## Rigorous sufficient conditions for index-guided modes in microstructured dielectric waveguides

Optics Express, Vol. 16, Issue 13, pp. 9261-9275 (2008)

http://dx.doi.org/10.1364/OE.16.009261

Acrobat PDF (177 KB)

### Abstract

We derive a sufficient condition for the existence of indexguided modes in a very general class of dielectric waveguides, including photonic-crystal fibers (arbitrary periodic claddings, such as “holey fibers”), anisotropic materials, and waveguides with periodicity along the propagation direction. This condition provides a rigorous guarantee of cutoff-free index-guided modes in any such structure where the core is formed by increasing the index of refraction (e.g. removing a hole). It also provides a weaker guarantee of guidance in cases where the refractive index is increased “on average” (precisely defined). The proof is based on a simple variational method, inspired by analogous proofs of localization for two-dimensional attractive potentials in quantum mechanics.

© 2008 Optical Society of America

## 1. Introduction

*index guiding*(or “total internal reflection”), in which a higher-index

*core*is surrounded by a lower-index

*cladding*

*ε*(

_{c}*ε*is the relative permittivity, the square of the refractive index in isotropic non-magnetic materials). A schematic of several such dielectric waveguides is shown in Fig. 1. In particular, we suppose that the waveguide is described by a dielectric function

*ε*(

*x*,

*y*,

*z*)=

*ε*(

_{c}*x*,

*y*,

*z*)+Δ

*ε*(

*x*,

*y*,

*z*) such that:

*ε*,

*ε*, and Δ

_{c}*ε*are periodic in

*z*(the propagation direction) with period

*a*(

*a*→ 0 for the common case of a waveguide with a constant cross-section); that the cladding dielectric function

*ε*is periodic in

_{c}*xy*(e.g. in a photonic-crystal fiber), with a homogeneous cladding (e.g. in a conventional fiber) as a special case; and the core is formed by a change Δ

*ε*in some region of the

*xy*plane, sufficiently localized that ∫|1/

*ε*-1/

*ε*|<∞ (integrated over the

_{c}*xy*plane and the unit cell in

*z*). This includes a very wide variety of dielectric waveguides, from conventional fibers [Fig. 1(a)] to photonic-crystal “holey” fibers [Fig. 1(b)] to waveguides with a periodic “grating” along the propagation direction [Fig. 1(c)] such as fiber-Bragg gratings and other periodic waveguides. We exclude metallic structures (i.e, we require

*ε*>0) and make the approximation of lossless materials (real

*ε*). We allow anisotropic materials. The case of substrates (e.g. for strip waveguides in integrated optics [12–14]) is considered in Sec. 5. We also consider only non-magnetic materials (relative permeability

*μ*=1), although a future extension to magnetic materials should be straightforward. Intuitively, if the refractive index is increased in the core, i.e. if Δ

*ε*is non-negative, then we might expect to obtain exponentially localized index-guided modes, and this expectation is borne out by innumerable numerical calculations, even in complicated geometries like holey fibers [2–5

2. P. Russell, “Photonic crystal fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

*ε*is strictly non-negative, is there a guided mode at

*every*wavelength, or is there the possibility of e.g. a long-wavelength cutoff (as was initially suggested in holey fibers [15

15. B. T. Kuhlmey, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre, “Microstructured optical fibers: where’s the edge?” Opt. Express **10**, 1285–1290 (2002). [PubMed]

16. S. Wilcox, L. Botten, C. M. de Sterke, B. Kuhlmey, R. McPhedran, D. Fussell, and S. Tomljenovic-Hanic, “Long wavelength behavior of the fundamental mode in microstructured optical fibers,” Opt. Express **13** (2005). [CrossRef] [PubMed]

*ε*is

*not*strictly non-negative, i.e. the core consists of partly increased and partly decreased index; it is known in such cases, e.g. in “W-profile fibers” [17

17. S. Kawakami and S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. **10**, 879–887 (1974). [CrossRef]

18. T. Okoshi and K. Oyamoda, “Single-polarization single-mode optical fibre with refractive-index pits on both sides of core,” Electron. Lett. **16**, 712–713 (80). [CrossRef]

9. A. Bamberger and A. S. Bonnet, “Mathematical analysis of the guided modes of an optical fiber,” SIAM J. Math. Anal. **21**, 1487–1510 (1990). [CrossRef]

*two*guided modes, and we will explore the consequences for single-polarization fibers in a subsequent paper. It turns out that all of these questions can be rigorously answered (in the sense of sufficient conditions for guidance) for the very general geometries considered in Fig. 1, without resorting to approximations or numerical computations.

