## Guidance in Kagome-like photonic crystal fibres II: perturbation theory for a realistic fibre structure |

Optics Express, Vol. 19, Issue 7, pp. 6957-6968 (2011)

http://dx.doi.org/10.1364/OE.19.006957

Acrobat PDF (2716 KB)

### Abstract

A perturbation theory is developed that treats a localised mode embedded within a continuum of states. The method is applied to a model rectangular hollow-core photonic crystal fibre structure, where the basic modes are derived from an ideal, scalar model and the perturbation terms include vector effects and structural difference between the ideal and realistic structures. An expression for the attenuation of the fundamental mode due to interactions with cladding modes is derived, and results are presented for a rectangular photonic crystal fibre structure. Attenuations calculated in this way are in good agreement with numerical simulations. The origin of the guidance in our model structure is explained through this quantitative analysis. Further perspectives are obtained through investigating the influence of fibre parameters on the attenuation.

© 2011 OSA

## 1. Introduction

1. L. Chen, G. J. Pearce, T. A. Birks, and D. M. Bird, “Guidance in Kagome-like photonic crystal fibres I: analysis of an ideal fibre structure,” Submitted to Opt. Express (2011). [PubMed]

1. L. Chen, G. J. Pearce, T. A. Birks, and D. M. Bird, “Guidance in Kagome-like photonic crystal fibres I: analysis of an ideal fibre structure,” Submitted to Opt. Express (2011). [PubMed]

1. L. Chen, G. J. Pearce, T. A. Birks, and D. M. Bird, “Guidance in Kagome-like photonic crystal fibres I: analysis of an ideal fibre structure,” Submitted to Opt. Express (2011). [PubMed]

## 2. Perturbation theory for the realistic model

### 2.1. Formulation of perturbation theory for the vector governing equation

*L*

^{0}denotes

**h**

*are the eigenvalues and eigenstates of the operator*

_{n}*L*

^{0}, corresponding to the square of the propagation constants and their associated fields, respectively. The subscript

*n*labels the modes including their polarisation. The vector governing equation corresponding to Eq. (1) in Ref. [1

*β*

^{2}and eigenvector

**H**now respectively denote the square of the propagation constant and field for the realistic system. The perturbation term,

*δL*(

**H**), is a linear vector function of

**H**. In our particular model structure,

*δL*(

**H**) takes the form where

*n*

^{2}is different from that in Ref. [1

*n*

^{2}is non-zero only at the intersections of the glass strips (where it takes the value

*n*and

_{a}*n*are refractive indices for air and glass respectively) and is zero elsewhere. The right-hand side of Eq. (3) consists of two parts: the first is the vector term in the governing equation; the second comes from the difference of the dielectric constant between the realistic and ideal model structures. We refer to them as the ‘vector term’ and ‘high-index term’, respectively.

_{g}**H**can be expressed by using the scalar solutions as basis functions:

**H**= ∑

_{n}a_{n}**h**

*, where*

_{n}*a*are expansion coefficients and

_{n}**H**includes

**h**

*in both the*

_{n}*x*and

*y*polarisations. Substituting this into Eq. (2) gives By multiplying

**h**

*and*

_{m}**h**

*have been analytically expressed using trigonometric functions. Moreover, the separation of the scalar fields along the*

_{n}*x*and

*y*axes reduces the area integrals in Eqs. (8) and (9) to one dimensional integrals. This makes the calculation of the matrix elements very straightforward and efficient. The details of the derivation are provided in Ref. [3].

*δL*in Eq. (5) can be classified into three types. The first are diagonal elements

_{mn}*δL*. These terms can be included in

_{mm}*L*

^{0}and cause a shift of the propagation constant for each mode. However, this effect does not lead to any interaction with other modes and therefore does not contribute to the loss. The diagonal terms of Eq. (5) can be rewritten as where

*m*, including both vector and high-index terms. As we shall see, these diagonal terms are substantially larger than the off-diagonal matrix elements. When considering cladding modes that have propagation constants close to the fundamental mode, it is these shifted eigenvalues that will be analysed. It is also these shifts that remove the bandgaps that are present in a scalar solution of the ideal model structure.

