Abstract

We study the cladding modes of photonic crystal fibers (PCFs) using a fully vectorial method. This approach enables us to analyze the modes and incorporate material dispersion in a straightforward fashion. We find the field flow lines, intensity distribution and polarization properties of these modes. The effective cladding indices of different PCFs are investigated in detail.

©2001 Optical Society of America

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References

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  1. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998).
    [Crossref]
  2. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999).
    [Crossref]
  3. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
    [Crossref] [PubMed]
  4. J. C. Knight, J. Boreng, T. A. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
    [Crossref] [PubMed]
  5. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
    [Crossref] [PubMed]
  6. M. Midrio, M. P. Singh, and C. G. Someda, “The space filling mode of holey fibers: an analytical vectorial solution,” J. Lightwave Technol. 18, 1031 (2000).
    [Crossref]
  7. A. A. Maradudin and A. R. McGurn, “Out of plane propagation of electromagnetic waves in two-dimensional periodic dielectric medium,” J. Mod. Opt. 41, 275–284 (1994).
    [Crossref]
  8. J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguidance by the photonic bandgap effect in optical fiber,” J. Opt. A—Pure Appl. Opt. 1, 477–482 (1999).
    [Crossref]
  9. E. Silvestre, M. V. Andres, and P. Andres, “Biorthonormal-basis method for the vector description of optical fiber modes,” J. Lightwave Technol. 16, 923–928 (1998).
    [Crossref]
  10. A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. 24, 276–278 (1999).
    [Crossref]
  11. G. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995).
  12. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell“Localized function method for modeling defect modes in 2-d photonic crystals,” J. Lightwave Technol.17, 2078–2081 (1999);T. M. Monro, D. J. Richardson, N.G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol.18, 50–56 (2000).
    [Crossref]

2000 (1)

1999 (4)

J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguidance by the photonic bandgap effect in optical fiber,” J. Opt. A—Pure Appl. Opt. 1, 477–482 (1999).
[Crossref]

A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. 24, 276–278 (1999).
[Crossref]

T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999).
[Crossref]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

1998 (3)

1997 (1)

1994 (1)

A. A. Maradudin and A. R. McGurn, “Out of plane propagation of electromagnetic waves in two-dimensional periodic dielectric medium,” J. Mod. Opt. 41, 275–284 (1994).
[Crossref]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995).

Allen, D. C.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

Andres, M. V.

Andres, P.

Barbeito, P. M.

J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguidance by the photonic bandgap effect in optical fiber,” J. Opt. A—Pure Appl. Opt. 1, 477–482 (1999).
[Crossref]

Barkou, S. E.

J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguidance by the photonic bandgap effect in optical fiber,” J. Opt. A—Pure Appl. Opt. 1, 477–482 (1999).
[Crossref]

Bennett, P. J.

T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999).
[Crossref]

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell“Localized function method for modeling defect modes in 2-d photonic crystals,” J. Lightwave Technol.17, 2078–2081 (1999);T. M. Monro, D. J. Richardson, N.G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol.18, 50–56 (2000).
[Crossref]

Birks, T. A.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998).
[Crossref]

J. C. Knight, J. Boreng, T. A. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[Crossref] [PubMed]

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell“Localized function method for modeling defect modes in 2-d photonic crystals,” J. Lightwave Technol.17, 2078–2081 (1999);T. M. Monro, D. J. Richardson, N.G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol.18, 50–56 (2000).
[Crossref]

Bjarklev, A.

J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguidance by the photonic bandgap effect in optical fiber,” J. Opt. A—Pure Appl. Opt. 1, 477–482 (1999).
[Crossref]

Boreng, J.

J. C. Knight, J. Boreng, T. A. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

Broderick, N. G. R.

Broderick, N.G. R.

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell“Localized function method for modeling defect modes in 2-d photonic crystals,” J. Lightwave Technol.17, 2078–2081 (1999);T. M. Monro, D. J. Richardson, N.G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol.18, 50–56 (2000).
[Crossref]

Broeng, J.

J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguidance by the photonic bandgap effect in optical fiber,” J. Opt. A—Pure Appl. Opt. 1, 477–482 (1999).
[Crossref]

Cregan, R. F.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

Ferrando, A.

Knight, J. C.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

J. C. Knight, J. Boreng, T. A. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[Crossref] [PubMed]

Mangan, B. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

Maradudin, A. A.

A. A. Maradudin and A. R. McGurn, “Out of plane propagation of electromagnetic waves in two-dimensional periodic dielectric medium,” J. Mod. Opt. 41, 275–284 (1994).
[Crossref]

McGurn, A. R.

A. A. Maradudin and A. R. McGurn, “Out of plane propagation of electromagnetic waves in two-dimensional periodic dielectric medium,” J. Mod. Opt. 41, 275–284 (1994).
[Crossref]

Midrio, M.

Miret, J. J.

Mogilevtsev, D.

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998).
[Crossref]

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell“Localized function method for modeling defect modes in 2-d photonic crystals,” J. Lightwave Technol.17, 2078–2081 (1999);T. M. Monro, D. J. Richardson, N.G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol.18, 50–56 (2000).
[Crossref]

Monro, T. M.

T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999).
[Crossref]

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell“Localized function method for modeling defect modes in 2-d photonic crystals,” J. Lightwave Technol.17, 2078–2081 (1999);T. M. Monro, D. J. Richardson, N.G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol.18, 50–56 (2000).
[Crossref]

Richardson, D. J.

