## Effective area of photonic crystal fibers

Optics Express, Vol. 10, Issue 7, pp. 341-348 (2002)

http://dx.doi.org/10.1364/OE.10.000341

Acrobat PDF (443 KB)

### 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

## 1 Introduction

*e*.

*g*. Refs. [1

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

3. J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. **21**, 1547–1549 (1996). [CrossRef] [PubMed]

4. J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding: errata,” Opt. Lett. **22**, 484–485 (1997). [CrossRef] [PubMed]

*d*arranged in a triangular lattice with pitch Λ, see Fig. 1. For a review of the basic operation we refer to Ref. [5

5. 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]

15. 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]

*d*<

*d*

^{*}~ 0.45Λ [5

5. 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]

^{*}. We demonstrate that the effective area of the second-order mode is a useful concept in determining this cut-off for a given hole size

*d*. For

*d*<

*d*

^{*}we recover the endlessly single-mode regime with λ

^{*}→ 0.

## 2 Eigen-modes and effective areas

**-field is governed by the general wave equation (see**

*H**e*.

*g*. Ref. [16])

*ε*is the dielectric function and

*ω*is the frequency of the harmonic mode,

*(*

**H**ω*,*

**r***t*) =

*(*

**H**_{ω}*)*

**r***e*

^{±iωt}.

*z*-axis) and for

*ε*(

*) =*

**r***ε*(

**r**_{⊥}) the solution is of the form

*is the transverse part of the*

**h**n*n*th eigen-mode and

*β*

_{n}is the corresponding propagation constant at frequency

*ω*. Often one will specify the free-space wavelength λ rather than the frequency

*ω*=

*c*(2

*π*/λ).

*n*th eigen-mode is given by [6]

*I*

_{n}(

**r**_{⊥}) = |

*(*

*h*n

**r**_{⊥})|

^{2}is the intensity distribution. It is easy to show that for a Gaussian mode

*(*

**h**

**r**_{⊥}) ∝

*e*

^{-(r⊥/ω)2}of width

*ω*the effective area is

*A*

_{eff}=

*πω*

^{2}. Applying Eq. (3) to close-to-Gaussian modes in some way corresponds to a Gaussian fit averaged over all angular directions. In Fig. 2 we illustrate this by an example. For

*d*/Λ = 0.3 and λ/Λ = 0.48 we compare the intensity

*I*of the fundamental mode (

*n*= 1 or 2) to the intensity

*I*

_{G}of the corresponding Gaussian with the width calculated from

*A*

_{eff}. As seen the over-all intensity distribution is reasonably described by the Gaussian with a width calculated from the effective area.

*e*.

*g*. Refs. [3

3. J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. **21**, 1547–1549 (1996). [CrossRef] [PubMed]

7. 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]

8. B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, and G. L. Burdge, “Cladding-Mode-Resonances in Air-Silica Microcstructed Optical Fibers,” J. Lightwave Technol. **18**, 1084–1100 (2000). [CrossRef]

### 2.1 Non-linearity coefficient

*γ*is given by [6]

*n*

_{2}is the nonlinear-index coefficient in the nonlinear part of the refractive index,

*δn*=

*n*

_{2}|

*|*

**E**^{2}. Knowledge of

*A*

_{eff}is thus an important starting point in the understanding of non-linear phenomena in PCFs. Due to the high index contrast between silica and air the PCF technology offers the possibility of a much tighter mode confinement (over a wide wavelength range) and thereby a lower effective area compared to standard-fiber technology. Furthermore, the microstructured cladding of the PCFs also allows for zero-dispersion engineering. For a recent demonstration of a highly nonlinear PCF with a zero-dispersion at λ = 1.55

*μ*m see Ref. [9].

