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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 8 — Apr. 19, 2004
  • pp: 1775–1779
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Predicting macrobending loss for large-mode area photonic crystal fibers

M. D. Nielsen, N. A. Mortensen, M. Albertsen, J. R. Folkenberg, A. Bjarklev, and D. Bonacinni  »View Author Affiliations


Optics Express, Vol. 12, Issue 8, pp. 1775-1779 (2004)
http://dx.doi.org/10.1364/OPEX.12.001775


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Abstract

We report on an easy-to-evaluate expression for the prediction of the bend-loss for a large mode area photonic crystal fiber (PCF) with a triangular air-hole lattice. The expression is based on a recently proposed formulation of the V-parameter for a PCF and contains no free parameters. The validity of the expression is verified experimentally for varying fiber parameters as well as bend radius. The typical deviation between the position of the measured and the predicted bend loss edge is within measurement uncertainty.

© 2004 Optical Society of America

1. Introduction

In solid-core photonic crystal fibers (PCF) the air-silica microstructured cladding (see Fig. 1) gives rise to a variety of novel phenomena [1

1. J. C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003). [CrossRef] [PubMed]

] including large-mode area (LMA) endlessly-single mode operation [2

2. T. A. Birks, J. C. Knight, and P. S. J. Russell, “Endlessly single mode photonic crystal fibre,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

]. Though PCFs typically have optical properties very different from that of standard fibers they of course share some of the overall properties such as the susceptibility of the attenuation to macro-bending.

Macrobending-induced attenuation in PCFs has been addressed both experimentally as well as theoretically/numerically in a number of papers [2

2. T. A. Birks, J. C. Knight, and P. S. J. Russell, “Endlessly single mode photonic crystal fibre,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

, 3

3. T. Sørensen, J. Broeng, A. Bjarklev, E. Knudsen, and S. E. B. Libori, “Macro-bending loss properties of photonic crystal fibre,” Electron. Lett. 37, 287–289 (2001). [CrossRef]

, 4

4. T. Sørensen, 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 for photonic crystal fibres,” IEE Proc.-Opt. 149, 206 (2002).

, 5

5. N. A. Mortensen and J. R. Folkenberg, “Low-loss criterion and effective area considerations for photonic crystal fibers,” J. Opt. A: Pure Appl. Opt. 5, 163–167 (2003). [CrossRef]

, 6

6. J. C. Baggett, T. M. Monro, K. Furusawa, V. Finazzi, and D. J. Richardson, “Understanding bending losses in holey optical fibers,” Opt. Commun. 227, 317–335 (2003). [CrossRef]

]. However, predicting bending-loss is no simple task and typically involves a full numerical solution of Maxwell’s equations as well as use of a phenomenological free parameter, e.g. an effective core radius. In this paper we revisit the problem and show how macro-bending loss measurements on high-quality PCFs can be predicted with high accuracy using easy-to-evaluate empirical relations.

2. Predicting macro-bending loss

Predictions of macro-bending induced attenuation in photonic crystal fibers have been made using various approaches including antenna-theory for bent standard fibers [3

3. T. Sørensen, J. Broeng, A. Bjarklev, E. Knudsen, and S. E. B. Libori, “Macro-bending loss properties of photonic crystal fibre,” Electron. Lett. 37, 287–289 (2001). [CrossRef]

, 4

4. T. Sørensen, 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 for photonic crystal fibres,” IEE Proc.-Opt. 149, 206 (2002).

], coupling-length criteria [2

2. T. A. Birks, J. C. Knight, and P. S. J. Russell, “Endlessly single mode photonic crystal fibre,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

, 5

5. N. A. Mortensen and J. R. Folkenberg, “Low-loss criterion and effective area considerations for photonic crystal fibers,” J. Opt. A: Pure Appl. Opt. 5, 163–167 (2003). [CrossRef]

], and phenomenological models within the tilted-index representation [6

6. J. C. Baggett, T. M. Monro, K. Furusawa, V. Finazzi, and D. J. Richardson, “Understanding bending losses in holey optical fibers,” Opt. Commun. 227, 317–335 (2003). [CrossRef]

]. Here, we also apply the antenna-theory of Sakai and Kimura [7

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

, 8

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

], but contrary to Refs. [3

3. T. Sørensen, J. Broeng, A. Bjarklev, E. Knudsen, and S. E. B. Libori, “Macro-bending loss properties of photonic crystal fibre,” Electron. Lett. 37, 287–289 (2001). [CrossRef]

, 4

4. T. Sørensen, 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 for photonic crystal fibres,” IEE Proc.-Opt. 149, 206 (2002).

