Confinement loss, including cladding material losseffects, in Bragg fibers

Jun-ichi Sakai* and Norihiro Nishida

Faculty of Science and Engineering, Ritsumeikan University, Noji-higashi, Kusatsu-city, Shiga 525-8577, Japan*Corresponding author: jsakai@se.ritsumei.ac.jp

Received September 7, 2010; revised December 1, 2010; accepted December 3, 2010;posted December 9, 2010 (Doc. ID 134737); published February 8, 2011

Confinement loss (CL), including cladding material losses, is comprehensively evaluated for TE, TM, and hybridmodes of hollow-core Bragg fibers. We show that the combined fiber loss tends to approach a loss depending onlyon CL Lconf for a small number N of cladding pairs, while it approaches a material loss limit value Lmat for large N.Transition point N tr is defined by a crossing point of the asymptotes of Lconf and Lmat. Lmat and N tr are investigatedas a function of cladding material losses, cladding index contrast, and core radius rc, mainly under the quarter-wave stack condition. Lmat decreases in inverse proportion to the third and first powers of rc for TE and the otherthree modes for sufficiently large rc. An increase in rc more effectively reduces Lmat than an increase in the clad-ding index contrast. For rc ¼ 2:0, 5.0, and 10:0 μm in the TE01 mode transmission, we attain fiber losses less thanabout 10−2, 10−3, and 10−4 times cladding material losses, respectively. © 2011 Optical Society of America

OCIS codes: 060.2270, 060.2310, 060.2400.

1. INTRODUCTIONHollow-core photonic bandgap (PBG) fibers (PBFs) have thepotential to achieve a loss lower than conventional silicafibers. Intrinsic optical loss factors include absorption, con-finement, scattering, and structural imperfection losses, andextrinsic factors include bend and splicing losses. The con-finement loss (CL) is caused by the finite thickness of theperiodic cladding region [1], yielding an insufficient opticalconfinement due to the Bragg diffraction originating fromthe periodicity in the cladding. The CL is a key factor for theappropriate design of PBFs because periodic cladding must beas thin as possible to reduce the fabrication process. Confine-ment and cladding material losses are closely related.

PBFs are primarily classified into two categories: one is ahollow-core PBF with air holes in the cladding and the other isa Bragg fiber that consists of an air core surrounded by per-iodic cladding with alternating high and low refractive indices.A low-loss PBF has been fabricated using a seven-cell core [2].The CL of hollow-core PBFs has been investigated using a full-vector finite-element method (FEM) [3]. The antiresonant fea-ture of an antiresonant reflecting optical waveguide has beenused as a core surround in hollow-core PBFs to reduce theCL [4].

The Bragg fiber has been fabricated using a chalcogenideglass polymer [5], an air–silica ring [6], and an air–polymerring structure [7]. The CL in the Bragg fiber has been evalu-ated by the transfer matrix method [8], Chew’s method [9,10],a mixed method of exact and asymptotic analyses [11], and amultilayer division method [12] that was developed to apply alow-index contrast [13,14] to high-index-contrast fibers. Theinfluence of surface mode on the CL spectrum was presentedin the Bragg fiber with a bridge structure using FEM [15]. Pre-vious papers barely considered cladding material losses ex-cept for fibers with a Si=Si3N4 structure [11]. On the otherhand, the material loss dependence of fiber loss has theoreti-cally been estimated [16] for the TE01 mode, which has the

lowest loss [8,11,12] among all modes, by neglecting the fielddecay due to material loss.

The purpose of the present paper is to offer a comprehen-sive study on the CL of various modes of hollow-core Braggfibers with cladding material losses by taking field decay intoconsideration. The CL, which can be calculated from the ima-ginary part of the propagation constant, is numerically evalu-ated here using the multilayer division method [12], wherematerial losses are treated in terms of the complex refractiveindex.

This paper proceeds as follows. Section 2 shows the pre-paration of CL evaluation, including a brief description ofthe Bragg fiber. Section 3 verifies the results obtained bythe present method and investigates the loss coefficient de-pendence of fiber loss. Section 4 presents numerical resultson the CL of various modes to elucidate its properties. In Sec-tion 5, we present the numerical results of the TE0μ mode indetail. Finally, Section 6 studies the PBG dependence of CLthat exhibits a unique property.

2. PREPARATION OF CL EVALUATION

A. Brief Description of a Bragg FiberThe Bragg fiber has a cylindrically symmetric microstructurewith a hollow core surrounded by periodic cladding (Fig. 1).The core index is nc, and its radius is rc. The cladding has afinite number N of layer pairs that consist of high na and lowindices nbðna > nb > ncÞ. Their corresponding layer thick-nesses are a and b, and the cladding period is Λ ¼ aþ b.The index of the cladding external layer is set to nex. The in-tensity absorption coefficients are designated by αa and αb forcladding layers a and b. The loss coefficient of the ex-ternal layer is not considered to concentrate on the effectof cladding loss.

