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Differential stress

2013-07-02T15:07:35Z

80.254.146.68:

[[File:Hexagonal hosohedron.png|thumb|The hexagonal [[hosohedron]], a regular map on the sphere with two vertices, six edges, six faces, and 24 flags.]]
In [[mathematics]], a '''regular map''' is a symmetric [[tessellation]] of a closed [[surface]]. More precisely, a regular map is a decomposition of a two-dimensional [[manifold]] such as a [[sphere]], [[torus]], or [[real projective plane]] into topological disks, such that every [[Flag (geometry)|flag]] (an incident vertex-edge-face triple) can be transformed into any other flag by a [[automorphism group|symmetry]] of the decomposition. Regular maps are, in a sense, topological generalizations of [[Platonic solids]]. The theory of maps and their classification is related to the theory of [[Riemann surface]]s, [[hyperbolic geometry]], and [[Galois theory]]. Regular maps are classified according to either: the [[genus (mathematics)|genus]] and [[orientability]] of the supporting surface, the [[Graph embedding |underlying graph]], or the [[automorphism group]].

==Overview==

Regular maps are typically defined and studied in three ways: topologically, group-theoretically, and graph-theoretically.

===Topological approach===
Topologically, a map is a [[CW complex |2-cell]] decomposition of a closed compact 2-manifold.

The genus g, of a map M is given by [[Euler characteristic|Euler's relation ]] <math> \chi (M) = |V| - |E| +|F| </math> which is equal to <math> 2 -2g </math> if the map is orientable, and <math> 2 - g </math> if the map is non-orientable. It is a crucial fact that there is a finite (non-zero) number of regular maps for every orientable genus except the torus.

===Group-theoretical approach===
Group-theoretically, the permutation representation of a regular map ''M'' is a transitive [[permutation group]] ''C'', on a set <math>\Omega</math> of [[Flag (geometry)|flags]], generated by a fixed-point free involutions ''r''0, ''r''1, ''r''2 satisfying (r0r2)2= I. In this definition the faces are the orbit of ''F'' = ''<''r0, ''r''1>, edges are the orbit of ''E'' = <''r''0, ''r''2>, and vertices are the orbit of ''V'' = <''r''1, ''r''2>. More abstractly, the automorphism group of any regular map is the non-degenerate, homomorphic image of a <2,m,n>-[[triangle group]].

===Graph-theoretical approach===
Graph-theoretically, a map is a cubic graph <math>\Gamma</math> with edges coloured blue, yellow, red such that: <math>\Gamma</math> is connected, every vertex is incident to one edge of each colour, and cycles of edges not coloured blue, have length 4. Note that <math>\Gamma</math> is the ''flag graph'' or ''graph encoded map (GEM)'' of the map, defined on the vertex set of flags <math>\Omega</math> and is not the skeleton G = (V,E) of the map. In general, |<math>\Omega</math>| = 4|E|.

A map M is regular iff Aut(M) [[Group action|acts]] [[Group action#Types_of_actions|regularly]] on the flags. Aut(''M'') of a regular map is transitive on the vertices, edges, and faces of ''M''. A map ''M'' is said to be reflexible iff Aut(''M'') is regular and contains an automorphism <math>\phi</math> that fixes both a vertex ''v'' and a face ''f'', but reverses the order of the edges. A map which is regular but not reflexible is said to be [[Chirality (mathematics)|chiral]].

==Examples==

* The [[great dodecahedron]] is a regular map with pentagonal faces in the orientable surface of genus 4.
* The [[Hemicube (geometry)|hemicube]] is a regular map of type {4,3} [[File:Hemicube2.PNG|thumb|The hemicube, a regular map.]]
* The [[hemi-dodecahedron]] is a regular map produced by pentagonal embedding of the Petersen graph in the projective plane.
* The p-[[hosohedron]] is a regular map of type {2, p}. Note that the hosohedron is non-polyhedral in the sense that it is not an [[abstract polytope]]. In particular, it doesn't satisfy the diamond property.
* The [[Dyck map]] is a regular map of 12 octagons on a genus-3 surface. Its underlying graph, the [[Dyck graph]], can also form a regular map of 16 hexagons in a torus.

