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

Transcription

1 Rigorous suffiient onditions for index-guided modes in mirostrutured dieletri waveguides Karen K. Y. Lee, Yehuda Avniel, and Steven G. Johnson Researh Laboratory of Eletronis, Massahusetts Institute of Tehnology, 77 Massahusetts Ave., Cambridge MA Abstrat: We derive a suffiient ondition for the existene of indexguided modes in a very general lass of dieletri waveguides, inluding photoni-rystal fibers (arbitrary periodi laddings, suh as holey fibers ), anisotropi materials, and waveguides with periodiity along the propagation diretion. This ondition provides a rigorous guarantee of utoff-free index-guided modes in any suh struture where the ore is formed by inreasing the index of refration (e.g. removing a hole). It also provides a weaker guarantee of guidane in ases where the refrative index is inreased on average (preisely defined). The proof is based on a simple variational method, inspired by analogous proofs of loalization for two-dimensional attrative potentials in quantum mehanis Optial Soiety of Ameria OCIS odes: ( ) Photoni rystal fibers; ( ) Guided waves; ( ) Fiber optis; ( ) Mirostrutured fibers. Referenes and links 1. A. W. Snyder and J. D. Love, Optial Waveguide Theory (Chapman and Hall, London, 1983). 2. P. Russell, Photoni rystal fibers, Siene 299, (2003). 3. A. Bjarklev, J. Broeng, and A. S. Bjarklev, Photoni Crystal Fibres (Springer, New York, 2003). 4. F. Zolla, G. Renversez, A. Niolet, B. Kuhlmey, S. Guenneau, and D. Felbaq, Foundations of Photoni Crystal Fibres (Imperial College Press, London, 2005). 5. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photoni Crystals: Molding the Flow of Light, 2nd ed. (Prineton Univ. Press, 2008). 6. R. Ramaswami and K. N. Sivarajan, Optial Networks: A Pratial Perspetive (Aademi Press, London, 1998). 7. C. Elahi, Waves in ative and passive periodi strutures: A review, Pro. IEEE 64, (1976). 8. S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, Guided and defet modes in periodi dieletri waveguides, J. Opt. So. Am. B 12, (1995). 9. A. Bamberger and A. S. Bonnet, Mathematial analysis of the guided modes of an optial fiber, SIAM J. Math. Anal. 21, (1990). 10. H. P. Urbah, Analysis of the domain integral operator for anisotropi dieletri waveguides, Journal on Mathematial Analysis 27 (1996). 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, (1989). 12. R. G. Hunsperger, Integrated Optis: Theory and Tehnology (Springer-Verlag, 1982). 13. B. E. A. Saleh and M. C. Teih, Fundamentals of Photonis (Wiley, 1991). 14. C.-L. Chen, Foundations for Guided-Wave Optis (Wiley, 2006). 15. B. T. Kuhlmey, R. C. MPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre, Mirostrutured optial fibers: where s the edge? Opt. Express 10, (2002). 16. S. Wilox, L. Botten, C. M. de Sterke, B. Kuhlmey, R. MPhedran, D. Fussell, and S. Tomljenovi-Hani, Long wavelength behavior of the fundamental mode in mirostrutured optial fibers, Opt. Express 13 (2005). # $15.00 USD Reeived 20 Mar 2008; revised 17 Apr 2008; aepted 18 Apr 2008; published 9 Jun 2008 (C) 2008 OSA 23 June 2008 / Vol. 16, No. 13 / OPTICS EXPRESS 9261

2 17. S. Kawakami and S. Nishida, Charateristis of a doubly lad optial fiber with a low-index inner ladding, IEEE J. Quantum Eletron. 10, (1974). 18. T. Okoshi and K. Oyamoda, Single-polarization single-mode optial fibre with refrative-index pits on both sides of ore, Eletron. Lett. 16, (80). 19. W. Eikhoff, Stress-indued single-polarization single-mode fiber, Opt. Lett. 7 (1982). 20. J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Mahesney, and R. E. Howard, A single-polarization fiber, IEEE J. Lightwave Tehnol. 1, (1983). 21. M. J. Messerly, J. R. Onstott, and R. C. Mikkelson, A broad-band single polarization optial fiber, IEEE J. Lightwave Tehnol. 9, (1991). 22. H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguhi, Absolutely single polarization photoni rystal fiber, IEEE Photon. Teh. Lett. 16, (2004). 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 elliptial entral air hole, IEEE J. Lightwave Tehnol. 23, (2005). 24. P. Kuhment, The Mathematis of Photoni Crystals, in Mathematial Modeling in Optial Siene, G. Bao, L. Cowsar, and W. Masters, eds., Frontiers in Applied Mathematis, pp (SIAM, Philadelphia, 2001). 25. D. R. Smith, D. C. Vier, T. Koshny, and C. M. Soukoulis, Eletromagneti parameter retrieval from inhomogeneous materials, Phys. Rev. E 71, 036,617 (2005). 26. P. Kuhment and B. Ong, On guided waves in photoni rystal waveguides, in Waves in Periodi and Random Media, vol. 339 of Contemporary Mathematis, pp (AMS, Providene, RI, 2003). 27. B. Simon, The bound state of weakly oupled Shrödinger operators in one and two dimensions, Ann. Phys. 97, (1976). 28. L. D. Landau and E. M. Lifshitz, Quantum Mehanis (Addison-Wesley, 1977). 29. H. Piq, Détermination et alul numérique de la première valeur propre d opérateurs de Shrödinger dans le plan, Ph.D. thesis, Université de Nie, Nie, Frane (1982). 30. E. N. Eonomou, Green s funtions in quantum physis (Springer, 2006). 31. C. Cohen-Tannoudji, B. Din, and F. Laloë, Quantum Mehanis (Hermann, Paris, 1977). 32. D. ter Haar, Seleted Problems in Quantum Mehanis (Aademi Press, New York, 1964). 33. E. Hewitt and K. Stromberg, Real and Abstrat Analysis (Springer, 1965). 34. J. D. Jakson, Classial Eletrodynamis, 3rd ed. (Wiley, New York, 1998). 35. J. A. Kong, Eletromagneti Wave Theory (Wiley, New York, 1975). 36. S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jaobs, and J. D. Joannopoulos, Roughness losses and volume-urrent methods in photoni-rystal waveguides, Appl. Phys. B 81, (2005). 37. P. Yeh, A. Yariv, and E. Marom, Theory of Bragg fiber, J. Opt. So. Am. 68, (1978). 1. Introdution In this paper, we present rigorous suffiient onditions for the existene of index-guided modes, inluding onditions for utoff-free modes, in a wide variety of dieletri waveguides from ordinary step-index fibers [1], to photoni-rystal holey fibers [2 5], and even fiber-bragg gratings [6] or other periodially modulated waveguides [5, 7, 8]. The dispersion relations of suh waveguides must almost always be omputed numerially, and so exat analytial theorems like the one derived here provide a foundation of ertainty that is not available in any other way. A rigorous theorem allows us to give a general answer (although not a neessary ondition) for questions suh as: if the waveguide ore has a mixture of higher- and lower-index regions, how muh higher-index material is enough for utoff-free guidane; and under what onditions do photoni-rystal fibers, like step-index fibers, have utoff-free guided modes? The theorem provides an absolute guarantee, with no alulation required, that stritly inreasing the refrative index to form the waveguide (e.g. filling in a hole of a holey fiber) yields a utoff-free guided mode. It also leads diretly to neessary onditions for single-polarization fibers, the subjet of another manusript urrently in prepration. Our work extends an earlier proof of guided modes for homogeneous-ladding, non-periodi, dieletri waveguides with isotropi [9] or anistropi [10] materials, and is losely related in spirit to proofs of the existene of bound modes in two-dimensional potentials for quantum mehanis [11]. The most ommon guiding mehanism in dieletri waveguides is index guiding (or total internal refletion ), in whih a higher-index ore is surrounded by a lower-index ladding ε (ε is the relative permittivity, the square of the refrative index in isotropi non-magneti materials). # $15.00 USD Reeived 20 Mar 2008; revised 17 Apr 2008; aepted 18 Apr 2008; published 9 Jun 2008 (C) 2008 OSA 23 June 2008 / Vol. 16, No. 13 / OPTICS EXPRESS 9262

