## Surface modes in air-core photonic band-gap fibers

Optics Express, Vol. 12, Issue 8, pp. 1485-1496 (2004)

http://dx.doi.org/10.1364/OPEX.12.001485

Acrobat PDF (1348 KB)

### Abstract

We present a detailed description of the role of surface modes in photonic band-gap fibers (PBGFs). A model is developed that connects the experimental observations of high losses in the middle of the transmission spectrum to the presence of surface modes supported at the core-cladding interface. Furthermore, a new PBGF design is proposed that avoids these surface modes and produces single-mode operation.

© 2004 Optical Society of America

## 1. Introduction

2. C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature **424**, 657–659 (2003). [CrossRef] [PubMed]

3. K. Saitoh and M. Koshiba, “Leakage loss and group velocity dispersion in air-core photonic bandgap fibers,” Opt. Express **11**, 3100–3109 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-3100. [CrossRef] [PubMed]

4. K. Saitoh, N. A. Mortensen, and M. Koshiba, “Air-core photonic band-gap fibers: the impact of surface modes,” Opt. Express **12**, 394–400 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394. [CrossRef] [PubMed]

## 2. Coupled-mode description of surface-mode losses

6. D. C. Allan, N. F. Borrelli, M. T. Gallagher, D. Müller, C. M. Smith, N. Venkataraman, J. A. West, P. Zhang, and K.W. Koch, “Surface modes and loss in air-core photonic bandgap fibers,” in *Photonic Crystal Materials and Devices* , Ali Adibi, Axel Scherer, and Shawn Yu Lin;, eds. Proc. SPIE **5000**, p. 161–174 (2003).

*supermodes*, can be described by linear superpositions of the core and surface modes. A loss term (-

*γA*

_{j}) is introduced into the coupled equations to address the proposed surface-mode loss mechanism. The loss rate

*γ*, characterizes that rate at which energy is coupled out of the surface modes and thus out of the coupled system. With this loss term, the coupled-mode equations are simply:

*A*

_{i}and

*A*

_{j}are the complex field amplitudes of the core or surface mode, respectively,

*κ*

_{ij}describes coupling between the modes,

*Δβ*

_{ij}is the difference in wavevectors between the two coupled modes, and

*z*is the distance along the fiber. The loss term represents a continuous loss of power from the surface mode due to assumed coupling to extended modes through small structural perturbations. This is motivated in more detail in Section 3.

*i*and surface mode

*j*are shown to have core-mode solutions that decay exponentially with an attenuation coefficient given by:

*γ*

_{j},

*κ*

_{ij}and

*Δβ*

_{ij}are in km

^{-1}. Here the wavelength dependence of

*Δβ*

_{ij}leads to an approximately Lorentzian loss spectrum. We assume that, in this approximation,

*κ*

_{ij}and

*γ*

_{j}are essentially independent of wavelength.

*Δβ*

_{ij}(

*λ*) has a minimum in the vicinity of each avoided crossing and thus the loss is greatest at these wavelengths. Near the avoided crossings the difference in propagation constants is shown to be approximately quadratic:

*Λ*is the pitch of the lattice,

*λ*

_{min,ij}is the wavelength of the avoided crossing,

*Δn*

_{min,ij}is the minimum difference in the supermode effective indices at the avoided crossing, and

*s*

_{ij}is the difference in slopes of the

*n*

_{eff}of the uncoupled modes (see dashed lines in Fig. 3(b)):

*ω̃*is the scaled frequency Λ/λ. The relationship between the coupling coefficient

*κ*

_{ij}and

*Δn*

_{min,ij}=

*κ*

_{ij}

*λ*

_{min,ij}/

*π*allows us to obtain values for the coupling coefficients from calculations of the avoided crossings such as the one shown in Fig. 3(b).

