Birefringence is an optical effect (manifesting as the observed difference in refractive indices for two orthogonally polarised light waves of the same wavelength and propagation direction). It is a consequence of aniaxial optical anisotropy. Anisotropy itself is a much more general notion and relates to the description of direction-dependent physical properties (optical, magnetic, mechanical etc)

@Aseya: Live is not always fair. If you could have learned Latin and Greek at school the idea would have been clear from the very beginning: birefringent 'refracting light in two ways'. Anisotopic: ' not treating all directions the same way'. Of course, Robin's explanation adds to this.

To complete what says Robin :

The sand and the schist possesses an anisotropie of conductivity. That is the conductivity do not be the same according to the propagation. They are anisotrope but not birefringents.

For example the ruby is birefringent. it possesses a normal axis of refractive index 1.770 and extraordinary axis 1.762. Its difference of index is equal to -0.008. The ellipsoïde of the index of the ruby has a flattened shape.

As Ulrich states, anisotropy refers to properties that depend on the spatial direction. In optics, a medium is said to be anisotropic, when its optical properties (absorption, dispersion) depend on the direction of propagation of the light.

However, with this definition, not all birefringent materials are anisotropic. An optically active substance, such as a solution of (natural) sugar in water exhibits circular dichroism (and circular birefringence), meaning that left- and right-handed circularly polarized beams are not absorbed (and dephased) in the same manner. Nontehless this substance is isotropic.

In optics, birefringence refers to substances whose index of refraction depends on the light polarization. If the indices are different for two orthogonal linear (circular) polarization states, one speaks of linear (circular) birefringence.

Yes, the polarization of the light, and its changes as the light propagates in matter, are interesting. But the last example, perfect at first sight, seems not completely correct. This is an example of what is called the optical activity but has nothing to do with anisotropy. Indeed, such a sample will exhibit the same behavior irrespectively what will be the direction of light propagation.

In birefringent solids one may expect *two* rays propagating inside the sample, often in two different directions (ordinary and extraordinary) when it is subjected to only one incident ray. As the angle between ordinary and extraordinary ray will depend on direction of incident light - we may speak of the (optical) anisotropy as well.

Marek,

just what I said! The sugar in water solution is isotropic, but exhibits (circular) dischroism.

Antoine,

The notion 'anisotropy' by no means include the sample size. This is why I questioned your example: the effect of optical activity - i.e. rotation of polarization - is proportional to the sample length being isotropic otherwise. Dichroism (or even trichroism) is different under this respect, and is inherently anisotropic. Yet, if we take into account that optical phenomena are usually wavelength dependent, we arrive at quite complicated picture. The same substance may be be optically inactive at one wavelength and may have different signs of optical activity at other wavelengths (the same applies to dichroism). I haven't heard about something similar when it comes to frequency dependent mechanical properties, but maybe I'm poorly informed. On the other hand, on microscopic level, phonons exhibit features usually attributed to photons. But maybe those last comments to go to far. One thing is sure: birefringence and anisotropy are not synonyms.

As I have to agree with all the previous, birefringence does refer to different refractive indexes for light polarized along different crystal axes. Anisotropy, or particularly anisotropic materials are, sometimes, referred to materials lacking inversion symmetry. I am not claiming that that's the only use of the term, but it is rather frequent, and thus should be mentioned. The direct outcome of such type of anisotropy is the ability of materials to produce even harmonics, as SHG (second harmonic generation). There is multitude of such crystals, i.e. b-BBO, Barium Titanate, LBO, KTP ...

Yuri,

I think you are confusing anisotropy or birefringence with nonlinearity. Higher harmonics may be generated by perfectly isotropic substances when the excitation is powerful enough. Dichroism occurs at arbitrarily low excitation.

Marek, I just pointed out that literature use for "anisotropy" is often refer to what indeed is a particular type of non-linearity. In fact, say a layer of oriented molecules would show a non-linearity (even orders), particularly due to the anisotropy. That is why, the two terms are often interchanged. Moreover, all the materials do exhibit odd orders of nonlinear optical response, but only anisotropic ones might have even orders nonlinearities as well.

Any way, I just wanted to remind that there is an additional meaning for the term.

Yuri,

Inversion symmetry refers to a system which has the same properties as its (point) mirror image. An object which has a chiral/helical structure lacks inversion symmetry, the same way as a substance composed chiral entitites, such as chiral molecules, does.

