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Gradient-index optics

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110:(bent) from their normal straight path. This is due to the variation of refractive index between the hot, less dense air at the surface of the road, and the denser cool air above it. The variation in temperature (and thus density) of the air causes a gradient in its refractive index, causing it to increase with height. This index gradient causes refraction of light rays (at a shallow angle to the road) from the sky, bending them into the eye of the viewer, with their apparent location being the road's surface. 282:– While point-by-point exposing the pre-designed structure an exposure dose is varied (scanning speed, laser power, etc.). This corresponds to spatially tunable monomer-to-polymer degree-of-conversion resulting to a different refractive index. The method is applicable to free-form micro-optical elements and multi-component optics. 301:, it involves a spherical index function and would be expected to be spherical in shape as well. This lens, however, is impractical to make and has little usefulness since only points on the surface and within the lens are sharply imaged and extended objects suffer from extreme aberrations. In 1905, 305:
used a dipping technique creating a gelatin cylinder with a refractive index gradient that varied symmetrically with the radial distance from the axis. Disk-shaped slices of the cylinder were later shown to have plane faces with radial index distribution. He showed that even though the faces of the
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The Earth's atmosphere acts as a GRIN lens, allowing observers to see the sun for a few minutes after it is actually below the horizon, and observers can also view stars that are below the horizon. This effect also allows for observation of electromagnetic signals from satellites after they have
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In imaging applications, GRIN lenses are mainly used to reduce aberrations. The design of such lenses involves detailed calculations of aberrations as well as efficient manufacture of the lenses. A number of different materials have been used for GRIN lenses including optical glasses, plastics,
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applications. Some years later several new techniques have been developed to fabricate lenses of the Wood type. Since then at least the thinner GRIN lenses can possess surprisingly good imaging properties considering their very simple mechanical construction, while thicker GRIN lenses found
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that focuses incident parallel rays of light onto a point on the opposite surface of the lens. This also limited the applications of the lens because it was difficult to use it to focus visible light; however, it had some usefulness in
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Zukauskas, Albertas; Matulaitiene, Ieva; Paipulas, Domas; Niaura, Gedinimas; Malinauskas, Mangirdas; Gadonas, Roaldas (2015). "Tuning the refractive index in 3D direct laser writing lithography: towards GRIN microoptics".
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ions in the glass are partially exchanged with lithium ones, with a larger amount of exchange occurring at the edge. Thus the sample obtains a gradient material structure and a corresponding gradient of the refractive
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of the lens varies from approximately 1.406 in the central layers down to 1.386 in less dense layers of the lens. This allows the eye to image with good resolution and low aberration at both short and long distances.
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The refractive index gradient of GRIN lenses can be mathematically modelled according to the method of production used. For example, GRIN lenses made from a radial gradient index material, such as
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lens were flat, they acted like converging and diverging lens depending on whether the index was a decreasing or increasing relative to the radial distance. In 1964, a posthumous book of
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The ability of GRIN lenses to have flat surfaces simplifies the mounting of the lens, which makes them useful where many very small lenses need to be mounted together, such as in
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The light path integral is able to characterize the path of light through the lens in a qualitative manner, such that the lens may be easily reproduced in the future.
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An axial gradient lens has been used to concentrate sunlight onto solar cells, capturing as much as 90% of incident light when the sun is not at an optimal angle.
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of a pool of water appearing on a road on a hot day. The pool is actually an image of the sky, apparently located on the road since light rays are being
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Antireflection coatings are typically effective for narrow ranges of frequency or angle of incidence. Graded-index materials are less constrained.
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A digest of technical papers presented at the Topical Meeting on Gradient Index Optical Imaging Systems, May 15-16, 1979, Rochester, New York
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Flores-Arias, M.T.; Bao, C.; Castelo, A.; Perez, M.V.; Gomez-Reino, C. (2006-10-15). "Crossover interconnects in gradient-index planar optics".
1855: 1151: 1089: 231:– Involving the deposition of different glass with varying refractive indexes, onto a surface to produce a cumulative refractive change. 1211: 764: 812: 622: 74:
typical of traditional spherical lenses. Gradient-index lenses may have a refraction gradient that is spherical, axial, or radial.
