Single parameter scalar variational analysis for single-mode fibers with smoothed-out and composite refractive index profiles: A comparative study

A number of single parameter variational approximations for single-mode graded-index fibers have been reported in the literature. One of these approximations was also proposed by us some time ago which describes the propagation characteristics of such fibers having power-law profile with a high degree of accuracy. Using these approximations we have computed the propagation constant and the normalized dispersion for single-mode fibers with smoothed-out and composite refractive index profiles which are of great current interest. We have shown that the single parameter approximation proposed by us earlier computes these propagation characteristics with a better degree of accuracy than all other single parameter approximations even for these special types of profiles.     

Modelisation de la propagation dans les fibres optiques

This paper presents a new method for investigating the effects of mode conversion due to microbends on the transmission characteristics (attenuation, frequency response ...) of multimode optical fibres. The method applies to both step-index and graded-index fibres and takes into account any dependence of the mode coupling coefficient upon propagation angle. The general theory of propagation with mode coupling is developed and we introduce the concept of modes. The propagation problem is then simplified by using hypotheses based on exerpimental results and the solution is obtained with the help of a computer. Practical results concerning the increase of attenuation due to microbends are presented, from which the precautions to be taken when cabling fibres are deduced. It is also shown that in every practical case encountered, the prediction of the total attenuation and of the frequency response of a given fibre, when the launching conditions are specified is obtained from analytical expressions which approximate well with the exact results obtained from the computer. All these results make the method powerful and very useful for system designers when evaluating the effect of introducing an optical fibre into a transmission system.     

Upper limit of bandwidth improvement by optical equalization of graded-index fibres

Optical equalization was suggested as a means for increasing the bandwidth of optical fibre communication links when non-optimal graded-index fibre profiles are available. In this paper it is shown that by proper choice of the length as well as of the index of the compensating fibre one can significantly increase the bandwidth capability of the link. An upper limit of this improvement is derived, which is much larger than that obtained with alternating fibres of equal lengths.     

Fully automatic system for the coupling alignment of optical fibres

To realize the low-loss connection of two optical fibres intended as a transmission medium for optical signals, very tight tolerances have to be observed on account of the small geometrical dimensions involved. The present work describes a fully automatic system for the alignment of optical fibres in thex-y planes relative to a light source, which has only a single intensity peak, so that maximum optical power is launched into the fibre. The light source can be, for instance, the radiating end of a fibre, a semi-conductor laser or a light-emitting diode. The optical power coupled into the fibre serves as the control signal. Three precision displacement stages driven by stepping motors serve to align the fibres. Various sequential control algorithms for optimal coupling are investigated with reference to theory. Two algorithms are tested with the aid of a computer with reference to the alignment accuracy and speed requirements which have to be met and the one shown by theoretical simulation to be the more favourable is selected for technical realization. The resulting system was used for numerous alignment operations for the coupling of two graded-index fibres. The alignment time is below 1 s and the coupling efficiency compared with manual coupling alignment efficiency (= 100%) is above 99%, corresponding to an additional loss of <0.05 dB.     

Thermal Diffusion Process Estimation for Fabrication of Graded Plastic Optical Fibre

Thermal diffusion of dopants is investigated in the process of generating the graded-index profile of plastic optical fibres. Because the diffusion coefficient in high polymers has been shown to depend strongly on dopant concentration, it is allowed in this work to vary with the radial coordinate of the multistep-core fibre. A novel multi-layer model is presented for solving the diffusion equation with the variable diffusion coefficient. It is solved by the finite difference method. The solution determines the dopant diffusion profile in the fibre. It is verified against a solution from the literature and two cases of fibres with diffused profiles. The application is demonstrated on two examples of graded-index plastic optical fibres, one originally with a two-step and the other with four-step core. The results indicate that closer to the core-cladding interface, the computed diffused profile with variable diffusion coefficient D is closer to target profile than the profile obtained with constant D for the same time of thermal process.     

Guided modes in fibres with parabolic-index core and homogeneous cladding

Mode propagation characteristics of parabolic-index fibres with a homogeneous cladding are studied by using vector wave analysis. By investigating the asymptotic behaviour of the eigenvalue equations and the field components at cut-off and far from cut-off, a comprehensive and consistent scheme for the designation of the various modes is established. Strong similarities with step index fibres are found for both the mode designation schemes and the degeneracy properties among the hybrid modes as well as the TE_0m and TM_0m modes. This suggests that the technique employed in this paper can be applied to other types of graded-index fibres. In addition, numerical results obtained by solving the eigenvalue equations for a wide range of parameters are presented, which verify the theoretical predictions. Comparisons with the infinite-core approximation (ICA) and the field decomposition technique (FDT) are given. It is shown that neither the ICA nor FDT are applicable for modes near cut-off.Work supported in part by the Department of Communications, Canada.     

