omni guide

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Low-loss asymptotically single-mode

propagation in large-core OmniGuide fibers

Steven G. Johnson, Mihai Ibanescu, M. Skorobogatiy,

Ori Weisb erg, Torkel D. Engeness, Marin Soljacic,

Steven A. Jacobs, J. D. Joannopoulos, and Yoel Fink

OmniGuide Communications, One Kendall Square, Building 100 #3,

Cambridge MA 02139

[email protected]

Abstract: We present the light-propagation characteristics of Om-niGuide fibers, which guide light by concentric multi-layer dielectricmirrors having the property of omnidirectional reflection. We showhow the lowest-loss TE01 mode can propagate in a single-mode fashionthrough even large-core fibers, with other modes eliminated asymptot-ically by their higher losses and poor coupling, analogous to hollowmetallic microwave waveguides. Dispersion, radiation leakage, materialabsorption, nonlinearities, bending, acircularity, and interface rough-ness are considered with the help of leaky modes and perturbationtheory, and both numerical results and general scaling relations arepresented. We show that cladding properties such as absorption andnonlinearity are suppressed by many orders of magnitude due to thestrong confinement in a hollow core, and other imperfections are tol-erable, promising that the properties of silica fibers may be surpassedeven when nominally poor materials are employed.c 2001 Optical Society of America

OCIS codes: (060.2310) Fiber optics; (230.1480) Bragg reflectors

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#37483 - $15.00 US Received November 09, 2001; Revised November 30, 2001


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15. R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Academic Press,London, London, 1998).

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38. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).39. M. Lohmeyer, N. Bahlmann, and P. Hertel, Geometry tolerance estimation for rectangular di-

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1 Introduction

Telecommunications has continued to push optical fibers towards ever-more demandingapplicationssuch as high bit rates, dense wavelength-division multiplexing (DWDM),and long distancesand in response there has been renewed interest in alternative fiberdesigns to lift fundamental limitations of silica fibers. A particularly exciting departurefrom traditional fibers are fibers based on photonic band gaps, forbidden frequencyranges in periodic dielectric structures that can confine light even in low-index or hollowregions [1]. Two main classes of fibers have emerged using photonic band gaps: photonic-crystal holey fibers that use a two-dimensional transverse periodicity [2], and Braggfibers1 that use a one-dimensional periodicity of concentric rings [313]. In this paper, westudy the propagation of light in a novel class of Bragg fibers: OmniGuide fibers witha hollow core, which use a multilayer cladding that exhibits omnidirectional reflectionin the planar limit [7,10,14].

In the following sections, we show how OmniGuide fibers bear strong resemblancesto the hollow metallic waveguides that are used in the microwave regime, confining a setof guided modes almost entirely within the hollow core with similar field patterns anddispersion characteristics. Because of this strong confinement, we prove that radiationleakage, material absorption, and nonlinearities from the cladding layers can be sup-pressed by many orders of magnitude. We also study imperfections, such as waveguidebends, acircularity (ellipticity), and surface roughness, and present both general expres-sions and numerical results for these effects. Moreover, like hollow metallic waveguides,we show that there is substantial loss discrimination between a single lowest-loss mode,

TE01, and other guided modesthis produces modal filtering that allows even a highly-multimode OmniGuide fiber to operate in an effectively single-mode fashion. Becausethis TE01 state is cylindrically symmetrical and non-degenerate, it has the additionalbenefit of immunity to polarization-mode dispersion (PMD) from fiber birefringence. Inthis way, we demonstrate that OmniGuide fibers have the potential to lift three majorphysical limitations on silica fibers: losses (currently 0.2 dB/km), nonlinearities, andPMD [15].

One of the hallmarks of photonic crystals is their flexibility, since their optical prop-erties are subject to deliberate structural and materials choices. OmniGuide fibers alsoembody this freedom, permitting a wide variety of layer-thickness designs to enhanceor inhibit specific characteristics (e.g. to tailor dispersion parameters), which we do notexplore here. Moreover, due to the suppression of cladding properties, a much broaderrange of materials is available for use than would normally be practical in low-loss optical

fibers. In this paper, we describe methods for understanding and designing OmniGuidefiber properties, and especially focus on general scaling laws, phenomena, and designtradeoffs that apply in such systems. For a particular example system, we arbitrarily

1Bragg fibers should not be confused with fiber Bragg gratingsthe former use a lateral indexmodulation for transverse optical confinement, while the latter use axial modulation for longitudinalconfinement and other effects.




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Metal

Hollow Metallic WaveguideDielectric Waveguide

R R

r

z

Fig. 1. (a) Hollow dielectric waveguide of radius R. Light is confined in the hollowcore by a multilayer dielectric mirror made of alternating layers with high (blue)and low (green) indices of refraction. (b) Hollow metallic waveguide of radius R.Light is confined in the hollow core by a metallic cylinder.

select an index contrast of 4.6 to 1.6 in the cladding, as has been used in several previ-ous publications [7,10,11,13].2 We explain how the properties scale with the indices inSec. 10, however, and our scaling laws, analytical techniques, and qualititative resultshold true for a wide range of alternative parameters.

2 Hollow Dielectric vs. Metallic Waveguides

In this section, we begin by introducing the basic structure and principles of operation forOmniGuide fibers, and develop an intuition for their behavior by an analogy with hollowmetallic waveguides [16,17]. One could also compare with other prior work, including thehybrid system of a metallic waveguide whose inner surface is coated with a multilayerBragg mirror [18, 19], as well as several other hollow waveguiding systems [20], butthe pure metallic guide provides the simplest foundation for understanding. In the leftpanel of Fig. 1, we depict a schematic of an OmniGuide fiber forming a hollow dielectricwaveguide. A hollow core (index of refraction unity) of radius R is surrounded by amultilayer cladding that consists of alternating layers having high and low indices ofrefraction. The high/low index layers are shown in blue/green. As discussed above, we

choose indices of refraction 4.6 and 1.6, with thicknesses here of 0.33a and 0.67a, where ais the thickness of one high/low bilayer (we select different thicknesses in Sec. 2.2). Oncea mode frequency is computed in units of c/a, the physical value of a is determinedvia a = for some desired operational wavelengh [1]. The radius R of the waveguidewill vary in the differing examples presented in this paper, from a minimum of 2a toa maximum of 30a. In the right panel of Fig. 1, we show a hollow metallic waveguide.The core is the same as that of the hollow dielectric waveguide, but a metal cylindernow replaces the multilayer cladding.

In the metallic case, light is confined in the core by the impenetrability of a near-perfect metal (nonexistent at optical frequencies)such confined modes for R = 2aare depicted in the right panel of Fig. 2. Dispersions relations like this one depicttwo conserved quantities: the axial wavenumber and the frequency . In cylindri-cal waveguides, modes can also be labeled by their angular momentum integer m. For

waveguides that lie along the z axis, the (z, t , ) dependence of the modes is then givenby: ei(zt+m). In a hollow metal tube, the eigenmodes are purely polarized as TM(Hz = 0) or TE (Ez = 0), and the -th mode of a given m is labeled TXm.

In the dielectric case, light is confined by the one-dimensional photonic band gap of

2The actual physical indices that we use are currently proprietary.




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0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

Frequency

(2c/a)

m = 1

m = 0

m = 2

m = 0, 1

m = 3

m = 2

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

Wavevector (2/a)

Frequency

(2c/a)

TE

TM

Wavevector (2/a)

R

Fig. 2. (Left) Projected band structure associated with the planar dielectric mirror.The blue regions correspond to (, ) pairs for which light can propagate within

the mirror. White and gray regions correspond to situations where light cannotpropagate within the mirror. The thick black line represents the light line ( = c).Shown in gray are the two omnidirectional frequency ranges of the mirror. (Right)Dispersion relations () of the lowest 7 modes supported by a hollow metallicwaveguide of radius R = 2a are plotted. TE/TM-polarized modes are shown inred/blue, and the modes have angular dependence eim. Note the degeneracy ofthe TE01 and the TM11 modes.

the multi-layer cladding, which is easy to analyze in the limit as the cladding becomesplanar. The one-dimensional gaps of the planar dielectric mirror structure as a functionof (the surface-parallel wavevector component) are depicted in the left panel of Fig. 2.In these gap regions, we expect the mirrors to behave similarly to a metal, and confinemodes strongly analogous to those of the metallic waveguide; this is verified below.Because every eigenmode has a finite, conserved m, the effective wavevector k = m/r

in the direction goes to zero for r . Without this fact, there would be no bandgaps in Fig. 2, since nonzero k would have to be projected onto the Bragg banddiagram. Also shown in this figure, as gray regions, are the ranges of omnidirectionalreflection: the frequencies at which any incident wave from air will be reflected bythe planar mirrors (and vice versa). Omnidirectional reflection per se is not strictlynecessary for guidance in these fibers, but its presence is strongly correlated with theregimes of large, polarization-independent gaps along the light line that are importantfor many of the properties studied in this paper.