The following is a complete list of regular maps in surfaces of positive [[Euler characteristic]]: the sphere and the projective plane (Coxeter 80).
{| class="wikitable"
|-
|Characteristic|| Genus|| [[Schläfli symbol]] || Group || Graph || Notes
|-
|2 || 0 || {p,2} || C2 × Dih''p'' || [[Cycle graph|C''p'']] || Dihedron
|-
|2 || 0 || {2,p} || C2 × Dih''p'' || ''p''-fold [[Complete graph|K2]] || Hosohedron
|-
|2 || 0 || {3,3} || Sym4 || [[Complete graph|K4]] || Tetrahedron
|-
|2 || 0 || {4,3} || C2 × Sym4 || [[Complete graph|K4]] [[Tensor product of graphs|×]] [[Complete graph|K2]] || Cube
|-
|2 || 0 || {3,4} || C2 × Sym4 || K2,2,2 || Octahedron
|-
|2 || 0 || {5,3} || C2 × Alt5 || || Dodecahedron
|-
|2 || 0 || {3,5} || C2 × Alt5 || [[Complete graph|K6]] [[Tensor product of graphs|×]] [[Complete graph|K2]] || Icosahedron
|-
|1 || - || {2p,2}/2 || Dih2''p'' || [[Cycle graph|C''p'']] || Hemidihedron
|-
|1 || - || {2,2p}/2 || Dih2''p'' || ''p''-fold [[Complete graph|K2]] || Hemihosohedron
|-
|1 || - || {4,3} || Sym4 || [[Complete graph|K4]] || Hemicube
|-
|1 || - || {3,4} || Sym4 || 2-fold [[Complete graph|K3]]|| Hemioctahedron
|-
|1 || - || {5,3} || Alt5 || [[Petersen graph]] || Hemidodecahedron
|-
|1 || - || {3,5} || Alt5 || [[Complete graph|K6]] || Hemi-icosahedron
|-
|}

== See also ==
*[[Topological graph theory]]
*[[Abstract polytope]]
*[[Planar graph]]
*[[Toroidal graph]]
*[[Graph embedding]]
*[[Regular tiling]]
*[[Platonic solid]]

== References ==
* {{citation
| last1 = Coxeter | first1 = H. S. M. | author1-link = Harold Scott MacDonald Coxeter
| last2 = Moser | first2 = W. O. J.
| edition = 4th
| isbn = 978-0-387-09212-6
| publisher = Springer Verlag
| series = Ergebnisse der Mathematik und ihrer Grenzgebiete
| title = Generators and Relations for Discrete Groups
| volume = 14
| year = 1980}}.
*{{citation
| last = van Wijk | first = Jarke J. | authorlink = Jack van Wijk
| doi = 10.1145/1531326.1531355
| journal = Proc. SIGGRAPH (ACM Transactions on Graphics)
| page = 12
| title = Symmetric tiling of closed surfaces: visualization of regular maps
| issue = 3
| url = http://www.win.tue.nl/~vanwijk/regularmaps_siggraph09.pdf
| volume = 28
| year = 2009}}.
*{{citation
| last1 = Conder | first1 = Marston | author1-link = Marston Conder
| last2 = Dobcsányi | first2 = Peter
| doi = 10.1006/jctb.2000.2008
| issue = 2
| journal = Journal of Combinatorial Theory, Series B
| pages = 224–242
| title = Determination of all regular maps of small genus
| volume = 81
| year = 2001}}.
*{{citation
| last = Nedela | first = Roman
| title = Maps, Hypermaps, and Related Topics
| url = http://www.savbb.sk/~nedela/CMbook.pdf
| year = 2007}}.
*{{citation
| last = Vince | first = Andrew
| contribution = Maps
| title = Handbook of Graph Theory
| year = 2004}}.
*{{citation
| last1 = Brehm | first1 = Ulrich
| last2 = Schulte | first2 = Egon
| contribution = Polyhedral Maps
| title = Handbook of Discrete and Computational Geometry
| year = 2004}}.

[[Category:Topological graph theory]]
[[Category:Discrete geometry]]

Rotational temperature

2013-05-10T10:11:15Z

80.254.146.4:

[[Image:Dispersion-dc.png|thumb|right|Refractive index profile of dispersion-compensating double-clad fiber. c:core, i:inner cladding, o:outer cladding.]]

