Abstract

We describe the differential multipole method, an extended multipole method used to calculate the modes of microstructured optical fibers with noncircular inclusions. We use a multipole expansion centered on each inclusion and a differential method to calculate the scattering properties of the individual inclusions. Representative results for a fiber with one ring of elliptical inclusions are presented, and a direct comparison is made with an existing method. The new method is also applied to a microstructured optical fiber with seven rings of elliptical inclusions, which is found, in effect, to support a single polarization of the fundamental mode.

© 2004 Optical Society of America

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References

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2004 (2)

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photonics Technol. Lett. 16, 182–184 (2004).
[CrossRef]

N. A. Issa, M. A. van Eijkelenborg, M. Fellew, F. Cox, G. Henry, and M. C. J. Large, “Fabrication and study of microstructured optical fibers with elliptical holes,” Opt. Lett. 29, 1336–1338 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (3)

2001 (3)

2000 (2)

1998 (1)

1997 (1)

1996 (1)

1994 (2)

K. M. Lo, R. C. McPhedran, I. M. Bassett, and G. W. Milton, “An electromagnetic theory of dielectric waveguides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
[CrossRef]

C. Chang and H. Chang, “Theory of the circular harmonics expansion method for multiple-optical-fiber system,” J. Lightwave Technol. 12, 415–417 (1994).
[CrossRef]

1980 (1)

1973 (1)

Bassett, I. M.

K. M. Lo, R. C. McPhedran, I. M. Bassett, and G. W. Milton, “An electromagnetic theory of dielectric waveguides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
[CrossRef]

Birks, T. A.

Botten, L. C.

Burdge, G. L.

Chang, C.

C. Chang and H. Chang, “Theory of the circular harmonics expansion method for multiple-optical-fiber system,” J. Lightwave Technol. 12, 415–417 (1994).
[CrossRef]

Chang, H.

C. Chang and H. Chang, “Theory of the circular harmonics expansion method for multiple-optical-fiber system,” J. Lightwave Technol. 12, 415–417 (1994).
[CrossRef]

Cox, F.

Eggleton, B. J.

Feit, M. D.

Fellew, M.

Fleck, J. J. A.

Henry, G.

Issa, N. A.

Kawanishi, S.

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photonics Technol. Lett. 16, 182–184 (2004).
[CrossRef]

Kerbage, C.

Knight, J. C.

Koyanagi, S.

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photonics Technol. Lett. 16, 182–184 (2004).
[CrossRef]

Kubota, H.

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photonics Technol. Lett. 16, 182–184 (2004).
[CrossRef]

Kuhlmey, B. T.

Large, M. C. J.

Li, L.

Lo, K. M.

K. M. Lo, R. C. McPhedran, I. M. Bassett, and G. W. Milton, “An electromagnetic theory of dielectric waveguides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
[CrossRef]

Martijn de Sterke, C.

Maystre, D.

McPhedran, R. C.

Milton, G. W.

K. M. Lo, R. C. McPhedran, I. M. Bassett, and G. W. Milton, “An electromagnetic theory of dielectric waveguides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
[CrossRef]

Nevière, M.

E. Popov and M. Nevière, “Maxwell equations in Fourier space: a fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. B 18, 2886–2894 (2001).
[CrossRef]

Osgood Jr., R. M.

Poladian, L.

Popov, E.

E. Popov and M. Nevière, “Maxwell equations in Fourier space: a fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. B 18, 2886–2894 (2001).
[CrossRef]

Ranka, J. K.

Renversez, G.

Robinson, O. A.

Russell, P. St. J.

Steel, M. J.

Stentz, A. J.

Tanaka, M.

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photonics Technol. Lett. 16, 182–184 (2004).
[CrossRef]

van Eijkelenborg, M. A.

Westbrook, P. S.

White, C. A.

White, T. P.

Wijngaard, W.

Windeler, R. S.

Yamaguchi, S.

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photonics Technol. Lett. 16, 182–184 (2004).
[CrossRef]

Appl. Opt. (1)

IEEE Photonics Technol. Lett. (1)

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photonics Technol. Lett. 16, 182–184 (2004).
[CrossRef]

J. Lightwave Technol. (5)

J. Opt. Soc. Am. (1)

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

J. Opt. Soc. Am. B (3)

Opt. Express (1)

Opt. Lett. (4)

Other (4)

M. Nevière and E. Popov, Light Propagation in Periodic Media (Marcel Dekker, New York, 2003).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

N. Issa, Optical Fibre Technology Centre, 206 National Innovation Centre, Australian Technology Park, Eveleigh New South Wales 1430, Australia, n.issa@oftc.usyd.edu.au (personal communication, 2003).

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Figures (4)

Fig. 1
Fig. 1

Schematic of the MOF considered. Λ is the center-to-center separation, a is the semimajor inclusion axis, and b is the semiminor inclusion axis, where b=ηa. The case of η=1 is denoted by the dashed and dotted circles and defines the escribed circle for each inclusion. The difference between the dashed and dotted circles is discussed in Section 3.

Fig. 2
Fig. 2

Comparison between a circular and a noncircular inclusion. The circles C1 and C2 touch the inside and outside of each inclusion, respectively, and have radii ri and re for the noncircular inclusion and radius r0 for the circular inclusion. The lth inclusion has refractive index nl and the background ne.

