Abstract

Confinement loss is comprehensively evaluated for TE, TM, and hybrid modes of Bragg fibers using a multilayer division method newly developed. We show the loss dependence on the core radius, wavelength, cladding index contrast, and the number of cladding pairs. The confinement loss is reduced in proportion to (a/b)2N and (n 2 b b/n 2 a a)2N for the TE and other three modes, respectively, with respect to cladding pairs N under the quarter-wave stack condition, with cladding high na and low indices nb and their corresponding thicknesses a and b. For sufficiently large core radius, the confinement loss decreases in inverse proportion to the third and first powers of core radius for the TE and other three modes, respectively. Low loss modes are the TE01, TE02, HE13, and TE03 modes in order of increasing confinement loss.

© 2008 Optical Society of America

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References

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  1. T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, and M. J. Steel, "Confinement losses in microstructured optical fibers," Opt.Lett. 26, 1660-1662 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [PubMed]
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    [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2007 (3)

2006 (2)

2005 (1)

2003 (3)

2002 (3)

2001 (2)

1987 (1)

J. Sakai and T. Kimura, "Bending loss of propagation modes in arbitrary-index profile optical fibers," Appl. Opt. 181499-1506 (1987).

1973 (1)

J. G. Dil and H. Blok, "Propagation of electromagnetic surface waves in a radially inhomogeneous optical waveguide," Opto-Electronics,  5415-428 (1973).
[CrossRef]

1961 (1)

Appl. Opt. (1)

J. Sakai and T. Kimura, "Bending loss of propagation modes in arbitrary-index profile optical fibers," Appl. Opt. 181499-1506 (1987).

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (4)

Opt. Express (7)

M. Yan and P. Shum, "Analysis of perturbed Bragg fibers with an extended transfer matrix method," Opt. Express,  14, 2596-2610 (2006), http://www.opticsexpress.org/abstract.cfm?URI=oe-14-7-2596.
[CrossRef] [PubMed]

W. Zhi, R. Guobin, L. Shuqin, L. Weijun and S. Guo, "Compact supercell method based on opposite parity for Bragg fibers," Opt. Express,  11, 3542-3549 (2003), http://www.opticsexpress.org/abstract.cfm?URI=oe-11-26-3542.
[CrossRef] [PubMed]

I. M. Bassett and A. Argyros, "Elimination of polarization degeneracy in round waveguide," Opt. Express,  10, 1342-1346 (2002), http://www.opticsexpress.org/abstract.cfm?URI=oe-10-23-1342.
[PubMed]

A. Argyros, "Guided modes and loss in Bragg fibers, "Opt. Express,  10, 1411-1417 (2002), http://www.opticsexpress.org/abstract.cfm?URI=oe-10-24-1411.
[PubMed]

Y. Xu, A. Yariv, J. G. Fleming, and S. Y. Lin, "Asymptotic analysis of silicon based Bragg fibers," Opt. Express,  11, 1039-1049 (2003), http://www.opticsexpress.org/abstract.cfm?URI=oe-11-9-1039
[CrossRef] [PubMed]

J. A. Monsoriu, E. Silvestre, A. Ferrando, P. Andrés and J. J. Miret, "Highindex-core Bragg fibers: dispersion properties," Opt. Express,  11, 1400-1405 (2003), http://www.opticsexpress.org/abstract.cfm?URI=oe-11-12-1400.
[CrossRef] [PubMed]

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, "Low-loss asymptotically single-mode propagation in large-core Omniguide fibers," Opt. Express,  9, 748-779 (2001), http://www.opticsexpress.org/abstract.cfm?URI=oe-9-13-748
[CrossRef] [PubMed]

Opt. Lett. (1)

Opt.Lett. (1)

T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, and M. J. Steel, "Confinement losses in microstructured optical fibers," Opt.Lett. 26, 1660-1662 (2001).
[CrossRef]

Opto-Electronics (1)

J. G. Dil and H. Blok, "Propagation of electromagnetic surface waves in a radially inhomogeneous optical waveguide," Opto-Electronics,  5415-428 (1973).
[CrossRef]

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Figures (16)

Fig. 1.
Fig. 1.