Bragg mirrors have different band-gaps for TE and TM polarizations, referringto fields purely parallel to the interface and fields with a normal component, respectively.(Both polarizations are shown in Fig. 2.) This corresponds to the waveguide TE andTM labels only for m = 0; all m = 0 modes have some nonzero Er component.

2.1 The modes in an OmniGuide fiberThe modes supported by any cylindrical waveguide, including metallic waveguides, Om-niGuide fibers, and traditional silica fibers, can be computed by the transfer-matrixmethod of [3]. Here, the longitudinal fields (Ez and Hz) of a given (m, , ) in an an-nular region of index nj are expanded in Bessel functions Jm(kjr) and Ym(kjr), with




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0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

Wavevector (2/a)

Frequency

(2

c/a)

R = 2.0 a

HE11

HE21

TM01

HE11

HE21

TM01

Fig. 3. Guided modes supported by a hollow OmniGuide fiber of radius R = 2a: redlines are for TE and HE modes, while blue is for TM and EH modes. In black is thelight line ( = c), and the solid blue regions represent the continuum of modesthat propagate within the multilayer cladding. Only the first three modes in eachband gap are labeled.

kj

n2j2/c2 2. At each interface, the coefficients are related by a 4 4 transfer

matrix that matches boundary conditions.3 The product of all these transfer matricesyields a single matrix relating the fields in the core to those in the outermost cladding.Then, by application of appropriate boundary conditions, the n wavevectors of thevarious modes can be found; this is discussed in more detail by Sec. 4 below.

Here, we are primarily interested in the modes that lie within the band gap of the

one-dimensional Bragg mirrors. Such modes must decay exponentially with r in thecladding, and therefore are truly guided modes in the limit of infinitely many claddinglayers (the case of finite layers is considered in Sec. 4). Most of these modes lie abovethe = c light line, and thus propagate within the hollow core in much the sameway as the modes of a metallic waveguide. It is also possible, however, for modes to liebeneath the light line and yet inside the band gap, in which case they are surface stateslocalized around the core/cladding interfacewe discuss this possibility further below.

In Fig. 3, we show the computed guided modes of the OmniGuide fiber with core-radius R = 2a and the abovementioned planar-mirror parameters. These modes are atnearly the same frequencies as the metallic waveguide modes of Fig. 2, with the one-dimensional bandgaps simply superimposed. In the dielectric waveguide, the modes areonly purely TE and TM for m = 0, but for m = 0 they are strongly TE-like or TM-like,and are called HE and EH, respectively. (When a mode enters the second gap, we add

a prime superscript.) Moreover, because these modes are so strongly confined withinthe core, and have the same orthogonality relationship as in a metal waveguide (see theappendix), the field patterns must also be nearly identical. We consider this analogy in

3The transfer matrix of Eq. (37) in [3] is erroneous for m = 0; we hope that this apparent typo-graphical mistake will be corrected by a future erratum.




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R

Fig. 4. An OmniGuide fiber with core radius R = 30a, the parameters that weemploy in the remainder of this paper. The omnidirectional mirror here comprises17 layers, starting with a high-index layer, with indices 4.6/1.6 and thicknesses0.22a/0.78a, respectively. (The omnidirectional mirror is surrounded by some coat-ing for mechanical support; this layer is not shown to scale.) We choose a = 0.434m,so that the lowest dissipation losses occur roughly at = 1.55m.

greater detail in a later publication [21], as well as in subsequent sections on losses and

bends.

2.2 A Large-core OmniGuide fiber

The above calculations yielded the modes of an OmniGuide fiber for a radius R = 2a.This small radius has the advantage of supporting only a few modes, which are easyto plot and understand in their entirety, and even has a single-mode frequency range.The analogy with metallic waveguides, however, tells us that this may not be the mostdesirable regime for fiber operation. In this section, we motivate the use of larger,ostensibly multi-mode cores for OmniGuide fibers, and describe the fiber parametersthat we will use for subsequent computations in this paper.

In metallic waveguides, the lowest-loss mode is TE01, and its ohmic losses decreaseas 1/R3 [16,17]. Moreover, the differential losses between TE01 and other modes createa modal-filtering effect that allows these waveguides to operate in an effectively single-mode fashion. On the other hand, for large core radii (high frequencies), losses becomedominated by scattering into other closely-spaced modes, especially into the degenerateTM11 mode via bends, and it was found that the optimal radius was in the range of4 11 [17].

Similar results hold for OmniGuide fibers: the lowest-loss mode is TE01, and manyof its losses fall off as 1/R3 (for much the same reasons as in metallic waveguides). Likethe metallic waveguides, and unlike silica fibers with their small material contrasts, wedemonstrate a strong modal-filtering effect based on the degree of confinement in thecore. Also as before, inter-modal scattering worsens with increasing R. (We show thatbending is less of a problem than in metallic waveguides, however, since the dielectriccladding breaks the TE01/TM11 degeneracy.) Thus, for the example OmniGuide fiberthat we use in the remainder of this paper, we choose a core radius of R = 30a, as

depicted in Fig. 4. The point of lowest TE01 dissipation losses that we compute inSec. 6.1 then lies at a frequency of = 0.28 2c/a, so if we make this correspond to thestandard = 1.55m of telecommunications, we have a = 0.434m. Equivalently, R =13.02m = 8.4, in the favorable range for the metallic-waveguide analogy.4 Throughout

4As a positive side-effect, we show that such a core radius brings chromatic dispersion down into arange comparable to that of single-mode silica fibers...although, because of the low nonlinearities, we




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20 0 20

30

20

10

0

10

20

30

x (a)

y(a)

EH11

20 0 20

30

20

10

0

10

20

30

x (a)

y(a)

TE01

Fig. 5. Transverse electric-field distributions in the OmniGuide fiber of Fig. 4 forthe TE01 mode (left) and the EH11 mode (right), which have = 0.27926 2/aand = 0.27955 2/a, respectively, at = 0.28 2c/a.

this paper, however, we will emphasize scaling laws with R in order to highlight theeffects of differing core sizes.

In order to choose the layer thickness, we employed an approximate quarter-wave

condition. It is well-known that, for normal incidence, a maximum band gap is obtainedfor a quarter-wave stack in which each layer has equal optical thickness /4: dhi/dlo =nlo/nhi [22]. Normal incidence, however, corresponds to = 0, whereas the modes ofinterest in the OmniGuide fiber lie almost on the = /c light line (in the limit ofR , the lowest-order modes are essentially plane waves propagating along z). Thus,we employ layer thicknesses determined by the quarter-wave condition along the lightline of air (similarly applied in [18]):

dhidlo

=

n2

lo 1

n2hi

1(1)

which yields dhi= 0.2176a and dlo = 0.7824a in this case. (Ref. [3] suggests an alternate

method that optimizes the structure for confinement of a given mode, but this yieldsessentially the same thicknesses as Eq. (1) for large R.)

As in the R = 2a case, the guided-modes of this R = 30a OmniGuide fiber can be la-beled by analogy to the modes of an equal-radius metallic waveguide. Two such modes inthe OmniGuide fiber, the lowest-loss TE01 and the linearly-polarized

5 EH11 (analogousto the TM11 mode in a metallic guide) are depicted in Fig. 5. The TE01 mode is circu-

larly symmetric and azimuthally polarized ( E)thus, unlike the doubly-degenerateEH11 mode (two orthogonal polarizations), TE01 cannot be split into two modes of dif-fering velocities by fiber imperfections, and is therefore immune to polarization-modedispersion (PMD). (See also Sec. 8.) Actually, the mode labeling in an OmniGuide fiberis more complex than in a metallic waveguide, because sometimes a mode will crossthe light line and become a surface state localized around the core/cladding interface.When that happens, another mode moves up and takes its place; for example, theTM01 mode crosses the light line at = 0.27 2c/a, while the TM02 mode continuouslytakes on the core field pattern of TM01. When in doubt, we label a mode as HE or EH

shall argue that such a correspondence is not strictly necessary.5With our choice of eim -dependence, the EH11 mode is actually circularly polarized; combined

with its degenerate m = 1 mode, however, one can instead express it as linearly polarized.




