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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 8 — Apr. 19, 2004
  • pp: 1769–1774
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Bloch method for the analysis of modes in microstructured optical fibers

Boris T. Kuhlmey, Ross C. McPhedran, and C. Martijn de Sterke  »View Author Affiliations


Optics Express, Vol. 12, Issue 8, pp. 1769-1774 (2004)
http://dx.doi.org/10.1364/OPEX.12.001769


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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

We discuss here a solution this challenge, which has proved effective within the context of studies using the multipole method [6

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

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]

]. It is not only present in the multipole code, which is available for free download [8], but could also be easily implemented in other methods which can deliver accurate solutions for the electromagnetic fields of modes in MOFs.

2. Bloch transform of MOF modes

We consider a MOF with 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 Bn (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 Ez and Hz , but in other methods they could be simply the complex values of Ez and Hz at a small number of points in each inclusion. We then define the Bloch transform for quantity Bn by

𝓑n(k)=l=1Niexp(ik·cl)Bn(cl).
(1)

If a mode consists of a superposition of N B Bloch waves with Bloch vectors kBm, so that its field distribution satisfies

V(r)=m=1NBexp(ikBm·r)vkBm(r),
(2)

where functions vkBm(r) have the periodicity of the lattice 𝓛, quantities Bn (c l) satisfy

Bn(cl)=m=1NBB̂nmexp(ikBm·cl),
(3)

where B^nm is the complex amplitude of the Bloch wave associated with Bloch vector kBm in the decomposition of Bn (c l). From Eqs. (1) and (3) it is then straightforward to see that 𝓑n (k) peaks when k=kBm.

Fig. 1. Field maps and total Bloch transform of a mode consisting essentially of a superposition of 6 Bloch waves. Note that the fields are depicted in the direct space (r-space), whereas the Bloch transform is in the reciprocal space (k-space): the white hexagon on the Bloch transform map depicts the edges of the first Brillouin zone. Here Λ=2.3 µm, λ=1.55 µm, d/Λ=0.15, and n silica=1.44402036.

Further, Eq. (1) shows that the Bloch transform has the periodicity of the reciprocal lattice 𝓛*. Indeed the reciprocal lattice is defined by all vectors G such that

c𝓛,G·c2π,
(4)

where ℤ is the set of all integers. Since the vectors cl form a subset of 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.

For a given mode, the essential information in the Bloch transform may be captured by plotting a single |𝓑n (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

𝓑T(k)=n1supk2(𝓑n(k))𝓑n(k).
(5)

3. Examples and basic properties of the Bloch transform

We now illustrate some of the properties of the Bloch transform in the context of two examples. In both the MOFs are made out of a cladding of hexagonally packed air inclusions in silica, the core being defined by a missing hole.

Fig. 2. Fundamental mode of two MOFs with different pitch, but with same d/Λ=0.3 and N r=8. The field distribution changes considerably between the two values of the pitch, but the Bloch transform remains a single peak centered on the origin. For all figures λ=1.55 µm and n silica=1.44402036.

We see that for 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 𝓑T (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.

Figure 2 shows a second example illustrating how helpful the Bloch transform becomes when trying to identify modes of MOFs having different yet comparable structures. Both lines of contour plots shown in Fig. 2 relate to the fundamental mode of a MOF with N r=8 holes of air inclusions in silica, with the same relative hole size d/Λ=0.3 and at same wavelength λ=1.55µm, 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.

This example illustrates well the Heisenberg-type property of the Bloch transform, which relates to its definition as a finite Fourier transform (cf Section 4). The more localized a mode is in real space, the more spread out is its Bloch transform in reciprocal space.

The most striking property of the Bloch transform is that the geometric distribution of the peaks - the “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

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]

] that the field distributions and the Fourier-Bessel coefficients of the same mode can vary considerably with varying fiber parameters, and that it can become extremely difficult to identify similar modes of different yet comparable structures. Our simulations have shown that the Bloch transform of a given mode keeps its shape regardless of the fiber parameters, and that this shape is indeed the most precise and convenient way to differentiate and define specific modes. It is difficult to prove this property without precisely defining a classification of modes, but it can be understood through the fact that the Bloch transform decomposes each mode in a natural basis for the structure. In fact we can use this property as an axiom, defining the nature of a mode through the properties of its Bloch transform.

4. Advanced properties

We here consider a mode consisting of a superposition of a finite number N B of Bloch waves so that Eq. (3) is satisfied.

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

Until now we have only interpreted the location of peaks of the Bloch transform. However, through a correct normalization of the Bloch transform, the numerical value of 𝓑n (k) can also be given a meaning.