3 (a) Cross setion of a waveguide (e.g. a onventional fiber) with a homogeneous ladding and an arbitrary-shape ore. (b) Cross setion of a photoni-rystal fiber with periodi ladding and arbitrary-shape ore. () A waveguide periodi in the propagation (z) diretion surrounded by a homogenous ladding. Fig. 1. Shematis of various types of dieletri waveguides in whih our theorem is appliable. Light propagates in the z diretion (along whih the struture is either uniform or periodi) and is onfined in the xy diretion by a higher-index ore ompared to the surrounding (homogeneous or periodi) ladding. A shemati of several suh dieletri waveguides is shown in Fig. 1. In partiular, we suppose that the waveguide is desribed by a dieletri funtion ε(x,y,z) =ε (x,y,z)+δε(x,y,z) suh that: ε, ε, and Δε are periodi in z (the propagation diretion) with period a (a 0 for the ommon ase of a waveguide with a onstant ross-setion); that the ladding dieletri funtion ε is periodi in xy (e.g. in a photoni-rystal fiber), with a homogeneous ladding (e.g. in a onventional fiber) as a speial ase; and the ore is formed by a hange Δε in some region of the xy plane, suffiiently loalized that 1/ε 1/ε < (integrated over the xy plane and the unit ell in z). This inludes a very wide variety of dieletri waveguides, from onventional fibers [Fig. 1(a)] to photoni-rystal holey fibers [Fig. 1(b)] to waveguides with a periodi grating along the propagation diretion [Fig. 1()] suh as fiber-bragg gratings and other periodi waveguides. We exlude metalli strutures (i.e, we require ε > 0) and make the approximation of lossless materials (real ε). We allow anisotropi materials. The ase of substrates (e.g. for strip waveguides in integrated optis [12 14]) is onsidered in Se. 5. We also onsider only non-magneti materials (relative permeability μ = 1), although a future extension to magneti materials should be straightforward. Intuitively, if the refrative index is inreased in the ore, i.e. if Δε is non-negative, then we might expet to obtain exponentially loalized index-guided modes, and this expetation is borne out by innumerable numerial alulations, even in ompliated geometries like holey fibers [2 5]. However, an intuitive expetation of a guided mode is far from a rigorous guarantee, and upon loser inspetion there arise a number of questions whose answers seem harder to guess with ertainty. First, even if Δε is stritly non-negative, is there a guided mode at every wavelength, or is there the possibility of e.g. a long-wavelength utoff (as was initially suggested in holey fibers [15], but was later ontradited by more areful numerial alulations [16])? Seond, what if Δε is not stritly non-negative, i.e. the ore onsists of partly inreased and partly dereased index; it is known in suh ases, e.g. in W-profile fibers [17] that there is the possibility of a long-wavelength utoff for guidane, but preisely how muh dereased-index # $15.00 USD Reeived 20 Mar 2008; revised 17 Apr 2008; aepted 18 Apr 2008; published 9 Jun 2008 (C) 2008 OSA 23 June 2008 / Vol. 16, No. 13 / OPTICS EXPRESS 9263