Perturbation analysis of finitely clad optical waveguides and couplers

In calculating the modal characteristics of a finitely clad optical waveguide, if the wavelength is much shorter than the core of the waveguide, then the cladding of the waveguide can be assumed to be infinite. On the other hand, if the wavelength is much longer than the core, the core can be ignored. These two approximations, although simple and accurate in the two respective extreme cases, do not provide adequate information in the transition region, where many tapered devices are operated. To overcome this a novel perturbation method is devised, in which these two approximations are combined in such a way that a result that is accurate in all cases can be obtained. Analytical expressions based on this method for describing the dispersion characteristics of finitely clad slab waveguides and circular step-index fibres are derived and compared with other conventional approximations. The combined use of this method and the coupled-mode theory for the analysis of optical couplers is also described.     

The effect of cladding loss in graded-index fibres

We present quantitative results for the effect of cladding absorption on the transmitted power and impulse response of a typical multimode, graded-index fibre that is excited by a Lambertian source. Power attenuation is due to the absorbing cladding material and the radiation loss of the tunnelling rays. Graded-index fibres are shown to be far less susceptible to the effect of cladding absorption than step-index fibres.     

Evaluation of the SSC/LHNC,SSCF and PY approximations for short ranged,anisotropic potentials

Three approximations for the orientational correlation function of molecular fluids—the single super chain/linearized hypernetted chain (SSC/LHNC), single super chain f-expansion (SSCF) and Percus-Yevick (PY) approximation—are evaluated in the dense liquid state for a fluid inter-acting with short ranged anisotropic interactions. These interactions are prototypical of the forces that determine the non-spherical shape of symmetric linear molecules. We find that SSC/LHNC is very poor, failing to predict many of the structural features present in the Monte Carlo (MC) simulation results. The PY and SSCF approximations produce much better results; however, neither approximation is completely satisfactory.     

Time dispersion effects of optical fibre joints

The time dispersion effects of a joint between two fibres are investigated. A general closed form expression is derived for the r.m.s. pulse width of a pair of different graded-index fibres separated by a joint. Numerical computations are reported for some cases of practical interest.     

Perturbation theory of optical fibres with power-law core profile

The characteristics of graded-index optical fibres with power-law core profile and homogeneous cladding have been studied using perturbation theory. Analytical expressions are derived for the cut-off frequency, propagation constant and, in the case of leaky modes, the attenuation coefficient of an arbitrary mode on such a fibre. For the case of monomode fibres, excellent agreement is found with existing theoretical results for the propagation constant of the HE₁₁ mode and cut-off frequency of the TE₀₁ mode.On leave from Department of Physics, IIT, New Delhi, India.     

A universal tunnelling coefficient for step- and graded-index multimode fibres

A single expression is presented for the power transmission coefficient of both step- and graded-index fibres, which is uniformly valid for all weakly tunnelling and refracting rays. Its accuracy is demonstrated in a comparison of the corresponding power attenuation coefficient for the step-index fibre with solutions to the electromagnetic boundary value problem.     

A variational analysis of single mode graded-index fibers

A variational method is presented to obtain the transverse electric field and the propagation constant of the fundamental mode of weakly guiding graded-index fibers. A comparison has also been made with results obtained by using other approximate methods and it has been shown that the present method gives extremely accurate results (in comparison to other methods) in the practical region of single mode operation and does not involve much computational effort.     

Bandwidths of single-mode and multimode optical fibre

An analysis is given of the effect of carrier spectral width on bandwidth of optical fibre transmission lines and a simple generalized expression is deduced relating carrier bandspread and maximum permissible pulse rate. It is shown that for the case of a semiconductor laser carrier source the bandwidth of a multimode fibre can be made to approach that expected from single-mode and graded-index fibres.     