Cross sectional view of the cylindrical symmetric fiber. ni , refractive index of the ith layer; ri , outer radius of the ith layer; r i1 and r i2, arbitrary radial coordinates within the ith layer; Ai -Di , amplitude coefficients of the ith layer; Di (r), representation matrix, Qi (r i2, r i1); displacement matrix between r=r i1 and r i2.

Fig. 2.
Fig. 2.

(Color online) Schematics of a Bragg fiber. rc , core radius; nc , refractive index of core; na and nb , indices of layers with thickness a and b, respectively; Λ=a+b, period in the cladding; nex, external layer index.

Fig. 3.
Fig. 3.

(Color online) Cladding pair dependence of confinement loss for TE, TM, and hybrid modes. λ 0=1.0 µm, rc =2.0 µm, na =2.5, and nb =n ex=1.5. Cladding thicknesses, a and b, are determined from Eq. (16) for each combination of na , nb , and UQWS.

Fig. 4.
Fig. 4.

(Color online) Core radius dependence of confinement loss for TE, TM, and hybrid modes. λ 0=1.0 µm, na =2.5, nb =n ex=1.5, and N=10. Cladding thicknesses, a and b, are determined from Eq. (16) for each combination of rc , na , and nb .

Fig. 5.
Fig. 5.

(Color online) Core radius dependence of confinement loss for TE and TM modes as a function of cladding high index. λ 0=1.0 µm, nb =n ex=1.5, and N=10. Cladding thicknesses, a and b, are determined from Eq. (16) for each combination of rc , na , and nb .

Fig. 6.
Fig. 6.

(Color online) Wavelength dependence of confinement loss for TE and TM modes. λ QWS=1.0 µm, na =2.5, nb =n ex=1.5, and N=10. All the modes are shown that exist at rc /λ 0=2.0.

Fig. 7.
Fig. 7.

(Color online) Wavelength dependence of confinement loss for hybrid modes. Parameters are the same as those in Fig. 6.

Fig. 8.
Fig. 8.

(Color online) Wavelength dependence of confinement loss for TE and TM modes as a function of cladding high index. λ QWS=1.0 µm, rc =2.0 µm, nb =n ex=1.5, and N=10. All the modes are shown that exist at λ 0=1.0 µm for each na .

Fig. 9.
Fig. 9.

(Color online) Wavelength dependence of confinement loss for TE01 and TM01 modes as a function of core radius. λ QWS=1.0 µm, na =2.5, nb =n ex=1.5, and N=10.

Fig. 10.
Fig. 10.

(Color online) Cladding high index dependence of confinement loss for several modes. λ 0=1.0 µm, rc =2.0 µm, nb =nex =1.5, and N=10. All the modes are shown that exist at rc /λ0=2.0 and na =2.5.

Fig. 11.
Fig. 11.

(Color online) Cladding pairs dependence of confinement loss for the TE01 mode as a function of core radius and cladding high index. λ 0=1.0 µm, and nb =n ex=1.5. Cladding thicknesses, a and b, are determined from Eq. (16) for each combination of rc and na . Dotted, solid, and dotted-dashed curves indicate na =2.5, 3.5, and 4.5, respectively.

Fig. 12.
Fig. 12.

(Color online) Wavelength dependence of confinement loss of the TE01 mode. λ QWS=1.0 µm, nb =n ex=1.5, and N=10. Cladding thicknesses, a and b, are determined from Eq. (16) at λ 0=1.0 µm for each combination of rc , na , and nb .

Fig. 13.
Fig. 13.

(Color online) Wavelength dependence of confinement loss of the TE01 mode. λ QWS=1.0 µm, rc =1.117 µm, and N=10. (a) nb =n ex=1.5. (b) na =3.5 and n ex=1.5.

Fig. 14.
Fig. 14.

(Color online)Wavelength dependence of confinement loss of the TE01 mode with three values of external layer index n ex. λ QWS=1.0 µm, rc =2.0 µm, na =2.5, nb =1.5, and N=10.

Fig. 15.
Fig. 15.

(Color online) Comparison in confinement loss between the present and transfer matrix methods. na =4.6, nb =1.6, n ex=nb , Λ=0.434 µm, rc =30Λ, a=0.22Λ, b=0.78Λ, and N=8. Solid curves indicate results obtained by the present method, and dotted curves indicate results obtained by the transfer matrix method.