*δL*

_{m0}

*and*

_{n}*δL*

_{nm0}, where

*m*

_{0}denotes

*m*

_{0}

*or*

_{x}*m*

_{0}

*, representing the*

_{y}*x*or

*y*polarised fundamental mode, and

*n*labels a cladding mode. Coupling of this type leads to light leaking from the fundamental mode into the cladding, giving rise to confinement loss. The third type,

*δL*

_{m0xm0y}and

*δL*

_{m0ym0x}which represent coupling between the two polarisations of the fundamental mode. In practice, however, these terms are zero due to symmetry.

### 2.2. Attenuation due to mode interactions

2. P. W. Anderson, “Localized magnetic states in metals,” Phys. Rev. **124**, 41–53 (1961). [CrossRef]

*δL*″ between different cladding modes, an analytic analysis can be performed. This approximation should be applicable because the attenuation is mainly caused by the interaction between the fundamental mode and cladding modes. The cladding-cladding terms are likely to be less important in affecting the overall magnitude of the attenuation.

*G*

_{m0x}

_{m0x}: Because of the symmetry of the two polarised fundamental modes, we have

*V*

_{m0x}

_{m0x}=

*V*

_{m0y}

_{m0y}and

*V*

_{m0x}

_{m0y}=

*V*

_{m0y}

_{m0x}. Equation (21) then becomes In Eq. (22), the term containing

*x*polarised fundamental mode.

*P*is the principal value and

*δ*(

*x*) is a Dirac delta function, the imaginary part of the perturbation terms in Eq. (23) can be expressed as Equation (25) is the key equation of this paper; it gives an analytic expression from which the attenuation of the fundamental mode due to interaction with cladding modes can be derived. It can be seen that the attenuation depends on two factors. One is the density of the cladding states which is expressed through the term

*m*

_{0}

*and*

_{x}*m*are labels of the

*x*polarised fundamental mode and cladding modes, respectively. The values of both of these parts can be calculated using Eqs. (8) to (10).

*σ*; a small value of

*σ*represents slight smoothing and a sharp peak in the plots. The value of

*σ*will be determined through convergence tests in the following.

## 3. Results for the model PCF structure

### 3.1. Mode distribution and interaction

*β*Λ)

^{2}for the dimensionless propagation constant and

*δL*

^{′}Λ

^{2}for the dimensionless perturbation matrix element.

*β*Λ)

^{2}, which is determined by Eq. (10). These shifted modes are the basis states for our perturbation theory. We calculate the diagonal shift for all the scalar modes of the ideal structure and then consider only the shifted modes which are close to the air-line in the calculation of attenuation.

*×*8 supercell with a normalised frequency

*k*

_{0}Λ = 40; this was analysed in detail in Ref. [1

*β*Λ)

^{2}less than 20). After the diagonal shift, the (

*β*Λ)

^{2}value for the fundamental guided mode is 1587.463, which is 0.342 greater than the unperturbed value. The nearest air-guided and glass-guided modes are separated from the fundamental mode by (

*β*Λ)

^{2}differences of 1.905 and 0.199, respectively.

*β*Λ)

^{2}are 0.604 and 222.707 for the air-guided and glass-guided modes, respectively. By comparison, the off-diagonal elements of the vector term are considerably smaller. Typical values are of order 10

^{−1}, 10

^{0}and 10

^{−3}respectively for air-air, glass-glass and air-glass interactions for the same polarised modes. For the interaction between different polarisations, they are of order 10

^{−5}, 10

^{0}and 10

^{−2}for the three groups. The high-index terms exhibit a rather different pattern. The average diagonal shifts are 0.001 and 32.705 for the air-guided and glass-guided modes, which are significantly smaller than for the vector term. For the off-diagonal elements of high-index term, typical values are 10

^{−5}, 10

^{0}and 10

^{−2}for air-air, glass-glass and air-glass mode interactions with the same polarisation; the interactions between differently polarised modes are identically zero. In general, we conclude that, apart from the diagonal term, the magnitude of the perturbation is relatively small, which gives us confidence in the validity of our perturbation theory. This also justifies our neglect of the second order terms in Eq. (22). In general the vector terms tend to be larger than the high-index terms, but neither can be neglected in the perturbation calculation.

