Parabolic Index Fibers Research Articles

Multimode parabolic index fiber integrated with spatiotemporal vertical cavity surface emitting laser sources for optical fiber system improvement

Abstract This study has demonstrated the multimode parabolic index fibers integrated with spatiotemporal VCSEL sources for the optical fiber system improvement. Average radial intensity is clarified against fiber radius variations with 0.1 % refractive difference index step. The Max base band Q factor form is demonstrated against the fiber refractive index step variations. The Max base band Q factor form variations are demonstrated against VCSEL diode bias current variations. The Max base band Q factor form variations are clarified against the fiber core radius variations. The Max base band Q factor form is simulated against the fiber cladding thickness variations. Multimode fiber calculation report at the wavelength 1550 nm for the light source (Pol. X = VCSEL LP [0,1]) is reported. The Max lighted base optical band form power is clarified versus time after parabolic index multimode fiber with 0.1 % refractive difference index step. The Max optical lighted base band power form is demonstrated against wavelength after parabolic index multimode fiber with 0.1 % refractive difference index step. The Max base Q band form factor is simulated after spatial APD photodetector receiver with 0.1 % refractive difference index step. The fiber base band mode form intensity in x and d directions is demonstrated within the fiber with 0.1 % refractive difference index step. The fiber base band encircled flux form is studied and simulated versus fiber radius variations with 0.1 % refractive difference index step.

Set of mathematical models for Bessel-Gauss beams coupling into the parabolic-index fiber under the influence of atmospheric turbulence and random jitter.

The expression of efficiency for Bessel-Gauss (BG) beams coupling into the parabolic fibers (PF) after passing through the Cassegrain antenna system is first derived. The effects of atmospheric turbulence and random jitter of the coupling lens on the efficiency are also taken into account to improve the practical applicability of our model. This article use a BG beam with a wavelength of 1550 mm and fiber with a core radius RF of 50 μm and a relative refractive index difference ζ of 0.01 for simulation testing. The optimal parameters of the antenna system are determined: the radius of the primary mirror and the secondary mirror is 8.33 cm and 1.25 cm, respectively. The coupling efficiency of BG beams of different orders reaches above 94% simultaneously when the lens's focal length is 7.8 cm. After taking into account the transmission efficiency of the antenna system, the system's total efficiency for BG beams of different orders averages 76.33%, when the transmission distance is 1 km. The results show that the same degree of turbulence and random jitter have different influences on the coupling efficiency of BG beams of different orders, and lower-order BG beams have better resistance to turbulence and jitter during propagation and coupling. Moreover, the effect of the guided mode field on the coupling efficiency and the resistance to turbulence varies for different values of mode radial index in the fiber p. The guided mode with p = 0 not only enables the BG beams of different orders to achieve the highest transmission efficiency in the coupling system almost simultaneously but also the random jitter and turbulence have less influence on the coupling efficiency of this mode. It means that the BG beams can have higher efficiency when coupled to the mode with p = 0 after long-distance transmission. This property of the fiber mode at p = 0 provides conditions for the simultaneous propagation of multiple BG beams in a parabolic fiber, which provides a theoretical basis for higher transmission capacity. This research work provides a theoretical model for the theoretical study of vortex beams and optical communication, which is beneficial for the design and application of vortex beams and has instructive meaning for practical engineering design.

A simple but accurate technique for prediction of confinement and normalized group delay parameters for propagation of first higher order mode in graded index fiber

Abstract This paper deals with expressions for simple but precise analyses of fractional modal power (FMP) inside the core of optical fibers, the excitation efficiency and the normalized group delay (NGD) for the first higher order (LP11) mode in step and parabolic index fibers both with and without Kerr type non linearity. To get the analytical results, we have employed simple power series Chebyshev expansion for the LP11 mode of the above mentioned fibers. At first, the analytical expression for linear case is found out and then by applying the method of iteration the propagation parameters are estimated when there is nonlinearity of the Kerr type. Here, some typical step and parabolic profile fibers have been used for our investigation. Our findings of confinement and group delay parameters perfectly match with the precise numerical findings made by the intricate finite element method (FEM). This implies the precision of our formalism. The study of nonlinear optical transmission systems will benefit from the findings.

Approximation of mode profile dispersion in the generalized nonlinear Schrödinger equation superior to the constant overlap approach for weakly guiding fibers

An efficient numerical approach to nonlinear propagation in waveguides based on an approximation of the overlap integrals’ frequency dependence is proposed. Two test cases are studied: a weakly guiding step-index fiber and a weakly guiding parabolic-index fiber, where the approximation is compared to the standard approach of neglecting the frequency dependence and calculating the overlaps at a center wavelength λ 0 , and benchmarked against a Gaussian quadrature approach, whose accuracy can be increased systematically. We believe that the combined accuracy and efficiency of the proposed approximation make it superior to the other two approaches in few-mode simulations.