] we make a full transformation of standard-fiber parameters such as Δ, W, and V [9

9. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).

] to fiber parameters appropriate to high-index contrast PCFs with a triangular arrangement of air holes. In the large-mode area limit we get (see Appendix)

αΛ186π1nSΛ2AeffλΛF(16π21nS2RΛ(λΛ)2VPCF3),F(x)=x12exp(x),
(1)

for the power-decay, P(z)=P(0)exp(-2α z), along the fiber. For a conversion to a dB-scale α should be multiplied by 20×log10(e) ≃8.686. In Eq. (1), R is the bending radius, A eff is the effective area [10

10. N. A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express 10, 341–348 (2002). URL http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341. [CrossRef] [PubMed]

], nS is the index of silica, and

VPCF=Λβ2βcl2
(2)

is the recently introduced effective V-parameter of a PCF [11

11. N. A. Mortensen, J. R. Folkenberg, M. D. Nielsen, and K. P. Hansen, “Modal cut-off and the V-parameter in photonic crystal fibers,” Opt. Lett. 28, 1879–1881 (2003). [CrossRef] [PubMed]

]. The strength of our formulation is that it contains no free parameters (such as an arbitrary core radius) and furthermore empirical expressions, depending only on λ/Λ and d/Λ, have been given recently for both A eff and V PCF [12

12. M. D. Nielsen, N. A. Mortensen, J. R. Folkenberg, and A. Bjarklev, “Mode-Field Radius of Photonic Crystal Fibers Expressed by the V-parameter,” Opt. Lett. 28, 2309–2311 (2003). [CrossRef] [PubMed]

, 13

13. M. D. Nielsen and N. A. Mortensen, “Photonic crystal fiber design based on the V-parameter,” Opt. Express 11, 2762–2768 (2003). URL http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-21-2762. [CrossRef] [PubMed]

].

From the function F(x) we may derive the parametric dependence of the critical bending radius R*. The function increases dramatically when the argument is less than unity and thus we may define a critical bending radius from x~1 where F~1/e. Typically the PCF is operated close to cut-off where V*PCF=π [11

11. N. A. Mortensen, J. R. Folkenberg, M. D. Nielsen, and K. P. Hansen, “Modal cut-off and the V-parameter in photonic crystal fibers,” Opt. Lett. 28, 1879–1881 (2003). [CrossRef] [PubMed]

] so that the argument may be written as

Fig. 1. Structural data for the LMA fibers which all have a cross-section with a triangular arrangement of air-holes running along the full length of the fiber.
Fig. 2. Macro-bending loss for the LMA-20 fiber for bending radii of R=8 cm (red, solid curve) and R=16 cm (black, solid curve). Predictions of Eq. (1) are also included (dashed curves).
π316π21nS2~14R*Λ(λΛ)2~1R*Λ3λ2
(3)

This dependence was first reported and experimentally confirmed by Birks et al. [2

2. T. A. Birks, J. C. Knight, and P. S. J. Russell, “Endlessly single mode photonic crystal fibre,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

] and recently a pre-factor of order unity was also found experimentally in Ref. [5

5. N. A. Mortensen and J. R. Folkenberg, “Low-loss criterion and effective area considerations for photonic crystal fibers,” J. Opt. A: Pure Appl. Opt. 5, 163–167 (2003). [CrossRef]

].

Fig. 3. Macro-bending loss for the LMA-25 fiber for bending radius of R=16 cm (solid curve). Predictions of Eq. (1) are also included (dashed curve).
Fig. 4. Macro-bending loss for the LMA-35 fiber for bending radius of R=16 cm (solid curve). Predictions of Eq. (1) are also included (dashed curve).

