先前研究中[JW Li, JS Hong, WT Chou. et al. Plasmonics (2018). https://doi.org/10.1007/s11468-018-0745-z]，利用坡印廷向量(Poynting vector)的x分量等於零(Sx = 0)定義漏斗形狀，探討時間平均的漏斗邊界如何隨狹縫寬度、薄膜厚度變化，其結果顯示漏斗形狀變化與穿透率變化一致。我們想知道漏斗形狀隨著金屬膜厚度改變的動態變化及對應的穿透率是否依舊一致，我們看到厚度為建設性干涉條件時漏斗面積隨時間震盪的週期與先前研究一致了，但破壞性干涉條件的週期改變了，我們想知道為什麼改變。
本研究主要探討二維金屬狹縫中往返波(round-trip wave)對漏邊界狀造成的相位變化。漏斗邊界是由Sx決定，而Sx由Ey與Hz決定。因此，我們除了觀察漏斗邊界的變化外，也觀察Ey與Hz如何隨著厚度變化進而改變Sx的相位。往返波造成的效應是由Fabry-Pérot共振條件決定，在狹縫內來回傳遞的往返波會與狹縫內以及狹縫入口上方的電磁場產生干涉改變相位。我們發現流入狹縫的能量流隨時間變化趨勢與漏斗面積隨時間變化的趨勢一致，但能量流增加或減少的量不一致，我們從時間平均的觀點出發，發現時間平均的漏斗面積大小與其面積內的平均能量流大小不一致。漏斗邊界的變化主要由Ey影響，Ey的變化受往返波以及靜電場影響，此外有能量回流到狹縫入口的現象。本研究經由對漏斗邊界相位變化的分析，進一步增進我們對漏斗效應機制的理解。

Previous studies [JW Li, JS Hong, WT Chou. et al. Plasmonics (2018). https://doi.org/10.1007/s11468-018-0745-z], using x component of poynting vector equal zero (Sx = 0) to define the shape of the funnel. It is discussed how the time-averaged funnel boundary varies with the slit width and film thickness. The results show that the funnel shape change is consistent with the change of transmittance. We want to know dynamic of the funnel shape varies with thickness. We see that the period of the funnel area, which the thickness is constructive interference condition, is consistent with the previous study, but destructive condition is not. We want to know why it changed.
This study focuses on the phase change caused by the round-trip wave in the two-dimensional metal slit on the funnel boundary. The funnel boundary is determined by Sx, and Sx is determined by Ey and Hz. Therefore, in addition to observing the change in the funnel boundary, we also observe how phase change of Ey and Hz by thickness. The effect of the round-trip wave is determined by the Fabry-Pérot resonance condition, and the round-trip wave interferes electromagnetic field at slit entrance. We found that the trend of energy flow into the slit over time is consistent with the trend of the funnel area over time, but the amount of increase or decrease in energy flow is inconsistent. From the time-average view, we find the time-averaged funnel boundary and its area is inconsistent with the time-average energy flow. The the funnel boundary is mainly affected by Ey. The Ey is affected by the round-trip wave, and there is a phenomenon that energy flows back to the slit entrance. This study promote our understanding of the funnel effect mechanism by analyzing the phase changes of the funnel boundary

1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
2. P. B. Catrysse, and B. A. Wandell, "Integrated color pixels in 0.18-µm complementary metal oxide semiconductor technology," Journal of the Optical Society of America A 20, 2293-2306 (2003).
3. T. Xu, Y.-K. Wu, X. Luo, and L. J. Guo, "Plasmonic nanoresonators for high-resolution colour filtering and spectral imaging," Nat. Commun. 1, 59 (2010).
4. N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub–Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534 (2005).
5. L. Novotny, and N. van Hulst, "Antennas for light," Nature Photonics 5, 83-90 (2011).
6. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, "Biosensing with plasmonic nanosensors," Nat. Mater. 7, 442-453 (2008).
7. J. A. Porto, F. J. García-Vidal, and J. B. Pendry, "Transmission Resonances on Metallic Gratings with Very Narrow Slits," Phys. Rev. Lett. 83, 2845-2848 (1999).
8. Y. Takakura, "Optical Resonance in a Narrow Slit in a Thick Metallic Screen," Phys. Rev. Lett. 86, 5601-5603 (2001).
9. J. Bravo-Abad, L. Martín-Moreno, and F. J. García-Vidal, "Transmission properties of a single metallic slit: From the subwavelength regime to the geometrical-optics limit," Physical Review E 69, 026601 (2004).
10. F. J. Garcia-Vidal, L. Martin-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon, "Transmission of light through a single rectangular hole in a real metal," Physical Review B 74, 4 (2006).
11. B. Sturman, E. Podivilov, and M. Gorkunov, "Transmission and diffraction properties of a narrow slit in a perfect metal," Physical Review B 82, 115419 (2010).
12. Y. Xie, A. Zakharian, J. Moloney, and M. Mansuripur, "Transmission of light through slit apertures in metallic films," Opt. Express 12, 6106-6121 (2004).
13. P. Lalanne, J. P. Hugonin, and J. C. Rodier, "Theory of Surface Plasmon Generation at Nanoslit Apertures," Phys. Rev. Lett. 95, 263902 (2005).
14. A. Y. Nikitin, D. Zueco, F. J. García-Vidal, and L. Martín-Moreno, "Electromagnetic wave transmission through a small hole in a perfect electric conductor of finite thickness," Physical Review B 78, 165429 (2008).
15. S.-H. Chang, and Y.-L. Su, "Mapping of transmission spectrum between plasmonic and nonplasmonic single slits. I: resonant transmission," J. Opt. Soc. Am. B 32, 38-44 (2015).
