{{Short description|Optical phenomenon}} thumb|Surface of wavevectors when three principal refractive indices are <math>1, 0.5, 1.5</math>. thumb|The surface of wavevectors. It has two sheets that intersect at 4 conoidal points. '''Conical refraction''' is an optical phenomenon in which a ray of light, passing through a biaxial crystal along certain directions, is refracted into a hollow cone of light. There are two possible conical refractions, one internal and one external. For internal refraction, there are 4 directions, and for external refraction, there are 4 other directions.
For internal conical refraction, a planar wave of light enters an aperture a slab of biaxial crystal whose face is parallel to the plane of light. Inside the slab, the light splits into a hollow cone of light rays. Upon exiting the slab, the hollow cone turns into a hollow cylinder.
For external conical refraction, light is focused at a single point aperture on the slab of biaxial crystal, and exits the slab at the other side at an exit point aperture. Upon exiting, the light splits into a hollow cone.
This effect was predicted in 1832 by William Rowan Hamilton<ref name=":2">{{Cite journal |last=Hamilton |first=William R. |date=1832 |title=Third Supplement to an Essay on the Theory of Systems of Rays |journal=The Transactions of the Royal Irish Academy |volume=17 |pages=v–144 |jstor=30078785 |issn=0790-8113}}</ref> and subsequently observed by Humphrey Lloyd in the next year.<ref name=":5">{{Cite journal |last=Lloyd |first=Humphrey |date=1831 |title=On the Phenomena Presented by Light in Its Passage along the Axes of Biaxal Crystals |journal=The Transactions of the Royal Irish Academy |volume=17 |pages=145–157 |jstor=30078786 |issn=0790-8113}}</ref> It was possibly the first example of a phenomenon predicted by mathematical reasoning and later confirmed by experiment.<ref name=":0">{{cite book |last1=Berry |first1=M. V. |title=Progress in Optics Volume 50 |date=2007-01-01 |volume=50 |pages=13–50 |editor-last=Wolf |editor-first=E. |chapter-url=https://www.sciencedirect.com/science/article/pii/S0079663807500028 |access-date=2024-04-23 |publisher=Elsevier |last2=Jeffrey |first2=M. R. |chapter=Chapter 2 Conical diffraction: Hamilton's diabolical point at the heart of crystal optics |doi=10.1016/S0079-6638(07)50002-8 |bibcode=2007PrOpt..50...13B |isbn=978-0-444-53023-3 }}</ref>
== History == {{see also|Physical crystallography before X-rays#Conical refraction}} The phenomenon of double refraction was discovered in the Iceland spar (calcite), by Erasmus Bartholin in 1669. was initially explained by Christiaan Huygens using a wave theory of light. The explanation was a centerpiece of his ''Treatise on Light'' (1690). However, his theory was limited to uniaxial crystals, and could not account for the behavior of biaxial crystals. inside the sphere.
In 1813, David Brewster discovered that topaz has two axes of no double refraction, and subsequently others, such as aragonite, borax, and mica, were identified as biaxial. Explaining this was beyond Huygens' theory.<ref name=":4">{{Cite journal |last=O'Hara |first=J. G. |date=1982 |title=The Prediction and Discovery of Conical Refraction by William Rowan Hamilton and Humphrey Lloyd (1832-1833) |journal=Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences |volume=82A |issue=2 |pages=231–257 |jstor=20489156 |issn=0035-8975}}</ref>
At the same period, Augustin-Jean Fresnel developed a more comprehensive theory that could describe double refraction in both uniaxial and biaxial crystals. Fresnel had already derived the equation for the wavevector surface in 1823, and André-Marie Ampère rederived it in 1828.<ref>Ampère, A. M. 1828. Mémoire sur la détermination de la surface courbe des ondes lumineuses dans un milieu dont l'élasticité est différente suivant les trois directions principales. Ann. Chim. Phys. 39, 113--45</ref> Many others investigated the wavevector surface of the biaxial crystal, but they all missed its physical implications. In particular, Fresnel mistakenly thought the two sheets of the wavevector surface are ''tangent'' at the singular points (by a mistaken analogy with the case of uniaxial crystals), rather than conoidal.<ref name=":4" />
William Rowan Hamilton, in his work on Hamiltonian optics, discovered the wavevector surface has four conoidal points and four tangent conics.<ref name=":2" /> These conoidal points and tangent conics imply that, under certain conditions, a ray of light could be refracted into a cone of light within the crystal. He termed this phenomenon "conical refraction" and predicted two distinct types: internal and external conical refraction, corresponding respectively to the conoidal points and tangent conics.
