Abstract

Modeling of microstructure fibers often involves severe computational bottlenecks, in particular when a design space with many degrees of freedom must be analyzed. Perturbative versions of numerical mode-solvers can substantially reduce the computational burden of problems involving automated optimization or irregularity analysis, where perturbations arise naturally. A basic theory is presented for perturbative multipole and boundary element methods, and the speed and accuracy of the methods are demonstrated. The specific optimization results in an elliptical-hole birefringent fiber design, with substantially higher birefringence than the intuitive unoptimized design.

© 2004 Optical Society of America

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References

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    [CrossRef] [PubMed]
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Bell Syst. Tech. J. (1)

P. Kaiser and H. W. Astle. �??Low-loss single-matrial fibres made from pure fused silica,�?? Bell Syst. Tech. J. 53, 1021�??39 (1974).

CLEO 2004, OSA Trends in Optics and Phot (1)

J. M. Fini. �??Perturbative modeling of irregularities in microstructure optical fibers,�?? In Conference on Lasers and Electro-Optics (CLEO), TOPS vol. 96, paper CThX6, (Optical Society of America, Washington, D.C., 2004).

European Conf. on Optical Comm. 2003 (1)

T. Hasegawa, T. Saitoh, D. Nishioka, E. Sasaoka, and T. Hosoya. �??Bend-insensitive single-mode holey fibre with SMF compatibility for optical wiring applications,�?? In European Conference on Optical Communications, paper We2.7.3, (2003).

J. Lightwave Technol. (3)

J. Lightwave. Technol. (1)

N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada. �??Boundary element method for analysis of holey optical fibers,�?? J. Lightwave. Technol. 21, 1787-92 (2003).
[CrossRef]

J. Opt. Soc. Am. B (1)

Jap. J. App. Phys. (1)

John Fini and Ryan Bise. �??Progress in fabrication and modeling of microstructured optical fiber,�?? Jap. J. App. Phys. 43, 5717�??5730 (2004).
[CrossRef]

Microwave Theory and Techniques (1)

Paul R. McIsaac. �??Symmetry-induced modal characteristics of uniform waveguides-I: Summary of results,�?? Microwave Theory and Techniques 23, 421-9 (1975).
[CrossRef]

Microwave. Theory and Technol. (1)

C. C. Su. �??A surface integral equations method for homogeneous optical fibres and coupled image lines of arbitrary cross sections,�?? Microwave. Theory and Technol., 33, 1114-9, (1985).
[CrossRef]

Nature (1)

Charlene M. Smith, Natesan Venkataraman, Michael T. Gallagher, Dirk Müller, James A. West, Nicholas F. Borrelli, Douglas C. Allan, and KarlW. Koch. �??Low loss hollow-core silica/air photonic bandgap fiber,�?? Nature, 424 657-9, (2003).
[CrossRef]

OFC 2002, OSA Trends in Optics and Phot. (1)

R. Bise, R. S. Windeler, et al. �??Tunable photonic band gap fiber,�?? In Optical Fiber Communications Conference (OFC), TOPS vol. 70, paper ThK3, (Optical Society of America, Washington, D.C., 2002).

OFC 2005 (1)

R. Bise and D. Trevor. �??Sol-gel-derived microstructured fibers: fabrication and characterization,�?? To appear in Optical Fiber Communications Conference (OFC), (Optical Society of America, Washington, D.C., 2005).

Opt. Express (7)

T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba. �??Hole-assisted lightguide fiber for large anomalous dispersion and low optical loss,�?? Opt. Express 9, 681-6, (2001).
[CrossRef] [PubMed]

Alexander Argyros, Ian M. Bassett, Martijn A. van Eijkelenborg, M.C.J. Large, Joseph Zagari, N.A.P. Nicorovici, Ross C. McPhedran, and C. Martijn de Sterke. �??Ring structures in microstructured polymer optical fibres,�?? Opt. Express 9, 813-20, (2001).
[CrossRef] [PubMed]

S. G. Johnson and J. D. Joannopoulos. �??Block-iterative frequency-domain methods for Maxwell�??s equations in a planewave basis,�?? Opt. Express 8, 173-190 (2001). software available at <a href=" http://ab-initio.mit.edu/mpb">http://ab-initio.mit.edu/mpb.</a>
[CrossRef] [PubMed]

T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith, and C. M. deSterke. �??Calculations of air-guiding modes in photonic crystal fibers using the multipole method,�?? Opt. Express, 9, 721-32 (2001).
[CrossRef]

I. K. Hwang, Y. J. Lee, and Y. H. Lee. �??Birefringence induced by irregular structure in photonic crystal fiber,�?? Opt. Express 11, 2799-2806 (2003).
[CrossRef] [PubMed]

James A. West, Charlene M. Smith, Nicholas F. Borrelli, Douglas C. Allan, and Karl W. Koch. �??Surface modes in air-core photonic band-gap fibers,�?? Opt. Express, 12, 1485-96 (2004).
[CrossRef]

S. G. Johnson, M. Ibanescu, et al. �??Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,�?? Opt. Express 9, 748-79 (2001).
[CrossRef] [PubMed]

Opt. Fiber Tech. (1)

F. Brechet, J. Marcou, et al. �??Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,�?? Opt. Fiber Tech. 6, 181-191 (2000).
[CrossRef]

