View source for Optical rotation

{{Short description|Rotation of the plane of linearly polarized light as it travels through a chiral material}}
{{Distinguish|circularly polarized light}}
[[File:Polarimeter (Optical rotation).svg|upright=1.5|thumb| Operating principle of a [[polarimeter]] for measuring optical rotation.
{{ordered list|
|Light source
|Unpolarized light
|[[Linear polarizer]]
|Linearly polarized light
|Sample tube containing molecules under study
|Optical rotation due to molecules
|Rotatable [[Polarimeter|linear analyzer]]
|Detector
}}]]

'''Optical rotation''', also known as '''polarization rotation''' or '''circular birefringence''', is the rotation of the orientation of the plane of [[polarization (waves)|polarization]] about the optical axis of [[linear polarization|linearly polarized]] light as it travels through certain materials. Circular birefringence and [[circular dichroism]] are the manifestations of '''optical activity'''. Optical activity occurs only in [[chiral]] materials, those lacking microscopic mirror symmetry. Unlike other sources of [[birefringence]] which alter a beam's state of polarization, optical activity can be observed in [[fluid]]s. This can include gases or solutions of [[chirality (chemistry)|chiral molecules]] such as sugars, molecules with helical [[secondary structure]] such as some proteins, and also [[Liquid crystal#Chiral phases or twisted nematics|chiral liquid crystals]].  It can also be observed in chiral solids such as certain crystals with a rotation between adjacent [[crystal]] planes (such as [[quartz]]) or [[Metamaterial#Chiral|metamaterial]]s.

When looking at the source of light, the rotation of the plane of polarization may be either to the right ('''dextrorotatory''' or '''dextrorotary''' — ''d''-rotary, represented by (+), clockwise), or to the left ('''levorotatory''' or '''levorotary''' — ''l''-rotary, represented by (−), counter-clockwise) depending on which [[stereoisomer]] is dominant. For instance, [[sucrose]] and [[camphor]] are ''d''-rotary whereas [[cholesterol]] is ''l''-rotary. For a given substance, the angle by which the polarization of light of a specified wavelength is rotated is proportional to the path length through the material and (for a solution) proportional to its concentration.

Optical activity is measured using a polarized source and [[polarimeter]]. This is a tool particularly used in the [[sugar industry]] to measure the sugar concentration of syrup, and generally in chemistry to measure the concentration or [[enantiomer|enantiomeric ratio]] of chiral molecules in solution. Modulation of a liquid crystal's optical activity, viewed between two sheet [[polarizer#Absorptive polarizers|polarizers]], is the principle of operation of [[liquid-crystal display]]s (used in most modern televisions and computer monitors).

== Forms ==
{{Redirect|Levo|the company|Levo (company)}}

'''Dextrorotation''' and '''laevorotation''' (also spelled '''levorotation''')<ref name="word">The first word component ''[[wikt:dextro-|dextro-]]'' comes from the [[Latin language|Latin]] word ''[[wiktionary:dexter|dexter]]'', meaning "right" (as opposed to left). ''[[wikt:laevo-|Laevo-]]'' or ''[[wikt:levo-|levo-]]'' comes from the Latin ''[[wiktionary:laevus|laevus]]'', meaning "left side".</ref><ref>The equivalent French terms are [[:fr:Dextrogyre (chimie)|''dextrogyre'']] and [[:fr:Lévogyre (chimie)|''lévogyre'']]. These are used infrequently in English, but sometimes they are, see e.g. [https://www.researchgate.net/publication/222099348_From_random_Screening_of_Chemical_Libraries_to_the_Optimization_of_FPP-competitive_Inhibitors_of_Farnesyl_Transferase Patrick Mailliet et al.], page 12 and 15.</ref> in [[chemistry]] and [[physics]] are the optical rotation of [[plane-polarized light]]. From the point of view of the observer, ''dextrorotation'' refers to clockwise or right-handed rotation, and ''laevorotation'' refers to counterclockwise or left-handed rotation.<ref>[https://chem.libretexts.org/Textbook_Maps/Organic_Chemistry_Textbook_Maps/Map%3A_Organic_Chemistry_(McMurry)/Chapter_05%3A_Stereochemistry_at_Tetrahedral_Centers/5.03_Optical_Activity#Polarimetry LibreTexts Chemistry – Polarimetry]</ref><ref>{{Cite book|url=http://apps.who.int/phint/|title=The International Pharmacopoeia|publisher=World Health Organization|year=2017|isbn=9789241550031|chapter=Determination of optical rotation and specific rotation|chapter-url=http://apps.who.int/phint/pdf/b/7.1.4.1.4-Determination-of-optical-rotation-and-specific-ro_.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://apps.who.int/phint/pdf/b/7.1.4.1.4-Determination-of-optical-rotation-and-specific-ro_.pdf |archive-date=2022-10-09 |url-status=live}}</ref>

