po50 v.2007/06/12 Prn:21/08/2007; 12:53 F:po50002.tex; VTEX/Jolanta p. 1 aid: 50002 pii: S0079-6638(07)50002-8 docsubty: REV 1 E. Wolf, Progress in Optics 50 1 © 2007 Elsevier B.V. 2 All rights reserved 2 3 3 4 4 5 5 6 6 7 Chapter 2 7 8 8 9 9 10 Conical diffraction: Hamilton’s diabolical point 10 11 at the heart of crystal optics 11 12 12 13 by 13 14 14 15 15 M.V. Berry, M.R. Jeffrey 16 16 17 17 H.H. Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 ISSN: 0079-6638 DOI: 10.1016/S0079-6638(07)50002-8 13 po50 v.2007/06/12 Prn:21/08/2007; 12:53 F:po50002.tex; VTEX/Jolanta p. 2 aid: 50002 pii: S0079-6638(07)50002-8 docsubty: REV 1 1 2 2 3 3 4 4 5 5 6 6 7 Contents 7 8 8 9 9 10 Page 10 11 § 1. Introduction . 15 11 12 12 § 2. Preliminaries: electromagnetism and the wave surface . 18 13 13 14 § 3. The diabolical singularity: Hamilton’s ray cone . 20 14 15 § 4. The bright ring of internal conical refraction . 23 15 16 16 17 § 5. Poggendorff’s dark ring, Raman’s bright spot . 26 17 18 § 6. Belsky and Khapalyuk’s exact paraxial theory of conical diffraction . 31 18 19 19 § 7. Consequences of conical diffraction theory . 34 20 20 21 § 8. Experiments . 41 21 22 § 9. Concluding remarks . 43 22 23 23 24 Acknowledgements . 45 24 25 Appendix 1: Paraxiality . 45 25 26 26 Appendix 2: Conical refraction and analyticity . 47 27 27 28 References . 48 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 14 po50 v.2007/06/12 Prn:21/08/2007; 12:53 F:po50002.tex; VTEX/Jolanta p. 3 aid: 50002 pii: S0079-6638(07)50002-8 docsubty: REV 1 1 2 2 3 3 4 4 5 5 6 6 7 § 1. Introduction. 7 8 8 9 The bicentenary of the birth in 1805 of William Rowan Hamilton (fig. 1) comes 9 10 at a time of renewed interest in the phenomenon of conical refraction (Born and 10 11 Wolf [1999], Landau, Lifshitz and Pitaevskii [1984]), in which a narrow beam of 11 12 12 light, incident along the optic axis of a biaxial crystal, spreads into a hollow cone 13 13 within the crystal, and emerges as a hollow cylinder. The prediction of the effect 14 14 in 1832 (Hamilton [1837]), and its observation by Lloyd (fig. 1) soon afterwards 15 15 (Lloyd [1837]), caused a sensation. For a non-technical account, see Lunney and 16 16 Weaire [2006]. 17 17 Our purpose here is to describe how the current understanding of conical re- 18 18 fraction has been reached after nearly two centuries of theoretical and experimen- 19 19 tal study. Although our treatment will be roughly historical, we do not adhere to 20 20 the practice, common in historical research, of describing each episode using only 21 21 22 sources and arguments from the period being studied. For the early history of con- 22 23 ical refraction, this has already been done, by Graves [1882] and O’Hara [1982]. 23 24 Rather, we will weave each aspect of the theory into the historical development in 24 25 ways more concordant with the current style of theoretical physics, hoping thus 25 26 to bring out connections with other phenomena in mathematics and physics. 26 27 There are many reasons why conical refraction is worth revisiting: 27 28 (i) It was an early (perhaps the first) example of a qualitatively new phe- 28 29 nomenon predicted by mathematical reasoning. By the early 1800s, it 29 30 was widely appreciated that mathematics is essential to understanding 30 31 the natural world. However, the phenomena to which mathematics had 31 32 been applied were already familiar (e.g., tides, eclipses, and planetary 32 33 orbits). Prediction of qualitatively new effects by mathematics may be 33 34 commonplace today, but in the 1830s it was startling. 34 35 (ii) With its intimate interplay of position and direction, conical refraction 35 36 was the first non-trivial application of phase space, and of what we now 36 37 call dynamics governed by a Hamiltonian. 37 38 (iii) Its observation provided powerful evidence confirming that light is a 38 39 transverse wave. 39 15 po50 v.2007/06/12 Prn:21/08/2007; 12:53 F:po50002.tex; VTEX/Jolanta p. 4 aid: 50002 pii: S0079-6638(07)50002-8 docsubty: REV 16 Conical diffraction and Hamilton’s diabolical point [2, § 1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 Fig. 1. Dramatis personae. 