## Determination of the mode reflection coefficient in air-core photonic bandgap fibers

Optics Express, Vol. 15, Issue 9, pp. 5342-5359 (2007)

http://dx.doi.org/10.1364/OE.15.005342

Acrobat PDF (1193 KB)

### Abstract

Using an eigenmode decomposition technique, we numerically determine the
backreflection coefficient of the modes of air-core photonic bandgap fibers for
flat terminations. This coefficient is found to be very small for the
fundamental air-guided mode, of the order of 10^{-5} to 10^{-6},
in contrast with the surface and bulk modes, which exhibit significantly higher
reflections, by about three to four orders of magnitude. For the Crystal Fibre
HC-1550-2 fiber, we find a reflection coefficient of
1.9×10^{-6} for an air termination, and approximately 3.3%
for a silica termination. We also find that the Fresnel approximation is ill
suited for the determination of the modal reflection coefficient, and instead
propose a more accurate new formula based on an averaged modal index.

© 2007 Optical Society of America

## 1. Introduction

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]

4. V. Dangui, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, “Phase sensitivity to temperature of
the fundamental mode in air-guiding photonic-bandgap
fibers,” Opt. Express **13**, 6669–6684
(2005). [CrossRef] [PubMed]

7. Crystal Fibre website, http://www.crystal-fibre.com

*n*

_{0}is characterized by a power reflection coefficient that is usually calculated by the well-known approximate Fresnel formula:

*n*is the effective index of the mode considered. The Fresnel approximation is equivalent to replacing the waveguide with a medium of uniform refractive index equal to the mode effective index, and to approximate the mode by a plane wave at normal incidence. As a result, the Fresnel approximation performs best for weakly-guiding waveguides with a small index contrast across the waveguide cross-section, for modes possessing a small numerical aperture (lower order modes), and for a strong refractive index contrast between the waveguide and the termination medium. Conversely, this approximation has proved to be inaccurate to evaluate the reflection coefficient of strongly guiding waveguides, [8] and it does not account for the spatial intensity distribution of the mode. Because PBFs are strongly guided waveguides, and because we are mostly concerned with the reflection of a leaky mode (the fundamental air-guided core mode) into air (low refractive index contrast between waveguide and termination medium), we expect the Fresnel approximation to be ineffective in this context, and that a more elaborate calculation will be required.

_{eff}## 2. Theory

*z*< 0 region (region I). The

*z*> 0 region (region II) is a semi-infinite medium of uniform index

*n*

_{0}, such as air. The PBF parameters are as follows: the cladding crystal periodicity is Λ, the core radius is

*R*, and the cladding air hole radius is

*ρ*. The transverse plane is hereafter defined as the interface plane terminating the PBF. We consider the reflection at this interface of a mode field propagating through the PBF in the +

*z*direction.

9. P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, “Towards a full vectorial and modal
technique for the analysis of integrated optics structures: The Radiation
Spectrum Method (RSM),” Opt. Commun. **140**, 128–145
(1997). [CrossRef]

*(*

**E**^{I}*) and can either be a core mode, a cladding mode, or a surface mode of the PBF, labeled*

**r***(*

**E**_{0}*). The reflected transverse electrical field, labeled*

**r***(*

**E**^{R}*), propagates in the -*

**r***z*direction and is written as a superposition of core and cladding mode fields of the PBF, which are collectively labeled

*(*

**E**_{m}*).*

**r***N*+1) modes, and

_{M}*a*is the amplitude of the signal reflected into mode

_{m}*m*. Furthermore, only modes of the same symmetry class as the incident mode will be present in the decomposition of the reflected field. For most application, we are concerned with the amount of light backreflected into the incident mode, which is labeled as

*m*= 0, since all other coefficients are much smaller (by many orders of magnitude). The intensity or power reflection coefficient into the fundamental mode is:

*z*direction is denoted

*(*

**E**^{T}*) and decomposed in a sum of plane waves propagating in free space. The transverse magnetic fields are decomposed in the same fashion. Using the continuity of the transverse electric and magnetic fields, we obtain the following relationships at the PBF interface plane:*

