Radio Science

A modal impedance-angle formalism: Schemes for accurate graded-index bent-slab calculations and optical fiber mode counting

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Abstract

[1] The propagation coefficients of guided modes along bent slabs and optical fibers may be calculated with the aid of a complex-power-flow, variational scheme. This scheme can be cast in the form of a Newton iterative scheme for a characteristic equation in terms of impedances rather than fields. Singularities of the associated Riccati equations for the impedances are circumvented via a generalized coordinate transformation, involving an angle for bent slabs, or a matrix of angles for optical fibers. Adopting the resulting impedance-angle formalism for bent slabs, numerical difficulties disappear, and accuracy ensues. The analysis of the vectorial optical-fiber problem benefits from its scalar bent-slab counterpart. In particular, the connection between the power flow and the impedance-angle formalism forms the physical underpinning for understanding that beyond a certain distance from the fiber core the derivative of the impedance matrix with respect to the propagation coefficient is positive definite. In turn, this provides the basis for the full-wave generalization for optical fibers of the mode-counting scheme developed in 1975 by Kuester and Chang for scalar wave propagation along a straight slab. With these key results, root-finding can be rendered more robust and efficient. A companion paper contains the necessary proofs for the complex-power-flow variational scheme and the mode-counting and mode-bracketing theorems.

 

1. Introduction

[2] To render problems in electromagnetics tractable, one may benefit from replacing Maxwell's equations by an equivalent system of field equations, e.g., the Marcuvitz-Schwinger equations in the case of waveguides. Criteria to assess the merits of such alternative systems may involve the accuracy, stability and robustness of the numerical schemes applicable to these systems. However, the physical interpretation should not be ignored, e.g., it seems prudent to base a numerical scheme on energy considerations.

[3] For midrange optical networks, multimode fibers are gaining ground owing to their cost-efficiency and ease of application. Regarding integrated optics, cross-talk and bending losses are key issues. For related optical waveguide problems, it is important that the modal propagation coefficients be evaluated accurately. Such field problems can sometimes be reformulated in terms of matrix Riccati differential equations for impedance-like quantities. Regarding Riccati equations, Schiff and Shnider [1999] deem the corresponding flow on a Grassmannian the fundamental quantity. Riccati equations have at most point-like singularities, a coordinate artifact that leaves this flow unaffected. However, standard techniques for ordinary differential equations (ODE's) fail to integrate through singularities. The Möbius schemes constructed by Schiff and Shnider are insensitive to singularities.

[4] Since high-precision, adaptive Möbius schemes with error control are not yet available, one may choose to avoid singularities altogether, e.g., an enlarged system of linear ODE's may be integrated, or an invariant embedding technique, or a coordinate transformation may be employed. Regarding Riccati equations for impedances originating from a system of linear ODE's for wave fields, these field problems form a natural example of an enlarged system of linear ODE's. To analyze mode propagation along a straight dielectric slab, Kuester and Chang [1975] applied an invariant embedding technique for numerical integration and employed a coordinate transformation owing to Prüfer [Godart, 1966] in order to express the number of modes in terms of the growth of an angle. This mode-counting algorithm is invaluable, as it facilitates robust root-finding. In a comprehensive overview of optical waveguide concepts, Vassallo [1991, p. 111] stated “Presumably this method can be extended to circular fibers, although we do not know of any work about such generalization.” Vassallo also devoted several sections to bent dielectric waveguides; in particular, he examined competing techniques for the approximation of bent-slab radiation losses.

[5] We investigate mode propagation along open circularly cylindrical graded-index waveguides, namely, the bent slab and the optical fiber. We present a complex-power-flow, variational scheme by which the modal propagation coefficients may be evaluated accurately. By reduction a Newton iterative scheme ensues. Its significance is that the associated characteristic equation does not depend on the field quantities, but only on their ratio. Thus, the use of impedances by Kuester and Chang acquires a complex-power-flow underpinning. For the bent slab, impedances are indispensable to attain adequate accuracy and dynamic range. The major contribution of this work comprises the full-wave generalization for optical fibers of the mode-counting scheme for the straight slab. The companion paper [De Hon, 2002] contains the necessary proofs.

