Abstract

The calculation of chromatic dispersion in silica-based photonic crystal fibers is discussed. A perturbative approach to the inclusion of material dispersion is presented, and it is shown that this method can give accurate predictions for a wide range of physical fiber dimensions on the basis of a single waveguide dispersion calculation. Furthermore, it is demonstrated that a proper choice of the transverse k vector in a supercell calculation can significantly improve the convergence of the results with respect to supercell size.

© 2003 Optical Society of America

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References

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  1. T. A. Birks, D. Mogilevtsev, J. C. Knight, and P. St. J. Russell, “Single material fibers for dispersion compensation,” in Optical Fiber Communication Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 109–111.
  2. T. M. Monro, D. J. Richardson, and N. G. Broderick, “Efficient modelling of holey fibers,” in Optical Fiber Communication Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 111–113.
  3. S. E. Barkou, J. Broeng, and A. Bjarklev, “Dispersion properties of photonic bandgap guiding fibers,” in Optical Fiber Communication Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 117–119.
  4. A. Ferrando, E. Silvestre, J. J. Miret, J. A. Monsoriu, M. V. Andrés, and P. St. J. Russell, “Designing a photonic crystal fibre with flattened chromatic dispersion,” Electron. Lett. 35, 325–327 (1999).
    [CrossRef]
  5. A. Ferrando, E. Silvestre, J. J. Miret, and P. Andrés, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. 25, 790–792 (2000).
    [CrossRef]
  6. A. Ferrando, E. Silvestre, P. Andrés, J. J. Miret, and M. V. Andrés, “Designing the properties of dispersion-flattened photonic crystal fibers,” Opt. Express 9, 687–697 (2001).
    [CrossRef] [PubMed]
  7. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
    [CrossRef]
  8. W. Pauli, Handbuch der Physik, 2nd. ed. (Springer, Berlin, 1933), Vol. 24, p. 162.
  9. H. Hellman, Einfuhrung in die Quantenchemie (Deuticke, Leipzig, 1937).
  10. R. P. Feynman, “Forces in molecules,” Phys. Rev. 56, 340 (1939).
    [CrossRef]
  11. J. W. Fleming, “Material dispersion in lightguide glasses,” Electron. Lett. 14, 326–328 (1978).
    [CrossRef]
  12. 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).
    [CrossRef] [PubMed]

2001 (2)

2000 (1)

1999 (1)

A. Ferrando, E. Silvestre, J. J. Miret, J. A. Monsoriu, M. V. Andrés, and P. St. J. Russell, “Designing a photonic crystal fibre with flattened chromatic dispersion,” Electron. Lett. 35, 325–327 (1999).
[CrossRef]

1993 (1)

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

1978 (1)

J. W. Fleming, “Material dispersion in lightguide glasses,” Electron. Lett. 14, 326–328 (1978).
[CrossRef]

1939 (1)

R. P. Feynman, “Forces in molecules,” Phys. Rev. 56, 340 (1939).
[CrossRef]

Alerhand, O. L.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Andrés, M. V.

A. Ferrando, E. Silvestre, P. Andrés, J. J. Miret, and M. V. Andrés, “Designing the properties of dispersion-flattened photonic crystal fibers,” Opt. Express 9, 687–697 (2001).
[CrossRef] [PubMed]

A. Ferrando, E. Silvestre, J. J. Miret, J. A. Monsoriu, M. V. Andrés, and P. St. J. Russell, “Designing a photonic crystal fibre with flattened chromatic dispersion,” Electron. Lett. 35, 325–327 (1999).
[CrossRef]

Andrés, P.

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Ferrando, A.

Feynman, R. P.

R. P. Feynman, “Forces in molecules,” Phys. Rev. 56, 340 (1939).
[CrossRef]

Fleming, J. W.

J. W. Fleming, “Material dispersion in lightguide glasses,” Electron. Lett. 14, 326–328 (1978).
[CrossRef]

Joannopoulos, J. D.

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).
[CrossRef] [PubMed]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Johnson, S. G.

Meade, R. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Miret, J. J.

Monsoriu, J. A.

A. Ferrando, E. Silvestre, J. J. Miret, J. A. Monsoriu, M. V. Andrés, and P. St. J. Russell, “Designing a photonic crystal fibre with flattened chromatic dispersion,” Electron. Lett. 35, 325–327 (1999).
[CrossRef]

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Russell, P. St. J.

A. Ferrando, E. Silvestre, J. J. Miret, J. A. Monsoriu, M. V. Andrés, and P. St. J. Russell, “Designing a photonic crystal fibre with flattened chromatic dispersion,” Electron. Lett. 35, 325–327 (1999).
[CrossRef]

Silvestre, E.