**r***a*.

_{m}11. V. Dangui, M. J. F. Digonnet, and G. S. Kino, “A fast and accurate numerical tool
to model the modal properties of photonic-bandgap
fibers,” Opt. Express **14**, 2979–2993
(2006). [CrossRef] [PubMed]

*. The non-zero coefficients are illustrated as blue dots, and the zero coefficients are left out (shown as white spaces). The upper left corner indicates the projection of PBF modes onto each other; the lower right corner shows the projection of plane waves onto each other. The remaining portions of the matrix show the projection of PBF modes onto plane waves propagating in air. The matrix size is 2082 × 2082 elements, of which nearly 950,000 are non-zero complex elements. This sheer number of non-zero elements makes it impossible to invert such a matrix accurately directly, so we resorted to an iterative method, described in Ref [12*

**A**12. H. . van der Vorst, “BI-CGSTAB: A fast and smoothly
converging variant of BI-CG for the solution of nonsymmetric linear
systems,” SIAM J. Sci. Stat. Comput. **13**, 631–644
(1992). [CrossRef]

## 3. Variation of the fundamental mode reflection coefficient across the PBF bandgap

*ρ*= 0.47Λ, and the core radius

*R*= 0.8Λ. Figure 4 represents the calculated bandgap of this fiber, as well as the dispersion curves of all of its core modes. The bandgap extends from

*λ*= 0.56Λ to

*λ*= 0.64Λ. Due to the choice of core radius, [13

13. H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, “Designing air-core photonic-bandgap
fibers free of surface modes,” IEEE J.
Quantum Eectron. **40**, 551–556
(2004). [CrossRef]

*λ*= 0.60Λ. The fundamental core modes have a very large fraction of their intensity (over 90 %) concentrated in air, which is the reason why a weak reflection at an air interface is expected.

*k*

_{0}, and thus in the radius of this circle. As the wavelength increases, the radius of the circle decreases, and thus fewer plane waves can be excited past the PBF termination plane, yielding a decrease in transmitted field. We therefore expect the fundamental-mode reflection to increase with wavelength. At the same time, since the circle boundary lies in a region where the Fourier transform intensity of the electric field is very weak (roughly 0.1% of its maximum value), we expect a weak dependence on wavelength.

*x*-polarized fundamental mode, defined as |

*a*

_{0}|

^{2}, when the fiber is terminated in air, calculated at several wavelengths across the bandgap. The reflection coefficient remains weak across the entire bandgap, its coefficient varying between ~0.6×10

^{-5}at the short-wavelength end of the bandgap and ~1.3×10

^{-5}at the long-wavelength end. We also observe a general trend towards an increase in the backreflection as the wavelength increases, which was predicted earlier on the basis of the smaller wavevectors associated with longer wavelengths: the mode energy is transmitted in air over a smaller range of wavevectors, and fewer plane waves can be excited, thus resulting in an increased reflection coefficient.

## 4. Reflection coefficient of various PBF modes

*ρ*= 0.47Λ) but increased the fiber core radius to

*R*= 2.0Λ, so that the fiber now supports a large number of core modes, as well as some surface modes. As is well known, the surface-mode energy is concentrated mostly in the silica at the core-cladding boundary. The core region of this fiber is shown in Fig. 8, together with the intensity contour lines of an exemplary surface mode.

_{0,1}-like fundamental modes, labeled HE

_{1,1}

^{x,y}, and the four LP

_{1,1}-like first-order modes, labeled TE

_{0,1}, HE

_{2,1}

^{x,y}, and TM

_{0,1}, respectively). These coefficients were calculated at

*λ*= 0.6Λ, which is about at the center of the bandgap, for a termination in air, and they represent reflection into the same backward-traveling mode as the incident mode. The reflection coefficient for the fundamental mode is 1.2×10

^{-5}, which is close to the value calculated for the previous (single-mode) fiber. The higher order core modes (LP

_{11}-like) have a significantly higher reflection coefficient, averaging 7×10

^{-5}. This is expected, since these modes have a higher fraction of their energy in the silica portions of the PBF, and hence are more strongly reflected. Note that the backreflection coefficients of the four LP

_{11}-like modes differ by as much as a factor of 3. This is due to the vectorial nature of the PBF waveguide, in which the strong index contrast between the air core and the silica cladding membranes lifts the near-degeneracy of these modes normally present in weakly-guiding fibers. The surface modes, which have an even higher fraction of their energy in silica, exhibit a much higher backreflection coefficient, in the range of 8×10

^{-3}. All of these core and surface modes have very close effective indices, in the vicinity of the refractive index of air, so the Fresnel approximation (Eq. 1) yields fairly similar reflection coefficients for all them (see Fig. 9). Clearly, and expectedly, the Fresnel approximation yields poor predictions.