4 light one higher-order guided mode frequeny ωh fundamental guided mode ω(β) 0.04 ε = 1 Δε = 11 h propagation onstant βh Fig. 2. Example dispersion relation of a simple 2d dieletri waveguide in air (inset) for the TM polarization (eletri field out of the plane), showing the light one, the light line, the fundamental (utoff-free) guided mode, and a higher-order guided mode with a utoff. regions does one need to have suh a utoff? Third, under some irumstanes it is possible to obtain a single-polarization fiber, in whih the waveguide is truly single-mode (as opposed to two degenerate polarization modes as in a ylindrial fiber) [18 23] our theorem an be extended, similar to Ref. 9, to a ondition for two guided modes, and we will explore the onsequenes for single-polarization fibers in a subsequent paper. It turns out that all of these questions an be rigorously answered (in the sense of suffiient onditions for guidane) for the very general geometries onsidered in Fig. 1, without resorting to approximations or numerial omputations. We will proeed as follows. First, in Se. 2, we review the mehanism of index guiding, state our result (a suffiient ondition for the existene of index-guided modes), and disuss some important speial ases. In Se. 3, we first prove this theorem for the simplified speial ase of a homogeneous ladding ε, where the proof is muh easier to follow. Then, in Se. 4, we generalize the proof to arbitrary periodi laddings, suh as for holey photoni-rystal fibers (with some algebrai details left to the appendix). In Se. 5, we disuss a few ontexts that go beyond the initial assumptions of our theorem: substrates, material dispersion, and finite-size effets. Finally, we offer some onluding remarks in Se. 6 disussing future diretions. 2. Statement of the theorem First, let us review the basi desription of the eigenmodes of a dieletri waveguide [1, 5]. In a waveguide as defined above, the solutions of Maxwell s equations (both guided and nonguided) an be written in the form of eigenmodes H(x,y,z)e iβ z iωt (via Bloh s theorem thanks to the periodiity in z) [5], where ω is the frequeny, β is the propagation onstant, and the magneti-field envelope H(x,y,z) is periodi in z with period a (or is independent of z in the ommon ase of a onstant ross setion, a 0). A plot of ω versus β for all eigenmodes is the dispersion relation of the waveguide, one example of whih is shown in Fig. 2. In the absene # $15.00 USD Reeived 20 Mar 2008; revised 17 Apr 2008; aepted 18 Apr 2008; published 9 Jun 2008 (C) 2008 OSA 23 June 2008 / Vol. 16, No. 13 / OPTICS EXPRESS 9264

5 of the ore (i.e. if Δε = 0), the (non-loalized) modes propagating in the infinite ladding form the light one of the struture [2 5]; and at eah real β there is a fundamental (minimum-ω) spae-filling mode at a frequeny ω (β ) with a orresponding field envelope H [2 5]. Suh a light one is shown as a shaded triangular region in Fig. 2. Below the light line ω (β ), the only solutions in the ladding are evanesent modes that deay exponentially in the transverse diretions [2 5, 24]. Therefore, one the ore is introdued (Δε 0), any new solutions with ω < ω must be guided modes, sine they are exponentially deaying in the ladding far from the ore: these are the index-guided modes (if any). Suh guided modes are shown as lines below the light one in Fig. 2: in this ase, both a lowest-lying ( fundamental ) guided mode with no low-frequeny utoff (although it approahes the light line asymptotially as ω 0) and a higher-order guided mode with a low-frequeny utoff are visible. Sine a mode is guided if ω < ω, we will prove the existene of a guided mode by showing that ω has an upper bound < ω, using the variational (min max) theorem for Hermitian eigenproblems [5]. [Modes that lie beneath the light one are not the only type of guided modes in mirostrutured dieletri waveguides. While all the guided modes in a traditional, homogeneous-ladding fiber lie below the light line and are onfined by the mehanism of index-guiding, there an also be bandgap-guided modes in photoni-rystal fibers [2 5]. These bandgap-guided modes lie above the ladding light line and are therefore not index-guided. Bandgap-guided modes always have a low-frequeny utoff (sine in the long-wavelength limit the struture an be approximated by a homogenized effetive medium that has no gap [25]). We do not onsider bandgap-guided modes in this work; suffiient onditions for suh modes to exist were onsidered by Ref. 26.] We will derive the following suffiient ondition for the existene of an index-guided mode in a dieletri waveguide at a given β : a guided mode must exist whenever D (ε 1 ε 1 ) D < 0, (1) where the integral is over xy and one period in z and D is the displaement field of the ladding s fundamental mode. From this, we an immediately obtain a number of useful speial ases: There must be a utoff-free guided mode if Δε 0 everywhere (i.e., if we only inrease the index to make the ore). For a homogeneous ladding (and isotropi media), there must be a utoff-free guided mode if (1/ε 1/ε ) < 0 (similar to the earlier theorem of Ref. 9, but generalized to inlude waveguides periodi in z and/or ores Δε that do not have ompat support). More generally, a guided mode has no long-wavelength utoff if eq. (1) is satisfied for the quasi-stati (ω 0, β 0) limit of D. Equation (1) an also be extended to a suffiient ondition for having two guided modes (or, equivalently, a neessary ondition for single-polarization guidane), when the ladding fundamental mode is doubly degenerate. We explore this generalization, analogous to a result in Ref. 9 for homogeneous laddings, in another manusript urrently being prepared. 3. Waveguides with a homogeneous ladding To illustrate the basi ideas of the proof in a simpler ontext, we will first onsider the ase of a homogeneous ladding (ε = onstant) and isotropi, z-invariant struures (ε is a salar funtion of x and y only). In doing so, we reprodue a result first proved (using a somewhat different approah) by [9] (although the latter result required Δε to have ompat support, # $15.00 USD Reeived 20 Mar 2008; revised 17 Apr 2008; aepted 18 Apr 2008; published 9 Jun 2008 (C) 2008 OSA 23 June 2008 / Vol. 16, No. 13 / OPTICS EXPRESS 9265

