## Abstract

Antiguiding, as opposed to positive index-contrast guiding (or index-guiding), in microstructured air-silica optical fibers is shown to have a significant influence on the fiber’s transmission property, especially when perturbations exist near the defect core. Antiguided modes are numerically analyzed in such fibers by treating the finite periodic air-silica composite (including the central defect) as the core and outer bulk silica region as the cladding. Higher-order modes, which can couple energy from the fundamental mode in the presence of waveguide irregularities, are predicted to be responsible for high leakage loss of realistic holey fibers. The modal property of an equivalent simple step-index antiguide model is also analyzed. Results show that approximation from a composite core waveguide to a simple step-index fiber always neglects some important modal characteristics.

© 2004 Optical Society of America

## 1. Introduction

Microstructured optical fibers (MOFs) made of air and pure silica [1] have attracted a significant amount of attention in recent years. Two mechanisms are identified as being responsible for waveguiding in such fibers: index-guiding [2] and photonic bandgap (PBG) guiding [3]. Index-guiding utilizes the positive index-contrast between the solid silica core and periodic silica-air cladding, whereas PBG-guiding utilizes the photonic forbidden bandgap to localize light in the core, or defect region. To date, most research has focused on index-guiding MOFs, owing to ease of fabrication and to the many revolutionary properties that they can provide. These properties include endlessly single-mode operation, tailored dispersion, very large or ultrasmall effective core area, and so on. [4]. PBG air-silica fibers, although they have many promising applications, have narrow working bandwidth, difficulty in fabrication, and other shortcomings. New materials need to be identified in order to make more practical PBG fibers [5]. In this paper, we will put our emphasis on the index-guiding fibers.

Figure 1 shows a commercial index-guiding MOF. The parameters are measured as follows: hole diameter *d*=2.3021*µm*, hole-to-hole distance *Λ*=6.3055*µm* [also refer to Fig. 2(b)]^{†}. Refractive indices for air and pure silica are assumed to be 1 and 1.45, respectively. The modal properties of such fiber have been numerically analyzed [6–12]. An interesting point to mention here is that an index-guiding MOF can be single-mode for an extended wavelength range, provided its *d* is relatively small with respect to *Λ*. For the fiber shown in Fig. 1(a), its effective *V* number (*V _{eff}*) [2] as a function of normalized frequency

*Λ/λ*is plotted in Fig. 1(b) (red curve), assuming the air-silica composite cladding extends infinitely far.

*V*here is calculated using

_{eff}${V}_{\mathit{eff}}=\frac{2\pi a}{\lambda}\sqrt{{{n}_{\mathit{silica}}}^{2}-{{n}_{\mathit{SFM}}}^{2}}.$

*a* is effective core radius. We used *a*=0.525Λ. *n _{SFM}* is index of the fundamental space filling mode (SFM) in a uniform 2D air-silica photonic crystal characterized by value

*d*and

*Λ*(hereafter, this 2D photonic crystal is referred as the PC) when wave is propagating along the air hole axis. This chosen

*a*value will give rise to single-mode cut-off

*V*valued at 2.18 [11]. The same figure also shows

_{eff}*V*curves for other fibers characterized by different

_{eff}*d/Λ*values. We may conclude from the plot that the fiber in Fig. 1(a) is endlessly single-mode.

† These two measured values, which are later used in our numerical study, have 4-decimal-point accuracy. It should be noted that current MOF fabrication technology has little control at such precision level.

However, recent experiments done by Eggleton *et al*. [13–15] have shown that when a fiber Bragg grating (FBG) is written in the core region of such a “single-mode” MOF, the transmission spectrum shows several dips other than only one dip. Each of these dips suggests one mode that is supported by the fiber. By comparing the spectrum with that obtained using a FBG written in a conventional single-mode fiber, they conclude that the dips at short wavelength region are due to coupling from core mode to backward-propagating “cladding modes”. Field patterns of these modes are experimentally recorded by tuning input wavelength at those dip positions. In this paper, we will show that these “cladding modes” are not actually comparable to those in conventional single-mode fibers for the fibers’ structural difference. These higher order modes are, in principle, caused by the antiguiding nature of the MOF, which is in turn due to finite periodic air-silica cladding size.