[[Image:High-power-dc.png|thumb|right|Refractive index profile of double-clad fiber for high power fiber lasers and amplifiers. c:core, i:inner cladding, o:outer cladding.]]

'''Double-clad fiber''' (DCF) is a class of [[optical fiber]] with a structure consisting of three layers of optical material instead of the usual two. The inner-most layer is called the ''[[Fiber optics#Principle of operation|core]]''. It is surrounded by the ''inner [[Cladding (fiber optics)|cladding]]'', which is surrounded by the ''outer cladding''. The three layers are made of materials with different [[refractive index|refractive indices]].

There are two different kinds of double-clad fibers. The first was developed early in optical fiber history with the purpose of engineering the [[Dispersion (optics)|dispersion]] of optical fibers. In these fibers, the core carries the majority of the light, and the inner and outer cladding alter the waveguide dispersion of the core-guided signal. The second kind of fiber was developed in the late 1980s for use with high power [[fiber amplifier]]s and [[fiber laser]]s. In these fibers, the core is doped with [[Active laser medium|active]] dopant material; it both guides and amplifies the signal light. The inner cladding and core together guide the [[Laser pumping|pump]] light, which provides the energy needed to allow amplification in the core. In these fibers, the core has the highest refractive index and the outer cladding has the lowest. In most cases the outer cladding is made of a [[polymer]] material rather than [[glass]].

==Dispersion-compensating fiber==

In double-clad fiber for dispersion compensation, the inner cladding layer has lower refractive index than the outer layer. This type of fiber is also called ''depressed-inner-cladding fiber'' and ''W-profile fiber'' (from the fact that a symmetrical plot of its refractive index profile superficially resembles the letter W).<ref name=kawakami>
{{cite journal
|author=S. Kawakami, S. Nishida
|year=1974
|title=Characteristics of a doubly clad optical fiber with a low-index inner cladding
|journal=[[IEEE Journal of Quantum Electronics]]
|volume=10 |issue=12 |pages=879–887
|bibcode=1974IJQE...10..879K
|doi=10.1109/JQE.1974.1068118
}}</ref>

This type of double-clad fiber has the advantage of very low [[microbend]]ing losses. It also has two zero-dispersion points, and low dispersion over a much wider [[wavelength]] range than standard singly clad fiber. Since the dispersion of such double-clad fibers can be engineered to a great extent, these fibers can be used for the compensation of [[chromatic dispersion]] in [[optical communications]] and other applications.

== Fiber for amplifiers and fiber lasers==
[[Image:Fl.svg|thumb|center|600px|Schematic diagram of cladding-pumped double-clad fiber laser]]
[[Image:OfsetDCF.png|thumb|right|Cross-section of circular DCF with offset core]]
[[Image:RectaDFC.png|thumb|right|Cross-section of DCF with rectangular inner cladding<ref name="Kouznetsov">
{{cite journal
|author=D. Kouznetsov, J. V. Moloney
|year=2003
|title=Highly efficient, high-gain, short-length, and power-scalable incoherent diode slab-pumped fiber amplifier/laser
|journal=[[IEEE Journal of Quantum Electronics]]
|volume=39 |issue=11 |pages=1452–1461
|bibcode=2003IJQE...39.1452K
|doi=10.1109/JQE.2003.818311
}}</ref>]]
In modern double-clad fibers for high power fiber amplifiers and lasers, the inner cladding has a higher refractive index than the outer cladding. This enables the inner cladding to guide light by [[total internal reflection]] in the same way the core does, but for a different range of wavelengths. This allows [[diode laser]]s, which have high power but low [[Radiance|brightness]], to be used as the optical pump source. The pump light can be easily coupled into the large inner cladding, and propagates through the inner cladding while the signal propagates in the smaller core. The doped core gradually absorbs the cladding light as it propagates, driving the amplification process. This pumping scheme is often called ''cladding pumping'', which is an alternative to the conventional ''core pumping'', in which the pump light is coupled into the small core. The invention of cladding pumping by a Polaroid fiber research team (H. Po, ''et al.'') revolutionized the design of fiber amplifiers and lasers.<ref name="Po">
{{cite conference
|author=H. Po, E. Snitzer, L. Tumminelli, F. Hakimi, N. M. Chu, T. Haw
|year=1989
|title=Doubly clad high brightness Nd fiber laser pumped by GaAlAs phased array
|booktitle=Proceedings of the Optical Fiber Communication Conference
|id=PD7
}}</ref> Using this method, modern fiber lasers can produce continuous power up to several kilowatts, while the signal light in the core maintains near [[Diffraction-limited system|diffraction-limited beam quality]].<ref name=jeong>
{{cite journal
|author=Y. Jeong, J. Sahu, D. Payne, J. Nilsson
|year=2004
|title=Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power
|url=http://www.opticsinfobase.org/abstract.cfm?id=81942
|journal=[[Optics Express]]
|volume=12 |issue=25 |pages=6088–6092
|bibcode=2004OExpr..12.6088J
|doi=10.1364/OPEX.12.006088
|pmid=19488250
}}</ref>

