## Reflection symmetry and mode transversality in microstructured fibers

Optics Express, Vol. 12, Issue 8, pp. 1497-1509 (2004)

http://dx.doi.org/10.1364/OPEX.12.001497

Acrobat PDF (666 KB)

### Abstract

We investigate the influence of reflection symmetry on the properties of the modes of microstructured optical fibers. It is found that structures with reflection symmetry tend to support non-degenerate modes which are closer in nature to the analogous TE and TM modes of circular step-index fibers, as compared with fibers with only rotational symmetry. Reflection symmetry induces modes to exhibit smaller longitudinal components and transverse fields which are more strongly reminiscent of the radial and azimuthal modes of circular fibers. The tendency towards “transversality” can be viewed as a result of the interaction of group theoretical restrictions on the mode profiles and minimization of the Maxwell Hamiltonian.

© 2004 Optical Society of America

## 1. Introduction

1. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides,” IEEE Transactions on Microwave Theory and Techniques **MTT-23(5)**, 421–433 (1975). [CrossRef]

*C*

_{∞v}(circular waveguide),

*C*

_{∞}(helical waveguide),

*C*(discrete rotational symmetry of order

_{nv}*n*with a reflection axis) or

*C*(discrete rotational symmetry of order

_{n}*n*with no reflection axis). Notably, the standard circular fiber is an instance of group

*C*

_{∞v}, and the familiar hexagonal photonic crystal fiber is an instance of

*C*

_{6v}.

2. M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. **26**, 488–490 (2001). [CrossRef]

2. M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. **26**, 488–490 (2001). [CrossRef]

3. S.-H. Kim and Y.-H. Lee, “Symmetry relations of two-dimensional photonic crystal cavity modes,” IEEE J. Quantum Electron. **39**, 1081–1085 (2003). [CrossRef]

4. A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Perturbation analysis of dispersion properties in photonic crystal fibers through the finite element method,” J. Lightwave Technol. **20**, 1433–1442 (2002). [CrossRef]

5. M. Koshiba and K. Saitoh, “Structural dependence of effective area and mode field diameter for holey fibers,” Opt. Express **11**, 1746–1756, http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-15-1746. [PubMed]

*rotational*symmetry is required for degeneracy. An axis of reflection, while virtually ubiquitous in waveguide designs is unnecessary. For instance, all three “satellite” fibers shown in Fig. 1 support degenerate fundamental modes, even though only the first two support reflection symmetry [2

2. M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. **26**, 488–490 (2001). [CrossRef]

_{6v}, B

_{6v}and C

_{6}respectively, where the subscripts serve as a reminder of their symmetry groups. The fibers have hole diameter-pitch ratios of

*d*/Λ=0.8 (large holes), and

*d*/Λ=0.2 (small holes), and glass index

_{s}*n*=1.45. For the three fibers, the small holes are aligned at angles of 0°, 30°, and 15° with respect to the large holes.

_{co}_{0}and TM

_{0}modes of the standard circular fiber. By transversality, we mean both the degree to which the longitudinal components are small, and the degree to which the transverse mode profiles resemble the profiles of truly transverse modes. Overall, we find that reflection symmetry drives the modes towards transversality. Finally, we provide an explanation for this behavior based on the interaction of symmetry constraints and energy minimization principles.

## 2. Field symmetries in modal solutions to Maxwell’s equations

*z*axis with propagation constant

*β*and frequency

*ω*.