Hamilton announced his discovery at the Royal Irish Academy on October 22, 1832. He then asked Humphrey Lloyd to prove this experimentally. Lloyd observed external conical refraction on 14 December with a specimen of aragonite from the Dollonds, which he published in February.<ref>Lloyd, Humphrey. "XXI. On the phænomena presented by light in its passage along the axes of biaxal crystals." ''The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science'' 2.8 (1833): 112-120.</ref> He then observed internal conical refraction and published it in March.<ref>Lloyd, Humphrey. "XXXIII. Further experiments on the phænomena presented by light in its passage along the axes of biaxal crystals." ''The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science'' 2.9 (1833): 207-210.</ref> Lloyd then combined both reports and added details into one paper.<ref name=":5" />
Lloyd discovered experimentally that the refracted rays are polarized, with polarization angle half that of the turning angle (see below), and told Hamilton about it, who then explained theoretically.
At the same time, Hamilton also exchanged letters with George Biddell Airy. Airy had independently discovered that the two sheets touch at conoidal points (rather than tangentially), but he was skeptical that this would have experimental consequences. He was only convinced after Lloyd's report.<ref name=":4" /><ref name=":6">{{Cite journal |date=January 1932 |title=Discovery of Conical Refraction by William Rowan Hamilton and Humphrey Lloyd (1833) |url=https://www.journals.uchicago.edu/doi/10.1086/346641 |journal=Isis |volume=17 |issue=1 |pages=154–170 |doi=10.1086/346641 |issn=0021-1753|url-access=subscription }}</ref>
This discovery was a significant victory for the wave theory of light and solidified Fresnel's theory of double refraction.<ref name=":0" /> Lloyd's experimental data are described in <ref name=":1" /> pages 350–355. <blockquote>The rays of the internal cone emerged, as they ought, in a cylinder from the second face of the crystal; and the size of this nearly circular cylinder, though small, was decidedly perceptible, so that with solar light it threw on silver paper a little luminous ring, which seemed to remain the same at different distances of the paper from the arragonite.<ref name=":3" /> </blockquote>In 1833, James MacCullagh claimed that it is a special case of a theorem he published in 1830<ref>{{Cite journal |last=Mac Cullagh |first=James |date=1830 |title=On the Double Refraction of Light in a Crystallized Medium, according to the Principles of Fresnel |journal=The Transactions of the Royal Irish Academy |volume=16 |pages=65–78 |jstor=30079025 |issn=0790-8113}}</ref> that he did not explicate, since it was not relevant to that particular paper.<ref>{{Cite journal |last=Flood |first=Raymond |author-link=Raymond Flood (mathematician) |date=2006 |title=Mathematics in Victorian Ireland |url=http://www.tandfonline.com/doi/abs/10.1080/17498430600964433 |journal=BSHM Bulletin: Journal of the British Society for the History of Mathematics |volume=21 |issue=3 |pages=200–211 |doi=10.1080/17498430600964433 |issn=1749-8430 |s2cid=122564180|url-access=subscription }}</ref> Augustin-Louis Cauchy discovered the same surface in the context of classical mechanics.<ref name=":3">Hamilton, William Rowan. "On some results of the view of a characteristic function in optics." ''British Association Report, Cambridge'' (1833): 360-370.</ref><blockquote>Somebody having remarked, "I know of no person who has not seen conical refraction that really believed in it. I have myself converted a score of mathematicians by showing them the cone of light". Hamilton replied, "How different from me! If I had seen it only, I should not have believed it. My eyes have too often deceived me. I believe it, because I have proved it."<ref name=":6" /></blockquote>thumb|External conical refraction thumb|Internal conical refraction
== Geometric theory == A note on terminology: The surface of wavevectors is also called the wave surface, the surface of normal slowness, the surface of wave slowness, etc. The index ellipsoid was called the surface of elasticity, as according to Fresnel, light waves are transverse waves in, in exact analogy with transverse elastic waves in a material.
=== Surface of wavevectors === For notational cleanness, define <math>a = n_x^{-2} - n_y^{-2}, b = n_y^{-2} - n_z^{-2}</math>. This surface is also known as Fresnel wave surface.
Given a biaxial crystal with the three principal refractive indices <math>n_x < n_y < n_z</math>. For each possible direction <math>\hat k</math> of planar waves propagating in the crystal, it has a certain group velocity <math>v_g(\hat k)</math>. The refractive index along that direction is defined as <math>n(\hat k) = c / v_g(\hat k)</math>.
