# Index ellipsoid

The index ellipsoid , also Fletcher ellipsoid after Lazarus Fletcher , is an indicatrix to describe the refraction of light (more precisely: the refractive indices , singular: - index , hence the name) in a birefringent crystal . Together with the Fresnel ellipsoid (after Augustin Jean Fresnel ), this ellipsoid enables a clear description of the propagation of light in matter.

## Basics

Construction of the ordinary (left) and extraordinary ray (right) according to Huygens. In the left case the wave surfaces are spheres, in the right case ellipsoids of revolution. The wave normals point in the same direction, but the directions of movement of the wave fronts are different.${\ displaystyle {\ vec {k}}}$${\ displaystyle {\ vec {P}}}$

The set of all points which - starting from a point-like excitation location - are reached simultaneously by a wave , form the wave surface of an elementary wave . The behavior of a plane wavefront can be explained by the Huygens principle :

An elementary wave emanates from every point on the wave front. The outer envelope of all wave surfaces of these elementary waves form the observable wave.

In an optically isotropic medium , the speed of propagation of light is identical in all directions, and the wave surfaces therefore correspond to spherical waves . The transition between two optically isotropic media can also be described with the help of Huygens' principle and leads to Snell's law of refraction .

If, on the other hand, a light beam is directed onto a calcite crystal ( anisotropic material), two light beams emerge. This phenomenon is called birefringence . While one ray follows Snell's law of refraction ( ordinary ray ), this does not apply to the second ( extraordinary ray ). The reason for this is that the speed of light in calcite depends on the direction of propagation and the direction of polarization of the light beam. While the wave surfaces of the ordinary ray are still spherical waves, the wave surfaces of the extraordinary ray are ellipsoids of revolution . The shape of the wave surface can be derived from the Fresnel ellipsoid ( Fresnel wave surface ).

If the Huygens construction is carried out with elliptical waves, the result is that the superposition of the wave surfaces leads to a plane wave again . The wave front of this plane wave no longer only moves in the direction of its normal , it can also move at an angle to it: the direction and speed of the wave normal (in the picture: k) and the beam direction (in the picture: P) no longer match.

For the wave normal, Snell's law of refraction still applies. For each ray the corresponding refractive indices can be determined with the help of the index ellipsoid.

## construction

In an optically anisotropic medium, the linear relationship between the electric field and the dielectric displacement must be written in a direction-dependent form, since these two vectors are generally no longer parallel to each other: ${\ displaystyle {\ vec {E}}}$ ${\ displaystyle {\ vec {D}}}$

${\ displaystyle {\ vec {D}} = \ varepsilon _ {0} {\ vec {\ vec {\ varepsilon}}} \ cdot {\ vec {E}},}$

where the dielectric constants form a symmetrical 2nd order tensor . ${\ displaystyle (\ varepsilon _ {i, j})}$

Correspondingly, the following applies to the inverse tensor of the dielectric modules : ${\ displaystyle \ eta _ {i, j}}$

${\ displaystyle {\ vec {E}} = {\ frac {1} {\ varepsilon _ {0}}} {\ vec {\ vec {\ eta}}} \ cdot {\ vec {D}}}$

These tensors each have three, in general, different eigenvalues , but agree in the position of their main axes.

In the form of the main axis they have the following form:

{\ displaystyle {\ begin {aligned} {\ vec {\ vec {\ eta}}} & = {\ begin {bmatrix} \ eta _ {1,1} & 0 & 0 \\ 0 & \ eta _ {2,2} & 0 \\ 0 & 0 & \ eta _ {3,3} \ end {bmatrix}} \\ {\ vec {\ vec {\ varepsilon}}} & = {\ begin {bmatrix} \ varepsilon _ {1,1} & 0 & 0 \\ 0 & \ varepsilon _ {2,2} & 0 \\ 0 & 0 & \ varepsilon _ {3,3} \ end {bmatrix}} \ end {aligned}}}

In their main axis system (x, y, z), tensors of the 2nd level can be represented with the eigenvalues as ellipsoids: ${\ displaystyle h_ {1}; \; h_ {2}; \; h_ {3}}$

${\ displaystyle h_ {1} x ^ {2} + h_ {2} y ^ {2} + h_ {3} z ^ {2} = 1}$.

