Editing Optical rotation (section)

== Theory ==

Optical activity occurs due to molecules dissolved in a fluid or due to the fluid itself only if the molecules are one of two (or more) [[stereoisomer]]s; this is known as an [[enantiomer]]. The structure of such a molecule is such that it is ''not'' identical to its [[mirror image]] (which would be that of a different stereoisomer, or the "opposite enantiomer"). In mathematics, this property is also known as [[chirality]]. For instance, a metal rod is ''not'' chiral, since its appearance in a mirror is not distinct from itself. However a screw or light bulb base (or any sort of [[helix]]) ''is'' chiral; an ordinary right-handed screw thread, viewed in a mirror, would appear as a left-handed screw (very uncommon) which could not possibly screw into an ordinary (right-handed) nut. A human viewed in a mirror would have their heart on the right side, clear evidence of chirality, whereas the mirror reflection of a doll might well be indistinguishable from the doll itself.

In order to display optical activity, a fluid must contain only one, or a preponderance of one, stereoisomer. If two enantiomers are present in equal proportions, then their effects cancel out and no optical activity is observed; this is termed a [[racemic]] mixture. But when there is an [[enantiomeric excess]], more of one enantiomer than the other, the cancellation is incomplete and optical activity is observed. Many naturally occurring molecules are present as only one enantiomer (such as many sugars). Chiral molecules produced within the fields of [[organic chemistry]] or [[inorganic chemistry]] are racemic unless a chiral reagent was employed in the same reaction.

At the fundamental level, polarization rotation in an optically active medium is caused by circular birefringence, and can best be understood in that way. Whereas [[birefringence|linear birefringence]] in a crystal involves a small difference in the [[phase velocity]] of light of two different linear polarizations, circular birefringence implies a small difference in the velocities between right and left-handed ''[[circular polarization]]s''.<ref name=fresnel-1822z /> Think of one enantiomer in a solution as a large number of little helices (or screws), all right-handed, but in random orientations. Birefringence of this sort is possible even in a fluid because the handedness of the helices is not dependent on their orientation: even when the direction of one helix is reversed, it still appears right handed. And circularly polarized light itself is chiral: as the wave proceeds in one direction the electric (and magnetic) fields composing it are rotating clockwise (or counterclockwise for the opposite circular polarization), tracing out a right (or left) handed screw pattern in space. In addition to the bulk [[refractive index]] which substantially lowers the phase velocity of light in any dielectric (transparent) material compared to the [[speed of light]] (in vacuum), ''there is an additional interaction between the chirality of the wave and the chirality of the molecules.'' Where their chiralities are the same, there will be a small additional effect on the wave's velocity, but the opposite circular polarization will experience an opposite small effect as its chirality is opposite that of the molecules.

Unlike linear birefringence, however, natural optical rotation (in the absence of a magnetic field) cannot be explained in terms of a local material [[permittivity]] tensor (i.e., a charge response that only depends on the local electric field vector), as symmetry considerations forbid this. Rather, circular birefringence only appears when considering nonlocality of the material response, a phenomenon known as [[spatial dispersion]].<ref name="landau">{{cite book
 |author1=L. D. Landau
 |author2-link=Evgeny Lifshitz
 |author2=E. M. Lifshitz
 |author3-link=Lev Pitaevskii
 |author3=L. P. Pitaevskii
 |year=1984
 |title=Electrodynamics of Continuous Media
 |edition=2nd |volume=8
 |publisher=[[Butterworth-Heinemann]]
 |isbn=978-0-7506-2634-7
 |pages=362–365
|author1-link= Lev Landau}}</ref> Nonlocality means that electric fields in one location of the material drive currents in another location of the material. Light travels at a finite speed, and even though it is much faster than the electrons, it makes a difference whether the charge response naturally wants to travel along with the electromagnetic wavefront, or opposite to it. Spatial dispersion means that light travelling in different directions (different wavevectors) sees a slightly different permittivity tensor. Natural optical rotation requires a special material, but it also relies on the fact that the wavevector of light is nonzero, and a nonzero wavevector bypasses the symmetry restrictions on the local (zero-wavevector) response. However, there is still reversal symmetry, which is why the direction of natural optical rotation must be 'reversed' when the direction of the light is reversed, in contrast to magnetic [[Faraday rotation]]. All optical phenomena have some nonlocality/wavevector influence but it is usually negligible; natural optical rotation, rather uniquely, absolutely requires it.<ref name="landau"/>

