This paper introduces a simple, analytical method for generalizing the behavior of bent, weakly-guided fibers and waveguides. It begins with a comprehensive study of the modes of the bent step-index fiber, which is later extended to encompass a wide range of more complicated waveguide geometries. The analysis is based on the introduction of a scaling parameter, analogous to the V-number for straight step-index fibers, for the bend radius. When this parameter remains constant, waveguides of different bend radii, numerical apertures and wavelengths will all propagate identical mode field distributions, except scaled in size. This allows the behavior of individual waveguides to be broadly extended, and is especially useful for generalizing the results of numerical simulations. The technique is applied to the bent step-index fiber in this paper to arrive at simple analytical formulae for the propagation constant and mode area, which are valid well beyond the transition to whispering-gallery modes. Animations illustrating mode deformation with respect to bending and curves describing polarization decoupling are also presented, which encompass the entire family of weakly-guided, step-index fibers.
©2007 Optical Society of America
Fiber-optic waveguides have long been a critical component of a wide range of photonic systems, utilized for communications [1,2], sensing [3,4], power delivery , and more recently optical sources and amplifiers [6,7]. Extensive information describing the behavior of optical fibers exists in the literature, based on both analytical models and numerical simulations. However, the vast majority of theoretical work is based on fibers that are assumed to be straight. To account for differences in fiber behavior caused by bending, perturbation methods are most often used [8,9], based on the properties of the straight fiber. These can work well for standard single-mode fibers, because the fields propagating along the fiber are only weakly distorted by bending. But for fibers supporting even a small number of modes, bending can greatly deform the mode fields [10–14], and render perturbation theory ineffective. In such cases the bent fiber must instead be studied through numerical simulation, for which a variety of techniques have been developed [15, 16]. Unfortunately both approaches have the drawback compared to simple analytical methods in that the calculated results apply to only a single particular fiber. Deducing the general behavior of bent fibers in relation to their many variable properties can therefore be time-consuming, and non-trivial.
This paper overcomes this limitation by introducing a simple, analytical method that allows the behavior of an individual bent waveguide to be extended to an entire family of similar designs. This is done by introducing conditions for which the mode fields propagating along one waveguide are essentially identical to those of innumerable others, except scaled spatially. Such an approach was taken by Gloge  years ago to generalize the behavior of straight, weakly-guided step-index fibers. This paper extends this powerful approach, however, to weakly-guided fibers that are circularly-bent and of arbitrary cross-section. The method also applies to both single-mode and multi-mode fibers. As such, it provides a general and relatively simple framework for understanding the behavior of bent, weakly-guided waveguides.
In order to illustrate this approach, the paper also presents a comprehensive study of the modes of the bent step-index fiber. Included are animations of bend-induced mode deformation that, unlike previous studies [10–11, 13, 16, 18–20], are generally applicable to all weakly-guided, step-index fibers. Universal curves and simple formulae for mode area and propagation constant are also developed, which differ from those in the literature [13, 18–21] in that they remain valid well beyond the transition to whispering gallery modes, and are self-consistent. A detailed analysis of polarization decoupling as a result of bending is also provided, in the form of universal curves. These results are then extended to a wide range of more complicated, though commonly used, fiber geometries. The end result is a wide-ranging study of the modes of many common bent fibers, facilitated by the framework introduced in this paper.
Section 2 begins by introducing general conditions for mode scalability in bent step-index fibers. These are derived analytically in Appendices A through C, and shown to be in excellent agreement with bent fiber simulations. Section 3 follows with a detailed study of the mode fields themselves. It begins with animations of bend-induced mode deformation, followed by a simple expression that predicts when the propagating fields transition to whispering gallery modes. Mode areas and propagation constants are considered next, leading to universal curves and simple empirical formulae. Section 4 follows by generalizing these results to more complicated waveguide geometries, and Section 5 concludes with a summary of results. Polarization coupling, which is strongly inhibited by bending, is discussed in Appendix D. A list of symbols used is presented in Appendix G.
2. Mode scalability in bent step-index fibers
It is well known that mode fields of the straight step-index fiber may be scaled in size by proper adjustment of the fiber properties [1, 17]. However, no such conditions have yet been shown to exist for bent optical fibers. This section overcomes this limitation by introducing general conditions for mode scalability in the bent step-index fiber. The predicted behavior is then confirmed through bent fiber simulations. The conditions for mode scalability presented here will later be extended to fibers of arbitrary cross-section in Section 4.
2.1 Mode scalability in straight fiber
In the straight step-index fiber, the primary factor determining the form of the mode field distributions is the V-number ,
where a is the fiber core radius, NA is the index-based numerical aperture,
ncore and nclad are the core and cladding refractive indices, and ko is the vacuum wavenumber, related to the vacuum wavelength λo by ko=2π/λo. For a given V-number, the transverse fields and the propagation characteristics of each fiber mode remain essentially the same, except in spatial extent, as core size, numerical aperture and wavelength are varied [1,17]. The modes of a given fiber therefore scale in size, without otherwise affecting their field distributions, while holding the V-number constant.
Mode scalability is often expressed through use of the normalized propagation constant,
where n eff(s)=βs/ko, βs is the modal propagation constant, and the subscript “s” denotes values specific to the straight fiber. This quantity is directly related to both the mode field distribution and its propagation characteristics . Thus, it is significant that for a given V-number and particular mode, bs remains essentially invariant. Although a weak dependence on NA exists , this is relatively inconsequential for weakly-guided fibers, for which NA 2≪2n2clad.