**r**

*=(*

_{t}*x*,

*y*). Then Maxwell’s equations allow us to express the transverse parts of the field in terms of the longitudinal components:

*n*(

**r**

*) is the local refractive index,*

_{t}*k*

_{0}=

*ω*/

*c*and

*n*(

**r**

*) is piecewise constant, we take the transverse divergence and curl of Eqs. (3) to find*

_{t}_{0}mode of a standard circular fiber, (see Fig. 2). The fields have the following properties:

*e*vanishes (by definition), and

_{z}*h*has no azimuthal dependence. Then Eqs. (4) require that

_{z}**h**

*is purely radially directed with ∇*

_{t}*×*

_{t}**h**

*=0, and*

_{t}**e**

*is purely azimuthally directed with ∇*

_{t}*·*

_{t}**e**

*=0, consistent with the profiles shown in Fig. 2(b-c).*

_{t}## 3. Review of symmetry classes

*C*

_{∞}guides the non-degenerate states are hybrid, and form a single class

*p*=1. (An example is an optical fiber with a short-pitch helical metal wire around the core [6

6. U. N. Singh, O. N. S. II, P. Khastgir, and K. K. Dey, “Dispersion characteristics of a helically cladded step-index optical fiber: an analytical study,” J. Opt. Soc. Am. B **12**, 1273–1278 (1995). [CrossRef]

*C*follow suit: the

_{n}*C*guides only support a single class

_{n}*p*=1 and even at large

*n*, the expansions for these modes always contain both

*e*and

_{z}*h*components, since the

_{z}*v*=0 terms in the expansion survive.

## 4. Comparison of circular fibers and MOFs

*BandSOLVE*by RSoft Design Group [7

7. RSoft Design Group, Inc. http://www.rsoftdesign.com.

^{16}plane waves per polarization. Note that the truncation of the plane wave expansion to a finite set of waves necessarily imposes a smoothing on the boundaries so that strictly speaking the results shown are not for perfectly discrete dielectric interfaces. Of course, with sufficient plane waves, the method converges to the results of other treatments that use discrete interfaces [2

**26**, 488–490 (2001). [CrossRef]

11. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express **11**, 1310–1321 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310. [CrossRef] [PubMed]

_{31}or EH

_{21}/HE

_{41}mode pair of circular fiber is split into separate states in a fiber with six-fold rotational symmetry.

## 5. Transverse profiles

*C*and

_{nv}*C*fibers. Both cases show the typical HE

_{n}_{11}pattern of an approximately uniformly polarized field with an intensity maximum at the core. Consequently, we will not examine the degenerate pair states further.

## 5.1. Quasi-transverse states

_{6v}. The field distributions show significant similarities with the TE

_{0}mode of the circular fiber—the

**h**

*field appears radial and the*

_{t}**e**

*field azimuthual. Indeed this type of mode, which formally belongs to the McIsaac class*

_{t}*p*=1 is customarily referred to as TE

_{0}in the MOF literature. The longitudinal

*h*field is also similar to the circular fiber mode, but of course it possesses six-fold symmetry rather than the continuous rotational symmetry in Fig. 2. The most obvious difference with the step-index fiber is the appearance of the longitudinal

_{z}*e*field which exhibits a six-fold set of oscillatory lobes circling the origin. One way to understand the appearance of a non-zero

_{z}*e*is to consider it as a source for the transverse fields in Eq. (3). In order for the transverse fields to satisfy the boundary conditions at the dielectric interfaces,

_{z}**h**

*and*

_{t}**e**

*can not be perfectly radial or azimuthal—both fields must possess a non-zero transverse curl and transverse divergence in the vicinity of the boundaries. This fact is illustrated in Fig. 5 which shows the quantity |∇*

_{t}*×*

_{t}**h**

*| for the*

_{t}*p*=1 mode. For the same mode in the circular fiber, this quantity vanishes. From the discussion following Eqs. (4), the non-vanishing of this quantity implies a nonzero

*e*. Thus the oscillatory

_{z}*e*component near the dielectric interfaces provides the extra freedom to the transverse fields needed to satisfy the boundary conditions. Note that this is consistent with the

_{z}*C*line in Table 1 which permits both longitudinal fields to be nonzero. All research to date suggests that all MOF modes are hybrid—there are no truly TE or TM states.

