Conical refraction

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 transformed 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 and subsequently observed by Humphrey Lloyd in the next year. It was possibly the first example of a phenomenon predicted by mathematical reasoning and later confirmed by experiment.

History

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 arragonite,* borax and mica were identified as biaxial.

In the early 19th century, Augustin-Jean Fresnel developed a more comprehensive theory that could describe double refraction in both uniaxial and biaxial crystals. However, his description of the wave surface in biaxial crystals was incomplete, overlooking certain key features.

William Rowan Hamilton, building upon Fresnel's work, mathematically analyzed the wave surface in biaxial crystals and discovered the presence of four conoidal cusps. These cusps implied 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.

Hamilton's prediction was met with skepticism by some, including George Biddell Airy, who had also studied the wave surface but had not realized its full implications. However, Humphrey Lloyd, working closely with Hamilton, conducted experiments using aragonite crystals and successfully observed both internal and external conical refraction. This discovery was a significant victory for the wave theory of light and solidified Fresnel's theory of double refraction. Lloyd's experimental data are described in pages 350–355.

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.

In 1833, James MacCullagh claimed that it is a special case of a theorem he published in 1830 that he did not explicate, since it was not relevant to that particular paper. Cauchy discovered the same surface in the context of classical mechanics.

is an early historical review coauthored by Lloyd. gives an updated historical review.

External conical refraction

Internal conical refraction

Theory

Surface of wavevectors

Given a biaxial crystal with the three principal refractive indices $n_{x}<n_{y}<n_{z}$. For each possible direction ${\hat {k}}$ of planar waves propagating in the crystal, it has a certain group velocity $v_{g}({\hat {k}})$. The refractive index along that direction is defined as $n({\hat {k}})=c/v_{g}({\hat {k}})$.

Define, now, the surface of wavevectors as the following set of points$$\{n({\hat {k}}){\hat {k}}:{\hat {k}}\in {\text{sphere of radius 1}}\}$$In general, there are two group velocities along each wavevector direction. To find them, draw the plane perpendicular to ${\hat {k}}$. 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, 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 (, page 346):$$(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})(n_{x}^{2}k_{x}^{2}+n_{y}^{2}k_{y}^{2}+n_{z}^{2}k_{z}^{2})-(n_{y}^{2}+n_{z}^{2})n_{x}^{2}k_{x}^{2}-(n_{z}^{2}+n_{x}^{2})n_{y}^{2}k_{y}^{2}-(n_{x}^{2}+n_{y}^{2})n_{z}^{2}k_{z}^{2}+(n_{x}n_{y}n_{z})^{2}=0$$or equivalently,$$\sum _{i}{\frac {n_{i}^{2}k_{i}^{2}}{\|k\|^{2}-n_{i}^{2}}}=0$$

[Proof]

The major and minor axes are the solutions to the constraint optimization problem: $${\begin{cases}k^{T}r&=0\quad &k\perp r\\r^{T}Mr&=1\quad &r{\text{ is on the index ellipsoid}}\\\mathrm {exr} (r^{T}r)&\quad &\|r\|{\text{is max/minimized}}\end{cases}}$$ where $M$ is the matrix with diagonal entries $n_{x}^{-2},n_{y}^{-2},n_{z}^{-2}$.

Since there are 3 variables and 2 constraints, we can use the Karush–Kuhn–Tucker conditions. That is, the three gradients ${\textstyle k,Mr,r}$ are linearly dependent.

