1368 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 9, MAY 1, 2010

Bend Loss in All-Solid Bandgap Fibers RevisitedYanfeng Li, Member, OSA, David M. Bird, and Tim A. Birks, Fellow, OSA

Abstract—The rate of decay of the fundamental guided modeinto the surrounding cladding in all-solid photonic bandgap fibersis investigated. Both a simple 1-D model and numerical calculationsof realistic fibers show that the decay rate varies asymmetricallyacross the bandgaps, with larger values near the upper edge of thebandgap. The implications for modeling bend loss in these fibersare discussed.

Index Terms—Bend loss, mode decay, numerical modeling, pho-tonic bandgap fiber, photonic crystal fiber.

I. INTRODUCTION

A LL-SOLID PHOTONIC BANDGAP fibers (ASPBFs),like hollow-core photonic bandgap fibers, guide light in

a low-index core due to the existence of a photonic bandgapin the two-dimensionally microstructured cladding [1]–[5].ASPBFs have attracted interest not only because of a varietyof practical applications [5]–[7] but also because their simplestructure allows for a comprehensive understanding of thephysics of bandgap guidance [8]–[13].

In [11], we presented an experimental and theoretical anal-ysis of bend loss in ASPBFs. We demonstrated and explaineda number of phenomena associated with bend loss, including avariation of susceptibility to loss between even- and odd-num-bered bandgaps, the existence of centripetal as well as cen-trifugal bend loss, and a greater susceptibility to bend loss to-ward the high frequency edge of band gaps. Although we areconfident that our analysis in [11] is qualitatively correct (it hasrecently been confirmed by extensive numerical calculations ofbend loss [14]), there is one quantitative aspect of that work thatwe now believe is oversimplified, namely the expression for thecritical bend radius given there in (2) (and here in (4)). It is ouraim in this paper to present a more detailed analysis of this ex-pression. Interestingly, the authors of [14] also question the va-lidity of this equation, but they present no further discussion ofthis point.

Our starting point is the analysis of various forms of loss inoptical waveguides given by Love [15]. Two length scales can be

Manuscript received January 13, 2010; revised March 08, 2010. First pub-lished March 15, 2010; current version published April 21, 2010. The work ofY. Li was supported by the China Scholarship Council’s Young Teachers’ Pro-gramme.

Y. Li is with the Ultrafast Laser Laboratory, College of Precision Instrumentand Optoelectronics Engineering and the Key Laboratory of OptoeloectronicsInformation and Technical Science, Tianjin University, Tianjin 300072, China(e-mail: yanfengli@tju.edu.cn).

D. M. Bird and T. A. Birks are with the Centre for Photonics and Photonic Ma-terials, Department of Physics, University of Bath, Bath BA2 7AY, U.K. (e-mail:d.bird@bath.ac.uk).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JLT.2010.2045638

defined that characterize the guidance: first, the coupling lengthbetween the guided mode and cladding modes

(1)

where is the free space wavelength of the light, and andare the effective indexes of the guided and cladding modes,

respectively; and second, the transverse decay length ofthe guided mode. The evanescent field of the guided mode de-cays exponentially into the cladding and it is assumed that thetransverse size of the mode is essentially determined by a termthat goes as , where is the radial coordinate. Loveshowed that the critical bend radius below which bend loss be-comes large can be expressed in terms of these length scales by

(2)

In a conventional step-index fiber, the mode decay rate isgiven by [16]

(3)

and (assuming the weakly guiding case ) the criticalbend radius becomes

(4)

In ASPBFs, the situation is more complex because of the ex-istence of cladding modes with both higher and lower effectiveindexes than the bandgap-guided mode. We showed in [11] thatbending can couple the guided mode into cladding modes at ei-ther the upper or lower band edge, and that the nearer band edge(in terms of effective index) dominates the coupling. This pic-ture is sufficient to explain qualitatively all of the experimentalobservations outlined earlier. To provide a more quantitative de-scription, we adapted the analysis given in (1)--(4); was re-placed by in both (1) and (3), where is the effectiveindex of the nearer band edge. This led directly to (2) forgiven in [11].