First we make explicit the value of the Bloch transform when k takes the value of one of the Bloch vectors kBm of the field:

𝓑n(kBm)=l=1Nij=1NBB̂njexp(i(kBmkBj)·cl),
(6)
=NiB̂nm+j=1,jmNBB̂njl=1Niexp(i(kBmkBj)·cl).
(7)

The analysis below holds when the last term in Eq. (7) can be neglected compared to the first. It can be shown that this is so when,

G𝓛*,jm,kBmkBj+GNi12Λ4.
(8)

The width of the FBZ being of the order of 2π/Λ, Eq.(8) is equivalent to having the distance between peaks, normalised to the width of the FBZ, greater than 2/(πNi1/2). We will see in Eq. (14) that the normalized width of the peaks is of the order of 1.91/(πNi1/2) so that Eq. (8) is in fact equivalent to the requirement that the peaks associated with the different Bloch factors do not overlap. In that case, then

𝓑n(kBm)NiB̂nm.
(9)

The value of the Bloch transform taken at a Bloch vector kBm gives therefore an approximation of the complex amplitude B^nm of the associated Bloch wave component in Eq. (3). Nevertheless, because of linearity, only relative amplitudes in that decomposition have a physical meaning, it is necessary to be able to compare the amplitudes to each other. Mathematically speaking, we aim at obtaining

B̂nm(Σj=1NBB̂nj2)12.
(10)

Rather than using Eq. (9) to evaluate the sum in Eq. (10), it is easier and numerically more precise to use Parseval like identities, whose derivations follow lines similar to the derivation of Parseval identites for Fourier series. We have

l=1NiBn(cl)2=1𝓐FBZFBZ𝓑n(k)2dkNii=1NBB̂ni2,
(11)

where 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

We have noted on the examples that the width of the peaks centered on the Bloch vectors kBm decreases with increasing cladding size. An analytic analysis of Bloch transform peaks shows that peaks corresponding to one Bloch wave of amplitude n and Bloch vector k B are of the form

𝓑n(k)=B̂nsin(N1(kkb)·u1Λ2)sin((kkb)·u1Λ2)sin(N2(kkb)·u2Λ2)sin((kkb)·u2Λ2),
(12)

where u i denote the unitary vectors along the directions defined by the elementary vectors defining 𝓛 and Ni 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

f(x)={sin(ax)sin(x)ifxmπ,maifx=mπ,m.
(13)

This function has main peaks of value a for x=mπ, m∈ℤ, and for large values of a has half-width points at x±1.91/a. The width δkm of the peaks of the Bloch transform along u m is thus given by

δkm21.91NmΛ.
(14)

The result to be retained from this analysis is that the width of a Bloch transform peak along a given direction varies as the inverse of cladding width along that same direction.

In the above analysis, the mode was assumed to be a superposition of a finite number of Bloch waves. When this is not the case, 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=2.3µm in Fig. 2), the magnitude of the Bn (c l) coefficients decays exponentially away from the defect. In that case only the Bn (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 Nm 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

We have found the Bloch transform discribed above to be extremely valuable for studies of MOF modes. Its ability to analyse modes in terms of resonating Bloch waves let us gain considerable insights into the physics of MOF modes, allowed us to identify modes without ambiguitiy and made a clear-cut distinction between cladding and defect modes possible. Although the method has been designed and tailored with the multipole method in mind, it is applicable within the framework of any method capable of generating accurate distributions of the electric and magnetic fields in MOFs. We consider its use will be of great advantage in design studies of MOFs for a range of advanced applications.

This work was produced with the assistance of the Australian Research Council under the ARC Centres of Excellence Program.

References and links

1.

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

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 424, 657–659 (2003). [CrossRef] [PubMed]

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]

5.

N. A. Mortensen, “Effective area of photonic crystal fibers”, Opt. Express 10, 341–348 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341 [CrossRef] [PubMed]

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]

8.

http://www.physics.usyd.edu.au/cudos/mofsoftware/

9.

T. A. Birks, J. C. Knight, and St. J. Russel, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22961–963 (1997). [CrossRef] [PubMed]

10.

M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators (Academic, New York, 1978).

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. MTT-23, 421–429 (1975). [CrossRef]

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

  1. P. St J. Russell, �??Photonic crystal fibers,�?? Science 299, 358-362 (2003). [CrossRef] [PubMed]
  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 424, 657-659 (2003). [CrossRef] [PubMed]
  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), <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]
  5. 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]
  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. B 19, 2322-2330 (2002), and �??Erratum,�?? J. Opt. Soc. Am. B 20, 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]
  8. <a href="http://www.physics.usyd.edu.au/cudos/mofsoftware/">http://www.physics.usyd.edu.au/cudos/mofsoftware/</a>.
  9. 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]
  10. M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators (Academic, New York, 1978).
  11. 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>.
  12. 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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