The lengths of the radius of the main axes of the ellipsoid are:

${\ displaystyle {\ frac {1} {\ sqrt {h_ {1}}}}; {\ frac {1} {\ sqrt {h_ {2}}}}; {\ frac {1} {\ sqrt {h_ {3}}}}}$

The ellipsoid that represents the dielectric module is the index ellipsoid. The main refractive indices can be calculated from its eigenvalues as follows:

{\ displaystyle {\ begin {aligned} n _ {\ alpha} = {\ frac {1} {\ sqrt {\ eta _ {1,1}}}} \\ n _ {\ beta} = {\ frac {1} {\ sqrt {\ eta _ {2,2}}}} \\ n _ {\ gamma} = {\ frac {1} {\ sqrt {\ eta _ {3,3}}}} \ end {aligned}} }

### Fresnel ellipsoid

The ellipsoid that represents the dielectric tensor is the Fresnel ellipsoid . The main speeds of light can be calculated from its eigenvalues as follows:

{\ displaystyle {\ begin {aligned} c _ {\ alpha} = {\ frac {1} {\ sqrt {\ varepsilon _ {1,1}}}} \, c \\ c _ {\ beta} = {\ frac {1} {\ sqrt {\ varepsilon _ {2,2}}}} \, c \\ c _ {\ gamma} = {\ frac {1} {\ sqrt {\ varepsilon _ {3,3}}}} \, c \ end {aligned}}}

Here is the speed of light in a vacuum. ${\ displaystyle c}$

## use

Index ellipsoid of an optically uniaxial crystal. The wave normal direction (k) and the intersection ellipse with the refractive indices n o and n a (= n e ) are shown in red

If a wave normal vector is placed at the origin of the index ellipsoid, the plane that is perpendicular to this vector and runs through the center of the ellipsoid intersects the index ellipsoid in such a way that the line of intersection is an ellipse :

• the main axes of this intersection ellipse indicate the directions of the electrical flux density of the ordinary and extraordinary rays
• the intercepts (radius of the main axes) indicate the corresponding refractive indices.

If the intersection ellipse is a circle, then all wave normals move in the same direction regardless of the polarization of the wave. Directions with this property are called optical axes .

### Fresnel ellipsoid

Using the same procedure, one can construct the corresponding cut ellipse for each beam direction on the Fresnel ellipsoid:

• the main axes of this intersection ellipse indicate the directions of the electric field of the ordinary and extraordinary rays
• the axis segments (radius of the main axes) indicate the associated speeds of light.

Plotting the radius of the ellipse in the beam direction and lets the beam direction to take any direction in space, we get a figure with two shells which the wave fronts of the ordinary and extraordinary beam describe (s. U. Links ).

## Propagation of light in dielectric media

According to Neumann's principle , all property tensors have a shape determined by the crystal system . In the case of 2nd level tensors there are three fundamentally different forms.

### Optically isotropic crystals

In a cubic crystal the eigenvalues ​​of all 2nd order tensors are the same. Both the index and Fresnel ellipsoid are therefore spheres. Consequently, the associated ellipses of intersection for all ray and wave normal directions are circles. The two shells of the jet surface coincide on a sphere. Therefore, all light rays behave identically regardless of their direction and polarization: cubic crystals are optically isotropic.

### Optically uniaxial crystals

In the whirling crystal systems ( trigonal , tetragonal and hexagonal ) the main axis of the tensor, which is also referred to as the optical axis, lies in the direction of the crystallographic c-axis; the other two main axes are perpendicular to it.

Accordingly, there are two different eigenvalues ​​or main refractive indices:

• ${\ displaystyle n _ {\ parallel}}$or (index for extraordinary / extraordinary)${\ displaystyle n_ {e}}$${\ displaystyle e}$
• ${\ displaystyle n _ {\ perp}}$or (index for ordinary ).${\ displaystyle n_ {o}}$${\ displaystyle o}$
1. For the crystal is called optically negative, the index and Fresnel ellipsoid are flattened ellipsoids of revolution .${\ displaystyle \ Delta n = n_ {e} -n_ {o} <0 \ Leftrightarrow n_ {e}
2. for the crystal is called optically positive, the index and Fresnel ellipsoid are elongated rotational ellipsoids.${\ displaystyle \ Delta n = n_ {e} -n_ {o}> 0 \ Leftrightarrow n_ {e}> n_ {o}}$

For the description, consider the ray in the main section, i.e. H. in the plane in which both the incident light beam and the optical axis lie. For each direction normal to the shaft, a semiaxis of the assigned cutting ellipse lies in the main plane, the other perpendicular to it.

• The length of the semi-axis perpendicular to the main plane is equal to n o for all directions .
• The length of the semiaxis lying in the main plane lies between n o and n e , depending on the angle of the shaft normal direction to the optical axis, with the extreme values:
• n e for angle 90 °
• n o for angle 0 °, i.e. H. if the wave normal lies in the direction of the optical axis, the intersection ellipse is a circle and both refractive indices are equal to n o .

The intersection ellipses of the Fresnel ellipsoid behave in the same way.