2. A Complex-Power-Flow Variational Scheme

[6] Electromagnetically, the materials are considered locally reacting, isotropic, radially inhomogeneous and, in the case of the optical fiber, lossless. The fields depend on time through the factor exp(jωt). We employ a normalized frequency equation image, where a either denotes the slab width, or the fiber core radius. The scalar free-space plane wave impedance is denoted as Z0 = equation image. In the companion paper, the occasional use of curvilinear coordinates is appropriate. In this paper, only the (contravariant) coordinates {x1, x2, x3} are reminiscent of tensor notation, i.e., these superscripts are indices, not powers. The coordinates x1 and x3 are associated with the radial direction and the direction of propagation, respectively. The radial coordinate is normalized to a, i.e., x1 = ρ = r/a. For the fiber and the bent slab, we choose {x1, x2, x3} = {ρ, φ, z} and {x1, x2, x3} = {ρ, y, θ}, respectively. Accordingly, the respective unit base vectors are {ur, uφ, uz} and {ur, uy, uθ}, entailing the correspondence {φ ↔ θ, z ↔ −y}. The unit vector along the direction of propagation is u = uz for the fiber and u = uθ for the slab. The permittivity and permeability and consequently the refractive index, n = equation image, depend on ρ. The radius of curvature normalized to a beyond which the medium is homogeneous, is denoted as ρc = rc/ac = 1 for the fiber). For ρ > ρc, we have μr = μre, εr = εre and n = ne. The symbols ∂ρ and ∂ν denote partial differentiation with respect to ρ and ν, respectively.

[7] A waveguide section equation image is thought to consist of an interior subdomain equation image, and an exterior one, equation image, separated by a cylindrical surface equation image with outward normal, n = ur (cf. Figure 1). A cross-sectional surface (x3 constant) is denoted by equation image. In equation image, we consider the complex-power interaction between progressive and regressive physical states, distinguished by the superscripts C and D and that depend on x3 through factors exp(∓jk3x3), respectively. Here k3 represents a trial (complex) propagation coefficient. Keeping the trial magnetic field continuous, we define an effective surface magnetic current distribution in terms of an electric-field mismatch across equation image, namely, equation image. In the companion paper, we derive the sequence

equation image

that can be rendered variational, yielding quadratic convergence to a modal propagation coefficient, provided that the initial estimate lies within its basin of convergence. The ratio in equation (1) may be interpreted as the transverse complex-power-flow interaction due to an effective source density, normalized to the complex-power-flow interaction through the waveguide cross-section. The invariance of the configuration with respect to both x2 and x3, implies that for the fiber we may choose the normalization of the fields associated with the progressive and regressive states C and D such that

equation image
equation image
equation image

where the azimuthal index m ∈ Z is fixed and ν denotes the normalized complex propagation coefficient. For the bent slab, we employ the correspondence {φ ↔ θ, z ↔ −y} and k3x3 = κθ = equation image, On equation image the radial field components follow from

equation image
Figure 1.

Depiction of the bent slab (a) and the optical fiber (b).

3. Radial Impedance Matrix for the Optical Fiber

[8] For the fiber we introduce the vectorial current and voltage amplitudes via

equation image

From Maxwell's equations we infer that

equation image

in which

equation image

In view of the field behavior about ρ = 0, we define the 2-vectors p and q via

equation image

To facilitate algebraic manipulations, we employ the 2 × 2 matrices equation image = (p1, p2) and equation image = (q1, q2), where the indices 1 and 2 are used to distinguish independent field solutions. Since the media are lossless, equation image and equation image are real. There is a pair of independent interior field solutions for which equation image and equation image remain finite and bounded away from zero as ρ ↓ 0. Likewise, there is a pair of independent exterior field solutions for which equation image and equation image exhibit exponential decay as ρ → ∞. The interior and exterior matrices at ρ = ρ1 are denoted as, equation imagei, equation imagei, equation imagee and equation imagee, respectively. The system of equations reads

equation image

where equation image denotes the 2 × 2 identity matrix. Owing to the symmetry of equation image and equation image, we infer from equation (8) that dρequation image = 0, which in view of the boundedness at ρ = 0 and the exponential decay for ρ → ∞, implies that equation image for both the interior and the exterior solutions. For the calculation of the number of modes and the modal propagation coefficients, the field quantities are superfluous. It suffices to evaluate the interior and exterior (normalized) radial impedance matrices, which in their generic form equation image are defined via equation image. At ρ = ρ1, the respective matrices are denoted as equation image. In view of equation (7), V and I are related via V = jZ0equation imageI. Using the symmetry equation image and equation (8), we infer that equation image is real and symmetric and satisfies the matrix Riccati differential equation [Schiff and Shnider, 1999],

