**Radio Science**

# A modal impedance-angle formalism: Rigorous proofs for optical fiber mode counting and bracketing

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## Abstract

[1] In a companion paper, a complex-power-flow variational scheme is applied to analyze mode propagation along open circularly cylindrical graded-index waveguides. It leads to a characteristic equation in terms of impedances rather than fields. The resulting impedance-angle formalism provides the basis for the full-wave generalization for optical fibers of the mode-counting scheme previously developed for a scalar wave propagation problem. The complex-power-flow variational scheme for bent waveguides is based on energy considerations. Hence, in its derivation, it is natural to consider a waveguide section (a volume) rather than a cross section (a surface). In the proof of the mode-counting and mode-bracketing theorems, the key issue is to show that the characteristic roots and the roots of the so-called separation function alternate. For general circularly cylindrical open waveguides, the required proofs are intricate. However, the special limiting cases in which the optical fiber is surrounded by electrically or magnetically perfectly conducting walls are tractable. To account for the general case, it appears to be necessary to regard a class of optical waveguide problems with a continuous transition from perfect electric conductor to perfect magnetic conductor boundary conditions via the situation pertaining to the actual exterior medium. Thus, a half-strip is constructed on which the so-called characteristic and separation graphs are seen to alternate. As spin-off, such a “sweep” might prove useful in the design of a fiber cladding.

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## 1. Introduction

[2] The primary purpose of the present paper is to provide proofs for the schemes presented in the companion paper [*De Hon and Bingle*, 2002], in which mode propagation along open circularly cylindrical graded-index waveguides is examined. To provide a unified framework for various configurations, the derivation of the complex-power-flow variational scheme for the determination of the modal propagation coefficients involves the analysis of a waveguide section in an orthogonal curvilinear system of coordinates.

[3] *Kuester and Chang* [1975] developed a mode-counting scheme pertaining to straight dielectric slabs. Due to the vectorial nature of the problem, the proof of its optical fiber counterpart is considerably more intricate; it involves a class of waveguide problems with a continuous transition from perfect electric conductor (PEC) to perfect magnetic conductor (PMC) boundary conditions. Given space limitations, extensive use is made of the theory presented in the companion paper; references to its equations, figures, etc., are preceded by a prefix “I.”

## 2. A Complex-Power-Flow Variational Scheme

[4] Electromagnetically, the materials are considered locally reacting, isotropic and radially inhomogeneous. We employ a right-handed orthogonal curvilinear system of coordinates {*x*^{1}, *x*^{2}, *x*^{3}}. By choice, the coordinates *x*^{1} and *x*^{3} are associated with the radial direction and the direction of propagation, respectively. The covariant base vectors **c**_{i} satisfy 〈**c**_{i}, **c**_{j}〉 = *g*_{ij}. Lowercase Roman and Greek indices take the values {1, 2, 3} and {1, 2}, respectively. Partial differentiation with respect to *x*^{p} is denoted as ∂_{p}. The summation convention applies. The antisymmetric tensor of rank three is defined through η^{mpk} = *g*^{−1/2}ϵ^{mpk}, where *g* = det(*g*_{ij}) and ϵ^{mpk} denotes the Levi–Civita symbol. The unit vector along the direction of propagation is **u** = **c**_{3} = **c**^{3}.

[5] A waveguide section is thought to consist of an interior subdomain and an exterior one, , separated by a cylindrical surface, (cf. Figure I:1). In we consider the complex-power-flow interaction between progressive and regressive physical states, distinguished by the superscripts C and D, respectively. The respective covariant field components depend on time *t* and on *x*^{3} via the factor exp(*j*ω*t*∓*jk*_{3}*x*^{3}), where *k*_{3} represents a trial, complex propagation coefficient. The fields comprise source-free solutions of Maxwell's equations on , e.g., for state C we have

with ∂_{3} → −*jk*_{3} [*Synge and Schild*, 1956, p. 227]. We ensure that for all trial states, the components of transverse to are continuous across while the components of transverse to are left free to jump across by an amount []_{i}^{e}. Let us multiply equations (1a) and (1b) by _{m}^{D} and _{k}^{D}, respectively, contract over the repeated indices, subtract the resulting equations, integrate over and integrate by parts to eliminate derivatives of _{m}^{C} across . This leads to the identity,

in which denote the covariant components of an effective surface magnetic current distribution. The states C and D serve as trial states for the respective modal progressive and regressive states A and B. Let the deviation of the associated trial fields from the modal fields be *O*(*h*). Since the modal fields satisfy the source-free counterpart (SFC) of Maxwell's equations on , substitution of the trial fields into the SFC of equation (2) for the modal fields A and B and subsequently using equation (2) as an identity, yields . Evaluation of successive approximations leads to