Define, now, the surface of wavevectors as the following set of points<math display="block">\{n(\hat k) \hat k : \hat k \in \text{sphere of radius 1}\}</math>In general, there are two group velocities along each wavevector direction. To find them, draw the plane perpendicular to <math>\hat k</math>. The indices are the major and minor axes of the ellipse of intersection between the plane and the index ellipsoid. At precisely 4 directions, the intersection is a circle (those are the axes where double refraction disappears, as discovered by Brewster, thus earning them the name of "biaxial"), and the two sheets of the surface of wavevectors collide at a conoidal point.
To be more precise, the surface of wavevectors satisfy the following degree-4 equation (,<ref name=":1">{{Cite book |last1=Preston |first1=Thomas |url=http://archive.org/details/theoryoflight00presrich |title=The theory of light |last2=Thrift |first2=William Edward |orig-date=1912 |date=1924 |location=London |publisher=Macmillan and Co. |via=University of California Libraries}}</ref> page 346):<math display="block">(k_x^2 + k_y^2 + k_z^2) (n_x^2k_x^2 + n_y^2k_y^2 + n_z^2k_z^2) - (n_y^2 + n_z^2) n_x^2 k_x^2 - (n_z^2 + n_x^2) n_y^2 k_y^2- (n_x^2 + n_y^2) n_z^2 k_z^2 + (n_xn_yn_z)^2 = 0 </math>or equivalently,<math display="block">\sum_i \frac{n_i^2 k_i^2}{ \|k\|^2 -n_i^2} = 0</math>{{hidden begin|style=width:100%|ta1=center|border=1px #aaa solid|title=[Proof]}} The major and minor axes are the solutions to the constraint optimization problem: <math display="block"> \begin{cases} k^T r &= 0 \quad & k\perp r \\ r^T M r &= 1 \quad &r\text{ is on the index ellipsoid} \\ \mathrm{exr}(r^T r) & \quad &\|r\| \text{is max/minimized} \end{cases} </math> where <math>M</math> is the matrix with diagonal entries <math>n_x^{-2}, n_y^{-2}, n_z^{-2}</math>.
Since there are 3 variables and 2 constraints, we can use the Karush–Kuhn–Tucker conditions. That is, the three gradients <math display="inline">k, Mr, r</math> are linearly dependent.
Let <math display="inline">Mr = \alpha k + \beta r</math>, then we have<math display="block"> 0 = \alpha k^T r = r^T M r - \beta r^T r \implies \beta = (r^T r)^{-1} </math>Plugging <math display="inline">r_x = \frac{\alpha k_x}{n_x^{-2} - \beta}, \dots</math> back to <math display="inline">k^T r = 0</math>, we obtain <math display="block"> \sum_i \frac{k_i^2}{n_i^{-2} - \|r\|^{-2}} = 0 </math>Let <math display="inline">\vec k</math> be the vector with the direction of <math display="inline">(k_x, k_y, k_z)</math>, and the length of <math display="inline">\|r\|</math>. We thus find that the equation of <math display="inline">\vec k</math> is <math display="block"> \sum_i \frac{k_i^2}{n_i^{-2} - \|\vec k\|^{-2}} = 0 </math>Multiply out the denominators, then multiply by <math display="inline">n_x^2n_y^2n_z^2</math>, we obtain the result. {{hidden end}}
In general, along a fixed direction <math display="inline">\hat k </math>, there are two possible wavevectors: The slow wave <math display="inline">n_+ \hat k</math> and the fast wave <math display="inline">n_- \hat k</math>, where <math display="inline">n_+</math> is the major semiaxis, and <math display="inline">n_-</math> is the minor.
Plugging <math display="inline">\vec k = n\hat k</math> into the equation of <math display="inline">\vec k</math>, we obtain a quadratic equation in <math display="inline">n^2</math>:<math display="block"> \left(\frac{k_x^2}{n_y^2n_z^2} + \cdots \right) n^4 - \left(\frac{k_x^2}{n_y^2} +\frac{k_x^2}{n_z^2} + \cdots \right) n^2 + 1 =0 </math>which has two solutions <math>n_-^2, n_+^2</math>. At exactly four directions, the two wavevectors coincide, because the plane perpendicular to <math display="inline">\hat k </math> intersects the index ellipsoid at a circle. These directions are <math display="inline">\hat k = (\pm \cos \theta, 0, \pm \sin \theta)</math> where <math display="inline">\sin^2 \theta = \frac{b}{a+b} </math>, at which point <math display="inline">n_- = n_+ = n_y</math>.
Expanding the equation of the surface in a neighborhood of <math display="inline">\vec k = ( n_y \cos \theta, 0, n_y \sin \theta)</math>, we obtain the local geometry of the surface, which is a cone subtended by a circle.
