Double-clad fiber
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216192369
2008-05-31T14:34:45Z
Domitori
990480
/* Spiral shape */ corrected number of figure
[[Image:OfsetDCF.png|thumb|right|Fig.1. Cross-section of circular DCF with offset core.]]
[[Image:RectaDFC.png|thumb|right|Fig.2. Cross-section of DCF with rectangular inner cladding
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]]
In [[fiber optics]], a '''double-clad fiber''' (or ''doubly'' clad fiber, or DCF) is an [[optical fiber]] that has a relatively small-diameter [[Fiber optics#Principle of operation|core]] and two layers of large-diameter [[cladding]]. Usually, both cladding layers have lower [[refractive index]] than the core, and the inner cladding layer has lower refractive index than the outer layer. This allows the inner cladding to carry [[transverse mode|multimode]] light of a different [[wavelength]] from that carried in the core of the fiber. 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). Mostly, DCF fibers are used as [[fiber laser]].
==Fiber lasers and optical amplifiers==
Double-clad fibers often are used for [[fiber laser]]s and [[optical amplifier]]s, because the core can be doped to act as the [[gain medium]] while the inner cladding layer carries a [[laser pumping|pump]] beam used to maintain the [[population inversion]] in the core. For such applications, the core may be [[single-mode fiber|single-mode]] or may be [[multimode fiber|multimode]] with a low [[numerical aperture]]. The use of double-clad fibers allows fiber lasers to be [[power scaling|scaled]] to higher powers than can otherwise be achieved.
The shape of the cladding is very important at narrow core and wide cladding. Circular symmetry in a double-clad fiber seems to be the worst solution for a fiber laser; in this case, many 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. Bedo|coauthors= W. Luthy, and H. P. Weber |
title=The effective absorption coefficient in double-clad fibers|
journal=[[Optics Communications]]|
volume=99| issue=| pages=331–335| year=1993|
url=
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVF-46JGTGD-M5&_user=10&_coverDate=06%2F15%2F1993&_alid=550903253&_rdoc=1&_fmt=summary&_orig=search&_cdi=5533&_sort=d&_docanchor=&view=c&_ct=1&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=c8a4c3ecc3d9a4e9ecb84f96cfef0333 |
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|
title=The absorption characteristics of circular, offset, and rectangular double-clad fibers|
author=A. Liu| coauthors= K. Ueda|
url=
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVF-497C4YV-BW&_user=10&_coverDate=12%2F15%2F1996&_alid=550869877&_rdoc=3&_fmt=summary&_orig=search&_cdi=5533&_sort=d&_docanchor=&view=c&_ct=3&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=688bbca25fdd98e29caadb676b003c1e
|
journal=[[Optics Communications]]|
volume=132|
year=1996|
pages= 511–518|
doi=10.1016/0030-4018(96)00368-9}}
</ref>, simulations of the paraxial propagation<ref name="Kouznetsov2">{{cite journal|
title=Efficiency of pump absorption in double-clad fiber amplifiers. 2: Broken circular symmetry|
author= Kouznetsov, D.|
coauthors=Moloney, J.V.| journal=[[JOSAB]]|
url=http://josab.osa.org/abstract.cfm?id=68991|
volume=39 | year=2003 | issue=6 | pages=1259–1263|
doi=10.1364/JOSAB.19.001259}}</ref>
and mode analysis<ref name="Kouznetsov3">{{cite journal|
title=Efficiency of pump absorption in double-clad fiber amplifiers.3:Calculation of modes|
author= Kouznetsov, D.| coauthors=Moloney, J.V.|
journal=[[JOSAB]]|
url=http://josab.osa.org/abstract.cfm?id=68997|
volume=19 |year=2003 | issue=6 | pages=1304–1309|
doi=10.1364/JOSAB.19.001304}}</ref>
give similar results.
== Chaotic fibers ==
[[Image:SpiralCladding.png|thumb|right|Fig.3. Spiral-shaped cladding (blue), its chunk (red), and 3 segments of a ray (green).]]
[[Image:ModesSpiralDCF4.jpg|400px|thumb|right|Fig.4. Modes of spiral-shaped DCF by <ref name="Kouznetsov3">{{cite journal|
title=Efficiency of pump absorption in double-clad fiber amplifiers.3:Calculation of modes|
author= Kouznetsov, D.| coauthors=Moloney, J.V.|
journal=[[JOSAB]]|
url=http://josab.osa.org/abstract.cfm?id=68997|
volume=19 |year=2003 | issue=6 | pages=1304–1309|
doi=10.1364/JOSAB.19.001304}}</ref>.