In case *dA* = *dx*^{1}*dx*^{2} is independent of *x*^{3}, the volume integral in equation (3) may be reduced to a surface integral over a cross-sectional surface , thus yielding equation (I:1).

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

[6] Henceforth, lossless media are considered. Partial differentiation with respect to the normalized propagation coefficient, ν and the angle χ are denoted as ∂_{ν} and ∂_{χ}, respectively. The analysis of the specific characteristic equation, *D*(ν) = 0, involves the separation and characteristic functions, respectively defined for ν > ν_{min} = , as

In view of Lemma I:1 and equation (I:29), σ(ν) denotes a nonsingular monotonically increasing function, implying the existence of a unique inverse ν = ν(σ) with range ν > ν_{min}. As a consequence, the number of zeroes of *S* in the arbitrary closed interval ν ∈ = [ν_{<}, ν_{>}] with < ν_{<} < ν_{>}, is given by

Note that the magnitude of the term between brackets in equation (4b) never exceeds unity. Hence, precluding double characteristic roots, the characteristic function must change sign in between consecutive zeroes of *S*. Double roots are covered by

**Lemma 1**A root ν = ν_{d}is a double root of a characteristic equation for fixed*m*, if and only if , with ℓ_{ψ},ℓ_{±}∈ . In view of equations (4a) and (4b) this is equivalent to*D*=*S*= 0.- Proof Let λ
_{1}and λ_{2}denote the eigenvalues of the real symmetric 2 × 2 matrix . Now, suppose that is a double root of det = 0 (with m fixed). Then, at we either haves λ_{1}= λ_{2}= 0, implying that , or, is a double zero of either λ_{1}or λ_{2}, implying that = 0, where**I**is the corresponding eigenvector. Since is symmetric positive definite, we have**I**^{T}> 0 and hence the latter option leads to a contradiction. In turn is equivalent to , with . In view of equations (4a) and (4b) this is equivalent to D = S = 0.

[7] For most fibers, double characteristic roots do not exist for fixed *m*. However, if ε_{r} and μ_{r} are proportional throughout, all *TM*_{0n} propagation coefficients coincide with their *TE*_{0n} counterparts. The combined properties of the impedance-angle formalism enable us to count the number of characteristic roots in , according to two theorems,

**Theorem 1**Every closed interval that has consecutive zeroes of the separation function as its endpoints, 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 in which

**Proof of Theorem 2**First, let us assume that ⌈σ_{<}⌉ ≤ ⌊σ_{>}⌋. Then, zeroes of the separation function ν_{⌈σ}_{<}_{⌉}= ν(⌈σ_{<}⌉) and ν_{⌊σ}_{>}_{⌋}= ν(⌊σ_{>}⌋) exist, such that ν_{<}≤ ν_{⌈σ}_{<}_{⌉}≤ ν_{⌊σ}_{>}_{⌋}≤ ν_{>}. Further, let us assume that all roots are simple, which, on account of Theorem 1 implies that the intervals [ν_{<}, ν_{⌈σ}_{<}_{⌉}] and [ν_{⌊σ}_{>}_{⌋},ν_{>}] contain at most one root each. Hence, there are no roots on [ν_{<}, ν_{⌈σ}_{<}_{⌉}] if and only if*D*(ν_{<})*D*(ν_{⌈σ}_{<}_{⌉}) > 0, or equivalently*N*_{−}= 1; otherwise*N*_{−}= 0. Likewise, there are no roots on [ν_{⌊σ}_{>}_{⌋}, ν_{>}] if and only if*D*(ν_{⌊σ}_{>}_{⌋})*D*(ν_{>}) > 0, or equivalently*N*_{+}= 0, otherwise*N*_{+}= 1. With reference to Theorem 1 and equation (5), we infer that there are ⌊σ_{>}⌋ − ⌈σ_{<}⌉ roots of*D*on [ν_{⌈σ}_{<}_{⌉}, ν_{⌊σ}_{>}_{⌋}], provided that all roots are simple. Combining these observations, we infer that the theorem holds, provided ⌈σ_{<}⌉ ≤ ⌊σ_{>}⌋ and all roots are simple.