*ε*, where the proof is much easier to follow. Then, in Sec. 4, we generalize the proof to arbitrary periodic claddings, such as for holey photonic-crystal fibers (with some algebraic details left to the appendix). In Sec. 5, we discuss a few contexts that go beyond the initial assumptions of our theorem: substrates, material dispersion, and finite-size effects. Finally, we offer some concluding remarks in Sec. 6 discussing future directions.

_{c}## 2. Statement of the theorem

**H**(

*x*,

*y*,

*z*)

*e*

^{iβz-iωt}(via Bloch’s theorem thanks to the periodicity in

*z*) [5], where

*ω*is the frequency,

*β*is the propagation constant, and the magnetic-field envelope

**H**(

*x*,

*y*,

*z*) is periodic in

*z*with period

*a*(or is independent of

*z*in the common case of a constant cross section,

*a*→ 0). A plot of

*ω*versus

*β*for all eigenmodes is the “dispersion relation” of the waveguide, one example of which is shown in Fig. 2. In the absence of the core (i.e. if Δ

*ε*=0), the (non-localized) modes propagating in the infinite cladding form the “light cone” of the structure [2–5

2. P. Russell, “Photonic crystal fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

*β*there is a fundamental (minimum-

*ω*) space-filling mode at a frequency

*ω*(

_{c}*β*) with a corresponding field envelope

**H**

_{c}[2–5

2. P. Russell, “Photonic crystal fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

*ω*(

_{c}*β*), the only solutions in the cladding are evanescent modes that decay exponentially in the transverse directions [2–5

2. P. Russell, “Photonic crystal fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

*ε*≠0), any new solutions with

*ω*<

*ω*must be guided modes, since they are exponentially decaying in the cladding far from the core: these are the index-guided modes (if any). Such guided modes are shown as lines below the light cone in Fig. 2: in this case, both a lowest-lying (“fundamental”) guided mode with no low-frequency cutoff (although it approaches the light line asymptotically as

_{c}*ω*→ 0) and a higher-order guided mode with a low-frequency cutoff are visible. Since a mode is guided if

*ω*<

*ω*, we will prove the existence of a guided mode by showing that

_{c}*ω*has an upper bound <

*ω*, using the variational (min–max) theorem for Hermitian eigenproblems [5].

_{c}2. P. Russell, “Photonic crystal fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

25. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous materials,” Phys. Rev. E **71**, 036,617 (2005). [CrossRef]

*β*: a guided mode

*must*exist whenever

*xy*and one period in

*z*and

**D**

_{c}is the displacement field of the cladding’s fundamental mode. From this, we can immediately obtain a number of useful special cases:

- There must be a cutoff-free guided mode if Δ
*ε*≥0 everywhere (i.e., if we only increase the index to make the core). - For a homogeneous cladding (and isotropic media), there must be a cutoff-free guided mode if ∫(1/
*ε*-1/*ε*)<0 (similar to the earlier theorem of Ref. 9_{c}, but generalized to include waveguides periodic in9. A. Bamberger and A. S. Bonnet, “Mathematical analysis of the guided modes of an optical fiber,” SIAM J. Math. Anal.

**21**, 1487–1510 (1990). [CrossRef]*z*and/or cores Δ*ε*that do not have compact support). - More generally, a guided mode has no long-wavelength cutoff if eq. (1) is satisfied for the quasi-static (
*ω*→ 0,*β*→ 0) limit of**D**_{c}.

*two*guided modes (or, equivalently, a necessary condition for single-polarization guidance), when the cladding fundamental mode is doubly degenerate. We explore this generalization, analogous to a result in Ref. 9

**21**, 1487–1510 (1990). [CrossRef]

## 3. Waveguides with a homogeneous cladding

*ε*(

*x*,

*y*) to be of the form:

*ε*is an an arbitrary change in

*ε*that forms the core of the waveguide. For convenience, we define a new function Δ by:

*ε*are that

*ε*be real and positive and that ∫|Δ|

*dxdy*be finite, as discussed above. Now, we wish to show that there must always be a (cutoff-free) guided mode as long as Δ

*ε*is “mostly positive,” in the sense that:

*ω*or

*β*, the existence of guided modes will hold at all frequencies (cutoff-free).

