Calculation of nonlinear modes guided by step-index fibers with finite cladding thickness

A circular optical fiber consisting of a nonlinear core bounded by a nonlinear cladding with a finite thickness is considered here. In this paper, the influence of the cladding radius in comparison with the core radius in the Power – effective index relations P(/k ₀) is examined. Numerical results indicate that the radius ratio between the cladding and the core is an important factor in changing the Power – effective index relations P(/k ₀) when the cladding has a stronger nonlinearity than that of the core.     

Curvature effects on leaky modes in weakly guiding optical fibres

The effect of curvature on leaky modes with low losses in weakly guiding optical fibres is investigated, within a theoretical model where the cladding (outer cladding in case of multiclad fibres) is extended to infinity. Such modes already radiate rays in the cladding when the fibre is straight and two cases can be considered according to whether this ray pattern is modified near the core area in the bent fibre (strong curvature) or is not modified (weak curvature). A general loss formula is given for the case of weak curvature. The rays which are emitted in the curvature plane, towards the inner side of the bend are captured later on by the core and induce an oscillating modulation of the leakage loss, with an amplitude varying asR ^–1/2 but limited to a few per cent in practical cases (for weak curvatures). The case of strong curvatures is analysed in the special case of the LP 01 mode in fibres with a large depressed inner cladding, through a 2-D ray method. It is found that the loss oscillations are rapidly damped while the average loss is considerably increased.     

Pulse broadening in near-optimum graded-index fibres

Impulse responses of a wide class of multimode optical fibres with near-optimum index profiles are investigated. The index profile is described by three parameters: the power law exponent, the magnitude and the width of a dip at the centre of the core. It is found that for overcompensated profiles (< _opt) a moderate dip decreases the r.m.s. dispersion. Leaky modes are found to improve the dispersion characteristics for fibres with a dip in the index profile. Impulse response measurements are interpretated to give information about the index profile.     

Calculation of cutoff frequency of arbitrary weak guidance optical waveguides by extended boundary condition method

The extended boundary condition method is applied to both circular and elliptical regions for calculating cutoff frequencies V_c of weakly guiding optical waveguides, which consist of a uniform cladding and a core with arbitrary shape and index profile. The frequency shift formula method is explained by means of the extended boundary condition method on circular regions. Comparing the method with analytic and other rigorous methods, we show that it is indeed a powerful tool. Numerical results confirm that the mode designation and arrangement order in elliptical and rectangular waveguides are, in general, dependent on the aspect ratio.     

Guided propagation in a step-index, multi-mode fiber: effect of index difference variation on allowable TM propagation constants

The effect of variation of core and cladding index difference, Δn=n_core−n_cladding, on allowable values for the guided mode transverse magnetic propagation constants within a step-index, multi-mode optical fiber is investigated. We use a iterative computational technique to calculate the propagation constants for modes inside and outside the core that satisfy the boundary conditions contained within the characteristic eigen-equation for the TM field components. Evidence of a strong dependence of the allowable propagation constants on changes of Δn is shown.     

Influence of temperature on the optical and structural properties along the diameter of optical fibres

Multiple-beam Fizeau fringes technique with an opto-thermal device is used to study the effect of temperature on the optical properties of Philips graded-index optical fibres. The refractive index profile of the optical fibre is measured at different temperatures. From these profiles the opto-thermal coefficient, the profile shape parameter α, the cladding/core maximum refractive index difference Δn and some guidance parameters of the optical fibre that play an important role in communication are determined. The variation of oscillation and dispersion energies along the diameter of the optical fibre (energy profile) are calculated at different temperatures. An empirical formula of the energy profile is obtained. Microinterferograms are given for illustrations.     

