Abstract

We propose analytical formulas for large-core low-contrast Bragg fibers that predict the leakage loss of their low-order core modes, whatever their polarizations are and whatever the number of rings in the cladding is, for the first time to our knowledge. We propose also a generalized analytical formulas, which encompasses the case of Bragg fibers with a nonperiodic cladding, by decomposing their cladding into its constitutive periodic elements, whose contributions are taken into account through a simple multiplication factor. These formulas are developed thanks to a perturbation approach and to the use of the asymptotic formulation of the field in the cladding and provide a good accuracy except in the vicinity of couplings between core modes and cladding modes.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. C. Baskiotis, D. Molin, G. Bouwmans, F. Gooijer, P. Sillard, Y. Quiquempois, and M. Douay, “Bend-induced transformation of the transmission window of a large-mode-area Bragg fibre,” in European Conference on Optical Communications (ECOC) (IEEE, 2008), paper Mo.4.B.2.
  4. D. Gaponov, P. Roy, S. Février, M. Likhachev, M. Salganskii, and M. Yashkov, “100 W from a photonic bandgap Bragg fiber laser,” in Conference on Lasers and Electro-Optics (CLEO) and Quantum Electronics and Laser Science Conference (QELS) (OSA, 2010), paper CTuC.
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    [CrossRef]
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2011 (1)

2010 (1)

2009 (2)

2008 (3)

2005 (1)

J.-I Sakai and P. Nouchi, “Propagation properties of Bragg fiber analyzed by a Hankel function formalism,” Opt. Commun. 249, 153–163 (2005).
[CrossRef]

2004 (1)

2003 (2)

Y. Xu, A. Yariv, J. Fleming, and S.-Y. Lin, “Asymptotic analysis of silicon based Bragg fibers,” Opt. Express 11, 1039–1049 (2003).
[CrossRef]

S. Février, P. Viale, F. Gérôme, P. Leproux, P. Roy, J.-M. Blondy, B. Dussardier, and G. Monnom, “Very large effective area singlemode photonic bandgap fibre,” Electron. Lett. 39, 1240–1242 (2003).
[CrossRef]

2002 (2)

2000 (1)

1993 (1)

J.-L. Archambault, R. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
[CrossRef]

1978 (1)

1977 (1)

1969 (1)

A. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide,” IEEE Trans. Microwave Theor. Tech. 17, 1130–1138 (1969).
[CrossRef]

1964 (1)

E. Marcatili and R. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Afshar, S.

Albin, S.

Archambault, J.-L.

J.-L. Archambault, R. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
[CrossRef]

Argyros, A.

Baskiotis, C.

C. Baskiotis, Y. Jaouën, R. Gabet, G. Bouwmans, Y. Quiquempois, M. Douay, and P. Sillard, “Microbending behavior of large-effective-area Bragg fibers,” Opt. Lett. 34, 3490–3492 (2009).
[CrossRef]

C. Baskiotis, D. Molin, G. Bouwmans, F. Gooijer, P. Sillard, Y. Quiquempois, and M. Douay, “Bend-induced transformation of the transmission window of a large-mode-area Bragg fibre,” in European Conference on Optical Communications (ECOC) (IEEE, 2008), paper Mo.4.B.2.

Bian, B.

Black, R.

J.-L. Archambault, R. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
[CrossRef]

Blondy, J.-M.

S. Février, P. Viale, F. Gérôme, P. Leproux, P. Roy, J.-M. Blondy, B. Dussardier, and G. Monnom, “Very large effective area singlemode photonic bandgap fibre,” Electron. Lett. 39, 1240–1242 (2003).
[CrossRef]

Bouwmans, G.

C. Baskiotis, Y. Jaouën, R. Gabet, G. Bouwmans, Y. Quiquempois, M. Douay, and P. Sillard, “Microbending behavior of large-effective-area Bragg fibers,” Opt. Lett. 34, 3490–3492 (2009).
[CrossRef]

C. Baskiotis, D. Molin, G. Bouwmans, F. Gooijer, P. Sillard, Y. Quiquempois, and M. Douay, “Bend-induced transformation of the transmission window of a large-mode-area Bragg fibre,” in European Conference on Optical Communications (ECOC) (IEEE, 2008), paper Mo.4.B.2.

Bures, J.

J.-L. Archambault, R. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
[CrossRef]

Douay, M.