Generalization of ray tracing in symmetric gradient-index media by Fermat's ray invariants.

Ray tracing in gradient-index (GRIN) media has been traditionally performed either by using the analytical or numerical solutions to the Eikonal equation or by creating a layered medium where Snell's law is calculated in each layer. In this paper, an exact general method to perform ray tracing in GRIN media is presented based on the invariants of the system as stated by Fermat's principle when the media presents symmetries. Its advantage, compared with other methods reported in the literature, relies on its easy implementation. Besides the GRIN distribution and the initial conditions of the incident ray, once the invariants of the system are stated the resulting math is simple to solve and interpret. To benchmark the algorithm, ray tracing in typical cases of GRIN media is calculated, finding minimal discrepancies between the analytical solutions and our simulations. The used media are axial refractive index and parabolic index fiber and lenses with spherical gradient-index symmetry, such as: Luneburg's, Gutman's, generalized Maxwell's Fish-eye, Eaton's, and concentrator lenses. Our method can be further applied to distributions with symmetries associated with other common curvilinear orthogonal coordinate systems, in particular to those associated to the separability of the Helmholtz equation that would allow us to investigate wave optics in these GRIN media with the associated geometries.

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.

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.

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.

A simple but accurate formalism for study of single-mode graded index fiber directional coupler in presence of Kerr nonlinearity

This paper presents performance of directional coupler consisting of two identical single-mode graded index fibers in presence as well as in absence of Kerr type nonlinearity. In this context, parabolic and step index fibers of some typical V numbers are used. Here, we employ a simple series expression for fundamental modal field for such fibers prescribed by Chebyshev formalism. Analytical expression of the coupling length of the directional coupler is formulated. Here, we apply method of iteration in order to take care of nonlinearity. The concerned evaluation involves little computation. Thus, the formalism developed for study of nonlinear directional coupler can be treated as an accurate but simple alternative to the existing complicated methods available in literature. This user friendly method will prove beneficial to the scientists and technologists working in the field of nonlinear optics.

Prediction of First Higher Order Modal Field for Graded Index Fiber in Presence of Kerr Nonlinearity

AbstractWe report evaluation of first higher order modal field for dual mode optical fiber having step and parabolic index profiles. The study is carried out both in absence as well as in presence of Kerr nonlinearity. The analysis is based on a simple iterative method involving Chebyshev formalism. Taking some typical step- and parabolic-index fibers as examples, we show that our results agree excellently with the exact results which can be obtained by applying rigorous methods. Thus, our simple formalism stands the merit of being considered as an accurate alternative to the existing cumbersome methods. The prescribed formalism provides scope for accurate estimation of different propagation parameters associated with first higher order mode in such kinds of fibers in presence of Kerr nonlinearity. The execution of formalism being user friendly, it will be beneficial to the system engineers working in the field of optical technology.

A simple method for study of effect of Kerr nonlinearity on effective core area, index of refraction and fractional modal power through the core of monomode graded index fibre

Using the simple power series expression for fundamental modal field derived by Chebyshev technique, we present investigation of some important propagation parameters like effective core area, index of refraction and fractional modal power guided through the core of single-mode graded index fibre in presence of Kerr nonlinearity. In absence of nonlinearity, the said power series expression leads to prescription of analytical expressions of the concerned propagation parameters. Employing those analytical expressions, we apply iterative method in order to evaluate the said parameters in presence of Kerr nonlinearity. Choosing some typical single-mode step and parabolic index fibres for our study, we verify that the results obtained by our simple formalism match excellently with the exact results which are obtainable by applying rigorous finite element technique. This excellent agreement attests to the accuracy of our formalism. Further, our formalism requires little computation in the context of evaluation of the said parameters. Accordingly, our formalism can be considered as a simple but accurate alternative to the existing complicated methods available in literature. Thus, this user-friendly but accurate formalism will benefit the system engineers in respect of selection of suitable fibre in which modal noise due to nonlinearity is minimum.

Evaluation of Splice Loss of Single-Mode Graded Index Fiber in Presence of Kerr Nonlinearity

We present here evaluation of splice loss of single-mode graded index fiber in presence of Kerr type nonlinearity by a simple iterative method involving Chebyshev technique. We restrict our analysis to two different cases of splice losses involving angular and transverse offsets because splices are highly tolerant for longitudinal separation. Our analysis by the said power series for two typical graded index fibers namely step index and parabolic index fibers provide excellent results. The evaluation of concerned parameters by our technique requires very little computations. Thus our formalism will prove user friendly to system engineers. Accordingly, this formalism may prove useful in many applications in optical technology involving graded index fibers.

