1188:
1180:
1630:, 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
1603:
1618:
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.
156:
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415:
756:
20:
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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
1617:
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
41:
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
1730:
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
557:
106:
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
388:
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
1725:
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
380:
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
1010:
1659:
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.
276:
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
1594:
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.
1167:. 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
1598:
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:
197:. 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
101:
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
392:
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
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1314:
798:
1586:
150:
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
1739:
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.
751:{\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)},}
345:
381:
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
1187:
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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
138:
within the visible crystal sections. This alignment can allow discrimination between minerals which otherwise appear very similar in plane polarized and cross polarized light.
1179:
1363:
1213:
389:
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.
42:
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).
1499:
365:
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.
1612:
107:
of two birefringent materials, an achromatic waveplate can be manufactured such that the spectral response of its phase retardance can be nearly flat.
1626:
A full-wave plate introduces a phase difference of exactly one wavelength between the two polarization directions, for one wavelength of light. In
1005:{\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)}.}
434:
is chosen so that the phase shift between polarization components is Γ = π. Now suppose a linearly polarized wave with polarization vector
376:, this is negligible compared to the variation in phase difference according to the wavelength of the light due to the fixed path difference (λ
171:
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
787:
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
291:
1109:
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
1475:
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
78:, which converts between different elliptical polarizations (such as the special case of converting from linearly polarized light to
401:
the refractive index changes in the second decimal place and true zero order plates are common for wavelengths above 1 μm.
1813:
1899:
1957:
1667:
1962:
1952:
1046:
denotes the polarization vector of the wave exiting the waveplate, then this expression shows that the angle between
175:
of the crystal is parallel to the surfaces of the plate. This results in two axes in the plane of the cut: the
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119:
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between two perpendicular polarization components of the light wave. A typical waveplate is simply a
110:
A common use of waveplates—particularly the sensitive-tint (full-wave) and quarter-wave plates—is in
59:
1732:
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are real. The effect of the quarter-wave plate is to introduce a phase shift term e =e =
373:
98:
71:
1611:
1462:{\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
1873:
1719:
1699:
1627:
1309:{\displaystyle (E_{f}\mathbf {\hat {f}} +E_{s}\mathbf {\hat {s}} )\mathrm {e} ^{i(kz-\omega t)},}
398:
135:
111:
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Stacking a series of different-order waveplates with polarization filters between them yields a
1809:
233:. This leads to a phase difference between the two components as they exit the crystal. When
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1581:{\displaystyle E(\mathbf {\hat {f}} +i\mathbf {\hat {s}} )\mathrm {e} ^{i(kz-\omega t)},}
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155:
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The sensitive-tint (full-wave) and quarter-wave plates are widely used in the field of
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1647:
362:
131:
1946:
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215:, 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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components of the wave, so that upon exiting the crystal the wave is now given by
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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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103:
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115:
1718:, in particular by allowing deduction of the shape and orientation of the
1638:. These plates are widely used in mineralogy to aid in identification of
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19:
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340:{\displaystyle \Gamma ={\frac {2\pi \,\Delta n\,L}{\lambda _{0}}},}
66:
wave travelling through it. Two common types of waveplates are the
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154:
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1606:
Composition of two linearly polarized waves, phase shifted by π/2
94:
1862:. Vol. 1. New York: John Wiley & Sons. p. 121.
1858:
Winchell, Newton Horace; Winchell, Alexander Newton (1922).
1829:
463:
is incident on the crystal. Let θ denote the angle between
1183:
Two waves differing by a quarter-phase shift for one axis
1666:. Either the filters can be rotated, or the waveplates
1493:, and the resulting wave upon exiting the waveplate is
1860:
Elements of
Optical Mineralogy: Principles and Methods
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1195:For a quarter-wave plate, the relationship between
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1702:. Addition of plates between the polarizers of a
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422:For a half-wave plate, the relationship between
70:, which rotates the polarization direction of
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1849:
1694:photographed using a petrographic microscope.
8:
1932:Encyclopedia of Laser Physics and Technology
1726:distinguish the orientation of the optical
1706:makes easier the optical identification of
27: Electric field parallel to optic axis
33: Electric field perpendicular to axis
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1654:Multiple-order vs. zero-order waveplates
1472:The wave is now elliptically polarized.
418:A wave passing through a half-wave plate
18:
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1797:
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1791:
1735:techniques to allow measurement of the
1682:Use in mineralogy and optical petrology
354:is the vacuum wavelength of the light.
251:, the extraordinary axis is called the
1591:and the wave is circularly polarized.
1722:within the visible crystal sections.
159:A waveplate mounted in a rotary mount
7:
1073:{\displaystyle \mathbf {\hat {p}} '}
1039:{\displaystyle \mathbf {\hat {p}} '}
255:and the ordinary axis is called the
122:makes the optical identification of
85:Waveplates are constructed out of a
1674:layers, to obtain a widely tunable
1160:{\displaystyle \mathbf {\hat {z}} }
1131:{\displaystyle \mathbf {\hat {f}} }
1102:{\displaystyle \mathbf {\hat {f}} }
780:{\displaystyle \mathbf {\hat {s}} }
543:{\displaystyle \mathbf {\hat {f}} }
514:{\displaystyle \mathbf {\hat {f}} }
485:{\displaystyle \mathbf {\hat {p}} }
456:{\displaystyle \mathbf {\hat {p}} }
1678:in optical transmission spectrum.
1622:Full-wave, or sensitive-tint plate
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714:
622:
569:
357:Waveplates in general, as well as
311:
295:
163:A waveplate works by shifting the
97:, or even plastic), for which the
14:
114:. Addition of plates between the
1830:"Mounted Achromatic Wave Plates"
1808:(4th ed.). pp. 352–5.
