本論文探討光波在雙軸晶體中的圓錐折射 (conical refraction) 現象。以介紹錐型折射歷史與應用開始，帶入Fresnel's equation of wave normals，推導 normal surface與 index ellipsoid，藉此定性分析內圓錐折射，接續推導分析外圓錐折射，並比較內、外圓錐折射的關係與差異。
利用Hamilton's principle分析圓錐折射的入射光束，並解釋圓錐折射成像上的Double Bright Rings, Poggendorff's Dark Ring and Raman Spot。詳細推導 Belskii 和 Khapalyuk 的精確近軸理論 (the Belskii-Khapalyuk's exact paraxial theory)，模擬以此為主要方法，接著介紹近似方法以快速求得光強度。詳細解釋線偏振入射光，成像偏振態分布不同於過往之研究結果。最後探討若晶體具有光學活性 (optical activity) 與磁光效應 (magneto-optical effect) 情形下，入射光的光強度分布。推演解釋過程中皆有模擬圖、手繪圖與之對應。
第四章、首先模擬歷史上圓錐折射相關數據，再比較光波入射勻向 (isotropic)、單軸 (uniaxial) 與雙軸 (biaxial) 晶體的差別；接著模擬在焦平面上，隨機偏振、特殊偏振入射光的光型成像，後模擬隨距離不同的變化；比較與探討多種不同狀況的光強度分布趨勢。延伸模擬探討晶體具光學活性 (optical activity) 情形下，不同偏振態、不同旋性入射光，其成像的光強度分布、與趨勢變化。

In this thesis, we analyze in detail the phenomena of conical refraction in biaxial crystals. In chapter 2, we introduce the history and applications of conical refraction in the beginning. After that, we derive the equations of normal surface and index ellipsoid via Fresnel′s equation of wave normal and the energy density of the light, respectively, to study the internal conical refraction. Likewise, we derive in detail the external conical refraction based on the principle of duality. Finally, we compare the internal with external conical refractions and sum them up.
In the chapter 3, we start with exploiting Hamilton′s principle to resolve the incident beam of conical refraction and explains double bright rings, Poggendorff′s dark circle and Raman spot of conical refractive image. We then derive the main formulas of Belskii-Khapalyuk′s exact paraxial theory, and discuss the approximation method, which is the main reference of the simulation method used in this thesis. Most importantly, for linearly polarized light, we report and explain an interesting new finding concerning the angular distribution of the polarization state of the refracted light that is different from the usual result got in previous researches. Finally, we discuss the light intensity distribution in chiral and magneto-optical crystals for various polarization states of incident light.
In chapter 4, we simulate conical-refraction-related phenomena after introducing the simulation parameters. The intensity patterns on the focal plane for non-polarized and specific polarized light are both simulated. We also study how the intensity and polarization of the refracted light changes in the space when different polarization states of the incident light are considered. We finally analyze how the refracted light intensity for unpolarized and specific polarized incident light changes when optical activity is present.

摘要 I
Abstract II
謝誌 III
目錄 IV
圖目錄 VI
表目錄 IX
一、緒論 1
1-1 圓錐折射的歷史 1
1-2 圓錐折射的實際應用 3
1-3 光波的偏振 4
1-3-1 偏振概念與Jones-Vector偏振表示式 4
二、雙折射轉換至圓錐折射 7
2-1 電磁波傳播在非勻向介質 8
2-1-1 Normal Surface (K-Surface) 9
2-1-2 Index Ellipsoid (Optical Indicatrix) 13
2-1-3 單軸晶體 (Uniaxial Crystal) 15
2-1-4 雙軸晶體 (Biaxial Crystal) 18
2-2 Hamilton惡魔奇點 20
2-2-1 Hamilton 射線圓錐的特別角 21
2-2-2 波面與射線面 25
2-3 內、外圓錐折射的研究 29
2-3-1 內圓錐折射 30
2-3-2 外圓錐折射 31
2-3-3 內、外圓錐折射小結 34
三、設定圓錐折射的模擬參數 36
3-1 圓錐折射的理論概要 36
3-1-1 雙亮環、 Poggendorff暗環與Raman Spot 36
3-1-2 光強度分布 41
3-2 Belskii and Khapalyuk的精確近軸理論 44
3-2-1 非手徵性雙軸晶體 44
3-2-2 錐形折射近似 51
3-2-3 偏振狀態的研析 54
3-3 手徵性或Faraday旋轉對圓錐折射的影響 57
3-3-1 手徵性圓錐折射 (Optical Activity) 57
3-3-2 磁光圓錐折射 (Faraday Rotation) 63
四、圓錐折射的模擬結果與分析 66
4-1 成像的關鍵性參數 68
4-2 非偏振入射光的光型成像 73
4-2-1 如何獲得清晰影像？ 73
4-2-2 當光傳播在不同距離 ？ 78
4-3 偏振入射光光型成像 86
4-3-1 焦平面成像 86
4-3-2 不同位置 成像圖 91
4-4 光波入射手徵性晶體的光型成像 95
4-4-1 非偏振入射光入射 95
4-4-2 非偏振入射光在不同位置的光型成像 97
4-4-3 偏振入射光入射 104
4-4-4 偏振入射光在不同位置的光型成像 106
五、結論與未來展望 114
5-1 結論 114
5-2 未來展望 116
參考文獻 117

[1] ‶Hamilton, William Rowan,″ Complete Dictionary of Scientific Biography. Encyclopedia. com. 27 Jun. (2019).