Electron. Lett. (2)

A. Ferrando, E. Silvestre, J. J. Miret, J. A. Monsoriu, M. V. Andrés, and P. St. J. Russell, “Designing a photonic crystal fibre with flattened chromatic dispersion,” Electron. Lett. 35, 325–327 (1999).
[CrossRef]

J. W. Fleming, “Material dispersion in lightguide glasses,” Electron. Lett. 14, 326–328 (1978).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. (1)

R. P. Feynman, “Forces in molecules,” Phys. Rev. 56, 340 (1939).
[CrossRef]

Phys. Rev. B (1)

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Other (5)

W. Pauli, Handbuch der Physik, 2nd. ed. (Springer, Berlin, 1933), Vol. 24, p. 162.

H. Hellman, Einfuhrung in die Quantenchemie (Deuticke, Leipzig, 1937).

T. A. Birks, D. Mogilevtsev, J. C. Knight, and P. St. J. Russell, “Single material fibers for dispersion compensation,” in Optical Fiber Communication Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 109–111.

T. M. Monro, D. J. Richardson, and N. G. Broderick, “Efficient modelling of holey fibers,” in Optical Fiber Communication Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 111–113.

S. E. Barkou, J. Broeng, and A. Bjarklev, “Dispersion properties of photonic bandgap guiding fibers,” in Optical Fiber Communication Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 117–119.

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Figures (6)

Fig. 1
Fig. 1

Sellmeier curve for the frequency-dependent dielectric constant in silica.

Fig. 2
Fig. 2

Convergence with supercell size of the guided-mode eigenfrequency in an index-guiding PCF having the air holes arranged in a triangular lattice structure, with d/Λ=0.6. The supercell consists of N×N elementary cells, where N is reported on the x axis. On the y axis, the normalized frequency is given. Squares are calculated at the Γ point, circles at k1, k2=(3/8, 1/8).

Fig. 3
Fig. 3

Dispersion curves for the guided mode of an index-guiding fiber having the cladding holes arranged on a triangular lattice with d/Λ=0.6 and Λ=0.9 μm. The three lower curves are calculated in the Γ point, and the two upper ones are calculated in (k1, k2)=(3/8, 1/8).

Fig. 4
Fig. 4

Dispersion curves for the guided mode of an index-guiding PCF with a triangular cladding hole lattice calculated either self-consistently or with the approximate schemes discussed in the text. The fiber has d/Λ=0.6, Λ=0.9 μm. The perturbative curve is calculated with ε0=(1.45)2, appropriate for λ01.05 μm, whereas the D=Dw+Dm curve was obtained with ε0=(1.4535)2, corresponding to λ00.8 μm.

Fig. 5
Fig. 5

Dispersion curves for the guided mode of a honeycomb PCF with a central hole calculated either self-consistently or with the approximate schemes discussed in the text. The structural parameters are dcl/Λ=0.64, dc=0.55dcl, and Λ=2.188 μm.

Fig. 6
Fig. 6

Perturbative and self-consistent dispersion curves for the guided mode of a honeycomb fiber with a central hole. The structural parameters are as in Fig. 5 except for Λ, which is here 0.866 μm.

Equations (30)

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H(r)=exp[i(βz-ωt)]H(x, y)
ω2c2=H, ΘHH, H,
ΘH=×1ε(r) ×H,
D=-ω22πcd2βdω2=ω22πcvg2dvgdω,
vg=dωdβ.
2ωc2 vg=ddβH, ΘHH, H=βH, ΘHH, H+vgεωεH, ΘHH, H.
vg=c Re[E*×H]zH,H-c22ωH,×ε(r)ωε2(r) ×HH,H vg.
vg=c Re[E*×H]zH,H-ω2D,  ln ε(r)ωEH,H vgvg0-ω2D,  ln ε(r)ωEH,H vg.
vg=vg01+ω2 Ed ln εω  ,
ωε=-ω2ε Ed.
ωscω01-Ed2ε0 Δε;
ε(ωsc)ε(ω0)-ω02ε0 EdΔε dεdωω0=ε0+Δε.
Δε=ε(ω0)-ε01+ω02ε0 Eddεdωω0,
ωscω01-Ed1-ε0ε(ω0)2ε0ε(ω0)+ω0Eddln εdωω0,
vg0ε=2ωεβ=-βω2ε Ed=-Ed2ε vg0-ω2εEdβ,
vgsc=vg0-12ε0Edvg0+ω0EdβΔε1+ωsc2 Ed ln εω  .
HR(r)=h(r-R),
Hn(r)=RanRh(r-R).
RRΘRR(anR)*anR=ω2c2ORR(anR)*anR,
ΘRR=h(r-R), Θh(r-R),
ORR=h(r-R), h(r-R).
Hk(r)=1NRexp(ikR)h(r-R),
ω02c2+t|R|=R0exp(ikR)=ωk2c21+O|R|=R0exp(ikR),
ω02c2=ΘRR,
t=ΘRR||R-R|=R0,
O=ORR||R-R|=R0.
R1=32, 12,R2=32, 12,
cos 2πk1+cos 2πk2+cos 2π(k1-k2)=0,
k=k1G1+k2G2,
RiGj=2πδij,

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