Further, there exists 4 planes, each of which is tangent to the surface at an entire circle (a ''trope'' ''conic'', as defined later). These planes have equation (,<ref name=":1" /> pages 349–350)<math display="block">k_x \sqrt{n_y^2 - n_x^2} \pm k_z \sqrt{n_z^2 - n_y^2} = \pm n_y \sqrt{n_z^2 - n_x^2} </math>or equivalently, <math display="inline">n_x k_x \sqrt{a} \pm n_zk_z\sqrt{b} = \pm n_xn_z \sqrt{a+b} </math>. and the 4 circles are the intersection of those planes with the ellipsoid<math display="block">(n_x^2 + n_y^2)k_x^2 + 2n_y^2k_y^2 + (n_z^2 + n_y^2)k_z^2 - (n_x^2 + n_z^2)n_y^2 = 0 </math>All 4 circles have radius <math>n_y^{-1}\sqrt{(n_y^2 - n_x^2) (n_z^2 - n_y^2)} = n_xn_z \sqrt{ab}</math>.
{{hidden begin|style=width:100%|ta1=center|border=1px #aaa solid|title=[Proof]}}By differentiating its equation, we find that the points on the surface of wavevectors, where the tangent plane is parallel to the <math display="inline">k_y</math> -axis, satisfies <math display="block">k_y ((n_x^2 + n_y^2)k_x^2 + 2n_y^2k_y^2 + (n_z^2 + n_y^2)k_z^2 - (n_x^2 + n_z^2)n_y^2) = 0</math> That is, it is the union of the <math display="inline">k_xk_z</math> -plane, and an ellipsoid.
Thus, such points on the surface of wavevectors has two parts: Every point with <math display="inline">k_y = 0</math>, and every point that intersects with the auxiliary ellipsoid <math display="block"> (n_x^2 + n_y^2)k_x^2 + 2n_y^2k_y^2 + (n_z^2 + n_y^2)k_z^2 - (n_x^2 + n_z^2)n_y^2 = 0 </math>
Using the equation of the auxiliary ellipsoid to eliminate <math display="inline">k_y^2</math> from the equation of the wavevector surface, we obtain another degree-4 equation, which splits into the product of 4 planes: <math display="block"> k_z \pm k_x \sqrt{\frac{n_y^2 -n_x^2}{n_z^2 - n_y^2}} \pm n_y \sqrt{\frac{n_z^2 -n_x^2}{n_z^2 - n_y^2}} </math>
Thus, we obtain 4 ellipses: the 4 planar intersections with the auxiliary ellipsoid. These ellipses all exist on the wavevector surface, and the wavevector surface has a tangent plane parallel to the <math display="inline">k_y</math> axis at those points. By direct computation, these ellipses are circles.
It remains to verify that the tangent plane is also parallel to the plane of the circle.
Let <math display="inline">P_0</math> be one of those 4 planes, and let <math display="inline">\vec k</math> be one point on the circle in <math display="inline">P_0</math>. If <math display="inline">k_y \neq 0</math>, then since the circle is on the surface, the tangent plane <math display="inline">P</math> to the surface at <math display="inline">\vec k</math> must contain the tangent line <math display="inline">l</math> to the circle at <math display="inline">\vec k</math>. Also, the plane <math display="inline">P</math> must also contain <math display="inline">l_y</math>, the line pass <math display="inline">\vec k</math> that is parallel to the <math display="inline">k_y</math> -axis. Therefore, the plane <math display="inline">P</math> is spanned by <math display="inline">l_y</math> and <math display="inline">l</math>, which is precisely the plane <math display="inline">P_0</math>. This then extends by continuity to the case of <math display="inline">k_y = 0</math>.{{hidden end}}
One can imagine the surface as a prune, with 4 little pits or dimples. Placing the prune on a flat desk, the prune would touch the desk at a point on a circle that covers a dimple.
In summary, the surface of wavevectors has singular points at <math display="inline">\hat k = (\pm \cos \theta, 0, \pm \sin \theta)</math> where <math display="inline">\theta = \arctan \sqrt{b/a} </math>. The special tangent plane to the surface touches it at two points that make an angle of <math>\arctan \frac{n_x}{n_z}\sqrt{b/a}</math> and <math>\arctan \frac{n_z}{n_x}\sqrt{b/a}</math>, respectively.<ref name=":7" />
The angle of the wave cone, that is, the angle of the cone of internal conical refraction, is <math display="inline">A_{internal} = \arctan n_y^2 \sqrt{ab}</math>.<ref name=":7" /> Note that the cone is an ''oblique'' cone. Its apex is perpendicular to its base at a point ''on'' the circle (instead of the center of the circle).