A Novel and Simple Formalism for Study of Effect of Kerr Nonlinearity on Petermann I and II Spot Sizes of Single-Mode-Graded Index Fiber

Abstract We present investigation of Petermann I and II spot sizes in the presence of Kerr nonlinearity. Our study is based on the simple power series formulation for fundamental modal field of single-mode-graded index fiber developed by Chebyshev formalism. Based on the said power series expression in the absence of nonlinearity, analytical expressions of the said spot sizes can be prescribed. Using the analytical expressions of the said spot sizes in the absence of nonlinearity, we apply iterative technique in order to predict the said propagation characteristics in presence of Kerr nonlinearity. In this context, we choose some typical single-mode step and parabolic index fibers. We show that the our results agree excellently with the exact results which can be obtained by using rigorous finite-element technique. This leads to verification of accuracy of our simple technique. Moreover, evaluation of the concerned parameters by our formalism involves little computation. Thus, our method provides an accurate but simple alternative to the existing rigorous methods in this context. Accordingly, this novel and simple formalism will prove user friendly to the system engineers in the field non linear optics.

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.

Robustness of Light-Transport Processes to Bending Deformations in Graded-Index Multimode Waveguides.

Light transport through a multimode optical waveguide undergoes changes when subjected to bending deformations. We show that optical waveguides with a perfectly parabolic refractive index profile are almost immune to bending, conserving the structure of propagation-invariant modes. Moreover, we show that changes to the transmission matrix of parabolic-index fibers due to bending can be expressed with only two free parameters, regardless of how complex a particular deformation is. We provide detailed analysis of experimentally measured transmission matrices of a commercially available graded-index fiber as well as a gradient-index rod lens featuring a very faithful parabolic refractive index profile. Although parabolic-index fibers with a sufficiently precise refractive index profile are not within our reach, we show that imaging performance with standard commercially available graded-index fibers is significantly less influenced by bending deformations than step-index types under the same conditions. Our work thus predicts that the availability of ultraprecise parabolic-index fibers will make endoscopic applications with flexible probes feasible and free from extremely elaborate computational challenges.

Prediction of Fundamental Modal Field for Graded Index Fiber in the Presence of Kerr Nonlinearity

Abstract The power series formulation for modal field of single-mode graded index fibers by Chebyshev technique has worked excellently in predicting accurately different propagation characteristics in simple fashion. Here we develop a simple iterative method involving Chebyshev formalism to predict the modal field of single-mode graded index fiber in the presence of Kerr-type nonlinearity. Taking step and parabolic index fibers as typical examples, we show that our results match excellently with the available exact results obtained vigorously. Thus, the reported technique can be considered as an accurate alternative to the existing cumbersome techniques. Accordingly, this formalism will be beneficial to the technologies for evaluation of modal noise in single-mode Kerr-type nonlinear graded index fibers.

Mismatch Considerations in Excitation of Single-Mode Circular Core Parabolic Index Fiber by Laser Diode via Upside Down Tapered Hemispherical Microlens on the Tip of the Fiber

AbstractWe report the theoretical investigation of the coupling optics involving laser diode to single-mode circular core parabolic index fiber via upside down tapered hemispherical microlens on the tip of the fiber in the presence of possible transverse and angular mismatches. Using the relevant

Laser Diode to Single-Mode Circular Core Parabolic Index Fiber Coupling via Upside-Down Tapered Hyperbolic Microlens on the Tip of the Fiber: Prediction of Coupling Optics by ABCD Matrix Formalism

Abstract We employ ABCD matrix formalism in order to investigate the coupling optics involving laser diode to single-mode circular core parabolic index fiber excitation via upside-down tapered hyperbolic microlens on the fiber tip. Analytic expressions for coupling efficiencies both in absence and in presence of transverse and angular mismatches are formulated. The concerned investigations are made for two practical wavelengths namely 1.3 µm and 1.5 µm. The execution of the prescribed formulations involves little computation. It has been found that the wavelength 1.5 µm is more efficient in respect of coupling. It is also seen that the present coupling device at both the wavelengths shows more tolerance with respect to angular mismatch. As regards tolerance with respect to transverse mismatch, the result is poor at both the wavelengths used. Consequently, it is desirable that designers should not to exceed transverse mismatch beyond 1 μm.

Tailoring frequency generation in uniform and concatenated multimode fibers.

We demonstrate that frequency generation in multimode parabolic-index fibers can be precisely engineered through appropriate fiber design. This is accomplished by exploiting the onset of a geometric parametric instability that arises from resonant spatiotemporal compression. By launching the output of an amplified Q-switched microchip laser delivering 400ps pulses at 1064nm, we observe a series of intense frequency sidebands that strongly depend on the fiber core size. The nonlinear frequency generation is analyzed in three fiber samples with 50μm, 60μm, and 80μm core diameters. We further demonstrate that by cascading fibers of different core sizes, a desired frequency band can be generated from the frequency lines parametrically produced in each section. The observed frequency shifts are in good agreement with analytical predictions and numerical simulations. Our results suggest that core scaling and fiber concatenation can provide a viable avenue in designing optical sources with tailored output frequencies.