We present the light-propagation characteristics of OmniGuide fibers, which guide light by concentric multi-layer dielectric mirrors having the property of omnidirectional reflection. We show how the lowest-loss TE01 mode can propagate in a single-mode fashion through even large-core fibers, with other modes eliminated asymptotically by their higher losses and poor coupling, analogous to hollow metallic microwave waveguides. Dispersion, radiation leakage, material absorption, nonlinearities, bending, acircularity, and interface roughness are considered with the help of leaky modes and perturbation theory, and both numerical results and general scaling relations are presented. We show that cladding properties such as absorption and nonlinearity are suppressed by many orders of magnitude due to the strong confinement in a hollow core, and other imperfections are tolerable, promising that the properties of silica fibers may be surpassed even when nominally poor materials are employed.
©2001 Optical Society of America
Telecommunications has continued to push optical fibers towards ever-more demanding applications—such as high bit rates, dense wavelength-division multiplexing (DWDM), and long distances—and in response there has been renewed interest in alternative fiber designs to lift fundamental limitations of silica fibers. A particularly exciting departure from traditional fibers are fibers based on photonic band gaps, forbidden frequency ranges in periodic dielectric structures that can confine light even in low-index or hollow regions . Two main classes of fibers have emerged using photonic band gaps: photonic-crystal “holey” fibers that use a two-dimensional transverse periodicity , and Bragg fibers1 that use a one-dimensional periodicity of concentric rings [3–13]. In this paper, we study the propagation of light in a novel class of Bragg fibers: “OmniGuide” fibers with a hollow core, which use a multilayer cladding that exhibits omnidirectional reflection in the planar limit [7, 10, 14].
In the following sections, we show how OmniGuide fibers bear strong resemblances to the hollow metallic waveguides that are used in the microwave regime, confining a set of guided modes almost entirely within the hollow core with similar field patterns and dispersion characteristics. Because of this strong confinement, we prove that radiation leakage, material absorption, and nonlinearities from the cladding layers can be suppressed by many orders of magnitude. We also study imperfections, such as waveguide bends, acircularity (ellipticity), and surface roughness, and present both general expressions 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 modes-this produces modal filtering that allows even a highly-multimode OmniGuide fiber to operate in an effectively single-mode fashion. Because this TE01 state is cylindrically symmetrical and non-degenerate, it has the additional benefit of immunity to polarization-mode dispersion (PMD) from fiber birefringence. In this way, we demonstrate that OmniGuide fibers have the potential to lift three major physical limitations on silica fibers: losses (currently ~0.2 dB/km), nonlinearities, and PMD.
One of the hallmarks of photonic crystals is their flexibility, since their optical properties are subject to deliberate structural and materials choices. OmniGuide fibers also embody this freedom, permitting a wide variety of layer-thickness designs to enhance or inhibit specific characteristics (e.g. to tailor dispersion parameters), which we do not explore here. Moreover, due to the suppression of cladding properties, a much broader range 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 OmniGuide fiber properties, and especially focus on general scaling laws, phenomena, and design tradeoffs that apply in such systems. For a particular example system, we arbitrarily select an index contrast of 4.6 to 1.6 in the cladding, as has been used in several previous publications [7, 10, 11, 13].2 We explain how the properties scale with the indices in Sec. 10, however, and our scaling laws, analytical techniques, and qualititative results hold 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 for OmniGuide fibers, and develop an intuition for their behavior by an analogy with hollow metallic waveguides [16,17]. One could also compare with other prior work, including the hybrid system of a metallic waveguide whose inner surface is coated with a multilayer Bragg mirror [18, 19], as well as several other hollow waveguiding systems , but the pure metallic guide provides the simplest foundation for understanding. In the left panel of Fig. 1, we depict a schematic of an OmniGuide fiber forming a hollow dielectric waveguide. A hollow core (index of refraction unity) of radius R is surrounded by a multilayer cladding that consists of alternating layers having high and low indices of refraction. 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 a is the thickness of one high/low bilayer (we select different thicknesses in Sec. 2.2). Once a mode frequency ν is computed in units of c/a, the physical value of a is determined via a=λν for some desired operational wavelengh λ . The radius R of the waveguide will vary in the differing examples presented in this paper, from a minimum of 2a to a 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 cylinder now 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=2a are depicted in the right panel of Fig. 2. Dispersions relations like this one depict two conserved quantities: the axial wavenumber β and the frequency ω. In cylindrical waveguides, modes can also be labeled by their “angular momentum” integer m. For waveguides that lie along the z axis, the (z,t,φ) dependence of the modes is then given by: e i(βz-ωt+mφ). In a hollow metal tube, the eigenmodes are purely polarized as TM (Hz =0) or TE (Ez =0), and the ℓ-th mode of a given m is labeled TX mℓ .
In the dielectric case, light is confined by the one-dimensional photonic band gap of the multi-layer cladding, which is easy to analyze in the limit as the cladding becomes planar. The one-dimensional gaps of the planar dielectric mirror structure as a function of β (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 confine modes strongly analogous to those of the metallic waveguide; this is verified below. Because every eigenmode has a finite, conserved m, the effective wavevector kφ =m/r in the direction goes to zero for r →∞. Without this fact, there would be no band gaps in Fig. 2, since nonzero kφ⊥ β would have to be projected onto the Bragg band diagram. Also shown in this figure, as gray regions, are the ranges of omnidirectional reflection: the frequencies at which any incident wave from air will be reflected by the planar mirrors (and vice versa). Omnidirectional reflection per se is not strictly necessary for guidance in these fibers, but its presence is strongly correlated with the regimes of large, polarization-independent gaps along the light line that are important for many of the properties studied in this paper.
Bragg mirrors have different band-gaps for “TE” and “TM” polarizations, referring to 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 and TM labels only for m≠0; all m≠0 modes have some nonzero Er component.
2.1 The modes in an OmniGuide fiber
The modes supported by any cylindrical waveguide, including metallic waveguides, OmniGuide fibers, and traditional silica fibers, can be computed by the transfer-matrix method of . Here, the longitudinal fields (Ez and Hz ) of a given (m, ω, β) in an annular region of index nj are expanded in Bessel functions Jm (kjr) and Ym (kjr), with
. At each interface, the coefficients are related by a 4×4 transfer matrix that matches boundary conditions.3 The product of all these transfer matrices yields a single matrix relating the fields in the core to those in the outermost cladding. Then, by application of appropriate boundary conditions, the βn wavevectors of the various 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 the cladding, and therefore are truly guided modes in the limit of infinitely many cladding layers (the case of finite layers is considered in Sec. 4). Most of these modes lie above the ω=cβ light line, and thus propagate within the hollow core in much the same way as the modes of a metallic waveguide. It is also possible, however, for modes to lie beneath the light line and yet inside the band gap, in which case they are surface states localized around the core/cladding interface—we discuss this possibility further below.