[2] Hamilton, W.R., ‶Theory of Systems of Rays,″ Trans. Royal Irish Acad. 15, 69-174 (1828).
[3] Hamilton, W.R., ‶Supplement to an Essay on the Theory of Systems of Rays,″ Trans. Royal Irish Acad. 16, 1-61 (1830).
[4] Hamilton, W.R., ‶Second Supplement to an Essay on the Theory of Systems of Rays,″ Trans. Royal Irish Acad. 16, 93-125 (1831).
[5] Hamilton W.R., ‶Third Supplement to an Essay on the Theory of Systems of Rays," Trans. Royal Irish Acad. 17, 1-144 (1837).
[6] Lloyd, H., ‶Further Experiments on the Phænomena presented by Light in its Passage along the Axes of Biaxal Crystals, ″ Philosophical Magazine, 3rd series, volume 2, pp. 207-210 (1833).
[7] Hamilton, W.R., ‶On a General Method of expressing the Paths of Light and of the Planets by the Coefficients of a Characteristic Function″ Dublin University Review and Quarterly Magazine, Vol. I, pp. 795-826 (1833).
[8] Lloyd, H., ‶On Conical Refraction″ Report of the Third Meeting of the British Association for the Advancement of Science held at Cambridge in, 370-373 (1833).
[9] Berry, M. V. & Jeffrey, M. R., ‶Conical diffraction: Hamilton's diabolical point at the
heart of crystal optics″ submitted to Progress in Optics 50 (2006).
[10] Poggendorff, J. C., ‶Ueber die konische Refraction, Pogg. Ann. 48, 461-462,″ (1839).
[11] Potter, R., ‶An examination of the phenomena of conical refraction in biaxial crystals,″
Phil. Mag. 8, 343-353 (1841).
[12] Voigt, W., ‶Bemerkung zur Theorie der konischen Refraktion,″ Phys. Z. 6, 672-673 (1905).
[13] Voigt, W., ‶Theoretisches unt Experimentalles zur Aufklaerung des optischen Verhaltens
aktiver Kristalle Ann,″ Phys. 18, 645-694 (1905).
[14] C. V. Raman, V. S. Rajagopalan, and T. M. K. Nedungadi, ‶Conical refraction in
naphthalene crystals,″ Proc. Indian Ins. Sci. A 14, 221-227 (1941).
[15] C. V. Raman, and T. M. K. Nedungadi, ‶Optical images formed by conical refraction,″
Nature 149(3785), 552-553 (1942).
[16] A. J. Schellt and N. Bloembergen, ‶Laser studies of internal conical diffraction. I.
Quantitative Comparison of experimental and theoretical conical intensity distribution in
aragonite,″ J Opt Soc Am. 68: 1093-1098 (1978).
[17] A. J. Schellt and N. Bloembergen, ‶Laser studies of internal conical diffraction. II.
Intensity patterns in an optically active crystal, α-iodic acid,″ J. Opt. Soc. Am., Vol. 68,
No. 8, August (1978).
[18] R. Indik and A. C. Newell, ‶Conical refraction and nonlinearity,″ Optics Express Vol.
14, Issue 22, pp. 10614-10620 (2006).
[19] Éamon Lalor, ‶An analytical approach to the theory of internal conical refraction,″ J. Math.
Phys. 13,449-454 (1972).
[20] A. M. Belskii and A. P. Khapalyuk, ‶Internal conical refraction of bounded light beams in
biaxial crystals,″ Opt Spectrosc (USSR) 44, 436-439 (1978).
[21] Warnick, K.F., Arnold, D.V., ‶Secondary dark rings of internal conical refraction,″ Phys.
Rev. E 55, 6092-6096 (1997).
[22] J. P. Fève, B. Boulanger and G. Marnier, ‶Experimental study of internal and external
conical refraction in KTP,″ Opt. Commun. 105, 243-252 (1994).