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are used, this equation is modified to incorporate the change in arc length for a spherical gradient, to each physical dimension:
828: 1769: 1565: 225:-rich glass is bombarded with neutrons to cause a change in the boron concentration, and thus the refractive index of the lens. 295:
suggested a lens whose refractive index distribution would allow for every region of space to be sharply imaged. Known as the
200:, in that all modes of the GRIN fibres propagate at the same speed, allowing for a higher temporal bandwidth for the fibre. 1681: 1658: 1514: 1289: 399:
relative to its value for any nearby curve joining the two points. The light path integral is given by the equation
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Keck D B and Olshansky R, "Optical Waveguide Having Optimal Index Gradient," U.S. Patent 3,904,268 (9 Sept. 1975).
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Mohr, R K; Wilder, J A; Macedo, P B; Gupta, P K (1979). "Graded index lenses by the molecular stuffing process".
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Hensler J R, "Method of Producing a Refractive Index Gradient in Glass," U.S. Patent 3,873,408 (25 Mar. 1975).
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Moore R S, "Plastic Optical Element Having Refractive Index Gradient," U.S. Patent 3,718,383 (Feb. 1973).
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An inhomogeneous gradient-index lens possesses a refractive index whose change follows the function
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separation of a specific glass causes pores to form, which can later be filled using a variety of
1701: 1623: 1371: 1361: 885: 735: 396: 177: 71: 335: 1814: 1779: 1749: 1706: 1643: 1628: 1540: 1431: 1260: 1217: 1207: 1157: 1147: 1034: 808: 770: 760: 243: 180:) are made with a radially-varying refractive index profile; this design strongly reduces the 115: 1570: 1560: 1555: 1325: 1252: 1184: 1070: 939: 897: 613: 321: 296: 273: 139: 95: 63: 1845: 1764: 1759: 1608: 1504: 1476: 1461: 1456: 1451: 1411: 1381: 307: 188:. The radial variation in refractive index allows for a sinusoidal height distribution of 170: 150: 1481: 1248: 1066: 943: 935: 1726: 1696: 1691: 1122: 392: 302: 269: 235: 83: 1839: 1633: 1590: 1496: 1471: 1366: 916: 311: 166: 135: 131: 67: 1799: 1754: 1711: 1401: 1376: 1330: 711: 388: 249: 189: 154: 17: 1256: 787:
Shirk J S, Sandrock M, Scribner D, Fleet E, Stroman R, Baer E, Hiltner A. (2006)
1824: 1789: 1509: 1391: 127: 134:. The flat surface also allows a GRIN lens to be easily optically aligned to a 1819: 1784: 1653: 1421: 1416: 1161: 803:
Tsiboulia, A B (2003). "Gradient Index (GRIN) Lenses". In Ronald G. Driggers.
1264: 1038: 595:{\displaystyle L=\int _{S_{o}}^{S}n(x,y,z){\sqrt {x'^{2}+y'^{2}+z'^{2}}}\,ds} 1686: 1585: 774: 316: 257: 162: 143: 107: 91: 1221: 1188: 1175:
Marchand, E.W. (1976). "Third-order aberrations of the photographic Wood".
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of the coordinates of the region of interest in the medium. According to
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is the most obvious example of gradient-index optics in nature. In the
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A gradient-index lens with a parabolic variation of refractive index (
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Another example of gradient index optics in nature is the common
39:). The lens focuses light in the same way as a conventional lens. 915:
Zhang, Jun-Chao; Xiong, Li-Min; Fang, Ming; He, Hong-Bo (2013).
1278: 1033:. paper WA1. Washington, D C: Optical Society of America. 196:. This differs from traditional optical fibres, which rely on 87: 759:(2nd ed.). Reading, Mass.: Addison-Wesley. p. 178. 690:{\displaystyle n_{r}=n_{o}\left(1-{\frac {Ar^{2}}{2}}\right)} 66:
of a material. Such gradual variation can be used to produce
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within the fibre, preventing the rays from leaving the
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or concentration of salts to give a varying gradient.