6 whereas we only require a weaker integrability ondition). Our proof, whih we generalize in the next setion, is losely inspired by a proof [11] of a related result in quantum mehanis, the fat that any attrative potential in two dimensions loalizes a bound state [11, 27 30]; we disuss this analogy in more detail below. That is, we take the dieletri funtion ε(x,y) to be of the form: ε(x,y)=ε + Δε(x,y), (2) where Δε is an an arbitrary hange in ε that forms the ore of the waveguide. For onveniene, we define a new funtion Δ by: Δ(x,y) ε 1 ε 1. (3) The only onstraints we plae on Δε are that ε be real and positive and that Δ dxdy be finite, as disussed above. Now, we wish to show that there must always be a (utoff-free) guided mode as long as Δε is mostly positive, in the sense that: Δ(x,y)dxdy < 0, (4) Sine Equation (4) is independent of ω or β, the existene of guided modes will hold at all frequenies (utoff-free). The foundation for the proof is the existene of a variational (min max) theorem that gives an upper bound for the lowest eigenfrequeny ω min. In partiular, at eah β, the eigenmodes H(x,y)e iβ z iωt satisfy a Hermitian eigenproblem [5]: where β 1 ε β H = ˆΘ β H = ω2 H, (5) 2 β + iβẑ, (6) with Equation (5) defining the linear operator ˆΘ β. In addition to the eigenproblem, there is also the transversality onstraint [5, 24]: β H = 0 (7) (the absene of stati magneti harges). From the Hermitian property of ˆΘ β, the variational theorem immediately follows [5]: ωmin 2 (β ) H ˆΘ β Hdxdy 2 = inf β H=0 H Hdxdy (8) That is, an upper bound for the smallest eigenvalue is obtained by plugging any trial funtion H(x, y), not neessarily an eigenfuntion, into the right-hand-side (the Rayleigh quotient ), as long as H is transverse [satisfies Equation (7)]. [Tehnially, we must also restrit ourselves to trial funtions where the integrals in Equation (8) are defined, i.e. the trial funtions must be in the appropriate Sobolev spae H( β ).] Conversely, if Equation (7) is not satisfied, it is easy to make the numerator of the right-hand-side (whih involved β H) zero, e.g. by setting H = ϕ + iβϕẑ for any ϕ(x, y), so transversality of the trial funtion is ritially important to obtaining a true upper bound. Now, we merely need to find a transverse trial funtion suh that the variational upper bound is below the light line of the ladding, whih will guarantee a guided fundamental mode. For a # $15.00 USD Reeived 20 Mar 2008; revised 17 Apr 2008; aepted 18 Apr 2008; published 9 Jun 2008 (C) 2008 OSA 23 June 2008 / Vol. 16, No. 13 / OPTICS EXPRESS 9266

7 homogeneous, isotropi ladding ε, the light line is simply ω 2 / 2 = β 2 /ε, and so the ondition for guided modes beomes: ε H ˆΘ β Hdxdy β 2 H Hdxdy 1 = ε ε β H 2 dxdy β 2 H 2 dxdy < 0, (9) where in the seond line we have integrated by parts. Although the basi idea of using the variational priniple to estimate eigenvalues is well known, the hallenge is to find an estimate that establishes loalization even for arbitrarily small Δε > 0 and/or for arbitrarily low frequenies. The problem of bound states in quantum mehanis is oneptually very similar. There, given a potential funtion V(x,y) in two dimensions with V <, one wishes to show that V < 0 (attrative) implies the existene of a bound state: an eigenfuntion of the Shrödinger 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 wavefuntion ψ for whih the Rayleigh quotient is negative in order to obtain a bound state. In one dimension, finding suh a trial funtion is simple for example, an exponentially deaying funtion e α x (or a Gaussian e αx2 ) will work for suffiiently small α and the proof is sometimes assigned as an undergraduate homework problem [32]. In two dimensions, however, finding a trial funtion is more diffiult in fat, no funtion of the form f (αr) (where r is the radius x 2 + y 2 ) will work (without more knowledge of the expliit solution for V) [11] and the earliest proofs of the existene of bound modes used more ompliated, non-variational methods [27, 30]. However, an appropriate trial funtion for a variational proof was eventually disovered [9, 29], and a simpler trial funtion e (r+1)α was later proposed independently by Yang and de Llano [11]. In the present eletromagneti ase, we found that the following trial funtion, inspired by the quantum ase above [11], works. That is, we an prove the existene of waveguided modes for a homogeneous ladding using the trial funtion, in ylindrial (r,φ) oordinates: H = ˆrγ osφ ˆφ (rγ) sinφ, (10) where γ = γ(r)=e 1 (r2 +1) α (11) for some α > 0, and (rγ) is the derivative with respet to r. Clearly, H in eq. (10) redues to an ˆx-polarized plane wave propagating in the ẑ diretion as α 0 (and hene γ 1). This is a key property of the trial funtion: in the limit of no loalization (α = 0, Δε = 0) it should reover a fundamental (lowest-ω) solution of the infinite ladding. Also, by onstrution, it satisfies the transversality ondition (7) (whih is why we hose this partiular form). We hose γ slightly differently from Ref. 11 for onveniene only (to make sure it is differentiable at the origin and goes to 1 for α 0). For future referene, the first two r derivatives of γ are: γ = 2αr(r 2 + 1) α 1 γ, (12) γ = 2α(r 2 + 1) α 1 γ [ 1 + 2αr 2 (r 2 + 1) α 1 + 2(1 α)r 2 (r 2 + 1) 1], (13) and are plotted along with γ in Fig. 3. What remains is, in priniple, merely a matter of algebra to verify that this trial funtion, for suffiiently small α, satisfies the variational ondition (9). In pratie, some are is required in appropriately bounding eah of the integrals and in taking the limits in the proper order, and we review this proess below. # $15.00 USD Reeived 20 Mar 2008; revised 17 Apr 2008; aepted 18 Apr 2008; published 9 Jun 2008 (C) 2008 OSA 23 June 2008 / Vol. 16, No. 13 / OPTICS EXPRESS 9267

