OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 16 — Aug. 8, 2005
  • pp: 6193–6201
« Show journal navigation

Modal Dynamics in Hollow-Core Photonic-Crystal Fibers with Elliptical Veins

Amit Hochman and Yehuda Leviatan  »View Author Affiliations


Optics Express, Vol. 13, Issue 16, pp. 6193-6201 (2005)
http://dx.doi.org/10.1364/OPEX.13.006193


View Full Text Article

Acrobat PDF (303 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Modal characteristics of hollow-core photonic-crystal fibers with elliptical veins are studied by use of a recently proposed numerical method. The dynamic behavior of bandgap guided modes, as the wavelength and aspect ratio are varied, is shown to include zero-crossings of the birefringence, polarization dependent radiation losses, and deformation of the fundamental mode.

© 2005 Optical Society of America

1. Introduction

In anisotropic crystals, such as ZnO and CdS, zero-crossings of the birefringence vs. wavelength curves were used to explain the existence of polariton absorption lines [11

11. J. J. Hopfield and D. G. Thomas, “Polariton absorption lines,” Phys. Rev. Lett. 15, 22–25 (1965). [CrossRef]

], and we shall show similar birefringence zero-crossings in birefringent hollow-core PCFs. Similar results have also been reported recently in [12

12. W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, “Dependence of mode characteristics on the central defect in elliptical hole photonic crystal fibers,” Opt. Express 11, 1966–1979 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-17-1966 [CrossRef] [PubMed]

].

2. Numerical method

The numerical method used in this study is a recently proposed source-model technique (SMT) [13

13. A. Hochman and Y. Leviatan, “Analysis of strictly bound modes in photonic crystal fibers by use of a source-model technique,” J. Opt. Soc. Am. A 21, 1073–1081 (2004). [CrossRef]

, 14

14. A. Hochman and Y. Leviatan, “Calculation of confinement losses in photonic crystal fibers by use of a source-model technique,” J. Opt. Soc. Am. B 22, 474–480 (2005). [CrossRef]

], which is adequate for the analysis of general dielectric waveguide geometries. The PCFs we analyzed have elliptical veins, the likes of which have been successfully manufactured recently [15

15. N. A. Issa, M. A. Van-Eijkelenborg, M. Fellew, F. Cox, G. Henry, and M. C. J. Large, “Fabrication and study of microstructured optical fibers with elliptical holes,” Opt. Lett. 29, 1336–1338 (2004). [CrossRef] [PubMed]

], and are depicted in Fig 1. Solid-core versions of these elliptical photonic-crystal fibers (EPCFs), as they have been tentatively dubbed [8

8. M. J. Steel and R. M. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. 19, 495–503 (2001). [CrossRef]

], have been analyzed recently by an adjustable boundary condition Fourier decomposition method (ABC-FDM) [16

16. N. A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibers,” J. Lightwave Technol. 21, 1005–1012 (2003). [CrossRef]

], and a differential multipole method (DMM) [10

10. S. Campbell, R. C. McPhedran, C. M. de Sterke, and L. C. Botten, “Differential multipole method for microstructured optical fibers,” J. Opt. Soc. Am. B 21, 1919–1928 (2004). [CrossRef]

]. Before proceeding to discuss hollow-core EPCFs, we present results obtained with the SMT for solid-core PCFs, and compare them with these published data.

Fig. 1. (a) Solid-core EPCF. The horizontal axis of the ellipse is kept fixed at 2.5μm, and the refractive index of the gray areas is 1.45. The lattice constant is denoted by Λ. (b) Hollow-core EPCF. The horizontal axis of the veins is kept fixed at 0.35Λ, and the horizontal axis of the core is kept fixed 0.57Λ. The refractive index of the gray areas is 1.4897.

2.1. Validation by comparison with previously published data

In Table 1, the effective indices of the fundamental modes that are approximately polarized in the x direction in the center of the PCF, are shown for a few solid-core EPCFs. The effective indices calculated by the ABC-FDM and DMM are quoted from Table 3 in [10

10. S. Campbell, R. C. McPhedran, C. M. de Sterke, and L. C. Botten, “Differential multipole method for microstructured optical fibers,” J. Opt. Soc. Am. B 21, 1919–1928 (2004). [CrossRef]

], and the geometry considered is shown in Fig. 1(a). Harmonic exp(jωt) time dependence is assumed throughout. The aspect ratio (vertical to horizontal axes ratio), is denoted by η, and takes on the values shown in the first column of the table. The excellent agreement in Re(n eff) reassures us of the correctness of our SMT implementation. The ABC-FDM results for Im(n eff) appear to be less accurate, as the agreement between the DMM and SMT results is very good.