## 5. Effect of PBF design parameters on the fundamental core mode reflection coefficient

13. H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, “Designing air-core photonic-bandgap
fibers free of surface modes,” IEEE J.
Quantum Eectron. **40**, 551–556
(2004). [CrossRef]

4. V. Dangui, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, “Phase sensitivity to temperature of
the fundamental mode in air-guiding photonic-bandgap
fibers,” Opt. Express **13**, 6669–6684
(2005). [CrossRef] [PubMed]

*ρ*= 0.47Λ) and at about the center of the bandgap (wavelength

*λ*= 0.6Λ). As expected, as the core radius increases, the reflection tends to decrease, except for radii where the PBF supports surface modes, such as for radii between 1.05Λ and 1.3Λ, and between 1.45Λ and 1.7Λ.[13

13. H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, “Designing air-core photonic-bandgap
fibers free of surface modes,” IEEE J.
Quantum Eectron. **40**, 551–556
(2004). [CrossRef]

4. V. Dangui, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, “Phase sensitivity to temperature of
the fundamental mode in air-guiding photonic-bandgap
fibers,” Opt. Express **13**, 6669–6684
(2005). [CrossRef] [PubMed]

*R*= 0.95 Λ, and is approximately equal to 2.9×10

^{-6}. Further reduction in the backreflection can be achieved by selecting a radius

*R*= 1.9Λ, where the backreflection is predicted to be 4.2×10

^{-7}, but at the expense of the PBF no longer being single-mode.

## 6. Effect of the termination medium refractive index on the reflection coefficient

*R*= 0.8Λ and

*ρ*= 0.47Λ, and a wavelength

*λ*= 0.6Λ, as a function of the termination refractive index. The reflections predicted by the Fresnel approximation are shown as a red curve. While both methods are in agreement for large termination medium indices, there are as expected significant differences for a termination medium index

*n*

_{0}close to the refractive index of air. When

*n*

_{0}is close to 1, the Fresnel approximation ceases to be valid and yields inaccurate answers, as we have seen before, by as much as 3 orders of magnitude. One interesting feature of these curves is the location of their minima. The Fresnel approximation predicts a minimum reflection for a termination medium of refractive index equal to

*n*< 1, while the calculated reflection exhibits a minimum for a termination medium index of approximately 1.005.

_{eff}## 7. Simplified formulation

*n*≤

_{modal}*n*, and the larger the fraction of energy the mode carries in air, the closer the averaged modal index is to 1. We show that the modal reflection coefficient can be well approximated by:

_{silica}*n*. The use of the averaged modal index also allows to account for the occurrence of a minimum in reflection for a termination medium with a refractive index larger than 1.

_{modal}*ρ*= 0.47Λ, and at a wavelength of

*λ*= 0.6Λ. The averaged modal index also shows sharp increases whenever the PBF supports surface modes, due to an enhanced overlap of the mode with the silica regions, [4

**13**, 6669–6684
(2005). [CrossRef] [PubMed]

*R*= 0.8Λ).

7. Crystal Fibre website, http://www.crystal-fibre.com

*λ*= 0.4Λ is shown in Fig. 16. There is very strong confinement of the mode field into the air core, with a field intensity overlap with silica now less than 1%. The Fourier transform intensity of this same mode's transverse electric field, plotted in logarithmic scale in the wavevector domain, is shown in Fig. 17. The black circle shows the boundary for all wavevectors such that

^{-6}for an air-termination.