*ω*

_{min}. In particular, at each

*β*, the eigenmodes

**H**(

*x*,

*y*)

*e*

^{iβz-iωt}satisfy a Hermitian eigenproblem [5]:

_{β}. In addition to the eigenproblem, there is also the “transversality” constraint [5, 24]:

_{β}, the variational theorem immediately follows [5]:

*any*“trial function”

**H**(

*x*,

*y*), not necessarily an eigenfunction, into the right-hand-side (the “Rayleigh quotient”), as long as

**H**is “transverse” [satisfies Equation (7)]. [Technically, we must also restrict ourselves to trial functions where the integrals in Equation (8) are defined, i.e.the trial functions must be in the appropriate Sobolev space

*H*(∇

_{β}×).] Conversely, if Equation (7) is not satisfied, it is easy to make the numerator of the right-hand-side (which involved ∇

_{β}×

**H**)

*zero*, e.g. by setting

**H**=∇

*φ*+

*iβφ*

**ẑ**for any

*φ*(

*x*,

*y*), so transversality of the trial function is critically important to obtaining a true upper bound.

*ε*, the light line is simply

_{c}*ω*

^{2}

_{c}/

*c*

^{2}=

*β*

^{2}/

*ε*, and so the condition for guided modes becomes:

_{c}*ε*>0 and/or for arbitrarily low frequencies.

*V*(

*x*,

*y*) in two dimensions with ∫|

*V*|<∞, one wishes to show that ∫

*V*<0 (attractive) implies the existence of a bound state: an eigenfunction of the Schrödinger operator -∇

^{2}+

*V*with eigenvalue (energy)<0. Again, this is a Hermitian eigenproblem and there is a variational theorem [31], so one merely needs to find some trial wavefunction

*ψ*for which the Rayleigh quotient is negative in order to obtain a bound state. In one dimension, finding such a trial function is simple—for example, an exponentially decaying function

*e*

^{-α|x|}(or a Gaussian

*α*—and the proof is sometimes assigned as an undergraduate homework problem [32]. In two dimensions, however, finding a trial function is more difficult—in fact, no function of the form

*f*(

*αr*) (where

*r*is the radius

*V*) [11

11. K. Yang and M. de Llano, “Simple variational proof that any two-dimensional potential well supports at least one bound state,” Am. J. Phys. **57**, 85–86 (1989). [CrossRef]

27. B. Simon, “The bound state of weakly coupled Schrödinger operators in one and two dimensions,” Ann. Phys. **97**, 279–288 (1976). [CrossRef]

**21**, 1487–1510 (1990). [CrossRef]

11. K. Yang and M. de Llano, “Simple variational proof that any two-dimensional potential well supports at least one bound state,” Am. J. Phys. **57**, 85–86 (1989). [CrossRef]

11. K. Yang and M. de Llano, “Simple variational proof that any two-dimensional potential well supports at least one bound state,” Am. J. Phys. **57**, 85–86 (1989). [CrossRef]

*r*,

*ϕ*) coordinates:

*α*>0, and (

*rγ*)′ is the derivative with respect to

*r*. Clearly,

**H**in eq. (10) reduces to an

**x̂**-polarized plane wave propagating in the

**ẑ**direction as

*α*→ 0 (and hence

*γ*→ 1). This is a key property of the trial function: in the limit of no localization (

*α*=0, Δ

*ε*=0) it should recover a fundamental (lowest-

*ω*) solution of the infinite cladding. Also, by construction, it satisfies the transversality condition (7) (which is why we chose this particular form). We chose

*γ*slightly differently from Ref. 11

**57**, 85–86 (1989). [CrossRef]

*α*→ 0). For future reference, the first two

*r*derivatives of

*γ*are:

*γ*in Fig. 3.

*α*, satisfies the variational condition (9). In practice, some care is required in appropriately bounding each of the integrals and in taking the limits in the proper order, and we review this process below.

