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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 10 — May. 19, 2003
  • pp: 1243–1251
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Resonances in microstructured optical waveguides

Natalia M. Litchinitser, Steven C. Dunn, Brian Usner, Benjamin J. Eggleton, Thomas P. White, Ross C. McPhedran, and C. Martijn de Sterke  »View Author Affiliations


Optics Express, Vol. 11, Issue 10, pp. 1243-1251 (2003)
http://dx.doi.org/10.1364/OE.11.001243


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Abstract

We propose a simple physical model that predicts the optical properties of a class of microstructured waveguides consisting of high-index inclusions that surround a low-index core. On the basis of this model, it is found that a large regime exists where transmission minima are determined by the geometry of the individual high-index inclusions. The locations of these minima are found to be largely unaffected by the relative position of the inclusions. As a result of this insight the difficult problem of analyzing the properties of complex structures can be reduced to the much simpler problem of analyzing the properties of an individual high-index inclusion in the structure.

© 2003 Optical Society of America

Microstructured optical waveguides are becoming of great importance in several emerging areas, including micrometer-scale novel photonic devices and integrated microcircuits [1

1. J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998). [CrossRef] [PubMed]

5

5. T. F. Krauss and R. M. De La Rue, “Photonic crystals at optical wavelengths - past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999). [CrossRef]

]. One variant of these structures is the microstructured optical fiber (MOF), or holey fiber. These structures are essentially dielectric cylinders with multiple inclusions running along their lengths [6

6. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999). [CrossRef] [PubMed]

,7

7. R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series, Technical Digest, Postconference Edition (Optical Society of America, Washington, D.C., 2002), pp. 466–468.

]. In these waveguides light is channeled through the core in one of two ways: total internal reflection (when the refractive index of the core is higher than that of the cladding), or coherent scattering from a microstructured cladding (typically when the refractive index of the core is lower than that of the cladding). The latter type can be subdivided into two categories: (1) waveguides with a solid core and high-index inclusions, and (2) those with air-core and air-holes.

Most theoretical studies of light propagation in MOF structures were performed numerically [8

8. S. E. Barkou, J. Broeng, and A. Bjarklev, “Silica-air photonic crystal fiber design that permits waveguiding by a true photonic bandgap effect,” Opt. Lett. 24, 46 (1999). [CrossRef]

11

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

], often via the widely used plane wave expansion method [8

8. S. E. Barkou, J. Broeng, and A. Bjarklev, “Silica-air photonic crystal fiber design that permits waveguiding by a true photonic bandgap effect,” Opt. Lett. 24, 46 (1999). [CrossRef]

]. Although powerful, these direct numerical simulations do not always provide an understanding of the physical mechanisms behind the light propagation. In this paper we put forth a simple physical model for the optical properties of a class of microstructured waveguides consisting of high-index inclusions that surround a low index core shown in Fig. 1. These waveguides have been investigated by a number of research groups, who have shown that they exhibit several novel properties including periodic spectral responses, strong dispersion and tunability [7

7. R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series, Technical Digest, Postconference Edition (Optical Society of America, Washington, D.C., 2002), pp. 466–468.

,12

12. M. A. Duguay, Y. Kukubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multiplayer structures,” Appl. Phys. Lett. 49, 13–15 (1986). [CrossRef]

,13

13. T. Baba and Y. Kukubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides—numerical results and analytical expressions,” IEEE J. Quantum Electron. QE-28, 1689–1700 (1992). [CrossRef]

]. Of these, the ability to tune the transmission spectra is of particular interest, as it has strong potential in new devices such as tunable filters and dispersion compensators.

Models to describe the guiding properties of a few specific microstructured optical waveguides have been given previously. In Ref. [14

14. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002). [CrossRef]

] the properties of a class of one-dimensional planar or concentric ring waveguides were studied in the context of antiresonant reflections from multiple cladding layers. It was shown that the spectral properties of these waveguides are principally determined by the index contrast and the thickness of the high-index layers, rather than by the lattice constant.

In this paper we present a general description of a class of microstructured fibers and waveguides with low index core and high index inclusions. We show that some of their important spectral properties can be described using a simple analytical formalism based on the normal modes of the high index inclusions. The formalism suggests that an important regime exists where the transmission minima of the entire waveguide structure are determined only by the geometry of individual high-index inclusions. As a result of this insight the complex and difficult problem of analyzing the propagation of light through the core can be reduced to the much simpler problem of analyzing the properties of an individual high-index inclusion. It should be noted that the structures described in Ref. [14

14. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002). [CrossRef]

] represent a subset of those to which the present analysis is applicable.

