The 1975 measurements of Gambling, Payne and Matsumura established a simple far-field method for extracting the normalized mode conversion coefficient DD of a step-index multimode optical fiber from Gloge's coupled-power-flow equation, and found DD in glass-core fibers to be roughly two orders of magnitude larger than in a liquid-core fiber, increasing further after heat treatment, with the difference attributed only qualitatively to fabrication-induced strains and inhomogeneities. The open core that survives fifty years of subsequent literature is twofold: (i) a quantitative microscopic attribution of DD in solid-core (glass) fibers — identifying which fabrication-induced defects and stress distributions dominate DD for given fibers, and how processing steps such as thermal treatment change it — at a level that would let DD be predicted from fiber fabrication rather than merely measured; and (ii) a delineation of the validity limits of the continuum, nearest-neighbor coupled-power-flow approximation underlying the closed-form extraction relations, i.e. how small a normalized frequency or how strong non-adjacent mode coupling may be before those relations fail. The third limitation acknowledged by the source work — independent experimental characterization of the angle-dependent attenuation coefficient A(θ)A(\theta) — has been directly met by later simultaneous-measurement and estimation methods (1978, 1981, 2006, 2013) and is no longer part of the open core.

Background

A step-index multimode optical fiber guides light in a core of constant refractive index surrounded by a lower-index cladding and supports many guided modes, which may be labeled by the angle θ\theta that their wave normals make with the fiber axis. Because modes at different angles propagate at different group velocities, the spread of modal power over θ\theta (intermodal dispersion) limits the transmission bandwidth of the fiber, and any mechanism that transfers power between modes — mode conversion, or mode mixing — controls how the launched distribution, and hence pulse spreading, evolves with propagation length. Mode conversion is caused by weak random perturbations of the waveguide: microbends (small random bends of the fiber axis, imposed for example by coating irregularities or external pressure), core-diameter and refractive-index fluctuations, and frozen-in mechanical stress. Gloge (1972) modeled this statistically: when the perturbations couple primarily adjacent modes and the mode spectrum is dense, the modal power P(Θ,z)P(\Theta,z) obeys a diffusion-type coupled-power-flow equation, ∂P/∂z=−A(Θ)P+(D/Θ) ∂/∂Θ (Θ ∂P/∂Θ)\partial P/\partial z = -A(\Theta)P + (D/\Theta)\,\partial/\partial\Theta\,(\Theta\,\partial P/\partial\Theta), where DD (in rad2 m−1\mathrm{rad^2\,m^{-1}}) is the normalized mode conversion coefficient and A(Θ)A(\Theta) is the angle-dependent attenuation coefficient. This form already contains the two approximations at issue here: the discrete mode spectrum is replaced by a continuum of propagation angles, which requires the normalized frequency V=2πa NA/λV = 2\pi a\,\mathrm{NA}/\lambda (core radius aa, numerical aperture NA, wavelength λ\lambda; VV sets how many guided modes the fiber supports) to be large enough that many closely spaced guided modes exist; and coupling is assumed to act only between nearest-neighbor modes in angle, which requires the perturbation spectrum to favor modes of nearly equal propagation constant. Gambling, Payne and Matsumura (1975) derived a closed-form solution of this equation for a collimated plane-wave input launched at a single core angle θ0\theta_0, and showed that the far-field output pattern (the angular distribution of the light emerging from the fiber end) changes from a ring to a filled disk beyond a transition angle θ0,m\theta_{0,m} obeying θ0,m2=4Dz\theta_{0,m}^2 = 4Dz, so that measuring the far field at two or more fiber lengths zz yields DD without any knowledge of A(Θ)A(\Theta). They verified the predicted square-root length scaling, measured D=3×10−6 rad2 m−1D = 3\times 10^{-6}\,\mathrm{rad^2\,m^{-1}} in an unstressed liquid-core fiber (increasing by up to about a factor of ten under transverse pressure), and found DD of order 3.03.0–7.4×10−4 rad2 m−17.4\times 10^{-4}\,\mathrm{rad^2\,m^{-1}} in glass-core fibers, increasing further after heat treatment — a difference they attributed only qualitatively to fabrication-induced strains and inhomogeneities. In the same period, Olshansky (1975) analyzed how mode coupling in graded-index fibers (fibers whose core index varies smoothly with radius rather than being constant) depends on fiber imperfection statistics, and Nagano and Kawakami (1980) extended mode-conversion-coefficient measurements to graded-index fibers. Later work measured DD together with mode-dependent loss (Miyagi et al. 1978; Nagano et al. 1981), re-solved the power-flow equation through equivalent stochastic formulations of the same diffusion-type dynamics, the Fokker–Planck and Langevin equation approaches (Savović and Djordjevich 2002), and generalized the model to angle-dependent diffusion and attenuation functions fitted to far-field patterns or impulse responses (the output pulse shapes following a short input pulse; Mateo et al. 2006; Mundus et al. 2013), but the microscopic attribution of DD in solid-core fibers has remained qualitative, and the quantitative validity limits of the continuum, nearest-neighbor approximation were never delineated.

