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.

 

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 {x1, x2, x3}. By choice, the coordinates x1 and x3 are associated with the radial direction and the direction of propagation, respectively. The covariant base vectors ci satisfy 〈ci, cj〉 = gij. Lowercase Roman and Greek indices take the values {1, 2, 3} and {1, 2}, respectively. Partial differentiation with respect to xp is denoted as ∂p. The summation convention applies. The antisymmetric tensor of rank three is defined through ηmpk = g−1/2ϵmpk, where g = det(gij) and ϵmpk denotes the Levi–Civita symbol. The unit vector along the direction of propagation is u = c3equation image = c3equation image.

[5] 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 (cf. Figure I:1). In equation image 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 x3 via the factor exp(jωtjk3x3), where k3 represents a trial, complex propagation coefficient. The fields comprise source-free solutions of Maxwell's equations on equation image, e.g., for state C we have

equation image
equation image

with ∂3 → −jk3 [Synge and Schild, 1956, p. 227]. We ensure that for all trial states, the components of equation image transverse to equation image are continuous across equation image while the components of equation image transverse to equation image are left free to jump across equation image by an amount [equation image]ie. Let us multiply equations (1a) and (1b) by equation imagemD and equation imagekD, respectively, contract over the repeated indices, subtract the resulting equations, integrate over equation image and integrate by parts to eliminate derivatives of equation imagemC across equation image. This leads to the identity,

equation image

in which equation image 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 equation image, 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 equation image. Evaluation of successive approximations leads to

equation image

In case dA = equation imagedx1dx2 is independent of x3, the volume integral in equation (3) may be reduced to a surface integral over a cross-sectional surface equation image, 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 > νmin = equation image, as

equation image
equation image

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 > νmin. As a consequence, the number of zeroes of S in the arbitrary closed interval ν ∈ equation image = [ν<, ν>] with equation image < ν< < ν>, is given by

equation image

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

  1. Lemma 1 A root ν = νd is a double root of a characteristic equation for fixed m, if and only if equation image, with ℓψ,ℓ±equation image. In view of equations (4a) and (4b) this is equivalent to D = S = 0.
  2. Proof Let λ1 and λ2 denote the eigenvalues of the real symmetric 2 × 2 matrix equation image. Now, suppose that equation image is a double root of det equation image = 0 (with m fixed). Then, at equation image we either haves λ1 = λ2 = 0, implying that equation image, or, equation image is a double zero of either λ1 or λ2, implying that equation image = 0, where I is the corresponding eigenvector. Since equation image is symmetric positive definite, we have IT equation image > 0 and hence the latter option leads to a contradiction. In turn equation image is equivalent to equation image, with equation image. 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 εr and μr are proportional throughout, all TM0n propagation coefficients coincide with their TE0n counterparts. The combined properties of the impedance-angle formalism enable us to count the number of characteristic roots in equation image, according to two theorems,

  1. 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.
  2. 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
    in which
    equation image
  1. 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>>] contain at most one root each. Hence, there are no roots on [ν<, ν⌈σ<] if and only if D<)D⌈σ<) > 0, or equivalently N = 1; otherwise N = 0. Likewise, there are no roots on [ν⌊σ>, ν>] if and only if D⌊σ>)D>) > 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<>] is simply given by N+N = 0 if D<)D>) > 0, and N+N = 1, otherwise.

  1. 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, equation image, is kept fixed, while the angle χ, introduced via
    equation image
    may take any value. For propagating modes we have ν > νmin. Hence, the half-strip
    equation image
    is considered the domain of analysis. Occasionally, the open domain equation image or its closure equation image is regarded, where equation image and equation image are boundary segments. The limiting χ-values 0 and π/2 correspond to PEC and PMC boundary conditions at ρ = ρ1, 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, ψi} are independent of χ. Inspection of the expression for equation imagee, given by equation (I:12), shows that equation imagee cot χ is independent of χ as well. Hence, ψe is also independent of χ. Introducing the angle η via

