66:
25:
128:
1965:
821:
629:
2006:
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 December 1822; printed in H. de
Senarmont, E. Verdet, and L. Fresnel (eds.),
428:
1079:
967:
512:
264:
816:{\displaystyle |\psi \rangle \ {\stackrel {\mathrm {def} }{=}}\ {\begin{pmatrix}\psi _{x}\\\psi _{y}\end{pmatrix}}={\begin{pmatrix}\cos \theta \exp \left(i\alpha _{x}\right)\\\sin \theta \exp \left(i\alpha _{y}\right)\end{pmatrix}}}
1476:
increases from zero, i.e., assumes positive values, the line evolves into an ellipse that is being traced out in the counterclockwise direction (looking in the direction of the propagating wave); this then corresponds to
1879:
1969:
1259:
305:
1659:
1129:
2011:, vol. 1 (1866), pp. 731–51; translated as "Memoir on the double refraction that light rays undergo in traversing the needles of quartz in the directions parallel to the axis",
1950:
1454:
1413:
234:, with their polarization planes at right angles to each other. Since the electric field can rotate clockwise or counterclockwise as it propagates, elliptically polarized waves exhibit
1899:
1346:
1831:
1569:
1505:
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Any fixed polarization can be described in terms of the shape and orientation of the polarization ellipse, which is defined by two parameters: axial ratio AR and tilt angle
1734:
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1801:
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depends on the handedness of the polarization. Positive indicates left hand polarization, while negative indicates right hand polarization, as defined by IEEE.
578:
146:
860:
traces out an ellipse in the x-y plane. The semi-major and semi-minor axes of the ellipse have lengths A and B, respectively, that are given by
830:. This is the most complete representation of polarized electromagnetic radiation and corresponds in general to elliptical polarization.
1770:. The axial ratio is the ratio of the lengths of the major and minor axes of the ellipse, and is always greater than or equal to one.
1527:
becomes negative from zero, the line evolves into an ellipse that is being traced out in the clockwise direction; this corresponds to
38:
1974:
1840:
263:
1211:
423:{\displaystyle \mathbf {E} (\mathbf {r} ,t)=\left|\mathbf {E} \right|\mathrm {Re} \left\{|\psi \rangle \exp \left\right\}}
1374:. This is the case of superposition of two simple harmonic motions (in phase), one in the x direction with an amplitude
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1074:{\displaystyle B=|\mathbf {E} |{\sqrt {\frac {1-{\sqrt {1-\sin ^{2}(2\theta )\sin ^{2}\beta }}}{2}}}}
962:{\displaystyle A=|\mathbf {E} |{\sqrt {\frac {1+{\sqrt {1-\sin ^{2}(2\theta )\sin ^{2}\beta }}}{2}}}}
507:{\displaystyle \mathbf {B} (\mathbf {r} ,t)={\hat {\mathbf {z} }}\times \mathbf {E} (\mathbf {r} ,t)}
208:
2033:"Chirality-induced polarization effects in the cuticle of scarab beetles: 100 years after Michelson"
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Comparison of
Elliptical Polarization with Linear and Circular Polarizations (YouTube Animation)
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1908:, the axial ratio equals 1 (or 0 dB) and the tilt angle is undefined. For the special case of
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to, the direction of propagation. An elliptically polarized wave may be resolved into two
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Arwin, Hans; Magnusson, Roger; Landin, Jan; Järrendahl, Kenneth (April 21, 2012).
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Alternatively, polarization can be represented as a point on the surface of the
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the semi-major axis makes with the x-axis. This angle can be calculated from
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2096:Électricité et Optique : La Lumière et les Théories Électrodynamiques
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in 1822, before the electromagnetic nature of light waves was known.
2017:
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Sinusoidal plane-wave solutions of the electromagnetic wave equation
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At a fixed point in space (or for fixed z), the electric vector
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1874:{\displaystyle \epsilon =\operatorname {arccot}(\pm AR)}
1185:. The orientation of the ellipse is given by the angle
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1254:{\displaystyle \tan 2\phi =\tan 2\theta \cos \beta }
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may be too technical for most readers to understand
2109:Animation of Elliptical Polarization (on YouTube)
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1762:
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1699:, the wave is left-circularly polarized, and when
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1481:; the semi-major axis is now oriented at an angle
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560:of the wave propagating in the +z direction, and
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1654:{\displaystyle A=B=|\mathbf {E} |/{\sqrt {2}}}
1124:{\displaystyle \beta =\alpha _{y}-\alpha _{x}}
2081:Théorie Mathématique de la Lumière, volume 1
2000:
1920:The reflected light from some beetles (e.g.
