Fourier decomposition method for mode characterization in metal-clad fiber with complex index profile

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Abstract

Fourier decomposition method in the cylindrical coordinates is used to characterize LP0x mode effective indexes in a step fiber with complex index profile under material dispersion, from the guided modes to the radiation mode. The method help sorting the modes in function of physical properties associated to a particular mode type LP0x, etc. The main findings of this work are: (1) The imaginary part of effective indexes of the guided modes and lower order cladding modes is several orders of magnitude smaller than the real part. (2) Both real and imaginary parts of metal-clad refractive indexes around the fiber have a strong influence on the real part of LP0x mode effective indexes. (3) In some regions of the wavelength, the fundamental mode LP01 in an aluminum-clad fiber disappears, but the higher order cladding modes still exist and propagate.

Introduction

Fiberoptic components [1], [2], [3], [4], fiberoptic laser amplifiers (EDFA) [5], and few emerging concept sensors in the fiberoptic system, whose refractive index profiles can be described in terms of complex functions, have attracted much interest recently. These elements are known to exhibit excellent properties for applications related to communication and sensing. Although the modes of a circular symmetry fiber can be evaluated with elementary analytical mathematical tools, more complicated situations must, in practice, be analyzed with numerical methods. Such a situation could occur, for example, when the index profile of the core or cladding layer in a step fiber becomes complex, thereby making the fiber absorbing or leaky, a case often met in fiber lasers, fiberoptic amplifiers, tunable fiberoptic filters, etc. In general, when the refractive index profile of a step fiber is real, the solutions of the eigenvalue equations are real and analytical [6]. But, if complex refractive index profile is introduced into the fiber, the solutions of the eigenvalue equations have the tendency to become complicated with no satisfying results reported so far.

Approximate and numerical methods have been reported in the papers, for evaluating the propagation characteristics of a fiber with complex index profile. The downhill method [7] attempts to determine the propagation constant of the guided mode, which has high model fidelity, in a single mode fiber with complex index profile. However, as far as the multimode fiber or single mode fiber with cladding modes are concerned, multi-hills and multi-valleys are introduced in the system. The method suffers from the intrinsic drawback that the roots may not be convergent to every mode, but to only one of the modes. Costantini et al. [1] and Adams [6] introduce another method under the assumption that the leak of the modes is significantly low. One of the consequences of this assumption is that the solutions of higher order cladding modes and radiation mode are not as accurate as those of lower order modes.

In order to solve the problem, new methods have been introduced and developed in the past decades. Sunanda et al. [5] used variational methods to analyze the complex fundamental mode field in the erbium-doped fiber amplifiers. The scalar wave equation has been solved by the Rayleigh–Ritz variational procedure for the propagation of the fundamental mode to yield the signal gain and pump absorption in the doped fiber with complex refractive index profile. In the near field scanning optical microscopy, TM/TE modal solutions of the finite element approach for submicron loss have been given by Themistos et al. [8], [9]. Metal clad optical fibers with submicron diameter are currently being used in the Near Field Scanning Optical Microscopy (NFSOM). Along with the presence of pure TE or TM modes, one also notes the presence of hybrid modes in the NFSOM structure. The propagation of the fundamental and higher order guided optical pure TE or TM modes in the metal cladding circular waveguides is studied, and expressions describing complex propagation and field profiles are developed, Prade et al. [10] discusses and classifies the optical guided modes that propagate in an ideal metallic dielectric structure possessing a step fiber symmetry. The intensity profile of the allowed guided modes is shown, and the geometric and frequency dispersion of the effective indexes of the modes is discussed as a function of the opto-geometrical parameters of the fibers. In spite of these effects, still some problems need to be investigated and which can be summarized as follows:

  • 1.

    Under the assumption that the imaginary part of the field is much smaller than the real part, the method reported in Sader [11] attempts to separate the real and imaginary parts of the complex scalar wave equation to obtain two real scalar equations, by neglecting the imaginary part of the field. However, no mathematical evidence has been given to corroborate this assumption.

  • 2.

    Optical fiber is regarded as a bent metal dielectric interface with optical properties similar to that of two layer planar waveguides. This assumption gives rise to two apparent difficulties [12]: (1) It introduces singularity at the pole so that appropriate pole condition should be imposed to ensure the desired smoothness in the Cartesian coordinates, and (2) even simple equations with constant coefficients in the Cartesian coordinates have variable coefficients of the form r±k in polar coordinates.

  • 3.

    Two-dimensional Galerkin method using a complete orthogonal set of sine functions [13], [14] can deal with a fiber with a complex index profile. Although by employing N terms of sine functions, it can theoretically compute N×N effective indexes of different modes, the determination of most of them (excepting LP0x, LP1x) is useless in practice, not to mention of the exponential increase in the computational complexity.

In the recent past, some new methods of Fourier decomposition, or the so-called Galerkin method, for the mode analysis in fiber have also been reported [13], [14]. Very recently, Kim et al. [15] developed an efficient dispersion method based on Galerkin method of Bessel function. The striking and significant results reported in [13], [14], [15] have led us to investigate Galerkin or Fourier Decomposition methods and mode characteristics in a metal-clad step fiber with a complex refractive index profile.

Section snippets

Fourier decomposition method in cylindrical coordinates for optical fiber

In a step fiber (see Fig. 1), each of the components of the electric and magnetic fields (Ez,Hz) must satisfy the Helmoholtz equation (1) in a cylindrical coordinate system [18], where refractive index n=n1, in the core (0⩽ra); n=n2, in the cladding (a<rb); n=n3, in the metal-clad around the fiber (b<r⩽(b+c)); n=n4=1.000275, in the surrounding medium of the fiber ((b+c)<rR); and k=2π/λ; with λ lying between 0.62 and 2.0 μm2Ur2+1rUr+1r22Uφ2+2Uz2+nneff2k2U=0.In Eq. (1), the complex

Simulation results

In order to characterize the mode distribution of a metal-clad step-index single-mode fiber with complex index profile, we choose a Be–Ge codoped fiber as in [6], [17]. The geometry of the fiber is a four-layer co-axial circular structure. The core radius a is 3.8 μm; the cladding radius b is 62.5 μm; and c is the thickness of the metal-clad around the fiber. The composition of the metal-clad is copper or aluminum. The complex refractive indexes of copper and aluminum related to material

Conclusion

To summarize, we have applied the Fourier Decomposition method in the cylindrical coordinates to transform the Helmholtz equation to a matrix eigenvalue problem. Following important results were obtained:

  • 1.

    The imaginary parts of the effective indexes of the guided modes and lower order cladding modes are several orders of magnitude smaller than the real parts, which corresponds to the conclusion reported in Sader [11].

  • 2.

    Both real and imaginary parts of the refractive indexes of metal-clad under

Acknowledgements

Yanzhou Zhou thanks the Scientific Research Foundation of Harbin Institute of Technology, PR China, for partial funding of this work (Grant HIT.2000.58).

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