The structures under investigation in this work are: (a) a planar waveguide with a low-index (n1) core surrounded by alternating layers of high (n2) and low (n1) indices; (b) a concentric ring waveguide composed of layers of similar indices; and (c) a MOF with a low-index core surrounded by high-index (n2) cylindrical inclusions. These are shown below in Fig. 1 (a), (b), and (c) respectively.

Fig. 1. Microstructured optical waveguide geometries (n2 > n1 in all three cases).

For the analysis of planar and ring structures we will consider the location of the resonances in the transmission spectrum; the latter quantity being defined as T(λ)=Pout(λ)/Pin(λ), where Pin and Pout are the total powers in the core of the waveguide into which a Gaussian beam has been launched. The predictions are compared to numerical calculations of the transmission spectrum, which can be easily obtained using the beam propagation method or similar techniques [9

9. R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000). [CrossRef]

,14

14. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002). [CrossRef]

,15

15. A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, “Analysis of spectral characteristics of photonic bandgap waveguides,” Opt. Express 10, 1320–1333 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1320 [CrossRef] [PubMed]

].

For the planar waveguide with step-index profile, the cutoff conditions (and the corresponding transmission minima) can be found analytically from kexd=πm, where kex=2πλn22n12 is a transverse component of the propagation vector, and given by:

λm=2dmn22n12,
(1)

When λ/a ≪ 1, k2x is identical to kex. From the equation for the lowest order mode in the low-index core we find that the condition λ/a ≪ 1 is equivalent to sin(θ1)=λ/(2n1 a) ≪1, which corresponds to our glancing angle assumption. For higher order core modes the condition Nλ/(2n1 a) ≪1 should be satisfied in order for Eq. (1) to remain valid.

The exact analytical solution for the ring waveguide (shown in Fig. 1(b)) is very complicated and can be found in reference [16

16. M. M. Z. Kharadly and J. E. Lewis, “Properties of dielectric-tube waveguides,” Proc. IEE 116, 214–224 (1969).

]. An approximate eigenvalue equation for the modes of this waveguide and an equation describing the cutoff conditions in weakly guiding approximation have been derived in Ref. [17

17. D. Marcuse and W. L. Mammel, “Tube waveguide for optical transmission,” Bell Syst. Tech. J. 52, 423–435 (1973).

]. It has been shown that in the limit of large core radius and small values of d/a the ring waveguide can be regarded as a slab waveguide that is rolled into a tube. In this limit the cutoff conditions for the modes of this waveguide approach those for a slab waveguide, i.e. the equation (1).

Clearly the simple model proposed here is valid only in the regime where the high-index inclusions support modes. Thus it is valid only for λd2n22n12 (so-called “short wavelength regime” [14

14. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002). [CrossRef]

,15

15. A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, “Analysis of spectral characteristics of photonic bandgap waveguides,” Opt. Express 10, 1320–1333 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1320 [CrossRef] [PubMed]

]).

When a wavelength approaches the cutoff condition for a particular mode of a high index layer (ring) predicted by Eq. (1), the layer becomes“transparent” and light escapes from the central core resulting in minima in the transmission spectrum. This condition can be viewed as the transverse frequency resonance of the high index layer. In Ref. [14

14. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002). [CrossRef]

] we used an antiresonant reflecting optical waveguide model to predict transmission resonances of the waveguides shown in Figs. 1(a) and 1(b). According to this model each layer in the multilayered cladding can be considered as a Fabry-Perot resonator. Narrow band resonances of this FP resonator correspond to transmission minima for the light propagating in the core, or “resonant” wavelengths of the low-index core waveguide. Wide antiresonances of the FP resonator correspond to a high transmission coefficient for the low-index core waveguide. In general, high transmission of the light in the low index core at a particular wavelength for a particular core mode can be guaranteed by proper choice of thicknesses of both high- and low-index layers such that they satisfy an antiresonant condition. The antiresonant condition for a high-index layer is given by kexd = π(2m+1)/2, m=0,1,2… and for a low-index layer by b=(2l+1)a/2, l=0,1,2… for fundamental core mode[12

12. M. A. Duguay, Y. Kukubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multiplayer structures,” Appl. Phys. Lett. 49, 13–15 (1986). [CrossRef]

]. In the structures considered in this paper b is typically less or equal a/2. This non-optimized thickness of low-index regions results in some fine structure in the high loss region in the vicinity of the transmission minima predicted by Eq. (1) and higher loss in “high transmission regions”. Throughout this paper we call the spectral regions between two minima “high transmission regions”, although the actual transmission may be not very high especially in single layer structures such as shown in Fig. 2(a).