2.2 Mode scalability in bent fiber
In the bent fiber, however, the situation is more complicated. Fiber curvature and bend-induced variations in the refractive index both tend to distort the mode field distributions [10,14], and ultimately push them away from the center of curvature, as illustrated in Figs. 5 and 6 of the following section. As a result, both the mode field distributions and their propagation characteristics depend not only on V-number, but also on the bend radius.
Since a complete discussion of mode scalability in bent fibers is rather lengthy, for brevity much of the analysis of this paper has been included as appendices. Appendices A and B derive general requirements for mode scaling in arbitrary, weakly-guided dielectric waveguides, for the straight and curved cases, respectively. These results are then used in Appendix C to show that for the step-index fiber, mode scaling will occur if both the V-number and the dimensionless ℜ-number, a normalized bend radius defined as
Weak-guidance (NA2≪2n2clad) has also been assumed. In effect, just as the modes of the straight fiber depend solely on the V-number, in the bent fiber they are determined by the two quantities V and ℜ. This point is proven mathematically in Appendix C, which shows that each mode’s normalized propagation constant remains invariant for a given combination of V and ℜ (within the accuracy of the weak-guiding approximation). Further justification for these claims is provided through bent fiber simulations in the following subsection. Note that for a fixed wavelength, mode scaling requires the core size to vary inversely with numerical aperture, while the bend radius changes much more rapidly, as NA -3.
2.3 Agreement with BPM simulation
In order to illustrate these points, a series of simulations of bent step-index fibers were performed for this paper using the beam propagation method (BPM) with conformal mapping [15,16]. This approach is common for mode solving, and has been shown to accurately predict bend loss in both single-mode and multimode fibers . A detailed description of the simulation procedure is provided in . Schematic diagrams of the bent fiber and its conformal mapped equivalent are given in Figs. 1 and 2.
Before considering simulation results in detail, it should first be noted that whereas in the straight fiber a mode’s linear velocity is spatially uniform, in the bent fiber it is the angular velocity which is conserved. This point is illustrated schematically in Fig. 2(a), which shows that the phase velocity c/neff increases while moving away from the center of curvature, such that the angular velocity remains constant (the behavior of the conformal mapped fiber in Fig. 2(b) is analogous, provided the coordinate transformation). It is therefore necessary to reference the linear propagation constant β to radial position in the bent fiber. This is typically done by defining β at the center of the fiber,
which ensures that the physical path length (along ρ=R) does not vary as the fiber is bent, neglecting any applied tension. For this paper, however, it is advantageous to discuss propagation in terms of a normalized angular propagation constant, defined here as
The added restriction in Eq. (7) compared to (3) accounts for the fact that in the bent fiber b would otherwise vary with position, as is evident from Fig. 2(a). This choice of reference also assures that all guided modes must fall within the range 0<b<1, as they do in the straight fiber. Since b is related to the linear propagation constant β through
it must reduce to the straight value bs at sufficiently large bend radii.
Though strictly speaking curved waveguides cannot support guided modes, since the fields must radiate to some extent , the term “guided” is used loosely here to apply to all modes for which 0<b<1. This criterion encompasses all propagating modes concentrated about the fiber core, and is thus analogous to that of the straight fiber. That each mode radiates as it propagates along the fiber also implies that its propagation constant β must be slightly complex. This has negligible impact on the results of this paper, however, provided that the loss remains reasonable .
The first step in the simulation process was to calculate the lowest-order modes of various step-index fibers, each with the same V-number, but a range of core sizes, numerical apertures, cladding refractive indices, wavelengths and bend radii. In each case the normalized angular propagation constants were then calculated, as defined for the bent fiber by Eq. (7). Figures 3(a) and 3(b) plot the normalized angular propagation constants calculated for the fundamental mode (LP01) of various bent fibers, each with the same V-number, 7.375, reasonable for high-power fiber amplifiers. As shown, the different curves overlapped when plotted versus ℜ, indicating that the normalized angular propagation constants were the same for each given ℜ-number. Similar behavior was also observed as wavelength and cladding refractive index were varied. Together, these results demonstrate the validity of the normalization presented in Eq. (4). Furthermore, each mode’s field distribution did not vary perceptibly, other than in size, for a given combination of V and ℜ. Examples of this are presented in Fig. 4. Together, these results illustrate that for the step-index fiber, the modes scale for a given combination of V and ℜ.
3. Modes of the bent step-index fiber
The theory introduced in the previous section greatly simplifies the analysis of the bent step-index fiber, effectively reducing it to a problem of only two variables. In essence, it allows the behavior of any one fiber to be taken as representative of all fibers for which V and ℜ are the same, and vice versa. This section utilizes these results to discuss the guided modes of the bent step-index fiber in general terms, and in particular, how they vary in response to bending. This provides not only a detailed analysis of the bent step-index fiber, but serves as an example for understanding the more complicated fiber geometries discussed later in Section 4.