Fig. 16.
Fig. 16.

(Color online) Comparison in confinement loss between the present and Chew’s methods. na =1.49, nb =1.17, n ex=na , a=0.2133 µm, b=0.346 µm, and N=16. Losses are shown for three core radii. Solid curves indicate results calculated by the present method, and dotted curves indicate results calculated by Chew’s method.

Tables (1)

Tables Icon

Table 1. Numerical Results on the Slope in Fig. 3

Equations (37)

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H z = [ A i H v ( 2 ) ( κ i r ) + B i H v ( 1 ) ( κ i r ) ] sin ( v θ + θ in ) ,
E z = [ C i H v ( 2 ) ( κ i r ) + D i H v ( 1 ) ( κ i r ) ] cos ( v θ + θ in ) ,
κ i k 0 [ n i 2 ( β k 0 ) 2 ] 1 2 .
( H z i E θ E z i H θ ) r = r = U t z D i ( r ) ( A i B i C i D i )
D i ( r ) ( d 11 d 12 0 0 d 21 d 22 d 23 d 24 0 0 d 33 d 34 d 41 d 42 d 43 d 44 ) .
d 11 = d 33 = H v ( 2 ) ( κ i r ) , d 12 = d 34 = H v ( 1 ) ( κ i r ) ,
d 21 = d 43 Y i 2 = ω μ 0 κ i H v ( 2 ) ( κ i r ) , d 22 = d 44 Y i 2 = ω μ 0 κ i H v ( 1 ) ( κ i r ) ,
d 23 = d 41 = v β κ i 2 r H v ( 2 ) ( κ i r ) , d 24 = d 42 = v β κ i 2 r H v ( 1 ) ( κ i r ) .
( A i B i C i D i ) = 1 U tz D i 1 ( r i 1 ) ( H z i E θ E z i H θ ) r = r i 1 .
( H z i E θ E z i H θ ) r = r i 2 = Q i ( r i 2 , r i 1 ) ( H z i E θ E z i H θ ) r = r i 1
Q i ( r i 2 , r i 1 ) D i ( r i 2 ) D i 1 ( r i 1 ) = ( q 11 i q 12 i q 13 i 0 q 21 i q 22 i q 23 i q 24 i q 31 i 0 q 33 i q 34 i q 41 i q 42 i q 43 i q 44 i ) .
q 11 i = q 33 i = π κ i r i 1 4 i [ H v ( 2 ) ( κ i r i 2 ) H v ( 1 ) ( κ i r i 1 ) H v ( 1 ) ( κ i r i 2 ) H v ( 2 ) ( κ i r i 1 ) ] ,
q 12 i = Y i 2 q 34 i = π κ i 2 r i 1 4 i ω μ 0 [ H v ( 2 ) ( κ i r i 2 ) H v ( 1 ) ( κ i r i 1 ) H v ( 1 ) ( κ i r i 2 ) H v ( 2 ) ( κ i r i 1 ) ] ,
q 13 i = Y i 2 q 31 i = π v β 4 i ω μ 0 [ H v ( 2 ) ( κ i r i 2 ) H v ( 1 ) ( κ i r i 1 ) H v ( 1 ) ( κ i r i 2 ) H v ( 2 ) ( κ i r i 1 ) ] ,
q 21 i = q 43 i Y i 2 = i π r i 1 4 ω ε 0 n i 2 { ( k 0 n i ) 2 ) [ H v ( 2 ) ( κ i r i 2 ) H v ( 1 ) ( κ i r i 1 ) H v ( 1 ) ( κ i r i 2 ) H v ( 2 ) ( κ i r i 1 ) ]
+ ( v β ) 2 κ i 2 r i 1 r i 2 [ H v ( 2 ) ( κ i r i 2 ) H v ( 1 ) ( κ i r i 1 ) H v ( 1 ) ( κ i r i 2 ) H v ( 2 ) ( κ i r i 1 ) ] } ,
q 22 i = q 44 i = i π κ i r i 1 4 [ H v ( 2 ) ( κ i r i 2 ) H v ( 1 ) ( κ i r i 1 ) H v ( 1 ) ( κ i r i 2 ) H v ( 2 ) ( κ i r i 1 ) ] ,