Problem Statement

Starting from the coupled-power-flow framework of Gloge (1972) and the far-field transition-angle measurement of Gambling, Payne and Matsumura (1975), establish a quantitative, predictive account of intermodal power coupling in step-index multimode fibers with solid (glass) cores, consisting of two linked components: (i) attribute the measured normalized mode conversion coefficient DD microscopically — determine which fabrication-induced perturbations (core-diameter and index fluctuations, microbends, frozen-in stress distributions, inhomogeneities) dominate DD for given solid-core fibers, quantitatively enough that DD can be predicted from the statistics of those perturbations, and explain how specific processing steps such as thermal treatment increase DD (the source paper measured glass-core DD roughly two orders of magnitude above a liquid-core fiber and only qualitatively blamed strains and inhomogeneities); and (ii) delineate the range of validity of the approximations under which the closed-form extraction relations, including the transition-angle law θ0,m2=4Dz\theta_{0,m}^2 = 4Dz, were derived — how small the normalized frequency VV (how few, how widely spaced modes) and how strong the coupling between non-adjacent modes may be before the continuum, nearest-neighbor coupled-power-flow solution and its transition-angle criterion fail quantitatively. Both components serve one objective: making the mode conversion coefficient, and hence the evolution of modal power and the associated bandwidth and loss behavior, predictable from how a solid-core fiber is made and treated, rather than only measurable after the fact. The third limitation acknowledged by the source work — independent experimental characterization of the angle-dependent attenuation coefficient A(θ)A(\theta) — is not part of this problem, because it has been achieved by later simultaneous-measurement and estimation methods (Miyagi et al. 1978; Nagano et al. 1981; Mateo et al. 2006; Mundus et al. 2013).

References

  1. W. A. Gambling, D. N. Payne, H. Matsumura, 'Mode conversion coefficients in optical fibers', Applied Optics 14(7), 1538–1542 (1975). DOI: 10.1364/ao.14.001538, https://doi.org/10.1364/ao.14.001538
  2. D. Gloge, 'Optical power flow in multimode fibers', Bell System Technical Journal 51(8), 1767–1783 (1972). DOI: 10.1002/j.1538-7305.1972.tb02682.x, https://doi.org/10.1002/j.1538-7305.1972.tb02682.x
  3. Robert Olshansky, 'Mode coupling effects in graded-index optical fibers', Applied Optics 14(4), 935–945 (1975). DOI: 10.1364/ao.14.000935, https://doi.org/10.1364/ao.14.000935
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  12. Alexandar Djordjevich, Svetislav Savović, 'Mode coupling in glass optical fibers and liquid-core optical fibers by three methods', Optics & Laser Technology 75, 29–33 (2015). DOI: 10.1016/j.optlastec.2015.06.013, https://doi.org/10.1016/j.optlastec.2015.06.013
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