8 1 0.5 γ γ and derivatives/α γ /α γ /α r Fig. 3. Plot of γ [eq. (11)], γ /α [eq. (12)], and γ /α [eq. (13)] versus r for α = 0.1. All three funtions go to zero for r, with no extrema other than those shown. We substitute the trial funtion (10) for H into the left-hand side of eq. (9): 1 ε ε(r) ( + iβẑ) H(r,φ) 2 d 2 r β 2 H 2 d 2 r ( ) 1 = ε + Δ(r) ẑ 1 [ ε r r (rh φ ) H ] r 2 + iβẑ H φ d 2 r β 2 H 2 d 2 r ( )( 1 sin 2 φ { [r(rγ) = ε + Δ(r) ] } 2 γ + β 2 ε r 2 H )d 2 2 r β 2 H 2 d 2 r ( ) 1 sin 2 φ ( = ε + Δ(r) 3rγ + r 2 γ ) 2 d 2 r + ε β 2 Δ(r) H 2 d 2 r ε r 2 (14) We proeed to show that the last line of the above expression is negative in the limit α 0, thus satisfying the ondition for the existene of bound modes. We first examine the seond term of eq. (14): lim β 2 Δ(r) H 2 d 2 r = β 2 Δ(r)d 2 r < 0. (15) α 0 The key fat here is that we are able to interhange the α 0 limit and the integral in this ase, thanks to Lebesgue s Dominated Convergene Theorem (LDCT) [33]: whenever the absolute value of the integrand is bounded above (for suffiiently small α)byanα-independent funtion with a finite integral, LDCT guarantees that the α 0 limit an be interhanged with the integral. In partiular, the absolute value of this integrand is bounded above by Δ multiplied by some onstant (sine H is bounded by a onstant: γ 1 and rγ is also easily seen to be bounded above for suffiiently small α), and Δ has a finite integral by assumption. Sine lim α 0 H 2 = 1, we obtain eq. (4), whih is negative by assumption. Now we must show that the remaining first term of eq. (14) goes to zero as α 0, ompleting our proof. This term is proportional to (ε 1 +Δ), but the Δ terms trivially go to zero by the same arguments as above: Δ allows the limit to be interhanged with the integration by LDCT, and as α 0 the γ and γ terms go to zero. The remaining ε 1 terms an be bounded above by a # $15.00 USD Reeived 20 Mar 2008; revised 17 Apr 2008; aepted 18 Apr 2008; published 9 Jun 2008 (C) 2008 OSA 23 June 2008 / Vol. 16, No. 13 / OPTICS EXPRESS 9268

9 sequene of inequalities as follows: 2π lim α r 2 = 16π lim 16π lim = 16π lim 8π lim α 0 α 0 0 α 0 0 α = 8πe 2 lim α 0 α sin 2 φ (3rγ + r 2 γ ) 2 rdrdφ α 2 r 3 (r 2 + 1) 2α 2 γ 2 [ 2 + αr 2 (r 2 + 1) α 1 +(1 α)r 2 (r 2 + 1) 1] 2 dr α 2 r(r 2 + 1) 2α 1 γ 2 [ 2 + α(r 2 + 1) α +(1 α) ] 2 dr α 2 t 4α 1 e 2 2t2α [ (3 α)+αt 2α ] 2 dt αue 2 2u [(3 α)+αu] 2 du [ 3 8 α α(3 α)+1 (3 α)2 4 ] = 0. (16) From the first to seond line, we substituted eqs. (12) and (13) and simplified. From the seond to third line, we bounded the integrand above by flipping negative terms into positive ones and replaing r 2 with r From the third to the fourth line, we made a hange of variables t 2 = r Then, from the fourth to fifth line, we made another hange of variable u = t 2α, and bounded the integral above by hanging the lower limit from u = 1tou = 0. The final integral an be performed exatly and goes to zero, ompleting the proof. 4. General periodi laddings In the previous setion we onsidered z-invariant waveguides with a homogeneous ladding and isotropi materials (for example, onventional optial fibers). We now generalize the proof in three ways, by allowing: transverse periodiity in the ladding material (photoni-rystal fibers), a ore and ladding that are periodi in z with period a (a 0 for the z-invariant ase), anisotropi ε and Δε materials (ε is a 3 3 positive-definite Hermitian matrix). In partiular, we onsider dieletri funtions of the form: ε(x,y,z)=ε (x,y,z)+δε(x,y,z), (17) where the ladding dieletri tensor ε (x,y,z) =ε (x,y,z + a) is z-periodi and also periodi in the xy plane (with an arbitrary unit ell and lattie), and the ore dieletri tensor hange Δε(x,y,z) =Δε(x,y,z + a) is z-periodi with the same period a. Both ε and the total ε must be positive-definite Hermitian tensors. As defined in eq. (3), we denote by Δ the hange in the inverse dieletri tensor. Similar to the isotropi ase, we require that Δ ij be finite for integration over the xy plane and one period of z, for every tensor omponent Δ ij. We also require that the omponents of ε 1 be bounded above. In the homogeneous-ladding ase, any light mode that lies beneath the (linear) light line of the ladding is guided. We have shown that suh a mode always exists, for all β, under the ondition of eq. (4), by showing that the variational upper bound on its frequeny lies below the light line. In the ase of a periodi ladding, the light line is the dispersion relation of the fundamental spae-filling mode of the ladding, whih orresponds to the lowest-frequeny mode at eah given propagation onstant β [2 5]. This light line is, in general, no longer straight, and there are mehanisms for guidane that are not available in the previous ase, # $15.00 USD Reeived 20 Mar 2008; revised 17 Apr 2008; aepted 18 Apr 2008; published 9 Jun 2008 (C) 2008 OSA 23 June 2008 / Vol. 16, No. 13 / OPTICS EXPRESS 9269