*I*

_{i}(

*z*=0), the overall loss spectrum on a fiber of length

*L*can be computed by:

*j*is over the surface modes that are coupled to core mode

*i*. The multimode nature of the launch leads to a complicated expression in which the resulting loss spectrum is dominated by the lowest-loss mode at each wavelength. Because of this, individual loss terms

*α*

_{ij}with large peak losses result in a total attenuation coefficient with significantly smaller peak attenuation and broader-than-expected tails.

*γ*be a free parameter that is assumed to be constant for all surface modes. Finally, we assume that the multiple core modes are launched equally.

*α*

_{ij}have very much larger peak losses than the cumulative loss coefficient. This is dramatically illustrated in Fig. 4(b). To achieve the observed peak surface-mode loss of ~240 dB/km,

*γ*must have a value of 4250 km

^{-1}or 37000 dB/km giving the

*α*

_{ij}peak values of 9000 dB/km. These values can be lowered significantly if we consider the impact of simple scale variations on the attenuation. If we ignore mode coupling, scale variations along the fiber simply shift the peak wavelengths of the

*α*

_{ij}. In this approximation, the fiber transmission can be thought of as a filter whose center wavelength varies along the fiber. For a linear fractional scale variation x along the fiber, this may be modeled by assuming that:

*γ*drops and the tails of the Lorentzians decrease as shown by the red curve in Fig. 4(a). The 4% variation used to model the red curve in Fig. 4(a) is comparable to the axial variation observed in various cross-sections of this fiber. We note that in the absence of mode-coupling this scale variation can be modeled as a linear system and any variation can be reduced to a monotonic function of fiber length.

## 3. Overlap calculations of core, surface and extended modes

*γ*, the surface modes have been shown to be leakier than core modes [4

4. K. Saitoh, N. A. Mortensen, and M. Koshiba, “Air-core photonic band-gap fibers: the impact of surface modes,” Opt. Express **12**, 394–400 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394. [CrossRef] [PubMed]

2. C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature **424**, 657–659 (2003). [CrossRef] [PubMed]

*γ*comes from a more traditional form of coupled-mode theory in which structural changes lead to perturbations in the modes. The electric field in a given cross-section can always be described by a local superposition of the guided and extended modes of that cross-section. If the structure changes, the modes will change, and the field must be re-expressed in this new basis of local modes. For small perturbations, the set of new modes can generally be matched one-for-one to the original set of modes with small changes in mode energy and mode structure, so core and surface modes of the unperturbed structure can be identified with corresponding core and surface modes of the perturbed structure. Loss is a consequence of the perturbation, leading to light being coupled from the guided core and surface modes to the extended or radiation modes. To investigate coupling loss we could calculate the modes of both the perturbed and unperturbed structure and simply re-express the original core mode in this new basis of modes. Alternatively we can turn to perturbation theory to calculate the coupling coefficients that described the coupling of light into these new modes of the structure.

10. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E **65**, 66611 (2002). [CrossRef]

11. M. Skorobogatiy, S. A. Jacobs, S. G. Johnson, and Y. Fink, “Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates,” Opt. Express **10**, 1227–1243 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1227. [CrossRef] [PubMed]

**E**

_{1}(

**r**,ω)=(

**E**

_{t1}, E

_{z1}), and

**E**

_{2}(

**r**,ω)=(

**E**

_{t2}, E

_{z2}) are the transverse and longitudinal electric-field components of the modes of the original structure at frequency

*ω*, and

*ε*(

**r**) and

*ε̃*(

**r**,z) are the dielectric constants of the original and perturbed structures, respectively. β

_{1}(ω) and β

_{2}(ω) are the propagation constants of the modes in the unperturbed structure. It is obvious that this integral is largest only when

**E**

_{1}(

**r**,ω) and

**E**

_{2}(

**r**,ω) have significant extent in the region of non-zero perturbation

*δε*(

**r**,z)=

*ε*(

**r**)-

*ε̃*(

**r**,z). In air-core PBGFs, perturbations to δε are greatest in the silica regions near the perimeters of the air holes.