equation image

For ρ ↓ 0, the interior radial impedance matrix behaves as equation image = equation image(0) + ρequation image′(0) + O2). Employing equations (6) and (9), evaluation of the matrix coefficients yields

equation image

The prime denotes differentiation with respect to the argument and

equation image

in which εr0, μr0 denote the constitutive coefficients at ρ = 0, and ε′r0 and μ′r0 the corresponding derivatives with respect to ρ. The evaluation of equation imagei is deferred to a later stage. The exterior radial impedance matrix is known analytically for ρ = ρ1 > 1, i.e.,

equation image

where equation image >0 and equation image. Prior to completing the fiber analysis, we discuss the bent slab, which allows a transparent introduction of the impedance-angle concept and the complex-power-flow connection.

4. Mode Propagation Along a Graded-Index Bent Slab

[9] For the bent slab, we analyze 2-D E-polarized modes. Duality accounts for H-polarization. Letting ∂y ≡ 0, I and V become scalar. Retaining the scalar current I, the nonvanishing field components are

equation image

where the complex (normalized) radial impedance Z corresponds to ρc−1(equation image)22 pertaining to the fiber. The relevant coupled system of ordinary differential equations reads

equation image
equation image

With reference to equations (2), (13), (14a), and (14b), it is readily verified that

equation image

As a consequence, equation (1) reduces to

equation image

which constitutes an elementary Newton iterative scheme for a characteristic equation of the form ZiZe = 0. Its significance is that the pertaining characteristic equation is unconventional, in that it does not depend on the field quantities, but only on their ratio. For numerical accuracy it is crucial to avoid calculating the fields for large radii of curvature, since they are proportional to Bessel or Hankel functions of very large complex order and real argument that are notoriously difficult to compute [Pennings, 1990]. By contrast, for the calculation of the impedance outside the slab only the logarithmic derivatives of the Bessel and Hankel functions are required, i.e.,

equation image

where {x, xc, κ} = equation imagen, ρcn, ρcν}. The impedance outside the slab is readily computed via continued-fraction expansions according to [Press et al., 1992, pp. 234–240]

equation image
equation image

in which αn+1 = κ2 − (2n + 1)2/4 and βn = 2(jx + n). For convergence, it is sufficient to chooseequation image in equation (18a) and x = x> >equation image in equation (18b). The derivatives with respect to the order, κ, follow from recurrences for RJ;n and RH;n,

equation image
equation image

Upon integrating equations (14a) and (14b), poles of Z may be encountered. To avoid this, we compute an angle Φ, defined via Z = tan Φ. A similar quantity has been used by Kuester and Chang [1975] to analyze graded-index planar slabs. The equations for Φ and ∂νΦ read

equation image
equation image

Given a trial value ν = ν(n), we evaluate equations (18b) and (19b) at an appropriate point x> in the complex x-plane. For equations (18a) and (19a), we select a suitable x = x<. Subsequently, we determine Φ and ∂νΦ at x<, x> and integrate equations (20a) and (20b) numerically, yielding the angles and their derivatives at either side of equation image, for use in equation (16).

[10] In a comparative study, Vassallo [1991, pp. 228–245] examined dielectric bent-slab H-polarization radiation losses. Our results are displayed in Figure 2 and Table 1. Vassallo considered (∂r2 + r−1rr−2Rc2κ2 + equation imagec0−2n2)V = 0, with n piecewise constant and Rc denoting the radius of curvature at the slab axis. Application of the “rectifying” transformations X = Rclog(r/Rc), or Y = rRc, results in straight-slab equations for V(X), or V(Y)equation image, respectively. The latter equation has led Marcuse [1976] and Vassallo to derive approximate equivalent-medium equations with n2eq = n2(1 + 2Y/Rc), and n2eq = n2(1 + 2(ν/n)2Y/Rc) with ν = equation image, respectively. The associated fields are proportional to Airy functions. To assess the respective solutions (M and V in Table 1), Vassallo developed a reference method for V(X) (VAS in Table 1), involving numerical integration of the straight-slab counterpart of equation (20a). At a suitably chosen outer boundary, x = Re(κ), he replaced equation image by a first-order Taylor expansion of an asymptotic approximation about x = κ. For x suitably small, he used the leading-order asymptotic expansion of Jκ/Jκ for ∣κ∣ > x ≫ 1. Although Vassallo's reference method is quite accurate, the predicted losses for weakly guiding bent slabs are slightly underestimated leading him to conclude that somehow Marcuse's equivalent-medium approach is always superior to his own. Usually, this is true; a possible explanation is that the centripetal field displacement effectively stretches the traversed path and hence the larger of the equivalent refractive indices prevails. However, we found that for weakly guiding slabs, their equivalent-medium techniques are, as they should be, roughly of the same accuracy.