Propagation of electromagnetic surface waves in a radially inhomogeneous optical waveguide

The propagation of electromagnetic surface waves along a radially inhomogeneous dielectric waveguide is investigated. The problem is formulated in terms of differential equations to be satisfied by the radially dependent parts of the electromagnetic field vectors. The dielectric waveguide is assumed to consist of a homogeneous cladding of infinite extent and a radially inhomogeneous core of higher permittivity. Numerical solutions of the differential equations in the core are obtained by two different methods, viz. by direct numerical integration and by substitution of an appropriate power series expansion. In the cladding the field is expressed in terms of modified Bessel functions. Imposing the boundary conditions at the interface of core and cladding, an equation for the unknown propagation coefficient is obtained. From this equation the propagation coefficients for the lower order modes are computed numerically. Numerical results are presented for some permittivity profiles of practical interest in single-mode transmission along optical fibres.Nomenclature a radius of the core - a _q vector coefficient in the power series expansion off() - A _i constants - A square matrix - b _q coefficient in the power series expansion of _r() - B square matrix - C square matrix - c _n unknown constant - d _n unknown constant - D() fundamental matrix - E,E _r,,z electric field vector and components - E,e _r,,z radially dependent parts ofe _r,,z - f solution vector - G _q square matrix - H,H _r,,z magnetic field vector and components - h _r,,z radially dependent parts ofE _r,,z - h reduced wavenumber - i radial mode number - j imaginary unit - k _0,m wave number - K _n modified Bessel function of the second kind and order n - n azimuthal mode number - t time - U normalized propagation constant - Z _m plane wave impedance of the cladding - r, , z cylindrical co-ordinates - p, q, s integers - propagation coefficient - increment - _0,m,r permittivity - normalized radiusr - ₀ wavelength in free space - ₀ permeability - angular frequency - d _r, differentiation with respect tor, Engineering and Professor H. J. Frankena of the Physics Department for their valuable discussions.     

Polarization properties of elliptical core non-hexagonal symmetry polymer photonic crystal fibre

In this paper, polarization properties and propagation characteristics of polymer photonic crystal fibres with elliptical core and non-hexagonal symmetry structure are investigated by using the full vectorial plane wave method. The results show that the birefringence of the fibre is induced by asymmetries of both the cladding and the core. Moreover, by adjusting the non-symmetrical ratio factor of cladding $\eta$ from 0.4 to 1 in step 0.1, we find the optimized design parameters of the fibre with high birefringence and limited polarization mode dispersion, operating in a single mode regime at an appropriate wavelength range. The range of wavelength approaches the visible and near-infrared which is consistent with the communication windows of polymer optical fibres.     

Practical application of the refracted near-field technique for the measurement of optical fibre refractive index profiles

Both the theoretical basis and experimental realization of the refracted near-field technique for the direct measurement of optical fibre profiles are presented. The technique requires minimal sample preparation, no computation and is applicable to both single and multimode fibres. Both the core and the cladding are profiled. After outlining the problems associated with other techniques, the use of this method for the measurement of fibre profile, numerical aperture and geometry is discussed. Leaky mode rejection and resolution are treated in detail. A fitting procedure for determining the-value of a profile is given. The experimental apparatus is fully discussed. Results are presented to illustrate both the applicability of the technique to single and multimode fibres and also the rejection of leaky modes. The experimental sensitivity is shown sufficient to reveal an index fluctuation having a wavelength < 1m and an amplitude of < 0.0001.     

Analysis of backscattered light from step-index optical fibers: Refractive index and radius ratio determination

A method has been suggested for the determination of the relative size of the core and cladding,c=a/b, and of the refractive indices of the core and cladding,n ₁ andn ₂, of step-index circular fibers by using the analysis of backscattered light. The method is suitable for fibers withc (1–1·5). Comparison with the methods used so far shows that the method extends the range of magnitude ofc for large-core fibers and raises the precision of determination ofc, n ₁ andn ₂.Paper presented at the 16th Microsymposium IUPAC, Prague July 12–16, 1976.I am thankful to Dr. B. Sedlacek for valuable comments and stimulating discussions.     

