page_title: Double-clad fiber
section_title: Filling factor
hierarchical_section_title: Double-clad fiber / Fiber for amplifiers and fiber lasers / Filling factor
context_page_description: Double-clad fiber 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 core. It is surrounded by the inner cladding, which is surrounded by the outer cladding. The three layers are made of materials with different 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 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 amplifiers and fiber lasers. In these fibers, the core is doped with active dopant material; it both guides and amplifies the signal light. The inner cladding and core together guide the 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.
context_section_description: 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
   is the cross-sectional area of the cladding
   is the radius of the core (which is taken to be circular)
   is the absorption coefficient of pump light in the core
   is the length of the double-clad fiber, and
   is a dimensionless adjusting parameter, which is sometimes called the "filling factor"; .
  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,  approaches unity. The value of  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 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  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 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. 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.
caption: Estimates of the pump efficiency in a double-clad fiber with 
    
      
        
          F
          =
          0.8
        
      
      {\displaystyle F=0.8}
    
   (blue) and 
    
      
        
          F
          =
          0.9
        
      
      {\displaystyle F=0.9}
    
   (red), discussed in[2] compared to the results of the ray tracing simulations[6](black curves).
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