Fig. 2. (a) Schematic of planar structure (n1=1.4, n2=1.8, d=3.437µm), (b) calculated transmission spectra for planar and ring structure of length L=5cm, (c) analytical modal cutoff condition, and (d) absolute value of the electric field in high index layer (vertical straight lines show the borders of high index layer).

Our analysis now shifts to that of the MOF shown in Fig. 1(c) (similar to the one in Ref. [7

7. R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series, Technical Digest, Postconference Edition (Optical Society of America, Washington, D.C., 2002), pp. 466–468.

]). We previously confirmed using the multipole method that the locations of the transmission minima are nearly the same for multiple rings of cylinders structure as for its single ring counterpart [18

18. T. P. White, R. C. McPhedran, C. Martijn de Sterke, N. M. Litchinitser, and B. J. Eggleton, “Resonance and scattering in microstructured optical fibers,” Opt. Lett. 27, 1977–1979 (2002). [CrossRef]

]. It was shown that the positioning of the inclusions around the low-index core has little effect on the transmission minima, only changing the actual value of the loss in the "high"-transmission region and the effective refractive index of the core mode. In this study we are only interested in the positions of transmission minima and therefore, we can again reduce a three-layer geometry (Fig. 1(c)) to a single-layer one, in this case a ring consisting of six cylinders. In Ref. [18

18. T. P. White, R. C. McPhedran, C. Martijn de Sterke, N. M. Litchinitser, and B. J. Eggleton, “Resonance and scattering in microstructured optical fibers,” Opt. Lett. 27, 1977–1979 (2002). [CrossRef]

] we showed that the transmission minima are determined by the scattering properties of individual cylinders, however it was not obvious how to relate the scattering cross section arguments with the properties of planar and ring type inclusions. Here we show that the transmission minima of any microstructured waveguide with low index core and high index inclusions are determined by the cutoffs for the modes of the high index inclusions, as long as these inclusions support the modes.

The resonant condition is found from the eigenmode equation for TE, TM and EH modes

Jl(kex a)=0

using the cosine approximation for Jl(kex a) (where Jl is the Bessel function of order l) when kex al, where a=d/2

kex d/2-lπ/2-π/4=(2ν-1)π/2,

and can be written as follows:

λm=2dn22n12m+12,
(2)

m=1,2,…

Again, we have assumed glancing angles of incidence.

It should be noted, that if we compare the resonant wavelengths obtained from Eqs. (1) and (2) for the same parameter d (see Fig. 1), we find that resonant wavelengths of the MOF with cylindrical inclusions are out-of-phase with those of the planar structures for the same d, n1, and n2. This means that wavelengths of minimum transmission in planar structures have high transmission in MOFs. This result follows from the cutoff (resonant) conditions (1) and (2) that can be written as kexd=mπ and kexd=(m+½)π respectively, where m is an integer number.

In summary, we found that a large regime exists where the positions of transmission minima of microstructured optical waveguides with low-index core and high-index inclusions can be found by calculating the cut-off wavelength for the modes of high-index regions, and thus depend only on the mode structure of those inclusions. The complex problem of analyzing the properties of complex microstructured waveguides is thereby reduced to the much simpler problem of analyzing an individual high-index inclusion. Analytical predictions were confirmed by numerical simulations.

This work was produced with the assistance of the Australian Research Council under the ARC Centre of Excellence program.

Fig. 3. (a) Transmission spectrum for fundamental mode of MOF shown in the inset (n1=1.44, n2=1.8, d=3.8µm). Straight dashed lines show analytical predictions (from Eq. (2)) for resonant wavelengths. (b) Analytical versus numerical predictions for resonant wavelengths for different d. (c) Longitudinal component of the Poynting vector Sz, and (d) Sz along X axis for two wavelengths. Straight dashed lines show the position of the high-index inclusion.

References and links

1.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998). [CrossRef] [PubMed]

2.

P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196–1201 (1978). [CrossRef]

3.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415–419 (2000). [CrossRef] [PubMed]

4.

B. J. Eggleton, C. Kerbage, P. S. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698–713 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-698 [CrossRef] [PubMed]

5.

T. F. Krauss and R. M. De La Rue, “Photonic crystals at optical wavelengths - past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999). [CrossRef]

6.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999). [CrossRef] [PubMed]

7.

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series, Technical Digest, Postconference Edition (Optical Society of America, Washington, D.C., 2002), pp. 466–468.

8.