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0740-3224/11/030379-08$15.00/0 © 2011 Optical Society of America

B. Calculation Method of CL with CladdingMaterial LossesAssume that the spatiotemporal dependence of the electro-magnetic fields is represented by Utz ¼ exp½iðωt − βzÞ�, whereω and β are the angular frequency and propagation constant,respectively. The propagation constant may be complex:β ¼ βR þ iβI . The CL, which arises from the finite thicknessof the periodic cladding, can be evaluated from imaginary partβI of propagation constant β as

CL½dB=km� ¼ −20ln 10

109 × βI ½μm−1�: ð1Þ

Here, we require βI < 0 in the present framework.Losses of cladding material are represented by complex

refractive index ~ni as

~ni ¼ ni − iki; ki ¼αIic2ω ði ¼ a; bÞ; ð2Þ

where ni denotes the normal index, ki is the extinction coeffi-cient of the cladding material, αIi is the intensity absorptioncoefficient of the cladding material, and c is the light velocityin the vacuum. Subscripts a and b indicate cladding layers aand b. Loss coefficient αi is αi½dB=km� ¼ 104 logðeÞ × αIi½m−1�.Equation (2) is available for αi ≪ 1010½dB=km� at wavelengthλ0 ¼ 1:0 μm and for αi ≪ 109½dB=km� at λ0 ¼ 10:0 μm.

If the periodic cladding of the Bragg fiber extends to infi-nity, its eigenvalue equation can be simplified using an asymp-totic expansion approximation [17]. However, if the periodiccladding is finite, then we must resort to troublesome numer-ical means to investigate its electromagnetic properties. For afinite number N of cladding pairs, we utilize the multilayerdivision method [12] to evaluate βI as well as βR.

The multilayer division method, which is applied only to acylindrically symmetric refractive index, is also available forthe present case, if the normal index ni is formally replaced bythe complex refractive index ~ni. The restriction of applicationrange yields a benefit that the number of parameters accom-panied by the boundary condition is greatly reduced in com-parison with versatile numerical methods, such as the FEMand the finite-difference time-domain method. The benefitis that the root finding is numerically stable even for the com-plex refractive index and higher-order modes. The mode ofBragg fibers with cladding material losses is easily designated(see Appendix A) near the quarter-wave stack (QWS) condi-tion, which is important for practical use due to excellentoptical confinement to the core.

C. Setting of ParametersFirst, wavelength λ0, refractive indices, and core radius rc areprescribed. Cladding thicknesses a and b are set to satisfy theQWS condition for the prescribed parameters as

aλ0

¼ 12

�4

�n2a − n2

c

�þ�UQWS

π

�2�λ0rc

�2�−1=2

; ð3Þ

which holds for a Bragg fiber with infinite periodic cladding[17].UQWS is a constant peculiar to the mode and relates to thezeroes of the Bessel function. In addition, b is obtained by anexpression where a and na are replaced by b and nb in Eq. (3).

Core index nc ¼ 1:0 and external layer index nex ¼ 1:5 areused (except in Subsection 3.A) throughout this paper. Thewavelength is set to λQWS ¼ 1:0 μm except for the data of Sub-section 3.A and Section 6. We assume here that material losscoefficients have no wavelength dependence.

3. CONFIRMATION OF EVALUATEDFIBER LOSSA. Verification of Evaluated Fiber Loss CharacteristicsCL, including cladding material losses, is first compared withthose derived previously. Fiber parameters are prescribed tobe the same as those in a theoretical estimation [16] forcomparison: λ0 ¼ 1:0 μm, na ¼ 2:5, nb ¼ nex ¼ 1:521, rc ¼1:1166 μm, and αa ¼ αb ¼ 10dB=km.

The combined fiber loss is semilogarithmically plotted inFig. 2 for the TE01 mode of a hollow-core Bragg fiber as a func-tion of N , the number of cladding pairs. The loss is reducedwith increasing N and converges to a constant value, Lmat ¼0:45dB=km. This convergence results from the fact that thefields are almost confined within the core and the periodiccladding, and the loss depends on only the material loss limitvalue Lmat. The converged value is slightly higher than a pre-viously estimated loss, 0:20dB=km [16], because the presentcalculation considers the field decay due to cladding materiallosses, but [16] neglects it. For small N , the combined fiberloss agrees with loss Lconf , where only the CL is considered.For later use, a cross-point of the asymptotes of Lmat and Lconf

is designated by N tr, the transition point. Lconf is dominant for

Fig. 1. Schematics of Bragg fiber: rc, core radius; nc, refractive indexof the core; na and nb, indices of layers with thicknesses a and b;Λ ¼ aþ b, period in cladding; nex, external layer index; N , numberof cladding pairs; αa and αb, loss coefficients of cladding layers aand b; na > nb > nc.

Fig. 2. Comparison of the combined fiber loss of the TE01 mode withfiber loss derived without field decay. Solid curve, combined fiberloss; dashed line, fiber loss derived without field decay [16];dotted–dashed line, CL alone [12]. λ0 ¼ 1:0 μm, na ¼ 2:5,nb ¼ nex ¼ 1:521, rc ¼ 1:1166 μm, and αa ¼ αb ¼ 10dB=km.