Isotropy refers to a system which is rotationally invariant, i.e., whose properties do not depend on the spatial direction (absorption/dispersion independent on the direction of light propagation in the case of optical isotropy, strain independent on the direction of applied stress in the case of mechanical isotropy, size independent on the direction along which is measured in the case of geometrical isotropy, etc).

Birefringence refers to a system whose dispersion depends on the light polarization. Linear birefringengence: the index of refraction differs for two (suitably chosen) orthogonal linear polarization directions. Circular birefringence: index differs for right and left-handed circular polarization.

Dichroism refers to a system whose absorption depends on the light polarization (linear dichroism: the transmission differs for two (suitably chosen) orthogonal linear polarization directions. Circular dichroism: transmission differs for right and left-handed circular polarization.

A substance which lacks inversion symmetry shows in general circular birefringence and circular dischroism.

Antonie, First of all thanks for the extensive review of terminology. I do fully agree with the definitions. The only point I wanted to make that, may be inaccurately, but nevertheless frequently the term of "anisotropy" is being used to say that some materials, most often crystals, lack the inversion symmetry. This argument is rather terminological, than scientific, as was my first comment.

Thanks for this discussion, it is good to go through the different terminology.

I would also like to comment on the use of the terms "anisotropy" versus "inversion symmetry". The latter leads to second-order NLO effects, the former does not, hence they should not be mixed.

As a simple example, liquid crystals are highly anisotropic but generally inactive when it comes to second-order NLO response because neighboring molecular stack in antiparallel fashion and cancel out the macroscopic second-order response (the material is centrosymmetric). To remove the inversion symmetry (or to make the material noncentrosymmetric) the neighboring dipoles have to be forced to adapt parallel orientation by applying, e.g. external electric field. This can bring about NLO activity but should not change much the anisotropy/birefringence of the sample.

Finally, such externally poled materials lack inversion symmetry but they are not optically active, i.e. they do not distinguish between RCP and LCP light. Optical activity requires chirality. Chiral materials lack inversion symmetry but materials lacking inversion symmetry are not necessarily chiral.

Interestingly, cubic magnetic crystals HAVE inversion symmetry (from purely geometric point of view), but exhibit magnetocrystalline anisotropy anyway.

Birefringence is a type of anisotropy.

Isotropy means that all physical and optical properties do not vary with angle at every spatial point in the material. Anisotropy means that some physical or optical processes do vary with angle at every spatial point in the material.

Birefringent means that the index of refraction of the material does vary with angle at every point in the material. Since the index of refraction is an optical property of the material, a birefringent material is automatically anisotropic.

In principle, it is possible that an anisotropic material may not be birefringent. However, this does not happen in practice. Most anisotropic materials are birefringent.

If you know of materials that are anisotropic in any mechanical property but not birefringent, then please tell us. This would be an exception to the rule.

@Robin: Most likely now RG folks will remove both yours and Ehsan's responses!

Anisotropy means dependence of physical (optical etc.) properties on direction. Birefringence means appearance of two waves with different velocities in material. David Rosen wrote “In principle, it is possible that an anisotropic material may not be birefringent. However, this does not happen in practice.” In crystals with a tensor of electrical permittivity ε and magnetic susceptibility μ (not equal 1) if ε is proportional to μ only one wave of arbitrary polarization can propagate with refractive index depending on the direction of propagation. Birefringence is consequence of the disproportion of ε and μ but is not the result of the material anisotropy.

@Marek: which means that magnetocrystalline anisotropy is not a linear response quantity. But it is usually discussed for magnetically ordered materials, which means that the (zero field) susceptibility diverges anyway. So we're way off the linear response reqime.

I wonder whether paramagnetic (cubic) metals with strong spin orbit interaction (e.g. Pt) possess anisotropic paramagnetic susceptibility. Should not be possible within linear response!?!