**H**into the left-hand side of eq. (9):

*α*→ 0, thus satisfying the condition for the existence of bound modes. We first examine the second term of eq. (14):

*α*→ 0 limit and the integral in this case, thanks to Lebesgue’s Dominated Convergence Theorem (LDCT) [33]: whenever the absolute value of the integrand is bounded above (for sufficiently small

*α*) by an

*α*-independent function with a finite integral, LDCT guarantees that the

*α*→ 0 limit can be interchanged with the integral. In particular, the absolute value of this integrand is bounded above by |Δ| multiplied by some constant (since |

**H**| is bounded by a constant: |

*γ*|≤1 and |

*rγ*′| is also easily seen to be bounded above for sufficiently small

*α*), and |Δ| has a finite integral by assumption. Since lim

_{α → 0}|

**H**|

^{2}=1, we obtain eq. (4), which is negative by assumption.

*α*→ 0, completing our proof. This term is proportional to (

*ε*

^{-1}

_{c}+Δ), but the Δ terms trivially go to zero by the same arguments as above: Δ allows the limit to be interchanged with the integration by LDCT, and as

*α*→ 0 the

*γ*′ and

*γ*″ terms go to zero. The remaining

*ε*

^{-1}

_{c}terms can be bounded above by a sequence of inequalities as follows:

*r*

^{2}with

*r*

^{2}+1. From the third to the fourth line, we made a change of variables

*t*

^{2}=

*r*

^{2}+1. Then, from the fourth to fifth line, we made another change of variable

*u*=

*t*

^{2α}, and bounded the integral above by changing the lower limit from

*u*=1 to

*u*=0. The final integral can be performed exactly and goes to zero, completing the proof.

## 4. General periodic claddings

*z*-invariant waveguides with a homogeneous cladding and isotropic materials (for example, conventional optical fibers). We now generalize the proof in three ways, by allowing:

- transverse periodicity in the cladding material (photonic-crystal fibers),
- a core and cladding that are periodic in
*z*with period*a*(*a*→ 0 for the*z*-invariant case), - anisotropic
*ε*and Δ_{c}*ε*materials (*ε*is a 3×3 positive-definite Hermitian matrix).

*ε*(

_{c}*x*,

*y*,

*z*)=

*ε*(

_{c}*x*,

*y*,

*z*+

*a*) is

*z*-periodic and also periodic in the

*xy*plane (with an arbitrary unit cell and lattice), and the core dielectric tensor change Δ

*ε*(

*x*,

*y*,

*z*)=Δ

*ε*(

*x*,

*y*,

*z*+

*a*) is

*z*-periodic with the

*same*period

*a*. Both

*ε*and the total

_{c}*ε*must be positive-definite Hermitian tensors. As defined in eq. (3), we denote by Δ the change in the inverse dielectric tensor. Similar to the isotropic case, we require that ∫|Δ

_{ij}| be finite for integration over the

*xy*plane and one period of

*z*, for every tensor component Δ

_{ij}. We also require that the components of

*ε*

^{-1}

_{c}be bounded above.

*always*exists, for all

*β*, under the condition of eq. (4), by showing that the variational upper bound on its frequency lies below the light line. In the case of a periodic cladding, the light line is the dispersion relation of the fundamental space-filling mode of the cladding, which corresponds to the lowest-frequency mode at each given propagation constant

*β*[2–5

2. P. Russell, “Photonic crystal fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

2. P. Russell, “Photonic crystal fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

*only*consider index-guided modes, which are guided because they lie below the light line. We will follow the same general procedure as in the previous section to derive the sufficient condition [eq. (1)] to guarantee the existence of guided modes. The homogeneous-cladding case is then a special case of this more general theorem, recovering eq. (4) (but generalizing it to

*z*-periodic cores), where in that case the cladding fundamental mode

**D**

_{c}is a constant and can be pulled out of the integral. The case of a

*z*-homogeneous fiber is just the special case

*a*→ 0, eliminating the

*z*integral in eq. (1).