### 3.2. Attenuation calculations

*σ*value should first be determined. For high precision, we want

*σ*sufficiently small so that only those states close to the fundamental mode are included. However, we also need an adequate number of these cladding states for computational accuracy. In this case, the size of supercell should be as large as possible. However, a problem arises if the supercell becomes too large. As discussed in the previous paper, modes are found by searching for identical field values at the centres of neighbouring supercells [1

*×*32 in size, even quadruple precision is insufficient. In the determination of

*σ*values, we have therefore chosen a set of supercells no larger than 32 × 32.

*β*Λ)

^{2}value on the right-hand side of Eq. (28) should be chosen to be that of the fundamental mode. However, to investigate convergence of the calculations it is convenient to plot the imaginary part of Δ(

*β*Λ)

^{2}for the fundamental mode as a function of (

*β*Λ)

^{2}. In general, we find that the air-guided modes have much higher density of states than the glass-guided modes; we therefore consider the air-guided and glass-guided modes separately. A larger

*σ*is required for the glass-guided modes to give a smooth density of states. Plots of the imaginary part of Δ(

*β*Λ)

^{2}for the fundamental mode are shown in Fig. 1 as a function of the smoothing width

*σ*and the size of the supercell.

*σ*that provides well converged results is 0.3. In this case the difference between 28 × 28 and 32 × 32 supercells is less than 1% over the whole range shown in Fig. 1. The glass-guided modes show a broader distribution over a wide range of (

*β*Λ)

^{2}, as shown in the right of Fig. 1. It is found that the smallest acceptable value of

*σ*is 10 for the largest supercell we have used. In this case the difference between 28 × 28 and 32 × 32 supercells is less than 2% over the range shown in Fig. 1.

*β*Λ)

^{2}at the propagation constant of the fundamental guided mode is non-zero only for the glass-guided modes. The density of states of air-guided modes is zero at this propagation constant and so these modes do not contribute to the attenuation. We conclude that interaction with the glass-guided modes determines attenuation in the high-transmission region. This should also be valid for other members of the class of PCFs that guide light due to weak coupling of modes, and thus provides quantitative support for the conclusions drawn for Kagome [6

6. F. Couny, F. Benabid, and P. S. Light, “Large-pitch Kagome-structured hollow-core photonic crystal fiber,” Opt. Lett. **31**, 3574–3576 (2006). [CrossRef] [PubMed]

8. A. Argyros and J. Pla, “Hollow-core polymer fibres with a kagome lattice: potential for transmission in the infrared,” Opt. Express **15**, 7713–7719 (2007). [CrossRef] [PubMed]

9. F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express **16**, 20626–20636 (2008). [CrossRef] [PubMed]

### 3.3. Frequency dependence of the attenuation

*β*, we now use it to analyse the dependence of the attenuation on frequency and fibre structure. At the sample frequency of

*k*

_{0}Λ = 40, the imaginary part of

*β*Λ for the fundamental mode is calculated to be 1.58

*×*10

^{−6}, indicating a low level of leakage.

*k*

_{0}Λ = 26 to 46 at a spacing of 0.2. Similar to that observed in Kagome and square-lattice hollow-core PCFs, the attenuation for rectangular hollow-core PCFs varies dramatically as a function of frequency. In the selected range, it generally shows a decreasing trend with increasing frequency. This feature can also be seen via three peaks at

*k*

_{0}Λ = 28.0, 33.2 and 37.4, for which the imaginary

*β*Λ declines from 5.1

*×*10

^{−5}to 1.3 × 10

^{−5}.