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2

0

2

4

6

8

10

E

(arb.units)

0 5 10 15 20 25 30 35 40 450.1

0

0.1

radius (a)

E

(arb.units)

coreboundary

Fig. 6. The (unnormalized) electric field E for the TE01 mode in the OmniGuidefiber of Fig. 4. The lower plot displays the same field, but with the vertical scaleexaggerated in order to show the field amplitude in the cladding. The field has anode near the core interface at R, and so the field amplitude in the cladding isdetermined by the slope at that point.

depending upon whether it is dominated by Hz or Ez at r = 0, respectively, and numberthe core modes according to their ordering above the light line.

3 Scaling Laws with Core Size

Because of the strong reflectivity of the dielectric mirrors, many of the mode propertiesare determined largely by the geometric size R of the core, within which the modes areconfined. Throughout this paper, we thereby derive scaling relations for the differentquantities computed, and in this section we lay the groundwork for those derivations

by presenting basic scalings of the fields and modes. These scaling relations are largelyindependent of details such as the precise index contrast that is used, so long as it issufficiently large for the metallic analogy to hold, and will provide a broad understandingof the advantages and tradeoffs of OmniGuide fiber structures. Later in the paper, weexhibit data to explicitly verify scaling relations derived from those in this section.

In particular, we will focus on the suppression of cladding phenomena for the TE0(especially TE01) modes of the fiber. The critical property of TE0 modes is that, byanalogy with a hollow metallic waveguide, they have a node in their electric field ( E)near r = R [16, 17] as depicted in Fig. 6. It then follows that the amplitude of theelectric field in the cladding is proportional to the slope of E at R. The form of Ein the core, however, is simply the Bessel function J1(r/R), where () is roughly the-th zero ofJ1. The slope at R is then (J0() J2()) /2R. Moreover, for our quarter-wave stack, E peaks near each of the nhi

nlo interfaces [3]. Thus, not including

any normalization of the J1 amplitude (i.e. E 1), we find that the unnormalizedE in the cladding scales as dhi/R. In addition, for most computations (such as theperturbation theory described in the appendix), one must normalize the power of the

field: this means dividing E by an additional factor proportional to

mode area R,




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and so:

normalized TE0 cladding E 1R2

. (2)

Moreover, the area of the field in the cladding is the perimeter R times some constant(penetration depth) that depends on the size of the band gap; combined with Eq. (2),we therefore find:

fraction of | E|2 in cladding for TE0 1R3 , (3)and from this we derive many other scaling relations. In contrast, for TM or mixed-polarization modes with an Er component, the unnormalized field amplitude in thecladding remains roughly constant with changing Rtheir fractional | E|2 in the claddingthen scales as only 1/R, so the cladding has a much greater effect on them.

By general phase-space arguments, the total number of modes in the core mustscale as the area R2. Moreover, in a metal waveguide, the dispersion relations look liken =

2/c2 2n/R2, where n are roots or extrema of Bessel functions. Therefore,

far from cutoff (R nc/),

mode separation 1R2

; (4)

this has important implications for many of the mode-coupling phenomena discussedlater. Unfortunately, can be somewhat more complicated in an OmniGuide fiber,due to the finite field penetration into the cladding and due to the transitions intosurface states and subsequent mode relabelings discussed in the previous section. Forexample, consider the case of the EH11 mode, which is degenerate with TE01 in themetallic limit [16,17]. The degeneracy is broken by the penetration of the fields into thecladding,6 inducing a small shift : by perturbation theory, is proportional to theamount of EH11 energy in the cladding, 1/R:

TE01 and EH11 mode separation 1R

. (5)

Testing this scaling relation numerically for = 1.55m, however, we find that thisseparation eventually scales as 1/R2 for R 40a. It turns out that the fundamental

HE11 mode has crossed the light line to become a surface state, and EH11 continuouslytransitions to being more HE11-like, thus scaling eventually as Eq. (4).

4 Leaky Modes and Radiation Loss

In the preceding discussion, we have neglected a point that may seem important: inreality, there will be only a finite number of cladding layers in the omnidirectionalmirror. Because of this, and the fact that the modes of interest lie above the light line ofthe outermost region, the field power will slowly leak out in a process akin to quantum-mechanical tunneling. This radiation loss, however, decreases exponentially with thenumber of cladding layers, and we quantify its smallness explicitly below. As has beenobserved elsewhere [11, 13], we show that only a small number of layers is required toachieve leakage rates well below 0.1 dB/km. Moreover, the radiation leakage strongly

differs between modes, inducing a modal-filtering effect [3, 6] that allows a large-coreOmniGuide fiber to operate in an effectively single-mode fashion.In the limit of infinitely many cladding layers, the modes in the OmniGuide core

are true confined modes, thanks to the band gap, with discrete real eigenvalues n

6Such degeneracy breaking can also be understood in terms of the phase shift of a TE/TM polarizedwave upon reflection from the omnidirectional mirror [21].




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(c.f. the appendix). For finitely many layers, modes are no longer truly confined (abovethe outermost light line), leading to a continuum of values with infinitely-extendedeigenstates [3]. The former confined modes become leaky resonances: superpositions ofreal- eigenmodes centered on n with a width proportional to the radiative decayrate n. Such resonances can be studied directly via the physical real- continuum, buta powerful alternative method is the leaky-mode technique, which employs a conceptualanalytic continuation from to in the complex plane to satisfy a boundary conditionof zero incoming flux [23]. The power decay rate n is then given by 2[n], where denotes the imaginary part; we have also verified the correctness of this decay rateagainst explicit real- superpositions as well as by beam-propagation methods (BPM)[24].

For an OmniGuide fiber, the leaky-mode method is as follows. The transfer-matrixformulation allows one to compute 2 2 matrices Mm(, ) that connect the TE andTM amplitudes at the origin to the amplitudes of the outgoing (+) or incoming ( )TE and TM cylindrical waves (Hankel functions) in the outermost region, as a functionof and for a given angular-momentum index m. (For m = 0, the two polarizationsdecouple [3].) For a leaky mode, we wish to impose the boundary condition of zeroincoming waves, so that there are no sources at ; such a solution exists wheneverthere is a zero eigenvalue of Mm. Therefore, we use the determinant:

fm(, ) det Mm(, ) , (6)so that the leaky mode is defined by fm(, n) = 0. Once such a n is found for a given, the corresponding eigenvector of Mm(,

n) yields the required mixed-polarizationamplitudes. With finitely many layers, the only real roots of fm lie below the lightline of the outermost region;7 above this light line, the incoming and outgoing flux areequal for real [3], corresponding to steady-state standing-wave patterns. The small

imaginary part of n above the light line yields the power decay rate n = 2[n].A similar leaky-mode method was previously used for Bragg fibers, albeit with only afirst-order Taylor-expansion approximation for [n] [5].

For all modes, the radiative decay decreases exponentially with increasing numbersof cladding layers, thanks to the exponential decay of fields in the Bragg band gap,

eventually to the point where other losses (e.g. absorption) dominate. At = 1.55mfor this structure, the TE/TM losses decrease by a factor of 10/5 per cladding bilayer,respectively. Because of the smaller TM band gap, the losses of mixed-polarization(m = 0) modes are eventually dominated by their less-confined TM components. InFig. 7, we display the computed radiation leakage rates for the lowest-loss TE01mode, the next-lowest loss TE02 mode, and the linearly-polarized EH11 mode to typifymixed-polarization modes. Like the absorption discussed later, these differential lossescreate a mode-filtering effect that allows the TE01 mode to operate as effectively single-mode, even for large-core OmniGuide fibers, a fact that was also noted in [3,6]. A similarprinciple was employed in hollow metallic waveguides [16,17] to select the TE01 mode.Note that, as in metallic waveguides, TE02 is not necessarily the mode of greatestconcernit is more important to consider modes that couple strongly to TE01 viaperturbations, such as those discussed in later sections. From Fig. 7, it is seen that with

only 17 cladding layers the TE01 mode has leakage rates well under 0.01 dB/km, andeven EH11 has decay lengths of meters, corresponding to [n]/[n] of 1013 and 109, respectively. Thanks to these low losses, the modes can be treated as truly

7Below the light line of the outermost region, the incoming-wave Hankel function instead becomesan exponentially-growing function, but its coefficient must be zero all the same.