In Fig. 3, we show the computed guided modes of the OmniGuide fiber with coreradius R=2a and the abovementioned planar-mirror parameters. These modes are at nearly the same frequencies as the metallic waveguide modes of Fig. 2, with the one-dimensional bandgaps simply superimposed. In the dielectric waveguide, the modes are only 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 within the core, and have the same orthogonality relationship as in a metal waveguide (see the appendix), the field patterns must also be nearly identical. We consider this analogy in greater detail in a later publication , 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 easy to 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 most desirable regime for fiber operation. In this section, we motivate the use of larger, ostensibly multi-mode cores for OmniGuide fibers, and describe the fiber parameters that we will use for subsequent computations in this paper.
In metallic waveguides, the lowest-loss mode is TE01, and its ohmic losses decrease as 1/R 3 [16, 17]. Moreover, the differential losses between TE01 and other modes create a 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 become dominated by scattering into other closely-spaced modes, especially into the degenerate TM11 mode via bends, and it was found that the optimal radius was in the range of 4λ-11λ .
Similar results hold for OmniGuide fibers: the lowest-loss mode is TE01, and many of its losses fall off as 1/R 3 (for much the same reasons as in metallic waveguides). Like the metallic waveguides, and unlike silica fibers with their small material contrasts, we demonstrate a strong modal-filtering effect based on the degree of confinement in the core. Also as before, inter-modal scattering worsens with increasing R. (We show that bending is less of a problem than in metallic waveguides, however, since the dielectric cladding breaks the TE01/TM11 degeneracy.) Thus, for the example OmniGuide fiber that 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 in Sec. 6.1 then lies at a frequency of ω≅0.28·2πc/a, so if we make this correspond to the standard λ=1.55µm of telecommunications, we have a=0.434µm. Equivalently, R=13.02µm=8.4λ, in the favorable range for the metallic-waveguide analogy. 4 Throughout this paper, however, we will emphasize scaling laws with R in order to highlight the effects 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 obtained for a “quarter-wave” stack in which each layer has equal optical thickness λ/4: d hi/d lo=n lo/n hi . Normal incidence, however, corresponds to β=0, whereas the modes of interest in the OmniGuide fiber lie almost on the β=ω/c light line (in the limit of R→∞, the lowest-order modes are essentially plane waves propagating along ẑ). Thus, we employ layer thicknesses determined by the quarter-wave condition along the light line of air (similarly applied in ):
which yields d hi≅0.2176a and d lo≅0.7824a in this case. (Ref.  suggests an alternate method that optimizes the structure for confinement of a given mode, but this yields essentially 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 labeled by analogy to the modes of an equal-radius metallic waveguide. Two such modes in the OmniGuide fiber, the lowest-loss TE01 and the linearly-polarized5 EH11 (analogous to the TM11 mode in a metallic guide) are depicted in Fig. 5. The TE01 mode is circularly symmetric and “azimuthally” polarized (‖ )—thus, unlike the doubly-degenerate EH11 mode (two orthogonal polarizations), TE01 cannot be split into two modes of differing velocities by fiber imperfections, and is therefore immune to polarization-mode dispersion (PMD). (See also Sec. 8.) Actually, the mode labeling in an OmniGuide fiber is more complex than in a metallic waveguide, because sometimes a mode will cross the 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, the TM01 mode crosses the light line at ω≅0.27·2πc/a, while the TM02 mode continuously takes on the core field pattern of TM01. When in doubt, we label a mode as HE or EH depending upon whether it is dominated by Hz or Ez at r=0, respectively, and number the 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 properties are determined largely by the geometric size R of the core, within which the modes are confined. Throughout this paper, we thereby derive scaling relations for the different quantities 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 largely independent of details such as the precise index contrast that is used, so long as it is sufficiently large for the metallic analogy to hold, and will provide a broad understanding of the advantages and tradeoffs of OmniGuide fiber structures. Later in the paper, we exhibit 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 TE0ℓ (especially TE01) modes of the fiber. The critical property of TE0φ modes is that, by analogy with a hollow metallic waveguide, they have a node in their electric field (Eφ ) near r=R [16, 17] as depicted in Fig. 6. It then follows that the amplitude of the electric field in the cladding is proportional to the slope of Eφ at R. The form of Eφ in the core, however, is simply the Bessel function J 1(ξr/R), where ξ(ω) is roughly the ℓ-th zero of J 1. The slope at R is then (J 0(ξ)-J 2(ξ))·ξ/2R. Moreover, for our quarter-wave stack, Eφ peaks near each of the n hi→n lo interfaces . Thus, not including any normalization of the J 1 amplitude (i.e. Eφ ~1), we find that the unnormalized Eφ in the cladding scales as d hi/R. In addition, for most computations (such as the perturbation theory described in the appendix), one must normalize the power of the field: this means dividing by an additional factor proportional to ,
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:
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 the cladding remains roughly constant with changing R—their fractional ||2 in the cladding then 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 must scale as the area R 2. Moreover, in a metal waveguide, the dispersion relations look like , where ξn are roots or extrema of Bessel functions. Therefore, far from cutoff (R≫ξnc/ω),
this has important implications for many of the mode-coupling phenomena discussed later. Unfortunately, Δβ can be somewhat more complicated in an OmniGuide fiber, due to the finite field penetration into the cladding and due to the transitions into surface states and subsequent mode relabelings discussed in the previous section. For example, consider the case of the EH11 mode, which is degenerate with TE01 in the metallic limit [16,17]. The degeneracy is broken by the penetration of the fields into the cladding, 6 inducing a small shift Δβ: by perturbation theory, Δβ is proportional to the amount of EH11 energy in the cladding, ~1/R:
Testing this scaling relation numerically for λ=1.55µm, however, we find that this separation eventually scales as 1/R 2 for R≳40a. It turns out that the “fundamental” HE11 mode has crossed the light line to become a surface state, and EH11 continuously transitions 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: in reality, there will be only a finite number of cladding layers in the omnidirectional mirror. Because of this, and the fact that the modes of interest lie above the light line of the outermost region, the field power will slowly leak out in a process akin to quantum-mechanical “tunneling.” This radiation loss, however, decreases exponentially with the number of cladding layers, and we quantify its smallness explicitly below. As has been observed elsewhere [11, 13], we show that only a small number of layers is required to achieve 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-core OmniGuide 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 βn (c.f. the appendix). For finitely many layers, modes are no longer truly confined (above the outermost light line), leading to a continuum of β values with infinitely-extended eigenstates . The former confined modes become leaky resonances: superpositions of real-β eigenmodes centered on βn with a width Δβ proportional to the radiative decay rate αn . Such resonances can be studied directly via the physical real-β continuum, but a powerful alternative method is the leaky-mode technique, which employs a conceptual analytic continuation from β to in the complex plane to satisfy a boundary condition of zero incoming flux . The power decay rate αn is then given by 2ℑ[n], where ℑ denotes the imaginary part; we have also verified the correctness of this decay rate against explicit real-β superpositions as well as by beam-propagation methods (BPM) .
