1199:
1191:
1641:, it is common to use a full-wave plate designed for green light (a wavelength near 540 nm). Linearly polarized white light which passes through the plate becomes elliptically polarized, except for that green light wavelength, which will remain linear. If a linear polarizer oriented perpendicular to the original polarization is added, this green wavelength is fully extinguished but elements of the other colors remain. This means that under these conditions the plate will appear an intense shade of red-violet, sometimes known as "sensitive tint". This gives rise to this plate's alternative names, the
1614:
1629:
delayed). If the input polarization is 45° to the fast and slow axis, the polarization on those axes are equal. But the phase of the output of the slow axis will be delayed 90° with the output of the fast axis. If not the amplitude but both sine values are displayed, then x and y combined will describe a circle. With other angles than 0° or 45° the values in fast and slow axis will differ and their resultant output will describe an ellipse.
167:
1698:
426:
767:
31:
158:
1669:
A multiple-order waveplate is made from a single birefringent crystal that produces an integer multiple of the rated retardance (for example, a multiple-order half-wave plate may have an absolute retardance of 37λ/2). By contrast, a zero-order waveplate produces exactly the specified retardance. This
1628:
The polarization of the incoming photon (or beam) can be resolved as two polarizations on the x and y axis. If the input polarization is parallel to the fast or slow axis, then there is no polarization of the other axis, so the output polarization is the same as the input (only the phase more or less
52:
Linearly polarized light entering a half-wave plate can be resolved into two waves, parallel and perpendicular to the optic axis of the waveplate. In the plate, the parallel wave propagates slightly slower than the perpendicular one. At the far side of the plate, the parallel wave is exactly half of
1741:
relative to crystal elongation – that is, whether the mineral is "length slow" or "length fast" – based on whether the visible interference colors increase or decrease by one order when the plate is added. Secondly, a slightly more complex procedure allows for a tint plate to be used in conjunction
568:
117:
of light, and the variation of the index of refraction. By appropriate choice of the relationship between these parameters, it is possible to introduce a controlled phase shift between the two polarization components of a light wave, thereby altering its polarization. With an engineered combination
399:
For a single waveplate changing the wavelength of the light introduces a linear error in the phase. Tilt of the waveplate enters via a factor of 1/cos θ (where θ is the angle of tilt) into the path length and thus only quadratically into the phase. For the extraordinary polarization the tilt also
1736:
In practical terms, the plate is inserted between the perpendicular polarizers at an angle of 45 degrees. This allows two different procedures to be carried out to investigate the mineral under the crosshairs of the microscope. Firstly, in ordinary cross polarized light, the plate can be used to
391:
in the denominator in the above equation). Waveplates are thus manufactured to work for a particular range of wavelengths. The phase variation can be minimized by stacking two waveplates that differ by a tiny amount in thickness back-to-back, with the slow axis of one along the fast axis of the
1021:
1670:
can be accomplished by combining two multiple-order wave plates such that the difference in their retardances yields the net (true) retardance of the waveplate. Zero-order waveplates are less sensitive to temperature and wavelength shifts, but are more expensive than multiple-order ones.
287:
Depending on the thickness of the crystal, light with polarization components along both axes will emerge in a different polarization state. The waveplate is characterized by the amount of relative phase, Γ, that it imparts on the two components, which is related to the birefringence
1605:
If the axis of polarization of the incident wave is chosen so that it makes a 0° with the fast or slow axes of the waveplate, then the polarization will not change, so remains linear. If the angle is in between 0° and 45° the resulting wave has an elliptical polarization.