equation image

we infer that

equation image

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

equation image

in terms of which we have

equation image
equation image

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

equation image
equation image
equation image

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

equation image

in which the respective normalized characteristic and separation functions,

equation image
equation image

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

  1. Lemma 2 There exists a ρ4 such that for ρ1 > ρ4 the following propositions hold:
    1. Precluding double characteristic roots, Theorem 1 holds for fibers with perfectly conducting electric (χ = 0) or magnetic (χ = π/2) walls.
    2. If separation graphs νΣ = νΣ1,χ), along which Σ = 0, exist on equation image, and if the endpoints are located on ∂equation imageχ, then the one that is nearest to ∂equation imageν is denoted as νΣ;1 (cf. Figures 1a and 1b). There may also be a single graph νΣ;0 with both endpoints on ∂equation imageν (cf. Figure 1b).
    3. If characteristic graphs (or graph segments) νΔ = νΔ1, χ), along which Δ = 0, exist on equation image and if they have at least one endpoint on ∂equation imageχ then the one(s) that is (are) closest to ∂equation imageν is (are) denoted as νΔ;1 (cf. Figures 1a and 1b). There may also be a single graph νΔ;0 with both endpoints on ∂equation imageν (cf. Figure 1b).
    4. If both νΔ;0 and νΔ;1 exist, then these graphs are separated by either νΣ;0 or νΣ;1.
    5. Precluding double characteristic roots, the separation and characteristic graphs are disjoint and induce a partition of equation image, 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.
Figure 1.

Possible locations of the characteristic and separation graphs indicated by solid anddashed lines, respectively.

[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.

  1. Proof of Proposition 1 of Lemma 2 In the PEC case, χ = 0, we have Δ = c+ic−i and Σ = s+i. Hence, precluding double characteristic roots, we have
    equation image
    equation image
    with ℓ0, ℓ1, ℓ2 ∈ Z, respectively. Let ν = ν1 be a root of Δ = 0, for which ϕ1i = ℓ1π and (ℓ2 − 1)π < ϕ2i < ℓ2π, i.e.,
    equation image
    On account of Lemma I:1 and equation (I:29), both ϕ1i and ϕ2i increase monotonically in between consecutive roots. Hence, at the next root, ν = ν2 > ν1, of Δ, we have either ℓ1π < ϕ1i2) < (ℓ1 + 1)π and ϕ2i2) = ℓ2π, or ϕ1i2) = (ℓ1 + 1)π and (ℓ2 − 1)π < ϕ2i2) < ℓ2π. This leads to
    equation image
    i.e., there exists a ν0 ∈ (ν1, ν2), such that Σ(ρ1, 0, ν0) = 0. We could equally well have started with ϕ2i1) = ℓ2π. 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.
  2. Proof of Proposition 2 of Lemma 2 Let us assume that separation graphs νΣ = νΣ1, χ) with ρ1 constant, exist on (a subset of) equation image, such that Σ(χ, νΣ1, χ)) = 0. In view of equation (16b), these graphs are symmetric, i.e., νΣ1, π/2 − χ) = νΣ1, χ). To determine the slope of a separation graph, it suffices to solve the system of equations {Σ = 0, dΣ/dχ = 0}, yielding dνΣ/dχ = −(∂χΣ/∂νΣ)∣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 ∂νΣ∣Σ=0 is bounded away from zero on equation image, and that
    equation image
    Invoking the implicit function theorem [Taylor, 1996, p. 13], we infer that if Σ = 0 at a single point of equation image\∂equation imageχ, the existence of a graph νΣ = νΣ1, χ) on (a subset of) equation image\∂equation imageχ is guaranteed. Approaching ∂equation imageχ the pertaining limits are regular, i.e., the existence extends to subsets of equation image. From equations (14a)(14c) we infer that for ρ1 > ρ2 = (equation imagene)−1(equation imagemequation imageequation imagem – 1equation image)1/2, there exists a unique ν = νη1) ∈ (νmin, ∞) such that cos2η|ν=νη = 0 and hence dνΣ/dχ = 0. Simple trigonometry reveals that for ν ≠ νη1) fixed, the equation Σ(ρ1,χ,ν) = 0 has either zero or two solutions for ρ1. In conclusion, for ρ1 > ρ2 and if νΣ1,0) < νη1), we have νΣ1,0) ≤ νΣ1,χ) ≤ νΣ1,equation image). The choice ρ1 > ρ2 serves to ensure that there are no separation graphs with endpoints on both ∂equation imageν and ∂equation imageχ, thus reducing the number of cases requiring treatment. So, for constant ρ1 > ρ2, every zero of the separation function on ∂equation imageχ is an endpoint of a unique graph νΣ. If such graphs with endpoints on ∂equation imageχ exist, then the one that is closest to ∂equation imageν is denoted as νΣ;1 (cf. Figure 1). There may be a graph νΣ;0 with both endpoints on ∂equation imageν (cf. Figure 1b). If that graph exists, it must be unique.
  3. Proof of Proposition 3 of Lemma 2 Next, let us assume that characteristic graphs νΔ = νΔ1, χ), with ρ1 constant, exist on (a subset of) equation image, such that Δ(χ, νΔ1, χ)) = 0. In general these graphs are asymmetric. To determine the slope of a graph νΔ, it suffices to solve the system of equations {Δ = 0, dΔ/dχ =0}, yielding dνΔ/dχ = −(∂χΔ/∂νΔ)∣Δ=0. Since we have precluded double characteristic roots, it is clear that Δ = 0 and ∂νΔ = 0 cannot be satisfied simultaneously, i.e., dνΔ/dχ remains bounded on equation image. Applying the implicit function theorem, we infer that if Δ = 0 is satisfied at a single point of the open domain equation image\∂equation imageχ, the existence of a graph νΔ = νΔ1,χ) on (a subset of) equation image\∂equation imageχ is guaranteed. Again, the existence extends to subsets of equation image. Note that unless νΔ is independent of χ, the equation Δ(ρ1, χ, ν) = 0 has at most two solutions for ρ1 and ν fixed. As a consequence, if a graph νΔ;0 with no endpoints on ∂equation imageχ exists, it is unique and characteristic graphs with endpoints on both ∂equation imageχ and ∂equation imageν 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 ρ1 across which the interior and exterior solutions are matched is irrelevant: the characteristic graphs are independent of ρ1. By contrast, the separation graphs move as ρ1 is varied. However, if for a particular value of ρ1 a separation graph νΣ is located in between two adjacent characteristic graphs νΔ, it must remain so for other values of ρ1, 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.