=== Surface of ray vectors === The surface of ray vectors is the polar dual surface of the surface of wavevectors. Its equation is obtained by replacing <math>n_i</math> with <math>n_i^{-1}</math> in the equation for the surface of wavevectors.<ref name=":2" /> That is,<math display="block">(r_x^2 + r_y^2 + r_z^2) (n_x^{-2}r_x^2 + n_y^{-2}r_y^2 + n_z^{-2}r_z^2) - (n_y^{-2} + n_z^{-2}) n_x^{-2} r_x^2 - (n_z^{-2} + n_x^{-2}) n_y^{-2} r_y^2- (n_x^{-2} + n_y^{-2}) n_z^{-2} r_z^2 + (n_xn_yn_z)^2 = 0 </math>All the results above apply with the same modification. The two surfaces are related by their duality: * The four special planes tangent to the surface of wavevectors on a circle correspond to the 4 conoidal points on the surface of ray vectors. * The four conoidal points on the surface of wavevectors correspond to the 4 planes tangent to the surface of ray vectors on a circle.
=== Approximately circular === In typical crystals, the difference between <math>n_x, n_y, n_z</math> is small. In this case, the conoidal point is approximately at the center of the tangent circle surrounding it. Thus, the cone of light (in both the internal and the external refraction cases) is approximately circular.
=== Polarization === [[File:Ellipso-KL-NP squashed.svg|thumb|Take the unit sphere. Each point on it is a possible direction of the wavevector. For each, plot the direction of the major and minor axes of the ellipse intersection. This results in two families of curves intersecting orthogonally on the unit sphere, with 4 singularities. The graph is topologically the same as that of the umbilical point of a generic ellipsoid.]] thumb|The index of the vector field is 1/2, which explains why the direction of polarization turns by half of <math>\phi</math>.
In the case of external conical refraction, we have one ray splitting into a cone of planar waves, each corresponding to a point on the tangent circle of the wavevector surface. There is one tangent circle for each of the four quadrants. Take the one with <math>k_x, k_z > 0</math>, then take a point on it. Let the point be <math>\vec k</math>.
To find the polarization direction of the planar wave in direction <math>\vec k</math>, take the intersection of the index ellipsoid and the plane perpendicular to <math>\vec k</math>. The polarization direction is the direction of the major axis of the ellipse intersection between the plane perpendicular to <math>\vec k</math> and the index ellipsoid.
Thus, the <math>\vec k</math> with the highest <math>k_z</math> corresponds to a light polarized parallel to the <math>k_y</math> direction, and the <math>\vec k</math> with the lowest <math>k_z</math> corresponds to a light polarized in a direction perpendicular to it. In general, rotating along the circle of light by an angle of <math>\phi</math> would rotate the polarization direction by approximately <math>\phi/2</math>.
This means that turning around the cone an entire round would turn the polarization angle by only half a round. This is an early example of the geometric phase.<ref>{{Cite journal |last1=Cohen |first1=Eliahu |last2=Larocque |first2=Hugo |last3=Bouchard |first3=Frédéric |last4=Nejadsattari |first4=Farshad |last5=Gefen |first5=Yuval |last6=Karimi |first6=Ebrahim |date=July 2019 |title=Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond |url=https://www.nature.com/articles/s42254-019-0071-1 |journal=Nature Reviews Physics |language=en |volume=1 |issue=7 |pages=437–449 |doi=10.1038/s42254-019-0071-1 |issn=2522-5820|arxiv=1912.12596 |bibcode=2019NatRP...1..437C }}</ref><ref>{{Cite journal |last=Berry |first=Michael |date=1990-12-01 |title=Anticipations of the Geometric Phase |url=https://pubs.aip.org/physicstoday/article/43/12/34/405932/Anticipations-of-the-Geometric-PhaseThe-notion |journal=Physics Today |language=en |volume=43 |issue=12 |pages=34–40 |doi=10.1063/1.881219 |bibcode=1990PhT....43l..34B |issn=0031-9228|url-access=subscription }}</ref> This geometric phase of <math>\pi</math> is observable in the difference of the angular momentum of the beam, before and after conical refraction.<ref>{{Cite journal |last1=Berry |first1=M V |last2=Jeffrey |first2=M R |last3=Mansuripur |first3=M |date=2005-11-01 |title=Orbital and spin angular momentum in conical diffraction |url=https://iopscience.iop.org/article/10.1088/1464-4258/7/11/011 |journal=Journal of Optics A: Pure and Applied Optics |volume=7 |issue=11 |pages=685–690 |doi=10.1088/1464-4258/7/11/011 |bibcode=2005JOptA...7..685B |issn=1464-4258|url-access=subscription }}</ref>
=== Algebraic geometry === {{Main|Wave surface}}
The surface of wavevectors is defined by a degree-4 algebraic equation, and thus was studied for its own sake in classical algebraic geometry.