T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999).
[Crossref]

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell“Localized function method for modeling defect modes in 2-d photonic crystals,” J. Lightwave Technol.17, 2078–2081 (1999);T. M. Monro, D. J. Richardson, N.G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol.18, 50–56 (2000).
[Crossref]

Roberts, P. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

Russell, P. St. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998).
[Crossref]

J. C. Knight, J. Boreng, T. A. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[Crossref] [PubMed]

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell“Localized function method for modeling defect modes in 2-d photonic crystals,” J. Lightwave Technol.17, 2078–2081 (1999);T. M. Monro, D. J. Richardson, N.G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol.18, 50–56 (2000).
[Crossref]

Silvestre, E.

Singh, M. P.

Someda, C. G.

Sondergaard, T.

J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguidance by the photonic bandgap effect in optical fiber,” J. Opt. A—Pure Appl. Opt. 1, 477–482 (1999).
[Crossref]

J. Lightwave Technol. (3)

J. Mod. Opt. (1)

A. A. Maradudin and A. R. McGurn, “Out of plane propagation of electromagnetic waves in two-dimensional periodic dielectric medium,” J. Mod. Opt. 41, 275–284 (1994).
[Crossref]

J. Opt. A—Pure Appl. Opt. (1)

J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguidance by the photonic bandgap effect in optical fiber,” J. Opt. A—Pure Appl. Opt. 1, 477–482 (1999).
[Crossref]

Opt. Lett. (3)

Science (2)

J. C. Knight, J. Boreng, T. A. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

Other (2)

G. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995).

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell“Localized function method for modeling defect modes in 2-d photonic crystals,” J. Lightwave Technol.17, 2078–2081 (1999);T. M. Monro, D. J. Richardson, N.G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol.18, 50–56 (2000).
[Crossref]

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Figures (7)

Fig. 1.
Fig. 1. Schematic of the triangular lattice of air holes in the cladding. The lattice pitch is Λ, and R is the radius of the air holes. Dielectric constants of air and silica are εa =1.0 and εb (λ), respectively.

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Fig. 2.
Fig. 2. Dispersions of the some lower cladding modes. The fundamental mode (FCM) is a two-fold degenerate mode. The next lower mode (HCM) includes six nearly degenerate modes. The material dispersion of the silica core is also shown in the figure.

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Fig. 3.
Fig. 3. Transverse magnetic fields of the two degenerate modes of the fundamental cladding mode.

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Fig. 4.
Fig. 4. Transverse magnetic field intensity of the fundamental cladding mode (x-polarized).

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Fig. 5.
Fig. 5. Dispersion curves calculated using the present vectorial method (solid lines) and the scalar method (dashed lines) for claddings of different air-hole radius R.

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Fig. 6.
Fig. 6. Effective cladding index as a function of λ/Λ and R/Λ. Λ is chosen to be 2.3 µm and the material dispersion of silica is considered in the calculations.

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Fig. 7.
Fig. 7. Calculated parameter V vs. wavelength for claddings of different pitches and air-hole radii. The red dashed line indicates the single-mode cutoff value of V (=2.405). (a) Lattice pitch Λ=2.3 µm. Results from the vectorial (scalar) method are shown by blue (black) solid lines. (b) Lattice pitch Λ=5.0 µm. Only the vectorial results are shown.

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Equations (14)

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H ( x , y , z ; t ) = [ H t ( x , y ) + H z ( x , y ) z ̂ ] exp ( i β z − i ω t )
∇ 2 H + k 2 ε H = − ∇ ( ln ε ) × ( ∇ × H ) ,
LH t = ( ∇ t 2 + k 2 ε ) H t + ∇ t ( ln ε ) × ( ∇ t × H t ) = β 2 H t ,
ε ( x , y ) = ∑ G ε ̂ ( G ) exp ( i G · x t ) ,
ln ε ( x , y ) = ∑ G κ ̂ ( G ) exp ( i G · x t ) ,
H j ( x , y ) = ∑ G H ̂ j ( G ) exp ( i G · x t ) , j = x , y .
∑ G ′ L G ′ G H ̂ ( G ′ ) = β 2 H ̂ ( G ) ,
[ L G ′ G ] u , v = [ − ∣ G ∣ 2 δ G ′ , G + k 2 ∑ G ″ ε ̂ ( G ″ ) ] δ u , v + [ Q G ′ G ] u , v , ( u , v = x , y )
[ Q G ′ G ] x , x = κ ̂ ( G − G ′ ) ( G y − G y ′ ) G y ′ ,
[ Q G ′ G ] x , y = − κ ̂ ( G − G ′ ) ( G y − G y ′ ) G x ′ ,
[ Q G ′ G ] y , x = − κ ̂ ( G − G ′ ) ( G x − G x ′ ) G y ′ ,
[ Q G ′ G ] y , y = κ ̂ ( G − G ′ ) ( G x − G x ′ ) G x ′ .
ε ̂ ( G ) = { ε b + ( ε a − ε b ) f , G = 0 ( ε a − ε b ) f 2 J 1 ( GR ) GR , G ≠ 0
κ ̂ ( G ) = { ln ε b + ( ln ε a − ln ε b ) f , G = 0 ( ln ε a − ln ε b ) f 2 J 1 ( GR ) GR , G ≠ 0 ,

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