Table 1. Effective indices of the fundamental x-polarized mode of a solid-core PCF.

table-icon
View This Table

3. Hollow-core EPCFs

While solid-core EPCFs have been studied by a number of authors, hollow-core EPCFs have hardly received any attention at all. In [17

17. W. Zhi, R. Guobin, and L. Shuqin, “Mode disorder in elliptical hole PCFs,” Opt. Fiber Technol.: Materials 10, 124–132 (2004). [CrossRef]

], a one-ring hollow-core EPCF was analyzed by a super-cell model which, however, could not take into account the radiation losses. For the modes studied in [17

17. W. Zhi, R. Guobin, and L. Shuqin, “Mode disorder in elliptical hole PCFs,” Opt. Fiber Technol.: Materials 10, 124–132 (2004). [CrossRef]

], the losses are anticipated to be relatively small, because they are guided by total internal reflection, so a super-cell analysis should be reasonable.

3.1. Modal Dynamics

To find bandgap-guided modes that carry most of their energy in the hollow core, we focused on a small portion of the complex n eff plane in the neighborhood of the light line, Re(n eff) = 1. We began with a PCF with circular veins, for which we calculated the bandgap for out-of-plane propagation by a plane-wave expansion method [18

18. A. A. Maradudin and A. R. McGurn, “Out of plane propagation of electromagnetic waves in a two-dimensional periodic dielectric medium,” J. Modern Opt. 41, 275–284 (1994). [CrossRef]

] (see Fig. 2), and then gradually decremented the aspect ratio. As in the solid-core example, the semimajor axis was kept fixed (see Fig. 1(b)).

Fig. 2. Bandgap for out-of-plane propagation shown shaded. The dispersion curve of the fundamental mode (black curve), and the light line (red horizontal line) are also indicated.

We use McIsaac’s [3

3. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides-I: Summary of results,” IEEE Trans. Microwave Theory and Techniques 23, 421–429 (1975). [CrossRef]

] symmetry-based mode numbering, where a mode that is approximately x-polarized in the PCF center is referred to as a p = 3 mode, and its counterpart, approximately y-polarized, is referred to as a p = 4 mode. The complex effective indices that correspond to modes of both polarizations are shown in Fig. 3, for decreasing aspect ratio, η. In the topmost panels the PCF has circular veins, and accordingly, most of the p = 3 modes have degenerate counterparts of symmetry class p = 4. There are some exceptions however, for example, the p = 4 mode at n eff = 0.975 - 0.01j; these are non-degenerate modes. McIsaac’s symmetry-based mode classification has a certain drawback, in that non-degenerate modes (referred to as p = 8) are also p = 4 modes, and non-degenerate p = 7 modes are also p = 3 modes. A different symmetry-based mode classification that circumvents this problem was proposed in [19

19. J. M. Fini, “Improved symmetry analysis of many-moded microstructure optical fibers,” J. Opt. Soc. Am. B 21, 1431–1436 (2004). [CrossRef]

], but we have stayed with the more widespread classification. These non-degenerate modes should maintain their polarization very well when propagating, but they may be difficult to excite because they are not the fundamental degenerate pair, which is shown in Figs. 4(a) and (b). Apart from that, the non-degenerate modes have zero light intensity at the PCF axis. We therefore focus only on p = 3 and p = 4 modes in this paper.

Fig. 3. Complex effective indices of modes in the neighborhood of the light line, for decreasing aspect ratio, η. The ratio of lattice constant to wavelength, Λ/λ, is 1.7.