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N < N tr, and Lmat is dominant for N > N tr. Lmat is a measureof achievable minimum fiber loss, and transition point N tr is ameasure of the N needed to maintain the loss as low aspossible.

For the TE01 mode of a hollow-core Bragg fiber with fourpairs of Si=Si3N4, the combined fiber loss was investigated bya mixed method of exact analysis and asymptotic expansionapproximation [11]. Cladding parameters were na ¼ 3:5 anda ¼ 0:11 μm for Si and nb ¼ 2:0 and b ¼ 0:21 μm for Si3N4.Other parameters were λ0 ¼ 1:55 μm, rc ¼ 7:5 μm, nc ¼ nex ¼1:0, and αb ¼ 0 dB=km. Only polysilicon loss αa was con-sidered in [11], which gave 1:2113 × 105 and 1:2123 ×105 dB=km for αa ¼ 105 and 106 dB=km, respectively. For cor-responding αa, fiber losses of 1:1963 × 105 and 1:1973 ×105 dB=km were obtained by the present method that utilizesno approximation. The proportional coefficient of fiber losswas about 1:0656 × 10−4 for the previous method, but it was1:0668 × 10−4 for the present method. Both methods providereasonable agreement.

B. Fiber Loss Dependence on Cladding Material LossesFigure 3 illustrates the combined fiber loss of the TE01 modeas a function of N and various cladding material losses.Material losses were taken for 11 combinations of αa andαb up to αa ¼ αb ¼ 108 dB=km. Parameters were λ0 ¼ 1:0 μm,na ¼ 2:5, nb ¼ nex ¼ 1:5, and rc ¼ 2:0 μm. Lconf is independentof the combinations of prescribed αa and αb values for smallN . For a sufficiently small N , Lconf obeys a formula:

Lconf ∝

� ða=bÞ2N for TE modeðn2

bb=n2aaÞ2N for TM and hybrid modes

ð4Þ

under the QWS condition [12]. As cladding material losses in-crease, Lmat increases, but N tr decreases. If different values ofαa and αb are interchanged, then the fiber loss with a smallvalue of αb is lower than that with a small αb. This is becausecladding layer thickness b is wider than layer thickness a un-der the QWS condition. A condition αa ≥ αb is desirable forobtaining a low-loss fiber, if possible.

Figure 4 shows material loss limit value Lmat and transitionpoint N tr drawn from Fig. 3 as a function of αa ¼ αb. Fiberparameters are identical to those in Fig. 3. It is evident fromFig. 4 that Lmat is proportional to αa for a fixed value of αb.Therefore, the value of LmatðαaÞ for a particular αa valuecan be estimated by

LmatðαaÞ ¼αa10

Lmatð10Þ; ð5Þ

where αa and the number inside parentheses indicate the losscoefficients in the decibel per kilometers unit. Here,Lmatð10Þ ¼ 7:22 × 10−2 dB=km and N trð10Þ ¼ 16:2 are obtainedfor the parameters given in Fig. 3. Proportional coefficientLmatð10Þ=10 is 7:22 × 10−3 in this case. The values of Lmatð10ÞandN trð10Þ depend on the core radius, the cladding index con-trast, and the wavelength, as shown later in Sections 4 and 5.Clearly, the level of the combined fiber loss can be keptrelatively low even if the cladding material losses are large.For example, one obtains a combined fiber loss of 7:2 ×103 dB=km for αa ¼ αb ¼ 1:0 × 106 dB=km. This property isuseful for obtaining a low-loss hollow-core Bragg fiber withsuch a large material loss as a polymer [7].

N tr decreases linearly with log αa in Fig. 4. This relationshipis derived as follows. Both Lmat and N tr satisfy Eq. (4) at thetransition point if the fiber parameters satisfy the QWS con-dition. From N tr for different αa values, we obtain

log½LmatðαaÞ=Lmatð10Þ� ¼ S½N trðαaÞ − N trð10Þ�: ð6Þ

Here, S indicates the slope in a semilogarithmic plot and canbe expressed as

S ¼�STE ≡ 2 logða=bÞ for TE modeSnon�TE ≡ 2 logðn2

bb=n2aaÞ for TM and hybrid modes

ð7Þ

from Eq. (4). For example, S ¼ −5:99 is obtained for the con-dition given in Fig. 3. The substitution of Eq. (5) into Eq. (6)leads to

N trðαaÞ ¼ N trð10Þ þlog αa − 1

S: ð8Þ

Fig. 3. Cladding pair dependence of the combined fiber loss for theTE01 mode as a function of the cladding material losses: λ0 ¼ 1:0 μm,na ¼ 2:5, nb ¼ nex ¼ 1:5, and rc ¼ 2:0 μm. a ¼ 0:1082μm,b ¼ 0:2157 μm, and Λ ¼ 0:3239 μm. Values shown in parentheses indi-cate cases in which αa and αb are different from each other.