For an OmniGuide fiber, the leaky-mode method is as follows. The transfer-matrix formulation allows one to compute 2×2 matrices (ω, β) that connect the TE and TM 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 function of ω and β for a given angular-momentum index m. (For m=0, the two polarizations decouple .) For a leaky mode, we wish to impose the boundary condition of zero incoming waves, so that there are no sources at ∞; such a solution exists whenever there is a zero eigenvalue of . Therefore, we use the determinant:
so that the leaky mode is defined by fm (ω, n)=0. Once such a n is found for a given ω, the corresponding eigenvector of (ω, n) yields the required mixed-polarization amplitudes. With finitely many layers, the only real roots of fm lie below the light line of the outermost region;7 above this light line, the incoming and outgoing flux are equal for real β , corresponding to steady-state standing-wave patterns. The small imaginary part of n above the light line yields the power decay rate αn =2[n]. A similar leaky-mode method was previously used for Bragg fibers, albeit with only a first-order Taylor-expansion approximation for ℑ[n] .
For all modes, the radiative decay α decreases exponentially with increasing numbers of 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 λ=1.55µm for 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. In Fig. 7, we display the computed radiation leakage rates α for the lowest-loss TE01 mode, the next-lowest loss TE02 mode, and the linearly-polarized EH11 mode to typify mixed-polarization modes. Like the absorption discussed later, these differential losses create 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 similar principle 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 greatest concern-it is more important to consider modes that couple strongly to TE01 via perturbations, 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, and even EH11 has decay lengths of meters, corresponding to ℑ[n]/ℜ[n] of ~10–13 and ~10-9, respectively. Thanks to these low losses, the modes can be treated as truly bound for most analyses (e.g. dispersion relations and perturbation theory), with the leakage rates at most included as an independent loss term.
The radiation losses are proportional to the field amplitude ||2 in the cladding, which goes like 1/R 4 for TE0ℓ from Eq. (2), multiplied by the surface area (which scales as R). Thus,
the same as the scaling of TE0ℓ ohmic losses in a hollow metallic waveguide [16, 17]. In contrast, because of their lack of a node near the boundary, TM and mixed-polarization radiation losses scale only as 1/R.
By conservation of energy, one can also express this αn as /P̃ n , where P ̃+ n is the radially-outgoing power of the leaky mode per unit length and P̃n is its forward-propagating power in the core. It has been suggested  that one instead use α′ n ≡ /Pn based on the powers at the real-βn resonance peak, but one can show that this formulation is too small by a factor of 4. In particular, denote the outgoing/incoming wave amplitudes at ω by S ±(β). From above, S - has a root at n≅βn +iα/2, so S-can be Taylor-expanded: S -(β)≅s·(β-βn -iα/2) for some s. The outgoing amplitude S + can then be inferred by time-reversal symmetry, S +(β)≅s*·(β-βn +iα/2), and so we see immediately that P ̃+ n ~|S +(βn +iα/2)|2≅4|S +(βn )|2~4 , and thus α≅4α′. (We have also verified this factor of 4 by explicit computation of fluxes.)
5 Group-velocity Dispersion
Given a dispersion relation β(ω), one important quantity is the group-velocity dispersion D (the rate at which pulses spread), canonically defined as :
in units of ps/(nm·km): the pulse-spreading (ps) per km of propagation per nm of Δλ. This can be computed exactly from the function f of Eq. (6), which defines the dispersion relation implicitly by f(ω, β)=0. The group velocity is thus given by v≡dω/dβ=-fβ /fω , where the subscripts denote partial differentiation, and one thereby finds that d 2 β/d ω 2=d(1/v)/dω=2fω fωβ /f 2β-fω ω/fβ - fβ β/ . We evaluate all of these derivatives analytically by differentiating the transfer matrices. We also considered material dispersion (n varying with ω) by the same methods, but we found that this has a negligible effect (due to the small field penetration into the cladding). For example, assuming that the cladding has the same dn/dω and /dω 2 as silica at 1.55µm, the contribution of material dispersion is less than 0.1 ps/(nm·km) over most of the bandwidth. What remains is the waveguide dispersion, which stems from the geometry of the core as well as the variable penetrability of the cladding, Goos-Hänchen shifts , etc., all of which are taken into account by differentiating f. The resulting dispersion as a function of wavelength is plotted in Fig. 8 for the TE01 mode of the R=30a OmniGuide fiber.
For comparison, we also plot the dispersion of the TE01 mode in an equal-radius hollow metallic waveguide, given by: D metal=-ω 2 ξ 2/(2πc 3 β 3 R 2), where and ξ=3.8317… is the first non-zero root of the J 1 Bessel function. Except near the edges of the band gap, the dispersion is very similar to that of the metallic waveguide—that is, D is dominated simply by the core shape—and D≅12 ps/nm·km at λ=1.55µm. From the metallic dispersion relation, we can also conclude that D~1/R 2.
As we discuss in the next section, the practical implications of dispersion in an OmniGuide fiber are quite different than in ordinary fibers, due to the absence of nonlinear effects. Because dispersion no longer interacts with nonlinearities, it can in principle be completely compensated after any distance of propagation, allowing one to put all dispersion 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 exacerbate four-wave mixing noise.
Another important consideration is the relative dispersion slope (RDS), as measured by (dD/dλ)/D; this quantity must ideally be matched in any dispersion-compensation system. For the OmniGuide fiber above, the RDS is around 0.0007nm-1. This is 15–30 times smaller than the RDS of contemporary TrueWave-RS (0.010nm-1) and LEAF (0.021nm-1) fibers, and smaller slopes are typically easier to achieve in dispersion-compensating fibers .
6 Suppression of Absorption and Nonlinearities
In this section, we compute the effect of absorption losses and nonlinearities in the cladding materials of an OmniGuide fiber. We show that these effects are strongly suppressed for the TE01 mode, allowing highly lossy and nonlinear materials to be employed—greatly broadening one’s choices for high-index materials. Moreover, we will see that there is the potential of greatly surpassing even the properties of silica fibers.
Absorption and nonlinearites correspond to tiny shifts Δε in the dielectric constant of the materials, and can therefore be treated by perturbation theory, as described in detail by the appendix. This common technique allows one to compute the shift Δβ due to a small perturbation, using only the unperturbed modes (computed earlier via transfer matrices). We have developed a new formulation of perturbation theory for use in this and subsequent sections, in explicit analogy with quantum mechanics, and use the Dirac notation Â|n〉=βn B̂|n〉 for the vectorial Maxwell eigenproblem in the modes |n〉. A perturbation is expressed as a shift ΔÂ in the eigen-operator, and the first-order shift in β is then Δ =⃗n|ΔÂ|n〉 from Eq. (43). The ΔÂ for a small Δε is given by Eq. (48) in the appendix.