The phase velocity of light in a medium is commonly expressed using the [[index of refraction]] ''n'', defined as the speed of light (in free space) divided by its speed in the medium. The difference in the refractive indices between the two circular polarizations quantifies the strength of the circular birefringence (polarization rotation),
: <math>\Delta n = n_\text{RHC} - n_\text{LHC}.</math>
While <math>\Delta n</math> is small in natural materials, examples of giant circular birefringence resulting in a negative refractive index for one circular polarization have been reported for chiral metamaterials.<ref>{{Cite journal| last = Plum| first =E.|author2=Zhou, J. |author3=Dong, J. |author4=Fedotov, V. A. |author5=Koschny, T. |author6=Soukoulis, C. M. |author7=Zheludev, N. I.  | title =Metamaterial with negative index due to chirality| journal =Physical Review B| volume =79| page =035407| year =2009| issue =3| doi =10.1103/PhysRevB.79.035407| arxiv =0806.0823| bibcode =2009PhRvB..79c5407P| s2cid =119259753| url =https://eprints.soton.ac.uk/65777/1/4174.pdf}}</ref>
<ref>{{Cite journal| last = Zhang| first =S.|author2=Park, Y.-S. |author3=Li, J. |author4=Lu, X. |author5=Zhang, W. |author6=Zhang, X.| title =Negative Refractive Index in Chiral Metamaterials| journal =Physical Review Letters| volume =102| page =023901| year =2009| issue =2| doi =10.1103/PhysRevLett.102.023901| pmid =19257274| bibcode =2009PhRvL.102b3901Z}}</ref>

The familiar rotation of the axis of ''linear'' polarization relies on the understanding that a linearly polarized wave can as well be described as the [[Superposition principle|superposition]] (addition) of a left and right circularly polarized wave in equal proportion. The phase difference between these two waves is dependent on the orientation of the linear polarization which we'll call <math>\theta_0</math>, and their electric fields have a relative phase difference of <math>2\theta_0</math> which then add to produce linear polarization:
: <math>\mathbf{E}_{\theta_0} = \frac{\sqrt{2}}{2} \big(e^{-i\theta_0} \mathbf{E}_\text{RHC} + e^{i\theta_0} \mathbf{E}_\text{LHC}\big),</math>
where <math>\mathbf{E}_{\theta_0}</math> is the [[electric field]] of the net wave, while <math>\mathbf{E}_\text{RHC}</math> and <math>\mathbf{E}_\text{LHC}</math> are the two circularly polarized [[Basis (linear algebra)|basis functions]] (having zero phase difference). Assuming propagation in the +''z'' direction, we could write <math>\mathbf{E}_\text{RHC}</math> and <math>\mathbf{E}_\text{LHC}</math> in terms of their ''x'' and ''y'' components as follows:
: <math>\mathbf{E}_\text{RHC} = \frac{\sqrt{2}}{2} (\hat{x} + i \hat{y}),</math>
: <math>\mathbf{E}_\text{LHC} = \frac{\sqrt{2}}{2} (\hat{x} - i \hat{y}),</math>
where <math>\hat{x}</math> and <math>\hat{y}</math> are unit vectors, and ''i'' is the [[imaginary unit]], in this case representing the 90-degree phase shift between the ''x'' and ''y'' components that we have decomposed each circular polarization into. As usual when dealing with [[phasor]] notation, it is understood that such quantities are to be multiplied by <math> e^{-i\omega t} </math> and then the actual electric field at any instant is given by the ''real part'' of that product.

