## Bloch method for the analysis of modes in microstructured optical fibers

Optics Express, Vol. 12, Issue 8, pp. 1769-1774 (2004)

http://dx.doi.org/10.1364/OPEX.12.001769

Acrobat PDF (1960 KB)

### Abstract

We discuss a transform technique for analyzing the wave vector content of microstructured optical fiber (MOF) modes, which is computationally efficient and gives good physical insight into the nature of the mode. In particular, if the mode undergoes a transition from a bound state to an extended state, this is evident in the spreading-out of its transform. The method has been implemented in the multipole formulation for finding MOF modes, but are capable of adaptation to other formulations.

© 2004 Optical Society of America

## 1. Introduction

6. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers I: formulation,” J. Opt. Soc. Am. B19, 2322–2330 (2002), and “Erratum,” J. Opt. Soc. Am. B20, 1581 (2003). [CrossRef]

7. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole formulation for microstructured optical fibers II: implementation and results,” J. Opt. Soc. Am. B **19**, 2331–2340 (2002). [CrossRef]

## 2. Bloch transform of MOF modes

*e.g*. photonic crystal cladding) have received most attention. Properties of such MOFs are closely linked to the band structure of their cladding, and popular models such as the effective index model for solid core MOFs are based on properties of the band structure [9

9. T. A. Birks, J. C. Knight, and St. J. Russel, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. **22**961–963 (1997). [CrossRef] [PubMed]

*N*

_{i}inclusions centered around position vectors

**c**

_{l}(

*l*∈[1..

*N*

_{i}]). We assume that position vectors

**c**

_{l}describe a subset of an infinite periodic lattice

*𝓛*. To form the Bloch transform of a given MOF mode, we choose a number of quantities

*B*

_{n}(

**c**

_{l}) characterizing the complex field amplitudes at each of the

*N*

_{i}inclusions. In the multipole formulation, it is natural to take these to be the amplitudes of the multipoles occurring in the expansions of

*E*

_{z}and

*H*

_{z}, but in other methods they could be simply the complex values of

*E*

_{z}and

*H*

_{z}at a small number of points in each inclusion. We then define the Bloch transform for quantity

*B*

_{n}by

*N*

_{B}Bloch waves with Bloch vectors

*𝓛*, quantities

*B*

_{n}(

**c**

_{l}) satisfy

*B*

_{n}(

**c**

_{l}). From Eqs. (1) and (3) it is then straightforward to see that

*𝓑*

_{n}(

**k**) peaks when

**k**=

*𝓛**. Indeed the reciprocal lattice is defined by all vectors

**G**such that

*𝓛*, adding any vector

**G**of

*𝓛** to

**k**in Eq. (1) leaves the result unchanged. It is hence sufficient to compute the Bloch transform in the first Brillouin zone (FBZ) associated with

*𝓛*.

*𝓑*

_{n}(

**k**)| as a function of

**k**. If this is not the case, it may be convenient to form the

*total Bloch transform*, by summing over the (appropriately normalized) transforms of all the representative quantities

## 3. Examples and basic properties of the Bloch transform

*N*

_{r}=4 the peaks in the Bloch transform are quite broad, and quite naturally become much narrower for

*N*

_{r}=10. For

*N*

_{r}=4, secondary peaks are not negligible; this exemplifies the importance of surface and defect effects for a system consisting of only 4 periodic layers. For

*N*

_{r}=10 on the contrary, peaks are well defined, and the importance of secondary peaks is less, suggesting that edge effects are becoming negligible. In both cases the main peaks are close to the edge of the first Brillouin zone, indicating that each Bloch component is close to a standing wave. We note that the exact position of the maxima of the peaks is not the same in both cases, but that the “overall shape” remains constant. We further note the predicted periodicity in the reciprocal space of

*𝓑*

^{T}(

**k**): the peaks outside the first Brillouin zone are replicates of the peaks inside the first Brillouin zone, and do not contain any additional information. Finally, we note that the Bloch transform has symmetry properties induced by the symmetry properties of the mode.

*N*

_{r}=8 holes of air inclusions in silica, with the same relative hole size

*d*/Λ=0.3 and at same wavelength λ=1.55

*µ*m, but with different values of the pitch. We see that the field patterns differ considerably, but that the Bloch transform remains similar for the two values of the pitch: there is only one peak centered on

**k**=0, only the width of the peak changes, being much narrower for the wider mode than for the well confined mode.

*cf*Section 4). The more localized a mode is in real space, the more spread out is its Bloch transform in reciprocal space.

*shape*” of the Bloch transform - is characteristic of a MOF mode, and is extremely stable when varying the wavelength or the fiber parameters. We found in our studies of mode transitions in MOFs [3

3. B. T. Kuhlmey, R. C. McPhedran, and C. M. de Sterke, “Modal cutoff in microstructured optical fibers,” Opt. Lett. , **27**, 1684–1686 (2002). [CrossRef]

4. B. T. Kuhlmey, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre, “Microstructured optical fibers: where’s the edge?,” Opt. Express ,**10**, 1285–1290 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285 [CrossRef] [PubMed]

## 4. Advanced properties

*N*

_{B}of Bloch waves so that Eq. (3) is satisfied.

### 4.1. Normalizing the Bloch transform: Bloch wave decomposition and Parseval identity

*𝓑*

_{n}(

**k**) can also be given a meaning.

**k**takes the value of one of the Bloch vectors

*π*/Λ, Eq.(8) is equivalent to having the distance between peaks, normalised to the width of the FBZ, greater than 2/(

*𝓐*

_{FBZ}is the area of the FBZ. Here the first identity is rigorous, and the second is valid when Eq. (8) is satisfied.