C. Baskiotis, Y. Jaouën, R. Gabet, G. Bouwmans, Y. Quiquempois, M. Douay, and P. Sillard, “Microbending behavior of large-effective-area Bragg fibers,” Opt. Lett. 34, 3490–3492 (2009).
[CrossRef]

C. Baskiotis, D. Molin, G. Bouwmans, F. Gooijer, P. Sillard, Y. Quiquempois, and M. Douay, “Bend-induced transformation of the transmission window of a large-mode-area Bragg fibre,” in European Conference on Optical Communications (ECOC) (IEEE, 2008), paper Mo.4.B.2.

Dussardier, B.

S. Février, P. Viale, F. Gérôme, P. Leproux, P. Roy, J.-M. Blondy, B. Dussardier, and G. Monnom, “Very large effective area singlemode photonic bandgap fibre,” Electron. Lett. 39, 1240–1242 (2003).
[CrossRef]

Feshchenko, R. M.

Février, S.

S. Février, P. Viale, F. Gérôme, P. Leproux, P. Roy, J.-M. Blondy, B. Dussardier, and G. Monnom, “Very large effective area singlemode photonic bandgap fibre,” Electron. Lett. 39, 1240–1242 (2003).
[CrossRef]

D. Gaponov, P. Roy, S. Février, M. Likhachev, M. Salganskii, and M. Yashkov, “100 W from a photonic bandgap Bragg fiber laser,” in Conference on Lasers and Electro-Optics (CLEO) and Quantum Electronics and Laser Science Conference (QELS) (OSA, 2010), paper CTuC.

Fleming, J.

Gabet, R.

Gaponov, D.

D. Gaponov, P. Roy, S. Février, M. Likhachev, M. Salganskii, and M. Yashkov, “100 W from a photonic bandgap Bragg fiber laser,” in Conference on Lasers and Electro-Optics (CLEO) and Quantum Electronics and Laser Science Conference (QELS) (OSA, 2010), paper CTuC.

Gérôme, F.

S. Février, P. Viale, F. Gérôme, P. Leproux, P. Roy, J.-M. Blondy, B. Dussardier, and G. Monnom, “Very large effective area singlemode photonic bandgap fibre,” Electron. Lett. 39, 1240–1242 (2003).
[CrossRef]

Gooijer, F.

C. Baskiotis, D. Molin, G. Bouwmans, F. Gooijer, P. Sillard, Y. Quiquempois, and M. Douay, “Bend-induced transformation of the transmission window of a large-mode-area Bragg fibre,” in European Conference on Optical Communications (ECOC) (IEEE, 2008), paper Mo.4.B.2.

Guo, S.

Hong, C.-S.

Jaouën, Y.

Lacroix, S.

J.-L. Archambault, R. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
[CrossRef]

Lee, R. K.

Leproux, P.

S. Février, P. Viale, F. Gérôme, P. Leproux, P. Roy, J.-M. Blondy, B. Dussardier, and G. Monnom, “Very large effective area singlemode photonic bandgap fibre,” Electron. Lett. 39, 1240–1242 (2003).
[CrossRef]

Likhachev, M.

D. Gaponov, P. Roy, S. Février, M. Likhachev, M. Salganskii, and M. Yashkov, “100 W from a photonic bandgap Bragg fiber laser,” in Conference on Lasers and Electro-Optics (CLEO) and Quantum Electronics and Laser Science Conference (QELS) (OSA, 2010), paper CTuC.

Lin, S.-Y.

Lu, J.

Marcatili, E.

E. Marcatili and R. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974).

Marom, E.

Molin, D.

C. Baskiotis, D. Molin, G. Bouwmans, F. Gooijer, P. Sillard, Y. Quiquempois, and M. Douay, “Bend-induced transformation of the transmission window of a large-mode-area Bragg fibre,” in European Conference on Optical Communications (ECOC) (IEEE, 2008), paper Mo.4.B.2.

Monnom, G.

S. Février, P. Viale, F. Gérôme, P. Leproux, P. Roy, J.-M. Blondy, B. Dussardier, and G. Monnom, “Very large effective area singlemode photonic bandgap fibre,” Electron. Lett. 39, 1240–1242 (2003).
[CrossRef]

Monro, T.

Monro, T. M.

Nouchi, P.

