**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.

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## 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(*j*ω*t*). We employ a normalized frequency , where *a* either denotes the slab width, or the fiber core radius. The scalar free-space plane wave impedance is denoted as *Z*_{0} = . In the companion paper, the occasional use of curvilinear coordinates is appropriate. In this paper, only the (contravariant) coordinates {*x*^{1}, *x*^{2}, *x*^{3}} are reminiscent of tensor notation, i.e., these superscripts are indices, not powers. The coordinates *x*^{1} and *x*^{3} are associated with the radial direction and the direction of propagation, respectively. The radial coordinate is normalized to *a*, i.e., *x*^{1} = ρ = *r*/*a*. For the fiber and the bent slab, we choose {*x*^{1}, *x*^{2}, *x*^{3}} = {ρ, φ, *z*} and {*x*^{1}, *x*^{2}, *x*^{3}} = {ρ, *y*, θ}, respectively. Accordingly, the respective unit base vectors are {**u**_{r}, **u**_{φ}, **u**_{z}} and {**u**_{r}, **u**_{y}, **u**_{θ}}, entailing the correspondence {φ ↔ θ, *z* ↔ −*y*}. The unit vector along the direction of propagation is **u** = **u**_{z} for the fiber and **u** = **u**_{θ} for the slab. The permittivity and permeability and consequently the refractive index, *n* = , depend on ρ. The radius of curvature normalized to *a* beyond which the medium is homogeneous, is denoted as ρ_{c} = *r*_{c}/*a* (ρ_{c} = 1 for the fiber). For ρ > ρ_{c}, we have μ_{r} = μ_{re}, ε_{r} = ε_{re} and *n* = *n*_{e}. The symbols ∂_{ρ} and ∂_{ν} denote partial differentiation with respect to ρ and ν, respectively.

[7] A waveguide section is thought to consist of an interior subdomain , and an exterior one, , separated by a cylindrical surface with outward normal, **n** = **u**_{r} (cf. Figure 1). A cross-sectional surface (*x*^{3} constant) is denoted by . In , we consider the complex-power interaction between progressive and regressive physical states, distinguished by the superscripts C and D and that depend on *x*^{3} through factors exp(∓*jk*_{3}*x*^{3}), respectively. Here *k*_{3} 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 , namely, . In the companion paper, we derive the sequence

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 *x*^{2} and *x*^{3}, 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

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 *k*_{3}*x*^{3} = κθ = , On the radial field components follow from

## 3. Radial Impedance Matrix for the Optical Fiber

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

From Maxwell's equations we infer that

in which

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

To facilitate algebraic manipulations, we employ the 2 × 2 matrices = (**p**_{1}, **p**_{2}) and = (**q**_{1}, **q**_{2}), where the indices 1 and 2 are used to distinguish independent field solutions. Since the media are lossless, and are real. There is a pair of independent interior field solutions for which and remain finite and bounded away from zero as ρ ↓ 0. Likewise, there is a pair of independent exterior field solutions for which and exhibit exponential decay as ρ → ∞. The interior and exterior matrices at ρ = ρ_{1} are denoted as, _{i}, _{i}, _{e} and _{e}, respectively. The system of equations reads

where denotes the 2 × 2 identity matrix. Owing to the symmetry of and , we infer from equation (8) that *d*_{ρ} = 0, which in view of the boundedness at ρ = 0 and the exponential decay for ρ → ∞, implies that 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 are defined via . At ρ = ρ_{1}, the respective matrices are denoted as . In view of equation (7), **V** and **I** are related via **V** = *jZ*_{0}**I**. Using the symmetry and equation (8), we infer that is real and symmetric and satisfies the matrix Riccati differential equation [*Schiff and Shnider*, 1999],

For ρ ↓ 0, the interior radial impedance matrix behaves as = (0) + ρ′(0) + *O*(ρ^{2}). Employing equations (6) and (9), evaluation of the matrix coefficients yields

The prime denotes differentiation with respect to the argument and

in which ε_{r0}, μ_{r0} denote the constitutive coefficients at ρ = 0, and ε′_{r0} and μ′_{r0} the corresponding derivatives with respect to ρ. The evaluation of _{i} is deferred to a later stage. The exterior radial impedance matrix is known analytically for ρ = ρ_{1} > 1, i.e.,

where >0 and . 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

where the complex (normalized) radial impedance *Z* corresponds to ρ_{c}^{−1}()_{22} pertaining to the fiber. The relevant coupled system of ordinary differential equations reads

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

As a consequence, equation (1) reduces to

which constitutes an elementary Newton iterative scheme for a characteristic equation of the form *Z*_{i} − *Z*_{e} = 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.,

where {*x*, *x*_{c}, κ} = {ρ*n*, ρ_{c}*n*, ρ_{c}ν}. The impedance outside the slab is readily computed via continued-fraction expansions according to [*Press et al.*, 1992, pp. 234–240]

in which α_{n+1} = κ^{2} − (2*n* + 1)^{2}/4 and β_{n} = 2(*jx* + *n*). For convergence, it is sufficient to choose in equation (18a) and *x* = *x*_{> }> in equation (18b). The derivatives with respect to the order, κ, follow from recurrences for *R*_{J;n} and *R*_{H;n},

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 ∂_{ν}Φ read

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 ∂_{ν}Φ at *x*_{<}, *x*_{>} and integrate equations (20a) and (20b) numerically, yielding the angles and their derivatives at either side of , 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 (∂_{r}^{2} + *r*^{−1}∂_{r} − *r*^{−2}*R*_{c}^{2}κ^{2} + *c*_{0}^{−2}*n*^{2})*V* = 0, with *n* piecewise constant and *R*_{c} denoting the radius of curvature at the slab axis. Application of the “rectifying” transformations *X* = *R*_{c}log(*r*/*R*_{c}), or *Y* = *r* − *R*_{c}, results in straight-slab equations for *V*(*X*), or *V*(*Y*), respectively. The latter equation has led *Marcuse* [1976] and Vassallo to derive approximate equivalent-medium equations with *n*^{2}_{eq} = *n*^{2}(1 + 2*Y*/*R*_{c}), and *n*^{2}_{eq} = *n*^{2}(1 + 2(ν_{∞}/*n*)^{2}*Y*/*R*_{c}) with ν_{∞} = , 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 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 *J*′_{κ}/*J*_{κ} 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.