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at varying intensities to give a refractive gradient.
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New York; London: Macmillan. p. 71. 1177:Journal of the Optical Society of America 670: 660: 643: 630: 624: 585: 576: 558: 540: 530: 500: 493: 488: 476: 440: 431: 424: 419: 407: 337: 450:{\displaystyle L=\int _{S_{o}}^{S}n\,ds} 879: 877: 875: 873: 871: 869: 747: 706:is the refractive index at a distance, 573: 555: 537: 58:covering optical effects produced by a 310:was published in which he described a 1104:Nivin, William Davidson, ed. (1890). 755:Hecht, Eugene; Zając, Alfred (1987). 7: 799: 797: 805:Encyclopedia of Optical Engineering 465:is the arc length of the curve. If 114:descended below the horizon, as in 25: 886:"Gradient-index optics: a review" 142:output, making it applicable for 1815:Radioactive waste vitrification 1770:Glass fiber reinforced concrete 242:is partially polymerized using 1142:Luneburg, Rudolf Karl (1964). 964:Irving, Michael (2022-06-28). 605:where prime corresponds to d/d 527: 509: 366: 348: 1: 1856:Glass engineering and science 1682:Chemically strengthened glass 1144:Mathematical Theory of Optics 1088:Maxwell, James Clerk (1854). 1055:Laser & Photonics Reviews 944:10.1088/1674-1056/22/4/044201 1515:Glass-ceramic-to-metal seals 1257:10.1016/j.optcom.2006.05.049 1206:. New York: Academic Press. 461:is the refractive index and 391:joining any two points of a 1202:Marchand, Erich W. (1978). 383:, the light path integral ( 1872: 1125:, Robert Williams (1905). 372:{\displaystyle n=f(x,y,z)} 229:Chemical vapour deposition 1546:Chemical vapor deposition 1467:Ultra low expansion glass 1357:Borophosphosilicate glass 884:Moore, Duncan T. (1980). 198:total internal reflection 1785:Glass-reinforced plastic 1447:Sodium hexametaphosphate 186:multi-mode optical fiber 176:Certain optical fibres ( 35:) with radial distance ( 1677:Anti-reflective coating 1551:Glass batch calculation 1432:Photochromic lens glass 725:is a positive constant. 155:optogenetic stimulation 1189:10.1364/JOSA.66.001326 1075:10.1002/lpor.201500170 691: 596: 451: 373: 40: 1810:Prince Rupert's drops 1659:Transparent materials 1619:Gradient-index optics 1427:Phosphosilicate glass 1237:Optics Communications 1204:Gradient index optics 692: 597: 467:Cartesian coordinates 452: 374: 256:ions. As a result of 30: 1775:Glass ionomer cement 1649:Photosensitive glass 1576:Liquidus temperature 1397:Fluorosilicate glass 902:10.1364/AO.19.001035 623: 475: 406: 336: 320:application e.g. in 298:Maxwell fisheye lens 280:Direct laser writing 1795:Glass-to-metal seal 1717:Self-cleaning glass 1639:Optical lens design 1249:2006OptCo.266..490F 1067:2015LPRv....9..706Z 936:2013ChPhB..22d4201Z 505: 436: 178:graded-index fibres 18:Gradient index lens 1780:Glass microspheres 1702:Hydrogen darkening 1624:Hydrogen darkening 1372:Chalcogenide glass 1362:Borosilicate glass 736:Graded-index fiber 687: 592: 484: 447: 415: 381:Fermat's principle 369: 41: 1833: 1832: 1750:Glass-coated wire 1722:sol–gel technique 1707:Insulated glazing 1644:Photochromic lens 1629:Optical amplifier 1581:sol–gel technique 1183:(12): 1326–1330. 