J.-I Sakai and P. Nouchi, “Propagation properties of Bragg fiber analyzed by a Hankel function formalism,” Opt. Commun. 249, 153–163 (2005).
[CrossRef]

Ouyang, G. X.

Quiquempois, Y.

C. Baskiotis, Y. Jaouën, R. Gabet, G. Bouwmans, Y. Quiquempois, M. Douay, and P. Sillard, “Microbending behavior of large-effective-area Bragg fibers,” Opt. Lett. 34, 3490–3492 (2009).
[CrossRef]

C. Baskiotis, D. Molin, G. Bouwmans, F. Gooijer, P. Sillard, Y. Quiquempois, and M. Douay, “Bend-induced transformation of the transmission window of a large-mode-area Bragg fibre,” in European Conference on Optical Communications (ECOC) (IEEE, 2008), paper Mo.4.B.2.

Robertson, I. D.

Rogowski, R.

Rowland, K.

Rowland, K. J.

Roy, P.

S. Février, P. Viale, F. Gérôme, P. Leproux, P. Roy, J.-M. Blondy, B. Dussardier, and G. Monnom, “Very large effective area singlemode photonic bandgap fibre,” Electron. Lett. 39, 1240–1242 (2003).
[CrossRef]

D. Gaponov, P. Roy, S. Février, M. Likhachev, M. Salganskii, and M. Yashkov, “100 W from a photonic bandgap Bragg fiber laser,” in Conference on Lasers and Electro-Optics (CLEO) and Quantum Electronics and Laser Science Conference (QELS) (OSA, 2010), paper CTuC.

Sakai, J.

Sakai, J.-I

J.-I Sakai and P. Nouchi, “Propagation properties of Bragg fiber analyzed by a Hankel function formalism,” Opt. Commun. 249, 153–163 (2005).
[CrossRef]

Salganskii, M.

D. Gaponov, P. Roy, S. Février, M. Likhachev, M. Salganskii, and M. Yashkov, “100 W from a photonic bandgap Bragg fiber laser,” in Conference on Lasers and Electro-Optics (CLEO) and Quantum Electronics and Laser Science Conference (QELS) (OSA, 2010), paper CTuC.

Schmeltzer, R.

E. Marcatili and R. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).

Shi, Y.

Sillard, P.

C. Baskiotis, Y. Jaouën, R. Gabet, G. Bouwmans, Y. Quiquempois, M. Douay, and P. Sillard, “Microbending behavior of large-effective-area Bragg fibers,” Opt. Lett. 34, 3490–3492 (2009).
[CrossRef]

C. Baskiotis, D. Molin, G. Bouwmans, F. Gooijer, P. Sillard, Y. Quiquempois, and M. Douay, “Bend-induced transformation of the transmission window of a large-mode-area Bragg fibre,” in European Conference on Optical Communications (ECOC) (IEEE, 2008), paper Mo.4.B.2.

Snyder, A.

A. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide,” IEEE Trans. Microwave Theor. Tech. 17, 1130–1138 (1969).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Viale, P.

S. Février, P. Viale, F. Gérôme, P. Leproux, P. Roy, J.-M. Blondy, B. Dussardier, and G. Monnom, “Very large effective area singlemode photonic bandgap fibre,” Electron. Lett. 39, 1240–1242 (2003).
[CrossRef]

Xu, Y.

Yariv, A.

Yashkov, M.

D. Gaponov, P. Roy, S. Février, M. Likhachev, M. Salganskii, and M. Yashkov, “100 W from a photonic bandgap Bragg fiber laser,” in Conference on Lasers and Electro-Optics (CLEO) and Quantum Electronics and Laser Science Conference (QELS) (OSA, 2010), paper CTuC.

Yeh, P.

Zhang, Y.

Zhang, Z.

Bell Syst. Tech. J. (1)

E. Marcatili and R. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).

Electron. Lett. (1)

S. Février, P. Viale, F. Gérôme, P. Leproux, P. Roy, J.-M. Blondy, B. Dussardier, and G. Monnom, “Very large effective area singlemode photonic bandgap fibre,” Electron. Lett. 39, 1240–1242 (2003).
[CrossRef]

IEEE Trans. Microwave Theor. Tech. (1)

A. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of a dielectric or optical waveguide,” IEEE Trans. Microwave Theor. Tech. 17, 1130–1138 (1969).
[CrossRef]

J. Lightwave Technol. (4)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

J.-I Sakai and P. Nouchi, “Propagation properties of Bragg fiber analyzed by a Hankel function formalism,” Opt. Commun. 249, 153–163 (2005).
[CrossRef]

Opt. Express (5)

Opt. Lett. (2)

Other (4)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974).

