Abstract

We consider the effective area A eff of photonic crystal fibers (PCFs) with a triangular air-hole lattice in the cladding. It is first of all an important quantity in the context of non-linearities, but it also has connections to leakage loss, macro-bending loss, and numerical aperture. Single-mode versus multi-mode operation in PCFs can also be studied by comparing effective areas of the different modes. We report extensive numerical studies of PCFs with varying air hole size. Our results can be scaled to a given pitch and thus provide a general map of the effective area. We also use the concept of effective area to calculate the “phase” boundary between the regimes with single-mode and multi-mode operation.

©2002 Optical Society of America

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References

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  13. D. Marcuse, “Loss analysis of sigle-mode fiber splices,” Bell Syst. Tech. J. 56, 703 (1977).
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    [Crossref] [PubMed]
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    [Crossref]
  19. J. Sakai, “Simplified bending loss formula for single-mode optical fibers,” Appl. Optics 18, 951–952 (1979).
    [Crossref]
  20. 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 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173
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2001 (1)

2000 (2)

1999 (2)

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]

B. J. Eggleton, P. S. Westbrook, R. S. Windeler, S. Spälter, and T. A. Strasser, “Grating resonances in air-silica microstructured optical fibers,” Opt. Lett. 24, 1460–1462 (1999).
[Crossref]

1997 (2)

1996 (1)

1979 (1)

J. Sakai, “Simplified bending loss formula for single-mode optical fibers,” Appl. Optics 18, 951–952 (1979).
[Crossref]

1978 (1)

J. Sakai and T. Kimura, “Bending loss of propagation modes in arbitrary-index profile optical fibers,” Appl. Optics 17, 1499–1506 (1978).
[Crossref]

1977 (2)

K. Petermann, “Fundamental mode microbending loss in graded index and w fibers,” Opt. Quantum Electron. 9, 167–175 (1977).
[Crossref]

D. Marcuse, “Loss analysis of sigle-mode fiber splices,” Bell Syst. Tech. J. 56, 703 (1977).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).

Atkin, D. M.

Barkou, S. E.

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]

Birks, T. A.

Bjarklev, A.

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]

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, “Highly Nonlinear Photonic Crystal Fiber with Zero-Dispersion at 1.55 μm”, OFC 2002 Postdeadline Paper, FA9–1.

T. Sørensen, N. A. Mortensen, J. Broeng, A. Bjarklev, T. P. Hansen, E. Knudsen, S. E. B. Libori, H. R. Simonsen, and J. R. Jensen, “Spectral macro-bending loss considerations on photonic crystal fibres,” IEE Proc.-Optoelectron., submitted.

Botton, L. C.

Broeng, J.

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]

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, “Highly Nonlinear Photonic Crystal Fiber with Zero-Dispersion at 1.55 μm”, OFC 2002 Postdeadline Paper, FA9–1.

T. Sørensen, N. A. Mortensen, J. Broeng, A. Bjarklev, T. P. Hansen, E. Knudsen, S. E. B. Libori, H. R. Simonsen, and J. R. Jensen, “Spectral macro-bending loss considerations on photonic crystal fibres,” IEE Proc.-Optoelectron., submitted.

N. A. Mortensen, J. R. Jensen, P. M. W. Skovgaard, and J. Broeng, “Numerical aperture of single-mode photonic crystal fibers,” preprint, http://arxiv.org/abs/physics/0202073

Burdge, G. L.

de Sterke, C. M.

Eggleton, B. J.

Ghatak, A. K.

A. K. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge University Press, Cambridge, 1998).

Hansen, K. P.

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, “Highly Nonlinear Photonic Crystal Fiber with Zero-Dispersion at 1.55 μm”, OFC 2002 Postdeadline Paper, FA9–1.

Hansen, T. P.

T. Sørensen, N. A. Mortensen, J. Broeng, A. Bjarklev, T. P. Hansen, E. Knudsen, S. E. B. Libori, H. R. Simonsen, and J. R. Jensen, “Spectral macro-bending loss considerations on photonic crystal fibres,” IEE Proc.-Optoelectron., submitted.

Jacobsen, C.

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, “Highly Nonlinear Photonic Crystal Fiber with Zero-Dispersion at 1.55 μm”, OFC 2002 Postdeadline Paper, FA9–1.

Jensen, J. R.

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, “Highly Nonlinear Photonic Crystal Fiber with Zero-Dispersion at 1.55 μm”, OFC 2002 Postdeadline Paper, FA9–1.