6.1 Absorption Loss
For absorption losses (also possibly including Rayleigh scattering), Δε is a small imaginary part added to ε, representing the (material-dependent) dissipation rate. Substituting Eq. (48) into Eq. (43), the resulting (purely imaginary) Δβ indicates the decay rate of the mode. Such a first-order perturbation method for dissipation losses, derived by various means, has often appeared in previous works [26–31]. The losses of a material are usually specified as a power dissipation rate α0 in units of e.g. dB/m; after some algebra, one finds that using Δε=iα 0·2c√ε/ω produces an ℑ[Δβ (1)] in Eq. (43) that is exactly the power dissipation rate α in the same units as α0.
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 materials have the same dissipation rate α0. Furthermore, we divide the computed dissipation rate α by α0, yielding the dimensionless, material-loss independent absorption suppression coefficient of the mode. This is done for each frequency ω across the band gap, yielding the plot in Fig. 9. Thus, the cladding losses are suppressed by more than four orders of magnitude over most of the bandwidth, a result that will be better understood from the scaling-law arguments in Sec. 6.3. For comparison, we also show the next-lowest loss TE02 mode, as well as the linearly-polarized EH11 (exemplifying the larger cladding penetration for modes with TM components).8 As was mentioned earlier, such a loss differential 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 the cladding region, will experience essentially the attenuation of the bulk cladding materials and will be even more strongly filtered out.) Moreover, even if the cladding has losses a thousand times greater than silica’s ~0.2 dB/km, the TE01 losses can be lower than the losses of silica fibers.
Another important problem in optical fibers is that of Kerr nonlinearities. Here, the index n of the material varies as a function of electric-field strength: n′≡n+n 2||2, where is the (small) “nonlinear index coefficient” of the material(s).10 Thus, to first order in n 2, one has:
Kerr nonlinearities cause several problems in fiber systems: self/cross-phase modulation (SPM/XPM), where the energy at one frequency shifts the β at the same/another frequency; and also four-wave mixing (FWM), in which energy at one frequency leaks into another frequency . SPM and XPM interact with dispersion to limit dispersion-compensation schemes, and FWM causes noise/crosstalk between channels. Our concern here is not to compute these effects per se, but rather to define the limits in which they may be neglected.
The strength of nonlinearities in a fiber is given by a nonlinear lengthscale L NL, defined as the inverse of the SPM phase shift Δβ; this is the lengthscale at which SPM and XPM become significant, and also appears as a scaling coefficient in the FWM noise . As is seen below, L NL is inversely proportional to the mode power P (to first order), so it is conventional to instead define the nonlinear strength γ≡1/(PL NL)=Δβ/P, which is a power-independent quantity proportional to the strength of nonlinear effects in the fiber.
In order to compute γ, we substitute Eq. (9) into Eqs. (48,43), solving for the first-order shift Δβ (1). This Δβ (1) is proportional to P through the field strength ||2 in Eq. (9). (An equivalent expression was derived by other means, for both the scalar and vectorial cases, in previous works [32–34].)
We now apply the above methods to compute the nonlinear strength γ of the TE01 mode in our OmniGuide fiber, assuming that the cladding materials all have some fixed n 2. Instead of choosing a particular material n 2, we instead calculate a cladding nonlinearity suppression factor—we divide γ by a γ0, with the latter computed by supposing that both the cladding and the core have the n 2 nonlinearity. The results, plotted in Fig. 10, show that the cladding nonlinearities are suppressed by more than eight 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 of the cladding, even for materials thousands of times more nonlinear than silica. Gases have Kerr constants almost 1000 times weaker than that of silica—combined with the fact that the core area here is almost 10 times larger than the effective area of a typical silica fiber, this implies nonlinearities in the OmniGuide fiber that are almost 10,000 times weaker than those of silica fibers. Such low nonlinearities would open dramatically new areas for fiber operation: for example, high powers, closely-spaced channels and/or low/zero dispersion without regard for FWM,11 use of non return-to-zero (NRZ) formats at high bit rates, and dispersion compensation at larger intervals without regard for SPM. A better understanding of this nonlinearity suppression can be found in the scaling laws derived in the following section.
6.3 Scaling Laws
The 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|ΔÂ|n〉 integral that determines the absorption loss has an integrand proportional to the fraction of ||2 in the cladding, which scales as 1/R 3 from Eq. (3), so:
This is a familiar result, since it is the same as the scaling of the TE01 ohmic dissipation losses in a hollow metallic waveguide. The scaling for the nonlinear strength γ is found by similar arguments. Here, however, there is an additional factor of ||2 from Eq. (9), and thus 1/R 4 from Eq. (2). The nonlinear strength γ of the cladding therefore scales like 1/R 7! The nonlinear strength γ0, when one adds nonlinearities to the core, scales inversely with the area R 2 as in an ordinary fiber, so:
It is because of these rapid 1/R 3 and 1/R 5 scalings that the cladding absorption and nonlinearities 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 suppression coefficients as a function of R in Fig. 11, superimposing the predicted scaling laws.
7 Waveguide Bends
A shallow bend in the waveguide axis, with unchanged waveguide cross-section12 and an instantaneous radius of curvature Rb ≫R, can be thought of as a small perturbation to the straight waveguide. The modes of the straight waveguide propagate almost unchanged around the bend, with a slight coupling between them due to the bending— such coupling, or scattering, is described via the z-dependent perturbation theory of the appendix. After computing the strength of such coupling terms, we first consider mode conversion due to coupling of guided modes for short bends in which they are effectively lossless. Then, in Sec. 7.3, we consider the effect of coupling to lossy modes over longer distances, 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 ΔÂ 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 the cross-section of a curved waveguide, we write Maxwell’s equations in cylindrical coordinates (ρ, θ, ζ), make the correspondence (x, y, z)→(Rb -ρ, ζ,Rθ) where -x̂ points to the center of curvature and x=0 bisects the waveguide, and recompute Eq. (38) of the appendix. Then, after some algebra, we find the mode-coupling coefficient:
As far as we can determine, this diagonal form for the bend-coupling operator was not previously known—although with some effort it can be shown to be equivalent to expressions derived by other means in earlier works, for example :
The simple form of Eq. (12), cast into cylindrical coordinates x=r cosφ, immediately yields a well-known result when the waveguide has cylindrical symmetry. Here, the modes can be chosen with a φ dependence of the form eimφ , so:
where Δm=m-m′. In other words, for a waveguide with a cylindrical symmetry, there is a selection rule: a bend can only directly couple modes with |Δm|=1 . From this selection rule, it follows that a mode cannot couple with itself, and the lowest-order correction to β is therefore of order 1/ , from Eq. (44) .
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 β since the index-contrast is small. In contrast, the OmniGuide fiber here is highly multi-mode, with many other guided modes at β values nearby to TE01 (within Δβ=10-3·2π/a); at the same time, the cladding modes (analogous to the radiation continuum) lie at distant β (Δβ≳0.1 · 2π/a) thanks to the large gap. Thus, since mode coupling varies inversely with Δβ, OmniGuide-fiber bend effects will be dominated by coupling/scattering into other guided modes in the core. In this case, we can employ Eq. (46) to compute the scattered power, which is therefore proportional to (Rb Δβ)-2. This scattered power will be dominated by the closest m=1 mode, and is significant when Rb approaches 1/Δβ.
