Triangular Index Fibers Research Articles

The effect of misalignment on the coupling optics involving laser diode and single-mode triangular index fiber with an upside down tapered hyperbolic microlens on its tip

Abstract This article contains theoretical investigation of the performance of the coupler comprising laser diode and mono-mode circular core triangular index fiber (TIF), which contains on its tip an upside down tapered hyperbolic microlens (UDTHM). Here, we take care of probable angular and transverse mismatches. In our derivation, we use the refractive index distribution in the tapering area, the ray equation, and the relation involving refraction by hyperbolic microlens. We report analytical formulation of coupling efficiencies in the presence of specified types of misalignments using a relevant ABCD refraction matrix for a UDTHM in the vertical plane. In the fields of lasers and fibers, we use Gaussian field distribution in our analysis. Our research is based on two widely employed wavelengths, 1.3 µm and 1.5 µm. In this study, we used three TIF with V numbers of 4.380, 3.511, and 1.924, respectively. Little computation is required to execute the prescribed formulations. This study surfaces the sensitivity of the coupler in presence of the aforementioned types of offsets. The found results will prove valuable to manufacturers working in the domain of UDTHM.

The influence of Kerr nonlinear effect on the signal and pump intensities in parabolic and triangular index EDFAs

Erbium-doped fiber amplifiers (EDFAs) is an integral part of all-optical communication system. Intensities of pump and signal within EDFAs is one of the significant parameters in its application perspective. Nonlinear effect on pump and signal may affect the performance aspect of EDFAs. In this work, we study intensity profile of pump and signal at different wavelengths of interest in EDFAs for the first higher order (LP11) excited mode considering the nonlinear impact. In this comparative analysis, EDFAs made up of some parabolic-index and triangular-index fibers of several V-numbers have been chosen. Our study involves the power series expression of the LP11 field prescribed by Chebyshev’s technique. A short computation is needed for achieving the objective by this method. Outcomes of this exercise are in superb conformity with those derived through the cumbersome finite element method (FEM) proving reliability of the Chebyshev’s method. The analysis is important in the context of use of dual mode optical fiber in EDFAs. The impact of Kerr nonlinearity on the performance of EDFAs involving dual mode optical fiber is also investigated. This analytical exercise through a handy and reliable mathematical instrument may be a dependable option for photonic EDFAs designers in practical realization of nonlinear EDFAs.

Analysis of splitting ratio of a symmetric directional coupler fabricated with triangular-index fibers in nonlinear condition-a design perspective

We report here one of the significant performance parameters of a 2 × 2 symmetric directional coupler, the splitting ratio, in the nonlinear conditions considering the coupler fabricated with two similar mono-mode triangular index fibers. Different sets of physical aspects like V-number, core-radius of the fibers used, core-to-core distance of separation between the fibers in coupler structure, operating wavelength, index of refraction of the core have been considered for evaluating the splitting ratio at different lengths of fused fiber coupler. This analytical exercise is an effort for extending an indication to the optical device design engineers. This has been worked out involving the reliable mathematical instrument, the Chebyshev technique. The change of behavior of the splitting ratio in different linear and nonlinear conditions and with different physical parameters has been studied rigorously. Our theoretical analysis goes superbly with that derived using the complicated and lengthy finite element technique. This exercise needs modest computation. Hence, this report on splitting ratio analysis of 2 × 2 symmetric directional coupler composed of two mono-mode triangular- index fibers will prove its feasibility as a ready reference in the area of nonlinear photonic design engineering.

Effect of misalignment on coupling efficiency in laser diode to single-mode circular core graded-index fiber coupling via cylindrical microlens on the fiber tip

Abstract We report here the first theoretical prediction of coupling efficiency of the laser diode to single-mode graded-index fiber coupling via cylindrical microlens on the tip of the fiber in presence of possible transverse and angular mismatches. Moderately efficient cylindrical microlens on the fiber is easily fabricable and graded-index fiber has less sensitivity to macro as well as micro bending. Thus this coupling device emerges as a potential candidate in the field of coupling optics. The analysis takes into consideration the allowable aperture provided by cylindrical microlens. Further, we shall consider here the coupling in the vertical plane only since the coupling in the horizontal plane has poor performance. Using the relevant ABCD matrix of refraction for a cylindrical lens in the vertical plane, we prescribe analytical expressions of the coupling efficiencies in presence of said kinds of misalignments. In our analysis, we use Gaussian field distribution for both the laser and fiber fields. We carry on our investigation for two commonly used wavelengths namely 1.3 and 1.5 µm. As an example of graded-index fiber, we use some typical step, parabolic, and triangular index fibers in our study. The execution of the prescribed formulations involves little computation. Thus the evaluation of the concerned coupling efficiencies is very much user friendly. The present analysis surfaces the tolerance factors of the coupling devices. Accordingly, the results will prove beneficial to designers and packagers in respect of the design of such coupler involving cylindrical microlens in the field of optimum launch optics.