T. Sørensen, N. A. Mortensen, J. Broeng, A. Bjarklev, T. P. Hansen, E. Knudsen, S. E. B. Libori, H. R. Simonsen, and J. R. Jensen, “Spectral macro-bending loss considerations on photonic crystal fibres,” IEE Proc.-Optoelectron., submitted.

N. A. Mortensen, J. R. Jensen, P. M. W. Skovgaard, and J. Broeng, “Numerical aperture of single-mode photonic crystal fibers,” preprint, http://arxiv.org/abs/physics/0202073

Joannopoulos, J. D.

Johnson, S. G.

Kerbage, C.

Kimura, T.

J. Sakai and T. Kimura, “Bending loss of propagation modes in arbitrary-index profile optical fibers,” Appl. Optics 17, 1499–1506 (1978).
[Crossref]

Knight, J. C.

Knudsen, E.

T. Sørensen, N. A. Mortensen, J. Broeng, A. Bjarklev, T. P. Hansen, E. Knudsen, S. E. B. Libori, H. R. Simonsen, and J. R. Jensen, “Spectral macro-bending loss considerations on photonic crystal fibres,” IEE Proc.-Optoelectron., submitted.

Libori, S. E. B.

T. Sørensen, N. A. Mortensen, J. Broeng, A. Bjarklev, T. P. Hansen, E. Knudsen, S. E. B. Libori, H. R. Simonsen, and J. R. Jensen, “Spectral macro-bending loss considerations on photonic crystal fibres,” IEE Proc.-Optoelectron., submitted.

Marcuse, D.

D. Marcuse, “Loss analysis of sigle-mode fiber splices,” Bell Syst. Tech. J. 56, 703 (1977).

McPhedran, R. C.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: molding the flow of light (Princeton University Press, Princeton, 1995).

Mogilevstev, D.

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]

Mortensen, N. A.

N. A. Mortensen, J. R. Jensen, P. M. W. Skovgaard, and J. Broeng, “Numerical aperture of single-mode photonic crystal fibers,” preprint, http://arxiv.org/abs/physics/0202073

T. Sørensen, N. A. Mortensen, J. Broeng, A. Bjarklev, T. P. Hansen, E. Knudsen, S. E. B. Libori, H. R. Simonsen, and J. R. Jensen, “Spectral macro-bending loss considerations on photonic crystal fibres,” IEE Proc.-Optoelectron., submitted.

Petermann, K.

K. Petermann, “Fundamental mode microbending loss in graded index and w fibers,” Opt. Quantum Electron. 9, 167–175 (1977).
[Crossref]

Petersson, A.

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, “Highly Nonlinear Photonic Crystal Fiber with Zero-Dispersion at 1.55 μm”, OFC 2002 Postdeadline Paper, FA9–1.

Russell, P. S. J.

Sakai, J.

J. Sakai, “Simplified bending loss formula for single-mode optical fibers,” Appl. Optics 18, 951–952 (1979).
[Crossref]

J. Sakai and T. Kimura, “Bending loss of propagation modes in arbitrary-index profile optical fibers,” Appl. Optics 17, 1499–1506 (1978).
[Crossref]

Simonsen, H. R.

T. Sørensen, N. A. Mortensen, J. Broeng, A. Bjarklev, T. P. Hansen, E. Knudsen, S. E. B. Libori, H. R. Simonsen, and J. R. Jensen, “Spectral macro-bending loss considerations on photonic crystal fibres,” IEE Proc.-Optoelectron., submitted.

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, “Highly Nonlinear Photonic Crystal Fiber with Zero-Dispersion at 1.55 μm”, OFC 2002 Postdeadline Paper, FA9–1.

Skovgaard, P. M. W.

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, “Highly Nonlinear Photonic Crystal Fiber with Zero-Dispersion at 1.55 μm”, OFC 2002 Postdeadline Paper, FA9–1.

N. A. Mortensen, J. R. Jensen, P. M. W. Skovgaard, and J. Broeng, “Numerical aperture of single-mode photonic crystal fibers,” preprint, http://arxiv.org/abs/physics/0202073

Sørensen, T.

T. Sørensen, N. A. Mortensen, J. Broeng, A. Bjarklev, T. P. Hansen, E. Knudsen, S. E. B. Libori, H. R. Simonsen, and J. R. Jensen, “Spectral macro-bending loss considerations on photonic crystal fibres,” IEE Proc.-Optoelectron., submitted.

Spälter, S.

Steel, M. J.