Moreover, we can be guided once again by the analogy with hollow metallic waveguides. There, as was discussed in Sec. 3, the TE01 mode is actually degenerate (Δβ=0) with another mode, TM11. The selection rule allows these two modes to couple, and therefore can produce significant scattering for any bend radius. Fortunately, the OmniGuide cladding breaks this degeneracy, but the strongest scattering is still commonly due to the EH11 mode (the analogue of TM11), which has a Δβ~10-4·2π/a~10cm-1 for R=30a. Actually, even in the OmniGuide fiber, EH11 intersects TE01 at a single degeneracy point (here, at λ=1.698µm)—the vicinity of this point must therefore be excluded from the usable bandwidth. From the scaling laws of Eqs. (4,5), however, we see that HE1ℓ modes may have smaller Δβ for large R—for the present structure, the HE11 and HE12 modes make a significant contribution (often 50%) to the bend losses for many wavelengths.
We compute the minimum bending radius R 0.1% for which the worst-case scattered power from Eq. (46) is 0.1%. As shown in the appendix, this is ostensibly independent of the number of turns—the scattered power of Eq. (46) simply oscillates with bending distance. There are additional loss mechanisms for large numbers of turns that we consider below, however. In this computation, we include the EH1ℓ and HE1ℓ modes for ℓ=1, 2, 3; the contribution of other modes is negligible. The results are plotted in Fig. 12. To find the losses for a different bend radius Rb , one would then scale 0.1% by (R 0.1%/Rb )2 (although this fails as the losses approach 100% and perturbation theory is no longer valid). Furthermore, given the Δβ scaling from Eq. (5) and the fact that the integrand of Eq. (12) yields an additional factor of R, we find that:
For large R, however, if coupling to HE modes with Δβ~1/R 2 begins to dominate, one has instead R 0.1%~R 3.
7.2.1 Long-distance bends: effects of lossy modes
Matters become more complex when the bend continues for a long distance, where one must take into account the lossiness of the modes. (Random variations in the bend could also be considered, by the techniques of Sec. 8.3.) In the derivation of Eq. (46), parasitic waves scattered at earlier positions interfere with waves scattered at later positions due to the phase mismatch Δβ. If the parasitic mode is lossy, however, the wave from the earlier position may have decayed away before it can interfere. Qualitatively, then, one
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 by computing its new complex from Eq. (44)—simply substituting the complex βn +iαn /2 into the equation, where αn is the power decay rate of |n〉. For Δα≪Δβ, this yields:
the lowest-order shift in the decay rate.13 This equation corresponds to taking the mean scattering losses from Eq. (46) once every 2/Δα n′n decay length. (If the modes both decay at the same rate, the field powers simply scale with distance and the interference proceeds as before.) When |n〉 is TE01, we will have α n ′ αn as in Figs. 7 and 9.
For example, at λ=1.55µm in our case of the R=30a OmniGuide fiber with 17 layers, the radiation decay length 2/α of EH11 is 60m from Fig. 7, with a bend radius R 0.1% of 20cm for 0.1% losses. Then, to induce a decay rate Δα(2) in TE01 of only 0.01 dB/km, one needs the average bending radius to be ≳40cm (neglecting the coupling 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 Bragg mirrors until the EH11 radiation losses were sufficiently low.
8 Elliptical and Uniform-Scaling Perturbations
In this section, we apply perturbation theory to treat elliptical and uniform-scaling perturbations of the fiber cross section, and develop a general form for the perturbation operator. To begin with, we consider the effect of a constant ellipticity/scaling along the length of the fiber. For doubly-degenerate linearly-polarized m=1 modes such as EH11, our formulation provides a quantitative way to understand ellipticity-induced birefringence and polarization mode dispersion (PMD). Here, however, in the case of m=0 singlet modes (such as TE01) in an OmniGuide fiber, the main effect of ellipticity is to increase radiation losses by coupling to other, much lossier, modes. Our expressions apply even for large index contrasts within the fiber, and the validity of the perturbation theory depends only on the amount of ellipticity/scaling.
There has been much previous research on estimating such quantities as the local birefringence induced by perturbations in the fiber profile, and such analyses can be found in many standard textbooks [23, 38]. Most such treatments, however, are geared toward understanding low-contrast, weakly guiding systems such as the ubiquitous silica fiber, and are not directly applicable to high-contrast systems such as OmniGuide fibers, as well as e.g. photonic-crystal fibers or lithographic integrated waveguides. While several approaches have been suggested for the perturbative treatment of such high index-contrast systems [23, 39–43] there is still no consensus on the best form for such an analysis.
Ellipticity and scaling correspond to moving dielectric-interface boundaries, and one way to analyze this would ostensibly be to find the change Δε and substitute it into the general formula of Eq. (48). However, as discussed in Sec. 8.2 and in analyzed in detail by , this formulation yields incorrect results (compared to exact calculations) for large contrasts Δε, due to the changing boundary conditions on the fields. Instead, we employ a different approach that maintains the field boundary conditions at the interfaces: we rescale the coordinates to xs =x(1+δx ), ys =y(1+δy ) and compute the modified Â of Eq. (38). In particular, we focus on two cases: a uniform scaling δx =δy , and an elliptical scaling δx =-δy . This change of coordinates alters the curl operations and induces a perturbation ΔÂ. Then, as before, we use z-independent perturbation theory to compute the corrected β of the perturbed eigenmodes, the birefringence, and the induced 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 that is capable of handling large index contrasts.
8.1 Elliptical perturbations
We first consider the case of a uniform elliptical perturbation via the rescaling δ≡δx =-δy , and define η≡δ/(1+δ)≅δ. In this case, we find after some cumbersome algebra  that the perturbation matrix elements take the following form in a cylindrical fiber:
where we have expressed the transverse fields in their cylindrical components, and the final Kronecker delta expresses a selection rule that only |Δm|=2 modes can directly couple under this form of perturbation.
In the case of a singlet m=0 mode, e.g. the TE01 operating mode of the OmniGuide fiber, the introduction of such ellipticity leads to only a second-order shift in the real part of β from Eq. (44), as well as increased losses from Eq. (16). Because a single mode can never be split into two modes, the birefringence and consequently the PMDf or any m=0 mode is zero.
Although the scaling of Eq. (17) is non-obvious, an empirically correct result is obtained here by postulating that an expression like Eq. (48) could be found even for large index contrasts: i.e. that the matrix element scales as the field integral over the perturbation Δε’s area. This area scales as δ · R, while the normalized field-amplitude | inside the perturbation scales as 1/R 2 for TE0ℓ from Eq. (2). On the other hand, this field amplitude is ~1/R for m≠0 modes. Thus, the following scaling is obtained:
8.1.1 Scaling relations and increased losses for TE0ℓ 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 α via Eqs. (16, 44). The latter equations are inversely proportional to the mode separation Δβ, which scales as 1/R 2 from Eq. (4). Thus:
We consider the imaginary part (the attenuation rate) separately via Eq. (16). There, one divides by Δβ 2; on the other hand, we showed that the radiative/dissipative decay rates α of the m≠0 modes scale as 1/R, and thus:
These two scaling relations are verified in Fig. 13, where we plot the real and imaginary parts of Δ (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 to mixed-polarization modes that have large radiation losses, due to the smaller TM band gap—this can be ameliorated, however, by increasing the number of cladding layers, as discussed below.