]]
In general, modes of a waveguide have [[scar]]s, 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|title=Modeling and optimization of double-clad fiber amplifiers using chaotic propagation of pump
| author= Leproux, P.| coauthors=S. Fevrier, V. Doya, P. Roy, and D. Pagnoux| journal=[[Optical Fiber Technology]]
| url=http://www.ingentaconnect.com/content/ap/of/2001/00000007/00000004/art00361
| volume=7 | year=2003 | issue=4 | pages=324–339|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.
<!--,<ref>http://josab.osa.org/abstract.cfm?id=68991</ref><ref>http://josab.osa.org/abstract.cfm?id=68997</ref>
-->
However, the scaring takes place even in chaotic fibers.
===Spiral shape===
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 [fig.3] of the spiral curve. The core placed in vicinity of this chunk is visited by all the rays more regularly, than in other chaotic fibers.
This behavior of rays has analogy in the wave optics. On the language of [[modes]], all the modes have non-zero
derivative in vicinity of the chunk, and cannot avoid the core placed there.
One example of modes is shown in fig.4. Although some of modes are scared and show wide voids, none of these voids cover the core.
The property of the DCF 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 it some angular momentum.
This transfer of angular momentum should be compensated with 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 the angular momentum transferredby the spiral part.
The interpretation with angular momentum indicates the reasonable size of the chunk. There is no reason to make this 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 is <math>R'(\phi)</math>, the larger fluctuations
of shape are allowed without breaking the condition <math>R'(\phi)>0</math>. Therefore, the size of the chunk should be of order of 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|
title=Boundary behaviour of modes of a Dirichlet Laplacian|
author=D.Kouznetsov|coauthors=J.Moloney|
url=http://www.metapress.com/content/be0lua88cwybywnl/?p=6fbbaafc684541b28a96403556968148&pi=6|
journal=[[Journal of Modern Optics]]|
volume=51|
year=2004|
pages= 1362–3044 }}
</ref>. Although this theorem is formulated for the core-less domain, it prohibits the modes avoiding the core. A mode avoiding the core, then, shold be similar to that of the core-less domain.
<!--http://www.metapress.com/content/be0lua88cwybywnl/?p=6fbbaafc684541b28a96403556968148&pi=6
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The stochastic optimization of the shape of cladding confirm that almost circular spiral realizes the best coupling of pump into the core <ref name="sto">{{cite journal
|title=Stochastic optimization of conventional and holey double-clad fibres
|author=I.Dristas|coauthors=T.Sun and K.T.V.Grattan
|url=http://www.iop.org/EJ/abst/1464-4258/9/4/016
|journal=[[Journal of Optics A]]
|volume=9
|year=2007
|pages= 1362–3044
|doi=10.1088/1464-4258/9/4/016 }}
</ref>.
== Filling factor ==
[[Image:FillingFactor.png|thumb|400px|Fig.5. 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">{{cite journal|title=Highly efficient, high-gain, short-length, and power-scalable incoherent diode slab-pumped fiber amplifier/laser| author= Kouznetsov, D.| coauthors=Moloney, J.V.| journal=[[IEEE Journal of Quantum Electronics]]| url=http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?tp=&arnumber=1242365&isnumber=27838|volume=39 | year=2003 | issue=11 | pages=1452–1461 | doi=10.1109/JQE.2003.818311}}
</ref>
compared to the results of the ray tracing simulations<ref name="Liu">{{cite journal|
title=The absorption characteristics of circular, offset, and rectangular double-clad fibers|
author=A. Liu| coauthors= K. Ueda|
url=
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVF-497C4YV-BW&_user=10&_coverDate=12%2F15%2F1996&_alid=550869877&_rdoc=3&_fmt=summary&_orig=search&_cdi=5533&_sort=d&_docanchor=&view=c&_ct=3&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=688bbca25fdd98e29caadb676b003c1e
|
journal=[[Optics Communications]]|
volume=132|
year=1996|
pages= 511–518|
doi=10.1016/0030-4018(96)00368-9}}
</ref>(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">{{cite journal|title=Highly efficient, high-gain, short-length, and power-scalable incoherent diode slab-pumped fiber amplifier/laser| author= Kouznetsov, D.| coauthors=Moloney, J.V.| journal=IEEE Journal of Quantum Electronics| url=http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?tp=&arnumber=1242365&isnumber=27838|volume=39 | year=2003 | issue=11 | pages=1452–1461 | doi=10.1109/JQE.2003.818311}}</ref>
:<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. We 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 combination of several exponentials; and the
only comparison with simulations (Fig.5) justifies this approximation. 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,
and values 0.8 and 0.9 are only empiric adjusitng parameters, which provide good agreement of the
simple estimate with numerical simulations for two specific classes of double-clad fibers, circular ofset and rectangular.