In order to determine the guided modes of the bent fiber, a semi-vector version of the imaginary-distance BPM was used , as described in detail in . This approach included polarization dependence in the mode calculations, but it did not account for polarization coupling. As a result, the analysis was limited to the linearly polarized (LP) fiber modes, rather than the exact hybrid (EH and HE) modes . We proceed however with the knowledge that hybrid modes may be approximated for weakly-guided waveguides by combinations of two oppositely polarized, degenerate LP modes . Furthermore, as detailed in Appendix D, significant bending tends to inhibit such polarization coupling, in which case the modes of the bent fiber reduce to essentially linearly polarized (LP) states.
Polarization dependence in the bent fiber was also simplified by two key points. The first was that shape birefringence in step-index fiber is orders of magnitude less than the bend-induced stress birefringence , which allows it to be neglected. The second was that birefringence does not appreciably affect the distribution of the LP mode fields, except possibly in cases of extreme birefringence approaching the index step itself. The remainder of the paper therefore presents modes and propagation constants calculated assuming zero material birefringence. These are indicative of both polarization states. It is quite straightforward to later account for the bend-induced birefringence, by adjusting the calculated propagation constants accordingly as discussed in Appendix D.
3.1 Mode field deformation
In order to illustrate the impact of bending on the fiber modes, a set of simulated mode profiles are presented in the animation in Fig. 5. Each frame plots the electric field magnitudes for the six lowest-order LP modes, for a particular value of ℜ, and the V-number 7.375. In each case the center of fiber curvature is located to the left of the figure, and the core-cladding boundary was indicated by the circular outline. The modal notation LPmne and LPmno was also adopted to differentiate between mode orientations which were even or odd, respectively, in the (vertical) direction normal to the plane of the bend, while conforming to the usual LPmn notation of the straight fiber . It is critical to note that Fig. 5 indicates how all weakly-guided fibers with the V-number 7.375 vary in response to bending. As will be shown momentarily, the mode progression is also representative of bent step-index fibers with different V-numbers as well.
To illustrate how the V-number influences the modes of the bent step-index fiber, Fig. 6 plots a similar animation to Fig. 5 for the case V=29.5. As shown, in the initial stages of transformation the modes shown were similar to those in Fig. 5. However, this occurred at much greater values of ℜ (for reference, ℜ=1000 corresponds to a bend radius of approximately 1.5m when NA=0.06 and λ0=1µm). With continued bending, all the modes transitioned to whispering gallery modes well before reaching cutoff, and thus disappearing from the figure. In this context, the term “whispering gallery mode” refers to a mode confined at the inside of the bend by the fiber curvature, rather than the core-cladding interface. This definition is based on the familiar disc resonator, in which whispering gallery modes propagate along the innermost edge of a cylindrical boundary, confined entirely by the surface curvature .
Together, the animations in Figs. 5 and 6 provide a representative picture of the variation in the lowest-order modes in the bent step-index fiber. That the initial stage of mode transformation was similar in the two figures is a key point, because it implies that similar behavior should also be expected for different V-numbers, although at different values of ℜ. Furthermore, since the modes in Fig. 6 transitioned fully to whispering gallery modes, which are relatively simply distributed, it is not difficult to envision their continued variation for larger values of V.
The similarity between Figs. 5 and 6 at different values of ℜ may be clarified by noting that in the following sections, fibers with different V-numbers will be shown to behave alike when maintaining the same value of ℜ(1-bs)/V. To illustrate this point, Fig. 7 displays modes of two fibers with different V-numbers, but the same value ℜ(1-bs)/V=1. In both cases the mode field distributions were similar, other than in their degree of confinement to the core. Comparable behavior was also observed for other values of ℜ(1-bs)/V. Thus, although the mode field scale exactly for a given pair V and ℜ (under the assumption of weak guiding), they are also remarkably similar for a given value of ℜ(1-bs)/V. Figures 5 and 6 are therefore reasonably representative of the step-index fiber in general. Extending them to other V-numbers simply requires the ℜ-number in each frame to be adjusted, such that ℜ(1-bs)/V is conserved.
3.2 Transition to whispering gallery modes
Inspection of the animations in Figs. 5 and 6 also indicates that the modes of the bent fiber may be grouped into two classes: perturbed modes at large bend radii, resembling those of the straight fiber, and whispering gallery modes at smaller radii, which fill only a fraction of the fiber core. It is clear that the most drastic field deformation occurs in the latter case. Establishing where the transition between the two cases takes place is therefore essential to understanding the bent step-index fiber.
where nstraight is the refractive index distribution of the straight fiber. This effect is shown schematically in Figs. 1 and 2. For adequately small bend radii, the sloping refractive index nbent falls below the modal effective index neff within the fiber core, as illustrated in Fig. 8. Since the mode fields must decay in regions where nbent<neff, this effectively confines the fields to a limited region of the core, where nbent>neff. The end result is a whispering mode, as shown in Fig. 8.
when measured through the center of the fiber (the latter condition for whispering gallery modes, and the former for perturbed). It follows that the transition between the whispering gallery and perturbed regimes occurs approximately when the widths Weffx and 2a are equal. Thus, using Eqs. (1) and (4), this transition may be shown to occur at the ℜ-number
In addition, it will be shown in Section 3.4 that the normalized angular propagation constant at this transition point, denoted btrans, is given approximately by
This definition is quite powerful, because it depends only on well-known properties of the straight fiber.