*β*, the eigenmodes

**H**(

*x*,

*y*,

*z*)

*e*

^{iβz-iωt}satisfy the same Hermitian eigenproblem (5) and transversality constraint (7) as before. We have a similar variational theorem to eq. (8) [5], except that, in the case of

*z*-periodicity, we now integrate over one period in

*z*as well as over

*x*and

*y*.

**H**such that this upper bound, called the “Rayleigh quotient” for

**H**, is below the light line

*ω*(

_{c}*β*)

^{2}/

*c*

^{2}. The corresponding condition on

**H**can be written [similar to eq. (9)]:

**57**, 85–86 (1989). [CrossRef]

*α*→ 0 (weak guidance) limit approaches the cladding fundamental mode

**H**

_{c}. As before, the trial function must be transverse (∇

_{β}·

**H**=0), which motivated us to write the trial function in terms of the corresponding vector potential. We denote by

**A**

_{c}the vector potential corresponding to the cladding fundamental mode

**H**

_{c}=∇

_{β}×

**A**

_{c}. In terms of

**A**

_{c}and

*γ*, our trial function is then:

**A**

_{c}to be Bloch-periodic (like

**H**

_{c}, since

**A**

_{c}also satisifies a periodic Hermitian generalized eigenproblem and hence Bloch’s theorem applies). (Alternatively, it is straightforward to show that the Coulomb gauge choice, ∇

_{β}·

**A**

_{c}=0, gives a Bloch-periodic

**A**

_{c}, by explicitly constructing the Fourier-series components of

**A**

_{c}in terms of those of

**H**

_{c}.) In contrast, our previous homogeneous-cladding trial function [eq. (10)] corresponds to a different gauge choice with an unbounded vector potential

- terms that contain Δ=
*ε*^{-1}-*ε*^{-1}_{c}, - terms that cancel due to the eigenequation (5),
- terms that have one first derivative of
*γ*, - terms that have (
*γ*′)^{2}, - terms that have
*γ*′*γ*″ or (*γ*″)^{2}.

*α*→ 0 limit; we distinguish them because category (v) turns out to be easier to analyze. There are no terms with

*γ*″ alone, as these can be integrated by parts to obtain category (iii) and (iv) terms. In the appendix, we provide an exhaustive listing of all the terms and how they combine as described above. In this section, we only outline the basic structure of this algebraic process, and explain why the category (iv) and (v) terms vanish as

*α*→ 0.

_{ij}| multiplied by some

*α*-independent constants, and |Δ

_{ij}| has a finite integral by assumption. (In particular, the

**A**

_{c}fundamental mode and its curls are bounded functions, being Bloch-periodic, and

*γ*and its first two derivatives are bounded for sufficiently small

*α*.) The result is precisely the left-hand side of eq. (1), which is negative by assumption.

**H***·(∇

_{β}×ε

^{-1}

_{c}γ∇

_{β}×

**H**

_{c}) (which comes from the term where the right-most curl falls on

**H**

_{c}rather than

*γ*) below:

**H**in terms of

**H**

_{c}. The first term of the last line above cancels

*iω*

_{c}∫

*γ*∇

*γ*·

**E**

_{c}×

**H***

_{c}, both of which will be exactly cancelled as described in the appendix, as well as some category (iv) and (v) terms.

*γ*′)

^{2}multiplied by some bounded function (a product of the various Bloch-periodic fields as well as the bounded

*ε*

^{-1}

_{c}). This integrand can then be bounded above by replacing the bounded function with the supremum

*B*of its magnitude, at which point the integral is bounded above by 2

*πB*∫

^{∞}

_{0}(

*γ*′)

^{2}

*rdr*. However, such integrands were among the terms we already analyzed in the homogeneous-cladding case, in eq. (16), and we explicitly showed that such integrals go to zero as

*α*→ 0.

*α*→ 0, similar to eq. (16), but a simpler proof of the same fact can be constructed via the LDCT condition. In particular, similar to the previous paragraph, after replacing bounded functions with their suprema we are left with cylindrical-coordinate integrands of the form

*γ*′

*γ*″

*r*and (

*γ*″)

^{2}

*r*. Both of these integrands, however, are bounded above by an

*α*-independent function with a finite integral, and hence LDCT allows us to put the

*α*→ 0 limit inside the integral and set the integrands to zero. Specifically, by inspection of eqs. (12) and (13), |

*γ*′

*γ*″|

*r*<4

*r*

^{2}(1+2+2)/(

*r*

^{2}+1)

^{2-δ}and (

*γ*″)

^{2}

*r*<4

*r*(1+2+2)

^{2}/(

*r*

^{2}+1)

^{2-δ}for

*α*<

*δ*/4, and both of these upper bounds have finite integrals, if we take

*δ*to be some number <1/2, since they decay faster than 1/

*r*.