1153:978-0-5203-2826-6 988:Sinai P, (1970). 924:Chinese Physics B 680: 583: 387:), taken along a 244:ultraviolet light 116:radio occultation 54:is the branch of 16:(Redirected from 1863: 1571:Ion implantation 1326:Glass transition 1299: 1292: 1285: 1276: 1269: 1268: 1232: 1226: 1225: 1199: 1193: 1192: 1172: 1166: 1165: 1139: 1133: 1132: 1119: 1113: 1111: 1101: 1085: 1079: 1078: 1049: 1043: 1042: 1026: 1020: 1017: 1011: 1008: 1002: 999: 993: 986: 980: 979: 977: 976: 961: 955: 954: 952: 950: 921: 912: 906: 905: 896:(7): 1035–1038. 881: 864: 863: 861: 860: 849: 843: 842: 840: 839: 825: 816: 801: 792: 785: 779: 778: 752: 696: 694: 693: 688: 686: 682: 681: 676: 675: 674: 661: 648: 647: 635: 634: 614:SELFOC Microlens 601: 599: 598: 593: 584: 582: 581: 580: 564: 563: 562: 546: 545: 544: 531: 504: 499: 498: 497: 456: 454: 453: 448: 435: 430: 429: 428: 378: 376: 375: 370: 182:modal dispersion 96:refractive index 64:refractive index 21: 1871: 1870: 1866: 1865: 1864: 1862: 1861: 1860: 1836: 1835: 1834: 1829: 1765:Glass electrode 1760:Glass databases 1737: 1731: 1669: 1663: 1595: 1529: 1505:Bioactive glass 1491: 1477:Vitreous enamel 1462:Thoriated glass 1457:Tellurite glass 1442:Soda–lime glass 1412:Gold ruby glass 1382:Cranberry glass 1335: 1309: 1303: 1273: 1272: 1234: 1233: 1229: 1214: 1201: 1200: 1196: 1174: 1173: 1169: 1154: 1141: 1140: 1136: 1128:Physical Optics 1121: 1120: 1116: 1103: 1102:(reprinted by: 1087: 1086: 1082: 1051: 1050: 1046: 1028: 1027: 1023: 1018: 1014: 1009: 1005: 1000: 996: 987: 983: 974: 972: 963: 962: 958: 948: 946: 919: 914: 913: 909: 883: 882: 867: 858: 856: 855:. 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Luneburg 288: 285: 284: 283: 277: 266: 247: 236:polymerisation 232: 226: 221:irradiation – 211: 208: 123: 120: 118:measurements. 79: 76: 44:Gradient-index 24: 14: 13: 10: 9: 6: 4: 3: 2: 1868: 1857: 1854: 1852: 1849: 1847: 1844: 1843: 1841: 1826: 1823: 1821: 1818: 1816: 1813: 1811: 1808: 1806: 1803: 1801: 1798: 1796: 1793: 1791: 1788: 1786: 1783: 1781: 1778: 1776: 1773: 1771: 1768: 1766: 1763: 1761: 1758: 1756: 1753: 1751: 1748: 1746: 1743: 1742: 1740: 1734: 1728: 1725: 1723: 1720: 1718: 1715: 1713: 1710: 1708: 1705: 1703: 1700: 1698: 1695: 1693: 1690: 1688: 1685: 1683: 1680: 1678: 1675: 1674: 1672: 1666: 1660: 1657: 1655: 1652: 1650: 1647: 1645: 1642: 1640: 1637: 1635: 1634:Optical fiber 1632: 1630: 1627: 1625: 1622: 1620: 1617: 1615: 1612: 1610: 1607: 1606: 1604: 1602: 1598: 1592: 1591:Vitrification 1589: 1587: 1584: 1582: 1579: 1577: 1574: 1572: 1569: 1567: 1564: 1562: 