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1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 310

4

103

102

101

100

101

102

103

104

105

wavelength (m)

leakymoderadiationloss

(dB/km)

TE01

TE02

EH11

Fig. 7. Radiation leakage through a finite number (17) of cladding layers in theOmniGuide fiber of Fig. 4. The lowest-loss mode is TE01 (solid blue) and the next-lowest is TE02 (red dots), while the linearly-polarized EH11 mode (black circles)typifies the higher losses for mixed-polarization modes due to the smaller TM bandgap.

bound for most analyses (e.g. dispersion relations and perturbation theory), with theleakage rates at most included as an independent loss term.

The radiation losses are proportional to the field amplitude | E|2 in the cladding,which goes like 1/R4 for TE0 from Eq. (2), multiplied by the surface area (which scalesas R). Thus,

TE0 radiation leakage 1/R3, (7)

the same as the scaling of TE0 ohmic losses in a hollow metallic waveguide [16,17]. Incontrast, because of their lack of a node near the boundary, TM and mixed-polarizationradiation losses scale only as 1/R.

By conservation of energy, one can also express this n as P+n / Pn, where P+n isthe radially-outgoing power of the leaky mode per unit length and Pn is its forward-propagating power in the core. It has been suggested [13] that one instead use n P+n /Pn based on the powers at the real-n resonance peak, but one can show that thisformulation is too small by a factor of 4. In particular, denote the outgoing/incoming

wave amplitudes at by S(). From above, S has a root at n = n + i/2, so Scan be Taylor-expanded: S() = s (n i/2) for some s. The outgoing amplitudeS+ can then be inferred by time-reversal symmetry, S+() = s ( n + i/2), andso we see immediately that

P+n |S+(n + i/2)|2 = 4|S+(n)|2 4P+n , and thus

= 4. (We have also verified this factor of 4 by explicit computation of fluxes.)




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1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 310

5

0

5

10

15

20

25

30

35

40

wavelength (m)

chromaticdispersionD(ps

/nm

km)

OmniGuide TE01

metal TE01

Fig. 8. Group-velocity (chromatic) dispersion of the TE01 mode in both the Om-niGuide fiber of Fig. 4 (solid blue) and a hollow metallic waveguide with the samecore radius (green circles).

5 Group-velocity Dispersion

Given a dispersion relation (), one important quantity is the group-velocity dispersionD (the rate at which pulses spread), canonically defined as [15]:

D 2

2c

d2

d2, (8)

in units of ps/(nm km): the pulse-spreading (ps) per km of propagation per nm of. This can be computed exactly from the function f of Eq. (6), which defines the

dispersion relation implicitly by f(, ) = 0. The group velocity is thus given by v d/d = f/f, where the subscripts denote partial differentiation, and one therebyfinds that d2/d2 = d(1/v)/d = 2ff/f2 f/f f2f/f3 . We evaluate all ofthese derivatives analytically by differentiating the transfer matrices. We also consideredmaterial dispersion (n varying with ) by the same methods, but we found that this hasa negligible effect (due to the small field penetration into the cladding). For example,assuming that the cladding has the same dn/d and d2n/d2 as silica at 1.55m,the contribution of material dispersion is less than 0.1 ps/(nm km) over most of thebandwidth. What remains is the waveguide dispersion, which stems from the geometryof the core as well as the variable penetrability of the cladding, Goos-Hanchen shifts [22],etc., all of which are taken into account by differentiating f. The resulting dispersionas a function of wavelength is plotted in Fig. 8 for the TE01 mode of the R = 30aOmniGuide fiber.

For comparison, we also plot the dispersion of the TE01 mode in an equal-radius hol-low metallic waveguide, given by: Dmetal = 22/(2c33R2), where =

2/c2 + 2/R2

and = 3.8317 is the first non-zero root of the J1 Bessel function. Except near theedges of the band gap, the dispersion is very similar to that of the metallic waveguidethat is, D is dominated simply by the core shapeand D = 12 ps/nm km at =1.55m. From the metallic dispersion relation, we can also conclude that D 1/R2.




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As we discuss in the next section, the practical implications of dispersion in an Om-niGuide fiber are quite different than in ordinary fibers, due to the absence of nonlineareffects. Because dispersion no longer interacts with nonlinearities, it can in principle becompletely compensated after any distance of propagation, allowing one to put all dis-persion compensation at the end of a fiber link, as well as to tolerate higher dispersions.Conversely, operating at or near a point of zero dispersion will no longer exacerbatefour-wave mixing noise.

Another important consideration is the relative dispersion slope (RDS), as measuredby (dD/d)/D; this quantity must ideally be matched in any dispersion-compensationsystem. For the OmniGuide fiber above, the RDS is around 0.0007nm1. This is 1530times smaller than the RDS of contemporary TrueWave-RS (0 .010nm1) and LEAF(0.021nm1) fibers, and smaller slopes are typically easier to achieve in dispersion-compensating fibers [25].

6 Suppression of Absorption and Nonlinearities

In this section, we compute the effect of absorption losses and nonlinearities in thecladding materials of an OmniGuide fiber. We show that these effects are stronglysuppressed for the TE01 mode, allowing highly lossy and nonlinear materials to beemployedgreatly broadening ones choices for high-index materials. Moreover, we willsee that there is the potential of greatly surpassing even the properties of silica fibers.

Absorption and nonlinearites correspond to tiny shifts in the dielectric constantof the materials, and can therefore be treated by perturbation theory, as described indetail by the appendix. This common technique allows one to compute the shift due to a small perturbation, using only the unperturbed modes (computed earlier viatransfer matrices). We have developed a new formulation of perturbation theory for usein this and subsequent sections, in explicit analogy with quantum mechanics, and usethe Dirac notation A |n = nB |n for the vectorial Maxwell eigenproblem in the modes|n. A perturbation is expressed as a shift A in the eigen-operator, and the first-ordershift in is then

(1)n = n|A|n from Eq. (43). The A for a small is given by

Eq. (48) in the appendix.

6.1 Absorption Loss

For absorption losses (also possibly including Rayleigh scattering), is a small imagi-nary part added to , representing the (material-dependent) dissipation rate. Substitut-ing Eq. (48) into Eq. (43), the resulting (purely imaginary) indicates the decay rateof the mode. Such a first-order perturbation method for dissipation losses, derived byvarious means, has often appeared in previous works [2631]. The losses of a materialare usually specified as a power dissipation rate 0 in units of e.g. dB/m; after somealgebra, one finds that using = i0 2c

/ produces an [(1)] in Eq. (43) that

is exactly the power dissipation rate in the same units as 0.Here, we calculate the losses of the TE01 mode in our example OmniGuide fiber by

assuming that the core is lossless, and that both the high and low-index cladding materi-als have the same dissipation rate 0. Furthermore, we divide the computed dissipationrate by 0, yielding the dimensionless, material-loss independent absorption suppres-sion coefficient of the mode. This is done for each frequency across the band gap,

yielding the plot in Fig. 9. Thus, the cladding losses are suppressed by more than fourorders of magnitude over most of the bandwidth, a result that will be better understoodfrom the scaling-law arguments in Sec. 6.3. For comparison, we also show the next-lowestloss TE02 mode, as well as the linearly-polarized EH11 (exemplifying the larger cladding




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1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 310

5

104

103

102

wavelength (m)

claddingabsorptionsuppr

ession

TE01

TE02

EH11

Fig. 9. Absorption losses due to the cladding materials the OmniGuide fiber (withcore radius 30a), as a fraction of the bulk cladding losses. The lowest-loss mode isTE01 (solid blue) and the next-lowest is TE02 (red dots), while the linearly-polarizedEH11 mode (black circles) typifies the higher losses for mixed-polarization modesdue to the smaller TM band gap.

penetration for modes with TM components).8 As was mentioned earlier, such a lossdifferential creates a mode-filtering effect that should allow single-mode-like operation.9

(Surface states and guided modes outside the band gap, i.e. propagating within thecladding region, will experience essentially the attenuation of the bulk cladding mate-rials and will be even more strongly filtered out.) Moreover, even if the cladding haslosses a thousand times greater than silicas 0.2 dB/km, the TE01 losses can be lowerthan the losses of silica fibers.