Arthur Cayley studied the surface in 1849.<ref>Cayley, A. "[http://www.numdam.org/item/JMPA_1846_1_11__291_0.pdf Sur la surface des ondes]." ''Journal de Mathématiques Pures et Appliquées'' 11 (1846): 291-296.</ref> He described it as a degenerate case of ''tetrahedroid quartic surfaces''. These surfaces are defined as those that are intersected by four planes, forming a tetrahedron. Each plane intersects the surface at two conics. For the wavevector surface, the tetrahedron degenerates into a flat square. The three vertices of the tetrahedron are conjugate to the two conics within the face they define. The two conics intersect at 4 points, giving 16 singular points.<ref>{{cite arXiv |last=Dolgachev |first=Igor |title=Kummer surfaces: 200 years of study |date=2019-10-16 |class=math.AG |eprint=1910.07650}}</ref>
In general, the surface of wavevectors is a Kummer surface, and all properties of it apply. For example:<ref>{{Cite book |last=Dolgachev |first=I. |title=Classical algebraic geometry: a modern view |date=2012 |publisher=Cambridge University Press |isbn=978-1-107-01765-8 |location=Cambridge; New York |chapter=10.3.3 Kummer surfaces}}</ref>
* It is projectively isomorphic to its dual surface. * There are at most 16 singular points. * Each ''trope'' of the surface corresponds to a singular point on its dual. Here, a ''trope'' is defined as a double-conic on the surface. In other words, it is where the intersection of the surface with a plane factors into a perfect square. * For each Kummer surface, there exists a two-dimensional family of lines, such that each point of the surface is tangent to two lines in the family.<ref>{{Citation |last1=Knörrer |first1=H. |author-link=Horst Knörrer |title=Arithmetik und Geometrie |volume=3 |pages=[https://archive.org/details/arithmetikundgeo0000unse/page/115 115–141]| url=https://neo-classical-physics.info/uploads/3/0/6/5/3065888/knorrer_-_the_fresnel_wave_surface.pdf |year=1986 |series=Math. Miniaturen |chapter=Die Fresnelsche Wellenfläche |location=Basel, Boston, Berlin |publisher=Birkhäuser |isbn=978-3-7643-1759-1 |mr=879281}}</ref>
More properties of the surface of wavevectors are in Chapter 10 of the classical reference on Kummer surfaces.<ref>{{Cite book |last=Hudson |first=R. W. H. T. (1875-1904 ) |url=http://archive.org/details/184605691 |title=Kummer's quartic surface |date=1905 |publisher=Cambridge [Eng.]: University Press |others=University of California Berkeley}}</ref>
Every linear material has a quartic dispersion equation, so its wavevector surface is a Kummer surface, which can have at most 16 singular points. That such a material might exist was proposed in 1910,<ref>{{Cite journal |last=Bateman |first=H. |date=1910 |title=Kummer's Quartic Surface as a Wave Surface |url=http://doi.wiley.com/10.1112/plms/s2-8.1.375 |journal=Proceedings of the London Mathematical Society |volume=s2-8 |issue=1 |pages=375–382 |doi=10.1112/plms/s2-8.1.375}}</ref> and in 2016, scientists made such a (meta)material, and confirmed it has 16 directions for conical refraction.<ref>{{Cite journal |last1=Favaro |first1=Alberto |last2=Hehl |first2=Friedrich W. |date=2016-01-22 |title=Light propagation in local and linear media: Fresnel-Kummer wave surfaces with 16 singular points |url=https://link.aps.org/doi/10.1103/PhysRevA.93.013844 |journal=Physical Review A |volume=93 |issue=1 |article-number=013844 |doi=10.1103/PhysRevA.93.013844|arxiv=1510.05566 |bibcode=2016PhRvA..93a3844F }}</ref>
== Diffraction theory == The classical theory of conical refraction was essentially in the style of geometric optics, and ignores the wave nature of light. Wave theory is needed to explain certain observable phenomena, such as Poggendorff rings, secondary rings, the central spot, and its associated rings. In this context, conical refraction is usually named "conical ''diffraction''" to emphasize the wave nature of light.<ref name=":7">{{Cite journal |last1=Berry |first1=M.V |last2=Jeffrey |first2=M.R |last3=Lunney |first3=J.G |date=2006-06-08 |title=Conical diffraction: observations and theory |url=https://royalsocietypublishing.org/doi/10.1098/rspa.2006.1680 |journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |volume=462 |issue=2070 |pages=1629–1642 |doi=10.1098/rspa.2006.1680 |bibcode=2006RSPSA.462.1629B |issn=1364-5021|url-access=subscription }}</ref><ref>{{Cite journal |last=Berry |first=M V |date=2004-04-01 |title=Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike |url=https://iopscience.iop.org/article/10.1088/1464-4258/6/4/001 |journal=Journal of Optics A: Pure and Applied Optics |volume=6 |issue=4 |pages=289–300 |doi=10.1088/1464-4258/6/4/001 |bibcode=2004JOptA...6..289B |issn=1464-4258|url-access=subscription }}</ref><ref name=":0" />
== Observations == The angle of the cone depends on the properties of the crystal, specifically the differences between its principal refractive indices. The effect is typically small, requiring careful experimental setup to observe. Early experiments used sunlight and pinholes to create narrow beams of light, while modern experiments often employ lasers and high-resolution detectors.