3. Experimental results

We have fabricated three LMA fibers by the stack-and-pull method and characterized them using the conventional cut-back technique. All three fibers have a triangular air-hole array and a solid core formed by a single missing air-hole in the center of the structure, see Fig. 1.

For the LMA-20 macro-bending loss has been measured for bending radii of R=8 cm and R=16 cm and the results are shown in Fig. 2. The predictions of Eq. (1) are also included. It is emphasized that the predictions are based on the empirical relations for A eff and V PCF provided in Refs. [12

12. M. D. Nielsen, N. A. Mortensen, J. R. Folkenberg, and A. Bjarklev, “Mode-Field Radius of Photonic Crystal Fibers Expressed by the V-parameter,” Opt. Lett. 28, 2309–2311 (2003). [CrossRef] [PubMed]

] and [13

13. M. D. Nielsen and N. A. Mortensen, “Photonic crystal fiber design based on the V-parameter,” Opt. Express 11, 2762–2768 (2003). URL http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-21-2762. [CrossRef] [PubMed]

] respectively and therefore do not require any numerical calculations. Similar results are shown in Figs. 3 and 4 for the LMA-25 and LMA-35 fibers, respectively.

4. Discussion and conclusion

The PCF, in theory, exhibits both a short and long-wavelength bend-edge. However, the results presented here only indicate a short-wavelength bend-edge. The reason for this is that the long-wavelength bend-edge occurs for λ≫Λ/2 [3

3. T. Sørensen, J. Broeng, A. Bjarklev, E. Knudsen, and S. E. B. Libori, “Macro-bending loss properties of photonic crystal fibre,” Electron. Lett. 37, 287–289 (2001). [CrossRef]

]. For typical LMA-PCFs it is therefor located in the non-transparent wavelength regime of silica.

In conclusion we have demonstrated that macro-bending loss measurements on high-quality PCFs can be predicted with good accuracy using easy-to-evaluate empirical relations with only d and Λ as input parameters. Since macro-bending attenuation for many purposes and applications is the limiting factor we believe that the present results will be useful in practical designs of optical systems employing photonic crystal fibers.

Appendix

The starting point is the bending-loss formula for a Gaussian mode in a standard-fiber [7

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

, 8

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

]

α=π81AeffρWexp(43RρΔV2W3)WRρ+V22ΔW
(4)

where A eff is the effective area, ρ is the core radius, R is the bending radius, and the standard-fiber parameters are given by [7

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

, 9

9. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).

]

Δ=sin2θc2,V=βρsinθc,W=ρβ2βcl2.
(5)

Substituting these parameters into Eq. (4) we get

αΛ182π3Λ2Aeff1βΛF(23RΛVPCF3(βΛ)2)
(6)

in the relevant limit where Rρ. Here, F and V PCF in Eqs. (1) and (2) have been introduced. For large-mode area fibers we make a further simplification for the isolated propagation constant; using that β=2πn eff/λ≃2πnS/λ we arrive at Eq. (1).

Acknowledgments

M. D. Nielsen acknowledges financial support by the Danish Academy of Technical Sciences.

References and links

1.

J. C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003). [CrossRef] [PubMed]

2.

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

3.

T. Sørensen, J. Broeng, A. Bjarklev, E. Knudsen, and S. E. B. Libori, “Macro-bending loss properties of photonic crystal fibre,” Electron. Lett. 37, 287–289 (2001). [CrossRef]

4.

T. Sørensen, 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 for photonic crystal fibres,” IEE Proc.-Opt. 149, 206 (2002).

5.

N. A. Mortensen and J. R. Folkenberg, “Low-loss criterion and effective area considerations for photonic crystal fibers,” J. Opt. A: Pure Appl. Opt. 5, 163–167 (2003). [CrossRef]

6.

J. C. Baggett, T. M. Monro, K. Furusawa, V. Finazzi, and D. J. Richardson, “Understanding bending losses in holey optical fibers,” Opt. Commun. 227, 317–335 (2003). [CrossRef]

7.

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

8.

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

9.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).

10.