Therefore the wave surface consists of two shells:

• a sphere with radius n o (the wavefront of the ordinary ray is spherical - regardless of the direction of the ray)
• an ellipsoid of revolution with semi-axes of lengths n o and n e ( axis of rotation ) (the wave front of the extraordinary ray is ellipsoid).

Apart from the points of contact , which each lie on a circle with the radius n o :

1. the ellipsoid completely in the sphere or
2. the sphere completely in the ellipsoid.

In summary: If a plane wave hits an optically uniaxial crystal, two rays are usually created.

• One of the two rays is polarized perpendicular to the main plane. In this beam, the direction of the beam and the normal direction of the wave coincide, as is the case with the beams in optically isotropic media. It follows Snell's law of refraction. This ray is therefore called the ordinary ray .
• The other beam is polarized in the main plane. With it, the normal direction of the wave and the direction of the beam do not usually coincide: Only the normal direction of the wave follows Snell's law of refraction, with a direction-dependent refractive index that deviates from the refractive index of the ordinary beam. The beam itself is shifted laterally compared to the "normal" direction of refraction. This ray is therefore called the extraordinary ray .
• If the light beam is radiated in the direction of the crystallographic c-axis, the ordinary and extraordinary beams behave identically: They have the same refractive index n o and the same beam speed. Therefore this axis is also called the optical axis . In an optically uniaxial system, the optical axis is a direction of optical isotropy.

### Optically biaxial crystals

In the orthorhombic , monoclinic and triclinic crystal system there are optically biaxial crystals. As a rule, two rays are obtained here , which behave like the extraordinary ray. They can be described according to the same principle, but the relationships are much more complicated. Therefore, only an overview of the most important features can be given here; for further information, reference is made to the specialist literature.

In optically biaxial crystals there are two optical axes and three main axes of different sizes (x, y, z) or main refractive indices . The main axes are chosen so that: ${\ displaystyle n _ {\ alpha}; \; n _ {\ beta}; \; n _ {\ gamma}}$

${\ displaystyle n_ {1,1} \,: = \, n _ {\ alpha} \; <\; n_ {2,2} \,: = \, n _ {\ beta} \; <\; n_ {3 , 3} \,: = \, n _ {\ gamma}}$

Their position to the crystal axes follows the Neumann principle . Since all three main axes are of different lengths, the index and Fresnel ellipsoid in these systems are each triaxial .

The optical axes can be found as follows: If the wave normal vector is rotated in the xz plane from the z to the x direction, all the resulting ellipses have a common main axis in the y direction with the length n β . The second main axis lies in the xz plane and runs through all values ​​between n α and n γ . Due to the above definition of the main axes, there must therefore be a direction in which this second main axis also has the length n β ; there must also be a corresponding direction between the z and x directions. These two directions are the optical axes or binormal . The angle between the binormal is halved by both the x and z axes.

• If the angle between the z-axis and the binormal is less than 45 °, the crystal is called optically biaxially positive
• if it is equal to 45 °, the crystal is called optically biaxially neutral
• if it is greater than 45 °, the crystal is optically biaxially negative .

The wave surface is a 4th order surface. It is a special form of a grief face . Here, too, there are two directions in which the two shells touch: the biradials . They are also in the xz plane, but not in the direction of the binormal. Therefore, in a two-axis system, the optical axis is not a direction of optical isotropy. This is the cause of the conical refraction, which is described in more detail below.

#### Inner conical refraction

A diaphragm is placed in front of a crystal plate cut perpendicular to a binormal so that only a thin beam of light can fall on the crystal perpendicular to the plate. If the crystal is then irradiated with an unpolarized light beam, a ring can be seen on a screen behind the crystal, the radius of which does not change with the distance from the crystal plate. Although the wave normals all remain in the direction of the binormal, the wave fronts shift - depending on their polarization direction - perpendicular to the wave normal. But since all wave normals remain parallel to each other, all rays emerge from the crystal perpendicular to the crystal surface and then continue to propagate parallel to each other. This effect is called inner conical refraction.

#### Outer conical refraction

A crystal plate that is cut perpendicular to a biradial is placed between two perforated diaphragms. The diaphragms are arranged in such a way that only those light rays leave the crystal that have moved in the crystal in the direction of the biradials. If the entrance aperture is irradiated with divergent light in such a way that all possible directions of polarization migrate through the crystal, a circle is also obtained on a screen behind the crystal, the radius of which increases with the distance from the crystal. The reason for this is that all polarization directions have the same beam speed and also move in the same direction in the crystal. But since they have different wave normals, they are broken differently on the crystal surfaces. Therefore, the entrance aperture must also be irradiated with divergent light. This effect is called outer conical refraction.