Kramers-Krönig relations in nonlinear optics

We review dispersion relations, which relate the real part of the optical susceptibility (refraction) to the imaginary part (absorption). We derive and discuss these relations as applied to nonlinear optical systems. It is shown that in the nonlinear case, for self-action effects the correct form for such dispersion relations is nondegenerate, i.e. it is necessary to use multiple frequency arguments. Nonlinear dispersion relations have been shown to be very useful as they usually only require integration over a limited frequency range (corresponding to frequencies at which the absorption changes), unlike the conventional linear Kramers-Krönig relation which requires integration over all absorbing frequencies. Furthermore, calculation of refractive index changes using dispersion relations is easier than a direct calculation of the susceptibility, as transition rates (which give absorption coefficients) are, in general, far easier to calculate than the expectation value of the optical polarization. Both resonant (generation of some excitation that is long lived compared with an optical period) and nonresonant instantaneous optical nonlinearities are discussed, and it is shown that the nonlinear dispersion relation has a common form and can be understood in terms of the linear Kramers-Krönig relation applied to a new system consisting of the material plus some perturbation. We present several examples of the form of this external perturbation, which can be viewed as the pump in a pump-probe experiment. We discuss the two-level saturated atom model and bandfilling in semiconductors among others for the resonant case. For the nonresonant case some recent work is included where the electronic nonlinear refractive coefficient,n ₂, is determined from the nonlinear absorption processes of two-photon absorption, Raman transitions and the a.c. Stark effect. We also review how the dispersion relations can be extended to give alternative forms for frequency summation which, for example, allows the real and imaginary parts of ⁽²⁾ to be related.     

Investigation of intermodal interference of LP01 and LP11 modes in the liquid-core optical fiber for temperature measurements

The intermodal interference of the LP₀₁ and LP₁₁ modes and determination of the equalization wavelength in the liquid-core optical fiber is presented. Theoretically was described the weakly guiding optical fiber with the constant core radius, where equalization wavelength is a function of the refractive indices of core and cladding. The dependence of equalization wavelength on refractive indices is employed for measurement of temperature. Temperature sensitivity using intermodal interference of modes LP₀₁ and LP₁₁ was documented in the liquid-core optical fiber consisted of fused silica as cladding and medicinal oil as a core. In the investigated temperature range the intermodal interference allows the temperature measurement with resolution of about 0.02 °C.     

Equal-phase-velocity propagation for the TE and TM modes of planar metal-clad optical waveguides

An asymmetrical four-layered metal-clad dielectric optical waveguide can support the TE_N and TM_N modes forN0 with the same phase-velocity if the core thickness,t ₃, and the buffer layer thickness,t ₂, are suitably chosen. Only metals that have a negative real part ₁ of their complex permittivity at optical frequencies are considered as the cladding material. It is shown that the application of a lossless approximation which set ₁=0 enables the development of a real expression which may be solved directly for the values oft ₂ andt ₃ that are required to achieve the equal-phase-velocity characteristic. Parametric plots for the guide dimensions are presented for representative waveguides and it is shown that an appropriate choice oft ₂ andt ₃ will reduce the absorption losses of the TE and TM modes to negligible levels. This property of the waveguide will find application in thin film optical devices such as polarization rotation modulators that require the simultaneous propagation of TE and TM polarized waves.     

Group velocity of modes and pulse distortion in dielectric optical waveguides

After a brief summary of the main factors causing pulse distortion in optical fibres, expressions are derived for phase velocity, group velocity and dispersion, that is velocity variation with frequency, of each mode. The analysis is carried out by making extensive use of the functiony(x) (y andx being the arguments of the first and second kind Bessel functions that appear in the solutions of Maxwell equations), that we call characteristic function. By using this function we obtain particularly simple and compact formulas that make easier the analysis of the behaviour of velocity and dispersion near and far from cut-off. It will also be seen that the ratio between the power propagating in the core,W ₁, and the power in the cladding,W ₂, for each mode takes a very simple form if expressed by means of the characteristic function: this fact permits a direct relation to be found between group velocity and power flow in an optical fibre. The study of the ratioW ₁/W ₂ will show that, contrary to expectation, for some modes, a substantial fraction of energy travels inside the core, even near cut-off. Expressions are given forW ₁/W ₂ at cut-off, and far from cut-off. Plots of velocity and dispersion are shown. On the grounds of the obtained results pulse distortion in optical fibres is discussed.     