A [[chemical compound]] that causes dextrorotation is ''dextrorotatory'' or ''dextrorotary'', while a compound that causes laevorotation is ''laevorotatory'' or ''laevorotary''.<ref name="Solomons, T.W. Graham 2008">{{cite book |last1=Solomons|first1= T.W. Graham | last2=Fryhle|first2=Graig B. | title = Organic Chemistry | edition = 9th | location = Hoboken | publisher = [[John Wiley & Sons, Inc.]] | year = 2008 | url = https://archive.org/details/organicchemistry0009edsolo_e2n2/page/195/mode/1up?q=dextrorotatory | page = 195 | isbn = 9780471684961 }}</ref> Compounds with these properties consist of [[Chirality (chemistry)|chiral]] molecules and are said to have optical activity. If a chiral molecule is dextrorotary, its [[enantiomer]] (geometric mirror image) will be laevorotary, and vice versa. Enantiomers rotate plane-polarized light the same number of degrees, but in opposite directions.

===Chirality prefixes===
{{Main|Chirality (chemistry)}}
{{See also|Absolute configuration|Aldose}}

A compound may be labeled as dextrorotary by using the "(+)-" or "''d''-" prefix. Likewise, a levorotary compound may be labeled using the "(−)-" or  "''l''-" prefix. The lowercase "''d''-" and "''l''-" prefixes are obsolete, and are distinct from the <small>[[Small caps|SMALL CAPS]]</small> "<small>D</small>-" and "<small>L</small>-" prefixes. The "<small>D</small>-" and "<small>L</small>-" prefixes are used to specify the enantiomer of chiral [[organic compounds]] in [[biochemistry]] and are based on the compound's [[absolute configuration]] relative to (+)-[[glyceraldehyde]], which is the <small>D</small>-form by definition. The prefix used to indicate absolute configuration is not directly related to the (+) or (−) prefix used to indicate optical rotation in the same molecule. For example, nine of the nineteen <small>L</small>-[[amino acid]]s naturally occurring in proteins are, despite the <small>L</small>- prefix, actually dextrorotary (at a wavelength of 589&nbsp;nm), and <small>D</small>-[[fructose]] is sometimes called "levulose" because it is levorotary.