While the use of in (1) is justified, its use in (3) is morequestionable. It is reasonable to assume that the decay rate intothe cladding depends on the mismatch between and ,but there is no a priori reason why the particular functional formgiven in (3) should be correct. Our aim in this paper is thereforeto investigate the decay rate in more detail in order to test thisassumption. We shall show that (3) does not provide a quantita-tively accurate description of in ASPBFs. This will be demon-strated in two ways. First, in Section II, we use a simple 1-Dmodel to investigate the behavior of the cladding decay rate inbandgap structures, which provides a counter-example to (3).

0733-8724/$26.00 © 2010 IEEE

LI et al.: BEND LOSS IN ALL-SOLID BANDGAP FIBERS REVISITED 1369

Fig. 1. Photonic DOS and decay rate for the 1-D model structure. Gray scalerepresents DOS of propagating states ������ � ���� �� and colors repre-sent decay rate of evanescent states � ��� ������ � ���� ���� ����� in thebandgaps. A cross-sectional view along the vertical line at � � � �� is plottedin Fig. 2.

Second, in Section III, we present numerical results for fora realistic ASPBF structure and compare them with those ob-tained from (3) with replaced by . We conclude inSection IV with a discussion of the implications of our resultsfor modeling bend loss in ASPBFs.

II. ANALYTIC 1-D MODEL

The general features of the photonic density of states (DOS)and the cladding decay rate for a bandgap structure can beobtained from an analysis of a scalar model consisting of a 1-Dperiodic array of delta-function layers [17]. The index distribu-tion is given by

(5)

where is the pitch, labels the layers of the structure, andis the “strength” of the delta-function layers relative to the

background index . As shown in the Appendix, it is straight-forward to obtain analytic solutions for this model, both for thepropagating states that contribute to the DOS and for the evanes-cent states that exist in the bandgap regions. These evanescentstates are characterized by an exponential decay, which can becalculated for any effective index at any normalized frequency

(where is the free space wavenumber). In a periodicstructure, it is convenient to use the normalized decay rate ,which represents the decay of the field per period. It is importantto note that this quantity can be defined and calculated withoutany reference to a defect in the lattice. A defect is requiredto obtain physically relevant states that exhibit this exponen-tial decay, but the rate of decay itself is a property only of thecladding structure.

The DOS and cladding decay rate for the model structure areshown in Fig. 1, where the background index andthe delta-function magnitude have been chosento match the realistic ASPBF structure, discussed in Section III.In Fig. 2, we show a cross-sectional view of Fig. 1 at .This figure also shows the decay rate that would be predicted

Fig. 2. Cross-sectional view of Fig. 1 at � � � ��. Solid blue and red linesrepresent the DOS and normalized decay rate, respectively. The approximationto �� given by (6) is shown by the black-dashed curves.

Fig. 3. ASPBF structure used in numerical analysis, consisting of high index�� � rods of diameter � in a lower index �� � background. The central defectis a rod whose diameter and index are � and � , respectively. � and �are the lattice directions of the supercell.

by the analysis presented in [11], i.e, (3) with replaced by:

(6)

Fig. 2 shows that (6) does not provide a good approximationto the real decay rate. While (6) predicts that behaves sym-metrically with respect to the upper and lower band edges, thereal decay rate is strongly asymmetric and is generally largertoward the edge of the upper, narrower band. Fig. 2 also showsthat (6) tends to overestimate the real decay rate, particularlytoward the lower band edge. We shall see in the next sectionthat these features are replicated by numerical calculations on amore realistic ASPBF structure.

III. NUMERICAL RESULTS

The structure of the ASPBF is shown in Fig. 3. Typically, amissing high-index rod acts as the fiber core but here, to allowtuning of the defect modes, we assume that the core defect is anadditional rod with diameter and index . Other param-eters are chosen to be as in [13]: ,and . The integral of in the direction of

is used to obtain the value of the delta-function strengthin the previous section.

The fully vectorial fixed-frequency plane-wave method de-scribed in [18] is used to calculate the photonic DOS of the

1370 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 9, MAY 1, 2010

Fig. 4. DOS for the ASPBF structure described in the text. The arrows schemat-ically show the tuning of the defect modes across the bandgaps at two chosennormalized frequencies � � � �� and � � � ��.