1746:Polarization (waves) § Parameterization
641:
368:
1295:. The ellipse collapses to a straight line
271:
53:Learn how and when to remove these messages
1736:, the wave is right-circularly polarized.
1449:{\displaystyle |\mathbf {E} |\sin \theta }
1408:{\displaystyle |\mathbf {E} |\cos \theta }
2016:
252:can be considered to be special cases of
183:Learn how and when to remove this message
165:Learn how and when to remove this message
149:, without removing the technical details.
110:Learn how and when to remove this message
16:Polarization of electromagnetic radiation
73:This article includes a list of general
1894:{\displaystyle \operatorname {arccot} }
1881:. The sign used in the argument of the
833:
517:for the magnetic field, where k is the
2121:
1341:{\displaystyle (A=|\mathbf {E} |,B=0}
256:. This terminology was introduced by
222:in any fixed plane intersecting, and
147:make it understandable to non-experts
2009:Oeuvres complètes d'Augustin Fresnel
1529:right-handed elliptical polarization
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59:
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1479:left-handed elliptical polarization
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79:it lacks sufficient corresponding
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2102:
1826:{\displaystyle 2\times \epsilon }
1564:{\displaystyle \beta =\pm \pi /2}
1500:{\displaystyle \phi \neq \theta }
34:This article has multiple issues.
1968: This article incorporates
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1981:General Services Administration
1912:, the axial ratio is infinite.
267:Elliptical polarization diagram
42:or discuss these issues on the
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1729:{\displaystyle \beta =-\pi /2}
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1598:{\displaystyle \theta =\pi /4}
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612:{\displaystyle |\mathbf {E} |}
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2021:, 2021 (open access); §§9–10.
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1796:{\displaystyle 2\times \tau }
1692:{\displaystyle \beta =\pi /2}
1367:{\displaystyle \phi =\theta }
285:electromagnetic wave equation
2057:10.1080/14786435.2011.648228
1915:
853:{\displaystyle \mathbf {E} }
7:
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1926:) is elliptical polarized.
1178:{\displaystyle \alpha _{y}}
1151:{\displaystyle \alpha _{x}}
283:plane wave solution of the
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546:{\displaystyle \omega =ck}
211:such that the tip of the
209:electromagnetic radiation
1904:For the special case of
1284:{\displaystyle \beta =0}
272:Mathematical description
228:linearly polarized waves
1348:) oriented at an angle
254:elliptical polarization
201:elliptical polarization
94:more precise citations.
2098:, via Internet Archive
2037:Philosophical Magazine
1976:Federal Standard 1037C
1970:public domain material
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1989: (in support of
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1763:{\displaystyle \tau }
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258:Augustin-Jean Fresnel
243:Circular polarization
2129:Polarization (waves)
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2094:H. Poincaré (1901)
2049:2012PMag...92.1583A
1946:Photon polarization
1910:linear polarization
249:linear polarization
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2043:(12): 1583–1599.
