一般光學透鏡的成像會受到繞射極限的影響，使得解析度無法小於二分之一的波長。這在電子微影製程或是微生物觀察等應用方面，都會造成限制。因此做出一個突破繞射極限的光學透鏡，便成為人類長久以來的夢想。本論文所探討的雙曲透鏡，便是一種可以突破此限制的光學結構。
電磁波雙曲透鏡是利用金屬與介電質，週期層狀排列的圓柱狀結構。它讓物體的成像能突破繞射極限並具有放大效果。利用等效介質的觀念能幫助我們預測模擬結果的主要特徵。本論文除了介紹雙曲透鏡的基本理論、模擬方法以及電磁波與聲波的類似性外，還針對電磁波雙曲透鏡的特性做各種探討。此外，本論文還利用負的動力學質量密度的概念，成功地設計出一個聲波版本的雙曲透鏡，並透過數值模擬驗證其功能。期望本文的研究結果能對超音波掃描或水底聲納的解析提升做出一點貢獻。

Conventional optical lens suffers from a physical restriction called diffraction limit. It not only makes the resolution of a lens unable to be lowered than half a wavelength, but also leads to other constraints in photolithography process and micro-organism observation. Therefore, making an optical lens which can beat the diffraction limit is one of mankind''s dreams. Recently, hyperlens was proposed as a solution to overcome the diffraction limit. In this thesis, we study the optical characteristics of hyperlens thoroughly and explore its resolution ability.
EM hyperlens is a cylindrical structure consisting of periodically arranged metallic and dielectric material layers. It makes the image of a subwavelength object magnified and breaks the diffraction limit. The main feature of simulation result can be predicted by using the concept of effective medium. In this paper, we introduce the basic theories, simulation methods, the analogies between the electromagnetic and acoustic wave systems, and discuss various properties of EM hyperlens as well. Besides, we also propose an acoustic hyperlens based on the concept of negative dynamic density. Its hyperlens properties are confirmed successfully via numerical simulations. We hope the research results in this thesis can make contributions to improving the resolution of ultrasonics or sonar systems in the future.

[1] Abbe, E.” Arch. Mikroskop.” Anat. 9, 413-420(1873)
[2] V. G. Veselago, “ The electrodynamics of substances with simultaneously negative values of εand μ,” Sov. Phys. Usp. 10, 509 (1968).
[3] J. B. Pendry, “Negative Refraction Makes a Perfect Lens”, Phys. Rev. Lett. 85, 3966 (2000)
[4] J. T. Shen and P. M. Platzman, “Near field imaging with negative dielectric constant lenses,” Appl. Phys. Lett. 80, 3286–3288 (2002).
[5] F. D. M. Haldane, “Electromagnetic surface modes at interfaces with negative refractive index make a "notquite-perfect" lens,” cond-mat/0206420 (2002).
[6] R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[7] Nicholas Fang,Zhaowei Liu,Ta-Jen Yen,Xiang Zhang,” Regenerating evanescent waves from a silver superlens”Optics Express, vol 11, issue 7, pp.682-687(2003)
[8] Nicholas Fang, Hyesog Lee, Cheng Sun, Xiang Zhang, “Sub–Diffraction-Limited Optical Imaging with a Silver Superlens”,Science vol 308. p534(2005)
[9] Thomas Taubner,Dmitriy Korobkin,Yaroslav Urzhumov,Gennady Shvets,Rainer Hillenbrand“Near-Field Microscopy Through a SiC Superlens” Science vol 313. p1595(2006)
[10] Zhaowei Liu, Ste´phane Durant, Hyesog Lee, Yuri Pikus, Nicolas Fang, Yi Xiong,
Cheng Sun, and Xiang Zhang, “Far-Field Optical Superlens”, Nano Letters 7, 403 (2007)
[11] Nicholas Fang and Xiang Zhang,” Imaging properties of a metamaterial superlens” APPLIED PHYSICS LETTERS, Vol.82, No.2 161 (2003)