1178:. For linearly polarized light, this is equivalent to saying that the effect of the half-wave plate is to rotate the polarization vector through an angle 2θ; however, for elliptically polarized light the half-wave plate also has the effect of inverting the light's
1609:
A circulating polarization can be visualized as the sum of two linear polarizations with a phase difference of 90°. The output depends on the polarization of the input. Suppose polarization axes x and y parallel with the slow and fast axis of the waveplate:
208:. The ordinary axis is perpendicular to the optic axis. The extraordinary axis is parallel to the optic axis. For a light wave normally incident upon the plate, the polarization component along the ordinary axis travels through the crystal with a speed
112:
is different for light linearly polarized along one or the other of two certain perpendicular crystal axes. The behavior of a waveplate (that is, whether it is a half-wave plate, a quarter-wave plate, etc.) depends on the thickness of the crystal, the
403:
A polarization-independent phase shift of zero order needs a plate with thickness of one wavelength. For calcite the refractive index changes in the first decimal place, so that a true zero order plate is ten times as thick as one wavelength. For
1478:
1325:
809:
1597:
161:
A wave in a uniaxial crystal will separate in two components, one parallel and one perpendicular to the optic axis, that will accumulate phase at different rates. This can be used to manipulate the polarization state of the
1750:
of the mineral. The optic angle (often notated as "2V") can both be diagnostic of mineral type, as well as in some cases revealing information about the variation of chemical composition within a single mineral type.
762:{\displaystyle \mathbf {E} \,\mathrm {e} ^{i(kz-\omega t)}=E\,\mathbf {\hat {p}} \,\mathrm {e} ^{i(kz-\omega t)}=E(\cos \theta \,\mathbf {\hat {f}} +\sin \theta \,\mathbf {\hat {s}} )\mathrm {e} ^{i(kz-\omega t)},}
356:
392:
other. With this configuration, the relative phase imparted can be, for the case of a quarter-wave plate, one-fourth a wavelength rather than three-fourths or one-fourth plus an integer. This is called a
1198:
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is chosen so that the phase shift between polarization components is Γ = π/2. Now suppose a linearly polarized wave is incident on the crystal. This wave can be written as
149:
within the visible crystal sections. This alignment can allow discrimination between minerals which otherwise appear very similar in plane polarized and cross polarized light.
1190:
1374:
1224:
400:
changes the refractive index to the ordinary via a factor of cos θ, so combined with the path length, the phase shift for the extraordinary light due to tilt is zero.
53:
a wavelength delayed relative to the perpendicular wave, and the resulting combination is a mirror-image of the entry polarization state (relative to the optic axis).
1510:
376:
formalism, which uses a vector to represent the polarization state of light and a matrix to represent the linear transformation of a waveplate or polarizer.
1623:
118:
of two birefringent materials, an achromatic waveplate can be manufactured such that the spectral response of its phase retardance can be nearly flat.
1637:
A full-wave plate introduces a phase difference of exactly one wavelength between the two polarization directions, for one wavelength of light. In
1016:{\displaystyle E(\cos \theta \,\mathbf {\hat {f}} -\sin \theta \,\mathbf {\hat {s}} )\mathrm {e} ^{i(kz-\omega t)}=E\mathrm {e} ^{i(kz-\omega t)}.}
445:
is chosen so that the phase shift between polarization components is Γ = π. Now suppose a linearly polarized wave with polarization vector
387:, this is negligible compared to the variation in phase difference according to the wavelength of the light due to the fixed path difference (λ
182:
crystal with a carefully chosen orientation and thickness. The crystal is cut into a plate, with the orientation of the cut chosen so that the
798:
lies along the waveplate's slow axis. The effect of the half-wave plate is to introduce a phase shift term e = e = −1 between the
302:
1120:
is −θ. Evidently, the effect of the half-wave plate is to mirror the wave's polarization vector through the plane formed by the vectors
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If the axis of polarization of the incident wave is chosen so that it makes a 45° with the fast and slow axes of the waveplate, then
89:, which converts between different elliptical polarizations (such as the special case of converting from linearly polarized light to
412:
the refractive index changes in the second decimal place and true zero order plates are common for wavelengths above 1 μm.
