4179 (2011).
[1.22] R. C. Shiu, Y. C. Lan, and C. M. Chen, Opt. Lett. 36, 4012 (2010).
[1.23] L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fanv, Phys. Rev. Lett. 103, 033902 (2009)
[1.24] G. D. Valle and S. Longhi, Opt. Lett. 35, 673 (2010).
[1.25] J. B. Pendry, D. Schurig, D. R. Smith, Science 312, 1780 (2006)
[1.26] I. Siddiqi and J. Clarke, Science 313, 1400 (2006)
[1.27] U. Leonhardt and T. Tyc, Science 323, 110 (2009)
[1.28] A. V. Kildishev and V. M. Shalaev, Opt. Lett. 33, 43 (2008).
[1.29] M. Heiblum and J. H. Harris, IEEE J. Quantum Electron. 11,75 (1975).
[1.30] Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, Nano Lett.10, 1991 (2010)
[1.31] Z. Liag and J. Li, Optics Express, 19 (18), 16821 (2011)
[1.32] N. I. Landy and W. J. Padilla, Optics Express, 17 (17), 14872 (2009)
[1.33] A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, Phys. Rev. B 82, 125430 (2010)
[1.34] A. Aubry, D. Y. Lei, A. I. Ferna´ndez-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, Nano Lett. 10, 2574–2579 (2010)
[1.35] D. Y. Lei, A. Aubry, S. A. Maier and J. B. Pendry, New J.Phys. 12, 093030 (2010)
[1.36] A. Aubry, D. Y. Lei, Stefan A. Maier, and J. B. Pendry, Phys. Rev. B 82, 205109 (2010)
[1.37] A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, Phys. Rev. Lett. 105, 233901 (2010)
[1.38] A. Aubry, D. Y. Lei, S. A. Maier, J. B. Pendry, ACS Nano 5, 3293 (2011)
[1.39] D. Y. Lei, A. Aubry, Y. Luo, S. A. Maier, and J. B. Pendry, ACS Nano 5, 3293 (2011)
[1.40] N. I. Zheludev, Science 328, 582 (2010)
[1.41] K. S. Yee, IEEE Trans. Antennas Propagat. AP-14, 302-307 (1966)
[1.42] Taflove, A., and S. C. Hagness. "Computational Electrodynamics: The Finite-Difference Time-Domain Method. 2005." Artech House, Norwood, MA.
[1.43] A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, Comput. Phys. Commun. 181, 687 (2010).
[2.1] S. Longhi (2009), “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243.
[2.2] C. M. de Sterke, J. E. Sipe, and L. A. Weller-Brophy (1991), “Electromagnetic Stark ladders in waveguide geometries,” Opt. Lett. 16, 1141.
[2.3] T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer (1999), “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752.
[2.4] R. Morandotii, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg(1999), “Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations,” Phys. Rev. Lett. 83, 4756.
[2.5] V. Agarwal, J. A. de Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil (2004), “Photon Bloch Oscillations in Porous Silicon Optical Superlattices,” Phys. Rev. Lett. 92, 097401.
[2.6] F. Bloch (1929), Z. Phys. 52, 555.
[2.7] G. Nenciu (1991), “Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians,” Rev. Mod. Phys. 63, 91.
[2.8] R. Davoyan, I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar (2009), “Plasmonic Bloch oscillations in chirped metal-dielectric structures,” Appl. Phys. Lett. 94, 161105.
[2.9] W. Lin, X. Zhou, and G. P. Wang (2007), “Spatial Bloch oscillations of plasmons in nanoscale metal waveguide arrays,” Appl. Phys. Lett. 91, 243113.
[2.10] G. Lenz, I. Talanina, and C. M. de Sterke (1999), “Bloch Oscillations in an Array of Curved Optical Waveguides,” Phys. Rev. Lett. 83, 963.
[2.11] Z. Jacob, L. V. Alekseyev, and E. Narimanov (2007), “Semiclassical theory of the hyperlens,” J. Opt. Soc. Am. A 24, A52.
[2.12] E. E. Narimanov and A. V. Kildishev (2009), “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106.
[2.13] Y. Huang, Y. Feng, and T. Jiang (2007), “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express 15, 11133-11141.
[2.14] A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson (2010), “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687.
[2.15] M. Heiblum and J. H. Harris (1975), “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11, 75.
[2.16] U. Leonhardt (2006), “Optical Conformal Mapping,” Science 312, 1777.
[2.17] J. B. Pendry, D. Schuring, and D. R. Smith (2006), “Controlling Electromagnetic Fields,” Science 312, 1780.
[2.18] Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang (2010), “Transformational Plasmon Optics,” Nano Lett. 10, 1991.
[2.19] L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fanv (2009), “Deep-Subwavelength Focusing and Steering of Light in an Aperiodic MetallicWaveguide Array,” Phys. Rev. Lett. 103, 033902.