Analysis of Kerr type nonlinear single-mode triangular index fiber directional coupler by a simple method

We present the coupling characteristics of directional coupler comprising two identical single-mode triangular index fibers both in the presence and the absence of Kerr nonlinearity. Triangular index fibers of some typical V numbers are chosen for our investigation. Our analysis is based on the simple series expression for the fundamental modal field for triangular index fiber developed by Chebyshev technique. Analytical expression of the coupling length in the absence as well as in the presence of Kerr nonlinearity is prescribed. Using the said expression of coupling length, we apply a method of iteration to evaluate the said coupling parameter in the presence of Kerr nonlinearity. We also show that our results match excellently with the exact results obtainable by rigorous finite-element technique. The execution of our formalism requires little computation. Thus our simple but accurate formalism to study the triangular index directional coupler will prove a suitable alternative to the existing rigorous methods in the field of nonlinear optics. This user-friendly method will prove beneficial to the scientists and technologists working in the field of nonlinear optics.

Laser diode to single-mode graded index fiber coupling via cylindrical microlens on the fiber tip: evaluation of coupling efficiency by ABCD matrix formalism

Abstract We use ABCD matrix formalism for estimation of the coupling optics involving laser diode to cylindrical lensed circular core graded index fiber excitation. The investigation has been carried on for some typical step, parabolic, and triangular index fibers as examples of graded index fiber. It is relevant to mention in this connection that we have shown that the coupling optics is excellent along the vertical plane only while the coupling along the horizontal plane produces poor performance. Analytic expressions for coupling efficiencies are prescribed. The investigations have been made for two commonly used wavelengths, namely 1.3 and 1.5 µm. Our formulations require little computation for the necessary execution. The wavelength of 1.3 µm has been found to be more efficient in respect of coupling. Our formalism takes care of the limited aperture allowed by the cylindrical lens and thus provides the realistic prediction of coupling optics. The results will prove extremely useful to the designers and packagers working with cylindrical microlens.

Influence of Kerr nonlinearity on group delay and modal dispersion parameters of single-mode graded index fibers: evaluation by a simple but accurate method

Abstract This paper reports simple but accurate analytical expressions of group delay and modal dispersion parameters for single-mode graded index fibers over a wide range of V numbers. The formulation employs power series expression for the fundamental modal field of graded index fiber derived by Chebyshev formalism. Choosing some typical step, parabolic and triangular index fibers as examples in our present study, we use the prescribed formulations to estimate group delay and modal dispersion parameters of those fibers both in presence and absence of Kerr nonlinearity. Iterative technique is applied for prediction of concerned propagation parameters in presence of Kerr nonlinearity. Our results show excellent agreement with the numerical exact ones both in absence and presence of Kerr nonlinearity. The exact results in case of Kerr nonlinearity are obtained using cumbersome finite element method. The execution of our accurate formalism involves little computation and is thus user friendly for technologists and researchers working in the field of nonlinear optical engineering.

Laser diode to single-mode triangular index fiber coupling via upside down tapered hyperbolic microlens on the fiber tip: Prediction of coupling efficiency by ABCD matrix formalism

We prescribe ABCD matrix of transmission relating to the upside down tapered hyperbolic micro lens drawn from the end of single-mode circular core triangular index fiber. The said prescription takes care of the refractive index profile in the tapered region, the ray trace equation and also the refraction formula at a hyperbolic interface. The analysis will also be helpful for the design of micro-optic image systems. Using the prescribed ABCD matrix of transmission, we formulate analytical expression for the coupling efficiency and employ it for prediction of the coupling optics. Here, we assume Gaussian distributions of the fields for both the source and the fiber. The execution of our formulation involves little computation and as such it is user friendly. We investigate the launch optics for two commonly used wavelengths such as 1.3 μm and 1.5 μm. This simple but accurate formalism will immensely benefit the designers and packagers concerned with launch optics.