For reference in the above expressions, the normalized angular propagation constants of various straight fiber modes are plotted as Fig. 19 in Appendix F. They may alternatively be found from the characteristic equation , which in terms of normalized quantities, for nclad≅ncore, is given by
Here J and K are Bessel and modified Bessel functions, the prime represents differentiation with respect to the argument, and m is the azimuthal mode number, corresponding to the subscript in LPmn. This equation may easily be solved numerically.
3.3 Effective mode area
The effective mode area, defined as the area where the fields surpassed the 1/e2 power level, is plotted for the fundamental mode of various bent fibers in Fig. 9. In each case the mode area was normalized to that of the straight fiber to demonstrate a clear trend. In the perturbation region, where ℜ>ℜtrans, the mode areas remained relatively constant as the fibers were bent. However, in the whispering gallery region (ℜ<ℜtrans), the mode areas decreased steadily with bending, following essentially the same path. For reference, the area of the guided portion of the core (where ncore was greater than neff) is also plotted in the figure. This illustrates that the reduction in mode area with bending followed the same trend as the area of the guided region, as expected.
The effective mode areas of the lowest-order modes of the bent fiber were also plotted in Fig. 10, for the V-number 29.5. These follow the same trend as the fundamental mode, although with some ripples caused by reorientation of the fields as the fiber was bent. From these curves it is clear that the quantity ℜtrans marks the approximate boundary of the whispering gallery mode region, where the effective mode areas begin to fall off. This was true of all modes and fibers simulated.
Figures 9 and 10 demonstrate not only when each mode’s area begins to fall off (at ℜtrans), but also how it varies with bending. In each case mode area tracks the size of the guided region, as shown. A mathematical expression for the size of the guided region is given in Appendix E. Such results are of particular importance for devices that rely on large mode areas for optimum performance, such as current high-power fiber lasers and amplifiers [28,29].
It is next useful to return to Eq. (13) and consider the relationship between ℜtrans and core size in detail. Note that the right side of Eq. (13) depends only on V-number for each given mode. It is thus a simple matter to determine ℜtrans uniquely for each value of V, with the aid of the characteristic Eq. (14). The whispering gallery transition may then be related to the area of the core, through the expression
as has been done in Fig. 11 for various LP0n modes of the step-index fiber. Each curve indicates that ℜtrans increases roughly as Acore3/2, while holding NA/λ0 constant. Other modes not shown in the figure also follow the same trend. Thus, as core size increases for a given mode and value of NA/λ0, the whispering gallery transition takes place at a correspondingly larger bend radius.
This limitation is considerably relaxed, however, with increasing mode order. Higher-order modes therefore allow much tighter bending without significant reductions in mode area. This is an added benefit compared to the usual motivation for using higher-order modes in large mode area (LMA) fibers: their reduced probability of inter-modal scattering [30, 31]. Such an approach has limitations, however, imposed by the restriction that ℜtrans must remain greater than 4V. This constraint is indicated in Fig. 11 by the dashed line, which marks the limit as btrans goes to zero (and thus bs goes to ½, through Eq. (13)). Along this dashed line, ℜtrans increases much more gradually than for each individual mode, as Acore1/2.
It is also worthwhile to consider how the numerical aperture enters into these results. This is accomplished by recasting Eq. (13) in terms of non-normalized quantities, which leads to an expression for the bend radius at the whispering gallery transition
This radius may be reduced, for a given core size, both by reducing bs (propagating higher-order modes), and by increasing the fiber NA. The latter conclusion is contrary to the conventional approach for obtaining large mode area, that of reducing the numerical aperture . However, it has the advantages that with increasing NA the modes become more strongly guided , less prone to inter-modal scattering [30–31], and scaleable to larger areas for a given bend radius. Its primary disadvantage is that that more modes are guided as the numerical aperture increases, so mode selection [28,31,33–35] becomes more challenging.
3.4 Propagation constants
The variation in the propagation constant of the bent step-index fiber is plotted in Fig. 12, where Δβ2=Re(β)2-βs2, for the lowest-order modes, and the V-numbers 2.36, 7.375, and 29.5 (the normalization on the vertical axis will be discussed in a moment). In each case, when ℜ>ℜtrans the propagation constant increased to second order or greater in R -1, as predicted by perturbation theory . However, in the whispering gallery region the propagation constant increased more gradually. This indicates that in addition to marking where mode area begins to fall off, ℜtrans is indicative of where simple perturbation theory breaks down.
Figure 12 also shows that when Δβ2 was normalized according to (k0NA)2(1-bs)/bs, the curves were relatively independent of the fiber V-number for each particular mode. Such a result is in keeping with the previous observation that modes are similar at a given value of ℜ(1-bs)/V, which is easily shown to be equal to 2ℜ/ℜtrans. Thus, in addition to the mode fields being similar for a given value of ℜ/ℜtrans, their propagation characteristics were as well. This leads to the conclusion that the curves in Fig. 12 are reasonably general, at least in the range tested where bs>ℜ.
That the curves in Fig. 12 all converge together in the whispering gallery region, regardless of mode or V-number, is remarkable. In order to clarify this trend, Figure 13(a) plots the same data from the whispering gallery region of Fig. 12, but on different axes. The dashed line in the figure marks where the two axes were equal, and matched the data quite well for the full range ℜ<ℜtrans. This implies that an approximate expression for the real part of β in the whispering gallery region is, for all modes,
Although this expression represents an empirical fit, it was remarkably accurate for all modes and fibers simulated.