The shape of the cladding is very important, especially when the core diameter is small compared to the size of the inner cladding. Circular symmetry in a double-clad fiber seems to be the worst solution for a fiber laser; in this case, many [[cladding mode|modes]] of the light in the cladding miss the core and hence cannot be used to pump it.<ref name=bedo>
{{cite journal
|author=S. Bedö, W. Lüthy, H. P. Weber
|year=1993
|title=The effective absorption coefficient in double-clad fibers
|journal=[[Optics Communications]]
|volume=99 |issue=5–6 |pages=331–335
|bibcode=1993OptCo..99..331B
|doi=10.1016/0030-4018(93)90338-6
}}</ref> In the language of [[geometrical optics]], most of the [[ray (optics)|rays]] of the pump light do not pass through the core, and hence cannot pump it.
[[Ray tracing (physics)|Ray tracing]],<ref name="Liu">
{{cite journal
|author=A. Liu, K. Ueda
|year=1996
|title=The absorption characteristics of circular, offset, and rectangular double-clad fibers
|journal=[[Optics Communications]]
|volume=132 |issue=5–6 |pages=511–518
|bibcode=1996OptCo.132..511L
|doi=10.1016/0030-4018(96)00368-9
}}</ref> simulations of the paraxial propagation<ref name="Kouznetsov2">
{{cite journal
|author=D. Kouznetsov, J. V. Moloney
|year=2003
|title=Efficiency of pump absorption in double-clad fiber amplifiers. II: Broken circular symmetry
|journal=[[Journal of the Optical Society of America B]]
|volume=39 |issue=6 |pages=1259–1263
|bibcode=2002JOSAB..19.1259K
|doi=10.1364/JOSAB.19.001259
}}</ref> and mode analysis<ref name="Kouznetsov3">
{{cite journal
|author=D. Kouznetsov, J. V. Moloney
|year=2003
|title=Efficiency of pump absorption in double-clad fiber amplifiers. III: Calculation of modes
|journal=[[Journal of the Optical Society of America B]]
|volume=19 |issue=6 |pages=1304–1309
|bibcode=2002JOSAB..19.1304K
|doi=10.1364/JOSAB.19.001304
}}</ref> give similar results.

=== Chaotic fibers ===
In general, modes of a waveguide have "scars", which correspond to the classical trajectories. The scars may avoid the core, then
the mode is not coupled, and it is vain to excite such a mode in the double-clad fiber amplifier. The scars can be distributed more or less uniformly in
so-called [[chaotic fiber]]s<ref name="Doya">
{{cite journal
|author=P. Leproux, S. Février, V. Doya, P. Roy, D. Pagnoux
|year=2003
|title=Modeling and optimization of double-clad fiber amplifiers using chaotic propagation of pump
|journal=[[Optical Fiber Technology]]
|volume=7 |issue=4 |pages=324–339
|bibcode=2001OptFT...7..324L
|doi=10.1006/ofte.2001.0361
}}</ref> have more complicated cross-sectional shape and provide more uniform distribution of [[intensity (physics)|intensity]] in the inner cladding, allowing efficient use of the pump light. However, the scarring takes place even in chaotic fibers.