Obviously, the simple estimate above fails, as the ofset 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>{{cite journal|
title=Boundary behaviour of modes of a Dirichlet Laplacian|
author=D.Kouznetsov|coauthors=J.Moloney|
url=http://www.metapress.com/content/be0lua88cwybywnl/?p=6fbbaafc684541b28a96403556968148&pi=6|
journal=[[Journal of Modern Optics]]|
volume=51|
year=2004|
pages= 1362–3044 }}
</ref>. 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 [[preform]] used to draw the fibers.
The [[power scaling]] of a fiber laser is limited by the
unwanted nonlinear effects, such as [[Brillouin scattering]] and the [[Raman conversion]]. Therefore
the powerful fiber laser should be short. However, for the efficient operation, the pump shold be absorbed in the core
along the short length; and the estimate (1) provides the optimistic estimate for this case. In particular, the higher step of the index of refraction from the inner cladding to the outher one allows to focus (and confine) the pump in a smaller inner cladding. 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
|title=Air-clad fibers: pump absorption assisted by chaotic wave dynamics?
|journal=[[Optics Express]]
|year=2007
|volume=15
|issue=14
|pages=8988–8996
|url=http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-14-8988
|doi=10.1364/OE.15.008988
}}</ref>. The estimate with filling factor gives an estimate, how shorter can be made an efficient DCF due to reduction of size of the inner cladding.
==Alternative structures==
For the good choice of the shape of the cladding, the filing factor <math>~F~</math> above approaches unity;
the following enhancement is possible at various kinds of tapering of the cladding
<ref name="taperedc">{{cite journal
|author=V. Filippov
|coauthors=Yu.Chamorovskii, J.Kerttula1, K.Golant, M.Pessa, and O.G.Okhotnikov
|title=Double clad tapered fiber for high power applications
|journal=[[Optics Express]]
|volume=16
|issue=3
|year=2008
|pages=1929–1944
|url=http://www.opticsexpress.org/abstract.cfm?URI=OPEX-16-3-1929
|doi=10.1364/OE.16.001929
}}
</ref>; non-convential shapes of such cladding are suggested
<ref name="slab">{{cite journal
|author=D.Kouznetsov
|coauthors=J.V.Moloney
|title=Slab delivery of incoherent pump light to double-clad fiber amplifiers: An analytic approach.
|journal=[[IEEE Journal of Quantum Electronics]]
|year=2004
|volume=40
|issue 4
|pages=378–383
|url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=28588&arnumber=1278604&count=15&index=6
|doi=10.1109/JQE.2004.824695
}}
</ref>.
The [[planar waveguide]]s with [[gain medium]] take an intermediate position between conventional [[solid-state lasers]] and the conventional double-clad lasers. The planar waveduide may confine multi-mode pump and
high-quality signal, allowing the efficient coupling of pump and the diffraction-limited output<ref name="bonner">{{cite journal
| url=
http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=17828&arnumber=823470&count=17&index=14
| author= C.L.Bonner
| coauthors=T. Bhutta, D.P.Shepherd, A.C.Tropper
| 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
| year=2000
| doi=10.1109/3.823470
}}</ref><ref name="Kouznetsov2">{{cite journal|title=Efficiency of pump absorption in double-clad fiber amplifiers. 2: Broken circular symmetry| author= Kouznetsov, D.| coauthors=Moloney, J.V.| journal=[[JOSAB]]| url=http://josab.osa.org/abstract.cfm?id=68991|volume=39 | year=2003 | issue=6 | pages=1259–1263|doi=10.1364/JOSAB.19.001259}}</ref>
==Other applications==
A double-clad fiber has the advantage of very low [[microbend]]ing losses. It also has two zero-[[dispersion (optics)|dispersion]] points, and low dispersion over a much wider [[wavelength]] range than a singly-clad fiber. Double-clad fibers can also be used for the compensation of [[chromatic dispersion]] in [[optical communications]] and other applications.
==Notes and references==
{{reflist}}
*[[Federal Standard 1037C]]
[[Category:Optical fiber]]