Towards efficient methods for the study of pattern formation in ferrofluid films

Hexagonal and labyrinthine patterns appear in thin ferrofluid films after application of a magnetic field perpendicular to the film. The pattern size and the stability of the hexagonal and labyrinthine structures can be predicted by free energy approaches. Several approximations are used in the literature to accelerate the calculation of the magnetic energy. They are usually based on the use of a uniform, average or constant magnetization. In the uniform approximation the magnetization at all points in the pattern is assumed to be equal to its value at the center of the stripes or cylinders in the labyrinthine or hexagonal patterns. Recent papers indicate that this approximation gives qualitatively wrong results. This is corroborated here by a comparison with accurate results. When a volume-averaged magnetization is used during the calculation of the demagnetization field, from which the magnetic energy is evaluated, the theoretical results are only slightly modified with respect to the accurate results. Thus, we can propose a new method which gives results in good agreement with the accurate values and accelerates the calculations by a factor of 1000. The influence of the approximations is explained by a study of the evolution of the demagnetization field in the patterns. This study indicates that the volume-averaged approximation might only be reliable for patterns with a homogeneously distributed magnetic fluid. Another approximation of a constant magnetization, which is widely used in the literature, assumes that the magnetization does not change during the pattern formation in contrast to the uniform and average approximations. A different way of computing the constant magnetization than that usually employed markedly improves the agreement with the accurate results. This is explained by the derivation of a direct relationship between the approximations of a constant and an average magnetization.Received: 28 October 2003, Published online: 2 March 2004PACS: 47.54. + r Pattern selection; pattern formation - 47.65. + a Magnetohydrodynamics and electrohydrodynamics - 77.84.Nh Liquids, emulsions, and suspensions; liquid crystals     

Evaluation of thesscf andpy approximations for quadrupolar fluids

Two approximations, the single super chainf-expansion (sscf), and Percus-Yevick (py) approximation, are evaluated for a molecular fluid in which the molecules interact with a pair potential, that is the sum of Lennard-Jones and quadrupole-quadrupole parts at two values of reduced quadrupole moment. These results are compared with Monte-Carlo results. Except for the harmonic coefficienth (222;r), thesscf approximation seems to be quite accurate for the lower value of quadrupole moment but at higher valuespy approximation produces much better results except forh(220;r).     

Bend loss calculation in single-mode graded-index fibers using variational fields

Calculation of bend loss for single-mode graded-index fibers utilizes the fundamental modal field. Using some of the single and two parameter scalar variational approximations available in the literature for this mode, the bend loss has been computed. The exact results are obtained using a well known numerical method. The accuracies of the results obtained from the use of these variational fields have been compared.     

Methods to determine surface energies of natural fibres: a review

To tailor the interaction across composite interfaces especially for the development of green composites, i.e. composites made completely from renewable materials, information about the fibre surfaces is required. We review the current state of the art of methods to determine the surface tension of natural fibres and discuss the advantages and disadvantages of techniques used. Although numerous techniques have been employed to characterise surface tension of natural fibres, it seems that commonly used wetting techniques are very much more affected by the non-ideal character of natural fibres. Inverse Gas Chromatography (IGC) is a much better suited technique to determine the surface energetic properties of natural fibres than wetting techniques. The surface tension of natural reinforcements, determined using IGC, was reported for nanosized bacterial cellulose as well as bamboo, cornhusk, flax, hemp and sisal, covering a wide range of cellulose content. The effect of methods to separate/extract fibres from the plants as well as of a few surface modification procedures on the fibre surface properties is also reviewed. The dispersive part of the natural fibre surface tension γ ^d _S varies from 32 to 61 mJ/m². The fibre surface tension increases with increasing cellulose content of natural fibres. We also found that a higher basicity (Donor Number, K _B to Acceptor Number, K _Aratio) was observed for fibres containing more cellulose. This may be reflective of higher crystalline cellulose content in the surfaces of the fibres, as only the ether linkage of the cellulose is labile for hydrogen bonding.     

Transmission properties of all-silica fibres for high-power Nd:YAG lasers

The transmission properties of two different types of all-silica fibres, step index and graded-index fibres, are investigated using a multimode high-power cw Nd:YAG rod laser. The reflections from the fibre surface back into the laser cavity affect the laser power and the beam parameters, waist diameter and divergence. A set-up for simultaneously measuring these values in front of and behind a fibre is presented. The maximum laser power transmitted by fibres is dependent upon the beam parameters, the fibre design and the surface conditions. An upper limit for the maximum divergence which can be coupled into a fibre with a certain numerical aperture without loss is given. The beam parameters behind the fibre, related to the input parameters, are presented.     

Intermodal dispersion measurements and interpretation in graded-index optical fibres

We describe in this paper a frequency-domain measurement configuration allowing the determination of intermodal dispersion, differential mode attenuation and mode-coupling effects in graded-index optical fibres. Attention is mainly given to intermodal dispersion which is obtained with a resolution of 5 ps km^–1 and a dynamic range of 15dB. Spatial (or modal) resolution is examined from both the theoretical and experimental point of view and the high degree of resolution of the experiment is illustrated. Interpretation of the results for deducing the index profile errors and correcting the preform manufacturing process shows the usefulness of the method for improving fibre quality.