====Spiral shape====
[[Image:SpiralCladding.png|thumb|right|Spiral-shaped cladding (blue), its chunk (red), and 3 segments of a ray (green).]]
[[Image:ModesSpiralDCF4.jpg|400px|thumb|right|Modes of spiral-shaped double-clad fiber.<ref name="Kouznetsov3" />]]
An almost-circular shape with small spiral deformation seems to be the most efficient for [[chaotic fiber]]s. In such a fiber, the [[angular momentum]] of a ray increases at each reflection from the smooth wall, until the ray hits the "chunk", at which the spiral curve is broken (see figure at right). The core, placed in vicinity of this chunk, is intercepted more regularly by all the rays compared to other chaotic fibers. This behavior of rays has an analogy in wave optics. In the language of [[Normal mode|modes]], all the modes have non-zero derivative in vicinity of the chunk, and cannot avoid the core if it is placed there. One example of modes is shown in the figure below and to the right. Although some of modes show scarring and wide voids, none of these voids cover the core.

The property of DCFs with spiral-shaped cladding can be interpreted as conservation of angular momentum. The square of the derivative of a mode at the boundary can be interpreted as pressure. Modes (as well as rays) touching the spiral-shaped boundary transfer some angular momentum to it. This transfer of angular momentum should be compensated by pressure at the chunk. Therefore, no one mode can avoid the chunk. Modes can show strong scarring along the classical trajectories (rays) and wide voids, but at least one of scars should approach the chunk to compensate for the angular momentum transferred by the spiral part.

The interpretation in terms of angular momentum indicates the optimum size of the chunk. There is no reason to make the chunk larger than the core; a large chunk would not localize the scars sufficiently to provide coupling with the core. There is no reason to locaize the scars within an angle smaller than the core: the small derivative to the radius makes the manufacturing less robust; the larger <math>R'(\phi)</math> is, the larger the fluctuations of shape that are allowed without breaking the condition <math>R'(\phi)>0</math>. Therefore, the size of the chunk should be of the same order as the size of the core.

More rigorously, the property of the spiral-shaped domain follows from the theorem about boundary behavior of modes of the [[Dirichlet Laplacian]].<ref name=boundary>
{{cite journal
|author=D. Kouznetsov, J. V. Moloney
|year=2004
|title=Boundary behaviour of modes of a Dirichlet Laplacian
|journal=[[Journal of Modern Optics]]
|volume=51 |pages=1362–3044
|bibcode=2004JMOp...51.1955K
|doi=10.1080/09500340408232504
}}</ref> Although this theorem is formulated for the core-less domain, it prohibits the modes avoiding the core. A mode avoiding the core, then, should be similar to that of the core-less domain.

Stochastic optimization of the cladding shape confirms that an almost-circular spiral realizes the best coupling of pump into the core.<ref name="sto">{{cite journal
|author=I. Dristas, T. Sun, K. T. V. Grattan
|year=2007
|title=Stochastic optimization of conventional and holey double-clad fibres
|journal=[[Journal of Optics A]]
|volume=9 |issue=4 |pages=1362–3044
|bibcode=2007JOptA...9..405D
|doi=10.1088/1464-4258/9/4/016
}}</ref>

=== Filling factor ===

[[Image:FillingFactor.png|thumb|Estimates of the pump efficiency in a double-clad fiber with <math>F=0.8
</math> (blue) and <math> F=0.9</math> (red), discussed in<ref name="Kouznetsov"/>
compared to the results of the ray tracing simulations<ref name="Liu" />(black curves).]]

The efficiency of absorption of pumping energy in the fiber is an important parameter of a double-clad fiber laser. In many cases this efficiency can be approximated with<ref name="Kouznetsov"/>
:<math>1- \exp\left( - F \frac{\pi r^2}{S}\alpha L \right) ,</math>
where
:<math>S</math> is the cross-sectional area of the cladding
:<math>r</math> is the radius of the core (which is taken to be circular)
:<math>\alpha</math> is the [[absorption coefficient]] of pump light in the core
:<math>L</math> is the length of the double-clad fiber, and
:<math>F</math> is a [[dimensionless number|dimensionless]] adjusting parameter, which is sometimes called the "filling factor"; <math>0<F<1</math>.

The filling factor may depend on the initial distribution of the pump light, the shape of the cladding, and the position of the core within it.