## 2. Numerical analysis of antiguided leaky modes

Antiguided modes appear in waveguides whose core index is smaller than its cladding index. Hollow optical fiber is a typical example [16]. Antiguided modes are leaky in general and they spill all over space. For simple structures whose geometry is easily describable mathematically, their modal property can be derived analytically. As the waveguide becomes complex, numerical method must be adopted. Under such circumstances, we need to consider proper boundary condition to truncate computation domain from an infinite one to a finite one. Scalar beam propagation method (BPM) with transparent boundary condition (TBC) [17] is chosen for our simulation. As long as modal polarization effect is not concerned, scalar approximation can be used to derive mode field distributions and their effective indices (*n _{eff}*) fairly accurately, provided the waveguide is formed by low-index contrast materials. The validity of scalar approximation for air-silica MOFs has been experimentally affirmed in [13–15].

The MOF under study is shown in Fig. 2(a). The fiber’s dimensional parameters are the same as those for Fig. 1(a). The central red-filled circle has a refractive index of *n _{0}*, and a diameter

*d*which is equal to all surrounding air holes’. This red region acts as a defect in the PC if

_{0}*n*≠1.0. Despite 5 rings of air-holes, we have included 18 more air-holes in the 6

_{0}^{th}ring in order to make the air-hole region more circular. Their inclusion does not change the waveguide’s C

_{6v}group symmetry [19]. Unlike usual way of defining core and cladding, we now treat the periodic air-silica composite (including red circle region) as core, and the bulk silica (white region) as cladding, which extends to infinity. We have neglected outer silica-air interface. This simplification will not affect our final result greatly since any light wandering into bulk silica region will eventually leak out of the outer silica-air interface; it will not be coupled back to the red circle region due to the woods of air cylinders. This point has been experimentally proven by Eggleton

*et al*. [13] where the transmission spectrum of a MOF FBG/LPG has not varied much when the MOF is immersed into index-matching liquid. Fig. 2(b) gives the zoom-in view of the fiber at the central defect region.

The MOF in Fig. 2(a) can be roughly approximated by a simple step-index (SSI) antiguide whose index profile is shown in Fig. 2(c). The refractive index value of its core (grey area) is given as the effective index of the longitudinally-propagating fundamental space-filling mode (SFM) in the PC at *λ*=1.55*µm*, i.e., *n*
_{1}=*n _{SFM}*=1.4443. This approximation is accurate only when the periodic air-silica composite in Fig. 2(a) functions the same as an infinitely-extended composite, which is not always true here, as later their supported modes will show. Also this approximation disregards the fine structure and symmetry of the composite core. Hence this simple model only qualitatively reflects the MOF’s property. Nevertheless, it can provide us physical insight of antiguiding nature in the MOF. Central red circle also acts as a defect with index

*n*

_{0}’=1.45 and diameter

*d*

_{0}’.

*d*

_{0}’ depends on

*n*

_{0}. When

*n*

_{0}=1.0,

*d*

_{0}’=0

*µm*, and both waveguides have a uniform core (one is composite, the other is single-material). When

*n*

_{0}=1.45,

*d*

_{0}’=1.05

*Λ*, and two waveguides both have a defect at the center: defect in the MOF is shown by the unit cell in Fig. 2(b); defect in the SSI antiguide [denoted in red in Fig. 2(c)] has the same area as that unit cell in the MOF.

For SSI antiguides, we choose analytical transfer-matrix method (a similar formulation for planar multilayer waveguide can be found in [18]), whose implementation is also based on scalar wave equation.