C. Baskiotis, D. Molin, G. Bouwmans, F. Gooijer, P. Sillard, Y. Quiquempois, and M. Douay, “Bend-induced transformation of the transmission window of a large-mode-area Bragg fibre,” in European Conference on Optical Communications (ECOC) (IEEE, 2008), paper Mo.4.B.2.

D. Gaponov, P. Roy, S. Février, M. Likhachev, M. Salganskii, and M. Yashkov, “100 W from a photonic bandgap Bragg fiber laser,” in Conference on Lasers and Electro-Optics (CLEO) and Quantum Electronics and Laser Science Conference (QELS) (OSA, 2010), paper CTuC.

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Figures (7)

Fig. 1.
Fig. 1.

Schematic of a Bragg fiber and notations.

Fig. 2.
Fig. 2.

Accuracy area (white area) of the asymptotic formulation of the field in the ring P of the cladding as a function of (nPnc) and of the internal radius of the ring rP1 [defined by the condition in Eq. (16)]. The positioning of the fiber A is also indicated.

Fig. 3.
Fig. 3.

Leakage loss spectrum of the fundamental core mode HE11 and the higher-order mode HE21 of the fiber A, of characteristics [rc=15μm, nc=1.4497, l1=2.6μm, n1=1.4786, n2=1.4506, l2=6μm], exhibiting 10 rings as computed with the formula in Eq. (34) and the matrix method.

Fig. 4.
Fig. 4.

(a) Leakage loss spectrum and (b) dispersion curve of the fundamental core mode of the fiber A exhibiting five rings in its cladding as computed with the analytical formulas in Eqs. (7) and (41), and the matrix method.

Fig. 5.
Fig. 5.

Schematic of the (a) modeling of a Bragg fiber presenting different periodic elements in its cladding and (b) waveguide used to determine the factor ΦElement2.

Fig. 6.
Fig. 6.

(a) Fundamental core-mode leakage loss spectrum of the fiber B, with five rings, of characteristics [rc=20μm, nc=n2=1.4497, l1=1.4μm, n1=1.4747, l2=11μm]; (b) the fiber A with five periods, both obtained with the analytical formulas and with the matrix method.

Fig. 7.
Fig. 7.

Fundamental core-mode leakage loss spectrum of the fiber A with five rings, in which the thickness of the fourth ring has been divided by two, obtained with the decomposition into the largest as possible constitutive elements, and with the matrix method.

Equations (58)

Equations on this page are rendered with MathJax. Learn more.

