自從Engheta於2005年開始發表一系列與RLC電路理論等效的光學奈米尺度元件後，開啟了對奈米光學電路的研究領域[1]。在奈米尺度的波導下的光學奈米電路，以一般的介電質等效為電容C，以金屬等效為電感L，波導的阻抗、金屬的損耗等效為電阻R[2]，再利用εnear zero(ENZ)材料在奈米尺度下對電磁波的高穿透率傳遞特性等效為nanoconnector來組成這些光電元件電路[1]。在整組光電元件組合大小遠小於入射光波波長的條件下，這些元件能類比於傳統電路理論，卻有更高的資料傳遞頻寬。
在本篇論文我們利用Finite-Difference Time-Domain(FDTD) 模擬來探討奈米光學電路中各種光學元件特性問題，進而分析奈米光電元件類比為電路理論的特性並與傳統電路公式比較。 我們先討論以介電質及Drude Model金屬組成RC、RL、RLC串聯、RLC並聯電路的穿透頻譜峰、谷值，並改變光學電路元件組合中材料的大小，以及在波導中不同位置來觀察穿透頻譜的峰值變化，並與二階電路的理論值作比較[3]。接著討論用金屬在電漿共振頻率時用來等效為ε near zero材料(ENZ)的光傳遞性，它會類比於電路中connector的效果，而讓大部分的光波在奈米尺度下轉彎穿透。另外， ENZ材料之穿透效果會受到面積大小限制，為了進一步提升nano-connector的穿透特性，我們利用金屬及介電值組合的meta-material結構來等效出ε及μnear zero的材料(EMNZ)，以突破這限制並達成更具發展性的應用。
接著，我們從根據Fabry-perot以及三層介質原理，反向地由穿透係數及反射係數來求一介質之等效介電係數以及透磁率。利用此Reversed-FP方法探討EMNZ meta-material結構時，發現此結構之穿透峰值產生原因並非Engheta教授於2007年發表於PRB的論文所述之ε及μ皆near zero的效果[4]，而是單純的等效阻抗匹配產生的穿透特性。這項發現推翻ENZ以及EMNZ的穿透理論，並可以延伸探討nano-circuits的理論的成立性。也為將來meta-material的穿透性質研究上，帶來新的觀點以及研究的方向。

In 2005 Engheta[1] proposed a revolutionary idea of applying the traditional lumped circuit theory to optical circuits at nano-scale by treating the displace current density D as the electric current I. This concept inspired a series of research on engineering meta-materials to function as optical nano-circuit devices. Under this new nano optical circuit theory, dielectric block, metal block, and material loss function as capacitor, inductor, and impedance, respectively in a sub-wavelength waveguide[2]. Stacking these basic elements forms optical devices analogy to the traditional RLC filter circuits. By further combining theεnear zero(ENZ) material, which is equivalent to the connector in traditional circuit, the optical phase in the transmitted signal can be preserved during propagation. Such new optical nano circuit with nano size devices operating at the optical wavelengths can provide much higher transmission bandwidth than the traditional electric circuits.
In this thesis, the concept of optical nano circuit theory was examined by Finite-Difference Time-Domain (FDTD) method. Filtering devices analogy to the traditional RLC circuits were analyzed and compared to the lumped circuit theory. First, we verified that the FDTD transmission spectrums of the optical nano circuit devices such as RC, RL, RLC parallel and RLC series circuits with simple dielectric and metal blocks in a sub-wavelength waveguide, match with the traditional lumped circuits theory[3]. We observed the shift of transmission peak by changing the size and the position of the lumped optical devices in the waveguide. Secondly, we modeled the optical nano connector withεnear zero(ENZ) material analogy to the wire in traditional circuit. Operating at plasma frequency, metal described by Drude model can function as ENZ material. The electromagnetic wave can be squeezed and almost 100% tunnel through narrow channel with arbitrary shape formed by such ENZ material without any phase change. However, the transmittance of ENZ material is limited by the overall area of the connector. In order to overcome such limit,εandμnear zero(EMNZ) material was introduced by inserting dielectric rods in the metal block as proposed by Engheta[].
To further understand the physical mechanism behind the EMNZ perfect transmission, we performed FDTD simulation to retrieve the effectiveε and μ of the EMNZ meta-material. Applying the Fabry-Perot model with 3-layered structure, effective impedance and refractive index can be obtained from the FDTD transmission and reflection coefficients. Consequently, effectiveε and μ can be uniquely determined. By analyzing EMNZ meta-material structure with such Reversed-FP method, we found that the 100% transmission was not caused by effective ε and μnear zero as first published by Engheta in 2007[4]. The real physical mechanism is the effective impedance match caused by such nano connector. This result could be used to verify the validity of optical nano-circuits theory and provide a new perspective on the future development of optical nano-circuits.

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[2] N.Engheta, A.Salandrino, A.Alu, “Circuit Elements at Optical Frequencies: Nanoinductors, Nanocapacitors, and Nanoresistors,” Phys. Rev. Lett., 95, 095504(2005)
[3] N.Engheta, M.E.Young, A.Alu, “Designs of Nanofilters for optical nanocircuits,” Phys. Rev. B, 77, 144107(2008)
[4] M.Silveirinha, N.Engheta, “Design of matched zero-index metamaterial using nonmagnetic inclusions in epsilon-near-zero media” Phys, Rev B, 75, 075119(2007)
[5] D.R.Smith, S.Schultz, P.Markos, C.M.Soukoulis “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients” Phys, Rev B, 65, 195104(2002)
[6] M.Silveirinha, N.Engheta, “Tunneling of Electromagnetic Energy through Subwavelength Channels and Bends using ε-Near-Zero Material,” Phys. Rev. Lett, 97, 157403(2006)
[7] M.Silveirinha, N.Engheta, A.Alu, M.G. Silveirinha, “Nanoinsulators and nanoconnectors for optical nanocircuits,” Journal of Applied Phys., 103, 064305(2008)
[8] N.Engheta, A.Alu, “Theory of Linear Chains of Meramaterial/Plasmonic Particles as Sub-Diffraction Optical Nanotransmission Lines” Phys. Rev B, 74, 205436()
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[11] B.Edward, A,Alu, M.Silveirinha, N.Engheta, “Reflectionless sharp bends and corners in waveguidesusing epsilon-near-zero effects” J.Appl. Phy, 105, 044905(2009)
[12] U S. Inan, A S. Inan “Electromagnetic waves” Prentice-Hall(2000), ISBN 0-201-36179-5