Fig. 5. Normalized Poynting vector component � (on natural log scale) ofthe fundamental guided mode at � � � �� along the supercell direction �(in units of �). The index of the central defect rod and the effective index of theguided mode are shown in the inset.

infinitely periodic triangular lattice with these parameters. TheDOS is shown in Fig. 4, where, as in Fig. 2, the gray scale rep-resents the density of propagating states and here the bandgapsare shown in red. The labels indicate the rod modes fromwhich the bands arise according to the ARROW model [8]–[10],[13]. The DOS shows how the rod modes broaden into claddingbands; between these modes, the cladding rods are antiresonant,expelling light and forming the bandgaps.

We choose to study the mode decay behavior at two normal-ized frequencies, and , in the first andsecond bandgaps, respectively. These bandgaps show the typ-ical behavior of odd- and even-numbered bandgaps [11], withthe floor of the first bandgap being determined by an rodmode, while that of the second is governed by an rod mode(and hence the bandgap is less deep). As mentioned earlier, wevary the parameters of the central defect so that we can tunethe index of the fundamental defect mode across the bandgaps,as shown schematically in Fig. 4. This allows us to build up amore complete picture of the decay rate, as was done for the1-D model structure in the previous section. The calculationsare performed using a 12 12 supercell.

Fig. 5 shows the -component of the normalized Poyntingvector of the fundamental defect mode at along thedirection for and different values of . The ef-fective index of each guided mode is also shown, in the inset.

Fig. 6. Mode decay rates �� as a function of effective index for (a) � � � ��

and (b) � � � ��. The vertical lines indicate the bandgap edges.

The plots clearly show an exponential decay of the mode intothe cladding, and it can be seen how the decay rate varies asthe guided mode crosses the bandgap. The decay rate steadilyincreases as and the effective index of the mode in-creases, until very close to the upper edge of the gap when themode profile suddenly changes and the decay rate decreases.

Similar calculations have been carried out for a range of de-fect parameters and at and .The parameters were chosen to tune the mode index across thebandgap that straddles the background index ateach frequency. (Since the mode index was always below theupper index of the band, Fig. 4, the modes we considerwere guided by a photonic bandgap and not by total internal re-flection.) The decay of the Poynting vector is fitted to a function

, where is the radial coordinate, to extractthe normalized decay rate . The fitted decay rates shown bythe red and blue points in Fig. 6 display two important features.First, the decay rate depends only on the effective index of thedefect mode. The red and blue points in Fig. 6 are derived fromcalculations that involve either a change of the defect diameter

LI et al.: BEND LOSS IN ALL-SOLID BANDGAP FIBERS REVISITED 1371

Fig. 7. Comparison of normalized critical bend radius � �� calculated from(2), using the numerical decay rate and the analytical decay rate given by (6),where the nearest band edge is used.

with the index fixed, or from a change of the defect index witha constant diameter. The points clearly lie on the same curve,showing that, as in the 1-D model, the decay rate is an intrinsicproperty of the cladding. Second, the asymmetry found in the1-D model is also seen in the realistic fiber structure, with thedecay rate being generally greater toward the upper edge of thebandgap. The black curves in Fig. 6 show the decay rate given bythe approximate expression (6). As in Fig. 2, it can be seen that(6) significantly overestimates the real decay rate, and it missesthe asymmetry of the real decay rate.

IV. DISCUSSION AND CONCLUSION

The key result from the previous two sections is that (6) is nota good approximation to the real modal decay rate. It followsthat (2) of our previous work [11] does not provide an accurateprediction of the critical bend radius. Our 1-D and 2-D calcula-tions give similar results, with the decay rate steadily increasingfrom the lower band edge and decreasing only very close tothe upper band edge. Since two waveguides with such differentdetailed structures share these properties, we are confident thatthey are representative of a wide class of ASPBF structures. Thedecay rate is also generally smaller than that given by (6).

As discussed in Section I, one motivation for analyzing thedecay rate is to model the bend loss in ASPBFs. We have there-fore used the ASPBF structure of Section III, with a missingrod as the central defect, to calculate the effective index of thefundamental guided mode across the first and second bandgaps.The critical bend radius (2) has been evaluated, with givenby (1) (with replaced by ) and given either by (6)or extracted from the numerical calculations. Plots of (inunits of ) are shown in Fig. 7. It can be seen that using (6) for

significantly overestimates the critical bend radius, particu-larly toward the lower frequency side of the bandgap where thediscrepancy can be larger than a factor of 2. This enhanced re-silience to bend loss at the lower frequency edge of the bandgapis consistent with the experimental results presented in [11].

In summary, we have shown that our previous work on bendloss in all-solid bandgap fibers, although qualitatively correctin describing the observed trends, in quantitative terms signif-icantly overestimates the susceptibility of these fibers to bendloss. We have shown that a toy model can provide a useful in-sight into the origin of this discrepancy, which lies in the be-havior of the modal decay rate into the cladding.

APPENDIX

Here, we outline the analysis that gives the DOS plot andmode decay rate for the model 1-D structure discussed inSection II. Consider a scalar model with pitch and indexdistribution given by (5). Let and let be the field,where . can be written as [19]

(A1)

where and and are constants. Thisform of is convenient because it behaves continuously ascrosses . is the propagation constant and is related to theeffective index by . The boundary conditions thatrelate and are

(A2)

(A3)

which can be derived by integrating the scalar wave equationacross the boundary [17].

Application of the boundary conditions leads to a transfer ma-trix relating the states in each successive layer, as shown in (A4)at the bottom of this page, where .

The matrix in (A4) has eigenvalues , where, if isreal, we have , where is the Bloch wavevector.is determined by

(A5)

where and . The function in (A5) hasthe following properties: 1) if , the waveis propagating and the field varies from period to period as

, where ; 2) if , the waveis evanescent and the field varies from period to period as

, where ; 3) if , thewave is evanescent and the field varies from period to period as

(A4)

1372 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 28, NO. 9, MAY 1, 2010

, where . thereforedetermines whether the wave is propagating or evanescent, and,in the latter case, gives the mode decay rate as a function ofthe propagation constant (represented by ), frequencyand “strength” of delta-function (both represented by ).

For propagating waves, we would like to calculate the DOS.Since and , if we evaluate theDOS with respect to intervals of for states equally spaced in

, we have

(A6)

Here, is obtained from (A5) and the differentiation is carriedout as follows:

(A7)

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Yanfeng Li graduated in 1999, and received the Ph.D. degree in optical engi-neering, in 2005, both from Tianjin University, Tianjin, China.

He was a Postdoctoral Fellow at Tianjin University, where he has been anAssociate Professor with Ultrafast Laser Laboratory, College of Precision In-strument and Optoelectronics Engineering since 2007, and where he is also withthe Key Laboratory of Optoeloectronics Information and Technical Science. Hiscurrent research interests include design of photonic crystal fibers and other de-vices for use in ultrafast laser science and terahertz photonics.

Dr. Li is a member of the Optical Society of America.

David M. Bird has been with the University of Bath since 1984, where he iscurrently a Professor of physics at the Centre for Photonics and Photonic Ma-terials, Department of Physics and is soon to take up the post of Dean of theFaculty of Science. At the Centre for Photonics and Photonic Materials, hehas involved on developing theoretical and computational methods for solvingMaxwell’s equations in structured dielectrics, with a particular emphasis onphotonic crystal fibers. His research interests include development and appli-cation of large-scale, first-principles electronic structure calculations, based ondensity functional theory.

Tim A. Birks has been with the University of Bath since 1996, where he is cur-rently a Professor at the Department of Physics and the Director of the Centrefor Photonics and Photonic Materials. Since 1994, he has been a Pioneer ofthe photonic crystal fiber, developing the original fabrication techniques. He isthe author or coauthor of more than 160 papers published in peer-reviewed in-ternational journals and conference proceedings. His research interests includethe postprocessing of photonic crystal fibers and understanding their guidancemechanisms, as well as continuing to find new applications for tapered conven-tional fibers.

Prof. Birks was a Fellow of the Optical Society of America in 2007, andcurrently holds a Leverhulme Research Fellowship.