Abstract

A rigorous full-vector finite element method is effectively applied to evaluating the effective area Aeff and the mode field diameter (MFD) of holey fibers (HFs) with finite cross sections. The effective modal spot size (a half of MFD), weff , is defined with the help of the second moment of the optical intensity distribution. The influence of hole diameter, hole pitch, operating wavelength, and number of rings of air holes on Aeff and weff is investigated in detail. As a result, it is shown that Aeff and weff are almost independent of the number of hole rings and that the relation Aeffweff2, which is frequently utilized in the conventional optical fibers, does not always hold, especially in smaller air-filling fraction and/or longer wavelength regions. In addition, we find that for HFs with large air holes operating at longer wavelengths, the mode profiles of the two linearly polarized fundamental modes are significantly different from each other, even though they are degenerate. Using the values of Aeff and weff obtained here, the beam divergence and the nonlinear phase shift are calculated and are compared with the earlier experimental results.

© 2003 Optical Society of America

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References

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Electron. Lett. (2)

K. Petermann, �??Constraints for fundamental-mode spot size for broadband dispersion-compensated singlemode fibers,�?? Electron. Lett. 19, 712-714, (1983).
[CrossRef]

K. Hayata, M. Koshiba, and M. Suzuki, �??Modal spot size of axially nonsymmetrical fibers,�?? Electron. Lett. 22, 127-129, (1986).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. Saitoh and M. Koshiba, �??Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers,�?? IEEE J. Quantum Electron. 38, 927-933, (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

N.A. Mortensen, J.R. Folken, P.M.W. Skovgaard, and J. Broeng, �??Numerical aperture of single-mode photonic crystal fibers,�?? IEEE Photon. Technol. Lett. 14, 1094-1096, (2002).
[CrossRef]

J.H. Lee, P.C. Teh, Z. Yusoff, M. Ibsen, W. Belardi, T.M. Monro, and D.J. Richardson, �??A holey fiberbased nonlinear thresholding devices for optical CDMA receiver performance enhancement,�?? IEEE Photon. Technol. Lett. 14, 876-878, (2002).
[CrossRef]

M. Koshiba and K. Saitoh, �??Polarization-dependent losses in actual holey fibers,�?? IEEE Photon. Technol. Lett. 15, 691-693, (2003).
[CrossRef]

M. Koshiba and K. Saitoh, �??Numerical verification of degeneracy in hexagonal photonic crystal fibers,�?? IEEE Photon. Technol. Lett. 13, 1313-1315, (2001).
[CrossRef]

IEICE Trans. Electron. (1)

T.A. Birks, J.C. Knight, B.J. Mangan, and P.St.J. Russell, �??Photonic crystal fibers: An endless variety,�?? IEICE Trans. Electron. E84-C, 585-592, (2001).

J. Lightwave Technol. (3)

Opt. Express (1)

Opt. Fiber Technol. (1)

J. Broeng, D. Mogilevstev, S.E. Barkou, and A. Bjarklev, �??Photonic crystal fibers: A new class of optical waveguides,�?? Opt. Fiber Technol. 5, 305-330, (1999).
[CrossRef]

Opt. Lett. (3)

SIAM Rev. (1)

J.W.H. Liu, �??The multifrontal method for sparse matrix solutions: theory and practice,�?? SIAM Rev. 34, 82- 109, (1992).
[CrossRef]

Other (3)

G. Agrawal, Nonlinear Fiber Optics, Academic Press (San Diego, CA), 2dn Edition (1995).

V. Finazzi, T.M. Monro, and D.J. Richardson, �??Confinement loss in highly nonlinear holey optical fibers,�?? Proc. Optical Fiber Commun. Conf. ThS4. (2002).

T.M. Monro, V. Finazzi, W. Belardi, K.M. Kiang, J.H. Lee, and D.J. Richardson, �??Highly nonlinear holey optical fibers: design, manufacture and device applications,�?? Proc. European Conf. Opt. Commun., Symposium 1.5. (2002).

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Figures (11)

Fig. 1.
Fig. 1.

Holey fiber with finite cross section.

Fig. 2.
Fig. 2.

Curvilinear hybrid edge/nodal element.

Fig. 3.
Fig. 3.

Effective area and confinement loss of holey fibers operating at the wavelength of (a) λ/Λ=0.2, (b) λ/Λ=0.4, (c) λ/Λ=0.6, (d) λ/Λ=0.8, (e) λ/Λ=1.0, and (f) λ/Λ=1.29.

Fig. 4.
Fig. 4.

Relationship between effective area and mode field diameter as a function of wavelength.

Fig. 5.
Fig. 5.

Intensity profile of (a) horizontally polarized and (b) vertically polarized modes in a holey fiber with d/Λ=0.3 at λ/Λ=0.2, where |Ex |2 in (a) and |Ey |2 in (b) are expressed in the intensity contours spaced by 1 dB.

Fig. 6.
Fig. 6.

Relationship among three kinds of modal spot sizes as a function of wavelength.

Fig. 7.
Fig. 7.

Intensity profile of (a) horizontally polarized and (b) vertically polarized modes in a holey fiber with d/Λ=0.9 at λ/Λ=1.2, where |Ex |2 in (a) and |Ey |2 in (b) are expressed in the intensity contours spaced by 1 dB.

Fig. 8.
Fig. 8.

Beam divergences of holey fibers.

Fig. 9.
Fig. 9.

Highly nonlinear holey fiber [13].

Fig. 10.
Fig. 10.

Effective area, modal spot size, and birefringence of a holey fiber shown in Fig. 9.

Fig. 11.
Fig. 11.

Nonlinear phase shift of a holey fiber shown in Fig. 9.

Equations (15)

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× ( [ s ] 1 × E ) k 0 2 n 2 [ s ] E = 0
E = [ E x E y E z ] = [ { U ( x , y ) } T { E t } e exp ( j β z ) { V ( x , y ) } T { E t } e exp ( j β z ) j β { N ( x , y ) } T { E z } e exp ( j β z ) ]
[ K ] { E } = β 2 [ M ] { E }
{ E } = [ { E t } { E z } ]
A eff = ( S E t 2 dx dy ) 2 S E t 4 dx dy
A eff = [ e e I e ( x , y ) dx dy ] 2 e e I e 2 ( x , y ) dx dy
I e ( x , y ) = { U ( x , y ) } T { E t } e 2 + { V ( x , y ) } T { E t } e 2 .
w x 2 = 4 S ( x x c ) 2 E t 2 dx dy S E t 2 dx dy = 4 e e ( x x c ) 2 I e ( x , y ) dx dy e e I e ( x , y ) dx dy
w y 2 = 4 S ( y y c ) 2 E t 2 dx dy S E t 2 dx dy = 4 e e ( y y c ) 2 I e ( x , y ) dx dy e e I e ( x , y ) dx dy
x c = S x E t 2 dx dy S E t 2 dx dy = e e x I e ( x , y ) dx dy e e I e ( x , y ) dx dy
y c = S y E t 2 dx dy S E t 2 dx dy = e e y I e ( x , y ) dx dy e e I e ( x , y ) dx dy .
w eff 2 = w x 2 + w y 2 2 .
L c Λ = 8.686 Im [ β Λ ] .
θ = tan 1 λ π w eff
ϕ SPM = 4 π n 2 P λ A eff

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