That such a simple expression exists for the propagation constant, despite wide variations in the field distributions, is related to the fact that the modes shift away from center of the fiber with bending. As the modes shift outward, they must travel further around the curve than if centered about the core, and the resulting path delay causes their propagation constants to increase accordingly. In the perturbation region this effect is relatively minor because the outward motion is constrained by the core-cladding interface. However, in the whispering gallery region it becomes the dominant effect.
The variation in the normalized angular propagation constant with bending is plotted in Fig. 13(b), for a variety of modes and V-numbers. Each point in the figure corresponds to the perturbation region (ℜ>ℜtrans), and falls along the line
with considerable accuracy. Such a result is consistent with the fact that β changes very little with bending for ℜ>ℜtrans, which causes neff to remain relatively constant at the center of the fiber. Thus, as the refractive index distribution tilts with bending, it is easily shown that to first order b must decrease according to Eq. (18).
This is an important result because when combined with Eq. (11), it leads to the simple expressions for btrans in Eq. (12), and ℜtrans in (13). Furthermore, Eq. (12) implies that the only modes capable of transitioning to whispering gallery modes before reaching cutoff (b=0) are those for which bs is greater than ½. This explains why analysis of the bent single-mode fiber is relatively simple: it is incapable of transitioning to a whispering gallery mode since bs is always less than 0.53 (see Fig. 19). Single-mode fibers are therefore reasonably well-described by perturbation theory (bs can be slightly larger than ½, but this is relatively inconsequential). Multimode fibers, on the other hand, and in particular those modes for which (1-bs) is small, exhibit more complicated behavior due to the existence of the whispering gallery transition.
This expression is useful for predicting bend loss in step-index fibers, as will be discussed in a future publication . It is also valuable for predicting mode areas since it is directly related to the size of the guided region (where neff<ncore), as discussed in Section 3.3.
4. Extension to other fiber geometries
Although the discussion thus far has focused on simple step-index fiber, much of the preceding analysis may also be applied to more complicated fiber geometries. The following subsections discuss how the results of Sections 2 and 3 may be extended to weakly-guided fibers of more complex geometry.
4.1 Mode scalability in arbitrary weakly-guided fibers
Section 2 introduced two conditions for mode scaling in bent step-index fibers: one corresponding to the bend radius (ℜ), and the other to the fiber cross-section (V). With more complicated fiber geometries analogous conditions also exist, such that the modes of most fibers are also generally scalable. These are derived in Appendices A and B, and summarized below. The only restriction inherent in the following conditions is that the fiber be weaklyguided (Δn≪n), from which it follows that it must also be slowly bent (Weffx≪R). Since this condition is easily satisfied, the following results cover a vast range of fiber geometries.
A fiber of arbitrary cross-section, supporting the transverse mode field distributions U T(r T), is first considered. Appendices A and B show that an infinite number of other fibers also exist that will support identical mode field distributions, except scaled in size by the factor M such that
where represents the scaled coordinate system
These fibers differ from the original only in that their refractive index profiles are also scaled, prior to bending, according to the expression
Here the function g1 describes the spatial variation in the fiber’s index profile, and nref is a spatially invariant reference index, though not necessarily the background. Both are known at M=1, as determined by the original fiber’s refractive index profile. The spatial variation in Eq. (22) is thus defined from the start. The form of n2ref(M) may be chosen arbitrarily, however, provided that the fibers remain weakly guided. For bent fibers the effective bend radius must also scale such that
where again the proportionality depends on the original fiber at M=1. These relations demonstrate how wavelength, refractive index, and bend radius are all interrelated, such that the mode field distributions may be scaled exactly for any weakly-guided fiber. Equations (22) and (23) are therefore analogous to those presented earlier for V and ℜ in the case of step-index fiber. However, they apply to all fiber geometries, under the assumptions of weakguiding and slow bending.
where the constant c1 depends upon the fiber geometry and the particular mode, and is presumably known for the original fiber. This expression is in turn analogous to the result from Section 2 that the normalized angular propagation constant will remain invariant for a given pair V and ℜ. Such is evident from Eqs. (7), (22) and (24). However, Eq. (24) represents the general form.
4.2 General condition for similar modes
In addition to the previous relations that describe how the modes will scale exactly in arbitrary bent fibers, there also exists an analogous expression to that of Section 3.1 for which the modes of the bent step-index fiber will be similar, but not exactly the same, for constant ℜ(1-bs)/V. In fibers of arbitrary cross-section, this condition generalizes to
where the function g2 is determined by the refractive index profile nstraight at M=1. This relation describes a family of bent fibers for which modes will be similar to each other from fiber to fiber, such as those illustrated in Fig. 7 at the end of Section 3.1. It is important to note, however, that Eq. (25) is in no way related to Eqs. (23) and (24), which pertain to mode scaling in its exact form. It is also worth noting that unlike Eq. (22), the magnitude of nstraight remains constant in Eq. (25).