Another important result for the radiation loss of the elliptically perturbed TE01 mode 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 of the unperturbed TE01 and HE21/EH21 modes, where the latter contribute the dominant coupling terms in Eq. (16) for Δα(2). As the number of layers in the mirror increase, the radiation losses of HE21 and other m=2 modes become exponentially greater than the 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-coupling to the m=2 modes, and moreover will have the same slope as the α of the TM modes.
8.2 Uniform-scaling perturbations
By the same methods, we can find the analogous perturbation matrix elements in the case of uniform rescaling of the fiber, δ≡δx =δy and η≡δ/(1+δ)≅δ. Although we do not use this result directly here, uniform scaling has the advantage of being easily checked against an exact transfer-matrix computation. We have done this, computing ∂β/∂η and ∂ 2 β/∂η 2 via Eqs. (43,44), and found that our perturbation theory yields the correct results for all modes. In contrast, using Eq. (48) to compute ∂β/∂η is found to produce the correct results only for pure TE m=0 modes—the reason for this is that the changing discontinuity condition on Er makes the unperturbed modes a poor basis for TM and mixed-polarization modes, causing significant errors in high index-contrast systems .
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 mode mixing, assuming that TE01 was adiabatically transformed into the analogous elliptical-fiber mode. As a fiber is drawn, however, there will be stochastic variations in the ellipticity along the fiber length, and this will create scattering losses in which power is transferred non-adiabatically from TE01 to other modes. Assuming that there is no systematic ellipticity, one can derive coupled-power equations averaged over an ensemble of possible stochastic ellipticity realizations. These equations for the transfer of power between modes take the form of a well known master equation . Defining Pn as a relative power in the n-th mode (with decay rate αn ), one has:
where the coupling (transition probability) Mnm is defined as follows. First, we separate the perturbation ΔÂ of Eq. (17) from the ellipticity δ by ΔÂ≡δ·ΔÂ, and suppose that the ellipticity δ(z) is a stochastic function with zero mean and an autocorrelation C(z)≡〈δ(ζ)δ(ζ-z)〉. Defining the Fourier transform Ξ(κ)≡∫C(z)e -iκzdz, we arrive at  Mnm ≡|〈n|Δâ|m〈|2Ξ(βn -βm ). One must choose a form for the autocorrelation function C(z), and here we assume that it is a Gaussian with rms ellipticity δ0 and a correlation length Lc (δ 0): C(z)= e-[z/Lc(δ0 )]2. For stochastic ellipticity from fiber drawing, a typical δ 0~1% , and it is likely that Lc ≳10cm.
The linear equations of Eq.(21) can be solved by finding their eigenvectors j and eigenvalues -j, where the effective decay rates α̃j are real as long as the autocorrelation C(z) is real-symmetric, and they can also be shown to be non-negative . 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 0. In the case of the OmniGuide fiber, elliptical perturbations of Eq. (17) couple the TE01 mode predominantly to HE21. Using these two modes and Eq.(21), we compute the effective loss rate 0 as a function of Lc for the OmniGuide fiber with δ0=1% at λ=1.55µm.15 The results, presented in Fig. 15, show that as long as Lc ≳1.6mm, the scattering into the 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/Δβ, the fractional power in the parasitic modes is less than 10-4.
9 Uncorrelated Roughness Losses
Another potential source of losses in any fiber is interface roughness and, since such effects worsen with increasing index contrast, they may be a practical concern in OmniGuide fibers. In this section, we make an order-of-magnitude estimate of roughness losses in an OmniGuide fiber, assuming uncorrelated small scatterers on the core/cladding interface. Fortunately, we find that the near-node in the electric field at the interface has the effect of greatly suppressing such roughness effects.
Suppose that we have a scatterer with volume Vs and a Δε given by the core/cladding contrast. Since it is small, we take to be roughly constant inside Vs , in which case the scatterer can be treated by the volume-current method  as a radiating dipole with moment =Δε Vs .16 Instead of exactly computing the lost radiation, we make several order-of-magnitude approximations. First, we assume that all radiated power is lost, since only negligible fraction should re-radiate into TE01 compared to all the other modes (Δβ and Δm are irrelevant for a point-like perturbation). Second, since the core is much larger than the wavelength, we neglect its finite size and consider a dipole radiating on a planar interface between semi-infinite regions. Third, we neglect the effect of the omnidirectional mirrors on one side of the dipole; generally, these are expected to reduce the radiated field, so this is a conservative approximation. Thus, in the end, we treat Vs as simply a dipole radiating in vacuum, with scattered power :
in dimensionless units (c=µ 0=ε 0=1) for simplicity.
This is the approximate power radiated by a single scatterer. For multiple scatterers distributed 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 have density σs (scatterers per interface area) then the power radiated per length is Ps ·2π Rσs . To convert this into a power-attenuation rate α, we divide it by the forward-propagating power P=〈n|B̂|n〉/4=1/4 (c.f. Eq. (42)) in our normalized field units:
where we have separated out the terms that are independent of our assumptions about the scatterer dimensions. Together, the parenthesized terms have units of 1/length5 in our units, and we can compute them as a “dimensionless” radiation “efficiency” factor ρs in our units of 1/a 5:
and from that compute α=ρs ·s, where s is the dimensionful measure of the scatterer quantity:
with a 10/ln 10 factor to convert to dB units (e.g. dB/km). If we take the scatterer size to be Vs ≅(10nm)3 and σs ≅0.1/µm2, this gives s≅0.03 dB/km for the a of the OmniGuide fiber. In Fig. 16, we plot ρs vs. λ for our fiber, taking as the field at the interface. The plot indicates a sharp dip towards ρs =0, due to the point where the node in the field lies directly on the interface-this feature is unphysical, since it indicates a breakdown of our assumption of constant in the scatterer. For comparison, we also plot ρs where ||2 is averaged over Δr=10nm. In either case, we see that ρs <10-3 over a wide bandwidth, providing a comfortable margin in both the scatterer parameters Vs and σs as well as our earlier approximations. In general, ||2 at the interface goes like 1/R 4 from Eq. (2), so the uncorrellated roughness losses are expected to scale as 1/R 3 (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, the same qualitative results hold for a wide range of indices as long as the confinement remains 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 analogy with the metallic waveguide remains valid when dp ≪R. Below, we derive scaling relations to predict how the OmniGuide-fiber properties will vary with the cladding indices (assuming dp ≪R).