10. N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada, “Boundary element method for analysis of holey optical fibers,” J. Lightwave Technol. **21**, 1787–1792 (2003). [CrossRef]

12. X. Wang, J. Lou, C. Lu, C.-L. Zhao, and W. Ang, “Modeling of pcf with multiple reciprocity boundary element method,” Opt. Express **12**, 961–966 (2004). [CrossRef] [PubMed]

*μ*m) [6

6. F. Couny, F. Benabid, and P. S. Light, “Large-pitch Kagome-structured hollow-core photonic crystal fiber,” Opt. Lett. **31**, 3574–3576 (2006). [CrossRef] [PubMed]

*μ*m) [9

9. F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express **16**, 20626–20636 (2008). [CrossRef] [PubMed]

*β*Λ is between 10

^{−6}and 10

^{−5}. The corresponding confinement loss varies from 0.58 dB/m to 5.8 dB/m (if we choose the pitch Λ = 15

*μ*m), in good agreement with the experimentally measured values.

7. F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science **318**, 1118–1121 (2007). [CrossRef] [PubMed]

8. A. Argyros and J. Pla, “Hollow-core polymer fibres with a kagome lattice: potential for transmission in the infrared,” Opt. Express **15**, 7713–7719 (2007). [CrossRef] [PubMed]

*β*=

*k*

_{0}, the resonance condition is given by

7. F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science **318**, 1118–1121 (2007). [CrossRef] [PubMed]

*n*and

_{g}*n*are the refractive indices for the glass and air, respectively,

_{a}*t*is the thickness of the glass strips, and

*j*is an integer. In the vicinity of a resonance the attenuation is found to increase; in Kagome hollow-core PCFs, a high loss region of width 200 nm separates the high transmission windows [7

7. F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science **318**, 1118–1121 (2007). [CrossRef] [PubMed]

*t*/Λ is 0.05; the lowest normalised resonance frequency is therefore

*k*

_{0}Λ = 56.2. The attenuations shown in Fig. 2 are located within the first high-transmission window (i.e. between

*j*= 0 and 1). In the following calculations, we focus on the guidance only within this window.

### 3.4. Effect of the fibre structure

*k*

_{0}Λ = 40. Similar peaks to those observed in Fig. 2 appear at

*t*/Λ = 0.030, 0.037 and 0.044; a comparison between the two plots shows that reducing the thickness of the glass struts leads to a shift of the attenuation feature towards a lower wavelength. For example, the highest frequency peak for

*t*/Λ = 0.05 is at

*k*

_{0}Λ = 37.4; for the structure with

*t*/Λ = 0.044, it moves to

*k*

_{0}Λ = 40.0. However, the overall appearance of the attenuation features remains unchanged.

*D*. The frequency

*k*

_{0}Λ and the thickness of glass strips

*t*/Λ are fixed at 40 and 0.05, respectively. The central defect is made by simultaneously moving the four glass strips nearest to the centre by a distance of between 0.03Λ and 0.185Λ.

*β*Λ vary more smoothly than those for the strut thickness, from 2.7 × 10

^{−7}to 4.7 × 10

^{−6}. The plots show a low-attenuation region, from

*D*= 1.1Λ to 1.15Λ, surrounded by two relatively high-loss areas. The minimum value of the imaginary

*β*Λ is about 2.8

*×*10

^{−7}(i.e. a loss of 0.16 dB/m for Λ = 15

*μ*m) at 1.11Λ. In this case, both the vector and the high-index perturbations are simultaneously suppressed.

*D*= 1.20Λ. They exhibit a very similar variation with frequency, but for the smaller core size, the attenuation characteristic is shifted towards a lower wavelength. This is similar to what happens for the thinner glass strips. A comparison of the magnitude in Fig. 4(b) shows that, for a larger central defect, the average value of the leakage is relatively smaller.