1824:
1910:
1968:
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1973:
1963:
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denotes the polarization vector of the wave exiting the waveplate, then this expression shows that the angle between
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of the crystal is parallel to the surfaces of the plate. This results in two axes in the plane of the cut: the
157:
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between two perpendicular polarization components of the light wave. A typical waveplate is simply a
121:
A common use of waveplates—particularly the sensitive-tint (full-wave) and quarter-wave plates—is in
70:
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are real. The effect of the quarter-wave plate is to introduce a phase shift term e =e =
384:
109:
82:
1622:
1473:{\displaystyle (E_{f}\mathbf {\hat {f}} +iE_{s}\mathbf {\hat {s}} )\mathrm {e} ^{i(kz-\omega t)}.}
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axes are the quarter-wave plate's fast and slow axes, respectively, the wave propagates along the
1884:
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1320:{\displaystyle (E_{f}\mathbf {\hat {f}} +E_{s}\mathbf {\hat {s}} )\mathrm {e} ^{i(kz-\omega t)},}
409:
146:
122:
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Stacking a series of different-order waveplates with polarization filters between them yields a
1820:
244:. This leads to a phase difference between the two components as they exit the crystal. When
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1592:{\displaystyle E(\mathbf {\hat {f}} +i\mathbf {\hat {s}} )\mathrm {e} ^{i(kz-\omega t)},}
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The sensitive-tint (full-wave) and quarter-wave plates are widely used in the field of
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1957:
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226:, while the polarization component along the extraordinary axis travels with a speed
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denote the propagation axis of the wave. The electric field of the incident wave is
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138:
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components of the wave, so that upon exiting the crystal the wave is now given by
806:
components of the wave, so that upon exiting the crystal the wave is now given by
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Creating circular polarization using a quarter-wave plate and a polarizing filter
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easier, in particular by allowing deduction of the shape and orientation of the
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126:
1729:, in particular by allowing deduction of the shape and orientation of the
17:
1649:. These plates are widely used in mineralogy to aid in identification of
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134:
30:
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101:
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351:{\displaystyle \Gamma ={\frac {2\pi \,\Delta n\,L}{\lambda _{0}}},}
77:
wave travelling through it. Two common types of waveplates are the
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424:
165:
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74:
1617:
Composition of two linearly polarized waves, phase shifted by π/2
105:
1873:. Vol. 1. New York: John Wiley & Sons. p. 121.
1869:
Winchell, Newton Horace; Winchell, Alexander Newton (1922).
1840:
474:
is incident on the crystal. Let θ denote the angle between
1194:
Two waves differing by a quarter-phase shift for one axis
1677:. Either the filters can be rotated, or the waveplates
1504:, and the resulting wave upon exiting the waveplate is
1871:
Elements of
Optical Mineralogy: Principles and Methods
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1206:For a quarter-wave plate, the relationship between
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1713:. Addition of plates between the polarizers of a
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433:For a half-wave plate, the relationship between
81:, which rotates the polarization direction of
1864:
1862:
1860:
1705:photographed using a petrographic microscope.
8:
1943:Encyclopedia of Laser Physics and Technology
1737:distinguish the orientation of the optical
1717:makes easier the optical identification of
38: Electric field parallel to optic axis
44: Electric field perpendicular to axis
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1665:Multiple-order vs. zero-order waveplates
1483:The wave is now elliptically polarized.
429:A wave passing through a half-wave plate
29:
1810:
1808:
1806:
1802:
1746:techniques to allow measurement of the
1693:Use in mineralogy and optical petrology
365:is the vacuum wavelength of the light.
262:, the extraordinary axis is called the
1602:and the wave is circularly polarized.
1733:within the visible crystal sections.
