## Analysis of light scattering from surface roughness in hollow-core photonic bandgap fibers |

Optics Express, Vol. 20, Issue 19, pp. 20980-20991 (2012)

http://dx.doi.org/10.1364/OE.20.020980

Acrobat PDF (8772 KB)

### Abstract

We present a theoretical method for analyzing radiation loss from surface roughness scattering in hollow-core photonic bandgap fibers (HC-PBGFs). We treat the scattering process as induced dipole radiation and combine statistical information about surface roughness, mode field distribution and fibre geometry to accurately describe the far-field scattering distribution and loss in fibers with an arbitrary cross-sectional distribution of air holes of any shape. The predicted angular scattering distribution, total scattering loss and the loss wavelength dependence are all shown to agree well with reported experimental data. Our method yields a simpler result than that obtained by more complex approaches and is to the best of our knowledge the first successful attempt to accurately describe roughness scattering in HC-PBGFs.

© 2012 OSA

## 1. Introduction

1. P. St. J. Russell, “Photonic crystal fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

4. M. N. Petrovich, F. Poletti, A. van Brakel, and D. J. Richardson, “Robustly single mode hollow core photonic bandgap fiber,” Opt. Express **16**, 4337–4346 (2008). [CrossRef] [PubMed]

*dB/km*[5

5. B. J. Mangan, L. Farr, A. Langford, P. J. Roberts, D. P. Williams, F. Couny, M. Lawman, M. Mason, S. Coupland, R. Flea, H. Sabert, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Low loss (1.7 dB/km) hollow core photonic bandgap fiber,” in Proceedings of Optical Fiber Communication Conference (2004), paper PDP24.

6. P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Williams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. S. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express **13**236–244 (2005). [CrossRef] [PubMed]

4. M. N. Petrovich, F. Poletti, A. van Brakel, and D. J. Richardson, “Robustly single mode hollow core photonic bandgap fiber,” Opt. Express **16**, 4337–4346 (2008). [CrossRef] [PubMed]

6. P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Williams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. S. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express **13**236–244 (2005). [CrossRef] [PubMed]

*et al.*[9

9. P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, T. A. Birks, J. C. Knight, and P. S. Russell, “Loss in solid-core photonic crystal fibers due to interface roughness scattering,” Opt. Express **13**, 7779–7793 (2005). [CrossRef] [PubMed]

*et al*[10

10. M.-C. Phan-Huy, J.-M. Moison, J. A. Levenson, S. Richard, G. Melin, M. Douay, and Y. Quiquempois, “Surface roughness and light scattering in a small effective area microstructured fiber,” J. Lightwave. Technol. **27**, 1597–1604 (2009). [CrossRef]

*et al.*[11

11. V. Dangui, M. J. F. Digonnet, and G. S. Kino, “Modeling of the propagation loss and backscattering in air-core photonic-bandgap fibers,” J. Lightwave Technol. **17**, 3783–3789 (2009). [CrossRef]

12. E. G. Rawson, “Theory Of scattering by finite dielectric needles illuminated parallel to their axes,” J. Opt. Soc. Am. **62**, 1284–1286 (1972). [CrossRef]

13. E. G. Rawson, “Analysis of scattering from fiber waveguides with irregular core surfaces,” Appl. Opt. **13**, 2370–2377 (1974). [CrossRef] [PubMed]

*et al.*[14

14. P. Mazumder, S. L. Logunov, and S. Raghavan, “Analysis of excess scattering in optical fibers,” J. Appl. Phys. **96**, 4042–4049 (2004). [CrossRef]

*rms*) roughness but different power density spectra can yield very different attenuation values.