@Kai: In paramagnets there is no (magnetic) anisotropy in absence of the external field. Magnetized paramagnet is, of course, optically anisotropic. But it is still isotropic in its responsivity to the external field, i.e. any orientation of magnetization may be achieved, no matter that even in paramagnets the magnetization is a non-linear function of applied field; there simply is no such thing like easy or hard direction of magnetization. However, if the temperature is low enough (below the so called blocking temperature), then the effects of crystal field enter the game. Due to its local symmetry hard/easy directions appear. In this blocked state, and in absence of external field the sample as a whole carries no net magnetic moment as the individual magnetic ions point now in 6 (crystalographicaly equivalent to [001]) or 8 directions (from [111] family). One of them is randomly selected when the interactions between magnetic ions is strong enough to produce spontaneous magnetization, what is only possible below Curie (ferromagnets) or Neel temperature (antiferromagnets). This happens in nanometer scale samples, spontaneously and uniformly magnetized to saturation. Bigger samples break down into magnetic domains and things become still more complicated. Besides, in magnetically ordered materials (ferromagnets) the response to external field is strongly non-linear and history dependent (hysteresis). Of course, there is clear difference between easy and hard direction of magnetization, even in very weak fields.

Concluding: linear response has nothing to do with magnetocrystalline anisotropy. Even worse: magnetically ordered bodies exhibit also the so called shape anisotropy ... And, yes, very thin layers of Pt are strongly anisotropic - just because of the shape anisotropy.

While it is true for the linear optical polarizability, cubic crystall structure is not equivalent to "isotropy of all physical quantities"! Consider the velocity of sound (longitudinal elastic wave propagation) in cubic symmetry. In general it is not isotropic but different in the low-index (high symmetry) directions [100], [110] and [111]. So, it is important to effectively check, whether a quantity considered actually actually belongs to the class of properties that are (to first order) isotropic in cubic crystals.

Exactly, the presence of inversion symmetry doesn't mean isotropy. The sound propagation in cubic crystals is an excellent example, some directions are even "forbidden" for sound waves propagation.

Kai Fauth sais:

So, it is important to effectively check, whether a quantity considered actually actually belongs to the class of properties that are (to first order) isotropic in cubic crystals.

Rosen response:

A material that is not isotropic to second order, but is isotropic to first order, is weakly anisotropic. Cubic crystals are not precisely isotropic.

The anisotropy of cubic crystals (i.e., cubic-lattice) makes a difference with regards to the effective-mass of the valence-hole. In a hypothetically isotropic crystal, the valence-hole band is split into an upper band and a lower band. The lower band is sometimes referred to as the split-off band. The effective mass-of the upper band is degenerate at all values of pseudo-momentum.

Anisotropy lifts the degeneracy of the upper band so that there are two effective-masses. The upper band becomes the heavy-hole band and the light hole band. The anisotropy splits the degeneracy even in a cubic-lattice crystal.

Birefringence is an optical effect (manifesting as the observed difference in refractive indices for two orthogonally polarised light waves of the same wavelength and propagation direction). It is a consequence of aniaxial optical anisotropy. Anisotropy itself is a much more general notion and relates to the description of direction-dependent physical properties (optical, magnetic, mechanical etc)

Correction for the previous answer: initially the ordinary and extraordinary waves propagated along the same direction, then the paths split. Also, uniaxial optical anisotropy.

The easiest way to think of this is that the the two different indices are the axis of an ellipsoid. When you align the axis of the ellipsoid in relation to the crystal structure you will find that the spacing of the atoms will be different along one axis compared to the other axis. This is the connection to between the material anisotropy and the refractive index. if you shine polarized light along the each axis, you will find the velocity to be different for each axis, so you have a different refractive index along each axis. If you rotate the crystal at some angle from this position, and do the same experiment you will find the refractive index to be different. You might want to do this to delay one polarization of light by a specific amount compared to another polarization. A 1/4 wave plate might be used to convert a linearly polarized light to a circularly polarized light.

It depends on how you use it. Orienting the axis so the reflection is lowest would help with light absorption. You should at least consider the refractive index orientation when putting an anti reflection coating on the material.

http://microscopy.berkeley.edu/courses/TLM/plm/birefr.html

Birefringent is when two axes are not the same internal refraction charateristics. For example the space in X is not as isotropic than Y axe. Isotropic is more general than birefringence. Isotropic means that all material caracterisitics are the same on X, Y Z, T, Spin, etc... Birefingence, therefore, is only when refraction is not the same on X and Y axes. Then you can calculate refraction indice n1=c/v1 on X axe and n2=c/v2 is refraction indice for Y axe. When you send an unique laser beam on the first face then a part of light propagates on X trajectory where n1.sin(theta1) define its internal angle. Then a second part of light propagates on Y axes with another angle n2.sin(theta2). Then  when light(s) "exit" from the birefringent material you got two "lights".