[8] Now, suppose that double roots do exist. If all double roots are located in the open interval (ν_{⌈σ}_{<}_{⌉}, ν_{⌊σ}_{>}_{⌋}), the number of roots on [ν_{⌈σ}_{<}_{⌉}, ν_{⌊σ}_{>}_{⌋}] is still given by ⌊σ_{>}⌋ − ⌈σ_{<}⌉, i.e., possible interior double roots are automatically incorporated. For a double root that coincides with ν_{⌈σ}_{<}_{⌉} or ν_{⌊σ}_{>}_{⌋}, the term ⌊σ_{>}⌋ − ⌈σ_{<}⌉ does not account for the multiplicity. However, in that case, the multiplicity is counted through *N*_{−} = 0 or *N*_{+} = 1, respectively. Finally, if there are no zeroes of the separation function in the interval [ν_{<}, ν_{>}], then ⌈σ_{<}⌉ = ⌊σ_{>}⌋ + 1. As a consequence, the number of roots in [ν_{<},ν_{>}] is simply given by *N*_{+} − *N*_{−} = 0 if *D*(ν_{<})*D*(ν_{>}) > 0, and *N*_{+} − *N*_{−} = 1, otherwise.

**Proof of Theorem 1**Initially, it is assumed that double roots do not occur. Only minor modifications are required to account for possible double roots. Instead of regarding a single configuration, a class of optical fibers with identical interior, but different exterior media is considered. In particular, the exterior refractive index, , is kept fixed, while the angle χ, introduced via may take any value. For propagating modes we have ν > ν_{min}. Hence, the half-strip is considered the domain of analysis. Occasionally, the open domain or its closure is regarded, where and are boundary segments. The limiting χ-values 0 and π/2 correspond to PEC and PMC boundary conditions at ρ = ρ_{1}, respectively. Precluding double solutions, the proof of the theorem for the PEC and PMC cases is straightforward; the continuous transition between these cases forms the elaborate part.

[9] Obviously, the interior angles {ϕ_{+i}, ϕ_{−i}, ψ_{i}} are independent of χ. Inspection of the expression for _{e}, given by equation (I:12), shows that _{e} cot χ is independent of χ as well. Hence, ψ_{e} is also independent of χ. Introducing the angle η via

we infer that

and hence η ∈ (0, π/2) is independent of χ. For brevity, we further define

in terms of which we have

In essence, the dependence of η on ν is captured by

The inequality follows from Lemma I:1 and equations (I:29), (13a) and (13b). Since Π > 0 on , the equations *D* = 0 and *S* = 0 are equivalent to Δ = 0 and Σ = 0, respectively, with

in which the respective normalized characteristic and separation functions,

are convenient functions on . The lemma below structures the remainder of the proof.

**Lemma 2**There exists a ρ_{4}such that for ρ_{1}> ρ_{4}the following propositions hold:- Precluding double characteristic roots, Theorem 1 holds for fibers with perfectly conducting electric (χ = 0) or magnetic (χ = π/2) walls.
- If separation graphs ν
_{Σ}= ν_{Σ}(ρ_{1},χ), along which Σ = 0, exist on , and if the endpoints are located on ∂_{χ}, then the one that is nearest to ∂_{ν}is denoted as ν_{Σ;1}(cf. Figures 1a and 1b). There may also be a single graph ν_{Σ;0}with both endpoints on ∂_{ν}(cf. Figure 1b). - If characteristic graphs (or graph segments) ν
_{Δ}= ν_{Δ}(ρ_{1}, χ), along which Δ = 0, exist on and if they have at least one endpoint on ∂_{χ}then the one(s) that is (are) closest to ∂_{ν}is (are) denoted as ν_{Δ;1}(cf. Figures 1a and 1b). There may also be a single graph ν_{Δ;0}with both endpoints on ∂_{ν}(cf. Figure 1b). - If both ν
_{Δ;0}and ν_{Δ;1}exist, then these graphs are separated by either ν_{Σ;0}or ν_{Σ;1}. - Precluding double characteristic roots, the separation and characteristic graphs are disjoint and induce a partition of , such that any two adjacent characteristic graphs are separated by a separation graph, while any two adjacent separation graphs are separated by a characteristic graph.