Figure 2.

Bending loss along a 90° bend of a weakly guiding bent slab for both polarizations; a = 4 μm, λ = 1000 nm, n = 1.503 in the core and n = 1.500 in the cladding.

Table 1. H-Polarization Bending Loss, −10πIm(κ)/log10,a Along a 90° Bend of a Bent Slab and Errors Resulting From Approximate Methodsb
Rc, mmReference Loss, dBErrors, %
VASneq (M)neq (V)
  • a

    In decibels.

  • b

    The parameters are listed in Figure 2.

103.079851502−0.07−0.570.73
150.239579118−0.17−0.871.03
200.015473242−0.27−1.271.32
250.000914202−0.37−1.561.63
300.000051323−0.78−1.971.90

5. Impedance Formulation for the Fiber Power Flow

[11] For the fiber I is real, whereas V is imaginary. Regarding equations (2b) and (3), we infer that equation image = {H*, −E*}exp(jkPxP), implying that

equation image

where S = equation imageE × H* ∈ equation image3 is the Poynting vector. In analogy with equation (15), we have 4u · Re(S) dA = ∂ρ[a2Z0equation imageIT(∂νequation image)I] dρ dφ. The total power flow, PPi + Pe, along u consists of the power constituents, Pi and De, flowing through equation image, (ρ < ρ1), and &#55349;&#56479;e, (ρ > ρ1), respectively, i.e.,

equation image

in which Ii and Ie respectively denote the interior and exterior vectorial current amplitudes at ρ = ρ1. By definition, a characteristic equation for ν (with m fixed) is an (arbitrary) equation, subject to the condition that its set of roots coincides with the set of normalized modal propagation coefficients {νmnm fixed, n = 1, …, nmax} (counting multiplicities). Since a modal field is a source-free solution to Maxwell's equations, imposing the continuity conditions across ρ = ρ1, i.e., Ii = Ie and equation imageiIi = equation imageeIe, is equivalent to demanding that (equation imageiequation imagee)Ii = 0 and Ie = Ii have a nontrivial solution. Hence, det(equation imageiequation imagee) = 0 constitutes a “field-free” characteristic equation in that it depends on impedances only.

[12] In view of the power interpretation of equation (22), one might surmise that ∂νequation imagei and −∂νequation imagee are positive definite regardless of the choice for ρ1. This conjecture is refuted upon regarding ∂νequation imagei for ρ1 ↓ 0 (cf. equation (10)). Nevertheless, for ρ1 large and ν ≠ νmn, ∂νequation imagei is asymptotic to ∂νequation imagei, where the matrix associated with the two unbounded solutions in the cladding, equation imagei, follows from equation (12) upon replacing Lm by equation image. A straightforward asymptotic analysis leads to the key result.

  • Lemma 1: For ρ1 > 1 sufficiently large and ν > ne, both ∂νequation imagei and −∂νequation imagee are real, symmetric, positive definite matrices, except at the isolated points ν = νmn, where ∂νequation imagei may be positive semidefinite.

6. Vectorial Impedance-Angle Formalism

[13] For the bent-slab problem, the field quantities are related via a scalar impedance. To circumvent problems related to possible poles and/or zeroes, we expressed the scalar impedance in terms of the tangent of an angle, Φ. Below we develop the vectorial generalization of this concept, namely, we introduce the symmetric matrix of angles,

equation image

and the associated sine and cosine matrices equation image = sin ϕ = equation image(sin Λ)equation imageT and equation image = cos ϕ = equation image(cos Λ)equation imageT, respectively. Next, we write

equation image

where equation image denotes a 2 × 2 matrix. It is straightforward to derive a closed-form expression for det equation image, showing that equation image is nonsingular throughout. Further, we introduce

equation image
equation image

and cast equation image in the following form:

equation image

where it should be noted that the substitution {δ, θ} → {−δ, θ ± π/2} leaves equation image unchanged. Employing equations (9) and (23)(26), one may derive the following generic system of coupled, nonlinear, first-order, ordinary differential equations for the angles,