Fig. 3
Fig. 3

Real part of the effective index of the fundamental modes and the FSMs of a seven-ring hexagonal MOF (inset) with elliptical inclusions oriented vertically with the major axis.

Fig. 4
Fig. 4

Imaginary part of the effective index on a log-linear scale of the fundamental modes of a seven-ring hexagonal MOF with elliptical inclusions oriented vertically with the major axis. At ν=0.65 the ratio of the losses of the guided x mode to the guided y mode is 0.02.

Tables (3)

Tables Icon

Table 1 Convergence of neff with Ma

Tables Icon

Table 2 Comparison between the DMM and the ABC Method for the Polarization of the Fundamental Mode with the Horizontal Ez Nodal Line

Tables Icon

Table 3 Comparison between the DMM and the ABC Method for the Polarization of the Fundamental Mode with the Vertical Ez Nodal Line

Equations (63)

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loss(dB/km)=20ln(10)2πλneffi×109,
Ezl=m=-MM[amE,lJm(kerl)+bmE,lHm(1)(kerl)]exp(imθl),
Ez=jlNcm=-MMbmE,jHm(1)(kerl)exp(imθl),
[I-R(H+S)]b=0,
Ezl=m=-MMcmE,lJm(klrl)exp(imθl).
Ezl=m=-MMez,m(r)exp(imθl),
vl(re)=Pelalbl,
vl(ri)=Pilcl,
vl(re)=vl(ri),
Pelalbl=Pilcl.
ddrvl(r)=Ovl(r).
vl(re)=Nlvl(ri).
vl(re)=Pelalbl=Nlvl(ri)=NlPilcl,
albl=Pel,-1NlPil(cl),
=Q1lQ2l(cl).
bl=Q2lQ1l,-1=Rlal.
(fg)n=m=-fn-mgm=m=-gn-mfm,
[fg]=[[f]][g].
[fg]=[[1/f]]-1[g],
×E=ik0K,
×K=-ik0(n2E),
N=(f)/|f|.
[n2E]=[n2ET+n2EN].
[n2E]=[[n2]][ET]+1n2-1[EN].
[n2E]=Qn2[E]=Qn2,rrQn2,rθ0Qn2,θrQn2,θθ000Qn2,zz[E].
[n2Er]=Qn2,rr[Er]+Qn2,rθ[Eθ],
iαθ[Kz]r-iβ[Kθ]=-ik0[n2Er]
[Er]=Qn2,rr-1{[n2Er]-Qn2,rθ[Eθ]},
=Qn2,rr-1β[Kθ]-1rMαθ[Kz]k0-Qn2,rθ[Eθ].
ddr[Ez][Kz][Eθ][Kθ]=O[Ez][Kz][Eθ][Kθ],
ddrvl(r)=Ovl(r),
W=C2|Ezlocal(θl)-EzWijngaard(θl)|dθlC2|EzWijngaard(θl)|dθl,
k0ncore>βr>βFSM.
Eθ(r, θ)=ik2βrEzθ-k0Kzr,
Kθ(r, θ)=ik2βrKzθ+k0n2Ezr,
ezl(ri)kzl(ri)eθl(ri)kθl(ri)=PilcE,lcK,l,
Pil=Ji00Ji-βki,2riMJi-ik0kiJiik0ni2kiJi-βki,2riMJi,
ezl(ri)kzl(ri)eθl(ri)kθl(ri)=PelaE,laK,lbE,lbK,l,
Pel=Je0He00Je0He-βke,2reMJe-ik0keJe-βke,2reMHe-ik0keHeik0ne2keJe-βke,2reMJeik0ne2keHe-βke,2reMHe.
aE,laK,lbE,lbK,l=Pe-1NlPicE,lcK,l=Q1Q2cE,lcK,l,
BEBK=Q2Q1-1AEAK=RlAEAK.
Qn2=Qn2,rrQn2,rθ0Qn2,θrQn2,θθ000Qn2,zz,
Qn2,rr=[[n2]][[Nθ2]]+1n2-1[[Nr2]],
Qn2,rθ=-[[n2]]-1n2-1[[NrNθ]],
Qn2,θr=-[[n2]]-1n2-1[[NrNθ]],
Qn2,θθ=[[n2]][[Nr2]]+1n2-1[[Nθ2]],
Qn2,zz=[[n2]].
O11=0,
O12=-iβrk0Qn2,rr-1M,
O13=-iβQn2,rr-1Qn2,rθ,
O14=iβ2k0Qn2,rr-1-ik0I,
O21=iβrk0 M,
O22=-irQn2,θrQn2,rr-1M,
O23=-iβ2k0I+ik0(Qn2,θθ-Qn2,θrQn2,rr-1Qn2,rθ),
O24=iβQn2,θrQn2,rr-1,
O31=0,
O32=ik0I-ir2k0MQn2,rr-1M,
O33=-1rI-irMQn2,rr-1Qn2,rθ,
O34=iβk0rMQn2,rr-1,
O41=ir2k0M2-ik0Qn2,zz,
O42=0,
O43=-iβrk0M,
O44=-1rI.

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