## 2. Formulation

*dV*= |

*f*(

*s*,

*z*)|

*dsdz*which is excited by the incident modal field. This approximation is justified by the fact that the

*rms*roughness resulting from frozen-in SCWs is of the order of 0.1

*nm*, much smaller than the wavelength of light. Since the normal component of the electric displacement

**D**and the parallel component of the electric field

**E**are continuous at all interfaces, the induced dipole moment is generally given as: where

*α*

_{||}and

*γ*

_{⊥}are the polarizability tensors [15

15. S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B **81**, 283–293 (2005). [CrossRef]

*et al.*[15

15. S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B **81**, 283–293 (2005). [CrossRef]

*α*and

*γ*for different dipole shapes and index contrasts. From their results, the effects of neglecting the exact shape of the dipole results in an error of less than 1% for silica-air interfaces at infrared wavelengths (index contrasts of ∼ 0.5 and ∼ 2). Further neglecting the effects of the discontinuity of the normal component of the electric field results in an additional error which is well below 5%. The polarizability additionally depends on the sign of the roughness defect. For example, when the perturbation is such that glass protrudes into air we speak of positive roughness, otherwise, the roughness is said to be negative. We take the polarizability

*α*

_{+}of a positive roughness “bump” as that of a glass sphere suspended in air and that of a negative one

*α*

_{−}as the polarizability of an air sphere in a glass background. The two differ in both sign and magnitude [15

15. S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B **81**, 283–293 (2005). [CrossRef]

*f*is random with zero average, we will neglect all the effects above and write the induced dipole moment in terms of the incident electric field at the interface as: where

*α*

_{0}=

*ε*

_{0}(|

*α*

_{+}| + |

*α*

_{−}|)/2 ≈ 0.715

*ε*

_{0}for silica and air,

*ε*

_{0}being the free-space permittivity.

*L*of the perturbed waveguide. This section is chosen so that it contains all the roughness spectral components of interest, but is also short enough for the incident field

*E*

_{0}to be approximately constant throughout. At point

*P*(

*ϑ*,

*ϕ*) (see Fig. 2) on a distant sphere of radius

*R*, the scattered field element from a single dipole located at the origin is [16]: where

*k*

_{0}is the free-space propagation constant. The total field at

*P*is obtained by adding together all the scattered field elements while taking into account the relative phases. Taking as reference a ray that would be scattered from the origin, the phase difference for a ray scattered from position (

*s*,

*z*) can be calculated with the help of Fig. 2 as:

*et al.*[14

14. P. Mazumder, S. L. Logunov, and S. Raghavan, “Analysis of excess scattering in optical fibers,” J. Appl. Phys. **96**, 4042–4049 (2004). [CrossRef]

*f*(

*s*,

_{i}*z*) to

*f*(

*z*) only. The

*z*–dependent part of the above expression can be rewritten as: where

*F*̃ is the spatial Fourier transform of the roughness distribution. Expanding the incident field at the scattering interface in terms of its components perpendicular (

*E*

_{0⊥}) and parallel(

*E*

_{0||}) to the

*Pz*plane and carrying out the vector product, Eq. (6) becomes:

**u**and

**v**are the unit vectors shown in Fig. 2. This result shows that the far-field distribution of scattered light is the product of a component depending on the roughness spectrum and a quantity that depends on the incident electric field at the interface, the geometry and optical properties of the fibre. Indeed, each spatial frequency component of the roughness scatters light in a specific direction

*ϕ*, a fact that conforms with the more complicated analysis of coupled-mode theory or volume-current methods [7, 8]. This also implies that only the spatial frequencies satisfying

*β*−

*k*

_{0}≤

*κ*≤

*β*+

*k*

_{0}contribute to radiation loss.

*R*(

*u*) and the power spectral density (PSD) which are a Fourier transform pair need to be introduced. The power spectral density of the roughness is defined as [26

26. J. M. Elson and J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. **69**31–47 (1979). [CrossRef]

*L*(

*ϕ*) the fraction of light scattered in the polar direction

*ϕ*that is actually lost. If

*P*

_{0}is the total optical power carried by the incident guided mode, the exponential loss coefficient is obtained by integration as:

*n*. The measured scattering angle

_{gl}*ϕ*is therefore obtained through Snell’s law

_{m}*n*cos

_{gl}*ϕ*= cos

_{m}*ϕ*.

## 3. Application to single-mode fibers

17. A. W. Snyder, “Radiation losses due to variations of radius on dielectric or optical fibers,” IEEE Trans. Microwave Theory Tech. **MT18**(9) 608–615 (1970). [CrossRef]

18. D. Marcuse, “Radiation losses of HE11 mode of a fiber with sinusoidally perturbed core boundary,” Appl. Opt. **14**, 3021–3025 (1975). [CrossRef] [PubMed]

*r*=

*a*+

*bsin*(2

*πz*/Λ) and refractive index

*n*

_{1}, and a cladding with refractive index

*n*

_{2}. When the sinusoidal perturbation is small enough to satisfy the assumptions in our treatment (

*b*≪

*λ*), the scattering loss suffered by the

*x*–polarized fundamental

*HE*

_{11}mode can be calculated using equation (11). Since the cladding is no longer air, the relative phase difference of equation (4) needs to be adjusted by replacing

*k*

_{0}with

*k*

_{0}

*n*

_{2}. For a pure sinusoidal perturbation over a long fiber section (

*L*≫ Λ), the power spectral density is Thus, not surprisingly, scattering only occurs in the direction that satisfies the phase-matching condition: Rather than showing the radiation pattern which is sharply peaked at this angle due to the delta PSD, it is more interesting to plot the loss coefficient as a function of the escaping angle when the spatial wavelength of the perturbation is changed. Such an example is shown in Fig. 3 where we have used

*n*

_{1}= 1.46,

*n*

_{2}= 1.458,

*a*= 5

*μm*and the wavelength of

*λ*= 1

*μm*is such that the waveguide is just below the cut-off of the first higher-order mode (

*V*= 2.4). To show the compatibility of our method with common numerical tools, the field components of the fundamental mode were calculated using a fully vectorial finite element method, and equation (8) was evaluated to obtain the scattered field. With

*L*(

*ϕ*) = 1 for all angles larger than the critical angle and

*L*(

*ϕ*) = 0 otherwise, the normalized scattering loss coefficient shown in Fig. 3 is identical to that obtained by Marcuse using a far more complex coupled mode theory [7]. This not only validates our simple approach but also shows that it can be implemented in combination with any arbitrary mode-solving numerical tool.

## 4. Roughness scattering in HC-PBGFs

19. F. Poletti, N. G. R. Broderick, D. J. Richardson, and T. M. Monro, “The effect of core asymmetries on the polarization properties of hollow core photonic bandgap fibers,” Opt. Express **13**9115–9124 (2005). [CrossRef] [PubMed]

### 4.1. Surface roughness due to frozen-in surface capillary waves

20. J. Jäckle and K. Kawasaki, “Intrinsic roughness of glass surfaces,” J. Phys. Condens. Matter **7**, 4351–4358 (1995). [CrossRef]

*κ*is the two dimensional surface wavevector,

*ρ*the glass density and

*g*the gravity constant.

*k*represents Boltzmann’s constant,

_{B}*T*the glass transition temperature and

_{g}*γ*the surface tension of the glass, which for pure silica are ∼ 1500

*K*and ∼ 1

*J.m*

^{−2}respectively.

6. P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Williams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. S. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express **13**236–244 (2005). [CrossRef] [PubMed]

*κ*| for the surface roughness to be physically acceptable, and gravity imposes another lower wavevector cut-off determined by the capillary length, which is of the order of ∼ 4

_{u}*mm*[21

21. T. Sarlat, A. Lelarge, E Søndergård, and D. Vandembroucq, “Frozen capillary waves on glass surfaces: an AFM study,” Eur. Phys. J. B **54**, 121–126 (2006). [CrossRef]

*z*which also requires the existence of low and high spatial frequency cut-offs to be physically acceptable. Since the high frequency cut-off is estimated to be of the order of a few molecular lengths (∼ 0.5

*nm*), it is of no particular concern here as the highest frequency

*β*+

*k*

_{0}contributing to scattering falls well below this value. In HC-PBGFs, the origins of a low-frequency cut-off are not yet known. Atomic Force microscopy (AFM) measurements on the inner surface of air-holes in HC-PBGFs and holey fibers revealed a roughness PSD consistent with frozen-in surface capillary waves of Eq. (16) between

*κ*= 0.2

*μm*

^{−1}and

*κ*= 30

*μm*

^{−1}but could not measure the low spatial frequency end of the spectrum [6

**13**236–244 (2005). [CrossRef] [PubMed]

10. M.-C. Phan-Huy, J.-M. Moison, J. A. Levenson, S. Richard, G. Melin, M. Douay, and Y. Quiquempois, “Surface roughness and light scattering in a small effective area microstructured fiber,” J. Lightwave. Technol. **27**, 1597–1604 (2009). [CrossRef]

*κ*= 0.1

_{c}*μm*

^{−1}have the same PSD value

*S*(

*κ*).

_{c}### 4.2. Angular distribution of scattered power and loss

22. N. V. Wheeler, M. N. Petrovich, R. Slavík, N. Baddela, E. Numkam Fokoua, J. R. Hayes, D. R. Gray, F. Poletti, and D. J. Richardson “Wide-bandwidth, low-loss, 19-cell hollow core photonic band gap fiber and its potential for low latency data transmission,” in Proceedings of Optical Fiber Communication Conference (2012), paper PDP5A.2.

*μm*and

*d*/Λ ∼ 0.975. Figure 4(a) shows plots of

*P*(

*ϕ*)/(2

_{m}*LP*

_{0}) as a function of scattering angle

*ϕ*for a number of guided modes within the fiber, calculated with the assumption that all scattered light is lost, or

_{m}*L*(

*ϕ*) = 1 for all

*ϕ*. The curve for the fundamental mode shows good qualitative agreement with the measured ARS data from Roberts et al, referring to a not too dissimilar fiber design [6

**13**236–244 (2005). [CrossRef] [PubMed]

### 4.3. Wavelength dependence of scattering loss

*λ*at which the lowest loss value occurs is proportionally scaled when an HC-PBGF with a given cross-section is drawn to different dimensions [23]. It has been shown experimentally that when a given fiber cross-section is drawn to different diameters, its minimum loss decreases as

_{c}*μm*where infrared absorption mechanisms begin to dominate. However, a theoretical understanding of this dependence has not hitherto been provided [6

**13**236–244 (2005). [CrossRef] [PubMed]

24. F. Benabid and P. J. Roberts, “Linear and nonlinear optical properties of hollow core photonic crystal fiber,” J. Mod. Opt. **2**, 87–124 (2011). [CrossRef]

*a*= 3.06 in the case of the low-loss HC-PBGF, as shown in Fig. 5. In general,

*a*depends on both the specific design of the fibre cross-section and the roughness PSD, and it has been found to assume values between 2.5 and 3.5 for a range of fibers we studied. This confirms that the scattering loss and its wavelength dependence are very much dependent on the design parameters.

**13**236–244 (2005). [CrossRef] [PubMed]

25. R. Amezcua-Correa, N. G. R. Broderick, M. N. Petrovich, F. Poletti, and D. J. Richardson, “Design of 7 and 19 cells core air-guiding photonic crystal fibers for low-loss, wide bandwidth and dispersion controlled operation,” Opt. Express **15**, 17577–17586 (2007). [CrossRef] [PubMed]

*F*was calibrated using measurements and simulations for the fibre in [22

22. N. V. Wheeler, M. N. Petrovich, R. Slavík, N. Baddela, E. Numkam Fokoua, J. R. Hayes, D. R. Gray, F. Poletti, and D. J. Richardson “Wide-bandwidth, low-loss, 19-cell hollow core photonic band gap fiber and its potential for low latency data transmission,” in Proceedings of Optical Fiber Communication Conference (2012), paper PDP5A.2.

*μm*so that

*F*= 0.0174

*μm*

^{−1}corresponds to 3.5

*dB/km*. As can be seen, this scalar quantity does not explain the observed wavelength dependence of the loss and is thus inaccurate in quantifying it. This can be understood by realizing that in addition to the fibre cross-section determining the field at the interfaces and how it changes with wavelength, when the operational wavelength is increased, the range of roughness spatial frequencies that contribute to the scattering also changes and becomes narrower. For example, at a wavelength of 1

*μm*the range of spatial frequencies required to calculate the scattering in air at angles

*ϕ*from 0 to 180° ranges from 0 to 12.55

*μm*

^{−1}while at a wavelength of 2

*μm*, it goes from 0 to 6.275

*μm*

^{−1}. Therefore only models that fully account for the roughness PSD like our method (and do not just use its

*rms*value like the

*F*–parameter) can predict the wavelength dependence of the loss.

## 5. The impact of roughness PSD

*κ*= 0.1

_{c}*μm*

^{−1}. In Fig. 6(a), we plot the computed scattering loss for our HC-PBGF as a function of the cut-off frequency at the wavelength of

*λ*= 1.55

_{c}*μm*. As

*κ*increases, the PSD

_{c}*S*(

*κ*) is lower for a broad range of frequencies, leading to the observed decrease in the scattering loss.

_{c}*κ*. A decrease in

_{c}*κ*corresponds to a higher PSD at the short spatial frequencies responsible for stronger scattering at longer wavelengths. As a result, the loss at longer wavelengths becomes higher than at shorter wavelengths, as can be readily seen in the figure. Additionally, we note that the scattering loss is directly proportional to the ratio

_{c}*T*/

_{g}*γ*which we have assumed is 1500

*K*/(

*J.m*

^{−2}), an increase in this ratio will therefore shift the scattering loss curve upwards.

*μm*

^{−1}, roughness from other sources may be present at frequencies below those practically measurable. It is well known that most realistic surfaces have roughness statistics that present one or more of three basic components: a long-range waviness, short-range random roughness and periodicity [26

26. J. M. Elson and J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. **69**31–47 (1979). [CrossRef]

11. V. Dangui, M. J. F. Digonnet, and G. S. Kino, “Modeling of the propagation loss and backscattering in air-core photonic-bandgap fibers,” J. Lightwave Technol. **17**, 3783–3789 (2009). [CrossRef]

*R*(

*u*) =

*σ*

^{2}

*exp*(−|

*u*|/

*L*) and

_{c}*L*being the correlation length and

_{c}*σ*the corresponding

*rms*roughness. The PSD arising from these auto-correlation functions are a Lorentzian

*L*at wavelength

_{c}*λ*= 1.55

*μm*, assuming

*σ*of 0.1

*nm*. In both cases, the loss increases with

*L*, peaks near

_{c}*L*4

_{c}*mm*and then decreases quickly, as typically observed with exponential or Gaussian roughness [27]. In a broad range of correlation lengths (70

*μm*− 3.7

*cm*) the scattering loss from such roughness components alone is higher than the measured value, which would suggest discarding this range. If however a long-range exponential roughness component with

*L*of the order of a few centimeters is present in our HC-PBGF in addition to SCWs, this leads to a predicted scattering loss value very close to the measured 3.5

_{c}*dB/km*. The contribution of this additional roughness component would be many orders of magnitude below that of SCWs over AFM-measurable spatial frequencies and would therefore be compatible with reported measurements, as can be seen in Fig. 7(b). Although clearly speculative, this confirms the importance of accurate roughness information in the low spatial frequency region to allow us making even more accurate loss predictions.

## 6. Conclusion

## Acknowledgments

## References and links

1. | P. St. J. Russell, “Photonic crystal fibers,” Science |

2. | R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Nature (London) |

3. | C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Muller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature (London) |

4. | M. N. Petrovich, F. Poletti, A. van Brakel, and D. J. Richardson, “Robustly single mode hollow core photonic bandgap fiber,” Opt. Express |

5. | B. J. Mangan, L. Farr, A. Langford, P. J. Roberts, D. P. Williams, F. Couny, M. Lawman, M. Mason, S. Coupland, R. Flea, H. Sabert, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Low loss (1.7 dB/km) hollow core photonic bandgap fiber,” in Proceedings of Optical Fiber Communication Conference (2004), paper PDP24. |

6. | P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Williams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. S. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express |

7. | D. Marcuse, |

8. | A. W. Snyder and J. D. Love, |

9. | P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, T. A. Birks, J. C. Knight, and P. S. Russell, “Loss in solid-core photonic crystal fibers due to interface roughness scattering,” Opt. Express |

10. | M.-C. Phan-Huy, J.-M. Moison, J. A. Levenson, S. Richard, G. Melin, M. Douay, and Y. Quiquempois, “Surface roughness and light scattering in a small effective area microstructured fiber,” J. Lightwave. Technol. |

11. | V. Dangui, M. J. F. Digonnet, and G. S. Kino, “Modeling of the propagation loss and backscattering in air-core photonic-bandgap fibers,” J. Lightwave Technol. |

12. | E. G. Rawson, “Theory Of scattering by finite dielectric needles illuminated parallel to their axes,” J. Opt. Soc. Am. |

13. | E. G. Rawson, “Analysis of scattering from fiber waveguides with irregular core surfaces,” Appl. Opt. |

14. | P. Mazumder, S. L. Logunov, and S. Raghavan, “Analysis of excess scattering in optical fibers,” J. Appl. Phys. |

15. | S. G. Johnson, M. L. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B |

16. | J. D. Jackson, |

17. | A. W. Snyder, “Radiation losses due to variations of radius on dielectric or optical fibers,” IEEE Trans. Microwave Theory Tech. |

18. | D. Marcuse, “Radiation losses of HE11 mode of a fiber with sinusoidally perturbed core boundary,” Appl. Opt. |

19. | F. Poletti, N. G. R. Broderick, D. J. Richardson, and T. M. Monro, “The effect of core asymmetries on the polarization properties of hollow core photonic bandgap fibers,” Opt. Express |

20. | J. Jäckle and K. Kawasaki, “Intrinsic roughness of glass surfaces,” J. Phys. Condens. Matter |

21. | T. Sarlat, A. Lelarge, E Søndergård, and D. Vandembroucq, “Frozen capillary waves on glass surfaces: an AFM study,” Eur. Phys. J. B |

22. | N. V. Wheeler, M. N. Petrovich, R. Slavík, N. Baddela, E. Numkam Fokoua, J. R. Hayes, D. R. Gray, F. Poletti, and D. J. Richardson “Wide-bandwidth, low-loss, 19-cell hollow core photonic band gap fiber and its potential for low latency data transmission,” in Proceedings of Optical Fiber Communication Conference (2012), paper PDP5A.2. |

23. | J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, |

24. | F. Benabid and P. J. Roberts, “Linear and nonlinear optical properties of hollow core photonic crystal fiber,” J. Mod. Opt. |

25. | R. Amezcua-Correa, N. G. R. Broderick, M. N. Petrovich, F. Poletti, and D. J. Richardson, “Design of 7 and 19 cells core air-guiding photonic crystal fibers for low-loss, wide bandwidth and dispersion controlled operation,” Opt. Express |

26. | J. M. Elson and J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. |

27. | D. Marcuse, “Mode conversion caused by surface imperfection of a dielectric slab waveguide,” Bell Syst. Tech. J. |

**OCIS Codes**

(060.2310) Fiber optics and optical communications : Fiber optics

(060.2400) Fiber optics and optical communications : Fiber properties

(290.5880) Scattering : Scattering, rough surfaces

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: June 15, 2012

Revised Manuscript: August 13, 2012

Manuscript Accepted: August 14, 2012

Published: August 29, 2012

**Citation**

Eric Numkam Fokoua, Francesco Poletti, and David J. Richardson, "Analysis of light scattering from surface roughness in hollow-core photonic bandgap fibers," Opt. Express **20**, 20980-20991 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-19-20980

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### References

- P. St. J. Russell, “Photonic crystal fibers,” Science299, 358–362 (2003). [CrossRef] [PubMed]
- R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Nature (London)285, 1537–1539 (1999).
- C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Muller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature (London)424, 657–659 (2003). [CrossRef]
- M. N. Petrovich, F. Poletti, A. van Brakel, and D. J. Richardson, “Robustly single mode hollow core photonic bandgap fiber,” Opt. Express16, 4337–4346 (2008). [CrossRef] [PubMed]
- B. J. Mangan, L. Farr, A. Langford, P. J. Roberts, D. P. Williams, F. Couny, M. Lawman, M. Mason, S. Coupland, R. Flea, H. Sabert, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Low loss (1.7 dB/km) hollow core photonic bandgap fiber,” in Proceedings of Optical Fiber Communication Conference (2004), paper PDP24.
- P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Williams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. S. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express13236–244 (2005). [CrossRef] [PubMed]
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