10 suh as bandgap guidane [2 5]. Bandgap-guided modes may exist above the light line and are, in general, not utoff-free beause the gap has a finite bandwidth. Here, we only onsider index-guided modes, whih are guided beause they lie below the light line. We will follow the same general proedure as in the previous setion to derive the suffiient ondition [eq. (1)] to guarantee the existene of guided modes. The homogeneous-ladding ase is then a speial ase of this more general theorem, reovering eq. (4) (but generalizing it to z-periodi ores), where in that ase the ladding fundamental mode D is a onstant and an be pulled out of the integral. The ase of a z-homogeneous fiber is just the speial ase a 0, eliminating the z integral in eq. (1). The proof is similar in spirit to that of the homogeneous-ladding ase. At eah β, the eigenmodes H(x,y,z)e iβ z iωt satisfy the same Hermitian eigenproblem (5) and transversality onstraint (7) as before. We have a similar variational theorem to eq. (8) [5], exept that, in the ase of z-periodiity, we now integrate over one period in z as well as over x and y. ωmin 2 (β ) H ˆΘ β H 2 = inf β H=0 H H. (18) As before, to prove the existene of a guided mode we will find a trial funtion H suh that this upper bound, alled the Rayleigh quotient for H, is below the light line ω (β ) 2 / 2. The orresponding ondition on H an be written [similar to eq. (9)]: H ˆΘ β H ω2 (β ) 2 H H < 0. (19) We onsidered a variety of trial funtions, inspired by the Yang and de Llano quantum ase [11], before finding the following hoie that allows us to prove the ondition (19). Similar to eq. (10), we want a slowly deaying funtion proportional to γ(r) =e 1 (r2 +1) α, from eq. (11), that in the α 0 (weak guidane) limit approahes the ladding fundamental mode H. As before, the trial funtion must be transverse ( β H = 0), whih motivated us to write the trial funtion in terms of the orresponding vetor potential. We denote by A the vetor potential orresponding to the ladding fundamental mode H = β A. In terms of A and γ, our trial funtion is then: H = β (γa )=γh + γ A. (20) For onveniene, we hoose A to be Bloh-periodi (like H, sine A also satisifies a periodi Hermitian generalized eigenproblem and hene Bloh s theorem applies). (Alternatively, it is straightforward to show that the Coulomb gauge hoie, β A = 0, gives a Bloh-periodi A, by expliitly onstruting the Fourier-series omponents of A in terms of those of H.) In ontrast, our previous homogeneous-ladding trial funtion [eq. (10)] orresponds to a different gauge hoie with an unbounded vetor potential A = 1 iβ ŷ+ β ψ, differing from a onstant vetor potential via the gauge funtion ψ = r iβ sinφ + e iβ z. Substituting eq. (20) into the left hand side of our new guidane ondition (19), we obtain five ategories of terms to analyze: (i) terms that ontain Δ = ε 1 ε 1, (ii) terms that anel due to the eigenequation (5), (iii) terms that have one first derivative of γ, (iv) terms that have (γ ) 2, # $15.00 USD Reeived 20 Mar 2008; revised 17 Apr 2008; aepted 18 Apr 2008; published 9 Jun 2008 (C) 2008 OSA 23 June 2008 / Vol. 16, No. 13 / OPTICS EXPRESS 9270

11 (v) terms that have γ γ or (γ ) 2. Category (i) will give us our ondition for guided modes, eq. (1), while ategory (ii) will be anelled exatly in eq. (19). We show in the appendix that all of the terms in ategory (iii) exatly anel one another. The terms in ategories (iv) and (v) all vanish in the α 0 limit; we distinguish them beause ategory (v) turns out to be easier to analyze. There are no terms with γ alone, as these an be integrated by parts to obtain ategory (iii) and (iv) terms. In the appendix, we provide an exhaustive listing of all the terms and how they ombine as desribed above. In this setion, we only outline the basi struture of this algebrai proess, and explain why the ategory (iv) and (v) terms vanish as α 0. Category (i) onsists only of one term: lim H ( β Δ β H ) α 0 = H ( ) β Δ β H ( β ) ( ) (21) = H Δ β H = ω2 2 D ΔD From the first to the seond line, we interhanged the limit with the integration, thanks to the LDCT ondition as in Se. 3, sine the magnitudes of all of the terms in the integrand are bounded above by the tensor omponents Δ ij multiplied by some α-independent onstants, and Δ ij has a finite integral by assumption. (In partiular, the A fundamental mode and its urls are bounded funtions, being Bloh-periodi, and γ and its first two derivatives are bounded for suffiiently small α.) The result is preisely the left-hand side of eq. (1), whih is negative by assumption. Next, we would like to anel ω2 2 H H by the eigen-equation (5). Thus, we examine the term H ( β ε 1 ) γ β H (whih omes from the term where the right-most url falls on H rather than γ) below: H ( β ε 1 = = = γ β H ) H (γ β ε 1 β H +( γ) ε 1 ) β H H γ ω2 2 H + H ω2 2 H H ( γ ε 1 β H ) H ω2 2 γ A + H ( γ ε 1 β H ) From the seond to the third lines, we used the eigenequation (5), and from the third to the fourth lines we used the definition (20) of H in terms of H. The first term of the last line above anels ω2 2 H H in eq. (19). The seond and third terms ontain two ategory (iii) terms: ω2 2 γh ( γ A ) and iω γ γ E H, both of whih will be exatly anelled as desribed in the appendix, as well as some ategory (iv) and (v) terms. The ategory (iv) integrands are all of the form (γ ) 2 multiplied by some bounded funtion (a produt of the various Bloh-periodi fields as well as the bounded ε 1 ). This integrand an then be bounded above by replaing the bounded funtion with the supremum B of its magnitude, at whih point the integral is bounded above by 2πB 0 (γ ) 2 rdr. However, suh # $15.00 USD Reeived 20 Mar 2008; revised 17 Apr 2008; aepted 18 Apr 2008; published 9 Jun 2008 (C) 2008 OSA 23 June 2008 / Vol. 16, No. 13 / OPTICS EXPRESS 9271 (22)