q 23 i = q 41 i = i π v β 4 κ i { [ H v ( 2 ) ( κ i r i 2 ) H v ( 1 ) ( κ i r i 1 ) H v ( 1 ) ( κ i r i 2 ) H v ( 2 ) ( κ i r i 1 ) ]
+ r i 1 r i 2 [ H v ( 2 ) ( κ i r i 2 ) H v ( 1 ) ( κ i r i 1 ) H v ( 1 ) ( κ i r i 2 ) H v ( 2 ) ( κ i r i 1 ) ] } ,
q 24 i = 1 Y i 2 q 42 i = 1 Y i 2 r i 1 r i 2 q 13 i .
( H z i E θ E z i H θ ) r = r i = F i ( r i , r 1 ) ( H z i E θ E z i H θ ) r = r 1 ,
F i ( r i , r 1 ) Q i ( r i , r i 1 ) Q i 1 ( r i 1 , r i 2 ) · · · Q 2 ( r 2 , r 1 )
= ( f 11 i f 12 i f 13 i f 14 i f 21 i f 22 i f 23 i f 24 i f 31 i f 32 i f 33 i f 34 i f 41 i f 42 i f 43 i f 44 i ) .
( s 11 s 12 s 13 0 s 21 s 22 s 23 s 24 s 31 0 s 33 s 34 s 41 s 42 s 43 s 44 ) ( A 1 A N + 1 C 1 C N + 1 ) = ( 0 0 0 0 ) ,
s i 1 = 2 f i 1 N J v ( κ 1 r 1 ) 2 f i 2 N ω μ 0 κ 1 J v ( κ 1 r 1 ) + 2 f i 4 N v β κ 1 2 r 1 J v ( κ 1 r 1 ) ( i = 1 4 ) ,
s 12 = s 34 = H v ( 2 ) ( κ N + 1 r N ) ,
s i 3 = 2 f i 3 N J v ( κ 1 r 1 ) + 2 f i 4 N ω ε 0 n 1 2 κ 1 J v ( κ 1 r 1 ) 2 f i 2 N v β κ 1 2 r 1 J v ( κ 1 r 1 ) ( i = 1 4 ) ,
s 22 = s 44 Y N + 1 2 = ω μ 0 κ N + 1 H v ( 2 ) ( κ N + 1 r N ) ,
s 24 = s 42 = v β κ N + 1 2 r N H v ( 2 ) ( κ N + 1 r N ) = v β κ N + 1 2 r N s 12 .
( A i B i C i D i ) = D i 1 ( r i ) F i ( r i , r 1 ) D 1 ( r 1 ) ( 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 ) ( A 1 A N + 1 C 1 C N + 1 )
P ω μ 0 β H z E z = ω μ 0 β ( s 12 s 23 s 13 s 22 ) + s 12 s 33 ( v β κ N + 1 2 r N ) ( s 12 s 21 s 11 s 22 ) + s 12 s 31 ( v β κ N + 1 2 r N )
λ 0 a = 2 [ 4 ( n a 2 n c 2 ) + ( U QWS π ) 2 ( λ 0 r c ) 2 ] 1 2 ,
L exp ( i 2 N K 1 S Λ ) = { ( a b ) 2 N for TE mode ( n b 2 b n a 2 a ) 2 N for TM mode
S = { S TE 2 log ( a b ) for TE mode S non TE 2 log ( n b 2 b n a 2 a ) for TM and hybrid modes .
A N + 1 A 1 = ( s 13 s 21 s 11 s 23 ) + ( s 13 s 31 s 11 s 33 ) ( v β κ N + 1 2 r N ) ( s 12 s 23 s 13 s 22 ) + s 12 s 33 ( v β κ N + 1 2 r N ) ,
C 1 A 1 = ( s 12 s 21 s 11 s 22 ) + s 12 s 33 ( v β κ N + 1 2 r N ) ( s 12 s 23 s 13 s 22 ) + s 12 s 33 ( v β κ N + 1 2 r N ) ,
C N + 1 A 1 = ( s 21 s 33 s 23 s 31 ) + ( s 13 s 31 s 11 s 33 ) ( s 22 s 12 ) ( s 12 s 23 s 13 s 22 ) + s 12 s 33 ( v β κ N + 1 2 r N ) .

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