*λ*= 0.4Λ, as a function of the termination medium index. The agreement is now excellent, the disagreement between the two approaches being at most 50% for all the points shown in the figure. Also, the averaged modal index model predicts that the minimum in the reflection occurs for a termination index of 1.0036, in good agreement with the prediction of the exact model (black dots), which falls somewhere between 1.003 and 1.004. The conclusion is that the averaged modal index approximation can be used for qualitative predictions when modeling a fiber with a low air-filling ratio (around 85%), and for quantitative predictions when modeling a fiber with an air-filling ratio of 90% or higher.

## 8. Conclusion

^{-5}to 10

^{-6}for the fundamental mode, a few 10

^{-3}for surface modes, and around 10

^{-2}for bulk modes. Furthermore we observed strong variations of the fundamental mode reflection as a function of core radius, due to the presence of surface modes, which enhance the overlap of the fundamental mode with the silica. For the Crystal Fibre HC-1550-2 PBF we predicted a reflection coefficient of 1.9×10

^{-6}for an air termination, and 3.3% for a silica termination, which would correspond to the splice reflection of a PBF with an index-guiding conventional silica optical fiber.

## Appendix: Eigenmode decomposition method

*(*

**E**^{I}*) and is assumed to be a mode (core or cladding mode) of the PBF,*

**r***(*

**E**_{0}*). The reflected transverse electrical field propagates in the negative*

**r***z*direction and is a superposition of PBF modes (core and cladding modes), and is denoted

*(*

**E**^{R}*). The modal decomposition of the reflected transverse electrical field is of the form:*

**r***a*

_{0}is the amplitude reflection coefficient of the incident mode considered into itself. The modes present in the decomposition of the reflected field are of the same symmetry class as the fundamental mode. The intensity reflection coefficient is:

*z*direction is denoted

*(*

**E**^{T}*) and decomposed in a sum of plane waves propagating in free space:*

**r***,*

**E**^{T}*s*is a normalization constant, and

*and*

**E**^{1}_{W,k}*denote the two amplitudes along orthogonal polarizations of a plane wave propagating in air, with a wavevector*

**E**^{2}_{W,k}*=[*

**k***k*

_{x}*k*

_{y}*k*] satisfying:

_{z}*a*and transmission coefficients

_{m}*b*:

_{k}_{M}+2N

_{W}) linear equations to compute the (N

_{M}+1)

*a*and 2N

_{m}_{W}

*b*coefficients. Equations (A11) and (A12) are derived from Eq. (A8), and Eq. (A13) and (A14) are derived from Eq. (A9).

_{k}*of size (1+N*

**A**_{M}+2N

_{W})×(1+N

_{M}+2N

_{W}), containing 1+N

_{M}+2N

_{W}+4(1+N

_{M})N

_{W}nonzero terms. The reflection coefficients

*are determined by solving an equation of the form:*

**ν***, which is used as a preconditioner to lower the conditioning number of*

**A***, and we thus solve:*

**A**11. V. Dangui, M. J. F. Digonnet, and G. S. Kino, “A fast and accurate numerical tool
to model the modal properties of photonic-bandgap
fibers,” Opt. Express **14**, 2979–2993
(2006). [CrossRef] [PubMed]

*with an error measured by:*

**ν**_{0}## Acknowledgments

## References and links

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

2. | J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J. P. Sandro, “Photonic crystals as optical
fibers-physics and applications,” Opt.
Mater. |

3. | R. S. Windeler, J. L. Wagener, and D. J. Giovanni, “Silica-air microstructured fibers: Properties and applications,” Optical Fiber Communications Conference, San Diego, CA, USA (1999). |

4. | V. Dangui, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, “Phase sensitivity to temperature of
the fundamental mode in air-guiding photonic-bandgap
fibers,” Opt. Express |

5. | D. M. Dagenais, K. P. Koo, and F. Bucholtz, “Fiber interferometry limitations due to parasitic optical cavities,” Lasers and Electro-Optics Society Conference, San Jose, CA USA (1993). |

6. | J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, “The coupling performance of photonic
crystal fibres in fibre stellar interferometry,”
Mon. Not. Roy. Astron. Soc. |