12 integrands were among the terms we already analyzed in the homogeneous-ladding ase, in eq. (16), and we expliitly showed that suh integrals go to zero as α 0. The ategory (v) integrands ould also be expliitly shown to vanish as α 0, similar to eq. (16), but a simpler proof of the same fat an be onstruted via the LDCT ondition. In partiular, similar to the previous paragraph, after replaing bounded funtions with their suprema we are left with ylindrial-oordinate integrands of the form γ γ r and (γ ) 2 r. Both of these integrands, however, are bounded above by an α-independent funtion with a finite integral, and hene LDCT allows us to put the α 0 limit inside the integral and set the integrands to zero. Speifially, by inspetion of eqs. (12) and (13), γ γ r < 4r 2 (1+2+2)/(r 2 + 1) 2 δ and (γ ) 2 r < 4r( ) 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, sine they deay faster than 1/r. In summary, we have shown that, if eq. (1) is satisfied, then the variational upper bound for our trial funtion [eq. (20)] is below the light line, and therefore an index-guided mode is guaranteed to exist. The speial ases of this theorem, as disussed in the introdution, immediately follow. 5. Substrates, dispersive materials, and finite-size effets In this setion, we briefly disuss several situations that lie outside of the underlying assumptions of our theorem: waveguides sitting on substrates, dispersive (ω-dependent) materials, and finite-size laddings. An optial fiber is ompletely surrounded by a single ladding material, but the situation is quite different in integrated optial waveguides. There, it is ommon to have an asymmetrial ladding, with air above the waveguide and a low-index material (e.g. oxide) below the waveguide, suh as in strip or ridge waveguides [12 14]. In suh ases, it is well known that the fundamental guided mode has a low-frequeny utoff even when the waveguide onsists of stritly nonnegative Δε [12, 14]. This does not ontradit our theorem beause we required the ladding to be periodi in both transverse diretions, whereas a substrate is not periodi in the vertial diretion. We have also assumed non-dispersive materials in our proof. What happens when we have more realisti, dispersive materials? Suppose that ε depends on ω but has negligible absorption (so that guided modes are still well-defined). For a given ω, we an onstrut a frequenyindependent ε struture mathing the atual ε at that ω, and apply our theorem to determine whether there are guided modes at ω. The simplest ase is when Δε 0 for all ω, in whih ase we must still obtain utoff-free guided modes. The theorem beomes more subtle to apply when Δε < 0 in some regions, beause not only must one perform the integral of eq. (1) to determine the existene of guided modes, but the ondition (1) is for a fixed β while the integrand is for a given frequeny, and the frequeny of the guided mode is unknown a priori. Finally, any real struture has a finite ladding. Both numerially and experimentally, this makes it diffiult to study the long-wavelength regime beause the modal diameter inreases rapidly with wavelength (i.e. the frequeny approahes the light line and the transverse deay rate beomes very slow) in fat, it seems likely that the modal diameter will inrease exponentially with the wavelength. In quantum mehanis (salar waves) with a potential well of depth V, the deay length of the bound mode inreases as e C/V when V 0, for some onstant C [11, 27]. In eletromagnetism, for the long wavelength limit, a homogenized effetive-medium ε desription of the struture beomes appliable [25], and in this effetive near-homogeneous limit the modes are desribed by a salar wave equation with a potential ω 2 Δ ε [34], and hene the quantum analysis should apply. Thus, by this informal argument, we would expet the modal diameter to expand proportional to e Cλ 2 for some onstant C (where λ = 2π/ω is the vauum wavelength), but a more expliit proof would be desirable. # $15.00 USD Reeived 20 Mar 2008; revised 17 Apr 2008; aepted 18 Apr 2008; published 9 Jun 2008 (C) 2008 OSA 23 June 2008 / Vol. 16, No. 13 / OPTICS EXPRESS 9272

13 6. Conluding remarks We have demonstrated suffiient onditions for the existene of utoff-free guided modes for general mirostrutured dieletri fibers, periodi in either or both the z diretion and in the transverse plane. The results are a generalization of previous results on the existene of suh modes in fibers with a homogeneous ladding index. Our theorem allows one to understand the guidane in many very ompliated strutures analytially, and enables one to rigorously guarantee guided modes in many strutures (espeially those where Δε 0 everywhere) by inspetion. There remain a number of interesting questions for future study, however, some of whih we outline below. Our eq. (1) is a suffiient ondition for index-guided modes, but it is ertainly not neessary in general: even when it is violated, one an have guided modes with a utoff (as for W-profile fibers [17] or waveguides on substrates [12, 14]), or other types of guided modes (suh as bandgap-guided modes [2 5]). However, these other types of guides modes in dieletri waveguides have a long-wavelength utoff, so one an pose the question: is eq. (1) a neessary ondition for utoff-free guided modes (where D is given by the long-wavelength limit of the ladding fundamental mode) in dieletri waveguides (as opposed to TEM modes in metalli oaxial waveguides, whih also have no utoff [35])? Based on theoretial reasoning and some numerial evidene, we suspet that the answer is no, but that it may be possible to modify eq. (1) to obtain a neessary ondition. In partiular, the variational theorem is losely related to first-order perturbation theory: if one has a small perturbation Δε and substitutes the unperturbed field into the Rayleigh quotient, the result is the first-order perturbation in the eigenvalue. However, when Δε is large, even if the volume of the perturbation is small, perturbation theory requires a orretion due to the eletri-field disontinuity at the interfae [36]. In the long-wavelength limit, perturbation theory is orreted by omputing the quasi-stati polarizability of the perturbation [36], and we onjeture that a similar orretion to our trial field may allow one to derive a neessary ondition for the absene of a utoff. Equation (1) is still a suffiient ondition (the variational theorem still holds even with a suboptimal trial funtion), but the preeding onsiderations predit that it will beome farther from a neessary ondition for the absene of a utoff as Δε is inreased, and this predition seems to be onfirmed by preliminary numerial experiments with W-profile fibers. Also using a variational approah, Ref. 26 proved a suffiient ondition for the existene of bandgap-guided modes in two dimensions. However, the ondition established in that work was fairly strong, requiring a minimum defet size to guarantee the existene of a guided mode, whereas numerial studies in one and two dimensions have suggested that no minimum defet size may be required [5], similar to the index-guided ase proved here. If there is, in fat, no minimum defet size for gap-guided modes in linear defets, it is possible that a trial funtion similar to the one here ould be applied to an approah like that of Ref. 26. Let us also mention five other interesting diretions to pursue. First, Ref. 9 atually proved a somewhat stronger ondition than eq. (1) for homogeneous laddings, in that they showed the existene of guided modes when the integral was 0 (and Δε > 0 in some region) rather than < 0 as in our ondition. Although the = 0 ase seems unlikely to be experimentally or numerially signifiant, we suspet 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). Seond, as disussed in Se. 5, it would be desirable to develop a suffiient ondition at a fixed 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 Se. 5, that the modal diameter should asymptotially inrease exponentially with the square of the wavelength. Fourth, it might be interesting to onsider the ase of Bragg fiber geometries onsisting of periodi sequenes of onentri layers [37], whih are not stritly periodi beause the layer urvature dereases with radius. # $15.00 USD Reeived 20 Mar 2008; revised 17 Apr 2008; aepted 18 Apr 2008; published 9 Jun 2008 (C) 2008 OSA 23 June 2008 / Vol. 16, No. 13 / OPTICS EXPRESS 9273