#### 2.1. Antiguiding of the MOF with a uniform composite core

First we let *n*
_{0}=1.0, i.e., the red circle in Fig. 2(a) represents an air hole (hereafter, we refer this MOF as MOF-1). The corresponding parameters in Fig. 2(b) are *n*
_{0}’=1.4443 and *d*
_{0}’=0*µ*m, which make the simple antiguide like a hollow optical waveguide. Modes of such waveguides can be calculated using a transfer-matrix method in cylindrical coordinates [18]. LP_{01}-, LP_{11}-, LP_{21}- and LP_{02}-like modes at *λ*=1.55*µm* are plotted in Fig. 3(a), (b), (c) and (d), respectively. Their *n _{eff}* values and corresponding leakage losses are summarized in Table 1.

Relatively large imaginary part in both *n _{eff}* values and mode field values suggest high leakage losses for these modes in such type of optical waveguides. This is also manifested by oscillating mode field tail at remote radial positions, which transports energy out from core region (mathematically, these leaky transverse mode profiles can be represented by Hankel functions with certain phase velocities along radial direction). The fundamental mode will only propagate about 5 centimeters before its power drops to 1 percent. For a SSI antiguide, though not shown here, it can be concluded that all modes are leaky. Fundamental mode always has the smallest loss. And the loss value decreases as: (1) index contrast between core and cladding increases; (2) operating wavelength decreases.

Upper plot in Fig. 4 gives the mode spectrum of MOF-1. The calculation is done by using scalar BPM. Correlation mode solver [20] has been used to derive the propagation constant and mode field distribution for each leaky mode. The fiber length is chosen as 2^{15}
*µm* for all similar calculations, unless otherwise specified.

Under symmetrical launching condition, only LP* _{0m}*-like modes are excited. Six dominant modes are given in Fig. 4. We observe that these modes can be grouped according to their corresponding SFMs in the PC. Mode-

*a*resembles the fundamental SFM of the PC when light propagates along

*z*direction. Its field envelope is close to a Gaussian curve, hence we call it LP

_{01}-like mode. Mode-

*b*and mode-

*c*are also derived from the fundamental SFM but with radial field nodes. They are LP

_{02}- and LP

_{03}-like modes, respectively. Mode-

*d, e*and

*f*are grouped under a higher-order SFM of the PC. They can be similarly treated as LP

_{01}-, LP

_{02}- and LP

_{03}-like modes under this SFM category. Effective indices and corresponding confinement losses are given in Table 2. It should be noted that loss for LP

_{01}-like mode in the higher-order SFM group (mode-

*d*) is smaller than that in the fundamental SFM group (mode-

*a*). The same thing happens for LP

_{02}- and LP

_{03}-like modes. Physically, such loss difference is due to the fact that high-order SFM experiences many more times of reflections in the PC than fundamental SFM. It can be concluded from Fig. 4 that: (1) the modes excited with symmetric launching condition can be either SFM-like with a Gaussian-shaped envelope, or with extra envelope nodes in the radial direction; (2) each mode excited has their field

*envelope*exhibiting full C

_{6v}group symmetry suggested by MOF-1’s core [19]; (3) each mode excited has their

*amplitude*exhibiting the symmetry not only suggested by the composite core, but also suggested by the composite itself. It is not difficult to imagine that all modes’ envelopes here tend to exhibit C

_{∞v}symmetry if more air-hole rings are included in the composite core (the composite core can be made infinitely close to a circular shape as the composite goes finer).

Only mode-*a* and *b* are comparable to modes we derived using SSI antiguide approximation. The difference in *n _{eff}* values between mode-a in Fig. 4 and that in Fig. 3 is 0.00033-0.000002

*i*. The difference in

*n*values between mode-

_{eff}*b*in Fig. 4 and mode-

*d*in Fig. 3 is 0.00042-0.000018

*i*. Both

*n*values and the mode field patterns show our approximation is generally accurate. SSI antiguide approximation neglects structural detail of MOF-1, hence it overlooks mode-

_{eff}*d, e*and

*f*here. But if we use the effective index of the PC’s SFM represented by mode-

*d*as

*n*

_{1}during our SSI antiguide approximation, we will get similar

*n*values corresponding to mode-

_{eff}*d, e*and

*f*here; mode-

*a, b*and

*c*will be overlooked instead.