Fig. 4. Material loss limit value Lmat and transition point N tr as afunction of the cladding absorption coefficient. Fiber parametersare the same as those in Fig. 3. αa ¼ αb.

J. Sakai and N. Nishida Vol. 28, No. 3 / March 2011 / J. Opt. Soc. Am. B 381

Equations (5) and (8) imply that the results for various com-binations of αa and αb are predicted by a result for a particularcombination of αa and αb for the same structure as long as thematerial losses are not so large. Note that these equations holdonly for a fiber with identical fiber parameters. It is expectedfrom these equations that we have Lmat ¼ 7:22 × 103 dB=kmand N tr ¼ 7:8 for αa ¼ αb ¼ 106 dB=km under the fiber param-eters shown in Fig. 3.

A linear relation between fiber loss and material loss holdseven for αa ≠ αb. We have Lmat ¼ 0:3689 and 0:4256dB=km fortwo combinations of αa ¼ 100dB=km and αb ¼ 10dB=km andαa ¼ 10dB=km and αb ¼ 100 dB=km, respectively. From thesedata, we can establish an empirical equation of material losslimit value

Lmatðαa; αbÞ ¼ Caαa þ Cbαb ð9Þ

with Ca ¼ 3:297 × 10−3 and Cb ¼ 3:927 × 10−3. Similarly, anexpression of the transition point is given by

N trðαa; αbÞ ¼ N trð10; 10Þ þ1Slog

Lmatðαa; αbÞLmatð10; 10Þ

; ð10Þ

where Lmatðαa; αbÞ in Eq. (9) is inserted into Eq. (10). Conse-quently, the combined fiber loss is presented for a certaincombination of αa ¼ αb ¼ 10dB=km, except as otherwisereferred to, hereafter.

4. NUMERICAL RESULTS ON COMBINEDFIBER LOSS OF VARIOUS MODESA. Dependence on the Number of Cladding PairsFigure 5 semilogarithmically plots the dependence of the com-bined fiber loss on the number of cladding pairs for variousmodes. Fiber parameters are λ0 ¼ 1:0 μm, na ¼ 2:5, nb ¼nex ¼ 1:5, rc ¼ 2:0 μm, and αa ¼ αb ¼ 10dB=km. We plot allthe TE0μ and TM0μ modes appearing at λ0 ¼ 1:0 μm, andHEνμ and EHνμ only with (ðνμÞ ¼ ð11Þ, (12), and (21) to avoidcomplexity. As N increases, the combined fiber loss of the in-dividual modes decreases and converges to a constant valuepeculiar to the mode. The TE01 mode in this case exhibits the

lowest loss among all modes as well as for a case where onlythe CL is considered [8,12]; hence, both Lmat and N tr are alsothe lowest in the TE01 mode.

For a large value of N in Fig. 5, the Lmat of TM0μ and EHνμmodes hardly depends on the mode numbers, but that of TE0μand HEνμ strongly depends on the mode numbers. This prop-erty can also be seen in the core radius dependence of theoptical power confinement factor [18]. These properties wellreflect the fact that the HE and EH modes respectively reduceto the TE and TM modes under the QWS condition [17]. Thereduction of the HE and EH modes to the TE and TM modescould not be found when only the CL was considered [12]. Fora small N , the combined fiber loss is nearly the same as the CLalone [12], whose slope can be represented by Eq. (7).

B. Characteristics of Material Loss Limit Value Lmat

The cladding index contrast dependence of material loss limitvalue Lmat is shown in Fig. 6 for various modes. Fiber param-eters are the same as those in Fig. 5 except for na. Thickness ais determined to satisfy the QWS condition, Eq. (3), for eachna. The number of cladding pairs is set to satisfy N > N tr. Lmat

decreases smoothly with the increase in na due to the im-provement in optical power confinement.

Figure 7 is a logarithmic plot of material loss limit valueLmat of various modes as a function of the core radius. Fiberparameters are the same as those in Fig. 5 except for rc. Clad-ding thicknesses a and b are determined to satisfy the QWScondition for each rc. An extremely large loss takes placeat the lower limits of rc, which correspond to the mode cutoff[17]. As rc increases, Lmat markedly decreases for small rc anddecreases at a constant rate for large rc. The Lmat of the TMand EH modes converges to an identical value despite modenumbers for large N . Lmat decreases in inverse proportion tothe third power of rc for the TE mode, whereas it decreases ina linear dependence on rc for the other three mode groups. Asimilar tendency has also been obtained in the CL [12] as wellas in the optical power confinement factor [19,20] of the Braggfiber. For rc ≥ 0:8 μm, the TE01 mode shows the lowest loss.