[10] The Proof of Proposition 1 in Lemma 2 is given first. Subsequently, Propositions 2–4 provide the setting in which Proposition 5, comprising the generalization of Proposition 1 for arbitrary χ ∈ (0, π/2), is rendered elementary.

**Proof of Proposition 1 of Lemma 2**In the PEC case, χ = 0, we have Δ =*c*_{+i}−*c*_{−i}and Σ = s_{+i}. Hence, precluding double characteristic roots, we have with ℓ_{0}, ℓ_{1}, ℓ_{2}∈ Z, respectively. Let ν = ν_{1}be a root of Δ = 0, for which ϕ_{1i}= ℓ_{1}π and (ℓ_{2}− 1)π < ϕ_{2i}< ℓ_{2}π, i.e., On account of Lemma I:1 and equation (I:29), both ϕ_{1i}and ϕ_{2i}increase monotonically in between consecutive roots. Hence, at the next root, ν = ν_{2}> ν_{1}, of Δ, we have either ℓ_{1}π < ϕ_{1i}(ν_{2}) < (ℓ_{1}+ 1)π and ϕ_{2i}(ν_{2}) = ℓ_{2}π, or ϕ_{1i}(ν_{2}) = (ℓ_{1}+ 1)π and (ℓ_{2}− 1)π < ϕ_{2i}(ν_{2}) < ℓ_{2}π. This leads to i.e., there exists a ν_{0}∈ (ν_{1}, ν_{2}), such that Σ(ρ_{1}, 0, ν_{0}) = 0. We could equally well have started with ϕ_{2i}(ν_{1}) = ℓ_{2}π. Now, recall that all characteristic roots being simple,*D*and hence Δ change sign in between consecutive zeroes of Σ. Hence, we conclude that the zeroes of Δ and Σ alternate for χ = 0. The PMC case is analogous.**Proof of Proposition 2 of Lemma 2**Let us assume that separation graphs ν_{Σ}= ν_{Σ}(ρ_{1}, χ) with ρ_{1}constant, exist on (a subset of) , such that Σ(χ, ν_{Σ}(ρ_{1}, χ)) = 0. In view of equation (16b), these graphs are symmetric, i.e., ν_{Σ}(ρ_{1}, π/2 − χ) = ν_{Σ}(ρ_{1}, χ). To determine the slope of a separation graph, it suffices to solve the system of equations {Σ = 0,*d*Σ/*d*χ = 0}, yielding*d*ν_{Σ}/*d*χ = −(∂_{χ}Σ/∂_{ν}Σ)∣_{Σ}_{=0}. On account of 0 < η < π/2, Lemma I:1 and equations (I:29), (4a), (13a), (13b), and (15), it turns out that ∂_{ν}Σ∣_{Σ=0}is bounded away from zero on , and that Invoking the implicit function theorem [*Taylor*, 1996, p. 13], we infer that if Σ = 0 at a single point of \∂_{χ}, the existence of a graph ν_{Σ}= ν_{Σ}(ρ_{1}, χ) on (a subset of) \∂_{χ}is guaranteed. Approaching ∂_{χ}the pertaining limits are regular, i.e., the existence extends to subsets of . From equations (14a)–(14c) we infer that for ρ_{1}> ρ_{2}= (*n*_{e})^{−1}(*m**m*– 1)_{1/2}, there exists a unique ν = ν_{η}(ρ_{1}) ∈ (ν_{min}, ∞) such that cos2η|_{ν=ν}_{η}= 0 and hence*d*ν_{Σ}/*d*χ = 0. Simple trigonometry reveals that for ν ≠ ν_{η}(ρ_{1}) fixed, the equation Σ(ρ_{1},χ,ν) = 0 has either zero or two solutions for ρ_{1}. In conclusion, for ρ_{1}> ρ_{2}and if ν_{Σ}(ρ_{1},0) < ν_{η}(ρ_{1}), we have ν_{Σ}(ρ_{1},0) ≤ ν_{Σ}(ρ_{1},χ) ≤ ν_{Σ}(ρ_{1},). The choice ρ_{1}> ρ_{2}serves to ensure that there are no separation graphs with endpoints on both ∂_{ν}and ∂_{χ}, thus reducing the number of cases requiring treatment. So, for constant ρ_{1}> ρ_{2}, every zero of the separation function on ∂_{χ}is an endpoint of a unique graph ν_{Σ}. If such graphs with endpoints on ∂_{χ}exist, then the one that is closest to ∂_{ν}is denoted as ν_{Σ;1}(cf. Figure 1). There may be a graph ν_{Σ;0}with both endpoints on ∂_{ν}(cf. Figure 1b). If that graph exists, it must be unique.**Proof of Proposition 3 of Lemma 2**Next, let us assume that characteristic graphs ν_{Δ}= ν_{Δ}(ρ_{1}, χ), with ρ_{1}constant, exist on (a subset of) , such that Δ(χ, ν_{Δ}(ρ_{1}, χ)) = 0. In general these graphs are asymmetric. To determine the slope of a graph ν_{Δ}, it suffices to solve the system of equations {Δ = 0,*d*Δ/*d*χ =0}, yielding*d*ν_{Δ}/*d*χ = −(∂_{χ}Δ/∂_{ν}Δ)∣_{Δ}_{=0}. Since we have precluded double characteristic roots, it is clear that Δ = 0 and ∂_{ν}Δ = 0 cannot be satisfied simultaneously, i.e.,*d*ν_{Δ}/*d*χ remains bounded on . Applying the implicit function theorem, we infer that if Δ = 0 is satisfied at a single point of the open domain \∂_{χ}, the existence of a graph ν_{Δ}= ν_{Δ}(ρ_{1},χ) on (a subset of) \∂_{χ}is guaranteed. Again, the existence extends to subsets of . Note that unless ν_{Δ}is independent of χ, the equation Δ(ρ_{1}, χ, ν) = 0 has at most two solutions for ρ_{1}and ν fixed. As a consequence, if a graph ν_{Δ;0}with no endpoints on ∂_{χ}exists, it is unique and characteristic graphs with endpoints on both ∂_{χ}and ∂_{ν}are absent. The converse holds too (cf. Figures 1a and 1b).