N. A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express 10, 341–348 (2002). URL http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341. [CrossRef] [PubMed]

11.

N. A. Mortensen, J. R. Folkenberg, M. D. Nielsen, and K. P. Hansen, “Modal cut-off and the V-parameter in photonic crystal fibers,” Opt. Lett. 28, 1879–1881 (2003). [CrossRef] [PubMed]

12.

M. D. Nielsen, N. A. Mortensen, J. R. Folkenberg, and A. Bjarklev, “Mode-Field Radius of Photonic Crystal Fibers Expressed by the V-parameter,” Opt. Lett. 28, 2309–2311 (2003). [CrossRef] [PubMed]

13.

M. D. Nielsen and N. A. Mortensen, “Photonic crystal fiber design based on the V-parameter,” Opt. Express 11, 2762–2768 (2003). URL http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-21-2762. [CrossRef] [PubMed]

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.2400) Fiber optics and optical communications : Fiber properties
(060.2430) Fiber optics and optical communications : Fibers, single-mode

ToC Category:
Research Papers

History
Original Manuscript: March 9, 2004
Revised Manuscript: April 7, 2004
Published: April 19, 2004

Citation
M. Nielsen, N. Mortensen, M. Albertsen, J. Folkenberg, A. Bjarklev, and D. Bonacinni, "Predicting macrobending loss for large-mode area photonic crystal fibers," Opt. Express 12, 1775-1779 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-8-1775


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References

  1. J. C. Knight, �??Photonic crystal fibres,�?? Nature 424, 847�??851 (2003). [CrossRef] [PubMed]
  2. T. A. Birks, J. C. Knight, and P. S. J. Russell, �??Endlessly single mode photonic crystal fibre,�?? Opt. Lett. 22, 961�??963 (1997). [CrossRef] [PubMed]
  3. T. Sørensen, J. Broeng, A. Bjarklev, E. Knudsen, and S. E. B. Libori, �??Macro-bending loss properties of photonic crystal fibre,�?? Electron. Lett. 37, 287�??289 (2001). [CrossRef]
  4. T. Sørensen, 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 for photonic crystal fibres,�?? IEE Proc.-Opt. 149, 206 (2002).
  5. N. A. Mortensen and J. R. Folkenberg, �??Low-loss criterion and effective area considerations for photonic crystal fibers,�?? J. Opt. A: Pure Appl. Opt. 5, 163�??167 (2003). [CrossRef]
  6. J. C. Baggett, T. M. Monro, K. Furusawa, V. Finazzi, and D. J. Richardson, �??Understanding bending losses in holey optical fibers,�?? Opt. Commun. 227, 317�??335 (2003). [CrossRef]
  7. J. Sakai and T. Kimura, �??Bending loss of propagation modes in arbitrary-index profile optical fibers,�?? Appl. Opt. 17, 1499�??1506 (1978). [CrossRef] [PubMed]
  8. J. Sakai, �??Simplified bending loss formula for single-mode optical fibers,�?? Appl. Opt. 18, 951�??952 (1979). [CrossRef] [PubMed]
  9. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).
  10. N. A. Mortensen, �??Effective area of photonic crystal fibers,�?? Opt. Express 10, 341�??348 (2002). <a href= " http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341</a>. [CrossRef] [PubMed]
  11. N. A. Mortensen, J. R. Folkenberg, M. D. Nielsen, and K. P. Hansen, �??Modal cut-off and the V�??parameter in photonic crystal fibers,�?? Opt. Lett. 28, 1879�??1881 (2003). [CrossRef] [PubMed]
  12. M. D. Nielsen, N. A. Mortensen, J. R. Folkenberg, and A. Bjarklev, �??Mode-Field Radius of Photonic Crystal Fibers Expressed by the V�??parameter,�?? Opt. Lett. 28, 2309�??2311 (2003). [CrossRef] [PubMed]
  13. M. D. Nielsen and N. A. Mortensen, �??Photonic crystal fiber design based on the V�??parameter,�?? Opt. Express 11, 2762�??2768 (2003). <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-21-2762">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-21-2762</a>. [CrossRef] [PubMed]

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