[12] Zubin Jacob, Leonid V. Alekseyev and Evgenii Narimanov,” Optical Hyperlens: Far-field imaging beyond the diffraction limit” Optics Express, vol 14, No.18 p.8247(2006)
[13] Zhaowei Liu, Hyesog Lee, Yi Xiong, Cheng Sun, Xiang Zhang,” Far-Field Optical Hyperlens Magnifying Sub-Diffraction-Limited Objects”, Science 315, 1686
(2007)
[14] Igor I. Smolyaninov, Yu-Ju Hung, Christopher C. Davis,” Magnifying Superlens in the Visible Frequency Range” Science vol. 315 (2007)
[15] Hu J.G,Cao Yong,Wang Pei,Lu,Y.H,Ming Hai,”Parameter Study on Far-Field Sub-Diffraction-Limited Hyperlens” Chin.Phys.Lett,Vol 25,No.11 4151 (2008)
[16] 張語軒，「雙曲透鏡之研究」，國立中央大學，碩士論文，民國97年。
[17] Zubin Jacob, Leonid V. Alekseyev, and Evgenii Narimanov, “Semiclassical theory of the hyperlens”, J.Opt. Soc. Am. A 24, 52 (2007)
[18] Alessandro Salandrino, Nader Engheta,”Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations” Physical Review B 74, 075103 (2006)
[19] Alexander V. Kildishev, Evgenii E. Narimanov,”Impedance-matched hyperlens” OPTICS LETTERS Vol.32, No.23(2007)
[20] Cheng-Ching Wang and Zhen Ye,” Spontaneous Emission in Cylindrical Periodically-Layered Structures” Phys. stat. sol. (a) 174, 527(1999)
[21] 欒丕綱、陳啟昌, “光子晶體-從蝴蝶翅膀到奈米光子學”，五南出版社，台灣，民國九十四年
[22] 黃建銘，「電磁波與聲波隱形斗篷之研究」，國立中央大學，碩士論文，民國97年。
[23] Ping Shenga, Jun Meia, Zhengyou Liub, Weijia Wena,”Dynamic mass density and acoustic metamaterials” Physica B 394, 256–261(2007)
[24] Zhengyou Liu, C. T. Chan, Ping Sheng,”Analytic model of phononic crystals with local resonances” Physical Review B 71, 014103 (2005)
[25] Jun Mei, Zhengyou Liu, Weijia Wen, Ping Sheng,” Effective dynamic mass density of composites” Physical Review B 76, 134205 (2007)
[26] Jun Mei, Zhengyou Liu, Weijia Wen, Ping Sheng,”Effective Mass Density of Fluid-Solid Composites” Phys. Review Lett. 96, 024301 (2006)
[27] Wei Wang, Hui Xing, Liang Fang, Yao Liu, Junxian Ma, Lan Lin, Changtao Wang, Xiangang Luo,” Far-field imaging device: planar hyperlens with
magnification using multi-layer metamaterial” Optics Express vol.16, No.25 21142(2008)
[28] Changtao Wang, Yanhui Zhao, Dachun Gan, Chunlei Du and Xiangang Luo,” Subwavelength imaging with anisotropic structure comprising alternately layered metal and dielectric films” Optics Express vol.16, No.6 4217(2008)
[29] HU Ji-Gang, WANG-Pei, LU Yong-Hua, MING Hai,CHEN Chun Chong, CHEN Jun-Xue,”Sub-di_raction-Limit Imaging in Optical Hyperlens”Chin.Phys.Lett. vol. 25, No.12 4439(2008)
[30] Alexander V. Kildishev, Uday K. Chettiar, Zubin Jacob, Vladimir M. Shalaev, Evgenii E. Narimanov,”Materializing a binary hyperlens design” APPLIED PHYSICS LETTERS 94, 071102(2009)
[31] Lin Chen and Guo Ping Wang,” Pyramid-shaped hyperlenses for threedimensional
subdiffraction optical imaging” Optics Express Vol.17, No.5 3903 (2009)
[32] Vladimir M. Shalaev,”Transforming Light” Science vol.322 384 (2008)
[33] Xianyu Ao and C. T. Chan,” Far-field image magnification for acoustic waves using anisotropic acoustic metamaterials” PHYSICAL REVIEW E 77,025601 (2008)