170:A waveplate mounted in a rotary mount
7:
1084:{\displaystyle \mathbf {\hat {p}} '}
1050:{\displaystyle \mathbf {\hat {p}} '}
266:and the ordinary axis is called the
133:makes the optical identification of
96:Waveplates are constructed out of a
1685:layers, to obtain a widely tunable
1171:{\displaystyle \mathbf {\hat {z}} }
1142:{\displaystyle \mathbf {\hat {f}} }
1113:{\displaystyle \mathbf {\hat {f}} }
791:{\displaystyle \mathbf {\hat {s}} }
554:{\displaystyle \mathbf {\hat {f}} }
525:{\displaystyle \mathbf {\hat {f}} }
496:{\displaystyle \mathbf {\hat {p}} }
467:{\displaystyle \mathbf {\hat {p}} }
1689:in optical transmission spectrum.
1633:Full-wave, or sensitive-tint plate
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725:
633:
580:
368:Waveplates in general, as well as
322:
306:
174:A waveplate works by shifting the
108:, or even plastic), for which the
25:
125:. Addition of plates between the
1841:"Mounted Achromatic Wave Plates"
1819:(4th ed.). pp. 352–5.
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296:of the crystal by the formula
1:
372:, can be described using the
379:Although the birefringence Δ
284:the situation is reversed.
201:, with index of refraction
190:, with index of refraction
27:Optical polarization device
1990:
1911:"Understanding Waveplates"
1948:Polarizers and Waveplates
1891:. University of Cambridge
1701:Thin crystalline film of
383:may vary slightly due to
50: The combined field
1786:Spatial light modulator
1715:petrographic microscope
153:Principles of operation
131:petrographic microscope
93:light and vice versa.)
69:device that alters the
1771:Photoelastic modulator
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1969:Polarization (waves)
1915:www.edmundoptics.com
1776:Polarization rotator
1731:optical indicatrices
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147:optical indicatrices
91:circularly polarized
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1645:or (less commonly)
110:index of refraction
1974:Optical components
1964:Optical mineralogy
1815:Hecht, E. (2001).
1711:optical mineralogy
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410:magnesium fluoride
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292:and the thickness
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100:material (such as
87:quarter-wave plate
83:linearly polarized
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849:
846:
843:
837:
834:
827:
824:
821:
818:
815:
784:
781:
758:
753:
750:
747:
744:
741:
738:
735:
732:
727:
722:
716:
713:
706:
703:
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688:
681:
678:
675:
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669:
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649:
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640:
635:
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623:
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608:
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582:
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547:
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518:
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489:
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442:
422:
419:
417:
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388:
362:
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347:
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336:
331:
327:
324:
320:
317:
311:
308:
281:
274:
255:
248:
241:
230:
223:
212:
205:
194:
154:
151:
46:
40:
34:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1986:
1975:
1972:
1970:
1967:
1965:
1962:
1961:
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1949:
1946:
1944:
1941:RP photonics
1940:
1937:
1936:
1932:
1916:
1912:
1906:
1903:
1890:
1886:
1885:"Tint plates"
1880:
1877:
1872:
1865:
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1792:
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1784:
1782:
1779:
1777:
1774:
1772:
1769:
1767:
1766:Fresnel rhomb
1764:
1762:
1759:
1758:
1754:
1752:
1749:
1745:
1740:
1734:
1732:
1728:
1724:
1723:thin sections
1720:
1716:
1712:
1704:
1699:
1692:
1690:
1688:
1684:
1680:
1676:
1671:
1664:
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1656:
1655:thin sections
1652:
1648:
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1626:
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1615:
1611:
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1603:
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1403:
1386:
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1371:
1370:
1369:
1367:
1363:
1359:
1355:
1348:
1341:
1337:
1333:
1314:
1306:
1303:
1300:
1297:
1294:
1288:
1261:
1257:
1253:
1236:
1232:
1221:
1220:
1219:
1213:
1209:
1200:
1192:
1185:
1183:
1181:
1077:
1043:
1010:
1002:
999:
996:
993:
990:
984:
956:
953:
947:
944:
941:
923:
920:
914:
911:
905:
902:
894:
891:
888:
885:
882:
876:
850:
847:
844:
841:
825:
822:
819:
813:
805:
801:
756:
748:
745:
742:
739:
736:
730:
704:
701:
698:
695:
679:
676:
673:
667:
664:
656:
653:
650:
647:
644:
638:
614:
611:
603:
600:
597:
594:
591:
585:
564:
440:
436:
427:
420:
415:
413:
411:
407:
401:
397:
395:
386:
382:
377:
375:
371:
366:
345:
338:
334:
329:
325:
318:
315:
309:
299:
298:
297:
295:
291:
285:
280:
273:
269:
265:
261:
254:
247:
240:
236:
229:
222:
218:
211:
204:
200:
193:
189:
188:ordinary axis
185:
181:
177:
168:
159:
152:
150:
148:
144:
140:
139:thin sections
136:
132:
128:
124:
119:
116:
111:
107:
103:
99:
94:
92:
88:
84:
80:
76:
72:
68:
64:
60:
32:
19:
1942:
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1914:
1905:
1893:. Retrieved
1888:
1879:
1870:
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1844:
1835:
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1672:
1668:
1646:
1642:
1636:
1627:
1620:
1608:
1604:
1601:
1501:
1494:
1487:
1485:
1482:
1365:
1361:
1360:between the
1357:
1350:
1343:
1339:
1335:
1331:
1329:
1211:
1207:
1205:
803:
799:
562:
438:
434:
432:
402:
398:
393:
380:
378:
374:Jones matrix
367:
360:
293:
289:
286:
278:
271:
267:
263:
252:
245:
238:
234:
227:
220:
216:
209:
202:
198:
191:
187:
180:birefringent
173:
120:
98:birefringent
95:
86:
78:
71:polarization
62:
58:
56:
1748:optic angle
1675:Lyot filter
416:Plate types
277: >
251: <
73:state of a
1958:Categories
1939:Waveplates
1921:2019-05-03
1850:2024-01-16
1826:0805385665
1797:References
1791:Zone plate
1739:indicatrix
1342:axis, and
1330:where the
1180:handedness
385:dispersion
370:polarizers
197:, and the
184:optic axis
127:polarizers
115:wavelength
18:Wave plate
1950:Animation
1687:pass band
1576:ω
1573:−
1545:^
1527:^
1457:ω
1454:−
1426:^
1398:^
1304:ω
1301:−
1273:^
1248:^
1163:^
1134:^
1105:^
1072:^
1038:^
1000:ω
997:−
969:^
957:θ
954:−
948:
936:^
924:θ
921:−
915:
892:ω
889:−
861:^
851:θ
848:
842:−
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826:θ
823:
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715:^
705:θ
702:
690:^
680:θ
677:
654:ω
651:−
625:^
601:ω
598:−
546:^
517:^
488:^
459:^
335:λ
323:Δ
319:π
307:Γ
268:slow axis
264:fast axis
59:waveplate
1889:DoITPoMS
1755:See also
1719:minerals
1651:minerals
1500: ≡
1493: =
1078:′
1044:′
532:, where
258:, as in
135:minerals
63:retarder
1895:Dec 31,
1781:Q-plate
1214:, and λ
441:, and λ
361:where λ
260:calcite
67:optical
1823:
1817:Optics
769:where
406:quartz
270:. For
102:quartz
65:is an
48:
42:
36:
1742:with
1727:rocks
1681:with
1659:rocks
176:phase
162:wave.
143:rocks
129:of a
75:light
1897:2016
1821:ISBN
1364:and
1349:and
1334:and
1149:and
1091:and
802:and
503:and
408:and
106:mica
1725:of
1721:in
1657:of
1653:in
1210:, Δ
1023:If
945:sin
912:cos
845:sin
820:cos
699:sin
674:cos
437:, Δ
141:of
137:in
104:or
61:or
1960::
1913:.
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671:(
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316:2
310:=
294:L
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288:Δ
282:o
279:n
275:e
272:n
256:o
253:n
249:e
246:n
242:e
239:n
237:/
235:c
231:e
228:v
224:o
221:n
219:/
217:c
213:o
210:v
206:e
203:n
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