Fig. 4. Deformation of the fundamental degenerate pair as the aspect ratio decreases. The p = 3 mode is shown in (a), (c), and (e) (AVI, 668KB); the p = 4 mode is shown in (b), (d), and (f) (AVI, 635KB). The value plotted is the intensity of light. The effective indices are: in (a) and (b), 0.9632–0.0075j, in (c) 0.9989–0.0078j, in (d) 1.0050–0.0179j, in (e) 1.0399–0.0095j, and in (f) 1.0618–0.0204j. [Media 1]
Fig. 5. Complex effective indices of modes that correspond to the fundamental degenerate pair in the circular-veins PCF. The effective indices were tracked while the aspect ratio η was slowly decremented by 0.002 each time. The difference in η between two markers is 0.01. The dotted lines connect points of equal η.

3.2. Adiabatic deformation of the fundamental mode

Each curve in Fig. 5 corresponds to a single mode in the sense that any two points on the curve can be connected by a continuous deformation of the structure. The mode profile, however, changes shape substantially along the curve. In Fig. 4, the intensity of light, normalized to unit overall intensity in the cross-section, is shown for a number of modes along the p = 3 curve of the fundamental pair. The mode begins strongly concentrated in the core, but as the veins become more elliptical it spreads over the cross-section. Eventually, the propagation constant crosses the light line, and the mode becomes evanescent in the air and confined mostly to the dielectric, as seen in Fig. 4(f). When the PCF is deformed, a different mode may become the fundamental, but if the PCF starts out as circular and is then deformed adiabatically, the fields in the PCF cross-section would actually follow the sequence shown in Fig. 4.

3.3. Wavelength dependence of the radiation losses and the birefringence

In Fig. 6(a), the wavelength dependence of Im(n eff) of the mode that is the fundamental mode in the circular-veins PCF is shown for three aspect ratios, including the circular case. For most of the wavelength range, the p = 3 mode is more lossy than the p = 4 mode, however, at the shorter wavelengths this order is reversed. When one more ring of veins is added, as shown in Fig. 6(b), the losses are reduced approximately by a factor of two. Here too, the p = 4 mode is less leaky than the p = 3 mode in the longer wavelengths, and the order is reversed at shorter wavelengths. This is quite an interesting feature, because it implies that the dominant polarization is determined by whether the wavelength is to the right or to the left of the intersection of the p = 3 and p = 4 curves.

Fig. 6. Wavelength dependence of (a) Im(n eff), (c) Re(n eff) and (e) the birefringence, ∆n eff ≜ Re(neffp=4 - neffp=3). In (b), (d) and (f), corresponding results for a PCF with one more ring of veins are shown. The mode is the fundamental mode in the circular-veins PCF.

The wavelength dependence of Re(n eff) and the birefringence are shown, respectively, in Figs. 6(c) and (e), for three aspect ratios, including the circular case. When one more ring of veins is added, as shown in Fig. 6(d), Re(n eff) is almost unchanged. This is as it should be in a well-confined mode, since the added ring of veins is added in a region where the fields are small and thus perturb the effective index only be slightly. On the other hand, the birefringence, which is the difference of two such perturbed curves may be altered significantly. As shown in Fig. 6(f), higher birefringence is obtained in this case. Considering the very small deviation from a circular geometry, the birefringence is quite high. For comparison, birefringence of the same order of magnitude (7 × 10-3,15 × 10-3) was found in [16

16. N. A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibers,” J. Lightwave Technol. 21, 1005–1012 (2003). [CrossRef]

], for a solid-core PCF with two rings of veins of aspect ratio of 0.7. It is interesting to note the zero-crossings of the birefringence, implying an accidental degeneracy at a specific wavelength. At this wavelength the polarizations may be easily coupled, and this might be utilized to make very sharp optical filters, as was suggested in [21

21. C. H. Henry, “Coupling of electromagnetic waves in CdS,” Phys. Rev. 143, 627–633 (1966). [CrossRef]

, 22

22. J. F. Lotspeich, “Iso-Idex coupled-wave electrooptic filter,” IEEE J. Quantum Electron. 15, 904–907 (1979). [CrossRef]

], in the context of anisotropic crystals.

4. Conclusion

Acknowledgments

The authors would like to thank Alon Ludwig for his assistance with the calculation of the bandgap for out-of-plane propagation.

References and links

1.

P. S. J. Russell, “Photonic Crystal Fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]

2.

A. M. Zheltikov, “Holey fibers,” Physics Uspekhi 170, 1203–1215 (2000). [CrossRef]

3.

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides-I: Summary of results,” IEEE Trans. Microwave Theory and Techniques 23, 421–429 (1975). [CrossRef]

4.