### 2.2 Numerical aperture

*ω*one has the standard approximate expression tan

*θ*≃ λ/

*πω*for the half-divergence angle

*θ*of the light radiated from the end-facet of the fiber [17]. The corresponding numerical aperture can then be expressed as

14. 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

### 2.3 Macro-bending loss

*α*the Sakai–Kimura formula [18

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

*P*, where

*A*

_{e}is the amplitude coefficient of the field in the cladding and

*P*the power carried by the fundamental mode. The Gaussian approximation gives

*P*= 1/

*A*

_{eff}[19

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

### 2.4 Splicing loss

*A*

_{eff,1}and

*A*

_{eff,2}will have a power transmission coefficient

*T*< 1 given approximately by

## 3 Numerical results

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 [CrossRef]

*ε*= 1 for the air holes and

*ε*= (1.444)

^{2}= 2.085 for the silica. Ignoring the frequency dependence of the latter the wave equation, Eq. (1) becomes scale-invariant [16] and all the results to be presented can thus be scaled to the desired value of Λ.

### 3.1 Fundamental mode

^{2}with a prefactor which depends slightly on the air hole size. In fact, scaling the results by a factor

*d*/Λ we find numerical evidence that

*d*/Λ is given (which is often the case during stack-and-pull production of PCFs) a given effective area can be obtained by scaling the fiber structure,

*i*.

*e*. the pitch Λ. The Corning SMF28 standard fiber has an effective area

*A*

_{eff}~ 86

*μ*m

^{2}at λ = 1550 nm and Fig. 3 suggests that a comparable effective area can be realized by the PCF technology using

*e*.

*g*.

*d*/Λ ~ 0.25 and Λ ~ 6.5

*μm*.

### 3.2 Second-order mode

*n*= 3) for a PCF with

*d*/Λ = 0.5 air holes. For short wavelengths the effective area is of the order Λ

^{2}with the mode confined to the core of the PCF. At long wavelengths the effective area diverges corresponding to a delocalized cladding mode. Even though the transition to a cladding mode is not abrupt the high slope of the effective area allows for a rather accurate introduction of a cut-off wavelength λ

^{*}. This cut-off wavelength is indicated by the crossing of the dashed line with the horizontal axis.

### 3.3 Single-mode versus multi-mode operation

*d*<

*d*

^{*}, they can even be endlessly single-mode [3

3. J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. **21**, 1547–1549 (1996). [CrossRef] [PubMed]

5. 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]

*d*

^{*}~ 0.45Λ was suggested from standard-fiber considerations.

*d*>

*d*

^{*}a second-order mode is guided for wavelengths shorter than a certain cut-off and above this cut-off the PCF is single-mode. This cut-off of the second-order mode can be found from studies of the corresponding effective area; below the cut-off the effective area

*A*

_{eff}is finite and comparable to Λ

^{2}and above the cut-off the effective area diverges (in a super-cell calculation it approaches the area of the super-cell).

*d*/Λ = 0.5 it is seen how we for wavelengths shorter than λ

^{*}~ 0.35Λ have a multi-mode PCF whereas the PCF is in the single-mode regime for λ > λ

^{*}. Carrying out the same analysis for different hole-sizes allows for constructing a “phase” diagram, see Fig. 5. The data points indicate calculated cut-off wavelengths and the solid line is a fit to the function

*α*≃ 1.34 and

*β*≃ 0.45.

*d*<

*d*

^{*}~ 0.45Λ [5

**5**, 305–330 (1999). [CrossRef]

*d*

^{*}but it should be emphasized that the numerical efforts needed to resolve the cut-off increase dramatically when

*d*approaches

*d*

^{*}.

## 4 Conclusion

*A*

_{eff}of photonic crystal fibers (PCFs) with a triangular air-hole lattice in the cladding. Based on extensive numerical studies of PCFs with varying air hole size we have constructed a map of the effective area which can be scaled to a desired value of the pitch. We have also utilized the concept of effective area to calculate the “phase” boundary between the regimes with single-mode and multi-mode operation. In this work we have studied PCFs with a triangular air-hole lattice cladding, but we emphasize that the approach applies to microstructured fibers in general.

## Acknowledgement

## References and links

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

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

3. | J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. |

4. | J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding: errata,” Opt. Lett. |

5. | J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, “Photonic crystal fibers: A new class of optical waveguides,” Opt. Fiber Technol. |

6. | G. P. Agrawal, |

7. | 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. |

8. | B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, and G. L. Burdge, “Cladding-Mode-Resonances in Air-Silica Microcstructed Optical Fibers,” J. Lightwave Technol. |

9. | 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 |

10. | T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botton, and M. J. Steel, “Confinement losses in microstructured optical fibers,” Opt. Lett. |

11. | K. Petermann, “Fundamental mode microbending loss in graded index and w fibers,” Opt. Quantum Electron. |

12. | 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. |

13. | D. Marcuse, “Loss analysis of sigle-mode fiber splices,” Bell Syst. Tech. J. |

14. | 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 |

15. | T. A. Birks, J. C. Knight, and P. S. J. Russell, “Endlessly single mode photonic crystal fibre,” Opt. Lett. |

16. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

17. | A. K. Ghatak and K. Thyagarajan, |

18. | J. Sakai and T. Kimura, “Bending loss of propagation modes in arbitrary-index profile optical fibers,” Appl. Optics |

19. | J. Sakai, “Simplified bending loss formula for single-mode optical fibers,” Appl. Optics |

20. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

**OCIS Codes**

(060.2430) Fiber optics and optical communications : Fibers, single-mode

(230.3990) Optical devices : Micro-optical devices

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 14, 2002

Revised Manuscript: April 2, 2002

Published: April 8, 2002

**Citation**

Niels Asger Mortensen, "Effective area of photonic crystal fibers," Opt. Express **10**, 341-348 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-7-341

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### References

- Opt. Express 9, 674�779 (2001), <a href="http://www.opticsexpress.org/issue.cfm?issue_id=124">http://www.opticsexpress.org/issue.cfm?issue_id=124</a> [PubMed]
- J. Opt. A: Pure Appl. Opt. 3, S103�S207 (2001).
- J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin, �All-silica single-mode optical fiber with photonic crystal cladding,� Opt. Lett. 21, 1547�1549 (1996). [CrossRef] [PubMed]
- J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin, �All-silica single-mode optical fiber with photonic crystal cladding: errata,� Opt. Lett. 22, 484�485 (1997). [CrossRef] [PubMed]
- 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]
- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).
- B. J. Eggleton, P. S.Westbrook, R. S.Windeler, S. Spalter, and T. A. Strasser, �Grating resonances in air-silica microstructured optical fibers,� Opt. Lett. 24, 1460�1462 (1999). [CrossRef]
- B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, G. L. Burdge, �Cladding-Mode-Resonances in Air-Silica Microcstructed Optical Fibers,� J. Lightwave Technol. 18, 1084�1100 (2000). [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 �m,� OFC 2002 Postdeadline Paper, (Optical Society of America, Washington, D.C., 2002) FA9-1.
- T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botton, and M. J. Steel, �Confinement losses in microstructured optical fibers,� Opt. Lett. 26, 1660�1662 (2001). [CrossRef]
- K. Petermann, �Fundamental mode microbending loss in graded index and w fibers,� Opt. Quantum Electron. 9, 167�175 (1977). [CrossRef]
- T. S�rensen, 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.
- D. Marcuse, �Loss analysis of sigle-mode fiber splices,� Bell Syst. Tech. J. 56, 703 (1977).
- N. A. Mortensen, J. R. Jensen, P. M. W. Skovgaard, and J. Broeng, �Numerical aperture of single-mode photonic crystal fibers,� preprint, <a href="http://arxiv.org/abs/physics/0202073">http://arxiv.org/abs/physics/0202073</a>
- 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]
- 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).
- J. Sakai and T. Kimura, �Bending loss of propagation modes in arbitrary-index profile optical fibers,� Appl. Opt. 17, 1499�1506 (1978). [CrossRef]
- J. Sakai, �Simplified bending loss formula for single-mode optical fibers,� Appl. Opt. 18, 951�952 (1979). [CrossRef]
- 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), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173</a> [CrossRef]

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