6.2 Nonlinearities

Another important problem in optical fibers is that of Kerr nonlinearities. Here, theindex n of the material varies as a function of electric-field strength: n n + n2| E|2,where n2 =

38n

(3) is the (small) nonlinear index coefficient of the material(s).10 Thus,to first order in n2, one has:

= 2nn2

E2 . (9)Kerr nonlinearities cause several problems in fiber systems: self/cross-phase modulation(SPM/XPM), where the energy at one frequency shifts the at the same/anotherfrequency; and also four-wave mixing (FWM), in which energy at one frequency leaks

8The strange shape of the EH11 curve in the plot is related to the fact that this mode labeling is

not exact for OmniGuide fibers; in fact, EH11 begins to make a transition to HE11-like behavior inthe middle of the band.

9Bessel-function afficionados can also show that the ratio of TE0 to TE0 losses is nearly (/)2,

where and are the corresponding roots ofJ1. This ratio is 3.35 for TE02 to TE01.10Often, one instead uses n = n+n2I, where I n| E|2 is the intensity of light; our analysis remains

the same except for the extra factor ofn.




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1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 310

9

108

107

106

wavelength (m)

claddingnonlinearitysuppression

Fig. 10. The TE01 modes suppression factor for cladding nonlinearities in the Om-niGuide fiber of Fig. 4, relative to nonlinearities that include the core.

into another frequency [15]. SPM and XPM interact with dispersion to limit dispersion-compensation schemes, and FWM causes noise/crosstalk between channels. Our concernhere is not to compute these effects per se, but rather to define the limits in which theymay be neglected.

The strength of nonlinearities in a fiber is given by a nonlinear lengthscale LNL,defined as the inverse of the SPM phase shift ; this is the lengthscale at which SPMand XPM become significant, and also appears as a scaling coefficient in the FWM noise[15]. As is seen below, LNL is inversely proportional to the mode power P (to first order),so it is conventional to instead define the nonlinear strength 1/(P LNL) = /P,which is a power-independent quantity proportional to the strength of nonlinear effectsin the fiber.

In order to compute , we substitute Eq. (9) into Eqs. (48,43), solving for the first-

order shift (1). This (1) is proportional to P through the field strength | E|2 inEq. (9). (An equivalent expression was derived by other means, for both the scalar andvectorial cases, in previous works [3234].)

We now apply the above methods to compute the nonlinear strength of the TE01mode in our OmniGuide fiber, assuming that the cladding materials all have somefixed n2. Instead of choosing a particular material n2, we instead calculate a claddingnonlinearity suppression factorwe divide by a 0, with the latter computed bysupposing that both the cladding and the core have the n2 nonlinearity. The results,plotted in Fig. 10, show that the cladding nonlinearities are suppressed by more thaneight orders of magnitude over much of the bandwidth. Thus, the nonlinearities of this

OmniGuide fiber will be dominated by the nonlinearities of air rather than those ofthe cladding, even for materials thousands of times more nonlinear than silica. Gaseshave Kerr constants almost 1000 times weaker than that of silicacombined with thefact that the core area here is almost 10 times larger than the effective area of a typicalsilica fiber, this implies nonlinearities in the OmniGuide fiber that are almost 10,000




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times weaker than those of silica fibers. Such low nonlinearities would open dramaticallynew areas for fiber operation: for example, high powers, closely-spaced channels and/orlow/zero dispersion without regard for FWM,11 use of non return-to-zero (NRZ) formatsat high bit rates, and dispersion compensation at larger intervals without regard forSPM. A better understanding of this nonlinearity suppression can be found in the scalinglaws derived in the following section.

6.3 Scaling LawsThe scaling laws for absorption loss and nonlinearities, as a function of core radius R,can be derived straightforwardly from the results in Sec. 3. In particular, the n|A|nintegral that determines the absorption loss has an integrand proportional to the fractionof | E|2 in the cladding, which scales as 1/R3 from Eq. (3), so:

TE0 cladding absorption 1R3

. (10)

This is a familiar result, since it is the same as the scaling of the TE01 ohmic dissipationlosses in a hollow metallic waveguide. The scaling for the nonlinear strength is foundby similar arguments. Here, however, there is an additional factor of | E|2 from Eq. (9),and thus 1/R4 from Eq. (2). The nonlinear strength of the cladding therefore scales

like 1/R

7

! The nonlinear strength 0, when one adds nonlinearities to the core, scalesinversely with the area R2 as in an ordinary fiber, so:

TE0 cladding nonlinearity

0 1

R5. (11)

It is because of these rapid 1/R3 and 1/R5 scalings that the cladding absorption andnonlinearities can be suppressed so strongly for TE01 in a large-core OmniGuide fiber.To demonstrate these scaling laws explicitly, we plot the absorption and nonlinear sup-pression coefficients as a function of R in Fig. 11, superimposing the predicted scalinglaws.

7 Waveguide Bends

A shallow bend in the waveguide axis, with unchanged waveguide cross-section12

andan instantaneous radius of curvature Rb R, can be thought of as a small perturba-tion to the straight waveguide. The modes of the straight waveguide propagate almostunchanged around the bend, with a slight coupling between them due to the bendingsuch coupling, or scattering, is described via the z-dependent perturbation theory of theappendix. After computing the strength of such coupling terms, we first consider modeconversion due to coupling of guided modes for short bends in which they are effectivelylossless. Then, in Sec. 7.3, we consider the effect of coupling to lossy modes over longerdistances, which causes an effectively increased loss rate for TE01.

7.1 The Bend Perturbation Operator

The unperturbed eigenmodes are coupled in a bend by the change A in the eigen-operator, produced by transforming the field-propagation equations into the curvilinear

coordinates of the bend. In order to find the propagation equations for the field on the11FWM noise is proportional to 2/(2+2), where the 2 term is usually negligible [15]. However,

with nonlinearities 10,000 times weaker than in silica, = 0.01 dB/km is sufficient to suppress FWMeven in the limit of = 0 (zero dispersion and/or channel spacing).12Bending can also create a stress-induced in the cladding material, which we do not consider

here since so little of our field penetrates into the cladding.




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10 20 3010

9

108

107

106

105

104

103

102

core radius R (a)

minimum

absorption/nonlinearity

suppression

absorption

absorption 1/R3

scalingnonlinearity

nonlinearity 1/R5

scaling

Fig. 11. Scaling of the cladding absorption and nonlinearity suppression factors fora core radius R varying from 7a to 30a (taking the minimum over the TE01 band ateach radius). Hollow squares/circles show the computed values, and the solid linesdisplay the values predicted by starting from the 30a value and applying the scalinglaws.

cross-section of a curved waveguide, we write Maxwells equations in cylindrical coordi-nates (,,) , make the correspondence (x, y, z) (Rb ,,R) where x points tothe center of curvature and x = 0 bisects the waveguide, and recompute Eq. (38) of theappendix. Then, after some algebra, we find the mode-coupling coefficient:

n A n = ExEyEzHzHyHz

x

cRb

ExEyEzHzHyHz

(12)As far as we can determine, this diagonal form for the bend-coupling operator wasnot previously knownalthough with some effort it can be shown to be equivalent toexpressions derived by other means in earlier works, for example [35]:

n A n

= n

x

Rb

Et Ht + Et Ht

(13)

iRb

EyH

z HyEz

.

The simple form of Eq. (12), cast into cylindrical coordinates x = r cos , immedi-

ately yields a well-known result when the waveguide has cylindrical symmetry. Here,the modes can be chosen with a dependence of the form eim, so:

n; m A n; m

= (r integral) 20

eim

ei + ei

d/4

= (r integral) m,1/2, (14)




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where m = mm. In other words, for a waveguide with a cylindrical symmetry, thereis a selection rule: a bend can only directly couple modes with |m| = 1 [35]. Fromthis selection rule, it follows that a mode cannot couple with itself, and the lowest-ordercorrection to is therefore of order 1/R2b , from Eq. (44) [36].