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 po50 v.2007/06/12 Prn:21/08/2007; 12:53 F:po50002.tex; VTEX/Jolanta p. 5 aid: 50002 pii: S0079-6638(07)50002-8 docsubty: REV 2, § 1] Introduction 17 1 (iv) Recently, it has become popular to study light through its singularities 1 2 (Berry [2001], Nye [1999], Soskin and Vasnetsov [2001]). In retrospect, 2 3 we see conical refraction as one of the first phenomena in singular po- 3 4 larization optics; another is the pattern of polarization in the blue sky 4 5 (Berry, Dennis and Lee [2004]). 5 6 (v) It was the first physical example of a conical intersection (diabolical 6 7 point) (Berry [1983], Uhlenbeck [1976], Berry and Wilkinson [1984]) 7 8 involving a degeneracy. Nowadays, conical intersections are popular in 8 9 theoretical chemistry, as spectral features indicating the breakdown of 9 10 the Born–Oppenheimer separation between fast electronic and slow nu- 10 11 clear freedoms (Cederbaum, Friedman, Ryaboy and Moiseyev [2003], 11 12 Herzberg and Longuet-Higgins [1963], Mead and Truhlar [1979]). By 12 13 analogy, conical refraction can be reinterpreted as an exactly solvable 13 14 model for quantum physics in the presence of a degeneracy. 14 15 (vi) The effect displays a subtle interplay of ray and wave physics. Although 15 16 its original prediction was geometrical (Sections 3 and 4), there are sev- 16 17 eral levels of geometrical optics (Sections 5 and 7), of which all except 17 18 the first require concepts from wave physics, and waves are essential to a 18 19 detailed understanding (Section 6). That is why we use the term conical 19 20 diffraction, and why the effect has taken so long to understand. 20 21 (vii) Analysis of the theory (Berry [2004b]) led to identification of an unex- 21 22 pected universal phenomenon in mathematical asymptotics: when expo- 22 23 nential contributions to a function compete, the smaller exponential can 23 24 dominate (Berry [2004a]). 24 25 (viii) There are extensions (Section 8) of the case studied by Hamilton, and 25 26 their theoretical understanding still presents challenges. Effects of chi- 26 27 rality (optical activity) have only recently been fully understood (Belsky 27 28 and Stepanov [2002], Berry and Jeffrey [2006a]), and further extensions 28 29 incorporate absorption (Berry and Jeffrey [2006b], Jeffrey [2007]) and 29 30 nonlinearity (Indik and Newell [2006]). 30 31 (ix) Conical diffraction is a continuing stimulus for experiments. Although 31 32 the fine details of Hamilton’s original phenomenon have now been ob- 32 33 served (Berry, Jeffrey and Lunney [2006]), predictions of new structures 33 34 that appear in the presence of chirality, absorption and nonlinearity re- 34 35 main untested. 35 36 (x) The story of conical diffraction, unfolding over 175 years, provides an 36 37 edifying contrast to the current emphasis on short-term science. 37 38 Although all results of the theory have been published before, some of our 38 39 ways of presenting them are original. In particular, after the exact treatment in 39 po50 v.2007/06/12 Prn:21/08/2007; 12:53 F:po50002.tex; VTEX/Jolanta p. 6 aid: 50002 pii: S0079-6638(07)50002-8 docsubty: REV 18 Conical diffraction and Hamilton’s diabolical point [2, § 2 1 Sections 2–4 we make systematic use of the simplifying approximation of parax- 1 2 iality. This is justified by the small angles involved in conical diffraction (Appen- 2 3 dix 1), and leads to what we hope are the simplest quantitative explanations of the 3 4 various phenomena. 4 5 5 6 6 7 § 2. Preliminaries: electromagnetism and the wave surface 7 8 8 9 Hamilton’s prediction was based on a singular property of the wave surface de- 9 10 scribing propagation in an anisotropic medium. Originally, this was formulated 10 11 in terms of Fresnel’s elastic-solid theory. Today it is natural to use Maxwell’s 11 12 electromagnetic theory. 12 13 For the physical fields in a homogeneous medium, we write plane waves with 13 14 wavevector k and frequency ω as 14 15 15 Re {Dk, Ek, Bk, Hk} exp i(k · r − ωt) , (2.1) 16 16 17 in which the vectors Dk, etc., are usually complex.