Let ${\textstyle Mr=\alpha k+\beta r}$, then we have$$0=\alpha k^{T}r=r^{T}Mr-\beta r^{T}r\implies \beta =(r^{T}r)^{-1}$$Plugging ${\textstyle r_{x}={\frac {\alpha k_{x}}{n_{x}^{-2}-\beta }},\dots }$ back to ${\textstyle k^{T}r=0}$, we obtain $$\sum _{i}{\frac {k_{i}^{2}}{n_{i}^{-2}-\|r\|^{-2}}}=0$$Let ${\textstyle {\vec {k}}}$ be the vector with the direction of ${\textstyle (k_{x},k_{y},k_{z})}$, and the length of ${\textstyle \|r\|}$. We thus find that the equation of ${\textstyle {\vec {k}}}$ is $$\sum _{i}{\frac {k_{i}^{2}}{n_{i}^{-2}-\|{\vec {k}}\|^{-2}}}=0$$Multiply out the denominators, then multiply by ${\textstyle n_{x}^{2}n_{y}^{2}n_{z}^{2}}$, we obtain the result.

In general, along a fixed direction ${\textstyle {\hat {k}}}$, there are two possible wavevectors: The slow wave ${\textstyle n_{+}{\hat {k}}}$ and the fast wave ${\textstyle n_{-}{\hat {k}}}$, where ${\textstyle n_{+}}$ is the major semiaxis, and ${\textstyle n_{-}}$ is the minor.

Plugging ${\textstyle {\vec {k}}=n{\hat {k}}}$ into the equation of ${\textstyle {\vec {k}}}$, we obtain a quadratic equation in ${\textstyle n^{2}}$:$$\left({\frac {k_{x}^{2}}{n_{y}^{2}n_{z}^{2}}}+\cdots \right)n^{4}-\left({\frac {k_{x}^{2}}{n_{y}^{2}}}+{\frac {k_{x}^{2}}{n_{z}^{2}}}+\cdots \right)n^{2}+1=0$$which has two solutions $n_{-}^{2},n_{+}^{2}$.

At exactly four directions, the two wavevectors coincide, because the plane perpendicular to ${\textstyle {\hat {k}}}$ intersects the index ellipsoid at a circle. These directions are ${\textstyle {\hat {k}}=(\pm \cos \theta ,0,\pm \sin \theta )}$ where ${\textstyle \sin ^{2}\theta ={\frac {n_{y}^{-2}-n_{z}^{-2}}{n_{x}^{-2}-n_{z}^{-2}}}}$, at which point ${\textstyle n_{-}=n_{+}=n_{y}}$.

Expanding the equation of the surface in a neighborhood of ${\textstyle {\vec {k}}=(n_{y}\cos \theta ,0,n_{y}\sin \theta )}$, 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 (, pages 349–350)$$k_{z}\pm k_{x}{\sqrt {\frac {n_{y}^{2}-n_{x}^{2}}{n_{z}^{2}-n_{y}^{2}}}}\pm n_{y}{\sqrt {\frac {n_{z}^{2}-n_{x}^{2}}{n_{z}^{2}-n_{y}^{2}}}}$$and the 4 circles are the intersection of those planes with the ellipsoid$$(n_{x}^{2}+n_{y}^{2})k_{x}^{2}+2n_{y}^{2}k_{y}^{2}+(n_{z}^{2}+n_{y}^{2})k_{z}^{2}-(n_{x}^{2}+n_{z}^{2})n_{y}^{2}=0$$

[Proof]

By differentiating its equation, we find that the points on the surface of wavevectors, where the tangent plane is parallel to the ${\textstyle k_{y}}$ -axis, satisfies $$k_{y}((n_{x}^{2}+n_{y}^{2})k_{x}^{2}+2n_{y}^{2}k_{y}^{2}+(n_{z}^{2}+n_{y}^{2})k_{z}^{2}-(n_{x}^{2}+n_{z}^{2})n_{y}^{2})=0$$ That is, it is the union of the ${\textstyle k_{x}k_{z}}$ -plane, and an ellipsoid.

Thus, such points on the surface of wavevectors has two parts: Every point with ${\textstyle k_{y}=0}$, and every point that intersects with the auxilliary ellipsoid $$(n_{x}^{2}+n_{y}^{2})k_{x}^{2}+2n_{y}^{2}k_{y}^{2}+(n_{z}^{2}+n_{y}^{2})k_{z}^{2}-(n_{x}^{2}+n_{z}^{2})n_{y}^{2}=0$$

Using the equation of the auxilliary ellipsoid to eliminate ${\textstyle k_{y}^{2}}$ from the equation of the wavevector surface, we obtain another degree-4 equation, which splits into the product of 4 planes: $$k_{z}\pm k_{x}{\sqrt {\frac {n_{y}^{2}-n_{x}^{2}}{n_{z}^{2}-n_{y}^{2}}}}\pm n_{y}{\sqrt {\frac {n_{z}^{2}-n_{x}^{2}}{n_{z}^{2}-n_{y}^{2}}}}$$

Thus, we obtain 4 ellipses: the 4 planar intersections with the auxilliary ellipsoid. These ellipses all exist on the wavevector surface, and the wavevector surface has tangent plane parallel to the ${\textstyle k_{y}}$ 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 ${\textstyle P_{0}}$ be one of those 4 planes, and let ${\textstyle {\vec {k}}}$ be one point on the circle in ${\textstyle P_{0}}$. If ${\textstyle k_{y}\neq 0}$, then since the circle is on the surface, the tangent plane ${\textstyle P}$ to the surface at ${\textstyle {\vec {k}}}$ must contain the tangent line ${\textstyle l}$ to the circle at ${\textstyle {\vec {k}}}$. Also, the plane ${\textstyle P}$ must also contain ${\textstyle l_{y}}$, the line pass ${\textstyle {\vec {k}}}$ that is parallel to the ${\textstyle k_{y}}$ -axis. Therefore, the plane ${\textstyle P}$ is spanned by ${\textstyle l_{y}}$ and ${\textstyle l}$, which is precisely the plane ${\textstyle P_{0}}$. This then extends by continuity to the case of ${\textstyle k_{y}=0}$.

One can imagine the surface as a prune, with 4 little pits or dimples. Putting the prune on a flat desk, the prune would touch the desk at a circle that covers up a dimple.

Surface of ray vectors

The surface of ray vectors is the projective dual surface of the surface of wavevectors. Its equation is obtained by replacing $n_{i}$ with $n_{i}^{-1}$ in the equation for the surface of wavevectors. That is,$$(r_{x}^{2}+r_{y}^{2}+r_{z}^{2})(n_{x}^{-2}r_{x}^{2}+n_{y}^{-2}r_{y}^{2}+n_{z}^{-2}r_{z}^{2})-(n_{y}^{-2}+n_{z}^{-2})n_{x}^{-2}r_{x}^{2}-(n_{z}^{-2}+n_{x}^{-2})n_{y}^{-2}r_{y}^{2}-(n_{x}^{-2}+n_{y}^{-2})n_{z}^{-2}r_{z}^{2}+(n_{x}n_{y}n_{z})^{2}=0$$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 $n_{x},n_{y},n_{z}$ is small. In this case, the conoidal point is approximately at the center of the tangent circle surrounding it, and thus, the cone of light (in both the internal and the external refraction cases) is approximately a circular cone.

Algebraic geometry

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. 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.

In general, the surface of wavevectors is a Kummer surface, and all properties of it apply. For example:

• 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.

More properties of the surface of wavevectors are in Chapter 10 of the classical reference on Kummer surfaces .

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.

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 refraction for small angles of incidence and has been used to analyze the detailed structure of the light patterns observed.
• 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.
• 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.
• Nonlinear optics: Nonlinear optical effects in biaxial crystals can interact with conical refraction, leading to complex and intriguing phenomena.

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, and in 2016, scientists made such a (meta)material, and confirmed it has 16 directions for conical refraction.

See also

• Birefringence
• Wave surface
• Crystal optics
• Polarization
• Caustics

References

• Jeffrey, M. R. (2007). Conical Diffraction: Complexifying Hamilton’s Diabolical Legacy (PhD thesis). University of Bristol.