R_{c}, mm | Reference Loss, dB | Errors, % | ||
---|---|---|---|---|

VAS | n_{eq} (M) | n_{eq} (V) | ||

- a
In decibels. - b
The parameters are listed in Figure 2.
| ||||

10 | 3.079851502 | −0.07 | −0.57 | 0.73 |

15 | 0.239579118 | −0.17 | −0.87 | 1.03 |

20 | 0.015473242 | −0.27 | −1.27 | 1.32 |

25 | 0.000914202 | −0.37 | −1.56 | 1.63 |

30 | 0.000051323 | −0.78 | −1.97 | 1.90 |

## 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 = {**H***, −**E***}exp(*jk*_{P}*x*^{P}), implying that

where **S** = **E** × **H*** ∈ ^{3} is the Poynting vector. In analogy with equation (15), we have 4**u** · Re(**S**) *d**A* = ∂_{ρ}[*a*^{2}*Z*_{0}**I**^{T}(∂_{ν})**I**] *d*ρ *d*φ. The total power flow, *P**P*_{i} + *P*_{e}, along **u** consists of the power constituents, *P*_{i} and *D*_{e}, flowing through , (ρ < ρ_{1}), and ��_{e}, (ρ > ρ_{1}), respectively, i.e.,

in which **I**_{i} and **I**_{e} respectively denote the interior and exterior vectorial current amplitudes at ρ = ρ_{1}. 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 {ν_{mn}∣*m* fixed, *n* = 1, …, *n*_{max}} (counting multiplicities). Since a modal field is a source-free solution to Maxwell's equations, imposing the continuity conditions across ρ = ρ_{1}, i.e., **I**_{i} = **I**_{e} and _{i}**I**_{i} = _{e}**I**_{e}, is equivalent to demanding that (_{i} − _{e})**I**_{i} = **0** and **I**_{e} = **I**_{i} have a nontrivial solution. Hence, det(_{i} − _{e}) = 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 ∂_{ν}_{i} and −∂_{ν}_{e} are positive definite regardless of the choice for ρ_{1}. This conjecture is refuted upon regarding ∂_{ν}_{i} for ρ_{1} ↓ 0 (cf. equation (10)). Nevertheless, for ρ_{1} large and ν ≠ ν_{mn}, ∂_{ν}_{i} is asymptotic to ∂_{ν}_{i}^{∞}, where the matrix associated with the two unbounded solutions in the cladding, _{i}^{∞}, follows from equation (12) upon replacing *L*_{m} by . A straightforward asymptotic analysis leads to the key result.

**Lemma 1**: For ρ_{1}> 1 sufficiently large and ν >*n*_{e}, both ∂_{ν}_{i}and −∂_{ν}_{e}are real, symmetric, positive definite matrices, except at the isolated points ν = ν_{mn}, where ∂_{ν}_{i}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,

and the associated sine and cosine matrices = sin **ϕ** = (sin Λ)^{T} and = cos **ϕ** = (cos Λ)^{T}, respectively. Next, we write

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

and cast in the following form:

where it should be noted that the substitution {δ, θ} → {−δ, θ ± π/2} leaves 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,

Employing equations (23) and (24) leads to

from which we infer that

If ±∂_{ν} 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 ∂_{ρ} of equation (28). For uniqueness, we select ϕ_{+}(0) ∈ ], ϕ_{−}(0) ∈ (0, π) and ψ(0) = θ(0) ∈ ]. A first-order Taylor expansion about ρ = 0 yields estimates for the interior angles at ρ = ρ_{1}, that are accurate to *O*(ρ_{0}^{2}). The interior angles at ρ = ρ_{0}, {ϕ_{±i}, ψ_{i}} are obtained via numerical integration of equations (27a), (27b), and (27c) starting at ρ_{0} sufficiently small. The exterior angles follow from equation (12). For uniqueness, we select ϕ_{+e} ∈ , ϕ_{−e} ∈ (0, π) and ψ_{e} ∈ .

## 7. The Mode-Counting and Mode-Bracketing Theorems

[14] In the impedance-angle formalism, the system of continuity conditions across ρ = ρ_{1} reads _{i}**W**_{i} = _{e}**W**_{e} and _{i}**W**_{i} = _{e}**W**_{e}. This is equivalent to (_{e}_{i} − _{e}_{i}) **W**_{i} = **0** and **W**_{e} = _{e}^{−1}_{i}**W**_{i} if it exists, or **W**_{e} = _{e}^{−1}_{i}**W**_{i}, otherwise. Upon defining the “characteristic function,” *D*(ν) = −2det(_{e}_{i} − _{e}_{i}), expansion leads to

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 ∈ = [ν_{<}, ν_{>}] with (μ_{re}ε_{re})^{$\frac{1}{2}$} < ν_{<} < ν_{>}. To this end, we introduce the ”separation function“

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 is given by

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 = [ν_{<}, ν_{>}] is given by 0 in which

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.

## 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.