14 Finally, as we mentioned in Se. 2, it is possible to extend the theorem to a ondition for two guided modes in many ases where the ladding fundamental mode is doubly degenerate, and we are urrently preparing another manusript desribing this result along with onditions for truly single-mode ( single-polarization ) waveguides. Aknowledgments We are grateful to M. Ghebrebrhan and G. Staffilani at MIT for helpful disussions. Appendix: All Rayleigh-quotient terms In this appendix, we provide an exhaustive listing of all the terms that appear when the trial funtion [eq. (20)] is substituted into eq. (19) (the ondition to be satisfied, a rearrangement of the Rayleigh quotient bound). Sine the terms that ontain Δ [ategory (i)] were already fully analyzed in Se. 4 (sine for these terms the limits ould be trivially interhanged), we onsider only the remaining terms involving ε (r). More speifially, the only non-trivial term to analyze is the Δ-free part of the left-most integral in eq. (19): H ( β ε 1 β H ) = H ( β ε 1 ) γ β H + H ( β ε 1 ) γ H (23) + H ( β ε 1 β ( γ A ) ). We have already seen, in eq. (22), that the first term breaks down into a term that anels ω 2 2 H H in eq. (19), via the eigen-equation, and two other terms. Removing the terms anelled by the eigenequation, and substituting i ω E for ε 1 β H (Ampère s law), we have: ω2 2 + H ( γ A )+ H β ε 1 [ ( H γ i ω E [ γ H + β ( γ A ) ] = ω2 2 [γh + γ A ] ( γ A ) i ω γ γ (E H ) i ω ( γ A ) (γ β ) [ ( γ E )+ H ε 1 γ H + β ( γ A ) ] + ( γ H ) ε 1 [ γ H + β ( γ A ) ] ( β ) [ + γ A ε 1 γ H + β ( γ A ) ] )] # $15.00 USD Reeived 20 Mar 2008; revised 17 Apr 2008; aepted 18 Apr 2008; published 9 Jun 2008 (C) 2008 OSA 23 June 2008 / Vol. 16, No. 13 / OPTICS EXPRESS 9274

15 = ω2 2 i ω + i ω γh ( γ A ) ω2 2 ( γ A ) ( γ E )+i ω γ ( β E ) ( γ A )+i ω + ( γ H ) ε 1 ( γ H ) ( + ( γ H ) ε 1 [ β ( γ A ) ] ) +.. γ A 2 i ω γ γ (E H ) γe ( γ H ) ( γ E ) ( γ A ) (24) ( β + ( γ A ) ) ( ε 1 β ( γ A ) ). Above, the first = step is obtained by substituting the trial funtion for H, integrating some of the β operators by parts, and distributing the derivatives of γh by the produt rule. The seond step is obtained by using Ampère s law again, ombined with integrations by parts and the produt rule;.. stands for the omplex onjugate of the preeding expression. Continuing, we obtain: = ω2 2 γh ( γ A ) ω2 2 γ A 2 2i ω γ γ R{E H } ( + i ω ) ( γ A ) ( γ E ) ω2 2 γh ( γ A ) ( + ( γ H ) ε 1 ( γ H )+ ( γ H ) ε 1 [ β ( γ A ) ] ) (25) +.. ( β + ( γ A ) ) ( ε 1 β ( γ A ) ) In obtaining this expression, we have grouped terms into omplex-onjugate pairs and used Faraday s law to replae β E with i ω H. At this point, we have two ω2 γh 2 ( γ A ) terms that exatly anel. All of the remaining terms, exept for i ω γ γ R{E H }, are multiples of two first or higher derivatives of γ, orresponding to ategory (iv) and (v) terms, whih we proved to vanish in Se. 4. The only remaining term is the E H term, in ategory (iii). This term is identially zero (for any α 0) beause it is purely imaginary, whereas all of the other terms are purely real and the overall expression must be real. More expliitly: 2i ω γ γ R{E H } = 2i ω γ 2 R{E H 2 } (26) = i ω (γ 2 R{E H } ) + i ω γ 2 (R{E H }) The first term of the last line is zero by the divergene theorem (transforming it into a surfae integral at infinity), sine γ 0 at infinity. For the seond term, the integrand is the divergene of the time-average Poynting vetor R{E H }, whih equals the time-average rate of hange of the energy density [34], whih is identially zero for any lossless eigenmode (suh as the ladding fundamental mode). # $15.00 USD Reeived 20 Mar 2008; revised 17 Apr 2008; aepted 18 Apr 2008; published 9 Jun 2008 (C) 2008 OSA 23 June 2008 / Vol. 16, No. 13 / OPTICS EXPRESS 9275