## 5. Substrates, dispersive materials, and finite-size effects

*ω*-dependent) materials, and finite-size claddings.

*ε*[12, 14

14. C.-L. Chen, *Foundations for Guided-Wave Optics* (Wiley, 2006). [CrossRef]

*ε*depends on

*ω*but has negligible absorption (so that guided modes are still well-defined). For a given

*ω*, we can construct a frequency-independent

*ε*structure matching the actual

*ε*at that

*ω*, and apply our theorem to determine whether there are guided modes at

*ω*. The simplest case is when Δ

*ε*≥0 for all

*ω*, in which case we must still obtain cutoff-free guided modes. The theorem becomes more subtle to apply when Δ

*ε*<0 in some regions, because not only must one perform the integral of eq. (1) to determine the existence of guided modes, but the condition (1) is for a fixed

*β*while the integrand is for a given frequency, and the frequency of the guided mode is unknown a

*priori*.

*exponentially*with the wavelength. In quantum mechanics (scalar waves) with a potential well of depth

*V*, the decay length of the bound mode increases as

*e*

^{C/V}when

*V*→ 0, for some constant

*C*[11

**57**, 85–86 (1989). [CrossRef]

27. B. Simon, “The bound state of weakly coupled Schrödinger operators in one and two dimensions,” Ann. Phys. **97**, 279–288 (1976). [CrossRef]

*description of the structure becomes applicable [25*ε ˜

25. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous materials,” Phys. Rev. E **71**, 036,617 (2005). [CrossRef]

*ω*

^{2}Δ

*[34], and hence the quantum analysis should apply. Thus, by this informal argument, we would expect the modal diameter to expand proportional to*ε ˜

*C*(where

*λ*=2

*πc*/

*ω*is the vacuum wavelength), but a more explicit proof would be desirable.

## 6. Concluding remarks

*z*direction and in the transverse plane. The results are a generalization of previous results on the existence of such modes in fibers with a homogeneous cladding index. Our theorem allows one to understand the guidance in many very complicated structures analytically, and enables one to rigorously guarantee guided modes in many structures (especially those where Δ

*ε*≥0 everywhere) by inspection. There remain a number of interesting questions for future study, however, some of which we outline below.

**21**, 1487–1510 (1990). [CrossRef]

*ε*>0 in some region) rather than <0 as in our condition. Although the =0 case seems unlikely to be experimentally or numerically significant, we suspect that a similar generalization should be possible for our theorem (reweighting the integrand to make it negative and then taking a limit as in Ref. 9

**21**, 1487–1510 (1990). [CrossRef]

*ω*rather than at a fixed

*β*, although we are not sure whether this is possible. Third, we would prefer a more rigorous version of the argument, in Sec. 5, that the modal diameter should asymptotically increase exponentially with the square of the wavelength. Fourth, it might be interesting to consider the case of “Bragg fiber” geometries consisting of “periodic” sequences of concentric layers [37

37. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. **68**, 1196–1201 (1978). [CrossRef]

*two*guided modes in many cases where the cladding fundamental mode is doubly degenerate, and we are currently preparing another manuscript describing this result along with conditions for truly single-mode (“single-polarization”) waveguides.

## Appendix: All Rayleigh-quotient terms

*ε*

_{c}(

**r**). More specifically, the only non-trivial term to analyze is the Δ-free part of the left-most integral in eq. (19):

*ε*

^{-1}

_{c}∇

_{β}×

**H**

_{c}(Ampère’s law), we have:

**H**, integrating some of the ∇

_{β}× operators by parts, and distributing the derivatives of

*γ*

**H**

_{c}by the product rule. The second step is obtained by using Ampère’s law again, combined with integrations by parts and the product rule; “c.c.” stands for the complex conjugate of the preceding expression. Continuing, we obtain:

_{β}×

**E**

_{c}with

*γ*, corresponding to category (iv) and (v) terms, which we proved to vanish in Sec. 4.