equation image
equation image
equation image

Employing equations (23) and (24) leads to

equation image

from which we infer that

equation image
equation image

If ±∂νequation image were positive semidefinite, some inequalities in equations (29a) and (29b) would become equalities. At ρ = 0, we evaluate ϕ(0) and the derivative with respect to ρ, ϕ′(0) by combining equations (6) and (10) and the counterpart for ∂ρequation image of equation (28). For uniqueness, we select ϕ+(0) ∈ equation image], ϕ(0) ∈ (0, π) and ψ(0) = θ(0) ∈ equation image]. A first-order Taylor expansion about ρ = 0 yields estimates for the interior angles at ρ = ρ1, that are accurate to O02). The interior angles at ρ = ρ0, {ϕ±i, ψi} are obtained via numerical integration of equations (27a), (27b), and (27c) starting at ρ0 sufficiently small. The exterior angles follow from equation (12). For uniqueness, we select ϕ+eequation image, ϕ−e ∈ (0, π) and ψeequation image.

7. The Mode-Counting and Mode-Bracketing Theorems

[14] In the impedance-angle formalism, the system of continuity conditions across ρ = ρ1 reads equation imageiWi = equation imageeWe and equation imageiWi = equation imageeWe. This is equivalent to (equation imageeequation imageiequation imageeequation imagei) Wi = 0 and We = equation imagee−1equation imageiWi if it exists, or We = equation imagee−1equation imageiWi, otherwise. Upon defining the “characteristic function,” D(ν) = −2det(equation imageeequation imageiequation imageeequation imagei), expansion leads to

equation image

Hence, the continuity conditions have a nontrivial solution, if and only if D(ν) = 0. We regard D(ν) = 0 as a suitable characteristic equation, because D is bounded, continuous and ”field-free.“ The main advantage, however, is that via the angles we are able to count the number of characteristic roots in an arbitrary closed interval ν ∈ equation image= [ν<, ν>] with (μreεre)$\frac{1}{2}$ < ν< < ν>. To this end, we introduce the ”separation function“

equation image

In view of Lemma 1 and equations (29a) and (29b), σ(ν) increases monotonically. As a consequence, the number of zeroes of the separation function in the closed interval equation image is given by

equation image

where ⌊σ>⌋ and ⌈σ<⌉ denote the largest and smallest integers, such that ⌊σ>⌋ ≤ σ> and ⌈σ<⌉ ≥ σ>, respectively. Precluding double roots, D(ν) changes sign in between consecutive zeroes of S(ν). In the companion paper, the following theorems are proven:

  • Theorem 1: Every closed interval that has consecutive zeroes of the separation function as its end-points, contains exactly one root (not counting multiplicities) of the characteristic equation.
  • Theorem 2: The number of roots (counting multiplicities) of the characteristic equation D(ν) = 0 in the closed interval equation image = [ν<, ν>] is given by
    equation image
    0 in which
    equation image

In view of Theorem 1, σ(ν), regarded as a mapping, is instrumental for mode bracketing. This is illustrated in Figure 3, where D(ν) is set against D(ν(σ)). Application of Theorem 2 has resulted in the mode distribution displayed in Figure 4.

Figure 3.

Profile of a dielectric optical fiber (top left) and the characteristic function, D, in its dependence on the normalized propagation coefficient, ν and the angle, σ, for the azimuthal index m = 12 and the vacuum wavelength λ0 = 1273.4 nm.

Figure 4.

Total number of modes, equation image, in the interval equation image = (ne, nmax], in its dependence on the azimuthal index m (fiber parameters as described in Figure 3).

8. Conclusion

[15] The complex-power-flow, variational scheme for the determination of the modal propagation coefficients formed the foundation for our analysis of mode propagation along bent slabs and optical fibers. From the bent-slab results, it became clear that upon choosing the proper impedance-based characteristic equation, numerical difficulties disappear, and accuracy ensues. Some scalar bent-slab expressions were amenable for generalization. Application of their vectorial, optical-fiber counterparts has led to the mode counting and bracketing theorems. With these key results, root-finding can be rendered more robust and efficient.

Acknowledgments

[16] The first and second authors thankfully acknowledge the fellowship grant from the Royal Netherlands Academy of Arts and Sciences, and the financial support received from Draka Fibre Technology BV, respectively.

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