T. A. Birks, J. C. Knight, and P. S. J. Russell, “Endlessly Single-mode Photonic Crystal Fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

5.

C. J. S. de Matos and J. R. Taylor, “Multi-kilowatt, all-fiber integrated chirped-pulse amplification system yielding 40× pulse compression using air-core fiber and conventional erbium-doped fiber amplifier,” Opt. Express 12, 405–409 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-405 [CrossRef] [PubMed]

6.

F. Benabid, J. C. Knight, G. Antonopoulos, and P. S. J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002). [CrossRef] [PubMed]

7.

J. Noda, K. Okamoto, and Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4, 1071–1089 (1986). [CrossRef]

8.

M. J. Steel and R. M. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. 19, 495–503 (2001). [CrossRef]

9.

K. Saitoh and M. Koshiba, “Photonic bandgap fibers with high birefringence,” IEEE Photonics Technol. Lett. 14, 1291–1293 (2002). [CrossRef]

10.

S. Campbell, R. C. McPhedran, C. M. de Sterke, and L. C. Botten, “Differential multipole method for microstructured optical fibers,” J. Opt. Soc. Am. B 21, 1919–1928 (2004). [CrossRef]

11.

J. J. Hopfield and D. G. Thomas, “Polariton absorption lines,” Phys. Rev. Lett. 15, 22–25 (1965). [CrossRef]

12.

W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, “Dependence of mode characteristics on the central defect in elliptical hole photonic crystal fibers,” Opt. Express 11, 1966–1979 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-17-1966 [CrossRef] [PubMed]

13.

A. Hochman and Y. Leviatan, “Analysis of strictly bound modes in photonic crystal fibers by use of a source-model technique,” J. Opt. Soc. Am. A 21, 1073–1081 (2004). [CrossRef]

14.

A. Hochman and Y. Leviatan, “Calculation of confinement losses in photonic crystal fibers by use of a source-model technique,” J. Opt. Soc. Am. B 22, 474–480 (2005). [CrossRef]

15.

N. A. Issa, M. A. Van-Eijkelenborg, M. Fellew, F. Cox, G. Henry, and M. C. J. Large, “Fabrication and study of microstructured optical fibers with elliptical holes,” Opt. Lett. 29, 1336–1338 (2004). [CrossRef] [PubMed]

16.

N. A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibers,” J. Lightwave Technol. 21, 1005–1012 (2003). [CrossRef]

17.

W. Zhi, R. Guobin, and L. Shuqin, “Mode disorder in elliptical hole PCFs,” Opt. Fiber Technol.: Materials 10, 124–132 (2004). [CrossRef]

18.

A. A. Maradudin and A. R. McGurn, “Out of plane propagation of electromagnetic waves in a two-dimensional periodic dielectric medium,” J. Modern Opt. 41, 275–284 (1994). [CrossRef]

19.

J. M. Fini, “Improved symmetry analysis of many-moded microstructure optical fibers,” J. Opt. Soc. Am. B 21, 1431–1436 (2004). [CrossRef]

20.

R. Sammut and A. W. Snyder, “Leaky modes on a dielectric waveguide: orthogonality and excitation,” Appl. Opt. 15, 1040–1044 (1976). [CrossRef] [PubMed]

21.

C. H. Henry, “Coupling of electromagnetic waves in CdS,” Phys. Rev. 143, 627–633 (1966). [CrossRef]

22.

J. F. Lotspeich, “Iso-Idex coupled-wave electrooptic filter,” IEEE J. Quantum Electron. 15, 904–907 (1979). [CrossRef]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

ToC Category:
Research Papers

History
Original Manuscript: March 8, 2005
Revised Manuscript: July 28, 2005
Published: August 8, 2005

Citation
Amit Hochman and Yehuda Leviatan, "Modal Dynamics in Hollow-Core Photonic-Crystal Fibers with Elliptical Veins," Opt. Express 13, 6193-6201 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-16-6193