A marked increase in Lmat for small rc of Fig. 7 can beexplained as follows. As rc decreases, each mode approachesthe guiding limit of the mode, namely, the zero of realpart βR of the propagation constant, and the optical power

Fig. 5. Cladding pair dependence of the combined fiber loss for var-ious modes: λ0 ¼ 1:0 μm, na ¼ 2:5, nb ¼ nex ¼ 1:5, rc ¼ 2:0 μm, andαa ¼ αb ¼ 10dB=km. Cladding layer thicknesses a and b are deter-mined from Eq. (3) for each mode. Two numbers in the figure indicateazimuthal ν and radial mode numbers μ in Figs. 5–9.

Fig. 6. Cladding index contrast dependence of material loss limitvalue Lmat for various modes. Fiber parameters are the same as thosein Fig. 5 except for na. Cladding thickness a is determined from Eq. (3)for each mode and na.

382 J. Opt. Soc. Am. B / Vol. 28, No. 3 / March 2011 J. Sakai and N. Nishida

confinement is reduced, suffering readily from an influence ofthe cladding material losses.

C. Characteristics of Transition Point Ntr

The cladding index contrast dependence of transition pointN tr

is shown in Fig. 8 for various modes. The fiber parameters arethe same as those in Fig. 5 except for na. Thickness a is deter-mined to satisfy the QWS condition shown in Eq. (3) for eachna.N tr is reducedwith increasingna because the optical powerconfinement is improved with increasing index contrast.Figure 8 shows thatN tr for everymodeapproachesnearly closevalues with increasing cladding index contrast. N tr is less thanabout 34.8 for na ≥ 2:5 among the modes shown.

Figure 9 depicts the core radius dependence of transitionpoint N tr for various modes. The fiber parameters are thesame as those in Fig. 5 except for rc. Cladding thicknessesa and b are prescribed based on the QWS condition for eachrc. As core radius rc increases, N tr decreases for the TE0μmode and increases for the TM0μ, HEνμ, and EHνμ modes be-cause fiber loss of the TE mode decreases faster than those ofother modes for the increase in N (Fig. 5). In addition, the Lmat

of the TE mode reduced faster than those of other modes for

the increase in rc (Fig. 7). Hence, the N tr of only the TE modedecreases with the increase in rc. This property is desirablefor the single-mode transmission of the TE01 mode.

Figure 9 shows that the N tr value converges to a constantvalue depending on each mode regardless of the mode num-bers with increasing rc. The smallest converging N tr of 15.6 isobtained for the TE mode. The converging N tr values are 36.1,34.8, and 34.8 for the TM, HE, and EHmodes, respectively. Forsufficiently large values of rc, almost the whole power of eachmode is confined to the center of the core. Then, the fields ofeach mode group are independent of mode numbers ν and μ.

The lower limits of rc in Lmat and N tr agree with each otherfor each mode (Figs. 7 and 9). As the core radius is reduced inFig. 9, N tr converges to an identical value of about 21.8 exceptfor the TM mode. This is because each mode approaches itsguiding limit with decreasing rc and most fields extend to thecladding, resulting in no mode dependence of the fields.Therefore, there is no mode dependence for small rc. Thesmallest N tr and rc of the TE0μ mode are in accordance withthose of the EH1μ mode with the same μ. This reflects the factthat these modes degenerate under the QWS condition.

5. NUMERICAL RESULTS ON COMBINEDFIBER LOSS OF THE TE0μ MODEWe see from the results of Section 4 that the TE01 mode ex-hibits the lowest fiber loss even when cladding material lossesare considered. Therefore, the characteristics of the TE0μmode will be scrutinized.

The dependence of the combined fiber loss of the TE01 modeis semilogarithmically plotted in Fig. 10 as a function of thenumber of cladding pairs, the cladding index contrast, andthe core radius. The fiber parameters are λ0 ¼ 1:0 μm, nb ¼nex ¼ 1:5, and αa ¼ αb ¼ 10dB=km. Cladding layer thick-nesses a and b are determined from Eq. (3) and a similar ex-pression concerning b. As N increases, the combined fiberloss decreases and converges to different constant valuesdepending on na and rc. The combined fiber loss decreaseswith an increase in na and rc because it improves opticalconfinement.

Material loss limit value Lmat of the TE0μ mode is illustratedin Fig. 11 as a function of core radius rc and the cladding index

Fig. 7. Core radius dependence of material loss limit value Lmat forvarious modes. Fiber parameters are the same as those in Fig. 5 ex-cept for rc. Cladding thicknesses a and b are determined from Eq. (3)for each mode and rc.

Fig. 8. Cladding index contrast dependence of transition point N trfor various modes. Fiber parameters are the same as those in Fig. 5except for na. Cladding thickness a is determined from Eq. (3) foreach mode and na.

Fig. 9. Core radius dependence of transition point N tr for variousmodes. Fiber parameters are the same as those in Fig. 5 except forrc. Cladding thicknesses a and b are determined from Eq. (3) for eachmode and rc.