7.2 Bends in an OmniGuide Fiber

In a conventional optical fiber, there is only a single guided mode, so all of the coupling

in a bend is to the radiation continuum, which lies at a nearby since the index-contrast is small. In contrast, the OmniGuide fiber here is highly multi-mode, withmany other guided modes at values nearby to TE01 (within = 103 2/a); at thesame time, the cladding modes (analogous to the radiation continuum) lie at distant ( 0.1 2/a) thanks to the large gap. Thus, since mode coupling varies inverselywith , OmniGuide-fiber bend effects will be dominated by coupling/scattering intoother guided modes in the core. In this case, we can employ Eq. (46) to compute thescattered power, which is therefore proportional to (Rb)

2. This scattered power willbe dominated by the closest m = 1 mode, and is significant when Rb approaches 1/.

Moreover, we can be guided once again by the analogy with hollow metallic waveg-uides. There, as was discussed in Sec. 3, the TE01 mode is actually degenerate ( = 0)with another mode, TM11. The selection rule allows these two modes to couple, andtherefore can produce significant scattering for any bend radius. Fortunately, the Om-

niGuide cladding breaks this degeneracy, but the strongest scattering is still commonlydue to the EH11 mode (the analogue of TM11), which has a 104 2/a 10cm1for R = 30a. Actually, even in the OmniGuide fiber, EH11 intersects TE01 at a singledegeneracy point (here, at = 1.698m)the vicinity of this point must therefore beexcluded from the usable bandwidth. From the scaling laws of Eqs. (4,5), however, wesee that HE1 modes may have smaller for large Rfor the present structure, theHE11 and HE12 modes make a significant contribution (often 50%) to the bend lossesfor many wavelengths.

We compute the minimum bending radius R0.1% for which the worst-case scatteredpower from Eq. (46) is 0.1%. As shown in the appendix, this is ostensibly independentof the number of turnsthe scattered power of Eq. (46) simply oscillates with bendingdistance. There are additional loss mechanisms for large numbers of turns that weconsider below, however. In this computation, we include the EH

1and HE

1modes

for = 1, 2, 3; the contribution of other modes is negligible. The results are plotted inFig. 12. To find the losses for a different bend radius Rb, one would then scale 0.1% by(R0.1%/Rb)

2 (although this fails as the losses approach 100% and perturbation theoryis no longer valid). Furthermore, given the scaling from Eq. (5) and the fact thatthe integrand of Eq. (12) yields an additional factor of R, we find that:

minimum bend Rb for 0.1% losses, R0.1% R2. (15)

For large R, however, if coupling to HE modes with 1/R2 begins to dominate,one has instead R0.1% R3.7.2.1 Long-distance bends: effects of lossy modes

Matters become more complex when the bend continues for a long distance, where onemust take into account the lossiness of the modes. (Random variations in the bend couldalso be considered, by the techniques of Sec. 8.3.) In the derivation of Eq. (46), parasiticwaves scattered at earlier positions interfere with waves scattered at later positions dueto the phase mismatch . If the parasitic mode is lossy, however, the wave from theearlier position may have decayed away before it can interfere. Qualitatively, then, one




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1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.30

10

20

30

40

50

60

70

80

wavelength (m)

min.

bendradiusR0.1

%f

or0.1

%l

oss(cm)

Fig. 12. Minimum bending radius R0.1% to achieve 0.1% worst-case scattering lossesfor the TE01 mode in the OmniGuide fiber of Fig. 4. Conversely, the losses for a givenbending radius Rb are 0.1% (R0.1%/Rb)

2. The sharp peak (actually a divergence)in R0.1% is due to the point of degeneracy between TE01 and EH11.

expects that the scattering will follow Eq. (46) for roughly a decay length of the mode,and will then restart once the original parasitic modes have decayed. Quantatitively,we estimate the effective decay rate of the hybridized curved-waveguide eigenmode bycomputing its new complex from Eq. (44)simply substituting the complex n+in/2into the equation, where n is the power decay rate of |n. For , this yields:

(2)

n = n=n

n A n

2

2nn(n

n) , (16)

the lowest-order shift in the decay rate.13 This equation corresponds to taking the meanscattering losses from Eq. (46) once every 2/nn decay length. (If the modes bothdecay at the same rate, the field powers simply scale with distance and the interferenceproceeds as before.) When |n is TE01, we will have n n as in Figs. 7 and 9.

For example, at = 1.55m in our case of the R = 30a OmniGuide fiber with17 layers, the radiation decay length 2/ of EH11 is 60m from Fig. 7, with a bendradius R0.1% of 20cm for 0.1% losses. Then, to induce a decay rate

(2) in TE01 ofonly 0.01 dB/km, one needs the average bending radius to be 40cm (neglecting thecoupling to modes besides EH11).

14 If this were too large, or if EH11 were more lossy(say, due to a smaller bandgap), then one could simply add more periods to the Braggmirrors until the EH11 radiation losses were sufficiently low.

13The fact that this decay rate is second-order in the perturbation 1/Rb was also found in hol-low metallic waveguides [16, 17, 37], by explicitly solving for the eigen-field correction in the curvedwaveguide.14In an undersea cable, the fiber spirals continuously within a central core, to prevent the fiber

from bearing any of the cable tensiona typical radius of bending curvature would be around 50cm(corresponding to 5% fiber slack and a 5cm core radius).




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8 Elliptical and Uniform-Scaling Perturbations

In this section, we apply perturbation theory to treat elliptical and uniform-scalingperturbations of the fiber cross section, and develop a general form for the perturbationoperator. To begin with, we consider the effect of a constant ellipticity/scaling alongthe length of the fiber. For doubly-degenerate linearly-polarized m = 1 modes suchas EH11, our formulation provides a quantitative way to understand ellipticity-inducedbirefringence and polarization mode dispersion (PMD). Here, however, in the case ofm = 0 singlet modes (such as TE01) in an OmniGuide fiber, the main effect of ellipticityis to increase radiation losses by coupling to other, much lossier, modes. Our expressionsapply even for large index contrasts within the fiber, and the validity of the perturbationtheory depends only on the amount of ellipticity/scaling.

There has been much previous research on estimating such quantities as the localbirefringence induced by perturbations in the fiber profile, and such analyses can befound in many standard textbooks [23,38]. Most such treatments, however, are gearedtoward understanding low-contrast, weakly guiding systems such as the ubiquitous sil-ica fiber, and are not directly applicable to high-contrast systems such as OmniGu-ide fibers, as well as e.g. photonic-crystal fibers or lithographic integrated waveguides.While several approaches have been suggested for the perturbative treatment of suchhigh index-contrast systems [23, 3943] there is still no consensus on the best form for

such an analysis.Ellipticity and scaling correspond to moving dielectric-interface boundaries, and oneway to analyze this would ostensibly be to find the change and substitute it into thegeneral formula of Eq. (48). However, as discussed in Sec. 8.2 and in analyzed in detail by[44], this formulation yields incorrect results (compared to exact calculations) for largecontrasts , due to the changing boundary conditions on the fields. Instead, we employa different approach that maintains the field boundary conditions at the interfaces: werescale the coordinates to xs = x(1 + x), ys = y(1 + y) and compute the modified

A of Eq. (38). In particular, we focus on two cases: a uniform scaling x = y, and anelliptical scaling x = y. This change of coordinates alters the curl operations andinduces a perturbation A. Then, as before, we use z-independent perturbation theoryto compute the corrected of the perturbed eigenmodes, the birefringence, and theinduced losses. To treat scaling or ellipticity that vary along the propagation direction,

z-dependent perturbation theory can be employed to calculate the coupling/scattering.We believe that this is the first general perturbative treatment for fiber ellipticity thatis capable of handling large index contrasts.

8.1 Elliptical perturbations

We first consider the case of a uniform elliptical perturbation via the rescaling x =y, and define /(1 + ) = . In this case, we find after some cumbersome algebra[44] that the perturbation matrix elements take the following form in a cylindrical fiber:

n; m A n; m

=

ErEHrH

c

ii

i

i

ErEHrH

m,m2, (17)

where we have expressed the transverse fields in their cylindrical components, and thefinal Kronecker delta expresses a selection rule that only |m| = 2 modes can directlycouple under this form of perturbation.

In the case of a singlet m = 0 mode, e.g. the TE01 operating mode of the OmniGuidefiber, the introduction of such ellipticity leads to only a second-order shift in the real




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part of from Eq. (44), as well as increased losses from Eq. (16). Because a single modecan never be split into two modes, the birefringence and consequently the PMD for anym = 0 mode is zero.