Sort:  Journal  |  Reset

References

  1. P. S. J. Russell, �??Photonic Crystal Fibers,�?? Science 299, 358�??362 (2003). [CrossRef]
  2. A. M. Zheltikov, �??Holey fibers,�?? Physics Uspekhi 170, 1203�??1215 (2000).
  3. P. R. McIsaac, �??Symmetry-induced modal characteristics of uniform waveguides-I: Summary of results,�?? IEEE Trans. Microwave Theory and Techniques 23, 421�??429 (1975).
  4. T. A. Birks, J. C. Knight, and P. S. J. Russell, �??Endlessly Single-mode Photonic Crystal Fiber,�?? Opt. Lett. 22, 961�??963 (1997).
  5. C. J. S. de Matos and J. R. Taylor, �??Multi-kilowatt, all-fiber integrated chirped-pulse amplification system yielding 40�? pulse compression using air-core fiber and conventional erbium-doped fiber amplifier,�?? Opt. Express 12, 405�??409 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-405">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-405</a> [CrossRef]
  6. F. Benabid, J. C. Knight, G. Antonopoulos, and P. S. J. Russell, �??Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,�?? Science 298, 399�??402 (2002). [CrossRef]
  7. J. Noda, K. Okamoto, and Y. Sasaki, �??Polarization-maintaining fibers and their applications,�?? J. Lightwave Technol. 4, 1071�??1089 (1986).
  8. M. J. Steel and R. M. Osgood, �??Polarization and dispersive properties of elliptical-hole photonic crystal fibers,�?? J. Lightwave Technol. 19, 495�??503 (2001). [CrossRef]
  9. K. Saitoh and M. Koshiba, �??Photonic bandgap fibers with high birefringence,�?? IEEE Photonics Technol. Lett. 14, 1291�??1293 (2002). [CrossRef]
  10. S. Campbell, R. C. McPhedran, C. M. de Sterke, and L. C. Botten, �??Differential multipole method for microstructured optical fibers,�?? J. Opt. Soc. Am. B 21, 1919�??1928 (2004).
  11. J. J. Hopfield and D. G. Thomas, �??Polariton absorption lines,�?? Phys. Rev. Lett. 15, 22�??25 (1965). [CrossRef]
  12. W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, �??Dependence of mode characteristics on the central defect in elliptical hole photonic crystal fibers,�?? Opt. Express 11, 1966�??1979 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-17-1966">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-17-1966</a>
  13. A. Hochman and Y. Leviatan, �??Analysis of strictly bound modes in photonic crystal fibers by use of a source-model technique,�?? J. Opt. Soc. Am. A 21, 1073�??1081 (2004). [CrossRef]
  14. A. Hochman and Y. Leviatan, �??Calculation of confinement losses in photonic crystal fibers by use of a sourcemodel technique,�?? J. Opt. Soc. Am. B 22, 474�??480 (2005). [CrossRef]
  15. N. A. Issa, M. A. Van-Eijkelenborg, M. Fellew, F. Cox, G. Henry, and M. C. J. Large, �??Fabrication and study of microstructured optical fibers with elliptical holes,�?? Opt. Lett. 29, 1336�??1338 (2004). [CrossRef]
  16. N. A. Issa and L. Poladian, �??Vector wave expansion method for leaky modes of microstructured optical fibers,�?? J. Lightwave Technol. 21, 1005�??1012 (2003). [CrossRef]
  17. W. Zhi, R. Guobin, and L. Shuqin, �??Mode disorder in elliptical hole PCFs,�?? Opt. Fiber Technol.: Materials 10, 124�??132 (2004).
  18. A. A. Maradudin and A. R. McGurn, �??Out of plane propagation of electromagnetic waves in a two-dimensional periodic dielectric medium,�?? J. Modern Opt. 41, 275�??284 (1994).
  19. J. M. Fini, �??Improved symmetry analysis of many-moded microstructure optical fibers,�?? J. Opt. Soc. Am. B 21, 1431�??1436 (2004). [CrossRef]
  20. R. Sammut and A. W. Snyder, �??Leaky modes on a dielectric waveguide: orthogonality and excitation,�?? Appl. Opt. 15, 1040�??1044 (1976).
  21. C. H. Henry, �??Coupling of electromagnetic waves in CdS,�?? Phys. Rev. 143, 627�??633 (1966). [CrossRef]
  22. J. F. Lotspeich, �??Iso-Idex coupled-wave electrooptic filter,�?? IEEE J. Quantum Electron. 15, 904�??907 (1979). [CrossRef]

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.

Supplementary Material


» Media 1: AVI (668 KB)     

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited