607:
36:
1784:
854:
259:
578:
Decomposition of field distributions into modes is useful because a large number of field amplitudes readings can be simplified into a much smaller number of mode amplitudes. Because these modes change over time according to a simple set of rules, it is also possible to anticipate future behavior of
1305:
323:
which has a significant longitudinal component to the propagated wave due to the inhomogeneity at the boundary of the dielectric substrate below the conductor and the air above it. In an optical fiber or other dielectric waveguide, modes are generally of the hybrid type.
1935:
1015:
804:
1779:
1840:
The overall intensity profile of a laser's output may be made up from the superposition of any of the allowed transverse modes of the laser's cavity, though often it is desirable to operate only on the fundamental mode.
318:
energy is normally transported in the fundamental TEM mode. The TEM mode is also usually assumed for most other electrical conductor line formats as well. This is mostly an accurate assumption, but a major exception is
575:, respectively. Fiber with a V-parameter of less than 2.405 only supports the fundamental mode (a hybrid mode), and is therefore a single-mode fiber whereas fiber with a higher V-parameter has multiple modes.
187:
amplitude at the walls of the waveguide, so the transverse pattern of the electric field of waves is restricted to those that fit between the walls. For this reason, the modes supported by a waveguide are
1946:
961:, and this may increase or decrease with the propagation of the beam, however the modes preserve their general shape during propagation. Higher order modes are relatively larger compared to the
460:
884:
mode is the lowest order. It is the fundamental transverse mode of the laser resonator and has the same form as a
Gaussian beam. The pattern has a single lobe, and has a constant
662:
1494:
1419:
1481:
1454:
1332:
565:
538:
487:
1390:
1361:
511:
1921:
1300:{\displaystyle E_{mn}(x,y,z)=E_{0}{\frac {w_{0}}{w}}H_{m}\left({\frac {{\sqrt {2}}x}{w}}\right)H_{n}\left({\frac {{\sqrt {2}}y}{w}}\right)\exp \left}
1837:
radians with respect to its horizontal or vertical neighbours. This is equivalent to the polarization of each lobe being flipped in direction.
656:
968:
mode, and thus the fundamental
Gaussian mode of a laser may be selected by placing an appropriately sized aperture in the laser cavity.
386:
314:
Hollow metallic waveguides filled with a homogeneous, isotropic material (usually air) support TE and TM modes but not the TEM mode. In
2056:
2006:
1908:
119:
53:
347:
is the number of half-wave patterns across the height of the waveguide. In circular waveguides, circular modes exist and here
327:
In rectangular waveguides, rectangular mode numbers are designated by two suffix numbers attached to the mode type, such as TE
100:
57:
72:
364:
392:
79:
183:
imposed on the wave by the waveguide. For example, a radio wave in a hollow metal waveguide must have zero tangential
46:
382:
234:. When two or more modes have an identical propagation constant along the waveguide, then there is more than one
137:
86:
226:
are imposed by a physical structure, a wave of a particular frequency can be described in terms of a transverse
378:
306:
2036:
2051:
1795:
mode corresponds to exactly the same fundamental mode as in the cylindrical geometry. Modes with increasing
979:
windows. In these lasers, transverse modes with rectangular symmetry are formed. These modes are designated
572:
68:
1815:
lobes present in the pattern. As before, higher-order modes have a larger spatial extent than the 00 mode.
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1888:
584:
193:
189:
598:
approximation for the field solution, treating it as if it contains only one transverse field component.