*γ*→ 0 at infinity. For the second term, the integrand is the divergence of the time-average Poynting vector ℜ{

**E**

_{c}×

**H***

_{c}}, which equals the time-average rate of change of the energy density [34], which is identically zero for any lossless eigenmode (such as the cladding fundamental mode).

## Acknowledgments

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7. | C. Elachi, “Waves in active and passive periodic structures: A review,” Proc. IEEE |

8. | S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B |

9. | A. Bamberger and A. S. Bonnet, “Mathematical analysis of the guided modes of an optical fiber,” SIAM J. Math. Anal. |

10. | H. P. Urbach, “Analysis of the domain integral operator for anisotropic dielectric waveguides,” Journal on Mathematical Analysis |

11. | K. Yang and M. de Llano, “Simple variational proof that any two-dimensional potential well supports at least one bound state,” Am. J. Phys. |

12. | R. G. Hunsperger, |

13. | B. E. A. Saleh and M. C. Teich, |

14. | C.-L. Chen, |

15. | B. T. Kuhlmey, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre, “Microstructured optical fibers: where’s the edge?” Opt. Express |

16. | S. Wilcox, L. Botten, C. M. de Sterke, B. Kuhlmey, R. McPhedran, D. Fussell, and S. Tomljenovic-Hanic, “Long wavelength behavior of the fundamental mode in microstructured optical fibers,” Opt. Express |

17. | S. Kawakami and S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. |

18. | T. Okoshi and K. Oyamoda, “Single-polarization single-mode optical fibre with refractive-index pits on both sides of core,” Electron. Lett. |

19. | W. Eickhoff, “Stress-induced single-polarization single-mode fiber,” Opt. Lett. |

20. | J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Macchesney, and R. E. Howard, “A single-polarization fiber,” IEEE J. Lightwave Technol. |

21. | M. J. Messerly, J. R. Onstott, and R. C. Mikkelson, “A broad-band single polarization optical fiber,” IEEE J. Lightwave Technol. |

22. | H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Tech. Lett. |

23. | M.-J. Li, X. Chen, D. A. Nolan, G. E. Berkey, J. Wang, W. A. Wood, and L. A. Zenteno, “High bandwidth single polarization fiber with elliptical central air hole,” IEEE J. Lightwave Technol. |

24. | P. Kuchment, “The Mathematics of Photonic Crystals,” in |

25. | D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous materials,” Phys. Rev. E |

26. | P. Kuchment and B. Ong, “On guided waves in photonic crystal waveguides,” in |

27. | B. Simon, “The bound state of weakly coupled Schrödinger operators in one and two dimensions,” Ann. Phys. |

28. | L. D. Landau and E. M. Lifshitz, |

29. | H. Picq, “Détermination et calcul numérique de la première valeur propre d’opérateurs de Schrödinger dans le plan,” Ph.D. thesis, Université de Nice, Nice, France (1982). |

30. | E. N. Economou, |

31. | C. Cohen-Tannoudji, B. Din, and F. Laloë, |

32. | D. ter Haar, |

33. | E. Hewitt and K. Stromberg, |

34. | J. D. Jackson, |

35. | J. A. Kong, |

36. | S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B |

37. | P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. |

**OCIS Codes**

(060.2310) Fiber optics and optical communications : Fiber optics

(130.2790) Integrated optics : Guided waves

(060.4005) Fiber optics and optical communications : Microstructured fibers

(060.5295) Fiber optics and optical communications : Photonic crystal fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: March 20, 2008

Revised Manuscript: April 17, 2008

Manuscript Accepted: April 18, 2008

Published: June 9, 2008

**Citation**

Karen K. Lee, Yehuda Avniel, and Steven G. Johnson, "Rigorous sufficient conditions for index-guided modes in microstructured dielectric waveguides," Opt. Express **16**, 9261-9275 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-13-9261