In particular, with each bilayer of cladding, the fields decrease by some attenuation factor κ(ω, β). For modes nearly on the ω=cβ light line, using “quarter-wave” bilayers given by Eq. (1), and defining , the mid-gap κ for TE/TM fields is [3, 22]:
The penetration depth dp of ||2 is then proportional to a/(1-κ 2). From Sec. 3, the TE0ℓ cladding ||~d hi/R 2=f hi a/R 2, where f hi≡d hi/a=ñlo/(ñlo+ñhi) from Eq. (1). We can then generalize Eq. (3) to:
The scaling of absorption loss is the same, except that from Sec. 6.1 it also proportional to n, whose average is ñ≡(n hi-n lo)f hi+n lo. Thus,
One can similarly find the scaling of nonlinearities (whose integrand is n||4):
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 this paper.
The number of cladding layers required to achieve a given radiation leakage rate for TE or TM modes scales inversely with log(1/κ te) or log(1/κ tm), respectively. So, for example, if one used indices of 1.459/1.450 (doped silica) [8, 9], ~2000 cladding layers would 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 TM gap along the light line (due to coupling such as in Sec. 8.1.1), and the fractional size of the TM gap on ω=cβ for our quarter-wave stack is :
where ω 0 is the mid-gap frequency:
The periodicity a changes with index as well, assuming a fixed vacuum λ. If λ is chosen to be near ω 0, and ν 0≡ω0a/(2πc), then a~λν 0.
11 Concluding Remarks
We have presented the propagation characteristics of the lowest-loss TE01 mode in an example OmniGuide fiber with a large core radius R=30a. Although it is impossible to consider all possible imperfections in such a fiber, we have studied radiation leakage, material absorption, nonlinearities, bends, acircularity, and roughness, and have shown that all of these are within acceptable bounds. Indeed, such properties as material absorption and roughness are suppressed by four or more orders of magnitude, holding forth the promise of optical fibers with substantially lower losses and negligible nonlinearities compared to conventional silica fibers—even when highly lossy and nonlinear materials are used for the multilayer cladding. (The same properties open the possibility of fibers for currently inaccessible wavelengths.) Almost all OmniGuide fiber properties are dominated by the geometry of the core, and for a wide range of R follow generalized scaling relations that we have derived, as summarized in Table 1. Even though such a large-core OmniGuide fiber is ostensibly highly multi-mode, the higher losses of the other modes creates a modal-filtering effect that allows the fiber to operate in a single-mode fashion as long as the differential losses are greater than the rate of coupling from imperfections. In this respect, and in many others, such a fiber is analogous to the hollow 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 as other topics such as input/output coupling and dispersion tailoring.
Many of our results were developed from a general perturbation theory, derived from an abstract eigenproblem formulation, that can be applied to arbitrary waveguiding systems. This perturbation theory is summarized in the following appendix, whereas the specific forms of the perturbation operator for particular imperfections were presented in the preceding sections.
Appendix: Waveguide Perturbation Theory
Having computed the exact eigenmodes of a perfect OmniGuide fiber via the transfer matrix method, we then considered the problem of various imperfections and perturbations 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 theory. We outline the development of such a perturbation theory in this appendix, and a later publication will consider the matter in greater detail .
Perturbation theory has been developed in many forms for Maxwell’s equations. Here, we derive it by first expressing Maxwell’s equations in waveguides as a (vectorial) generalized eigenproblem, and then appropriating the perturbation theory from the standard abstract-algebraic formulation developed for quantum mechanics . Particular waveguide perturbations are then simply a matter of finding the change in the eigen-operator induced by the perturbation. This approach to perturbation theory has the advantage of separating the details of Maxwell’s equations and the particular system from the generic eigenproblem algebra, but has not been widely used for the full-vector electromagnetic waveguide problem. Earlier work has preferred other methods, which in most cases yield equivalent results to our unified approach, but we have also found some novel generalizations and formulations.
In the following section, we use the Dirac notation of abstract state kets |ψ〉 and linear operators Â, where the state in this case represents the electromagnetic fields. The inner product 〈φ|ψ〉 is defined below.
The Maxwell Waveguide Eigenproblem
Here, we derive a general, abstract formulation for the propagation equation of electromagnetic states in a waveguide along the ẑ direction. Furthermore, we show that the modes of the waveguide satisfy a Hermitian generalized eigenproblem in β, which will then yield orthogonality relations, perturbation theory, and other results. An alternative eigenproblem in ω is described in , and another eigenproblem formulation for use in plasma physics was presented in 
Consider the source-free Maxwell’s equations for fields with a definite-frequency time-dependence e -iωt. In this case, the and fields can be cast purely in terms of their transverse (ẑ and ŷ) components E⃗ t and B⃗ t [23, 38], since:
With this substitution, and moving all of the z derivatives to one side in Maxwell’s equations, the field equations take on the form:
where we define Â, B̂, and |ψ〉 below. Eq. (35) plays the role of Schrödinger’s equation from quantum mechanics, with z taking the place of time. |ψ〉 is (abstractly) the electromagnetic field pattern as a function of z, and (concretely), in this case, is represented by the 4-component column vector:
where we define the inner product of two states |ψ〉 and |ψ′〉 by:
and the integral is over the waveguide cross-section at a given z. Â and B̂ are the Hermitian operators (under this inner product):
Note that Â is positive-definite (for ω > 0), but B̂ is not; more on this below. Also, Â is Hermitian if ε is purely real—in physical systems, ε has a small imaginary part corresponding to material absorption, but we add this feature a posteriori in Sec. 6.1 via perturbation theory.
To start with, we consider the ideal case where the waveguide is uniform along z, in which case the Bloch-Floquet theorem  implies that the states of the system can be chosen with a z-dependence of the form eiβz for some wavenumber β. These are the eigenmodes of the waveguide, i.e. the “normal modes” :
Substituted into Eq. (35), they satisfy the Hermitian generalized eigenproblem:
These states must satisfy the usual properties of Hermitian eigenproblems, albeit somewhat modified from quantum mechanics by the presence of a non-positive-definite B̂ :
• The eigenvalues β are real as long as 〈ψ|B̂|ψ〉≠0. (Because B̂ is not positive-definite, this is not always true, leading to evanescent modes with complex or imaginary β.)
• Orthogonality: for two eigenstates |ψ〉 and |ψ′〉 with eigenvalues β and β′ such that β*≠β′, 〈ψ|B̂|ψ′〉=0.
• The eigenstates generally form a complete basis for the system.
• For localized modes (e.g. guided modes of a waveguide), the eigenvalues are a discrete sequence βn with eigenstates |n〉.
The orthonormality relation, expressed explicitly in terms of the transverse fields, becomes:
which is proportional to the forward-propagating power for 〈ψ|B̂|ψ〉. The fact that eigenmodes are orthogonal under this metric has previously been derived from the Lorentz reciprocity theorem for electromagnetism , whereas here it follows automatically from the general properties of Hermitian eigenproblems.
In this paper, we focus primarily on the propagating (real β), guided (discrete β) modes of the waveguide. We normalize them so that 〈n|B̂|n〉=1. When one expands a field in the eigenstates, in principle the continuum and evanescent modes are required as well, but we shall show that in the cases of interest here, only a few guided modes are needed for acceptable accuracy.
