Double-clad fiber
In fiber optics, a double-clad fiber (or doubly clad fiber, or DCF) is an optical fiber that has a relatively small-diameter 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 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 lasers and optical amplifiers, because the core can be doped to act as the gain medium while the inner cladding layer carries a pump beam used to maintain the population inversion in the core. For such applications, the core may be single-mode or may be multimode with a low numerical aperture. The use of double-clad fibers allows fiber lasers to be 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.
In the language of geometrical optics, most of the rays of the pump light do not pass through the core, and hence cannot pump it. 
Ray tracing, simulations of the paraxial propagation
and mode analysis
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 fibers
have more complicated cross-sectional shape and provide more uniform distribution of intensity in the inner cladding, allowing efficient use of the pump light.
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 fibers.
In such a fiber, the angular momentum of a ray increases at each reflection from the smooth wall, until the
ray hits the chunk  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 R'(\phi), the larger fluctuations
-f shape are allowed without breaking the condition R'(\phi)>0. 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. 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.
The stochastic optimization of the shape of cladding confirm that almost circular spiral realizes the best coupling of pump into the core .

# Filling factor
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
where
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
-nly 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, ~F~ approaches unity. The value of ~F~ 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 ~F~ approaches unity especially quickly in the spiral-shaped cladding, due to the special boundary behavior of the modes of the Dirichlet Laplacian. 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
. 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 ~F~ above approaches unity;
the following enhancement is possible at various kinds of tapering of the cladding
; non-convential shapes of such cladding are suggested
The planar waveguides 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

# Other applications
A double-clad fiber has the advantage of very low microbending losses. It also has two zero-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
-  S. Bedo (1993). "The effective absorption coefficient in double-clad fibers". Optics Communications. 99: 331335. doi:10.1016/0030-4018(93)90338-6. Unknown parameter |coauthors= ignored (help).mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"\"""\"""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{display:none;font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
-  Jump up to: 2.0 2.1 A. Liu (1996). "The absorption characteristics of circular, offset, and rectangular double-clad fibers". Optics Communications. 132: 511518. doi:10.1016/0030-4018(96)00368-9. Unknown parameter |coauthors= ignored (help)
-  Jump up to: 3.0 3.1 Kouznetsov, D. (2003). "Efficiency of pump absorption in double-clad fiber amplifiers. 2: Broken circular symmetry". JOSAB. 39 (6): 12591263. doi:10.1364/JOSAB.19.001259. Unknown parameter |coauthors= ignored (help)
-  Jump up to: 4.0 4.1 Kouznetsov, D. (2003). "Efficiency of pump absorption in double-clad fiber amplifiers.3:Calculation of modes". JOSAB. 19 (6): 13041309. doi:10.1364/JOSAB.19.001304. Unknown parameter |coauthors= ignored (help)
-   Leproux, P. (2003). "Modeling and optimization of double-clad fiber amplifiers using chaotic propagation of pump". Optical Fiber Technology. 7 (4): 324339. doi:10.1006/ofte.2001.0361. Unknown parameter |coauthors= ignored (help)
-  Jump up to: 6.0 6.1 D.Kouznetsov (2004). "Boundary behaviour of modes of a Dirichlet Laplacian". Journal of Modern Optics. 51: 13623044. Unknown parameter |coauthors= ignored (help)
-  I.Dristas (2007). "Stochastic optimization of conventional and holey double-clad fibres". Journal of Optics A. 9: 13623044. doi:10.1088/1464-4258/9/4/016. Unknown parameter |coauthors= ignored (help)
-  Jump up to: 8.0 8.1 Kouznetsov, D. (2003). "Highly efficient, high-gain, short-length, and power-scalable incoherent diode slab-pumped fiber amplifier/laser". IEEE Journal of Quantum Electronics. 39 (11): 14521461. doi:10.1109/JQE.2003.818311. Unknown parameter |coauthors= ignored (help)
-  N.A.Mortensen (2007). "Air-clad fibers: pump absorption assisted by chaotic wave dynamics?". Optics Express. 15 (14): 89888996. doi:10.1364/OE.15.008988.
-  V. Filippov (2008). "Double clad tapered fiber for high power applications". Optics Express. 16 (3): 19291944. doi:10.1364/OE.16.001929. Unknown parameter |coauthors= ignored (help)
-  D.Kouznetsov (2004). "Slab delivery of incoherent pump light to double-clad fiber amplifiers: An analytic approach". IEEE Journal of Quantum Electronics. 40: 378383. doi:10.1109/JQE.2004.824695. Text "issue 4
" ignored (help); Unknown parameter |coauthors= ignored (help)
-  C.L.Bonner (2000). "Double-clad structures and proximity coupling for diode-bar-pumped planar waveguide lasers". IEEE Journal of Quantum Electronics. 36 (2): 236242. doi:10.1109/3.823470. Unknown parameter |coauthors= ignored (help)
- Federal Standard 1037C