[11] For the location of the roots of the characteristic equation, the normalized radius ρ_{1} across which the interior and exterior solutions are matched is irrelevant: the characteristic graphs are independent of ρ_{1}. By contrast, the separation graphs move as ρ_{1} is varied. However, if for a particular value of ρ_{1} a separation graph ν_{Σ} is located in between two adjacent characteristic graphs ν_{Δ}, it must remain so for other values of ρ_{1}, since otherwise a point would exist for which *D* = *S* = 0; on account of Lemma 1 that point would also comprise a (precluded) double root of the characteristic equation.

**Proof of Proposition 4 of Lemma 2**Regarding the graph ν_{Δ;1}, let (χ_{c}, ν_{c}) denote the endpoint of ν_{Δ;1}that is closest to ∂_{ν}(cf. Figure 1c). We choose ρ_{1}> ρ_{2}, and define_{Δ;0}= {(χ, ν)∣(χ, ν) ∈ , ν_{min}< ν < ν_{c}, and ν > ν_{Δ;0}(χ) if ν_{Δ;0}> ν_{min}}, displayed as the shaded area in Figure 1c. Now, assume that in breach of Proposition 4, there is no (segment of a) separation graph ν_{Σ;0}in_{Δ;0}. Then, examination of equations (16a) and (16b) on yields Since ∂_{ν}Δ = −*s*_{+i}(∂_{ν}ϕ_{+i±}∂_{ν}ϕ_{−1}) at (χ_{c}, ν_{c}), we infer from Lemma I:1 and equations (I:25b) and (I:29) that sgn(∂_{ν}Δ) = −sgn(s_{+i}) for {χ = χ_{c}, ν ↑ ν_{c}}, i.e., tanϕ_{+i}↓ 0 as ν↑ν_{c}. Hence, equation (21) leads to tanϕ_{+i}> 0 on_{Δ;0}, which in view of equations (14a) and (14b) implies that a ρ_{3}> 0 exists such that 0 ≤ lim_{ν ↓ ν}_{min}tanϕ_{+i}tan2η < 1 for ρ > ρ_{3}, i.e., lim_{ν ↓ ν}_{min}Σ = 0 has two solutions χ_{±}∈ [0, π/2], which satisfy*s*_{2χ}_{±}= tanϕ_{+i}tan 2η and*c*_{2χ}_{±}= ±(1 − s_{2η}^{2}/*c*_{+i}^{2})^{−1/2}/*c*_{2η}. Having assumed violation of Proposition 4, the corresponding graph ν_{Σ;0}must lie in between ν_{min}and ν_{Δ;0}(i.e., below_{Δ;0}). However, at the endpoints of ν_{Σ;0}, we have , which, in the absence of double roots, implies that Regarding equations (14a) and (14b) leads to the conclusion that a ρ_{4}> max(ρ_{2},ρ_{3}) exists, such that η is small enough to ensure that c_{+i}Δ > 0 for ρ_{1}> ρ_{4}, χ = χ_{±}and ν ↓ ν_{min}. This means that given equation (21), there must be (segments of) a separation graph in_{Δ;0}.**Proof of Proposition 5 of Lemma 2**Recall that there is a characteristic root in between consecutive zeroes of Σ. Tracking the graphs between boundary points, it is concluded on account of Propositions 1–4 of Lemma 2 that Proposition 5 is valid.

[12] Next, assume that there is a double characteristic root at (χ_{d}, ν_{d}), or, equivalently *D* = *S* = 0 at (χ_{d}, ν_{d}). Let **u**_{χ} and **u**_{ν} denote unit vectors that point in the positive χ- and ν- directions, respectively. On account of Lemmas I:1 and 1 and equations (I:29) and (16b), we have (−1)^{ℓ+}∂_{ν}Σ = *c*_{+e}^{−1}(∂_{ν}ϕ_{+i} − ∂_{ν}ϕ_{+e}) > 0. Hence, the vector that is tangent to ν_{Σ} at (χ_{d}, ν_{d}) and has a positive χ- component, is given by

Obviously, ∂_{ν}Δ = 0 at (χ_{d}, ν_{d}). It may be shown that ∂_{χ}Δ = 0 at (χ_{d}, ν_{d}) as well, implying that for a local analysis about a double root, one should investigate

where _{d} denotes the Hessian. For *N* > 0, the number of characteristic roots above or below a point of ν_{Σ} changes across χ = χ_{d} (cf. Figure 2a). This would impede efforts to accommodate for double roots. Instead, if *N* < 0 were to hold universally (cf. Figure 2c), and if one assigns one of the double roots to the region above ν_{Σ}, while assigning the other root to the region below that graph, then the double root generalization is elementary. If *N* = 0, further analysis involving higher-order derivatives of Σ and Δ would be demanded (cf. Figure 2b). The calculation of *N* requires diligence, eventually resulting in *N* = −16*c*_{+e}^{2}*P*_{0}*P*_{1}, in which

where *R*_{i} = −(sin2ϕ_{2i})/(sin2ϕ_{1i}) and (…)^{1/2} > 0. On account of c_{+e} > 0, Lemma I:1 and both energy-based angular inequalities in equation (I:29), we infer that *N* < 0.

## 4. Final Notes

[13] Since energy is contained in volumes and power flows through areas, it is natural to commence the derivation of a complex-power-flow variational scheme for bent waveguides, by regarding a waveguide section rather than a cross section. In the construction of the proof for the mode-counting and mode-bracketing theorems, it was necessary to regard a class of waveguide problems with a continuous transition from PEC to PMC boundary conditions, via the situation pertaining to the actual exterior medium. As spin-off, such a “sweep” might prove useful in the design of a fiber cladding.

## Acknowledgments

[14] The author thankfully acknowledges the fellowship grant from the Royal Netherlands Academy of Arts and Sciences. The author would like to thank Martijn van Beurden for many stimulating discussions.