4.3 Comparison to step-index fiber
That mode scaling is possible in all weakly-guided fibers suggests that certain behavior discussed in Section 3 may also be extended to more complex fiber geometries. With more complicated refractive index profiles, however, the fiber NA given by Eq. (2) is no longer valid (here the NA is based on the index step, rather than far field output, which is an important distinction ). Consequently, the quantities V, ℜ and bs are no longer defined. Nevertheless, much of the previous analysis may still be applied to an important class of fibers such as those in Fig. 14, provided some clarification.
Although the fiber cross-sections in Fig. 14 differ substantially, each has the same index step at the same position, |ρ-R|=a. For the moment we also consider modes with similar values of neff, as shown, such that their fields are guided in the region from -a to a. Since perturbation theory indicates that β should vary to second order or greater in R -1 for all fibers symmetric in the ρ-direction , and in this regard the bent fiber has been shown to follow perturbation theory until reaching the whispering gallery transition, it follows that neff will remain relatively constant at the center of each fiber prior to this transition. Thus, as the fibers in Fig. 14 are bent, their effective indices will pivot about the fiber center as shown. The result is that each mode in the figure transitions to a whispering gallery mode at essentially the same bend radius. The only disparity stems from the fact that neff should vary somewhat from fiber to fiber due to differences in the refractive index profiles.
These examples demonstrate that the behavior of more complicated fiber geometries is often closely related to that of the simple step-index fiber. The primary difference, as far as the whispering gallery transition is concerned, is that the refractive index step at the boundary that confines the mode must be used when computing V, ℜ and bs, in this case |ρ-R|=a. Equation (13) will then express the bend radius where the size of the guided region begins to vary. This point corresponds to the whispering gallery transition in all examples of Fig. 14, at which the modes are no longer entirely confined by the index step. However, it is not necessarily true of all fiber geometries.
In general, beyond ℜtrans the modes and propagation constants will vary differently from fiber to fiber due to differences in their refractive index profiles. In light of this, the simple step-index fiber merely serves as a guide for the other fibers in Fig. 14. Nonetheless, behavior should be quite comparable for the fibers in Figs. 14(a) and 14(b), and to a lesser extent that in 14(c). This is due to the fact that the guided region remains identical same in each case.
This paper has introduced a general and relatively simple framework for understanding the behavior of bent, weakly-guided fibers and waveguides. Much of the analysis has dealt specifically with the bent step-index fiber, resulting in a rather comprehensive study of its modal behavior. Some of the more important points are summarized as follows:
1) Just as the modes of the straight step-index fiber depend solely on the V-number, in the bent fiber they are determined by the two quantities V and ℜ. The mode field distributions all scale exactly as V and ℜ are held constant, and their normalized angular propagation constants remain the same.
2) The modes of fibers with different values of V and/or ℜ are similar, but not exactly the same, when bent such that ℜ/ℜtrans is conserved.
3) The transition to a whispering gallery mode occurs approximately at the ℜ-number ℜtrans.
4) Prior to the whispering gallery transition, mode areas and angular propagation constants are both relatively unaffected by bending. Thereafter they depend primarily on ℜ/ℜtrans, regardless of the particular field distribution.
5) Modes for which bs<½reach cutoff (b=0) before reaching the whispering gallery mode transition.
6) The bend radius of whispering gallery transition may be reduced, for a given core area, both by reducing bs (propagating higher-order modes), and by increasing the fiber NA.
7) The lowest-order fiber modes transition to linearly polarized states when bent to the whispering gallery transition, and beyond (see Appendix D). Although exceptions are possible, they are unlikely under the assumption of weak-guiding.
8) All quantities necessary for computing V, ℜ and ℜtrans are those of the straight fiber, and are thus well known.
In addition, much of the preceding analysis for the bent step-index fiber may be extended to the specific group of common fiber designs indicated in Fig. 14.
More generally, this paper has shown that the modes of any bent, weakly-guided waveguide may be considered as representative of those of an entire family of waveguides, for which two scaling conditions apply: one for the waveguide cross section, and the other for the bend radius. This allows the behavior of a single waveguide to be extended to the entire group of similar designs, and as such, greatly simplifies the analysis of bent, weakly-guided fibers and waveguides.
Appendix A: General conditions for mode scaling
It is well-known that the modes of the straight, weakly-guided step-index fiber may be scaled in size, without otherwise affecting the transverse fields, while maintaining the same V-number from fiber to fiber. The purpose of this appendix is to introduce analogous conditions for straight, weakly-guided waveguides in general. Such conditions, when satisfied, allow the modes of one particular waveguide to be extended to an entire family of other waveguides, with different mode sizes, refractive indices and wavelengths.
The electromagnetic wave equation may be expressed in terms of electric field in a general dielectric medium as
The left side of this expression is the homogenous wave equation, while the right side accounts for variations in the dielectric with position, and reduces to the usual boundary conditions at abrupt interfaces. An identical expression also holds for the magnetic field, but with ε and µ reversed. If it is assumed that the dielectric does not vary in the z-direction, and supports at least one guided mode, then each of the guided modes may be expressed in the form
where is the coordinate transverse to the z-direction. By combining Eqs. (A1–A2), it follows that the guided modes must satisfy the relations
for their transverse and longitudinal vector components, respectively.
The boundary conditions on the right in (A3-A4) determine the propagation constant β of each guided mode. They also cause the normal components of the fields to become discontinuous across discrete boundaries, and lead to coupling between polarizations. The latter two conditions make it impossible for a second waveguide with different material properties to support exactly the same modes, or even spatially scaled versions, except for the trivial cases in which √ε and √µ are either exactly the same as the original, or scale inversely with waveguide size. In any other situation, the field discontinuity must necessarily be altered. However, for weakly-guided waveguides , in which the relative variation in ε and µ is small (Δε≪2ε, Δµ≪2µ), the field discontinuity at the boundaries is negligible. This allows the scalar form of the wave equation,
to instead be used, subject to the conditions that U and ∇×U must remain continuous. In this weakly-guided regime, it is possible for different waveguides to support otherwise identical mode field distributions, except scaled in size. For the remainder of the appendices, weakly-guided waveguides are therefore assumed. In cases of stronger guiding, the mode fields are often similar to those approximated by (A5), so that the following analysis may also be loosely applicable.
We first assume that the dielectric distribution given by ε(r T), µ(r T) forms a waveguide, which supports at least one guided mode U(r T). If this field distribution is to be scaled in size by the factor M, such that
then it follows from (A5) and (A6) that U’(r T’) will also satisfy conditions as a guided mode, with propagation constant β’, in a waveguide with different material properties ε(r T’), µ(r T’) given by
Thus, if the waveguide cross-section is scaled according to (A8), then the waveguide modes will scale according to (A6), and vice-versa. Furthermore, by normalizing the mode fields with respect to power flow, as in Eq. (D4), Eq. (A6) simplifies to
Equation (A8) may also be rewritten in terms of refractive index as
where g3 is a function of rT’, but not of M. By differentiating Eq. (A11) with respect to M, then with respect to rT’, and integrating, it may easily be shown that n2 and n2eff must each vary with M according to
Here the function g1 describes the spatial variation in the waveguide’s index profile, nref is a spatially invariant reference index (though not necessarily the background), and c1 is a constant. Each are presumably known for M=1, as determined from the original refractive index profile. The spatial variation in Eq. (A12) is thus specified for all M. The form of n2ref(M), however, can be chosen arbitrarily. Together, Eqs. (A12–A13) denote the necessary conditions for mode scaling in an arbitrary, weakly-guided waveguide.
Appendix B: Mode scaling in arbitrary curved waveguides
For a waveguide curved along a circular arc of radius R, the conformal mapping technique  allows the curved waveguide to be represented by an equivalent straight waveguide, by replacing its refractive index distribution n with the distribution
Here the distance x is measured outward from the center of the waveguide, as shown in Fig. 1. The exponential term in (B1) is purely geometric in nature, and accounts for the increased optical path length along the waveguide with distance from the center of curvature.
The modes of the curved waveguide will scale in size according to (A9–A10), only if the mapped index distribution in (B1) is of the form (A12). This condition may be easily satisfied if the refractive index of the straight waveguide also scales according to (A12). Then the conformal-mapped refractive index becomes
where relatively slow curvature (x≪R) has been assumed. Other than the trivial case in which the entire refractive index distribution scales inversely with M (and linearly with R), there is only one situation in which (B2) takes the form of (A12). This occurs when
as might occur, for example, when the background refractive index is held constant, while scaling only the index variation. The waveguide modes will then satisfy the scaling conditions (A9–A10) provided that the bend radius is also scaled according to
where the proportionality constant depends on the bend radius at M=1. In practice, nref may be chosen as a constant with respect to M, and thus dropped from the preceding expression.
Equations (A12), (B3) and (B4) therefore denote the necessary conditions for mode scaling in an arbitrary curved, weakly-guided waveguide. Restrictions on refractive index are identical to those for the straight waveguide. The only additional constraints are that the bend radius must be scaled accordingly, and much larger than the mode width in the x-direction.
For bent waveguides, bending the material also leads to physical changes in the refractive index due to elasto-optic effects . This is easily accounted for by replacing the bend radius R in the preceding expressions by the effective bend radius Reff, as discussed in [14,22], and expressed for silica glass by Eq. (5).
Appendix C: Mode scaling in step-index fibers
In the case of step index fiber, Eq. (A12) may be simplified to the following restriction on numerical aperture
This is equivalent to Eq. (1), and implies that for mode scaling to occur, either the wavelength or the fiber NA must vary. Furthermore, it follows from Eq. (B4) that the modes of the bent fiber will scale provided that the effective bend radius varies according to
and (C1) is satisfied. This in turn is equivalent to Eq. (4). Note as well that Eqs. (A12) and (A13) imply that the normalized propagation constants given by Eqs. (3) and (7) also remain invariant as the modes are scaled, for straight and curved step-index fibers, respectively.
Appendix D: Polarization decoupling in bent step-index fibers
D.1 Non-degeneracy of even and odd modes
Inspection of Fig. 12 leads to a somewhat subtle point: that the mode’s orientation, i.e. whether it is even or odd with respect to the plane of the bend, can significantly affect how its angular propagation constant varies in the bent fiber. This is most evident for the LP11 mode, whose angular propagation constant changes less for the even orientation than for the odd. To help clarify this effect, Fig. 15 plots the difference between the normalized angular propagation constants of various even and odd mode pairs. This shows that bending tends to break the degeneracy of the LPmne and LPmno states, so that, unlike in the straight fiber, they do not share the same propagation constant. It is important to note that this effect is entirely due to differences in mode orientation, rather than differences in polarization, so is unrelated to material birefringence in the bent fiber. The splitting becomes more pronounced with decreasing bend radius, except in limited regions where bLPe-bLPo passes through zero, as shown for the LP21 mode. Normalizing the vertical axis of the figure by (1-bs) provided curves that were, for the most part, independent of V-number, and therefore reasonably general.