13. 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 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173. [CrossRef] [PubMed]

_{12}|

^{2}is almost an order of magnitude larger than |C

_{13}|

^{2}or |C

_{23}|

^{2}. The calculation also predicts that |C

^{12}|

^{2}is peaked at the avoided crossing where the two supermodes have a strong surface-mode component and that the coupling decreases rapidly away from the avoided crossing. The second plot in Fig. 6(b) shows the coefficients |C

_{ii}|

^{2}. In perturbation theory these terms give the first order correction to the propagation constant and are an indication of the overlap of the mode with the perturbation. From Fig 6(b) we see that when the supermodes are surface-mode-like they see significant impact from the perturbation but when they are core-mode-like the impact is reduced by nearly three orders of magnitude. Only when the perturbation impacts both modes “1” and “2” does the coupling occur as shown by |C

_{12}|

^{2}.

*γ*, we believe that we presented a plausibility argument that this loss rate is significantly higher than that due to direct coupling between the core and extended modes. This is an important factor in understanding the nature of the surface-mode loss mechanism. We expect that for higher air-filling fractions and/or larger core diameters the coupling will decrease as the core modes become confined more tightly to the core region thus decreasing the overlap with the surface modes.

## 4. Fiber design without surface modes

_{d}=1.06Λ. As the core radius increases, these surface modes continue through the gap interacting with the core modes as described in the preceding sections. At a radius of about 1.36Λ the surface modes have disappeared but they soon reappear at R

_{d}=1.46Λ.

_{d}=1.0Λ, 1.4Λ and 1.8Λ. A further simplification in Fig. 10(a) plots the highest core-confined energy for each defect radius. This clearly reveals PBGF designs that significantly reduce the impact of surface modes. The ideal single-mode design space is between R

_{d}~0.9Λ and 1.1Λ, just below the defect radii often found in experimental fibers [2

2. C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature **424**, 657–659 (2003). [CrossRef] [PubMed]

_{d}~1.35Λ and 1.45Λ but in this region the fiber becomes multimode, a property which is undesirable in most low-loss fiber applications.

_{d}~2.5Λ [14]. Although the entire spectrum is not shown in Ref. 14, the sharpness of the surface-mode features is in agreement with the coupling and supermode arguments presented in Section 3. The lower loss is attributed to reduced mode coupling because less modal energy is contained in the glass. This suggestion is in agreement with the results shown in Fig. 10(b) in which both the energy outside of the core and the energy in the glass are plotted against the core radius for the fundamental core mode. Although the presence of the surface-mode coupling regions makes an exact power-law scaling hard to assess, the energy outside of the core appears to follow a 1/

## 5. Summary

## References and links

1. | N. Venkataraman, M.T. Gallagher, D. Müller, Charlene M. Smith, J. A. West, K. W. Koch, and J. C. Fajardo, “Low-Loss (13 dB/km) Air-Core Photonic Band-Gap Fibre”, |

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

3. | K. Saitoh and M. Koshiba, “Leakage loss and group velocity dispersion in air-core photonic bandgap fibers,” Opt. Express |

4. | K. Saitoh, N. A. Mortensen, and M. Koshiba, “Air-core photonic band-gap fibers: the impact of surface modes,” Opt. Express |

5. | D. C. Allan, et al., |

6. | D. C. Allan, N. F. Borrelli, M. T. Gallagher, D. Müller, C. M. Smith, N. Venkataraman, J. A. West, P. Zhang, and K.W. Koch, “Surface modes and loss in air-core photonic bandgap fibers,” in |

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

8. | P. Yeh, |

9. | D. C. Allan, N.F. Borrelli, J. C. Fajardo, K. W. Koch, and J. A. West. Corning Incorporated “Optimized defects in band-gap waveguides,” U.S. Pat. Appl. 20020136516-A1. February 4 2002. |