In order to get modes other than LP_{0m}-like, we need to excite MOF-1 at an offset position. Fig. 5 shows the mode spectrum and its corresponding dominant modes when the excitation source is launched at *x*=2.5×*Λ* and *y*=0*µm*. There are three groups of modes, represented by mode-*a, d* and *e*. Each of them corresponds to a different SFM of the PC. Mode-*a* is the same as that in Fig. 4. Mode-*d* and *e* correspond to higher-order SFMs. Mode-*d* only has reflection symmetry about plane defined by *y*=0*µm* and *z*=0*µm*; such mode cannot be excited by symmetric launching since its amplitude does not fulfill C_{6v} symmetry though its envelope does. Mode-*e* is actually degenerate to mode-*d* in Fig. 4 since they share the same SFM of the PC, despite there is a π phase difference along nearest-neighbor direction between them. Such degeneracy makes modal analysis in a composite-core fiber more difficult. Besides three LP_{01}-like modes, LP_{11}- and LP_{21}-like modes are also excited. They have, respectively, one and two nodal planes along azimuthal direction. Losses for six modes displayed are given in Table 3. Similarly, modes in higher-order SFM group generally have smaller losses. We can conclude from Fig. 5 that, except that LP_{01}-like mode under fundamental SFM group, modes (either their amplitude or their amplitude envelope) excited by an offset launching do not fully support symmetry of the composite core; modes excited tend to have variations in the azimuthal direction.

Mode-*b* and *c* here are comparable to those in Fig. 3. Two LP_{11}-like modes have a *n _{eff}* difference of 0.00025-0.00000091

*i*, and two LP

_{21}-like modes have a

*n*difference of 0.000396-0.0000278

_{eff}*i*.

#### 2.2. Antiguiding of the MOF with a defective cmposite core

Now we turn our attention to the fiber similar to that in Fig. 1(a), in which *n _{0}*=1.45 (hereafter, we refer this MOF as MOF-2). Though modal study for such fiber has appeared in many publications, none of them has noticed its antiguiding nature. Following our convention, the periodic air-silica composite is still treated as the fiber’s core, but now with a defect in the center. And surrounding bulk-silica is the cladding. Its equivalent simple antiguide model [Fig. 2(c)] has the parameters

*n*’=1.45 and

_{0}*d*’=1.05Λ, with all other parameters unchanged.

_{0}Figure 6 shows four least leaky modes for this antiguide. Their modal *n _{eff}* values are given in Table 4. Comparing to Fig. 3, the LP

_{01}-like fundamental mode has changed significantly in appearance due to lifted-index region at the center, which attracts most of the mode energy. The fundamental mode has a loss of 3.4675e-6

*cm*

^{-1}, which is almost negligible. But nevertheless, small oscillating mode field tail can still be observed for this “bounded” mode (see inset in its 1D plot). This is due to evanescent wave coupling between central lifted-index region and outer silica region.

Figure 7 gives the mode spectrum and two dominant modes (mode-*a* and *e*) when MOF-2 is excited with a symmetrically launched source. Their mode fields all fully support C_{6v} group symmetry. Whereas mode-*a* can find its analogue in SSI antiguide model [Fig. 6(a)], mode-*e* has no equivalent as such. These excited modes do not belong to any SFM of the PC. Instead, they belong to the defect modes of the PC. Such modes only reflect the symmetry of the defect. Unless the mode is extending to bulk silica cladding region, symmetry of the composite waveguide core is not evident in these modes. Mode-*b*, *c* and *d* shown in the spectrum are comparatively too lossy to be captured by the correlation mode solver. But as we will show later, they are dominating modes when the excitation is launched asymmetrically. *n _{eff}* values for mode-

*a*and

*b*are 1.446889+1.633e-9

*i*and 1.437310+7.426e-5

*i*, corresponding to a loss of 1.3239e-4

*cm*

^{-1}and 6.0205

*cm*

^{-1}, respectively.

Comparing mode-*a* and that shown in Fig. 6, the difference in their real part of *n _{eff}* values is 0.000768. Their mode fields are both Gaussian-like. Unlike MOF-1, we cannot get

*n*value of mode-

_{eff}*e*by just using another

*n*

_{o}’ value for SSI antiguide approximation.

Figure 8 shows the mode spectrum when MOF-2 is excited with an offset source. Mode-*a, b, c, d* dominate the spectrum. From their 2D mode field plots, we observe that mode-*a, b, c, d* belong to the fundamental SFM group. They are called LP_{11}-, LP_{21}-, LP_{31}-, LP_{22}-like modes, respectively. Mode-*e*, which is the same as that in Fig. 7, is still excited, but with a very small power fraction. Mode-*f* and *g* are LP_{01}- and LP_{11}-like, belonging to a high-order SFM group. This SFM represented by mode-*f* can roughly be considered as the same SFM represented by mode-*e* in Fig. 5. The difference in *n _{eff}* values between the two representative modes is only 0.000005-0.00000298

*i*, despite the perturbation caused by different

*n*

_{0}in two waveguides. However, the slight difference in modal appearance should be attributed to the lifted central index

*n*

_{0}.

Modes excited in Fig. 8 have *n _{eff}* values and their corresponding losses given in Table 5. Mode-

*a*is comparable to mode-

*b*in Fig. 6. They have a n

_{eff}difference at 0.00035-0.00000693

*i*. Mode-

*b*is comparable to mode-

*d*in Fig. 6. They have a n

_{eff}difference at 0.000394+0.00002711

*i*. Indices for mode-

*a*and

*b*have no much difference from mode-

*b*and

*c*in Fig. 5. This is due to the fact that mode field at the center must be zero as azimuthal mode number is greater than 1. Hence the lifted index there has little contribution to the modal effective index.

Together with two modes shown in Fig. 7, we can see that except mode-*a* in Fig. 7, all other modes which can exist in MOF-2 are extremely lossy. They can only propagate for about 1–4cm in this fiber before their power drops by 2 orders. So effectively, as far as optical communication system is concerned, MOF-2 is a single-mode fiber, and it remains single-mode for an extended wavelength range for reason explained in [2]. Following convention, we call mode-*a* in Fig. 7 as fundamental mode for MOF-2. All other modes are called higher-order leaky modes.

## 3. Discussion on antiguided leaky modes in MOFs

Higher-order leaky modes in MOF-2 are not equivalent to cladding modes as present in conventional step-index single-mode fibers (SISMF). The difference between them can be understood from their origins and their influence on optical devices and optical transmission systems.

In SISMF, core mode is the fundamental mode fully supported by doped-core and pure-silica cladding, and cladding modes are caused by reflection at the air-clad interface (we can understand these modes by treating the whole fiber as guiding core and surrounding air as cladding). Due to positive index-contrast between the Germanium-doped core and pure silica cladding in SISMF, light tends to settle into the core mode. Cladding mode cannot be excited easily if the light source is intentionally coupled into the core region within an acceptance angle. Possible irregularities in fiber cable and interconnects, which act as mode scramblers, can redistribute energy from propagating core mode into cladding modes. Current SISMF fabrication technology allows little variation on core shape along axial direction. And state-of-the-art fiber splicers can splice fibers with negligible loss. So cladding modes in SISMF do not pose a big problem in current optical communication system. And though cladding modes may appear at launching end or some other irregularity sites, they can easily be stripped off by applying an outer layer of high-index jacket around the fiber.

The effect of cladding mode becomes significant when we are analyzing fiber-based devices like FBG. When FBG is written in the doped core, the grating acts as discontinuities in the waveguide. It will perturb and thus redistribute energy among all possible modes, including cladding modes. If any mode has the propagation constant that fulfills the phase matching condition of the grating [21], it will get reflected. Those modes which did not get reflected will have their energy coupled back into the core mode after a short distance. Energy loss caused by FBG reflection can be observed clearly as sudden dips in the grating’s transmission spectrum [15]. In most of the cases, we want to reflect only one channel of information (say at *λ*) by using one uniform FBG. *λ* here is the fundamental Bragg wavelength. Coupling from forward-propagating core mode to backward-propagating core mode happens at this Bragg wavelength. But if the cladding mode comes into play (especially when the index modulation in the FBG is big or the grating is tilted), channels in slightly shorter wavelength region (say at *λ’*) may also experience reflection. *λ’* here is a higher-order Bragg wavelength. Coupling from forward-propagating core mode to backward-propagating cladding mode happens at this wavelength. Such cladding mode resonances in SISMF give us undesired losses in wavelength-division-multiplexing (WDM) transmission system. But such losses can always be eliminated by employing special-designed cladding-mode suppression fibers [22] for FBG fabrication.