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558:angular frequency
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1507:. Similarly, if
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2038:
2034:
2027:
2019:
2014:
2010:
2003:
1999:
1992:
1986:
1982:
1978:
1977:
1971:
1961:
1960:
1952:
1949:
1947:
1944:
1942:
1941:Fresnel rhomb
1939:
1937:
1934:
1933:
1927:
1925:
1924:
1913:
1911:
1907:
1902:
1888:
1865:
1862:
1859:
1853:
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1820:
1817:
1814:
1806:
1790:
1787:
1784:
1776:
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1757:
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1723:
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1715:
1712:
1709:
1706:
1686:
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1678:
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1672:
1664:
1646:
1640:
1621:
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1615:
1612:
1592:
1588:
1584:
1581:
1578:
1558:
1554:
1550:
1547:
1544:
1541:
1532:
1530:
1514:
1494:
1491:
1488:
1480:
1463:
1443:
1440:
1437:
1402:
1399:
1396:
1361:
1358:
1355:
1335:
1332:
1329:
1326:
1308:
1305:
1294:
1278:
1275:
1272:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1208:
1207:
1206:
1192:
1170:
1166:
1143:
1139:
1116:
1112:
1108:
1103:
1099:
1095:
1092:
1065:
1059:
1056:
1051:
1047:
1040:
1037:
1031:
1026:
1022:
1018:
1015:
1010:
1007:
985:
982:
975:
974:
973:
953:
947:
944:
939:
935:
928:
925:
919:
914:
910:
906:
903:
898:
895:
873:
870:
863:
862:
861:
831:
829:
808:
801:
795:
791:
787:
783:
779:
776:
773:
770:
767:
759:
753:
749:
745:
741:
737:
734:
731:
728:
725:
719:
714:
709:
701:
697:
687:
683:
676:
651:
638:
626:
625:
624:
622:
585:
583:
567:
559:
540:
537:
534:
531:
524:
523:
522:
520:
498:
495:
479:
462:
456:
453:
433:
432:
416:
411:
406:
402:
399:
396:
393:
390:
386:
382:
378:
374:
371:
365:
356:
343:
335:
331:
325:
322:
302:
301:
300:
298:
294:
290:
286:
282:
279:
265:
261:
259:
255:
251:
250:
245:
244:
239:
237:
233:
229:
225:
221:
218:describes an
217:
214:
210:
206:
202:
198:
187:
184:
169:
166:
158:
155:February 2010
148:
144:
138:
135:This article
133:
124:
123:
114:
111:
103:
100:November 2018
93:
89:
83:
82:
76:
71:
62:
61:
56:
54:
47:
46:
41:
40:
35:
30:
21:
20:
2040:
2036:
2026:
2008:
2002:
1985:the original
1975:
1936:Ellipsometry
1921:
1919:
1903:
1772:
1749:
1533:
1528:
1478:
1264:
1084:
971:
837:
828:Jones vector
825:
586:
555:
516:
275:
253:
247:
241:
240:
205:polarization
200:
194:
179:
161:
152:
136:
106:
97:
78:
50:
43:
37:
36:Please help
33:
2087:(1892) via
1991:MIL-STD-188
295:fields is (
92:introducing
1957:References
519:wavenumber
281:sinusoidal
75:references
39:improve it
1916:In nature
1860:±
1854:
1845:ϵ
1821:ϵ
1818:×
1805:longitude
1791:τ
1788:×
1758:τ
1716:π
1713:−
1707:β
1679:π
1673:β
1585:π
1579:θ
1551:π
1548:±
1542:β
1515:β
1495:θ
1492:≠
1489:ϕ
1464:β
1444:θ
1441:
1403:θ
1400:
1362:θ
1356:ϕ
1273:β
1249:β
1246:
1240:θ
1234:
1225:ϕ
1219:
1193:ϕ
1167:α
1140:α
1113:α
1109:−
1100:α
1093:β
1060:β
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1041:θ
1032:
1019:−
1011:−
948:β
945:
929:θ
920:
907:−
792:α
780:
774:θ
771:
750:α
738:
732:θ
729:
698:ψ
684:ψ
642:⟩
639:ψ
621:amplitude
532:ω
480:×
474:^
400:ω
397:−
375:
369:⟩
366:ψ
278:classical
236:chirality
45:talk page
2123:Category
2085:Volume 2
2065:13988658
1930:See also
1837:, where
1835:latitude
293:magnetic
289:electric
287:for the
2079:(1889)
2045:Bibcode
2018:4745976
2015::
1833:as the
1803:as the
1777:, with
1665:. When
1456:. When
619:is the
580:is the
556:is the
220:ellipse
203:is the
141:Please
88:improve
2063:
2013:Zenodo
1889:arccot
1851:arccot
1085:where
672:
645:
224:normal
216:vector
77:, but
2061:S2CID
1972:from
972:and
587:Here
2083:and
1807:and
1571:and
1158:and
291:and
276:The
246:and
2053:doi
1534:If
1438:sin
1397:cos
1265:If
1243:cos
1231:tan
1216:tan
1048:sin
1023:sin
936:sin
911:sin
777:exp
768:sin
735:exp
726:cos
372:exp
230:in
207:of
195:In
145:to
2125::
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2051:.
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2039:.
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