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### References

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- S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, "Guided and defect modes in periodic dielectric waveguides," J. Opt. Soc. Am. B 12, 1267-1272 (1995). [CrossRef]
- A. Bamberger and A. S. Bonnet, "Mathematical analysis of the guided modes of an optical fiber," SIAM J. Math. Anal. 21, 1487-1510 (1990). [CrossRef]
- H. P. Urbach, "Analysis of the domain integral operator for anisotropic dielectric waveguides," J. Math. Anal. 27, (1996).
- K. Yang and M. de Llano, "Simple variational proof that any two-dimensional potential well supports at least one bound state," Am. J. Phys. 57, 85-86 (1989). [CrossRef]
- R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, 1982).
- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991). [CrossRef]
- C.-L. Chen, Foundations for Guided-Wave Optics (Wiley, 2006). [CrossRef]
- B. T. Kuhlmey, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre, "Microstructured optical fibers: where???s the edge?" Opt. Express 10, 1285-1290 (2002). [PubMed]
- S. Wilcox, L. Botten, C. M. de Sterke, B. Kuhlmey, R. McPhedran, D. Fussell, and S. Tomljenovic-Hanic, "Long wavelength behavior of the fundamental mode in microstructured optical fibers," Opt. Express 13 (2005). [CrossRef] [PubMed]
- S. Kawakami and S. Nishida, "Characteristics of a doubly clad optical fiber with a low-index inner cladding," IEEE J. Quantum Electron. 10, 879-887 (1974). [CrossRef]
- T. Okoshi and K. Oyamoda, "Single-polarization single-mode optical fibre with refractive-index pits on both sides of core," Electron. Lett. 16, 712-713 (80). [CrossRef]
- W. Eickhoff, "Stress-induced single-polarization single-mode fiber," Opt. Lett. 7 (1982). [CrossRef] [PubMed]
- J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Macchesney, and R. E. Howard, "A single-polarization fiber," IEEE J. Lightwave Technol. 1, 370-374 (1983). [CrossRef]
- M. J. Messerly, J. R. Onstott, and R. C. Mikkelson, "A broad-band single polarization optical fiber," IEEE J. Lightwave Technol. 9, 817-820 (1991). [CrossRef]
- H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, "Absolutely single polarization photonic crystal fiber," IEEE Photon. Tech. Lett. 16, 182-184 (2004). [CrossRef]
- M.-J. Li, X. Chen, D. A. Nolan, G. E. Berkey, J. Wang,W. A. Wood, and L. A. Zenteno, "High bandwidth single polarization fiber with elliptical central air hole," IEEE J. Lightwave Technol. 23, 3454-3460 (2005). [CrossRef]
- P. Kuchment, "The Mathematics of Photonic Crystals," in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, and W. Masters, eds., Frontiers in Applied Mathematics, pp. 207-272 (SIAM, Philadelphia, 2001).
- D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous materials," Phys. Rev. E 71, 036,617 (2005). [CrossRef]
- P. Kuchment and B. Ong, "On guided waves in photonic crystal waveguides," in Waves in Periodic and Random Media, vol. 339 of Contemporary Mathematics, pp. 105-115 (AMS, Providence, RI, 2003).
- B. Simon, "The bound state of weakly coupled Schrödinger operators in one and two dimensions," Ann. Phys. 97, 279-288 (1976). [CrossRef]
- L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Addison-Wesley, 1977).
- H. Picq, "Détermination et calcul numérique de la premiére valeur propre d???opérateurs de Schrödinger dans le plan," Ph.D. thesis, Université de Nice, Nice, France (1982).
- E. N. Economou, Green???s functions in quantum physics (Springer, 2006).
- C. Cohen-Tannoudji, B. Din, and F. Lalo¨e, Quantum Mechanics (Hermann, Paris, 1977).
- D. ter Haar, Selected Problems in Quantum Mechanics (Academic Press, New York, 1964).
- E. Hewitt and K. Stromberg, Real and Abstract Analysis (Springer, 1965).
- J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).
- J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1975).
- S. G. Johnson, M. L. Povinelli, M. Solja??i??, A. Karalis, S. Jacobs, and J. D. Joannopoulos, "Roughness losses and volume-current methods in photonic-crystal waveguides," Appl. Phys. B 81, 283-293 (2005). [CrossRef]
- P. Yeh, A. Yariv, and E. Marom, "Theory of Bragg fiber," J. Opt. Soc. Am. 68, 1196-1201 (1978). [CrossRef]

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