_{nv}*p*=2 which for the circular fiber is the TM

_{0}mode. For this mode, we find very similar profiles to those in Fig. 4 but with the roles of

**E**and

**H**reversed—

**e**

*is largely radial,*

_{t}**h**

*is largely azimuthal, and*

_{t}*h*shows oscillatory behavior. Since the plots are similar to those for the

_{z}*p*=1 mode with the labels reversed, we do not include them here. We have also calculated the field profiles for the same two modes in A

_{6v}. For each mode and field, the profiles are very similar to those of the first fiber.

## 5.2. Removing reflection symmetry

_{6}which does not have reflection symmetry. As with the previous fiber, the first six modes consist of two degenerate pairs, and two non-degenerate states, modes 2 and 5, both of which belong to the class

*p*=1. Figure 6 shows the field components for mode 5. While there are similarities to the earlier profiles, these fields are clearly quite different in nature. (We show the

*second*of the two

*p*=1 modes, because it resembles the profiles in Fig. 4 somewhat more closely than does the first

*p*=1 mode.) For the transverse fields, the distinction between radial and azimuthal fields is now completely lost; instead both fields have a spiral shape. The spiral character is also seen in the rather beautiful longitudinal field profiles. Moreover, whereas as in the C

_{6v}mode in Fig. 4,

*h*was essentially a central peak, while the “minor” field

_{z}*e*had an oscillatory character for the

_{z}*C*

_{6}mode, both

*h*and

_{z}*e*have a mixed oscillatory and peaked character.

_{z}## 6. Degree of transversality

*C*

_{6v}and

*C*

_{6}fibers are similar in structure to the corresponding transverse modes of step-index circular fibers, those of the

*C*

_{6v}fibers are considerably closer to the circular fiber modes. However, since neither the

*C*

_{6v}or

*C*

_{6}fibers support genuinely transverse modes, it is necessary to determine whether the greater transverse character of the

*shape*of the

*C*

_{6v}modes is reflected in the actual strength of the longitudinal components.

**H**” indicates the transversality

*f*[

_{z}**H**] for the magnetic field of the TE mode. For the two

*C*

_{6v}fibers, (Fig. 7a and b), we can clearly distinguish “major” components (TE,

**H**, and TM,

**E**) and “minor” components (TE,

**E**, and TM,

**H**) Note that the minor components are substantially smaller than the major components. Indeed, for fiber B

_{6v}, the minor components are a factor of ten smaller than the major components and for both fibers, the minor components are approximately equal or smaller to the transversality of the fundamental HE

_{11}modes. On the other hand, for fiber C

_{6}, (Fig.7c), the distinction between minor and major components is much less sharp, particularly at higher frequencies. Thus in this case, the TE/TM designation is truly inappropriate and we simply label the modes as two instances of the

*p*=1 class.

## 7. Systematic studies

*C*symmetry [Fig. 8(a-b)], consisted of concentric rings of holes. In the innermost ring,

_{nv}*n*holes are removed. The fibers with

*C*symmetry [Fig. 8(c-d)] consisted of

_{n}*n*spirals. Such fibers have not been seen in practice yet, but could be easily created using polymer materials [12

12. M. van Eijkelenborg, M. Large, A. Argyros, J. Zagari, S. Manos, N. A. Issa, I. M. Bassett, S. C. Fleming, R. C. McPhedran, C. M. de Sterke, and N. A. P. Nicorovici, “Microstructured polymer optical fibre,” Opt. Express **9**, 319–327 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-7-319. [CrossRef] [PubMed]

_{0}and TM

_{0}like profiles. The

*C*modes again exhibit many characteristics of circular fibers while the spiral fibers show much less distinction between TE and TM-like states.