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285:of the crystal by the formula
1:
361:, can be described using the
368:Although the birefringence Δ
273:the situation is reversed.
190:, with index of refraction
179:, with index of refraction
16:Optical polarization device
1979:
1900:"Understanding Waveplates"
1937:Polarizers and Waveplates
1880:. University of Cambridge
1690:Thin crystalline film of
372:may vary slightly due to
39: The combined field
1775:Spatial light modulator
1704:petrographic microscope
142:Principles of operation
120:petrographic microscope
82:light and vice versa.)
58:device that alters the
1760:Photoelastic modulator
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1958:Polarization (waves)
1904:www.edmundoptics.com
1765:Polarization rotator
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136:optical indicatrices
80:circularly polarized
1733:interference figure
1634:or (less commonly)
99:index of refraction
1963:Optical components
1953:Optical mineralogy
1804:Hecht, E. (2001).
1700:optical mineralogy
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89:material (such as
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804:
773:
770:
747:
742:
739:
736:
733:
730:
727:
724:
721:
716:
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705:
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695:
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689:
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677:
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664:
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632:
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579:
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564:
536:
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507:
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478:
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431:
411:
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377:
351:
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336:
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270:
263:
244:
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194:
183:
143:
140:
35:
29:
23:
15:
13:
10:
9:
6:
4:
3:
2:
1975:
1964:
1961:
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1938:
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1930:RP photonics
1929:
1926:
1925:
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1905:
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1895:
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1879:
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1874:"Tint plates"
1869:
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1768:
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1758:
1756:
1755:Fresnel rhomb
1753:
1751:
1748:
1747:
1743:
1741:
1738:
1734:
1729:
1723:
1721:
1717:
1713:
1712:thin sections
1709:
1705:
1701:
1693:
1688:
1681:
1679:
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1644:thin sections
1641:
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1221:
1210:
1209:
1208:
1202:
1198:
1189:
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1174:
1172:
1170:
1066:
1032:
999:
991:
988:
985:
982:
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936:
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912:
909:
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894:
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883:
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871:
865:
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836:
833:
830:
814:
811:
808:
802:
794:
790:
745:
737:
734:
731:
728:
725:
719:
693:
690:
687:
684:
668:
665:
662:
656:
653:
645:
642:
639:
636:
633:
627:
603:
600:
592:
589:
586:
583:
580:
574:
553:
429:
425:
416:
409:
404:
402:
400:
396:
390:
386:
384:
375:
371:
366:
364:
360:
355:
334:
327:
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318:
314:
307:
304:
298:
288:
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284:
280:
274:
269:
262:
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243:
236:
229:
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218:
211:
207:
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193:
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182:
178:
177:ordinary axis
174:
170:
166:
157:
148:
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128:thin sections
125:
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108:
105:
100:
96:
92:
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83:
81:
77:
73:
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65:
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1903:
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1877:
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1833:
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1661:
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1616:
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1597:
1593:
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1490:
1483:
1476:
1474:
1471:
1354:
1350:
1349:between the
1346:
1339:
1332:
1328:
1324:
1320:
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1200:
1196:
1194:
792:
788:
551:
427:
423:
421:
391:
387:
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367:
363:Jones matrix
356:
349:
282:
278:
275:
267:
260:
256:
252:
241:
234:
227:
223:
216:
209:
205:
198:
191:
187:
180:
176:
169:birefringent
162:
109:
87:birefringent
84:
75:
67:
60:polarization
51:
47:
45:
1737:optic angle
1664:Lyot filter
405:Plate types
266: >
240: <
62:state of a
1947:Categories
1928:Waveplates
1910:2019-05-03
1839:2024-01-16
1815:0805385665
1786:References
1780:Zone plate
1728:indicatrix
1331:axis, and
1319:where the
1169:handedness
374:dispersion
359:polarizers
186:, and the
173:optic axis
116:polarizers
104:wavelength
1939:Animation
1676:pass band
1565:ω
1562:−
1534:^
1516:^
1446:ω
1443:−
1415:^
1387:^
1293:ω
1290:−
1262:^
1237:^
1152:^
1123:^
1094:^
1061:^
1027:^
989:ω
986:−
958:^
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943:−
937:
925:^
913:θ
910:−
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850:^
840:θ
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812:
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691:
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669:θ
666:
643:ω
640:−
614:^
590:ω
587:−
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506:^
477:^
448:^
324:λ
312:Δ
308:π
296:Γ
257:slow axis
253:fast axis
48:waveplate
1878:DoITPoMS
1744:See also
1708:minerals
1640:minerals
1489: ≡
1482: =
1067:′
1033:′
521:, where
247:, as in
124:minerals
52:retarder
1884:Dec 31,
1770:Q-plate
1203:, and λ
430:, and λ
350:where λ
249:calcite
56:optical
1812:
1806:Optics
758:where
395:quartz
259:. For
91:quartz
54:is an
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1716:rocks
1670:with
1648:rocks
165:phase
151:wave.
132:rocks
118:of a
64:light
1886:2016
1810:ISBN
1353:and
1338:and
1323:and
1138:and
1080:and
791:and
492:and
397:and
95:mica
1714:of
1710:in
1646:of
1642:in
1199:, Δ
1012:If
934:sin
901:cos
834:sin
809:cos
688:sin
663:cos
426:, Δ
130:of
126:in
93:or
50:or
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277:Δ
271:o
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245:o
242:n
238:e
235:n
231:e
228:n
226:/
224:c
220:e
217:v
213:o
210:n
208:/
206:c
202:o
199:v
195:e
192:n
184:o
181:n
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