A Simple but Accurate Method for Estimation of the Effect of Kerr Nonlinearity on Confinement and Excitation of the Fundamental Mode in Single Mode Graded Index Fiber

This paper presents prediction of the fractional modal power and excitation of the fundamental mode inside the core of a single-mode graded index fiber, both in presence and absence of Kerr nonlinearity. Our analysis is based on simple power series expression for fundamental mode of graded index fiber, which has been formulated by applying Chebyshev technique. In absence of nonlinearity, analytical expressions for the concerned propagation parameters are prescribed. Employing these analytical expressions, method of iteration is applied for evaluation of the said parameters in presence of Kerr nonlinearity. In this context, some typical step, parabolic and triangular index fibers are used for our study. In absence of nonlinearity, our results show excellent match with the exact results available in literature. In presence of nonlinearity, the results found by our formalism also agree excellently with the exact results which can be obtained by application of rigorous finite element technique. Thus, this simple but accurate method will prove extremely useful for the system users.

Study of coupling optics involving graded index fiber excitation via upside down tapered parabolic microlens on the fiber tip

Based on the prescribed ABCD matrix for the transmission of paraxial rays, we analyze theoretically laser diode to single mode graded index fiber excitation via an upside-down tapered parabolic microlens on the fiber tip. In this context, we use some typical step, parabolic and triangular index fibers. The ABCD transformation matrices for upside down tapered parabolic microlens on the tips of step-index, triangular-index and parabolic-index fibers are employed in our analysis. In this formalism, we employ Gaussian field distributions for both the source and the fiber. In our study, we use here two different laser diodes emitting wavelengths 1.3 μm and 1.5 μm respectively. The execution of the prescribed formalism requires little computation. Such formalism is expected to generate tremendous interest for study of coupling optics related to such kind of coupler.

Mismatch considerations in laser diode to single-mode circular core triangular index fiber excitation via upside down tapered hemispherical microlens on the fiber tip

We present a theoretical investigation of coupling efficiency of a laser diode to a single-mode circular core triangular index fiber coupling via upside-down tapered hemispherical microlens on the tip of the fiber in presence of possible transverse and angular offsets. Using the relevant ABCD matrix formalism for such type of coupler, we prescribe analytical formulations of coupling efficiencies in presence of the said mismatches. Using the prescribed expressions, we evaluate the coupling efficiencies in presence of transverse as well as angular misalignments. The concerned execution involves little computation. The results found will benefit the designers and packagers working in the field of optimum launch optics.

Laser diode to single-mode triangular-index fiber excitation via upside down hemispherical microlens on the fiber tip: Prescription of ABCD matrix of transmission and estimation of coupling efficiency

We briefly introduce the upside-down taper lens drawn from the end of single-mode circular core triangular-index fiber and analyse it. Based on paraxial approximation, we formulate ABCD transformation matrix of such fiber end. The said prescription involves modelling of refractive index profile in tapered region, ray trace equation and also the ray refraction formula at a spherical surface. The analysis can be employed for design of micro-optic image systems and also for study of coupling optics involving such tapered lensing device. Employing the prescribed ABCD matrix of transmission, we formulate analytical expression for coupling efficiency and employ them to predict the concerned coupling optics. In this context, Gaussian field distributions for both the source and the fiber are assumed. In our present study, we use two practical wavelengths namely 1.3μm and 1.5μm. This simple but accurate method will prove beneficial to the designers and packagers working with the field of launch optics.

Aspect ratio based nonlinear effects in spot size dependent propagation characteristics of trapezoidal index single mode fiber

We investigate the nonlinear propagation characteristics of single mode trapezoidal index fibers involving the exact numerical linear and nonlinear spot sizes and their relative change with respect to aspect ratio. From comparison of normalized spot sizes in absence and presence of Kerr nonlinearity, it is seen that nonlinear effect is more pronounced in lower values of normalized frequency for each aspect ratio. Further, for near cut-off propagation, the triangular index fiber excels in performance in respect of percentage change of spot sizes and fractional core power.

A Simple but Accurate Method for Analytical Estimation of Splice Loss in Single-Mode Triangular Index Fibers for Different V Numbers Including the Low Ones

AbstractWe report formulation of simple but accurate analytical expressions at the splice for both angular and transverse mismatches in case of single mode triangular index fibers. Here, we employ the simple series expression for fundamental modal field of such fibers. The analysis takes care of large

Bending-loss studies of a single-mode triangular-index fiber with a depressed cladding ring with a vector-mode method

Attention is directed to the bending-loss properties of a triangular-index fiber with a depressed cladding ring, which can be advantageously used to obtain either a dispersion-shifted or a dispersion-flattened feature. Both cases are studied to search for the set of ring parameters that yield the lowest bending loss while ensuring a low dispersion. Near-optimal designs are proposed and compared with other well-known dispersion-shifted and dispersion-flattened fiber designs. The numerical methods used in our computations are the vector-mode power-series expansion method for fields and dispersion calculations and the volume-current method for bending-loss predictions.