[23] Jeffrey, M. R., ‶Conical Diffraction: Complexifying Hamilton’s Diabolical Legacy,″ Ph.D.
Thesis, University of Bristol (2007).
[24] O’Dwyer, D. P. et al. ‶Generation of continuously tunable fractional optical orbital angular
momentum using internal conical diffraction,″ Opt. Express 18, 16480-16485 (2010).
[25] D. P. O’Dwyer, ‶Optical trapping using cascade conical refraction of light,″ Optics Express
Vol. 20, No. 19 (2012).
[26] O’Dwyer, D. P., Phelan, C. F., Ballantine, K. E., Rakovich, Y. P., Lunney, J. G., and
Donegan, J. F., ‶Conical diffraction of linearly polarised light controls the angular position
of a microscopic object,″ Opt. Express, Vol. 18, pp. 27319-27326 (2010).
[27] Phelan, C. F., Donegan, J. F., and Lunney, J. G., ‶Generation of a radially polarized light
beam using internal conical diffraction,″ Opt. Express, Vol. 19, pp. 21793-21802 (2011).
[28] https://www.laserfocusworld.com/optics/article/16549780/conical-refraction-becomes-
practical
[29] Shih, H., Bloembergen, N., ‶Conical refraction in second harmonic generation,″ Phys. Rev.
184, pp. 895-904 (1969).
[30] VCT, Home page of Vision Crystal Technology, http://www.vct-ag.com
[31] A. Peinado, A. Turpin, A. Lizana, E. Fernández, J. Mompart, and J. Campos, ‶Conical
refraction as a tool for polarization metrology,″ Opt. Lett. 38, 4100-4103 (2013).
[32] Robert Perceval Graves., ‶Life of Sir William Rowan Hamilton,″ Chapter XIII, Vol. 1,
Hodges Figgis, Dublin (1882).
[33] A. Yariv, P. Yeh, ‶Optical Waves in Crystals: Propagation and Control of Laser Radiation″
Wiley Interscience (2002).
[34] B. E. A. Saleh ‶Fundamentals of Photonics″ Second Edition.
[35] M. V. Berry, and M. R. Jeffrey, ‶Chiral conical diffraction,″ J. Opt. A 8, 363-372 (2006).
[36] Landau, L.D., Lifshitz, E.M., Pitaevskii, L.P., ‶Electrodynamics of Continuous Media,″
Pergamon, Oxford (1984).
[37] Berry, M.V., Jeffrey, M.R., Lunney, J.G., ‶Conical diffraction: observations and theory,″
Proc. R. Soc. A 462, 1629-1642 (2006).
[38] Yu. P. Mikhailichenko, ‶Conical refraction: Experiments and large-scale demonstrations,″
Russian Physics Journal, Vol. 50, No. 8, pp 788-795, August (2007).
[39] James G. Lunney and Denis Weaire, ‶The ins and outs of conical refraction,″ School of
Physics, Trinity College Dublin, Ireland
[40] Born, M., Wolf, E., ‶Principles of Optics,″ seventh ed., Pergamon, London (1999).
[41] Mikhailychenko Y P ‶Large scale demonstrations on conical refraction,″ Russian Physics
Journal, Vol. 50, No. 8 (2007).
[42] Berry, M.V, ‶Conical diffraction asymptotics: fine structure of Poggendorff rings and axial
spike,″ J. Optics A 6, 289-300 (2004b).
[43] G. N. Watson, ‶A Treatise on the Theory of Bessel Functions,″ 2nd Edition, Cambridge
University Press, Cambridge (1944).
[44] A. Belafhal, ‶Theoretical intensity distribution of internal conical refraction,″ Opt. Commun. 178(4-6), 257-265 (2000).
[45] E. V. Kuznetsov and A. M. Merzlikin, ‶Conical refraction in a magneto-optical biaxial
crystal,″ J. Opt. 19 (2017).
[46] Portigal, D. L. & Burstein, E., ‶Effect of Optical Activity or Faraday Rotation on Internal
Conical Refraction″ J.Opt.Soc.Amer. 62, 859-864 (1972).
[47] Amin Abdolvand, Keith G. Wilcox, Todor K. Kalkandjiev and Edik U. Rafailov, ‶Conical
refraction Nd:KGd(WO4)2 laser″ Optics Express Vol. 18, Issue 3, pp. 2753-2759 (2010).
[48] Todor Kirilov Kalkandjiev ‶Conical refraction: an experimental introduction″ Proceedings
of SPIE - The International Society for Optical Engineering, May (2008).
[49] Jeffrey, M.R., ‶The spun cusp complexified: complex ray focusing in chiral conical diffraction,″ J. Opt. A. 9, 634-641 (2007).