1561:Glass melting 1559: 1557: 1556:Glass forming 1554: 1552: 1549: 1547: 1544: 1542: 1539: 1538: 1536: 1532: 1526: 1523: 1521: 1518: 1516: 1513: 1511: 1508: 1506: 1503: 1502: 1500: 1498: 1494: 1488: 1485: 1483: 1480: 1478: 1475: 1473: 1472:Uranium glass 1470: 1468: 1465: 1463: 1460: 1458: 1455: 1453: 1452:Soluble glass 1450: 1448: 1445: 1443: 1440: 1438: 1435: 1433: 1430: 1428: 1425: 1423: 1420: 1418: 1415: 1413: 1410: 1408: 1405: 1403: 1400: 1398: 1395: 1393: 1390: 1388: 1385: 1383: 1380: 1378: 1375: 1373: 1370: 1368: 1367:Ceramic glaze 1365: 1363: 1360: 1358: 1355: 1353: 1350: 1348: 1345: 1344: 1342: 1338: 1332: 1329: 1327: 1324: 1322: 1319: 1318: 1316: 1312: 1307: 1300: 1295: 1293: 1288: 1286: 1281: 1280: 1277: 1266: 1262: 1258: 1254: 1250: 1246: 1242: 1238: 1231: 1228: 1223: 1219: 1215: 1209: 1205: 1198: 1195: 1190: 1186: 1182: 1178: 1171: 1168: 1163: 1159: 1155: 1149: 1145: 1138: 1135: 1130: 1129: 1124: 1118: 1115: 1109: 1108: 1099: 1095: 1091: 1084: 1081: 1076: 1072: 1068: 1064: 1060: 1056: 1048: 1045: 1040: 1036: 1032: 1025: 1022: 1016: 1013: 1007: 1004: 998: 995: 991: 985: 982: 971: 967: 960: 957: 945: 941: 937: 933: 930:(4): 044201. 929: 925: 918: 911: 908: 903: 899: 895: 891: 887: 880: 878: 876: 874: 872: 870: 866: 854: 848: 845: 834: 830: 824: 822: 818: 814: 813:9780824742508 810: 806: 800: 798: 794: 790: 784: 781: 776: 772: 768: 762: 758: 751: 748: 741: 737: 734: 733: 729: 724: 717: 713: 709: 704: 700: 683: 677: 671: 667: 663: 657: 654: 650: 644: 640: 636: 631: 627: 619: 618: 617: 615: 610: 608: 589: 586: 577: 569: 565: 559: 551: 547: 541: 533: 524: 521: 518: 515: 512: 506: 501: 494: 490: 485: 481: 478: 471: 468: 464: 460: 444: 441: 437: 432: 425: 421: 416: 412: 409: 402: 401: 400: 398: 394: 390: 386: 382: 363: 360: 357: 354: 351: 345: 342: 339: 327: 325: 323: 318: 313: 309: 304: 300: 299: 294: 286: 281: 278: 275: 271: 267: 263: 259: 255: 251: 248: 245: 241: 238:– An organic 237: 233: 230: 227: 224: 220: 217: 216: 215: 209: 207: 204: 201: 199: 195: 191: 187: 183: 179: 174: 172: 168: 167:zinc selenide 164: 158: 156: 152: 149: 145: 141: 138:, to produce 137: 133: 129: 121: 119: 117: 111: 109: 105: 100: 97: 93: 89: 85: 77: 75: 73: 69: 65: 61: 57: 53: 49: 45: 38: 34: 29: 19: 1851:Fiber optics 1800:Porous glass 1755:Safety glass 1712:Porous glass 1670:modification 1618: 1482:Wood's glass 1402:Fused quartz 1377:Cobalt glass 1331:Supercooling 1240: 1236: 1230: 1203: 1197: 1180: 1176: 1170: 1143: 1137: 1127: 1117: 1106: 1097: 1093: 1083: 1058: 1054: 1047: 1030: 1024: 1015: 1006: 997: 992:. 