S. E. Barkou, J. Broeng, and A. Bjarklev, “Silica-air photonic crystal fiber design that permits waveguiding by a true photonic bandgap effect,” Opt. Lett. 24, 46 (1999). [CrossRef]

9.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000). [CrossRef]

10.

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002). [CrossRef]

11.

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

12.

M. A. Duguay, Y. Kukubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multiplayer structures,” Appl. Phys. Lett. 49, 13–15 (1986). [CrossRef]

13.

T. Baba and Y. Kukubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides—numerical results and analytical expressions,” IEEE J. Quantum Electron. QE-28, 1689–1700 (1992). [CrossRef]

14.

N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002). [CrossRef]

15.

A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, “Analysis of spectral characteristics of photonic bandgap waveguides,” Opt. Express 10, 1320–1333 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1320 [CrossRef] [PubMed]

16.

M. M. Z. Kharadly and J. E. Lewis, “Properties of dielectric-tube waveguides,” Proc. IEE 116, 214–224 (1969).

17.

D. Marcuse and W. L. Mammel, “Tube waveguide for optical transmission,” Bell Syst. Tech. J. 52, 423–435 (1973).

18.

T. P. White, R. C. McPhedran, C. Martijn de Sterke, N. M. Litchinitser, and B. J. Eggleton, “Resonance and scattering in microstructured optical fibers,” Opt. Lett. 27, 1977–1979 (2002). [CrossRef]

19.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

OCIS Codes
(050.2230) Diffraction and gratings : Fabry-Perot
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(230.3990) Optical devices : Micro-optical devices
(230.7370) Optical devices : Waveguides

ToC Category:
Research Papers

History
Original Manuscript: April 17, 2003
Revised Manuscript: May 11, 2003
Published: May 19, 2003

Citation
Natalia M. Litchinitser, Steven C. Dunn, Brian Usner, Benjamin J. Eggleton, Thomas P. White, Ross C. McPhedran, and C. Martijn de Sterke, "Resonances in microstructured optical waveguides," Opt. Express 11, 1243-1251 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-10-1243


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References

  1. J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476-1478 (1998). [CrossRef] [PubMed]
  2. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196-1201 (1978). [CrossRef]
  3. M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415-419 (2000). [CrossRef] [PubMed]
  4. B. J. Eggleton, C. Kerbage, P. S. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698-713 (2001), <a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-698">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-698</a> [CrossRef] [PubMed]
  5. T. F. Krauss and R. M. De La Rue, “Photonic crystals at optical wavelengths - past, present and future,” Prog. Quantum Electron. 23, 51-96 (1999). [CrossRef]
  6. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537-1539 (1999). [CrossRef] [PubMed]
  7. R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” in Optical Fiber Communication Conference, Vol. 70 of OSA Trends in Optics and Photonics Series, Technical Digest, Postconference Edition (Optical Society of America, Washington, D.C., 2002), pp. 466-468.
  8. S. E. Barkou, J. Broeng, and A. Bjarklev, “Silica-air photonic crystal fiber design that permits waveguiding by a true photonic bandgap effect,” Opt. Lett. 24, 46 (1999). [CrossRef]
  9. R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, , “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000). [CrossRef]
  10. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322- 2330 (2002). [CrossRef]
  11. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. Martijn de Sterke, and R. C. McPhedran, “Multipole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B 19, 2331-2340 (2002). [CrossRef]
  12. M. A. Duguay, Y. Kukubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multiplayer structures,” Appl. Phys. Lett. 49, 13-15 (1986). [CrossRef]
  13. T. Baba and Y. Kukubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides—numerical results and analytical expressions,” IEEE J. Quantum Electron. QE-28, 1689-1700 (1992). [CrossRef]
  14. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592-1594 (2002). [CrossRef]
  15. A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, “Analysis of spectral characteristics of photonic bandgap waveguides,” Opt. Express 10, 1320-1333 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1320 [CrossRef] [PubMed]
  16. M. M. Z. Kharadly and J. E. Lewis, “Properties of dielectric-tube waveguides,” Proc. IEE 116, 214-224 (1969).
  17. D. Marcuse and W. L. Mammel, “Tube waveguide for optical transmission,” Bell Syst. Tech. J. 52, 423-435 (1973).
  18. T. P. White, R. C. McPhedran, C. Martijn de Sterke, N. M. Litchinitser, and B. J. Eggleton, “Resonance and scattering in microstructured optical fibers,” Opt. Lett. 27, 1977-1979 (2002). [CrossRef]
  19. J. A. Stratton, Electromagnetic theory (McGraw-Hill, New York, 1941).

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