Although the scaling of Eq. (17) is non-obvious, an empirically correct result isobtained here by postulating that an expression like Eq. (48) could be found even forlarge index contrasts: i.e. that the matrix element scales as the field integral over theperturbation s area. This area scales as R, while the normalized field-amplitudeE inside the perturbation scales as 1/R2 for TE0 from Eq. (2). On the other hand,this field amplitude is 1/R for m = 0 modes. Thus, the following scaling is obtained:

TE0 A m = 2

R2

. (18)

8.1.1 Scaling relations and increased losses for TE0 modes

Given the matrix element of Eq. (17) and the scaling relation of Eq. (18), we can find

the form of the shift in the complex propagation constant and the attenuation rate via Eqs. (16, 44). The latter equations are inversely proportional to the mode separation, which scales as 1/R2 from Eq. (4). Thus:

(2) 2R2 . (19)We consider the imaginary part (the attenuation rate) separately via Eq. (16). There,one divides by 2; on the other hand, we showed that the radiative/dissipative decayrates of the m = 0 modes scale as 1/R, and thus:

(2) = 2

(2) 2R

. (20)

These two scaling relations are verified in Fig. 13, where we plot the real and imaginaryparts of (2) as a function of core radius R for the OmniGuide TE01 mode with 1%ellipticity. The large additional loss induced by the ellipticity comes from the coupling tomixed-polarization modes that have large radiation losses, due to the smaller TM bandgapthis can be ameliorated, however, by increasing the number of cladding layers, asdiscussed below.

Another important result for the radiation loss of the elliptically perturbed TE01mode is its scaling with respect to the number of layers in the omnidirectional mirror.This scaling is depicted in Fig. 14. For comparison, we also show the radiation losses ofthe unperturbed TE01 and HE21/EH21 modes, where the latter contribute the dominantcoupling terms in Eq. (16) for (2). As the number of layers in the mirror increase,the radiation losses of HE21 and other m = 2 modes become exponentially greater thanthe TE01 losses, due to the smaller TM band gap. Thus, for a large number of layers,TE01 losses are expected to be totally dominated by the losses via ellipticity-couplingto the m = 2 modes, and moreover will have the same slope as the of the TM modes.

8.2 Uniform-scaling perturbations

By the same methods, we can find the analogous perturbation matrix elements in thecase of uniform rescaling of the fiber, x = y and /(1 + ) = . Although wedo not use this result directly here, uniform scaling has the advantage of being easilychecked against an exact transfer-matrix computation. We have done this, computing/ and 2/2 via Eqs. (43,44), and found that our perturbation theory yields the




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0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

x 103

3.5

3

2.5

2

1.5

1

0.5

x 106

1/R2c

(1/m)2

wavenumbershift(2) (1/m)

0.025 0.03 0.035 0.04 0.045 0.05

0.1

0.15

0.2

0.25

0.3

1/Rc (1/m)

wavenumbershift

(2) (dB/km)

=1%

=1%

Fig. 13. Scaling of the ellipticity-induced phase shift [(2)] and loss (2) forthe OmniGuide-fiber TE01 mode at = 1.55m as a function of the core radius R,varying from 2080a. The abscissa is the expected scaling form of 1/R2 and 1/R,respectively. The amount of ellipticity is = 1% and the cladding has 17 layers.

5 10 15 20 25 3010

8

106

104

102

100

102

104

106

108

Number of cladding layers

radiationlosses+

(2)(

dB/km)

unperturbed HE21

unperturbed TE01

ellipticity

2%

1%

0.5%

EH21

Fig. 14. The estimated radiative loss +(2) of the TE01 mode at = 1.55m forthe OmniGuide fiber of Fig. 4 with an elliptical perturbation, plotted versus thenumber of cladding layers for three ellipticities: = 0.5%, 1%, 2%. Red circles indic-tate the losses for the perturbed fibers, while solid blue lines above and below arethe losses of the unperturbed HE21 and TE01 modes, respectively. As the numberof layers increases, the losses become dominated by coupling to HE 21, due to theweaker band gap for TM polarizations.




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correct results for all modes. In contrast, using Eq. (48) to compute / is found toproduce the correct results only for pure TE m = 0 modesthe reason for this is thatthe changing discontinuity condition on Er makes the unperturbed modes a poor basisfor TM and mixed-polarization modes, causing significant errors in high index-contrastsystems [44].

8.3 Nonuniform stochastic elliptical perturbations

Elliptical deformations of an OmniGuide fiber can cause two sorts of losses. Above,we estimated the losses that come from the ellipticity per se, due to the induced modemixing, assuming that TE01 was adiabatically transformed into the analogous elliptical-fiber mode. As a fiber is drawn, however, there will be stochastic variations in theellipticity along the fiber length, and this will create scattering losses in which poweris transferred non-adiabatically from TE01 to other modes. Assuming that there is nosystematic ellipticity, one can derive coupled-power equations averaged over an ensembleof possible stochastic ellipticity realizations. These equations for the transfer of powerbetween modes take the form of a well known master equation [45]. Defining Pn as arelative power in the n-th mode (with decay rate n), one has:

Pnz

= nPn +

mMnm(Pm Pn), (21)

where the coupling (transition probability) Mnm is defined as follows. First, we separatethe perturbation A of Eq. (17) from the ellipticity by A a, and supposethat the ellipticity (z) is a stochastic function with zero mean and an autocorrelationC(z) ()( z). Defining the Fourier transform () C(z)eizdz, we arriveat [45] Mnm | n|a|m |2(nm). One must choose a form for the autocorrelationfunction C(z), and here we assume that it is a Gaussian with rms ellipticity 0 and a

correlation length Lc(0): C(z) = 20e[z/Lc(0)]

2

. For stochastic ellipticity from fiberdrawing, a typical 0 1% [46], and it is likely that Lc 10cm.

The linear equations of Eq.(21) can be solved by finding their eigenvectors pj andeigenvalues j , where the effective decay rates j are real as long as the autocorrelationC(z) is real-symmetric, and they can also be shown to be non-negative [45]. These

j are

the effective decay rates, and after a long distance the decay will be dominated by the

smallest eigenvalue 0 and its corresponding equilibrium power distribution p0. In thecase of the OmniGuide fiber, elliptical perturbations of Eq. (17) couple the TE01 modepredominantly to HE21. Using these two modes and Eq.(21), we compute the effectiveloss rate 0 as a function ofLc for the OmniGuide fiber with 0 = 1% at = 1.55m.15The results, presented in Fig. 15, show that as long as Lc 1.6mm, the scattering intothe the lossier HE21/EH21 modes yields TE01 losses 0 < 0.01 dB/km. In the insert,we plot the relative power in the modes as a function of Lc. For Lc 1.6mm 1/,the fractional power in the parasitic modes is less than 104.

9 Uncorrelated Roughness Losses

Another potential source of losses in any fiber is interface roughness and, since such ef-fects worsen with increasing index contrast, they may be a practical concern in OmniGu-

ide fibers. In this section, we make an order-of-magnitude estimate of roughness lossesin an OmniGuide fiber, assuming uncorrelated small scatterers on the core/claddinginterface. Fortunately, we find that the near-node in the electric field at the interfacehas the effect of greatly suppressing such roughness effects.

15We have also solved the equations with 16 modes and found that the main conclusions are the sameas in the case of two-mode coupling, with merely a more complicated curve shape.




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0 200 400 600 800 1000 1200 1400 1600 1800 200010

4

103

102

101

100

101

102

Lc

(m)

effectivelosses(dB/km)

0 500 1000 150010

6

104

102

100

Lc

(m)

relativepowerinparasiticmod

es

EH21HE

21

1% ellipticity

TE01

Fig. 15. Enhanced loss of the TE01 mode at = 1.55m from randomly-varyingellipticity with rms 0 = 1%, as a function of the ellipticity correlation length Lc, dueto coupling with the HE21 and EH21 modes. The inset shows the fractional scatteredpower in the parasitic HE21/EH21 modes as a function of Lc. If the correlationlength exceeds 1.6mm, the induced losses become smaller than 0.01 dB/km andthe fractional parasitic power is less than 104.

Suppose that we have a scatterer with volume Vs and a given by the core/cladding

contrast. Since it is small, we take E to be roughly constant inside Vs, in which casethe scatterer can be treated by the volume-current method [47] as a radiating dipole

with moment p = EVs.16 Instead of exactly computing the lost radiation, we make

several order-of-magnitude approximations. First, we assume that all radiated poweris lost, since only negligible fraction should re-radiate into TE01 compared to all the

other modes ( and m are irrelevant for a point-like perturbation). Second, sincethe core is much larger than the wavelength, we neglect its finite size and consider adipole radiating on a planar interface between semi-infinite regions. Third, we neglectthe effect of the omnidirectional mirrors on one side of the dipole; generally, these areexpected to reduce the radiated field, so this is a conservative approximation. Thus, inthe end, we treat Vs as simply a dipole radiating in vacuum, with scattered power [38]:

Ps =c2

00

12

c

4|p|2 =

4

12

EVs2 , (22)in dimensionless units (c = 0 = 0 = 1) for simplicity.