  1. Proof of Proposition 4 of Lemma 2 Regarding the graph νΔ;1, let (χc, νc) denote the endpoint of νΔ;1 that is closest to ∂equation imageν (cf. Figure 1c). We choose ρ1 > ρ2, and define equation imageΔ;0 = {(χ, ν)∣(χ, ν) ∈ equation image, νmin < ν < νc, and ν > νΔ;0(χ) if νΔ;0 > νmin}, displayed as the shaded area in Figure 1c. Now, assume that in breach of Proposition 4, there is no (segment of a) separation graph νΣ;0 in equation imageΔ;0. Then, examination of equations (16a) and (16b) on equation image yields
    equation image
    Since ∂νΔ = −s+i(∂νϕ+i±νϕ−1) at (χc, νc), we infer from Lemma I:1 and equations (I:25b) and (I:29) that sgn(∂νΔ) = −sgn(s+i) for {χ = χc, ν ↑ νc}, i.e., tanϕ+i ↓ 0 as ν↑νc. Hence, equation (21) leads to tanϕ+i > 0 on equation imageΔ;0, which in view of equations (14a) and (14b) implies that a ρ3 > 0 exists such that 0 ≤ limν ↓ νmin tanϕ+itan2η < 1 for ρ > ρ3, i.e., limν ↓ νminΣ = 0 has two solutions χ± ∈ [0, π/2], which satisfy s± = tanϕ+i tan 2η and c± = ±(1 − s2/c+i2)−1/2/c. Having assumed violation of Proposition 4, the corresponding graph νΣ;0 must lie in between νmin and νΔ;0 (i.e., below equation imageΔ;0). However, at the endpoints of νΣ;0, we have equation image, which, in the absence of double roots, implies that
    equation image
    Regarding equations (14a) and (14b) leads to the conclusion that a ρ4 > max(ρ23) exists, such that η is small enough to ensure that c+iΔ > 0 for ρ1 > ρ4, χ = χ± and ν ↓ νmin. This means that given equation (21), there must be (segments of) a separation graph in equation imageΔ;0.
  2. 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 (χd, νd), or, equivalently D = S = 0 at (χd, νd). Let uχ and uν 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+νΣ = c+e−1(∂νϕ+i − ∂νϕ+e) > 0. Hence, the vector that is tangent to νΣ at (χd, νd) and has a positive χ- component, is given by

equation image

Obviously, ∂νΔ = 0 at (χd, νd). It may be shown that ∂χΔ = 0 at (χd, νd) as well, implying that for a local analysis about a double root, one should investigate

equation image

where equation imaged denotes the Hessian. For N > 0, the number of characteristic roots above or below a point of νΣ changes across χ = χd (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 νΣ, 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 = −16c+e2P0P1, in which

equation image
equation image

where Ri = −(sin2ϕ2i)/(sin2ϕ1i) 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.

Figure 2.

Possible local behavior of the characteristic and separation graphs about a double characteristic root at (χd, νd).

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.

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