141:
618:
In a laser with cylindrical symmetry, the transverse mode patterns are described by a combination of a
623:
231:
1860:
1488:
976:
835:
223:
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are integers labeling the radial and angular mode orders, respectively. The intensity at a point
238:
possible in order to describe a wave with that propagation constant (for instance, a non-central
180:
1395:
996:
being the horizontal and vertical orders of the pattern. The electric field pattern at a point
2002:
1920:
U.S. Navy-Marine Corps
Military Auxiliary Radio System (MARS), NAVMARCORMARS Operator Course,
1904:
1855:
580:
173:
93:
1422:
799:{\displaystyle I_{pl}(\rho ,\varphi )=I_{0}\rho ^{l}\left^{2}\cos ^{2}(l\varphi )e^{-\rho }}
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145:
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pattern of the radiation in the plane perpendicular (i.e., transverse) to the radiation's
1774:{\displaystyle I_{mn}(x,y,z)=I_{0}\left({\frac {w_{0}}{w}}\right)^{2}\left^{2}\left^{2}}
1865:
579:
the field distribution. These simplifications of complex field distributions ease the
496:
184:
2045:
1819:
1426:
885:
619:
590:
The modes in typical low refractive index contrast fibers are usually referred to as
372:
315:
247:
243:
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1924:, Figure 1-38.—Various modes of operation for rectangular and circular waveguides.
17:
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1803:
show lobes appearing in the horizontal and vertical directions, with in general
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235:
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35:
1783:
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27:
Electromagnetic wave with oscillations perpendicular to the direction of travel
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No magnetic field in the direction of propagation. These are sometimes called
279:
No electric field in the direction of propagation. These are sometimes called
212:
208:
149:
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205:
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153:
297:
because there is only an electric field along the direction of propagation.
283:
because there is only a magnetic field along the direction of propagation (
1972:
343:
is the number of half-wave patterns across the width of the waveguide and
230:(or superposition of such modes). These modes generally follow different
850:
is the spot size of the mode corresponding to the
Gaussian beam radius.
971:
In many lasers, the symmetry of the optical resonator is restricted by
957:
The overall size of the mode is determined by the
Gaussian beam radius
303:
Non-zero electric and magnetic fields in the direction of propagation.
273:
Neither electric nor magnetic field in the direction of propagation.
1943:
EE 247: Introduction to
Optical Fiber Communications, Lecture Notes
1782:
852:
605:
257:
169:
161:
351:
is the number of full-wave patterns along the circumference and
385:. To determine the number of modes in a step-index fiber, the
242:
laser mode can be equivalently described as a superposition of
892:
show concentric rings of intensity, and modes with increasing
29:
204:
Unguided electromagnetic waves in free space, or in a bulk
930:, is a special case consisting of a superposition of two
355:
is the number of half-wave patterns along the diameter.
215:; these can be described as TEM modes as defined below.
896:
show angularly distributed lobes. In general there are
1936:"Lecture 3: Wave Optics Description of Optical Fibers"
395:
377:
The number of modes in an optical fiber distinguishes
1497:
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1398:
1369:
1340:
1313:
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665:
546:
519:
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for a beam propagating along the z-axis is given by
262:
Field patterns of some commonly used waveguide modes
1421:are the waist, spot size, radius of curvature, and
60:. Unsourced material may be challenged and removed.
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1475:
1448:
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455:{\textstyle V=k_{0}a{\sqrt {n_{1}^{2}-n_{2}^{2}}}}
454:
266:Modes in waveguides can be classified as follows:
196:for the boundary conditions of a given waveguide.
1945:. Stanford University. p. 8. Archived from
594:(linear polarization) modes, which refers to a
287:is the conventional symbol for magnetic field).
8:
2001:, pp. 71–79, Elsevier Academic Press, 2006,
1977:Encyclopedia of Laser Physics and Technology
1787:Rectangular transverse mode patterns TEM(mn)
192:. The allowed modes can be found by solving
1922:Chapter 1, Waveguide Theory and Application
659:) from the centre of the mode is given by:
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1491:. The corresponding intensity pattern is
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610:Cylindrical transverse mode patterns TEM(
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211:, can be described as a superposition of
120:Learn how and when to remove this message
1903:, pp.52-53, London: Edward Arnold 1971
1881:
888:across the mode. Modes with increasing
908:spots in the mode pattern (except for
270:Transverse electromagnetic (TEM) modes
307:Planar transmission line § Modes
148:direction. Transverse modes occur in
7:
2037:Detailed descriptions of laser modes
58:adding citations to reliable sources
1999:Fundamentals of Optical Waveguides
179:Transverse modes occur because of
25:
1889:"Transverse electromagnetic mode"
1456:is a normalization constant; and
857:Cylindrical transverse mode with
1934:Kahn, Joseph M. (Sep 21, 2006).