This is the approximate power radiated by a single scatterer. For multiple scatterersdistributed along the waveguide in an uncorrelated fashion, so that their radiation is

incoherent, one can simply add the power radiated from each one. If the scatterers havedensity s (scatterers per interface area) then the power radiated per length is Ps2Rs.To convert this into a power-attenuation rate , we divide it by the forward-propagating

16This is an approximation for large ; the correct moment is generally lesssee e.g. the induceddipole moment of a spherical scatterer in [38].




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1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 310

8

107

106

105

104

103

102

101

wavelength (m)

roughnessradiationefficienc

ys

(1/a5)

using E field at interface

using mean E over r=10nm

Fig. 16. Roughness radiation efficiency s, as defined by Eq. (24) for the TE01mode of the OmniGuide fiber of Fig. 4, based on the | E|2 at the core/claddinginterface (blue dots) or averaged over a 10nm interval (red circles). Approximateroughness losses are computed by multiplying s with a dimensionful quantity sthat is dependent on the scatterer quantity; e.g. s = 0.03 dB/km for one 10nmscatterer every 10m2.

power P = n|B|n /4 = 1/4 (c.f. Eq. (42)) in our normalized field units:

=Ps 2Rs

P=

4

12

E2 2R 4 sV2s , (23)where we have separated out the terms that are independent of our assumptions aboutthe scatterer dimensions. Together, the parenthesized terms have units of 1/length5 inour units, and we can compute them as a dimensionless radiation efficiency factors in our units of 1/a

5:

s 24R

3

E2 , (24)and from that compute = s s, where s is the dimensionful measure of the scattererquantity:

s sV2s

a5 10

ln10, (25)

with a 10/ ln 10 factor to convert to dB units (e.g. dB/km). If we take the scatterersize to be Vs = (10nm)3 and s = 0.1/m2, this gives s = 0.03 dB/km for the a of theOmniGuide fiber. In Fig. 16, we plot s vs. for our fiber, taking E as the field at theinterface. The plot indicates a sharp dip towards s = 0, due to the point where the nodein the field lies directly on the interfacethis feature is unphysical, since it indicates abreakdown of our assumption of constant E in the scatterer. For comparison, we alsoplot s where | E|2 is averaged over r = 10nm. In either case, we see that s < 103over a wide bandwidth, providing a comfortable margin in both the scatterer parametersVs and s as well as our earlier approximations. In general, | E|2 at the interface goes




27/32

like 1/R4 from Eq. (2), so the uncorrellated roughness losses are expected to scale as1/R3 (just like the radiation-leakage and absorption losses).

10 Scaling Laws with Index Contrast

Although the computations in this paper assumed an index contrast of 4.6/1.6, thesame qualitative results hold for a wide range of indices as long as the confinementremains strong. The index contrast manifests itself in the decay rate of the field into the

cladding, which determines the characteristic penetration depth dp, and the analogywith the metallic waveguide remains valid when dp R. Below, we derive scalingrelations to predict how the OmniGuide-fiber properties will vary with the claddingindices (assuming dp R).

In particular, with each bilayer of cladding, the fields decrease by some attenuationfactor (, ). For modes nearly on the = c light line, using quarter-wave bilayersgiven by Eq. (1), and defining n n2 1, the mid-gap for TE/TM fields is [3,22]:

te= nlonhi , (26)

tm=

n2lo

nhi

n2

hinlo> te. (27)

The penetration depth dp of | E|2 is then proportional to a/(1 2). From Sec. 3, theTE0 cladding | E| dhi/R2 = fhia/R2, where fhi dhi/a = nlo/(nlo + nhi) fromEq. (1). We can then generalize Eq. (3) to:

fraction of

| E|2 in cladding for TE0

f2hi(1 2te)

aR

3. (28)

The scaling of absorption loss is the same, except that from Sec. 6.1 it also proportionalto n, whose average is n (nhi nlo)fhi + nlo. Thus,

TE0 cladding absorption nf2

hi(1

2te)

a

R3

. (29)

One can similarly find the scaling of nonlinearities (whose integrand is n| E|4):

TE0 cladding nonlinearity

0

nf4hi

(1 4te) a

R

5. (30)

We have verified that these scalings are accurate over a wide range of index contrasts,and corresponding relations can also be found for the other quantities computed in thispaper.

The number of cladding layers required to achieve a given radiation leakage rate forTE or TM modes scales inversely with log(1/te) or log(1/tm), respectively. So, forexample, if one used indices of 1.459/1.450 (doped silica) [8, 9], 2000 cladding layerswould be required to achieve the same TE radiation-leakage rates as in Fig. 7.

The bandwidth also varies with index contrast. One is limited by the size of the TMgap along the light line (due to coupling such as in Sec. 8.1.1), and the fractional sizeof the TM gap on = c for our quarter-wave stack is [22]:

tm0

=4

sin1

n2

hinlo n2lonhi

n2hinlo + n2lonhi

, (31)




28/32

where 0 is the mid-gap frequency:

0 =nlo + nhi4nlonhi 2ca . (32)

The periodicity a changes with index as well, assuming a fixed vacuum . If is chosento be near 0, and 0 0a/(2c), then a 0.

11 Concluding Remarks

We have presented the propagation characteristics of the lowest-loss TE01 mode in anexample OmniGuide fiber with a large core radius R = 30a. Although it is impossibleto consider all possible imperfections in such a fiber, we have studied radiation leakage,material absorption, nonlinearities, bends, acircularity, and roughness, and have shownthat all of these are within acceptable bounds. Indeed, such properties as material ab-sorption and roughness are suppressed by four or more orders of magnitude, holdingforth the promise of optical fibers with substantially lower losses and negligible nonlin-earities compared to conventional silica fiberseven when highly lossy and nonlinearmaterials are used for the multilayer cladding. (The same properties open the possibilityof fibers for currently inaccessible wavelengths.) Almost all OmniGuide fiber propertiesare dominated by the geometry of the core, and for a wide range of R follow gener-

alized scaling relations that we have derived, as summarized in Table 1. Even thoughsuch a large-core OmniGuide fiber is ostensibly highly multi-mode, the higher lossesof the other modes creates a modal-filtering effect that allows the fiber to operate in asingle-mode fashion as long as the differential losses are greater than the rate of couplingfrom imperfections. In this respect, and in many others, such a fiber is analogous to thehollow metallic waveguides that have been extensively studied in the microwave regime.In future publications, we will consider many of these issues in greater detail, as well asother topics such as input/output coupling and dispersion tailoring.

Many of our results were developed from a general perturbation theory, derived froman abstract eigenproblem formulation, that can be applied to arbitrary waveguidingsystems. This perturbation theory is summarized in the following appendix, whereas thespecific forms of the perturbation operator for particular imperfections were presentedin the preceding sections.

quantity scaling

typical mode separation 1/R2, or 1/R with EH11group-velocity dispersion D 1/R2

leakage, absorption, and roughness losses 1/R3

cladding nonlinearity suppression 1/R5

minimum bend radius R0.1% for 0.1% bend losses R2 R3

ellipticity- losses from mode mixing 2/R

Table 1. Scaling relations with core radius R for TE01 in OmniGuide fibers.

Appendix: Waveguide Perturbation Theory

Having computed the exact eigenmodes of a perfect OmniGuide fiber via the transfermatrix method, we then considered the problem of various imperfections and perturba-tions to this idealized situation, such as absorption, nonlinearities, bending, ellipticity,and so on. In principle, one could compute the exact solutions under these conuiditions,but since the effects are small, they can be more efficiently estimated via perturbation




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theory. We outline the development of such a perturbation theory in this appendix, anda later publication will consider the matter in greater detail [48].

Perturbation theory has been developed in many forms for Maxwells equations.Here, we derive it by first expressing Maxwells equations in waveguides as a (vecto-rial) generalized eigenproblem, and then appropriating the perturbation theory from thestandard abstract-algebraic formulation developed for quantum mechanics [49]. Partic-ular waveguide perturbations are then simply a matter of finding th

omni guide

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