kP=(2πλ)nP2neff2.
αBragg(dB/m)=10ln(10)PrPz
Pr=12θ=02π[EθHz*EzHθ*]r=RRdθ
Pz=12θ=02πr=0[ErHθ*EθHr*]rdrdθ,
αBragg=αAntiguide·PrBraggPrAntiguide,
V=2πλrcn12neff2.
2πλrcnc2Re(neff)2=u,
u={uν1,μ,ifHEν,μmodeuν+1,μ,ifEHν,μ,TE0,μorTM0,μmode,
αAntiguide=(40πλln(10))(λ2π)2×da·u2ncrc3(2πλ)2(n12nc2)+u2rc2,
da={1,ifTE0,μmoden12nc2,ifTM0,μmode12(1+n12nc2),if hybrid mode.
[EzHθHzEθ](rP1rrP)=MP(r)[APBPCPDP].
MP(r)=[JνYν00iωε0nP2kPJviωε0nP2kPYviνβkP2rJνiνβkP2rYν00JνYνiνβkP2rJνiνβkP2rYνiωμ0kPJviωμ0kPYv],
MPas(r)=[MPTM(r)00MPTE(r)],
MPS(r)=2πkPr×[eikP(rrP1)eikP(rrP1)ωsPkPeikP(rrP1)ωsPkPeikP(rrP1)],
sP={μ0,ifS=TEε0nP2,ifS=TM.
kP2·rP2(4ν21)(4ν29)128.
kP=(2πλ)2(nP2nc2)+uν1,μ2rc2.
PrHE,EH=PrTE+PrTM,
PrTE=ωμ0kout2|Cout|2andPrTM=ωε0nout2kout2|Aout|2.
PzHE,EH=PzTE+PzTM.
αHE,EH=PrHE,EH2PzTE,TM=12(αTE+αTM).
Mc(rc)W0=M1(rc)W1.
[AoutAntiguideCoutAntiguide]=πkoutrc2Jν(kcrc)[A0C0].
W1Bragg=T·WoutBragg,
T=p=1NrMp1(rp)Mp+1(rp).
Tas=k1kout·[TPlanar-TM00TPlanar-TE],
T1S=[X1SUN1UN2Y1SUN1Y1S*UN1X1S*UN1UN2],
X1S=eik1l1×[cos(k2l2)i2(d1k2k1+1d1k1k2)sin(k2l2)],
Y1S=eik1l1[i2(d1k2k11d1k1k2)sin(k2l2)],
UN=sin((N+1)KΛ)sin(KΛ),
KΛ=arccos(Re(X1S)),
d1={(n1/n2)2,ifS=TM1,ifS=TE.
πk1rc2Jν(kcrc)A0=[(X1S+Y1S*)UN1UN2]AoutBragg.
α2N=αAntiguide|(X1S+Y1S*)UN1UN2|2.
T2=[T2TM00T2TE],
W1=L1W2,L1=M11(rc+l1)M2(rc+l1).
L1=12k1k2×[(1+k1k21d1)eik1l1(1k1k21d1)eik1l1(1k1k21d1)eik1l1(1+k1k21d1)eik1l1].
T2N+1=L1T2.
πk1rc2Jν(kcrc)A0=k1k2[(aX2+a*Y2*)UN1aUN2]AoutBragg,
a=cos(k1l1)id2k1k2sin(k1l1).
α(2N+1)=d2(k1/k2)αAntiguide|(aX2S+a*Y2S*)UN1aUN2|2.
Pr1=ΦElement1·PrAntiguide1,
ΦElement1=αElement1αAntiguide1.
Woutj=MElementjWoutj1,
MElementj=T=1NjLT1,
Prj=ΦElementj·Prj1,
αBragg=αAntiguidej·j=1NeΦElementj.
Φ1ring=k1k21d(k1S1)2+d(k2C1)2,
Φ1period=k12k22k12(1dk1S1S2k2C1C2)2+k22(dk2C1S2+k1S1C2)2,
Mout(r)=[HνIHνII00iωε0nP2kPHνIiωε0nP2kPHνIIiνβkP2rHνIiνβkP2rHνI00HνIHνIIiνβkP2rHνIiνβkP2rHνIIiωμ0kPHνIiωμ0kPHνII],
{Eθ=iA0νβkc2rJν(kcr)iC0ωμ0kcJv(kcr)Hr=iA0ωε0nc2νkc2rJν(kcr)+iC0βkcJv(kcr),
{Hθ=iA0ωε0nc2kcJv(kcr)+iC0νβkc2rJν(kcr)Er=iA0βkcJv(kcr)+iC0ωμ0νkc2rJν(kcr).
PZTE=π|A0|2ωε0nc2βkc2r=0(νkcr)2(Jν(kcr))2rdr+π(A0C0*νβ2kc3+A0*C0ω2nc2νc2kc3)r=0Jν(kcr)Jv(kcr)dr+π|C0|2ωμ0βkc2r=0(Jv(kcr))2rdr.
PZTM=π|A0|2ωε0nc2βkc2r=0(Jv(kcr))2rdr+π(A0C0*νβ2kc3+A0*C0ω2nc2νc2kc3)r=0Jν(kcr)Jν(kcr)dr+π|C0|2ωμ0βkc2r=0(νkcr)2(Jν(kcr))2rdr.
(νkcr)2(Jν(kcr))2=(Jν(kcr)+Jν+1(kcr))2.
PZTEPZTM=π(|A0|2ωε0nc2βkc2|C0|2ωμ0βkc2)×r=0(2Jv(kcr)Jν+1(kcr)+Jν+12(kcr))rdr.
2Jv(kcr)=Jν1(kcr)Jν+1(kcr).
r=0(2Jv(kcr)Jν+1(kcr)+Jν+12(kcr))rdr=r=0(Jν1(kcr)Jν+1(kcr))rdr.

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