_{nv}*n*=3 to

*n*=7 we adjusted the spacing of the rings and holes in order to match the effective index and mode areas of corresponding modes to within 3–4%. This was done essentially by trial and error. The transversality of the modes as defined by Eqs. (5) was then calculated as before. We show results for the case of

*n*=4 which are typical. For the concentric ring

*C*

_{4v}fiber, we chose a scale length Λ=4 and hole diameter

*d*/Λ=3/9. The first ring of holes had a radius of 2.75Λ, and the radii of subsequent rings increased in steps of 0.75Λ. The holes were equally spaced around each ring. For the spiral

*C*

_{4}fiber, the hole sizes were the same and successive rotations of the spirals were approximately Λ apart. Fig. 9 shows the fraction of each field in both the TE and TM-like modes for the

*C*

_{4}and

*C*

_{4v}fibers. The curves labeled “major components” represent

*h*for the TE-like mode and

_{z}*e*for the TM-like mode. The “minor components” denote

_{z}*h*for the TM-like mode and

_{z}*e*for TE-like mode. Note that the major components are very similar for both the

_{z}*C*

_{4}and

*C*

_{4v}fibers. However, the minor components for the

*C*

_{4}fiber are a factor six or so larger than in the

*C*

_{4v}fiber. Once again, therefore the fiber with reflection symmetry clearly exhibits a much greater degree of transversality in its quasi-transverse modes.

## 8. Discussion

*C*fibers, the expansion of the minor components is a pure sine series. It is easily seen that as a result, the minor field must vanish along the lines

_{nv}*ϕ*=2

*mπ*/

*n*for integer

*m*. This is apparent in images such as Figs. 4b and 8b. Now, recall from the discussion following Eqs. (4) that if one longitudinal component vanishes, then the transverse fields have either a vanishing transverse divergence or transverse curl. Thus along the n nodal lines, the non-degenerate states in

*C*fibers recover the exact field structure of the plain circular fiber—locally, the transverse fields are purely radial or azimuthal. On the other hand for the

_{nv}*C*fibers, the minor longitudinal field is less restricted, consisting of a complex Fourier series. As can be seen from Figs. 6b and 8d, while not forbidden by the longitudinal expansion, nodal lines do not appear. The difference in transversality can then be understood in terms of a simple energy argument. In a variational formulation of the wave equation [13], eigenmodes act to minimize their energy by reducing field curvature as far as possible and maximizing the fraction of energy in high index regions. This is completely true in the scalar problem, and remains the dominant effect in the vector treatment. In our problem of waveguides with rotational symmetry, the group theory places essentially no constraints on the modes of the

_{n}*C*fibers, which can then seek out an optimum lowest energy configuration. In the

_{n}*C*fibers however, the group theory constrains the field structure along the nodal lines. Away from the nodal lines, the field is free to adopt any structure but since the field must be pure radial/azimuthal along the nodal lines, there must be a significant energy penalty due to field curvature if it is not approximately radial/azimuthal everywhere. In fact, by this argument we can predict an increasing degree of transversality as the order of rotational symmetry, (and thus number of reflection lines increases), since the field is constrained in more locations. This is confirmed in Fig. 10 which shows the transversality (using a slightly different measure) as a function of rotational order for fibers with a single ring of circular holes, again matched to have similar mode areas and effective indices. A small value represents a highly transverse mode. The transverse fields are also shown for the

_{nv}*C*

_{3v}and

*C*

_{9v}instances of this structure, with the latter clearly showing a more ideally transverse structure.

## 9. Conclusion

## Appendix

*f*.

## Acknowledgments

## References and links

1. | P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides,” IEEE Transactions on Microwave Theory and Techniques |

2. | M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. |

3. | S.-H. Kim and Y.-H. Lee, “Symmetry relations of two-dimensional photonic crystal cavity modes,” IEEE J. Quantum Electron. |

4. | A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Perturbation analysis of dispersion properties in photonic crystal fibers through the finite element method,” J. Lightwave Technol. |

5. | M. Koshiba and K. Saitoh, “Structural dependence of effective area and mode field diameter for holey fibers,” Opt. Express |