The <small>D</small>- and <small>L</small>- prefixes describe the molecule as a whole, as do the (+) and (−) prefixes for optical rotation. In contrast, the (''R'')- and (''S'')- prefixes from the [[Cahn–Ingold–Prelog priority rules]] characterize the [[absolute configuration]] of each specific chiral [[stereocenter]] with the molecule, rather than a property of the molecule as a whole. A molecule having exactly one chiral stereocenter (usually an [[asymmetric carbon]] atom) can be labeled (''R'') or (''S''), but a molecule having multiple stereocenters needs more than one label. For example, the essential amino acid [[Threonine|<small>L</small>-threonine]] contains two chiral stereocenters and is written (2''S'',3''S'')-threonine. There is no strict relationship between the R/S, the <small>D</small>/<small>L</small>, and (+)/(−) designations, although some correlations exist. For example, of the naturally occurring amino acids, all are <small>L</small>, and most are (''S''). For some molecules the (''R'')-enantiomer is the dextrorotary (+) enantiomer, and in other cases it is the levorotary (−) enantiomer. The relationship must be determined on a case-by-case basis with experimental measurements or detailed computer modeling.<ref name=Stephens>See, for example, {{cite journal | doi = 10.1002/chir.10270 | title = Determination of absolute configuration using calculation of optical rotation | year = 2003 | last1 = Stephens | first1 = P. J. | last2 = Devlin | first2 = F. J. | last3 = Cheeseman | first3 = J. R. | last4 = Frisch | first4 = M. J. | last5 = Bortolini | first5 = O. | last6 = Besse | first6 = P. | journal = Chirality | volume = 15 | pages = S57–64 | pmid = 12884375}}</ref>

== History ==
[[Image:TartrateCrystal.svg|thumb|The two asymmetric crystal forms, dextrorotatory and levorotatory, of [[tartaric acid]].]]
[[Image:Sucrose solution and polaroid (optical activity).jpg|thumb|Sucrose solution concentration measuring experiment, demonstrating optical rotation.]]
The rotation of the orientation of [[linear polarization|linearly polarized]] light was first observed in 1811 in [[quartz]] by French physicist [[François Arago]].<ref>Arago (1811) [https://babel.hathitrust.org/cgi/pt?id=ucm.5326746608;view=1up;seq=103 "Mémoire sur une modification remarquable qu'éprouvent les rayons lumineux dans leur passage à travers certains corps diaphanes et sur quelques autres nouveaux phénomènes d'optique"] (Memoir on a remarkable modification that light rays experience during their passage through certain translucent substances and on some other new optical phenomena), ''Mémoires de la classe des sciences mathématiques et physiques de l'Institut Impérial de France'', 1st part : 93–134.</ref> In 1820, the English astronomer [[John Herschel|Sir John F.W. Herschel]] discovered that different individual quartz crystals, whose crystalline structures are mirror images of each other (see illustration), rotate linear polarization by equal amounts but in opposite directions.<ref>Herschel, J.F.W. (1820) [https://babel.hathitrust.org/cgi/pt?id=nyp.33433004518324;view=1up;seq=87 "On the rotation impressed by plates of rock crystal on the planes of polarization of the rays of light, as connected with certain peculiarities in its crystallization,"] ''Transactions of the Cambridge Philosophical Society'', '''1''' :  43–51.</ref> [[Jean Baptiste Biot]] also observed the rotation of the axis of polarization in certain liquids<ref>Biot, J. B. (1815) [https://www.biodiversitylibrary.org/item/26553#page/196/mode/1up "Phenomene de polarisation successive, observés dans des fluides homogenes"] (Phenomenon of successive polarization, observed in homogeneous fluids), ''Bulletin des Sciences, par la Société Philomatique de Paris'', 190–192.</ref> and vapors of organic substances such as [[turpentine]].<ref>Biot (1818 & 1819) "Extrait d'un mémoire sur les rotations que certaines substances impriment aux axes de polarisation des rayons lumineux" (Extract from a memoir on the [optical] rotations that certain substances impress on the axes of polarization of light rays), ''Annales de Chimie et de Physique'', 2nd series, '''9''' :  [https://babel.hathitrust.org/cgi/pt?id=hvd.hx3dvg;view=1up;seq=384 372–389]; '''10''' :  [https://babel.hathitrust.org/cgi/pt?id=ien.35556014127617;view=1up;seq=67 63–81]; for Biot's experiments with turpentine vapor (''vapeur d'essence de térébenthine''), see pp. 72–81.</ref> In 1822, [[Augustin-Jean Fresnel]] found that optical rotation could be explained as a species of [[birefringence]]: whereas previously known cases of birefringence were due to the different speeds of light polarized in two perpendicular planes, optical rotation was due to the different speeds of right-hand and left-hand circularly polarized light.<ref name=fresnel-1822z>A. Fresnel, "Mémoire sur la double réfraction que les rayons lumineux éprouvent en traversant les aiguilles de cristal de roche suivant les directions parallèles à l'axe", read 9&nbsp;December 1822; printed in H.&nbsp;de Senarmont, E.&nbsp;Verdet, and L.&nbsp;Fresnel (eds.), ''Oeuvres complètes d'Augustin Fresnel'', vol.&nbsp;1 (1866), pp.{{nnbsp}}731–751; translated as "Memoir on the double refraction that light rays undergo in traversing the needles of quartz in the directions parallel to the axis", {{Zenodo|4745976}}, 2021 (open&nbsp;access); especially §13.</ref> Simple [[polarimeter]]s have been used since this time to measure the concentrations of simple sugars, such as [[glucose]], in solution. In fact one name for <small>D</small>-glucose (the biological isomer), is ''dextrose'', referring to the fact that it causes linearly polarized light to rotate to the right or [[wiktionary:dexter|dexter]] side. In a similar manner, levulose, more commonly known as [[fructose]], causes the [[plane of polarization]] to rotate to the left. Fructose is even more strongly levorotatory than glucose is dextrorotatory. [[Inverted sugar syrup|Invert sugar syrup]], commercially formed by the [[hydrolysis]] of [[sucrose]] syrup to a mixture of the component simple sugars, fructose, and glucose, gets its name from the fact that the conversion causes the direction of rotation to "invert" from right to left.