Poggendorff observed two rings separated by a thin dark band.<ref>{{Cite journal |last=Poggendorff |date=January 1839 |title=Ueber die konische Refraction |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.18391241104 |journal=Annalen der Physik |volume=124 |issue=11 |pages=461–462 |doi=10.1002/andp.18391241104 |bibcode=1839AnP...124..461P |issn=0003-3804}}</ref> This was explained by Voigt.<ref>{{Cite journal |last=Voigt |first=W. |date=January 1906 |title=Bemerkungen zur Theorie der konischen Refraktion |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.19063240103 |journal=Annalen der Physik |volume=324 |issue=1 |pages=14–21 |doi=10.1002/andp.19063240103 |bibcode=1906AnP...324...14V |issn=0003-3804}}</ref> See Born and Wolf, section 15.3, for a derivation.
Potter observed in 1841<ref>{{Cite journal |last=Potter |first=R. |date=May 1841 |title=LVII. An examination of the phænomena of conical refraction in biaxal crystals |url=https://www.tandfonline.com/doi/full/10.1080/14786444108650311 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |volume=18 |issue=118 |pages=343–353 |doi=10.1080/14786444108650311 |issn=1941-5966|url-access=subscription }}</ref> certain diffraction phenomena that were inexplicable with Hamilton's theory. Specifically, if we follow the two rings created by the internal conic refraction, then the inner ring would contract until it becomes a single point, while the outer ring expands indefinitely. A satisfactory explanation required later developments in diffraction theory.<ref>{{Cite thesis |last=Jeffrey |first=M. R. |title=Conical Diffraction: Complexifying Hamilton's Diabolical Legacy |date=2007 |degree=PhD |publisher=University of Bristol}}</ref>
Conical refraction has also been observed in transverse sound waves in quartz.<ref>{{Cite journal |last1=McSkimin |first1=H. J. |last2=Bond |first2=W. L. |date=1966-03-01 |title=Conical Refraction of Transverse Ultrasonic Waves in Quartz |url=https://pubs.aip.org/jasa/article/39/3/499/618773/Conical-Refraction-of-Transverse-Ultrasonic-Waves |journal=The Journal of the Acoustical Society of America |volume=39 |issue=3 |pages=499–505 |bibcode=1966ASAJ...39..499M |doi=10.1121/1.1909918 |issn=0001-4966 |url-access=subscription}}</ref>
== Modern developments ==
The study of conical refraction has continued since its discovery, with researchers exploring its various aspects and implications. Some recent work includes:
* '''Paraxial theory''': This theory provides a simplified description of conical diffraction for small angles of incidence and has been used to analyze the detailed structure of the light patterns observed.<ref>{{Cite journal |last1=Sokolovskii |first1=G. S. |last2=Carnegie |first2=D. J. |last3=Kalkandjiev |first3=T. K. |last4=Rafailov |first4=E. U. |date=2013-05-06 |title=Conical Refraction: New observations and a dual cone model |url=https://opg.optica.org/oe/abstract.cfm?uri=oe-21-9-11125 |journal=Optics Express |volume=21 |issue=9 |pages=11125–11131 |doi=10.1364/OE.21.011125 |issn=1094-4087|doi-access=free |pmid=23669969 |bibcode=2013OExpr..2111125S }}</ref> * '''Chiral crystals''': The inclusion of optical activity (chirality) in the crystal leads to new phenomena, such as the transformation of the conical cylinder into a "spun cusp" caustic.<ref>{{Cite journal |last1=Berry |first1=M V |last2=Jeffrey |first2=M R |date=2006-05-01 |title=Chiral conical diffraction |url=https://iopscience.iop.org/article/10.1088/1464-4258/8/5/001 |journal=Journal of Optics A: Pure and Applied Optics |volume=8 |issue=5 |pages=363–372 |doi=10.1088/1464-4258/8/5/001 |bibcode=2006JOptA...8..363B |issn=1464-4258|url-access=subscription }}</ref> * '''Absorption and dichroism''': The presence of absorption in the crystal significantly alters the behavior of light, leading to the splitting of diabolical points into pairs of branch points and affecting the emergent light patterns.