Dispersion flattening in a W fiber

An exhaustive and simple method for dispersion flattening in double-clad single-mode fibers is presented. The idea is to generate all such fibers with a certain cutoff wavelength and then find, by one-dimensional minimization and direct inspection, fibers yielding minimum dispersion. As examples, the root mean square chromatic dispersion over the wavelength range (1.25, 1.60 µm) is minimized for a W fiber and also for a triangular-index fiber with a depressed inner cladding. The material dispersion is included through a Sellmeier refractive-index formula. The full vector solution of Maxwell's equations is used. It is found, as is well known, that the W fiber is capable of dispersion flattening. The validity of the numerical results presented depends on the validity of the approximate refractive-index model used.

Minimization of the chromatic dispersion over a broad wavelength range in a single-mode optical fiber

The effective refractive index as a function of vacuum wavelength is approximated by Lagrange interpolation polynomials. The rms value of the chromatic dispersion is then calculated analytically. Itis demonstrated that use of fourth-degree polynomials is far more efficient than the use of second-degree polynomials. The rms value of the chromatic dispersion over the wavelength range (1.25 µm, 1.60 µm) is calculated and minimized for step-index fibers, triangular index fibers, and α-power fibers. The full vector solution of Maxwell's equations is used. The error induced by the approximate refractive-indexmodel is found to be negligible at the point of minimum dispersion.

Dispersion studies of a single-mode triangular-index fiber with a trench by the vector mode analysis

A Runge-Kutta procedure was used for investigating the dispersion characteristics of single-mode triangular-index fibers with a trench in the cladding. Numerical results on the total dispersion, its sensitivities to the core radius, wavelength, index difference variations, and its influence on the fiber's bandwidth-length product are presented. Some interesting features, due to the presence of a trench, are pointed out. One salient feature is the appearance of dispersion flattening over a certain wavelength range near 1.55 microm for b/a = 2 and c/a > 2.7 (a = core radius, b = inner trench radius, c = outer trench radius) where the total dispersion is low, yielding a bandwidth-length product as high as 10(4) GHz . km. The presence of a trench has not been investigated previously.

Analysis of mode field radius calculated from single-mode fiber far-field radiation patterns

We have investigated the angular range and mesh size required to calculate accurate mode field radii (MFR) from single-mode fiber far-field radiation patterns using the inverse Hankel transform. A computer aided fiber modeling program was used to calculate near-field mode distributions for both step-index and triangular index fibers at 1.55 μm. These distributions were then transformed to the far-field and used as input "data" for the inverse transform studies. Experimental far-field data measured on fibers similar to those modeled were also used. We find that the MFR is insensitive to mesh sizes less than 0.6° but the range of far-field data required depends upon the fiber design and the criteria used to determine the MFR. This study provides guidelines for calculating accurate MFR from experimental data when using the Hankel transform.

Microdeformation losses of single-mode fibers

In this paper, we consider the scattering losses of single-mode fibers that are caused by microdeformations such as microbends of the fiber axis and random fluctuations of the fiber core diameter. Since very little is known about the statistics of microdeformations of actual fibers, we assume that the autocorrelation functions of random bends and random core diameter fluctuations are Gaussian, characterized by the rms deviation and the correlation length of the random function. We consider single-mode fibers with step, parabolic, and triangular (linear) refractive-index profiles and reach the following conclusions: (1) Whereas for equal (large) mode radii the microbending losses of all three fiber types are the same, losses due to random core diameter fluctuations can be three times as high in step-index fibers as in triangular-index fibers. Since triangular-index fibers have sometimes been observed to have lower scattering losses than step-index fibers, one might conclude that, in these cases, excess losses may be caused by random radius fluctuations rather than by microbends. (2) Radial refractive-index ripples, which tend to be present in the deposited claddings of single-mode fibers, seem unlikely to be a major source of microdeformation losses. (3) The wavelength dependence of microdeformation losses depends strongly on the value of the correlation length of the Gaussian autocorrelation function of the fiber deformations. If the correlation length is of the same order of magnitude as the fiber radius, the losses are only slightly wavelength dependent. For very long correlation lengths the losses are very much smaller (for the same rms variation of the random functions), but they become strongly wavelength dependent, increasing sharply with increasing wavelength.