J. Sakai and N. Nishida Vol. 28, No. 3 / March 2011 / J. Opt. Soc. Am. B 383

contrast. The fiber parameters are the same as those in Fig. 10.The number of cladding pairs is fixed at N ¼ 40, which is al-ways larger than N tr for all cases. An extremely large loss atsmall rc results from the mode cutoff, as pointed out in Fig. 7.Lmat decreases with increases of na and rc. The third-powerdependence of Lmat on rc can be seen despite na here as wellas in Fig. 7. For example, for na ¼ 2:5 and rc ¼ 2:0 μm of theTE01 mode, we have Lmat ¼ 7:22 × 10−2 dB=km. For the samerc, we have Lmat ¼ 2:84 × 10−2 and 1:57 × 10−2 dB=km for na ¼3:5 and 4.5, respectively. For a fixed value of na ¼ 2:5, we ob-tain Lmat ¼ 4:54 × 10−3 and 5:66 × 10−4 dB=km for rc ¼ 5:0 and10:0 μm, respectively. An increase in the core radius effec-tively reduces the fiber loss.

A plot of transition point N tr of the TE0μ mode is shown inFig. 12 as a function of core radius rc and the cladding indexcontrast. The fiber parameters are the same as those in Fig. 10.For a small value of rc, N tr increases with increased radialmode number μ. At minimum rc corresponding to the guidinglimit of the individual mode, theN tr of the TE0μ mode becomesidentical despite the mode number μ, although it depends onna. For instance, N tr ¼ 21:8, 13.7, and 10.8 are obtained atna ¼ 2:5, 3.5, and 4.5, respectively. N tr is reduced with in-

creasing rc and na and converges to constant values with in-creasing rc regardless of the mode number, as expected fromFig. 9. For example, the converging values ofN tr are 15.6, 10.6,and 8.7 for na ¼ 2:5, 3.5, and 4.5, respectively.

6. WAVELENGTH DEPENDENCE OFCOMBINED FIBER LOSSBoth Lmat and N tr show their minimum values for the TE01

mode among all modes in the hollow-core Bragg fiber. TheTE01 mode can be used as a propagation mode in the Braggfiber because other modes decay faster. The wavelength de-pendence of the combined fiber loss of the TE01 mode is in-vestigated here.

A. Photonic Band Dependence of Combined Fiber LossWe investigate the wavelength dependence of the combinedfiber loss over a wide range of wavelengths. We set na ¼ 2:5,nb ¼ nex ¼ 1:5, rc ¼ 2:0 μm, and N ¼ 17. The fiber parametersare first prescribed to satisfy the QWS condition at λQWS ¼1:0 μm, and then we vary the wavelength by firmly fixingthe remaining parameters. In particular, our attention is fo-cused on the performance near λQWS=ℓ, where ℓ is an integer.

The PBG dependence of the combined fiber loss of the TE01

mode of the hollow-core Bragg fiber is plotted in Fig. 13 forthree kinds of αa ¼ αb. We find three low-loss regions nearλQWS=ℓ with odd ℓ. It is natural that the minima decreases withdecreasing αa. The minimum values decrease with increasingodd ℓ. For example, for αa ¼ αb ¼ 1:0 dB=km, the minimumlosses are 2:96 × 10−2, 1:91 × 10−3, and 7:16 × 10−4 dB=km forℓ ¼ 1, 3, and 5, respectively. Wavelengths λmin;ℓ exhibitingthe minimum loss are 0.99, 0.327, and 0:195 μm for identicalℓ values. Cladding material losses significantly influence fiberminimum loss, but have little influence on the PBG width. Thecombined fiber losses for even ℓ hardly depend on claddingmaterial losses because their large fiber losses depend onthe CL only. The values of effective index βR=k0 were0.95237, 0.988, 0.995, 0.997, and 0.998 for ℓ ¼ 1 to 5, respec-tively and, thus, show relatively high values despite differentvalues of ℓ. For ℓ ¼ 1 estimated from the infinite periodic clad-ding and QWS case [17], we have the effective index βR∞=k0 ¼ 0:95238, indicating an approximate equality between ef-fective indices, namely, propagation constants, described inAppendix A.

Fig. 10. Cladding pair dependence of combined fiber loss for theTE01 mode as a function of several na and rc: λ0 ¼ 1:0 μm,nb ¼ nex ¼ 1:5, and αa ¼ αb ¼ 10dB=km.

Fig. 11. Core radius dependence of material loss limit value Lmat forthe TE0μ mode as a function of the cladding index contrast. Fiber pa-rameters are the same as those in Fig. 10. N ¼ 40.

Fig. 12. Core radius dependence of transition point N tr for the TE0μmode as a function of the cladding index contrast. Fiber parametersare the same as those in Fig. 10.

384 J. Opt. Soc. Am. B / Vol. 28, No. 3 / March 2011 J. Sakai and N. Nishida

The PBG dependence of the combined fiber loss can be ex-plained by considering the in-phase condition at the core–cladding boundary for waves reflected from each claddinginterface [21]. If the phase of the reflected waves is in phase(out of phase), then the optical confinement becomes excel-lent (bad) for an odd (even) ℓ as a consequence of the con-structive (destructive) interference. Therefore, a low-lossregion is obtained for an odd ℓ, but an extremely large loss isobtained for an even ℓ. In this case, it is essential that the ef-fective index be nearly unchanged despite ℓ, as shown above.

A significant loss difference between odd- and even-numbered PBGs has also been observed in the bend loss ofan all-solid silica PBF [22]. The PBG order dependence ofbend loss has semiquantitatively been explained by a picturebased on the overlap strength between the modes of neighbor-ing rods or by evaluating the critical bend radius [22].

B. Wavelength Dependence of Material LossLimit Value Lmat

Material loss limit value Lmat is shown in Fig. 14 for the TE01

mode as a function of wavelength, core radius rc, and claddingindex contrast. Cladding material losses are prescribed atαa ¼ αb ¼ 10dB=km. Only the fundamental PBG is shownhere. Fiber parameters are first prescribed to satisfy the QWScondition at λQWS ¼ 1:0 μm, and then the wavelength is variedby keeping the remaining parameters fixed. Lmat decreaseswith increasing rc and na. For example, for na ¼ 3:5 andrc ¼ 2:0 μm, the minimum fiber loss of 2:57 × 10−2 dB=km isobtained at λmin;1 ¼ 0:920 μm, which shifts from λQWS. If rcis chosen to be 2.0, 5.0, and 10:0 μm, then we have fiber lossesless than 10−1, 10−2, and 10−3 dB=km, respectively, for the fun-damental PBG. These fiber losses are less than about 10−2,10−3, and 10−4 times of cladding material losses, respectively.On the other hand, the PBG width has a stronger dependenceon na than on rc.

7. CONCLUSIONThe CL, including cladding material losses, was comprehen-sively studied for the TE, TM, and hybrid modes of hollow-core Bragg fibers mainly under the QWS condition using

the multilayer division method. In particular, detailed numer-ical data were presented for the TE01 mode. If the core radiusis set to 2.0, 5.0, and 10:0 μm under appropriate fiber design,then we have fiber losses below about 10−2, 10−3, and 10−4

times of cladding material losses, respectively.As number N of the cladding pairs increases, the combined

fiber loss of all modes decreases and converges to a constantloss Lmat peculiar to cladding material losses and the mode.For small N , the combined fiber loss tends to approach a lossdetermined only by CL Lconf . Material loss limit value Lmat de-creases with increasing core radius and cladding index con-trast. Lmat decreases in inverse proportion to the third andfirst powers of the core radius for the TE and the other threemodes for a sufficiently large core radius. Transition point N tr

is defined by a crossing point of the asymptotes of Lconf andLmat. Lmat and N tr for various cladding material losses can beestimated by those for particular material losses.

Odd-numbered PBGs exhibit a low-fiber loss while even-numbered PBGs exhibit an extremely large fiber loss, if fiberparameters are prescribed to satisfy the QWS condition at thefundamental PBG. Fiber losses at the fundamental PBG de-crease with increasing cladding index contrast and coreradius. The width of the fundamental PBG strongly dependson the core radius, but hardly on the cladding index contrast.

APPENDIX A: MODE DESIGNATION ANDPROPAGATION CONSTANT EVALUATIONFor a Bragg fiber with infinite periodic cladding, the real pro-pagation constant βR∞ of any mode can be expressed in a sim-ple analytical form under the QWS condition [17]. For a finitenumber N of periodic cladding pairs with no material loss, weassume that the propagation constant β be represented byβRN þ iβIN . Then, the lateral propagation constant κi of theith layer is treated as complex. The βRN and βIN are simulta-neously obtained by solving the eigenvalue equation in themultilayer division method [12], and ∣βIN ∣ is markedly lowerthan ∣βRN ∣. The βRN is close to the βR∞ for the same fiber pa-rameters and wavenumber as those of the infinite case, aslong as N is not extremely small. For the present case, wherethe cladding material loss αi is included in terms of the

Fig. 13. Wavelength dependence of the combined fiber loss of theTE01 mode as a function of the cladding material losses: na ¼ 2:5,nb ¼ nex ¼ 1:5, rc ¼ 2:0 μm, and N ¼ 17. Cladding layer thicknessesa and b are determined from Eq. (3) at λQWS ¼ 1:0 μm. Only thewavelength varied; the remaining parameters were kept fixed.

Fig. 14. Wavelength dependence of material loss limit value Lmat ofthe TE01 mode as a function of na and rc: nb ¼ nex ¼ 1:5,αa ¼ αb ¼ 10dB=km, and N ¼ 40. Cladding layer thicknesses a andb are determined from Eq. (3) at λQWS ¼ 1:0 μm for each combinationof na and rc. Only the wavelengths varied; the remaining parameterswere kept fixed.