[2.20] J. A. Arnaud (1976), Beam and Fiber Optics, New York: Academic Press.
[2.21] X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan (2006) , “All-Angle Broadband Negative Refraction of Metal Waveguide Arrays in the Visible Range: Theoretical Analysis and Numerical Demonstration,” Phys. Rev. Lett. 97, 073901.
[3.1] F. Bloch (1928), Z. Phys. 52, 555.
[3.2] G. Nenciu (1991), “Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians,” Rev. Mod. Phys. 63, 91.
[3.3] C. M. de Sterke and J. E. Sipe (1991), “Electromagnetic Stark ladders in waveguide geometries,” Opt. Lett. 16, 1141.
[3.4] T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer (1999), “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752.
[3.5] R. Morandotii, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg(1999), “Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations,” Phys. Rev. Lett. 83, 4756.
[3.6] V. Agarwal, J. A. de Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil (2004), “Photon Bloch Oscillations in Porous Silicon Optical Superlattices,” Phys. Rev. Lett. 92, 097401.
[3.7] H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel (2006), “Visual Observation of Zener Tunneling,” Phys. Rev. Lett. 96, 023901.
[3.8] S. Longhi (2008), “Optical Bloch Oscillations and Zener Tunneling with Nonclassical Light,” Phys. Rev. Lett. 101, 193902.
[3.9] F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann (2009), “Bloch-Zener Oscillations in Binary Superlattices,” Phys. Rev. Lett. 102, 076802.
[3.10] C. Zener (1934), “A Theory of the Electrical Breakdown of Solid Dielectrics,” Proc. R. Soc. London A 145, 523.
[3.11] R. Davoyan, I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar (2009), “Plasmonic Bloch oscillations in chirped metal-dielectric structures,” Appl. Phys. Lett. 94, 161105.
[3.12] W. Lin, X. Zhou, and G. P. Wang (2007), “Spatial Bloch oscillations of plasmons in nanoscale metal waveguide arrays,” Appl. Phys. Lett. 91, 243113.
[3.13] R. C. Shiu, Y. C. Lan, and C. M. Chen (2010), “Plasmonic Bloch oscillations in cylindrical metal–dielectric waveguide arrays,” Opt. Lett. 35, 4012.
[3.14] W. Lin, Y. Gu, and G. P. Wang (2008), “Zener tunneling in plasmonic metal gap waveguide superlattices,” Appl. Phys. Lett. 93, 133118.
[3.15] J. K. Kim, A. N. Noemaun, F. W. Mont, D. Meyaard, E. F. Schubert, D. J. Poxson, H. Kim, C. Sone, and Y. Park (2008), “Elimination of total internal reflection in GaInN light-emitting diodes by graded-refractive-index micropillars,” Appl. Phys. Lett. 93, 221111.
[3.16] A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson (2010), “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687.
[3.17] L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fanv (2009), “Deep-Subwavelength Focusing and Steering of Light in an Aperiodic MetallicWaveguide Array,” Phys. Rev. Lett. 103, 033902.
[4.1] R. Davoyan, I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar (2009), “Plasmonic Bloch oscillations in chirped metal-dielectric structures,” Appl. Phys. Lett. 94, 161105.
[4.2] W. Lin, X. Zhou, and G. P. Wang (2007), “Spatial Bloch oscillations of plasmons in nanoscale metal waveguide arrays,” Appl. Phys. Lett. 91, 243113.
[4.3] R. C. Shiu, Y. C. Lan, and C. M. Chen (2010), “Plasmonic Bloch oscillations in cylindrical metal–dielectric waveguide arrays,” Opt. Lett. 35, 4012.
[4.4] R. C. Shiu and Y. C. Lan (2011), “Plasmonic Zener tunneling in metal–dielectric waveguide arrays,” Opt. Lett. 36, 4179.
[4.5] X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-Angle Broadband Negative Refraction of Metal Waveguide Arrays in the Visible Range: Theoretical Analysis and Numerical Demonstration,” Phys. Rev. Lett. 97, 073901 (2006).
[4.6] H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer (2006), “Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices,” Phys. Rev. Lett. 96, 053903.
[4.7] S. Longhi (2008), “Optical Bloch Oscillations and Zener Tunneling with Nonclassical Light,” Phys. Rev. Lett. 101, 193902.
[4.8] F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte and A. Tünnermann (2009), “Bloch-Zener Oscillations in Binary Superlattices,” Phys. Rev. Lett. 102, 076802.
[4.9] M. J. Zheng, Y. S. Chan, and K. W. Yu (2011), “Photonic Bloch–dipole–Zener oscillations in binary parabolic optical waveguide arrays,” J. Opt. Soc. Am. B 28, 1339.
[4.10] A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson (2010), “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687.
[4.11] L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fanv (2009), “Deep-Subwavelength Focusing and Steering of Light in an Aperiodic MetallicWaveguide Array,” Phys. Rev. Lett. 103, 033902.