6. | U. N. Singh, O. N. S. II, P. Khastgir, and K. K. Dey, “Dispersion characteristics of a helically cladded step-index optical fiber: an analytical study,” J. Opt. Soc. Am. B |

7. | RSoft Design Group, Inc. http://www.rsoftdesign.com. |

8. | D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Localized function method for modeling defect modes in 2-D photonic crystals,” J. Lightwave Technol. |

9. | T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. |

10. | T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, and M. J. Steel, “Confinement losses in microstructured optical fibres,” Opt. Lett. |

11. | R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express |

12. | M. van Eijkelenborg, M. Large, A. Argyros, J. Zagari, S. Manos, N. A. Issa, I. M. Bassett, S. C. Fleming, R. C. McPhedran, C. M. de Sterke, and N. A. P. Nicorovici, “Microstructured polymer optical fibre,” Opt. Express |

13. | C. Vassallo, |

**OCIS Codes**

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

(060.2400) Fiber optics and optical communications : Fiber properties

**ToC Category:**

Focus Issue: Photonic crystals and holey fibers

**History**

Original Manuscript: January 13, 2004

Revised Manuscript: February 12, 2004

Published: April 19, 2004

**Citation**

Michael Steel, "Reflection symmetry and mode transversality in microstructured fibers," Opt. Express **12**, 1497-1509 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-8-1497

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### References

- P. R. McIsaac, �??Symmetry-induced modal characteristics of uniform waveguides,�?? IEEE Transactions on Microwave Theory and Techniques MTT-23(5), 421�??433 (1975). [CrossRef]
- M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, �??Symmetry and degeneracy in microstructured optical fibers,�?? Opt. Lett. 26, 488�??490 (2001). [CrossRef]
- S.-H. Kim and Y.-H. Lee, �??Symmetry relations of two-dimensional photonic crystal cavity modes,�?? IEEE J. Quantum Electron. 39, 1081�??1085 (2003). [CrossRef]
- A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, �??Perturbation analysis of dispersion properties in photonic crystal fibers through the finite element method,�?? J. Lightwave Technol. 20, 1433�??1442 (2002). [CrossRef]
- M. Koshiba and K. Saitoh, �??Structural dependence of effective area and mode field diameter for holey fibers,�?? Opt. Express 11, 1746�??1756, <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-15-1746">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-15-1746</a>. [PubMed]
- U. N. Singh, O. N. S. II, P. Khastgir, and K. K. Dey, �??Dispersion characteristics of a helically cladded step-index optical fiber: an analytical study,�?? J. Opt. Soc. Am. B 12, 1273�??1278 (1995). [CrossRef]
- RSoft Design Group, Inc. <a href="http://www.rsoftdesign.com">http://www.rsoftdesign.com</a>.
- D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, �??Localized function method for modeling defect modes in 2-D photonic crystals,�?? J. Lightwave Technol. 17(11), 2078�??2081 (1999). [CrossRef]
- T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, �??Modeling large air fraction holey optical fibers,�?? J. Lightwave Technol. 18(1), 50�??56 (2000). [CrossRef]
- T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, and M. J. Steel, �??Confinement losses in microstructured optical fibres,�?? Opt. Lett. 26, 488�??490 (2001). [CrossRef]
- R. Guobin,W. Zhi, L. Shuqin, and J. Shuisheng, �??Mode classification and degeneracy in photonic crystal fibers,�?? Opt. Express 11, 1310�??1321 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310</a>. [CrossRef] [PubMed]
- M. van Eijkelenborg, M. Large, A. Argyros, J. Zagari, S. Manos, N. A. Issa, I. M. Bassett, S. C. Fleming, R. C. McPhedran, C. M. de Sterke, and N. A. P. Nicorovici, �??Microstructured polymer optical fibre,�?? Opt. Express 9, 319�??327 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-7-319">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-7-319</a>. [CrossRef] [PubMed]
- C. Vassallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991).

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