J. Sakai and N. Nishida Vol. 28, No. 3 / March 2011 / J. Opt. Soc. Am. B 385

complex refractive index ~ni, the real part βR is close to the βRNfor the same fiber parameters and N as those of the finite Ncase, as long as the material loss is not so large (see the sec-ond paragraph of Subsection 2.B). The imaginary part βI ap-proaches the βIN with decreasing N , as shown in Fig. 3. It isalso possible for us to discriminate between HE and EHmodes by using the sign of real part of the P parameter[12] even in the present case. The imaginary part βI of eachmode correctly designated is evaluated in this way.

ACKNOWLEDGMENTSThe authors express their thanks to Y. Suzuki for helping withpart of the computation.

REFERENCES1. T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, and

M. J. Steel, “Confinement losses in microstructured opticalfibers,” Opt. Lett. 26, 1660–1662 (2001).

2. C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Muller, J. A.West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-losshollow-core silica/air photonic bandgap fiber,” Nature 424,657–659 (2003).

3. K. Saitoh and M. Koshiba, “Leakage loss and group velocity dis-persion in air-core photonic bandgap fibers,” Opt. Express 11,3100–3109 (2003).

4. P. J. Roberts, D. P. Williams, J. Mangan, H. Sabert, F. Couny, W.J. Wadsworth, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Rea-lizing low loss air core photonic crystal fibers by exploiting anantiresonant core surround,” Opt. Express 13, 8277–8285(2005).

5. B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y.Fink, “Wavelength-scalable hollow optical fibers with largephotonic bandgaps for CO2 laser transmission,” Nature 420,650–653 (2002).

6. G. Vienne, Y. Xu, C. Jakobsen, H. Deyerl, J. B. Jensen, T.Sørensen, T. P. Hansen, Y. Huang, M. Terrel, R. K. Lee, N. A.Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv,“Ultra-large bandwidth hollow-core guiding in all-silica Braggfibers with nano-supports,” Opt. Express 12, 3500–3508 (2004).

7. A. Argyros, M. A. van Eijkelenborg, M. C. J. Large, and I. M.Bassett, “Hollow-core microstructured polymer optical fiber,”Opt. Lett. 31, 172–174 (2006).

8. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D.Engeness, M. Soljačić, S. A. Jacobs, J. D. Joannopoulos,and Y. Fink, “Low-loss asymptotically single-mode propagationin large-core Omniguide fibers,” Opt. Express 9, 748–779(2001).

9. I. M. Bassett and A. Argyros, “Elimination of polarizationdegeneracy in round waveguides,” Opt. Express 10, 1342–1346(2002).

10. A. Argyros, “Guided modes and loss in Bragg fibres, “Opt.Express 10, 1411–1417 (2002).

11. Y. Xu, A. Yariv, J. G. Fleming, and S. Y. Lin, “Asymptotic analysisof silicon based Bragg fibers,” Opt. Express 11, 1039–1049(2003).

12. J. Sakai and H. Niiro, “Confinement loss evaluation based on amultilayer division method in Bragg fibers,” Opt. Express 16,1885–1902 (2008).

13. J. G. Dil and H. Blok, “Propagation of electromagnetic surfacewaves in a radially inhomogeneous optical waveguide,” Opto-electronics 5, 415–428 (1973).

14. J. Sakai and T. Kimura, “Bending loss of propagation modes inarbitrary-index profile optical fibers,” Appl. Opt. 17, 1499–1506(1978).

15. M. Foroni, D. Passaro, F. Poli, A. Cucinotta, S. Selleri, J.Lægsgaard, and A. O. Bjarklev, “Guiding properties of silica/air hollow-core Bragg fibers,” J. Lightwave Technol. 26,1877–1884 (2008).

16. J. Sakai, “Optical loss estimation in a Bragg fiber,” J. Opt. Soc.Am. B 24, 763–772 (2007).

17. J. Sakai, “Hybrid modes in a Bragg fiber: general properties andformulas under the quarter-wave stack condition,” J. Opt. Soc.Am. B 22, 2319–2330 (2005).

18. J. Sakai, J. Sasaki, and K. Kawai, “Optical power confinementfactor in a Bragg fiber: 2. Numerical results,” J. Opt. Soc. Am.B 24, 20–27 (2007).

19. W. Zhi, R. Guobin, L. Shuqin, L. Weijun, and S. Guo, “Compactsupercell method based on opposite parity for Bragg fibers,”Opt. Express 11, 3542–3549 (2003).

20. J. Sakai, “Optical power confinement factor in a Bragg fiber:1. Formulation and general properties,” J. Opt. Soc. Am. B24, 9–19 (2007).

21. J. Sakai and Y. Suzuki, “Equivalence between in-phase and anti-resonant reflection conditions in Bragg fiber and its applicationto antiresonant reflecting optical waveguide-type fibers,” J. Opt.Soc. Am. B 28, 183–192 (2011).

22. T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M.Bird, “Bend loss in all-solid bandgap fibres,” Opt. Express 14,5688–5698 (2006).

386 J. Opt. Soc. Am. B / Vol. 28, No. 3 / March 2011 J. Sakai and N. Nishida