D.2 Material birefringence
Material birefringence in the bent fiber, induced by stress-optic effects, also causes the propagation constants of oppositely polarized modes to shift by the amount 
where the subscripts x and y refer to the polarization direction, afiber is the fiber radius, ν is Poisson’s ratio, and p11 and p12 are strain-optic coefficients. In silica glass, (p12-p11)(1+ν) is approximately equal to 0.18 . For the purpose of comparison, Eq. (D1) may be rewritten in terms of normalized quantities as
where silica glass has been assumed.
Figure 16 compares this polarization-dependent shift in b to that calculated previously for different mode orientations. It is clear that for typical fibers (NA~0.1, afiber/a~10) there are regions where either effect can dominate.
D.3 Polarization decoupling
A consequence of broken degeneracy in the bent fiber is that velocity matching between pairs of even and odd LP modes, which is necessary for strong polarization-coupling , is inhibited. For example, in the straight fiber it is coupling between oppositely polarized LPmne and LPmno modes for m>0 that gives rise to the hybrid HE(m+1)n and EH(m-1)n modes. To estimate the degree of polarization coupling in the bent fiber, an approach from reference  was used, and applied to the simulated mode field distributions of this paper.
This approach approximates the hybrid mode fields, U H, as linear combinations of orthogonally-polarized LP states
each normalized according to
The fractional power in the minor polarization (the LP state with the least power) was then determined from the following relations, adapted from 
The line integral in (D6) is performed along the core-cladding interface.
Figures 17 and 18 plot the fractional power in the minor polarization calculated for various simulated modes of the bent step-index fiber, corresponding to the two cases in which mode orientation (bLPo-bLPe) and material birefringence (Δ(by-bx)), respectively, dominate in Eq. (D6). Since there were actually four very similar hybrid modes to each LPmn mode, comprised of different combinations of polarization and orientation, their results were averaged to obtain each curve. It should also be noted that in both figures the approximation Pminor/Ptotal≅0.25/Λ2 was used, which limits accuracy as Pminor/Ptotal approaches/exceeds ½.
In the situation where mode orientation dominates (|bLPo-bLPe|≫|Δ(by-bx)|), note that the absolute value term on the right in Eq. (D6) remains invariant for any pair V and ℜ. When sufficiently small, the power in the minor polarization therefore scales approximately as (NA/ncore)4 while V and ℜ are held constant. Though the values plotted in Fig. 17 correspond to the case NA/ncore=0.0656, they may thus be scaled accordingly. Furthermore, though Fig. 17 covers only a limited sample of fiber designs, normalizing its vertical axis by V-2 resulted in curves that were reasonably independent of V-number.
Universal curves were also obtained for the case in which birefringence dominates (|bLPo-bLPe|≪|Δ(by-bx)|), as shown in Fig. 18. In this extreme the power in the minor polarization scaled approximately as (a/afiber)4, and the curves were otherwise independent of both V-number and the fiber numerical aperture.
Since Figs. 17 and 18 provide different values for Pminor/Ptotal, depending upon whether birefringence or mode orientation dominates, it should be noted that the correct value is the lesser of the two. An exception occurs, however, if neither effect dominates such that |bLPo-bLPe|≅|Δ(by-bx)|. In such a case, birefringence and mode orientation can counteract each other, and thus restore degeneracy between appropriately polarized even and odd states. This should occur only over a very limited range of ℜ/ℜtrans, however, so represents the exception rather than the rule. Figure 17 provides an (analogous) example, in which the power in the minor polarization spikes for the LP21 mode as bLPe-bLPo crosses through zero.
Together, Figs. 17 and 18 provide a relatively general picture of polarization decoupling in the bent step-index fiber. They indicate that polarization decoupling begins at a threshold bend radius, which is different for each mode, and proceeds rapidly thereafter. They also show that for weakly-guided fibers (NA2≪2nclad2), the lowest-order modes all reduce to essentially linearly polarized states upon reaching the whispering gallery transition and beyond. In such cases the LP modes presented in Section 3.1 therefore represent the true modes of the bent step-index fiber.
Appendix E: Size of the guided region
As discussed in Sections 3.2 and 3.3, the size of the guided region of the core (where neff<ncore) varies significantly with bending in the whispering gallery region. It follows from geometry and a simple integration that the area of this region is given by
where Weffx is expressed in Eq. (10), and the ± sign is positive when Weffx>a and negative otherwise. Equation (10) may be simplified, however, with the aid of Eqs. (1–3), (13) and (19). This leads to the approximate expression
Appendix F: Normalized propagation constants in straight step-index fiber
For reference, the normalized propagation constants of various modes of the straight step-index fiber are plotted versus V-number in Fig. 19.
The author would like to thank James H. Cole, Carl A. Villarruel, and Frank Bucholtz for many illuminating discussions, and Jeff Salzano for assisting with the multimedia. This research was performed while the author held a National Research Council Research Associateship Award at the U.S. Naval Research Laboratory, Washington, DC.
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