7. | Crystal Fibre website, http://www.crystal-fibre.com |

8. | D. Khalil, “Reflection at the end of strongly
guiding dielectric waveguide,”
Opto-Electronics and Communications Conference, Yokohama,
Japan , |

9. | P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, “Towards a full vectorial and modal
technique for the analysis of integrated optics structures: The Radiation
Spectrum Method (RSM),” Opt. Commun. |

10. | D. Marcuse, |

11. | V. Dangui, M. J. F. Digonnet, and G. S. Kino, “A fast and accurate numerical tool
to model the modal properties of photonic-bandgap
fibers,” Opt. Express |

12. | H. . van der Vorst, “BI-CGSTAB: A fast and smoothly
converging variant of BI-CG for the solution of nonsymmetric linear
systems,” SIAM J. Sci. Stat. Comput. |

13. | H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, “Designing air-core photonic-bandgap
fibers free of surface modes,” IEEE J.
Quantum Eectron. |

14. | 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 band-gap fibers,” in |

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

16. | J. A. West, C. M. Smith, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Surface modes in air-core photonic
band-gap fibers,” Opt. Express |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(060.2310) Fiber optics and optical communications : Fiber optics

(120.5700) Instrumentation, measurement, and metrology : Reflection

**ToC Category:**

Photonic Crystal Fibers

**History**

Original Manuscript: August 16, 2006

Revised Manuscript: November 6, 2006

Manuscript Accepted: November 7, 2006

Published: April 18, 2007

**Citation**

Vinayak Dangui, Michel J. Digonnet, and Gordon S. Kino, "Determination of the mode reflection coefficient in air-core photonic bandgap fibers," Opt. Express **15**, 5342-5359 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5342

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### References

- 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]
- J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J. P. Sandro, "Photonic crystals as optical fibers-physics and applications," Opt. Mater. 11, 143-151 (1999). [CrossRef]
- R. S. Windeler, J. L. Wagener, and D. J. Giovanni, "Silica-air microstructured fibers: Properties and applications," Optical Fiber Communications Conference, San Diego, CA, USA (1999).
- V. Dangui, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, "Phase sensitivity to temperature of the fundamental mode in air-guiding photonic-bandgap fibers," Opt. Express 13, 6669-6684 (2005).H [CrossRef] [PubMed]
- D. M. Dagenais, K. P. Koo, and F. Bucholtz, "Fiber interferometry limitations due to parasitic optical cavities," Lasers and Electro-Optics Society Conference, San Jose, CA, USA (1993).
- J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, "The coupling performance of photonic crystal fibres in fibre stellar interferometry," Mon. Not. Roy. Astron. Soc. 368, 203-210 (2006).
- Crystal Fibre website, http://www.crystal-fibre.com
- D. Khalil, "Reflection at the end of strongly guiding dielectric waveguide," Opto-Electronics and Communications Conference, Yokohama, Japan, 1, 352-353 (2002).
- P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, "Towards a full vectorial and modal technique for the analysis of integrated optics structures: The Radiation Spectrum Method (RSM)," Opt. Commun. 140, 128-145 (1997). [CrossRef]
- D. Marcuse, Theory of dielectric optical waveguides, (Academic Press, 1974).
- V. Dangui, M. J. F. Digonnet, and G. S. Kino, "A fast and accurate numerical tool to model the modal properties of photonic-bandgap fibers," Opt. Express 14, 2979-2993 (2006).H [CrossRef] [PubMed]
- H. . van der Vorst, "BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems," SIAM J. Sci. Stat. Comput. 13, 631-644 (1992). [CrossRef]
- H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, "Designing air-core photonic-bandgap fibers free of surface modes," IEEE J. Quantum Eectron. 40, 551-556 (2004). [CrossRef]
- 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 band-gap fibers," in Photonic Crystal Materials and Devices, A. Adibi, A. Scherer, S. Yu Lin, eds., Proc. SPIE 5000,161-174 (2003). [CrossRef]
- 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). [CrossRef] [PubMed]
- J. A. West, C. M. Smith, N. F. Borrelli, D. C. Allan, and K. W. Koch, "Surface modes in air-core photonic band-gap fibers," Opt. Express 12, 1485-1496 (2004). [CrossRef] [PubMed]

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