513:is the fiber's core radius, and
34:
45:needs additional citations for
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290:Transverse magnetic (TM) modes
276:Transverse electric (TE) modes
1:
954:with respect to one another.
365:Equilibrium mode distribution
2023:(5th ed.). p. 158.
250:which are described below).
2078:
362:
2057:Electromagnetic radiation
1414:{\displaystyle \zeta (z)}
585:fiber-optic communication
383:single-mode optical fiber
138:electromagnetic radiation
626:. The modes are denoted
389:needs to be determined:
379:multi-mode optical fiber
248:Laguerre-Gaussian modes
218:However in any sort of
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244:Hermite-Gaussian modes
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1476:{\displaystyle H_{k}}
1451:
1449:{\displaystyle E_{0}}
1416:
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1358:
1329:
1327:{\displaystyle w_{0}}
1302:
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801:
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560:{\displaystyle n_{2}}
535:
533:{\displaystyle n_{1}}
508:
484:
482:{\displaystyle k_{0}}
457:
261:
232:propagation constants
142:electromagnetic field
2021:Principles of Lasers
1971:Paschotta, Rüdiger.
1495:
1460:
1433:
1396:
1385:{\displaystyle R(z)}
1367:
1356:{\displaystyle w(z)}
1338:
1311:
1016:
926:mode, the so-called
663:
544:
517:
497:
466:
393:
54:improve this article
2019:Svelto, O. (2010).
1861:Laser beam profiler
973:polarizing elements
836:Laguerre polynomial
737:
624:Laguerre polynomial
449:
431:
236:modal decomposition
224:boundary conditions
194:Maxwell's equations
181:boundary conditions
1822:of each lobe of a
1789:
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1489:Hermite polynomial
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569:refractive indices
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18:Modal distribution
1901:Wave Transmission
1856:Longitudinal mode
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657:polar coordinates
581:signal processing
506:{\displaystyle a}
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174:optical resonator
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69:"Transverse mode"
16:(Redirected from
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1487:-th physicist's
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1979:. RP Photonics
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1899:F. R. Connor,
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1035:
1027:
1024:
1020:
1009:
1005:
1001:
985:
978:
974:
969:
955:
952:
944:
937:
929:
928:doughnut mode
922:
912:
905:
901:
887:
877:
873:
864:
860:
855:
851:
837:
832:
818:
814:
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791:
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753:
748:
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733:
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688:
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674:
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667:
658:
652:
648:
632:
625:
621:
620:Gaussian beam
613:
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593:
588:
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582:
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552:
548:
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436:
432:
427:
422:
418:
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407:
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399:
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388:
384:
380:
374:
373:Cladding mode
370:
366:
358:
356:
354:
350:
346:
342:
337:
331:
325:
322:
317:
316:coaxial cable
309:
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166:optical fiber
163:
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124:
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113:
110:November 2009
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
2020:
2014:
1998:
1997:K. Okamoto,
1993:
1981:. Retrieved
1976:
1966:
1954:. Retrieved
1947:the original
1942:
1929:
1916:
1900:
1895:
1884:
1839:
1833:
1826:
1817:
1810:
1806:
1790:
1007:
1003:
999:
983:
970:
956:
950:
942:
935:
927:
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611:
591:
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577:
376:
352:
348:
344:
340:
335:
329:
326:
313:
304:
300:Hybrid modes
294:
284:
280:
265:
217:
203:
178:
164:waves in an
133:
131:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
1851:Normal mode
947:), rotated
369:Mode volume
213:plane waves
150:radio waves
146:propagation
2046:Categories
1877:References
842:and index
491:wavenumber
363:See also:
321:microstrip
254:Waveguides
209:dielectric
154:microwaves
80:newspapers
1731:−
1721:
1634:−
1624:
1400:ζ
1290:ζ
1263:−
1251:−
1169:−
1161:
945:= 1, 2, 3
880:, the TEM
838:of order
792:ρ
789:−
778:φ
769:
742:ρ
709:ρ
689:φ
683:ρ
587:systems.