10, 99-104 989: 984: 973:. Retrieved 969: 959: 947:. Retrieved 927: 923: 910: 893: 889: 857:. Retrieved 847: 836:. Retrieved 832: 804: 788: 783: 756: 750: 722: 715: 712:optical axis 707: 702: 698: 611: 606: 604: 462: 458: 389:ray of light 384: 331: 297: 290: 250:Ion exchange 213: 205: 202: 175: 159: 147: 128:photocopiers 125: 122:Applications 112: 101: 81: 51: 47: 43: 42: 36: 32: 1825:Glass fiber 1790:Glass cloth 1534:Preparation 1510:CorningWare 1392:Flint glass 1387:Crown glass 1340:Formulation 710:, from the 322:Selfoc rods 293:J C Maxwell 210:Manufacture 72:aberrations 1840:Categories 1820:Windshield 1654:Refraction 1614:Dispersion 1422:Milk glass 1417:Lead glass 1162:1149437946 975:2022-06-28 859:2021-07-11 838:2021-07-11 789:NRL Review 742:References 397:stationary 303:R. W. Wood 157:in brain. 140:collimated 1687:Corrosion 1586:Viscosity 1541:Annealing 1265:0030-4018 1039:489755284 970:New Atlas 658:− 486:∫ 417:∫ 317:microwave 291:In 1854, 258:diffusion 163:germanium 144:endoscopy 108:refracted 92:human eye 78:In nature 1805:Pre-preg 1609:Achromat 1352:Bioglass 1347:AgInSbTe 791:pp 53–61 775:13761389 730:See also 697:, where 574:′ 556:′ 538:′ 457:, where 234:Partial 132:scanners 60:gradient 1736:Diverse 1668:Surface 1525:Zerodur 1245:Bibcode 1222:4497777 1100:: 9–11. 1063:Bibcode 932:Bibcode 287:History 254:lithium 240:monomer 219:Neutron 148:in vivo 86:of the 62:of the 1846:Optics 1738:topics 1601:Optics 1407:GeSbTe 1314:Basics 1263:  1220:  1210:  1160:  1150:  1037:  949:13 May 811:  773:  763:  757:Optics 393:medium 328:Theory 265:index. 262:sodium 169:, and 104:mirage 94:, the 68:lenses 56:optics 52:optics 1520:Macor 1487:ZBLAN 1321:Glass 1306:Glass 920:(PDF) 395:, is 274:salts 270:Phase 223:Boron 184:of a 136:fiber 1261:ISSN 1218:OCLC 1208:ISBN 1158:OCLC 1148:ISBN 1123:Wood 1035:OCLC 951:2016 809:ISBN 771:OCLC 761:ISBN 312:lens 194:core 190:rays 153:and 130:and 84:lens 82:The 48:GRIN 1253:doi 1241:266 1185:doi 1071:doi 940:doi 898:doi 88:eye 1842:: 1259:. 1251:. 1239:. 1216:. 1181:66 1179:. 1156:. 1096:. 1092:. 1069:. 1057:. 968:. 938:. 928:22 926:. 922:. 894:19 892:. 888:. 868:^ 831:. 820:^ 796:^ 769:. 714:; 607:s. 324:. 260:, 173:. 165:, 50:) 1298:e 1291:t 1284:v 1267:. 1255:: 1247:: 1224:. 1191:. 1187:: 1164:. 1112:) 1098:9 1077:. 1073:: 1065:: 1059:9 1041:. 978:. 953:. 942:: 934:: 904:. 900:: 862:. 841:. 815:. 777:. 723:A 719:o 716:n 708:r 703:r 699:n 684:) 678:2 672:2 668:r 664:A 655:1 651:( 645:o 641:n 637:= 632:r 628:n 590:s 587:d 578:2 570:z 566:+ 560:2 552:y 548:+ 542:2 534:x 528:) 525:z 522:, 519:y 516:, 513:x 510:( 507:n 502:S 495:o 491:S 482:= 479:L 463:S 459:n 445:s 442:d 438:n 433:S 426:o 422:S 413:= 410:L 385:L 367:) 364:z 361:, 358:y 355:, 352:x 349:( 346:f 343:= 340:n 46:( 37:x 33:n 20:)

Index

Gradient index lens

optics
gradient
refractive index
lenses
aberrations
lens
eye
human eye
refractive index
mirage
refracted
radio occultation
photocopiers
scanners
fiber
collimated
endoscopy
calcium imaging
optogenetic stimulation
germanium
zinc selenide
sodium chloride
graded-index fibres
modal dispersion
multi-mode optical fiber
rays
core
total internal reflection

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