In 1849, [[Louis Pasteur]] resolved a problem concerning the nature of [[tartaric acid]].<ref>Pasteur, L. (1850) [https://babel.hathitrust.org/cgi/pt?id=hvd.hx3dy7;view=1up;seq=66 "Recherches sur les propriétés spécifiques des deux acides qui composent l'acide racémique"] (Researches on the specific properties of the two acids that compose the racemic acid), ''Annales de chimie et de physique'', 3rd series, '''28''' :  56–99; see also appendix, pp. 99–117.</ref> A solution of this compound derived from living things (to be specific, [[Lees (fermentation)|wine lees]]) rotates the plane of [[Polarization (waves)|polarization]] of light passing through it, but tartaric acid derived by [[chemical synthesis]] has no such effect, even though its reactions are identical and its elemental composition is the same. Pasteur noticed that the crystals come in two asymmetric forms that are mirror images of one another. Sorting the crystals by hand gave two forms of the compound: Solutions of one form rotate polarized light clockwise, while the other form rotate light counterclockwise.  An equal mix of the two has no polarizing effect on light. Pasteur deduced that the molecule in question is asymmetric and could exist in two different forms that resemble one another as would left- and right-hand gloves, and that the organic form of the compound consists of purely the one type.

In 1874, [[Jacobus Henricus van 't Hoff]]<ref>van 't Hoff, J.H. (1874) [https://babel.hathitrust.org/cgi/pt?id=hvd.32044106337231;view=1up;seq=479 "Sur les formules de structure dans l'espace"] (On structural formulas in space),  ''Archives Néerlandaises des Sciences Exactes et Naturelles'', '''9''' :  445–454.</ref> and [[Joseph Achille Le Bel]]<ref>Le Bel, J.-A. (1874) [https://babel.hathitrust.org/cgi/pt?id=hvd.hc1j13;view=1up;seq=345 "Sur les relations qui existent entre les formules atomiques des corps organiques et le pouvoir rotatoire de leurs dissolutions"] (On the relations that exist between the atomic formulas of organic substances and the rotatory power of their solutions), ''Bulletin de la Société Chimique de Paris'', '''22''' : 337–347.</ref> independently proposed that this phenomenon of optical activity in carbon compounds could be explained by assuming that the 4 saturated chemical bonds between carbon atoms and their neighbors are directed towards the corners of a regular tetrahedron. If the 4 neighbors are all different, then there are two possible orderings of the neighbors around the tetrahedron, which will be mirror images of each other. This led to a better understanding of the three-dimensional nature of molecules.