<ref>{{Cite journal |last1=Berry |first1=M V |last2=Jeffrey |first2=M R |date=2006-12-01 |title=Conical diffraction complexified: dichroism and the transition to double refraction |url=https://iopscience.iop.org/article/10.1088/1464-4258/8/12/003 |journal=Journal of Optics A: Pure and Applied Optics |volume=8 |issue=12 |pages=1043–1051 |doi=10.1088/1464-4258/8/12/003 |bibcode=2006JOptA...8.1043B |issn=1464-4258|url-access=subscription }}</ref> * '''Nonlinear optics''': Nonlinear optical effects in biaxial crystals can interact with conical refraction, leading to complex and intriguing phenomena.<ref>{{Cite journal |last1=Bloembergen |first1=N. |last2=Shih |first2=H. |date=May 1969 |title=Conical refraction in nonlinear optics |journal=Optics Communications |volume=1 |issue=2 |pages=70–72 |doi=10.1016/0030-4018(69)90011-x |bibcode=1969OptCo...1...70B |issn=0030-4018}}</ref> * '''Applications''': Conical refraction had found applications in optical trapping, free-space optical communications, polarization metrology, super-resolution imaging, two-photon polymerization, and lasers.<ref>{{Cite journal |last1=Turpin |first1=Alex |last2=Loiko |first2=Yury V. |last3=Kalkandjiev |first3=Todor K. |last4=Mompart |first4=Jordi |date=September 2016 |title=Conical refraction: fundamentals and applications |url=https://onlinelibrary.wiley.com/doi/10.1002/lpor.201600112 |journal=Laser & Photonics Reviews |volume=10 |issue=5 |pages=750–771 |doi=10.1002/lpor.201600112 |bibcode=2016LPRv...10..750T |issn=1863-8880|hdl=10803/322801 |hdl-access=free }}</ref> == See also ==
* Birefringence * Wave surface * Crystal optics * Polarization * Caustics * Keller cone
== External links ==
* [https://web.archive.org/web/20060514134205/http://www.demophys.tsu.ru/Original/Hamilton/Hamilton.html Images of conical refractions with rings of radius larger than one meter, through monocrystal of rhombic sulfur. By Yu. P. Mikhailichenko] * Images of the Fresnel wave surface,<ref>{{Citation |last=Knörrer |first=H. |author-link=Horst Knörrer |title=Die Fresnelsche Wellenfläche |date=1986 |work=Arithmetik und Geometrie: Vier Vorlesungen |pages=115–141 |editor-last=Knörrer |editor-first=Horst |place=Basel |publisher=Birkhäuser |language=de |doi=10.1007/978-3-0348-5226-5_4 |isbn=978-3-0348-5226-5 |editor2-last=Schmidt |editor2-first=Claus-Günther |editor3-last=Schwermer |editor3-first=Joachim |editor3-link=Joachim Schwermer |editor4-last=Slodowy |editor4-first=Peter}}</ref><ref>{{Cite journal |last=Rowe |first=David E. |author-link=David E. Rowe |date=2018-03-01 |title=On Models and Visualizations of Some Special Quartic Surfaces |journal=The Mathematical Intelligencer |language=en |volume=40 |issue=1 |pages=59–67 |doi=10.1007/s00283-017-9773-3 |issn=1866-7414}}</ref> (pages 470–485 <ref>{{Citation |last1=Schaefer |first1=Clemens |title=Band 3, Teil 1 Elektrodynamik und Optik |date=2011-12-22 |url=https://www.degruyter.com/document/doi/10.1515/9783111421698/html |access-date=2024-04-25 |publisher=De Gruyter |language=de |doi=10.1515/9783111421698 |isbn=978-3-11-142169-8|url-access=subscription }}</ref>).
== References == {{Reflist}}
{{DEFAULTSORT:Refractive Index}} Category:Polarization (waves) Category:Optical mineralogy Category:Refraction Category:Optical quantities Category:Physical optics