The exponential behavior of the efficiency of absorption of pump in the core is not obvious. One could expect that some modes of the cladding (or some rays) are better coupled to the core than others; therefore, the "true" dependence could be a combination of several exponentials. Only comparison with simulations justifies this approximation, as shown in the figure above and to the right. In particular, this approximation does not work for circular fibers, see the initial work by Bedo et all, cited below.
For chaotic fibers, <math>F</math> approaches unity. The value of <math>F</math> can be estimated by [[numerical analysis]] with propagation of waves, expansion by modes or by geometrical optics [[Ray tracing (physics)|ray tracing]], and values 0.8 and 0.9 are only empirical adjusting parameters, which provide good agreement of the simple estimate with numerical simulations for two specific classes of double-clad fibers: circular offset and rectangular. Obviously, the simple estimate above fails when the offset parameter becomes small compared to the size of cladding.

The filling factor <math>F</math> approaches unity especially quickly in the spiral-shaped cladding, due to the special boundary behavior of the modes of the [[Dirichlet Laplacian]].<ref name=boundary /> Designers of double-clad fiber look for a reasonable compromise between the optimized shape (for the efficient couplung of pump into the core) and the simplicity of the manufacturing of the [[Optical fiber#Preform|preform]] used to draw the fibers.

The [[power scaling]] of a fiber laser is limited by unwanted nonlinear effects such as [[stimulated Brillouin scattering]] and [[stimulated Raman scattering]]. These effects are minimized when the fiber laser is short. For efficient operation, however, the pump should be absorbed in the core along the short length; the estimate above applies in this optimistic case. In particular, the higher the step in refractive index from inner to outer cladding, the better-confined the pump is. As a limiting case, the index step can be of order of two, from glass to air.<ref name="air">
{{cite journal
|author=N. A. Mortensen
|year=2007
|title=Air-clad fibers: pump absorption assisted by chaotic wave dynamics?
|url=https://www.opticsinfobase.org/oe/viewmedia.cfm?uri=oe-15-14-8988&seq=0
|journal=[[Optics Express]]
|volume=15 |issue=14 |pages=8988–8996
|arxiv=0707.1189
|bibcode=2007OExpr..15.8988M
|doi=10.1364/OE.15.008988
}}</ref> The estimate with filling factor gives an estimate of how short an efficient double-clad fiber laser can be, due to reduction in size of the inner cladding.

===Alternative structures===

For good cladding shapes the filling factor <math>F</math>, defined above, approaches unity; the following enhancement is possible at various kinds of tapering of the cladding;<ref name="taperedc">
{{cite journal
|author=V. Filippov, Yu. Chamorovskii, J. Kerttula1, K. Golant, M. Pessa, O. G. Okhotnikov
|year=2008
|title=Double clad tapered fiber for high power applications
|url=https://www.opticsinfobase.org/oe/viewmedia.cfm?uri=oe-16-3-1929&seq=0
|journal=[[Optics Express]]
|volume=16 |issue=3 |pages=1929–1944
|bibcode=2008OExpr..16.1929F
|doi=10.1364/OE.16.001929
|pmid=18542272
}}</ref> non-conventional shapes of such cladding are suggested.<ref name="slab">
{{cite journal
|author=D. Kouznetsov, J. V. Moloney
|year=2004
|title=Slab delivery of incoherent pump light to double-clad fiber amplifiers: An analytic approach
|journal=[[IEEE Journal of Quantum Electronics]]
|volume=40 |issue=4 |pages=378–383
|bibcode=2004IJQE...40..378K
|doi=10.1109/JQE.2004.824695
}}</ref>

[[Planar waveguide]]s with an active gain medium take an intermediate position between conventional [[solid-state laser]]s and double-clad fiber lasers. The planar waveguide may confine a multi-mode pump and a high-quality signal beam, allowing efficient coupling of the pump, and diffraction-limited output.<ref name="Kouznetsov2" /><ref name="bonner">
{{cite journal
|author=C. L. Bonner, T. Bhutta, D. P. Shepherd, A. C. Tropper
|year=2000
|title=Double-clad structures and proximity coupling for diode-bar-pumped planar waveguide lasers
|journal=[[IEEE Journal of Quantum Electronics]]
|volume=36 |issue=2 |pages=236–242
|bibcode=2000IJQE...36..236B
|doi=10.1109/3.823470
}}</ref>

==Notes and references==

{{reflist}}

{{DEFAULTSORT:Double-Clad Fiber}}
[[Category:Optical fiber]]