Given a Hermitian eigenproblem, we can now directly employ the perturbation-theory formulations derived for quantum mechanics  in order to find the effects of a small perturbation ΔÂ. Such perturbation theory takes on two equivalent forms: z-independent and z-dependent. The former expresses the new eigenstates and eigenvalues as a power-series expansion in ΔÂ. The latter expands a propagating state |ψ〉 in the basis of the unperturbed states |n〉, with z-varying expansion coefficients. Such a z-dependent expansion is often called “coupled-mode theory” in electromagnetism [23, 45].
The main results from the z-independent theory that we will employ here are that the first- and second-order corrections to an eigenvalue βn are given by:
where Δβnn ≡βn -βn ′. Because of the smallness of the perturbations we consider, we generally only use the second-order correction if the first-order correction is zero by some symmetry. A minor modification must be made in the case of degenerate (equal-β) modes, such as ±m pairs in cylindrical fibers; in this case, the degeneracy is typically broken by ΔÂ, and the first-order splitting is given by diagonalizing the ΔÂ operator in the basis of the degenerate modes.
In z-dependent perturbation theory, a state |ψ〉 in the perturbed waveguide is expanded in the unperturbed eigenstates |n〉 with coefficients cn (z), satisfying:
These equations are exact if all modes are included in the expansion. In the most common case, one starts with only one mode cn (0)=1 and wishes to find the lost power |cn ′(z)|2 in “parasitic” (scattered) modes |n′〉 after some distance z. Eq. (45) can be approximately solved in that case to lowest-order for a uniform perturbation ΔÂ, yielding:
Thus, the lost power oscillates rather than growing steadily-this is due to destructive interference in the scattering, as long as Δβ±0. This fortunate interference is stymied, however, if the modes are lossy or if ΔÂ contains some non-uniform disorder, a fact that we considered with Eq. (16) and in Sec. 8.3.
For perturbations that vary smoothly in z, one can alternatively expand in the instantaneous eigenstates |n(z)〉 and βn (z) of Â(z), in which case the coefficients obey:
One can see, qualitatively, that changes slow compared to the lengthscale of 1/Δβ should cause small scattering and induce adiabatic evolution of the modes; this is proved rigorously in .17
Although both Eq. (44) and Eq. (46) in principle require an infinity of modes to be included, the Δβ in the denominators implies that one usually need only consider a few modes with nearby β. This is especially true in the case of large-core OmniGuide fibers, in which the close β spacing of the guided modes causes a few couplings to dominate, whereas the cladding (Bragg mirror) modes are at distant β. As a rule of thumb, if a perturbation has a characteristic lengthscale L (e.g. the bend radius Rb ), only modes with Δβ≲1/L contribute significantly.
Weak Dielectric Perturbations
One of the most general and important forms of waveguide pertubation is when the dielectric function ε is changed by a small Δε. In this case, we simply compute ΔÂ by substituting ε+Δε into Eq. (38) and subtracting the unperturbed Â. The resulting expression is inconvenient to evaluate directly because it contains derivatives. Since perturbation theory only employs ΔÂ inside inner products 〈n′|ΔÂ|n〉, however, we derive a simpler formulation by integrating by parts and then returning to the six-component field-representation via Eqs. (33–34). This yields:
Here, we have intentionally dropped terms of O(Δε 2) or less, even though our general approach ostensibly lacks that limitation (perturbation theory should remain valid as long as Δε yields a small integral 〈n′|ΔÂ|n〉). The reason for this, as we discussed in Sec. 8.2, is that for large Δε the changing boundary conditions on the fields cause the naïve ΔÂ to be incorrect. This small-Δε formula is sufficient to compute absorption losses and nonlinearities, but not for ellipticity or other moving-boundary problems in systems with large index contrasts.
Eq. (48), in conjunction with the first-order shift Δ =〈n|ΔÂ|n〉/〈n|B̂|n〉, has been derived elsewhere by various methods [23, 35, 45]. However, with the exception of , it has not been recognized that this expression needs to be corrected for large Δε; we further investigate this limitation in .
|1||Bragg fibers should not be confused with fiber Bragg gratings—the former use a lateral index modulation for transverse optical confinement, while the latter use axial modulation for longitudinal confinement and other effects.|
|2||The actual physical indices that we use are currently proprietary.|
|3||The transfer matrix of Eq. (37) in  is erroneous for m≠0; we hope that this apparent typo-graphical mistake will be corrected by a future erratum.|
|4||As a positive side-effect, we show that such a core radius brings chromatic dispersion down into a range comparable to that of single-mode silica fibers...although, because of the low nonlinearities, we shall argue that such a correspondence is not strictly necessary.|
|5||With our choice of eimφ φ-dependence, the EH11 mode is actually circularly polarized; combined with its degenerate m=-1 mode, however, one can instead express it as linearly polarized.|
|6||Such degeneracy breaking can also be understood in terms of the phase shift of a TE/TM polarized wave upon reflection from the omnidirectional mirror .|
|7||Below the light line of the outermost region, the incoming-wave Hankel function instead becomes an exponentially-growing function, but its coefficient must be zero all the same.|
|8||The 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 in the middle of the band.|
|9||Bessel-function afficionados can also show that the ratio of TE0ℓ to TE0ℓ′ losses is nearly (ξℓ/ξℓ′)2, where ξℓ and ξℓ′ are the corresponding roots of J1. This ratio is ~3.35 for TE02 to TE01.|
|10||Often, one instead uses n′=n+n2I, where I~n|E⃗|2 is the intensity of light; our analysis remains the same except for the extra factor of n.|
|11||FWM noise is proportional to γ2/(Δβ2+α2), where the α2 term is usually negligible . However, with nonlinearities 10,000 times weaker than in silica, α=0.01 dB/km is sufficient to suppress FWM even in the limit of Δβ=0 (zero dispersion and/or channel spacing).|
|12||Bending can also create a stress-induced Δε in the cladding material, which we do not consider here since so little of our field penetrates into the cladding.|
|13||The fact that this decay rate is second-order in the perturbation 1/Rb was also found in hollow metallic waveguides [16, 17, 37], by explicitly solving for the eigen-field correction in the curved waveguide.|
|14||In an undersea cable, the fiber spirals continuously within a central core, to prevent the fiber from bearing any of the cable tension—a typical radius of bending curvature would be around 50cm (corresponding to 5% fiber slack and a 5cm core radius).|
|15||We have also solved the equations with 16 modes and found that the main conclusions are the same as in the case of two-mode coupling, with merely a more complicated curve shape.|
|16||This is an approximation for large Δε; the correct moment is generally less—see e.g. the induced dipole moment of a spherical scatterer in .|
|17This adiabaticity was also considered in , where it was erroneously claimed that an analogue of Eq. (41) could be expressed as an ordinary (non-generalized) Hermitian eigenproblem with the same field-state representation (leading to an incorrect orthogonality relation).|
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