Strasser, T. A.

Thyagarajan, K.

A. K. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge University Press, Cambridge, 1998).

Westbrook, P. S.

White, C. A.

White, T. P.

Windeler, R. S.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: molding the flow of light (Princeton University Press, Princeton, 1995).

Appl. Optics (2)

J. Sakai and T. Kimura, “Bending loss of propagation modes in arbitrary-index profile optical fibers,” Appl. Optics 17, 1499–1506 (1978).
[Crossref]

J. Sakai, “Simplified bending loss formula for single-mode optical fibers,” Appl. Optics 18, 951–952 (1979).
[Crossref]

Bell Syst. Tech. J. (1)

D. Marcuse, “Loss analysis of sigle-mode fiber splices,” Bell Syst. Tech. J. 56, 703 (1977).

IEE Proc. (1)

T. Sørensen, N. A. Mortensen, J. Broeng, A. Bjarklev, T. P. Hansen, E. Knudsen, S. E. B. Libori, H. R. Simonsen, and J. R. Jensen, “Spectral macro-bending loss considerations on photonic crystal fibres,” IEE Proc.-Optoelectron., submitted.

J. Lightwave Technol. (1)

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. (5)

Opt. Quantum Electron. (1)

K. Petermann, “Fundamental mode microbending loss in graded index and w fibers,” Opt. Quantum Electron. 9, 167–175 (1977).
[Crossref]

Other (7)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: molding the flow of light (Princeton University Press, Princeton, 1995).

A. K. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge University Press, Cambridge, 1998).

N. A. Mortensen, J. R. Jensen, P. M. W. Skovgaard, and J. Broeng, “Numerical aperture of single-mode photonic crystal fibers,” preprint, http://arxiv.org/abs/physics/0202073

Opt. Express9, 674–779 (2001), http://www.opticsexpress.org/issue.cfm?issue id=124
[PubMed]

J. Opt. A: Pure Appl. Opt.3, S103–S207 (2001).

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, “Highly Nonlinear Photonic Crystal Fiber with Zero-Dispersion at 1.55 μm”, OFC 2002 Postdeadline Paper, FA9–1.

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Figures (5)

Fig. 1.
Fig. 1. Schematic of end-facet of an all-silica photonic crystal fiber. The microstructured cladding consists of air holes of diameter d arranged in a triangular lattice with pitch Λ. The silica core guiding the light is formed by the “missing” air hole (indicated by the dashed circle).

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Fig. 2.
Fig. 2. Comparison of the real intensity I( r ⊥) to the Gaussian intensity IG ( r ⊥) with a width obtained from A eff. The upper left panel shows IG /I, the upper right panel shows I, the lower right panel shows IG , and the lower left panel shows the dielectric function ε with d/Λ = 0.3. The fields are calculated at λ/Λ = 0.48.

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Fig. 3.
Fig. 3. The upper panel shows the effective area as a function of wavelength for different hole sizes. The lower panel shows the same data as in the upper panel, but with the vertical axis scaled by the factor d/Λ and the horizontal axis scaled by the factor (d/Λ)-1.

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Fig. 4.
Fig. 4. The effective area of the second-order mode for d/Λ = 0.5. For short wavelengths A eff,3 ~ Λ2 and the mode is guided in the core. For high wavelengths the mode becomes a cladding mode with effective area approaching the area of the super-cell used in the calculation. The crossing of the dashed line with the horizontal axis indicates the cut-off for the second order mode.

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Fig. 5.
Fig. 5. Diagram illustrating “phases” with single-mode and multi-mode operation. The data points are cut-off values obtained from analysis of the form indicated in Fig. 4.

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Equations (8)

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∇ × [ ε ( r ) ∇ × H ω ( r ) ] = ( ω c ) 2 H ω ( r )
H ω ( r ) = ∑ n α n h n ( r ⊥ ) e ± i β n ( ω ) z
A eff , n ( λ ) = [ ∫ d r ⊥ I n ( r ⊥ ) ] 2 ∫ d r ⊥ I n 2 ( r ⊥ ) ,
γ = n 2 ω c A eff = n 2 2 π λ A eff
NA = sin θ ≃ ( 1 + π A eff λ 2 ) − 1 / 2 .
T ≈ 4 A eff , 1 A eff , 2 ( A eff , 1 + A eff , 2 ) 2
A eff ∝ ( Λ d ) × Λ 2 + O ( λ d ) ,
f ( x ) = α ( x − x 0 ) β , x 0 = 0.45 ,

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