10. | S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E |

11. | M. Skorobogatiy, S. A. Jacobs, S. G. Johnson, and Y. Fink, “Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates,” Opt. Express |

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

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

14. | B. J. Mangan, L. Farr, A. Langford, P. J. Roberts, D. P. Williams, F. Couny, M. Lawman, M. Mason, S. Coupland, R. Flea, and H. Sabert, “Low loss (1.7 dB/km) hollow core photonic bandgap fiber,” |

**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(060.2280) Fiber optics and optical communications : Fiber design and fabrication

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

**ToC Category:**

Focus Issue: Photonic crystals and holey fibers

**History**

Original Manuscript: March 2, 2004

Revised Manuscript: March 28, 2004

Published: April 19, 2004

**Citation**

James West, Charlene Smith, Nicholas Borrelli, Douglas Allan, and Karl Koch, "Surface modes in air-core photonic band-gap fibers," Opt. Express **12**, 1485-1496 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-8-1485

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### References

- N. Venkataraman, M.T. Gallagher, D. Müller, Charlene M. Smith, J. A. West, K. W. Koch, and J. C. Fajardo, �??Low-Loss (13 dB/km) Air-Core Photonic Band-Gap Fibre�??, Proceedings of ECOC 2002 (Copenhagen, Denmark, 2002) PD1.1.
- C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan and K. W. Koch, "Low-loss hollow-core silica/air photonic bandgap fibre," Nature 424, 657-659 (2003). [CrossRef] [PubMed]
- K. Saitoh and M. Koshiba, "Leakage loss and group velocity dispersion in air-core photonic bandgap fibers," Opt. Express 11, 3100-3109 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-3100">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-3100</a>. [CrossRef] [PubMed]
- K. Saitoh, N. A. Mortensen, and M. Koshiba, "Air-core photonic band-gap fibers: the impact of surface modes," Opt. Express 12, 394-400 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394</a>. [CrossRef] [PubMed]
- D. C. Allan, et al., Photonic Crystals and Light Localization in the 21st Century, C. M. Soukoulis (ed.), (Kluwer Academic Press, The Netherlands, 2001), pp. 305-320. [CrossRef]
- D. C. Allan, N. F. Borrelli, M. T. Gallagher, D. Müller, C. M. Smith, N. Venkataraman, J. A. West, P. Zhang, K.W. Koch, �??Surface modes and loss in air-core photonic bandgap fibers,�?? in Photonic Crystal Materials and Devices, Ali Adibi, Axel Scherer, and Shawn Yu Lin;, eds. Proc. SPIE 5000, p. 161-174 (2003).
- J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, N.J., 1995), pp. 73-76.
- P. Yeh, Optical Waves in Layered Media, (John Wiley & Sons, New York, N.Y., 1988) pp. 337-345.
- D. C. Allan, N.F. Borrelli, J. C. Fajardo, K. W. Koch, and J. A. West. Corning Incorporated �??Optimized defects in band-gap waveguides,�?? U.S. Pat. Appl. 20020136516-A1. February 4 2002.
- S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, �??Perturbation theory for Maxwell�??s equations with shifting material boundaries,�?? Phys. Rev. E 65, 66611 (2002). [CrossRef]
- M. Skorobogatiy, S. A. Jacobs, S. G. Johnson, and Y. Fink, �??Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates,�?? Opt. Express 10, 1227-1243 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1227">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1227</a>. [CrossRef] [PubMed]
- A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic Publishers, Boston, MA, 2000), Eq. 31-50a.
- 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 (2001), <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] [PubMed]
- B. J. Mangan, L. Farr, A. Langford, P. J. Roberts, D. P. Williams, F. Couny, M. Lawman, M. Mason, S. Coupland, R. Flea and H. Sabert, �??Low loss (1.7 dB/km) hollow core photonic bandgap fiber,�?? Proceedings of OFC 2004, (OSA, Los Angeles, CA, 2004) PDP24.

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