In 1945, Charles William Bunn<ref>{{cite book |last=Bunn |first= C. W.|year=1945 |title=Chemical Crystallography |location=New York |publisher=Oxford University Press |page=88}}</ref> predicted optical activity of achiral structures, if the wave's propagation direction and the achiral structure form an experimental arrangement that is different from its mirror image. Such optical activity due to [[Chirality (electromagnetism)#Extrinsic 3d chirality|extrinsic chirality]] was observed in the 1960s in liquid crystals.<ref>{{cite journal|author=R. Williams|doi=10.1103/PhysRevLett.21.342|title=Optical Rotatory Effect in the Nematic Liquid Phase of p-Azoxyanisole|journal=Physical Review Letters|volume=21|page=342|year=1968|issue=6|bibcode=1968PhRvL..21..342W }}</ref><ref>{{cite journal|author=R. Williams|doi=10.1063/1.1671194|title=Optical-rotary power and linear electro-optic effect in nematic liquid crystals of p-azoxyanisole|journal=Journal of Chemical Physics|volume=50|page=1324|year=1969|issue=3|bibcode=1969JChPh..50.1324W }}</ref>

In 1950, [[Sergey Vavilov]]<ref>{{cite book |last=Vavilov |first= S. I.|year=1950 |title=Mikrostruktura Sveta (Microstructure of Light) |location=Moscow |publisher=USSR Academy of Sciences Publishing }}</ref> predicted optical activity that depends on the intensity of light and the effect of nonlinear optical activity was observed in 1979 in [[lithium iodate]] crystals.<ref>{{Cite journal| last = Akhmanov| first =S. A.|author2=Zhdanov, B. V. |author3=Zheludev, N. I. |author4=Kovrigin, A. I. |author5=Kuznetsov, V. I. | title =Nonlinear optical activity in crystals| journal =JETP Letters| volume =29| page =264| year =1979}}</ref>

Optical activity is normally observed for transmitted light. However, in 1988, M. P. Silverman discovered that polarization rotation can also occur for light reflected from chiral substances.<ref>{{Cite journal| last = Silverman| first =M.|author2=Ritchie, N. |author3=Cushman, G. |author4=Fisher, B. | title =Experimental configurations using optical phase modulation to measure chiral asymmetries in light specularly reflected from a naturally gyrotropic medium | journal =Journal of the Optical Society of America A| volume =5| page =1852| year =1988| issue =11| doi =10.1364/JOSAA.5.001852 | bibcode =1988JOSAA...5.1852S}}</ref> Shortly after, it was observed that chiral media can also reflect left-handed and right-handed circularly polarized waves with different efficiencies.<ref>{{Cite journal| last = Silverman| first =M.|author2=Badoz, J. |author3=Briat, B.| title =Chiral reflection from a naturally optically active medium  | journal =Optics Letters| volume =17| page =886| year =1992| issue =12| doi =10.1364/OL.17.000886 | pmid =19794663| bibcode =1992OptL...17..886S}}</ref> These phenomena of specular circular birefringence and specular circular dichroism are jointly known as specular optical activity. Specular optical activity is very weak in natural materials.

In 1898 [[Jagadish Chandra Bose]] described the ability of twisted artificial structures to rotate the polarization of [[microwave]]s.<ref>{{Cite journal|last = Bose| first =Jagadis Chunder| title =On the Rotation of Plane of Polarisation of Electric Waves by a Twisted Structure| year =1898| doi =10.1098/rspl.1898.0019| jstor =115973|journal = Proceedings of the Royal Society|volume = 63| issue =389–400| pages =146–152| s2cid =89292757}}</ref> Since the early 21st century, the development of artificial materials has led to the prediction<ref>{{Cite journal| last = Svirko| first =Y.|author2=Zheludev, N. I. |author3=Osipov, M.| title =Layered chiral metallic microstructures with inductive coupling| journal =Applied Physics Letters| volume =78| page =498| year =2001| issue =4| doi =10.1063/1.1342210| bibcode =2001ApPhL..78..498S}}</ref> and realization<ref>{{Cite journal| last = Kuwata-Gonokami| first =M.|author2=Saito, N. |author3=Ino, Y. |author4=Kauranen, M. |author5=Jefimovs, K. |author6=Vallius, T. |author7=Turunen, J. |author8=Svirko, Y. | title =Giant Optical Activity in Quasi-Two-Dimensional Planar Nanostructures| journal =Physical Review Letters| volume =95| page =227401| year =2005| issue =22| doi =10.1103/PhysRevLett.95.227401| pmid =16384264| bibcode =2005PhRvL..95v7401K}}</ref><ref>{{Cite journal| last = Plum| first =E.|author2=Fedotov, V. A. |author3=Schwanecke, A. S. |author4=Zheludev, N. I. |author5=Chen, Y. | title =Giant optical gyrotropy due to electromagnetic coupling| journal =Applied Physics Letters| volume =90| page =223113| year =2007| issue =22| doi =10.1063/1.2745203| bibcode =2007ApPhL..90v3113P}}</ref> of chiral metamaterials with optical activity exceeding that of natural media by orders of magnitude in the optical part of the spectrum. Extrinsic chirality associated with oblique illumination of metasurfaces lacking two-fold rotational symmetry has been observed to lead to large linear optical activity in transmission<ref>{{Cite journal| last = Plum| first =E.|author2=Fedotov, V. A. |author3=Zheludev, N. I.  | title =Optical activity in extrinsically chiral metamaterial| journal =Applied Physics Letters| volume =93| page =191911| year =2008| issue =19| doi =10.1063/1.3021082 | arxiv =0807.0523| bibcode =2008ApPhL..93s1911P| s2cid =117891131| url =https://eprints.soton.ac.uk/65831/1/4221.pdf}}</ref> and reflection,<ref>{{Cite journal| last = Plum| first =E.|author2=Fedotov, V. A. |author3=Zheludev, N. I.  | title =Specular optical activity of achiral metasurfaces| journal =Applied Physics Letters| volume =108| page =141905| year =2016| issue =14| doi =10.1063/1.4944775 | bibcode =2016ApPhL.108n1905P| hdl =10220/40854| url =https://eprints.soton.ac.uk/389739/1/specular%2520optical%2520activity%25207rev.pdf}}</ref> as well as nonlinear optical activity exceeding that of lithium iodate by 30 million times.<ref>{{Cite journal| last = Ren| first =M.|author2=Plum, E. |author3=Xu, J. |author4=Zheludev, N. I. | title =Giant nonlinear optical activity in a plasmonic metamaterial| journal =Nature Communications| volume =3| page =833| year =2012| doi =10.1038/ncomms1805 | pmid =22588295| bibcode =2012NatCo...3..833R| doi-access =free}}</ref>

== 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_{RHC}-n_{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| 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}  (e^{-i\theta_0}   \mathbf{E}_{RHC}+e^{i\theta_0}\mathbf{E}_{LHC})  \, \, ,</math>

where <math>\mathbf{E}_{\theta_0}</math> is the [[electric field]] of the net wave, while <math>\mathbf{E}_{RHC}</math> and <math>\mathbf{E}_{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}_{RHC}</math> and <math>\mathbf{E}_{LHC}</math> in terms of their ''x'' and ''y'' components as follows:
:<math>\mathbf{E}_{RHC} = \frac{\sqrt{2}}{2} (\hat{x} + i \hat{y}) </math>

:<math>\mathbf{E}_{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}_{RHC}</math> and <math>\mathbf{E}_{LHC}</math> into the equation for  <math>\mathbf{E}_{\theta_0}</math> we obtain:
:<math>\mathbf{E}_{\theta_0}=   \frac{\sqrt{2}}{2}  (e^{-i\theta_0}   \mathbf{E}_{RHC}+e^{i\theta_0}\mathbf{E}_{LHC}) \, \, </math>

:: <math>=  \frac{1}{2}  (\hat{x} (e^{-i\theta_0} + e^{i\theta_0}) +
 \hat{y} i (e^{-i\theta_0} - e^{i\theta_0}))  \, \, </math>

:: <math>=   \hat{x} \cos(\theta_0) +  \hat{y} \sin(\theta_0) </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}_{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}_{out}=   \frac{\sqrt{2}}{2}  ( e^{-i\Delta\theta}  e^{-i\theta_0}   \mathbf{E}_{RHC}+e^{i\Delta\theta} e^{i\theta_0}\mathbf{E}_{LHC}) \, \, . </math>
Using similar math as above we find:
:<math>\mathbf{E}_{out}=   \hat{x} \cos(\theta_0 +\Delta\theta) +  \hat{y} \sin(\theta_0+\Delta\theta) </math>
thus 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.

== Applications ==
For a pure substance in solution, if the color and path length are fixed and the [[specific rotation]] is known, the observed rotation can be used to calculate the concentration. This usage makes a [[Polarimetry|polarimeter]] a tool of great importance to those trading in or using sugar syrups in bulk.

==Comparison to the Faraday effect==
Rotation of light's [[plane of polarization]] may also occur through the [[Faraday effect]] which involves a static [[magnetic field]]. However, this is a distinct phenomenon that is not classified as "optical activity." Optical activity is reciprocal, i.e. it is the same for opposite directions of wave propagation through an optically active medium, for example clockwise polarization rotation from the point of view of an observer. In case of optically active isotropic media, the rotation is the same for any direction of wave propagation. In contrast, the Faraday effect is non-reciprocal, i.e opposite directions of wave propagation through a Faraday medium will result in clockwise and anti-clockwise polarization rotation from the point of view of an observer. Faraday rotation depends on the propagation direction relative to that of the applied magnetic field. All compounds can exhibit polarization rotation in the presence of an applied magnetic field, provided that (a component of) the magnetic field is oriented in the direction of light propagation. The Faraday effect is one of the first discoveries of the relationship between light and electromagnetic effects.

== See also ==
{{div col|colwidth=30em}}
* [[Cryptochirality]]
* [[Specific rotation]]
* [[Circular dichroism]]
* [[Birefringence]]
* [[Geometric phase]]
* [[Polarization (waves)|Polarization]]
* [[Chirality (chemistry)]]
* [[Chirality (electromagnetism)]]
* [[Polarization rotator]]
* [[Hyper Rayleigh Scattering Optical Activity]]
* [[Raman optical activity]] (ROA)
{{div col end}}

== References ==
{{Reflist}}

==Further reading==
* Eugene Hecht, ''Optics'', 3rd ed., Addison-Wesley, 1998, {{ISBN|0-201-30425-2}}
* [[Akhlesh Lakhtakia]], [http://ebooks.worldscinet.com/ISBN/9789812812100/toc.shtml Beltrami Fields in Chiral Media], World Scientific, Singapore, 1994
* A step by step tutorial on [http://people.bath.ac.uk/vkv23/English/RechercheT2a.htm Optical Rotation]
* Morrison. Robert. T, and Boyd. Robert. N, ''Organic Chemistry'' (6th ed.), Prentice-Hall Inc (1992).

{{Chiral synthesis}}
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[[Category:Polarization (waves)]]
[[Category:Stereochemistry]]