## 4. Conclusion

## Acknowledgments

## References and links

1. | L. Chen, G. J. Pearce, T. A. Birks, and D. M. Bird, “Guidance in Kagome-like photonic crystal fibres I: analysis of an ideal fibre structure,” Submitted to Opt. Express (2011). [PubMed] |

2. | P. W. Anderson, “Localized magnetic states in metals,” Phys. Rev. |

3. | L. Chen, “Modelling of photonic crystal fibres,” Ph.D. thesis, University of Bath (2009). |

4. | S. Davison and M. Stesliska, |

5. | E. Economou, |

6. | F. Couny, F. Benabid, and P. S. Light, “Large-pitch Kagome-structured hollow-core photonic crystal fiber,” Opt. Lett. |

7. | F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science |

8. | A. Argyros and J. Pla, “Hollow-core polymer fibres with a kagome lattice: potential for transmission in the infrared,” Opt. Express |

9. | F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express |

10. | N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada, “Boundary element method for analysis of holey optical fibers,” J. Lightwave Technol. |

11. | T.-L. Wu and C.-H. Chao, “Photonic crystal fiber analysis through the vector boundary-element method: effect of elliptical air hole,” IEEE Photon. Technol. Lett. |

12. | X. Wang, J. Lou, C. Lu, C.-L. Zhao, and W. Ang, “Modeling of pcf with multiple reciprocity boundary element method,” Opt. Express |

13. | Y. Wang, F. Couny, P. Roberts, and F. Benabid, “Low loss broadband transmission in optimized core-shape kagome hollow-core pcf,” in “Lasers and Electro-Optics (CLEO) and Quantum Electronics and Laser Science Conference (QELS), 2010 Conference on,” (2010), pp. 1–2. |

**OCIS Codes**

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

(060.2400) Fiber optics and optical communications : Fiber properties

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: January 27, 2011

Manuscript Accepted: March 9, 2011

Published: March 25, 2011

**Citation**

Lei Chen and David M. Bird, "Guidance in Kagome-like photonic crystal fibres II: perturbation theory for a realistic fibre structure," Opt. Express **19**, 6957-6968 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6957

Sort: Year | Journal | Reset

### References

- L. Chen, G. J. Pearce, T. A. Birks, and D. M. Bird, “Guidance in Kagome-like photonic crystal fibres I: analysis of an ideal fibre structure,” Submitted to Opt. Express (2011). [PubMed]
- P. W. Anderson, “Localized magnetic states in metals,” Phys. Rev. 124, 41–53 (1961). [CrossRef]
- L. Chen, “Modelling of photonic crystal fibres,” Ph.D. thesis, University of Bath (2009).
- S. Davison and M. Stesliska, Basic Theory of Surface States (Oxford U. Press, 1992).
- E. Economou, Green’s Functions in Quantum Physics (Springer-Verlag, 1990).
- F. Couny, F. Benabid, and P. S. Light, “Large-pitch Kagome-structured hollow-core photonic crystal fiber,” Opt. Lett. 31, 3574–3576 (2006). [CrossRef] [PubMed]
- F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 1118–1121 (2007). [CrossRef] [PubMed]
- A. Argyros and J. Pla, “Hollow-core polymer fibres with a kagome lattice: potential for transmission in the infrared,” Opt. Express 15, 7713–7719 (2007). [CrossRef] [PubMed]
- F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express 16, 20626–20636 (2008). [CrossRef] [PubMed]
- N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada, “Boundary element method for analysis of holey optical fibers,” J. Lightwave Technol. 21, 1787–1792 (2003). [CrossRef]
- T.-L. Wu and C.-H. Chao, “Photonic crystal fiber analysis through the vector boundary-element method: effect of elliptical air hole,” IEEE Photon. Technol. Lett. 16, 126–128 (2004). [CrossRef]
- X. Wang, J. Lou, C. Lu, C.-L. Zhao, and W. Ang, “Modeling of pcf with multiple reciprocity boundary element method,” Opt. Express 12, 961–966 (2004). [CrossRef] [PubMed]
- Y. Wang, F. Couny, P. Roberts, and F. Benabid, “Low loss broadband transmission in optimized core-shape kagome hollow-core pcf,” in “Lasers and Electro-Optics (CLEO) and Quantum Electronics and Laser Science Conference (QELS), 2010 Conference on,” (2010), pp. 1–2.

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.