Opt. Lett. (4)

Photon. Technol. Lett. (1)

M. Koshiba and K. Saitoh. �??Polarization-dependent confinement losses in actual holey fibers,�?? Photon. Technol. Lett. 15, 691-3 (2003).
[CrossRef]

Phys. Rev. E (1)

Steven G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink. �??Perturbation theory for Maxwell�??s equations with shifting material boundaries,�?? Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

S. A. Diddams, D. J. Jones, et al. �??Direct link between microwave and optical frequencies with a 300THz femtosecond laser comb,�?? Phys. Rev. Lett., 84, 5102-5 (2000).
[CrossRef] [PubMed]

Other (1)

William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Numerical recipes in C, the art of scientific computing. (Cambridge University Press, New York, 1992).

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Figures (7)

Fig. 1.
Fig. 1.

For many realistic fibers, irregularity in hole geometry is small, but large enough to seriously impact optical properties such as birefringence. A perturbative approach is then natural.

Fig. 2.
Fig. 2.

Standard multipole and boundary-element methods require many large-matrix operations for each geometry, since there are many effective index values in the search. A perturbative approach needs very few large-matrix operations for each perturbed geometry.

Fig. 3.
Fig. 3.

Test of perturbative method for random hole irregularities. The fiber has three rings of cladding holes with spacing 2 microns, diameter 1 micron, and index 1. The wavelength was 1630 nm and the substrate index was set to 1.45. The geometric perturbations consisted of independently displacing six holes (σx =σy = .02Λ), and placing the remaining 30 holes to preserve sixfold rotational symmetry. Dashed lines indicate expected error trends (blue and pink), the unperturbed loss value (red), and ideal agreement between standard and perturbative methods (black).

Fig. 4.
Fig. 4.

For a birefringent fiber with elliptical holes, four holes of the inner ring were initially misaligned. An automated optimization of birefringence adjusts orientations of these four holes, ultimately aligning them with the orientation of the fixed holes. This is an optimization test with an intuitive optimum design. Modes are calculated at wavelength λ = 1.17Λ.

Fig. 5.
Fig. 5.

A consistency check confirms that the estimates of birefringence perturbation agree with non-perturbative results at each step in the optimization.

Fig. 6.
Fig. 6.

Birefringence is plotted for two optimizations where all 18 holes are free to rotate. Both initially x-oriented and y-oriented holes arrive at equivalent fiber geometries with a substantial improvement in birefringence. Again λ = 1.17Λ.

Fig. 7.
Fig. 7.

Orientation angles for the six inner holes are plotted versus optimization step. All holes are initially oriented to ϕj = 90° and ϕj = 0 for the left and right optimizations respectively. Both optimizations are converging to equivalent geometries, rotated 120 degrees from each other. The optimal perturbations at each step approximately maintain point-reflection symmetry about the origin (from which “hidden” curves can be inferred).

Equations (21)

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M ( n eff , λ ) v = 0 .
M ( n ) = M ( n eff = n ; λ , p )
M ( n ) = M 0 ( n ) + δ M 1 ( n ) + δ 2 M 2 ( n ) +
M 1 ( n ) = M δ M ( n ; λ , p 0 + ε p 1 ) M ( n ; λ , p 0 ε p 1 ) 2 ε .
v = v 0 + δ v 1 + δ 2 v 2 +
n = n 0 + δ n 1 + δ 2 n 2 +
[ M 0 ( n 0 + δ n 1 + ) + M 1 ( n 0 + ) + ] [ v 0 + δ v 1 + ] = 0 ,
δ n 1 = u 0 δ M 1 ( n 0 ) v 0 u 0 M 0 ( n 0 ) v 0
M 0 ( n 0 ) v 0 = 0 .
M 0 ( n 0 ) δ v 1 + [ δ n 1 M 0 ( n 0 ) + δ M 1 ( n 0 ) ] v 0 = 0 ,
u 0 [ δ n 1 M 0 ( n 0 ) + δ M 1 ( n 0 ) ] v 0 = 0 .
M 0 p M 0 ( n 0 ) = I v ̂ 0 v ̂ 0 .
δ v 1 = M 0 p [ δ n 1 M 0 ( n 0 ) + δ M 1 ( n 0 ) ] v 0 .
[ M 0 ( n 0 ) ] [ δ 2 v 2 ] + [ δ n 1 M 0 ( n 0 ) + δ M 1 ( n 0 ) ] [ δ v 1 ] + [ M 2 ] [ v 0 ] = 0
M 2 = ( δ 2 v 2 ) M 0 ( n 0 ) + 1 2 ( δ n 1 ) 2 M 0 ( n 0 ) + δ M 1 ( n 0 ) ( δ n 1 ) + δ 2 M 2 ( n 0 )
v 0 = Bx .
A [ δ n 1 M 0 ( n 0 ) + δ M 1 ( n 0 ) ] Bx = 0 ,
A δ M 1 ( n 0 ) Bx = δ n 1 A M 0 ( n 0 ) Bx ,
[ p n Bi ] j = n slow ϕ j n fast ϕ j .
p m + 1 = p m + [ Δ p ] m .
[ Δ p ] m = δ p n Bi p n Bi .

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