433:−
305:See also
220:waveguide
206:isotropic
190:quantized
168:and in a
158:waveguide
1845:See also
975:such as
573:cladding
567:are the
387:V number
339:, where
240:Gaussian
1983:Jan 26,
1973:"Modes"
1791:The TEM
1483:is the
940:modes (
915:). The
489:is the
295:E modes
281:H modes
94:scholar
2005:
1956:27 Jan
1907:
1392:, and
1307:where
949:360°/4
846:, and
806:where
635:where
602:Lasers
596:scalar
462:where
371:, and
222:where
96:
89:
82:
75:
67:
1950:(PDF)
1939:(PDF)
1820:phase
1809:+ 1)(
988:with
886:phase
869:With
381:from
333:or TM
170:laser
162:light
101:JSTOR
87:books
2003:ISBN
1985:2015
1958:2015
1905:ISBN
1818:The
1813:+ 1)
1799:and
992:and
861:=2,
655:(in
639:and
540:and
228:mode
152:and
73:news
1824:TEM
1718:exp
1621:exp
1158:exp
981:TEM
963:TEM
932:TEM
917:TEM
913:= 0
906:+1)
878:= 0
811:= 2
760:cos
628:TEM
246:or
172:'s
136:of
56:by
2048::
1975:.
1941:.
1827:mn
1793:00
1429:;
1363:,
1334:,
984:mn
965:00
882:00
874:=
865:=1
820:,
631:pl
612:pl
592:LP
493:,
367:,
336:mn
330:mn
176:.
132:A
2009:.
1987:.
1960:.
1911:.
1834:π
1811:n
1807:m
1805:(
1801:n
1797:m
1767:2
1762:]
1757:)
1750:2
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1714:)
1709:w
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1686:n
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1677:[
1670:2
1665:]
1660:)
1653:2
1649:w
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1628:(
1617:)
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1595:(
1589:m
1585:H
1580:[
1573:2
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1563:w
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1548:(
1541:0
1537:I
1533:=
1530:)
1527:z
1524:,
1521:y
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1512:(
1507:n
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1500:I
1485:k
1469:k
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1403:(
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1377:z
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1371:R
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1266:j
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1240:R
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1172:(
1165:[
1154:)
1149:w
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1140:2
1132:(
1126:n
1122:H
1117:)
1112:w
1108:x
1103:2
1095:(
1089:m
1085:H
1079:w
1074:0
1070:w
1062:0
1058:E
1054:=
1051:)
1048:z
1045:,
1042:y
1039:,
1036:x
1033:(
1028:n
1025:m
1021:E
1010:)
1008:z
1006:,
1004:y
1002:,
1000:x
998:(
994:n
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921:i
919:0
911:l
904:p
902:(
900:l
898:2
894:l
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876:l
872:p
863:l
859:p
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844:l
840:p
828:p
823:L
817:w
815:/
813:r
809:ρ
785:e
781:)
775:l
772:(
764:2
754:2
749:]
745:)
739:(
734:l
729:p
725:L
720:[
713:l
703:0
699:I
695:=
692:)
686:,
680:(
675:l
672:p
668:I
653:)
651:φ
649:,
647:r
645:(
641:l
637:p
614:)
553:2
549:n
526:1
522:n
501:a
475:0
471:k
446:2
441:2
437:n
428:2
423:1
419:n
413:a
408:0
404:k
400:=
397:V
353:n
349:m
345:n
341:m
310:.
285:H
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
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