Leaky modes in MOF-2 originate from the antiguiding nature between the composite air-silica core and pure-silica cladding. These modes are inherently leaky. Such type of optical fiber, unless rigorously fabricated, will experience relatively high transmission loss, since any imperfection near the center will excite many leaky modes. Most of these leaky modes have very small overlapping ratio with the fundamental mode, hence this energy redistribution is irreversible. This suggests that using fiber like MOF-2 as a long-distance optical signal transmission medium is not feasible. Increasing number of air-hole rings in design and improving precision technology during fabrication are effective ways to reduce such leakage loss.

When FBG is written in the lifted-index region in MOF-2, its transmission spectrum has several dips, each corresponding to a different mode whose propagation constant phase-matches with the FBG [15]. By tuning source wavelength at one of the dips, fundamental mode and higher-order leaky modes can be recorded. Whereas cladding mode resonances in SISMF can be eliminated by using special fibers during FBG fabrication, dips corresponding to higher-order leaky mode in MOF-2’s transmission spectrum cannot be easily mitigated (one possible way could be to increase number of air hole rings in MOF designs). In addition, due to short distances between fundamental Bragg resonance and those higher-order Bragg resonances [15], one FBG might reflect more than one channel in WDM systems (but with only one peak appearing on the reflection spectrum since higher-order modes are all leaky). That limits the usage of MOF in FBG devices, or may bring some new ways of applying FBG devices for aught we know.

Some people utilized the fact that all modes in an antiguide are leaky to make a vertical-cavity surface-emitting laser (VCSEL), which has very high output power but remains single-mode [23]. The high power is achieved by a big core area, and single-modeness is achieved by making higher-order modes far lossier than the fundamental one (only fundamental mode survives after gain competition). So theoretically speaking, we can use structure of index profile like MOF-2 in VCSEL cavity design, but with reduced number of air-hole rings and an optimized dimension so as to increase the loss discrimination between fundamental mode and higher-order modes. Some preliminary work has recently been published in [24].

## 4. Conclusion

Modes in a composite-core fiber have been described in general. The antiguiding effect exists in microstructured fibers with a finite number of air-hole rings. All modes for such a waveguide are shown to be leaky. Although the fundamental mode has relatively small leakage loss, its energy could be redistributed into higher-order modes, which then dissipate very quickly, in the presence of waveguide irregularities. Such a leakage mechanism can prevent the MOF studied from being employed as a long-haul transmission medium, unless we increase the number of air-hole rings and decrease the thickness of bulk silica surrounding the air-hole rings. The effect of these higher-order leaky modes on the FBG transmission spectrum is also discussed. Possible employment of an MOF-like index profile with a reduced number of rings of air-holes in a high-power VCSEL cavity design is also suggested. Antiguided modes in such MOFs are compared with those in its approximated model with a simple step-index profile. Since the approximation involves treating a composite material as a homogeneous one, a similar method does not apply to modes guided by photonic bandgap, where the composite plays the role of opening the bandgap. Though scalar BPM is relatively efficient in deriving the modal property of such waveguides, full characterization of such fiber’s modal property needs a numerical or semianalytical method based on full-vector formulation.

## Acknowledgments

M.Yan would like to thank J. Zhang for discussion on FBG, H. Liu for discussion on VCSEL, and Dr. Le Binh for valuable comments on the manuscript. He also would like to thank the Singapore government for providing financial support for his first degree and current research work.

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