### 4.2. Width of the Bloch transform peaks: Heisenberg-like uncertainty

*B̂*

_{n}and Bloch vector

**k**

_{B}are of the form

**u**

_{i}denote the unitary vectors along the directions defined by the elementary vectors defining

*𝓛*and

*N*

_{i}the number of inclusions along these same directions (so that

*N*

_{i}is of the order of

*N*

_{1}

*N*

_{2}). The peaks of the Bloch transform along each direction are therefore of the same type as the function

*a*for

*x*=

*mπ, m*∈ℤ, and for large values of

*a*has half-width points at

*x*⋍

*mπ*±1.91/

*a*. The width

*δk*

_{m}of the peaks of the Bloch transform along

**u**

_{m}is thus given by

*e.g*. when the mode is a surface or defect state, the relation has to be modified. In the case of a localized defect mode (

*e.g*. the mode for Λ=2.3

*µ*m in Fig. 2), the magnitude of the

*B*

_{n}(

**c**

_{l}) coefficients decays exponentially away from the defect. In that case only the

*B*

_{n}(

**c**

_{l}) coefficients associated with inclusions close to the defect contribute significantly to the Bloch transform. The analytical analysis leading to Eq. (14) shows that the Heisenberg relation (14) remains true in these cases if

*N*

_{m}is replaced by the number of inclusions on which the mode’s fields are significant, so that (14) is in fact a relation between the spatial extent of the mode and the width of the Bloch transform peaks.

## 5. Discussion and conclusions

## References and links

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

2. | 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 |

3. | B. T. Kuhlmey, R. C. McPhedran, and C. M. de Sterke, “Modal cutoff in microstructured optical fibers,” Opt. Lett. , |

4. | B. T. Kuhlmey, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre, “Microstructured optical fibers: where’s the edge?,” Opt. Express , |

5. | N. A. Mortensen, “Effective area of photonic crystal fibers”, Opt. Express |

6. | T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers I: formulation,” J. Opt. Soc. Am. B19, 2322–2330 (2002), and “Erratum,” J. Opt. Soc. Am. B20, 1581 (2003). [CrossRef] |

7. | B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole formulation for microstructured optical fibers II: implementation and results,” J. Opt. Soc. Am. B |

8. | |

9. | T. A. Birks, J. C. Knight, and St. J. Russel, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. |

10. | M. Reed and B. Simon, |

11. | G. Allaire, C. Conca, and M. Vanninathan, “The Bloch Transform and applications,” 29th Congress of Numerical Analysis, ESAIM: Proceedings 3, 65–84 (1998), http://www.edpsciences.org/articlesproc/Vol.3/conca/conca.htm |

12. | P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides-I: Summary of results,” IEEE Trans. Microwave Theory Tech. |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(060.2310) Fiber optics and optical communications : Fiber optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 2, 2004

Revised Manuscript: April 6, 2004

Published: April 19, 2004

**Citation**

Boris Kuhlmey, Ross McPhedran, and C. de Sterke, "Bloch method for the analysis of modes in microstructured optical fibers," Opt. Express **12**, 1769-1774 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-8-1769

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### References

- P. St J. Russell, �??Photonic crystal fibers,�?? Science 299, 358-362 (2003). [CrossRef] [PubMed]
- 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 424, 657-659 (2003). [CrossRef] [PubMed]
- B. T. Kuhlmey, R. C. McPhedran, and C. M. de Sterke, �??Modal cutoff in microstructured optical fibers,�?? Opt. Lett., 27, 1684-1686 (2002). [CrossRef]
- B. T. Kuhlmey, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre, �??Microstructured optical fibers: where�??s the edge?,�?? Opt. Express,10, 1285-1290 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285</a>. [CrossRef] [PubMed]
- N. A. Mortensen, �??Effective area of photonic crystal fibers�??, Opt. Express 10, 341-348 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341</a>. [CrossRef] [PubMed]
- T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, �??Multipole method for microstructured optical fibers I: formulation,�?? J. Opt. Soc. Am. B 19, 2322-2330 (2002), and �??Erratum,�?? J. Opt. Soc. Am. B 20, 1581 (2003). [CrossRef]
- B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke and R. C. McPhedran, �?? Multipole formulation for microstructured optical fibers II: implementation and results,�?? J. Opt. Soc. Am. B 19, 2331-2340 (2002). [CrossRef]
- <a href="http://www.physics.usyd.edu.au/cudos/mofsoftware/">http://www.physics.usyd.edu.au/cudos/mofsoftware/</a>.
- T. A. Birks and J. C. Knight and St. J. Russel, �??Endlessly single-mode photonic crystal fiber,�?? Opt. Lett. 22 961-963 (1997). [CrossRef] [PubMed]
- M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators (Academic, New York, 1978).
- G. Allaire, C. Conca and M. Vanninathan,�??The Bloch Transform and applications,�?? 29th Congress of Numerical Analysis, ESAIM: Proceedings 3, 65-84 (1998), <a href="http://www.edpsciences.org/articlesproc/Vol.3/conca/conca.htm">http://www.edpsciences.org/articlesproc/Vol.3/conca/conca.htm</a>.
- P. R. McIsaac, �??Symmetry-induced modal characteristics of uniform waveguides-I: Summary of results,�?? IEEE Trans. Microwave Theory Tech. MTT-23, 421-429 (1975). [CrossRef]

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