Substituting these expressions for <math>\mathbf{E}_\text{RHC}</math> and <math>\mathbf{E}_\text{LHC}</math> into the equation for <math>\mathbf{E}_{\theta_0},</math> we obtain
: <math>\begin{align}
 \mathbf{E}_{\theta_0} &= \frac{\sqrt{2}}{2} \big(e^{-i\theta_0} \mathbf{E}_\text{RHC} + e^{i\theta_0} \mathbf{E}_\text{LHC}\big) \\
 &= \frac{1}{2} \big[\hat{x} \big(e^{-i\theta_0} + e^{i\theta_0}\big) + \hat{y} i \big(e^{-i\theta_0} - e^{i\theta_0}\big)\big] \\
 &= \hat{x} \cos\theta_0 + \hat{y} \sin\theta_0.
\end{align}</math>
The last equation shows that the resulting vector has the ''x'' and ''y'' components in phase and oriented exactly in the <math>\theta_0</math> direction, as we had intended, justifying the representation of any linearly polarized state at angle <math>\theta</math> as the superposition of right and left circularly polarized components with a relative phase difference of <math>2\theta</math>. Now let us assume transmission through an optically active material which induces an additional phase difference between the right and left circularly polarized waves of <math>2\Delta \theta</math>. Let us call <math>\mathbf{E}_\text{out}</math> the result of passing the original wave linearly polarized at angle <math>\theta</math> through this medium. This will apply additional phase factors of <math>-\Delta \theta</math> and <math>\Delta \theta</math> to the right and left circularly polarized components of <math>\mathbf{E}_{\theta_0}</math>:
: <math>\mathbf{E}_\text{out} = \frac{\sqrt{2}}{2} \big(e^{-i\Delta\theta} e^{-i\theta_0} \mathbf{E}_\text{RHC} + e^{i\Delta\theta} e^{i\theta_0} \mathbf{E}_\text{LHC}\big).</math>
Using similar math as above, we find
: <math>\mathbf{E}_\text{out} = \hat{x} \cos(\theta_0 + \Delta\theta) + \hat{y} \sin(\theta_0 + \Delta\theta),</math>
describing a wave linearly polarized at angle <math>\theta_0 + \Delta\theta</math>, thus rotated by <math>\Delta\theta</math> relative to the incoming wave <math>\mathbf{E}_{\theta_0}.</math>

We defined above the difference in the refractive indices for right and left circularly polarized waves of <math>\Delta n</math>. Considering propagation through a length ''L'' in such a material, there will be an additional phase difference induced between them of <math>2\Delta \theta</math> (as we used above) given by
: <math>2\Delta \theta = \frac{\Delta n L2\pi}{\lambda},</math>
where <math>\lambda</math> is the wavelength of the light (in vacuum). This will cause a rotation of the linear axis of polarization by <math>\Delta \theta</math> as we have shown.

In general, the refractive index depends on wavelength (see [[Dispersion (optics)|dispersion]]) and the differential refractive index <math>\Delta n</math> will also be wavelength dependent. The resulting variation in rotation with the wavelength of the light is called [[optical rotatory dispersion]] (ORD). ORD spectra and [[circular dichroism]] spectra are related through the [[Kramers–Kronig relation]]s. Complete knowledge of one spectrum allows the calculation of the other.

So we find that the degree of rotation depends on the color of the light (the yellow sodium D line near 589&nbsp;nm [[wavelength]] is commonly used for measurements) and is directly proportional to the path length <math>L</math> through the substance and the amount of circular birefringence of the material <math>\Delta n</math> which, for a solution, may be computed from the substance's [[specific rotation]] and its concentration in solution.

Although optical activity is normally thought of as a property of fluids, particularly [[aqueous solutions]], it has also been observed in crystals such as [[quartz]] (SiO<sub>2</sub>). Although quartz has a substantial linear birefringence, that effect is cancelled when propagation is along the [[Optic axis of a crystal|optic axis]]. In that case, rotation of the plane of polarization is observed due to the relative rotation between crystal planes, thus making the crystal formally chiral as we have defined it above. The rotation of the crystal planes can be right or left-handed, again producing opposite optical activities. On the other hand,  [[amorphous]] forms of [[silica]] such as [[fused quartz]], like a racemic mixture of chiral molecules, has no net optical activity since one or the other crystal structure does not dominate the substance's internal molecular structure.