16. S.-H. Chang, and Y.-L. Su, "Mapping of transmission spectrum between plasmonic and nonplasmonic single slits. II: nonresonant transmission," J. Opt. Soc. Am. B 32, 45-51 (2015).
17. F. J. García-Vidal, and L. Martín-Moreno, "Transmission and focusing of light in one-dimensional periodically nanostructured metals," Physical Review B 66, 155412 (2002).
18. S. Astilean, P. Lalanne, and M. Palamaru, "Light transmission through metallic channels much smaller than the wavelength," Opt. Commun. 175, 265-273 (2000).
19. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, "Mimicking Surface Plasmons with Structured Surfaces," Science 305, 847-848 (2004).
20. H. T. Liu, and P. Lalanne, "Microscopic theory of the extraordinary optical transmission," Nature 452, 728-731 (2008).
21. F. van Beijnum, C. Retif, C. B. Smiet, H. T. Liu, P. Lalanne, and M. P. van Exter, "Quasi-cylindrical wave contribution in experiments on extraordinary optical transmission," Nature 492, 411-414 (2012).
22. T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, "Enhanced light transmission through a single subwavelength aperture," Opt. Lett. 26, 1972-1974 (2001).
23. F. J. Garcı́a-Vidal, L. Martı́n-Moreno, H. J. Lezec, and T. W. Ebbesen, "Focusing light with a single subwavelength aperture flanked by surface corrugations," Appl. Phys. Lett. 83, 4500 (2003).
24. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of Highly Directional Emission from a Single Subwavelength Aperture Surrounded by Surface Corrugations," Phys. Rev. Lett. 90, 167401 (2003).
25. G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, "Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths," Scientific reports 4, 6333 (2014).
26. X. Zhang, L. Yan, Y. Guo, W. Pan, B. Luo, and X. Luo, "Enhanced Far-Field Focusing by Plasmonic Lens Under Radially Polarized Beam Illumination," Plasmonics 11, 109-115 (2016).
27. F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, "Multiple Paths to Enhance Optical Transmission through a Single Subwavelength Slit," Phys. Rev. Lett. 90, 213901 (2003).
28. D. A. Thomas, and H. P. Hughes, "Enhanced optical transmission through a subwavelength 1D aperture," Solid State Commun. 129, 519-524 (2004).
29. H. Shi, X. Dong, Y. Lv, and C. Du, "Multi-interaction of surface wave between subwavelength grooves surrounding a single metallic slit," Applied Physics B 95, 345-350 (2009).
30. J. S. Hong, and K. R. Chen, "Light diffraction by a slit and grooves with a point source model based on wave dynamics," Phys. Rev. A 96, 13 (2017).
31. K. R. Chen, "Focusing of light beyond the diffraction limit of half the wavelength," Opt. Lett. 35, 3763-3765 (2010).
32. K. R. Chen, W. H. Chu, H. C. Fang, C. P. Liu, C. H. Huang, H. C. Chui, C. H. Chuang, Y. L. Lo, C. Y. Lin, H. H. Hwung, and A. Y. G. Fuh, "Beyond-limit light focusing in the intermediate zone," Opt. Lett. 36, 4497-4499 (2011).
33. G. Subramania, S. Foteinopoulou, and I. Brener, "Nonresonant Broadband Funneling of Light via Ultrasubwavelength Channels," Phys. Rev. Lett. 107, 163902 (2011).
34. A. V. Goncharenko, K. Y. Kim, J. S. Hong, and K. R. Chen, "Complex Mechanism of Enhanced Optical Transmission Through a Composite Coaxial/Circular Aperture," Plasmonics 7, 417-426 (2012).
35. J.-S. Hong, A. E. Chen, and K.-R. Chen, "Modulated light transmission through a subwavelength slit at early stage," Opt. Express 23, 9901-9910 (2015).
36. Y. Zeng, Y. Fu, X. S. Chen, W. Lu, and H. Agren, "Dynamics of the damping oscillator formed by the collective generation of surface polaritons for extraordinary light transmission through subwavelength hole arrays in thin metal films," Physical Review B 76, 8 (2007).
37. J. Wuenschell, and H. K. Kim, "Excitation and propagation of surface plasmons in a metallic nanoslit structure," IEEE Trans. Nanotechnol. 7, 229-236 (2008).
38. Y. Xi, Y. S. Jung, and H. K. Kim, "Interaction of light with a metal wedge: the role of diffraction in shaping energy flow," Opt. Express 18, 2588-2600 (2010).
39. F. Pardo, P. Bouchon, R. Haïdar, and J.-L. Pelouard, "Light Funneling Mechanism Explained by Magnetoelectric Interference," Phys. Rev. Lett. 107, 093902 (2011).
40. D. C. Adams, S. Inampudi, T. Ribaudo, D. Slocum, S. Vangala, N. A. Kuhta, W. D. Goodhue, V. A. Podolskiy, and D. Wasserman, "Funneling Light through a Subwavelength Aperture with Epsilon-Near-Zero Materials," Phys. Rev. Lett. 107, 133901 (2011).
41. J.-W. Li, J.-S. Hong, W.-T. Chou, D.-J. Huang, and K.-R. Chen, "Light Funneling Profile During Enhanced Transmission Through a Subwavelength Metallic Slit," Plasmonics (2018).
42. Y. Kane, "Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
43. R. Courant, K. Friedrichs, and H. Lewy, "Über die partiellen Differenzengleichungen der mathematischen Physik," Mathematische Annalen 100, 32-74 (1928).
44. H. A. Haus, and J. R. Melcher, Electromagnetic fields and energy (Prentice Hall, 1989).