Unified and generalized Fresnel numbers

The concept of Fresnel number is discussed and expressions are derived for misaligned optical systems. For the case of perfectly aligned optical elements, the usual number,N, based on a Fresnel zone concept is found to be given byN = (a ²/)(D ₁/B ₁ +A ₂/B ₂), whereB ₁,D ₁ andB ₂,A ₂ are the transfer matrix elements of the optical systems before and after a circular aperture of radiusa respectively. A modified definition of the Fresnel number is proposed for Gaussian beam propagation. This parameterN _G, is related to the complex beam parameter and may be represented by the angle = tan^–1 N _G, found in the familiar Collins chart and its dual representation. A general relation for the transformation of this Fresnel number is found. The expressions for Gaussian beam transformation are thus simplified, since the waist-waist transform is given byN _G1 =N _G2 = 0. Finally, these two kinds of Fresnel numbers are written as tensors when applied to cases involving elliptical apertures, astigmatic beams and nonsymmetrical systems.     

Study of optical fibre structures by atomic force microscopy

A novel method using atomic force microscopy (AFM) to study optical fibre structures at the fibre end-face has been successfully developed. The doping concentration profiles of fibres revealed by differential etching speeds in a saturated solution of ammonium bifluoride at room temperature (25°C) were obtained from AFM topographic images. The superior spatial resolution of AFM made it possible to resolve concentric structures a hundred times smaller than the feature, due to the difference in the known refractive index (n) of 1×10^-3. Fibres with small core diameters and anisotropic structures, such as polarization-maintaining fibres, were studied with ease.     

Dual-mode optical fibres with zero intermodal dispersion

An exact analytical approach to the synthesis of the refractive-index profile of dual-mode optical fibres with zero intermodal dispersion is presented, and the computer-aided, trial-and-error synthesis of the optimal profile is described. The synthesis is based on inverse scattering theory. Self-consistency of free potentials corresponding to individual modes was attained by numerical methods. The characteristics of the optical fibres with synthetized profiles were analysed, the optimization constraints were specified, and an optimum profile was found. The results can be used for the design of broadband optical fibres.Notation (r), (r) Single and double derivatives - C _l(r) Determinant of a matrix det - k _l (r) Vector - k _l (r) _l (r) The scalar products of the vectors - K _l(wr) The modified Bessel function of the second kind - _l(wr) The modified Bessel function of the first kind - J _l(uaR) The Bessel function of the first kind     

Theoretical analysis of acoustic-optic frequency shifter by dual mode fiber

The frequency shifting between two modes in an optical fiber resulting from acoustic wave propagation has been studied theoretically. Our theoretical results confirm the earlier experimental work. The polarization effect introduced by acoustic waves has been examined as well. The analysis shows that the polarization characteristic and symmetry property of the modes are not preserved in the coupling process of acoustic-optic frequency shifting for arbitrary ratio of the fiber cladding radius to the acoustic wavelengthd/ _a, in contrast to stable mechanical microbending, and that for the givend/ _a = 0.397, the polarized light signal at frequency ₂, shifted from that at frequency ₁, is quasi-linearly polarized for any acoustic-optic interaction length. However, the polarization effect can be neglected when the fiber cladding radius is much smaller than the acoustic wavelength.     

Extension de la representation scalaire du champ aux fibres optiques à gaine finie

Résumé Le concept de mode LP introduit dans la théorie du guidage faible est étendu aux fibres optiques à gaine finie, pour les modes de gaine dont l'indice effectif reste voisin de l'indice de la gaine. Une étude plus précise montre ensuite comment les modes LP_mn dégénérés se scindent en modes HE_{m+1, n} et EH_{m–1, n} non dégénérés. On indique comment écrire la condition aux limites sur l'interface fibre/air pour la fonction scalaire qui décrit le champ; à l'ordre 0 (modes LP), il suffit d'annuler cette fonction.

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The LP mode concept, which was introduced in the weak guiding theory is extended to optical fibers with finite cladding, for cladding modes the effective index of which remains close to the cladding refractive index. A more accurate study shows how the degenerate LP_mn modes split into nondegenerate HE_{m+1, n} and EH_{m–1, n} modes. The boundary condition to be satisfied by the scalar field on the air/fibre interface is derived; for zeroth order (LP modes), this scalar field must be zero on this interface.