Abstract

We present the light-propagation characteristics of OmniGuide fibers, which guide light by concentric multi-layer dielectric mirrors having the property of omnidirectional reflection. We show how the lowest-loss TE01 mode can propagate in a single-mode fashion through even large-core fibers, with other modes eliminated asymptotically by their higher losses and poor coupling, analogous to hollow metallic microwave waveguides. Dispersion, radiation leakage, material absorption, nonlinearities, bending, acircularity, and interface roughness are considered with the help of leaky modes and perturbation theory, and both numerical results and general scaling relations are presented. We show that cladding properties such as absorption and nonlinearity are suppressed by many orders of magnitude due to the strong confinement in a hollow core, and other imperfections are tolerable, promising that the properties of silica fibers may be surpassed even when nominally poor materials are employed.

© 2001 Optical Society of America

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References

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2001 (1)

Y. Xu and A. Yariv, “Asymptotic analysis of Bragg fibers and dielectric coaxial fibers,” In Proc. SPIE, A. Dutta, A. A. S. Awwal, N. K. Dutta, and K. Okamoto, eds., 4532, 191–205 (2001).
[Crossref]

2000 (8)

J. A. Harrington, “A review of IR transmitting, hollow waveguides,” Fiber Integr. Opt. 19, 211–227 (2000).
[Crossref]

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, and H. Damsgaard, “Dispersion compensating fibers,” Optical Fiber Tech. 6, 164–180 (2000).
[Crossref]

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Elec. Lett. 36, 514–515 (2000).
[Crossref]

F. Brechet, P. Leproux, P. Roy, J. Marcou, and D. Pagnoux, “Analysis of bandpass filtering behavior of singlemode depressed-core-index photonic bandgap fibre,” Elec. Lett. 36, 870–872 (2000).
[Crossref]

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]

Y. Xu, R. K. Lee, and A. Yariv, “Asymptotic analysis of Bragg fibers,” Opt. Lett. 25, 1756–1758 (2000).
[Crossref]

T. Kawanishi and M. Izutsu, “Coaxial periodic optical waveguide,” Opt. Express 7, 10–22 (2000), http://www.opticsexpress.org/oearchive/source/22933.htm.
[Crossref] [PubMed]

D. Q. Chowdhury, “Comparison between optical fiber birefringence induced by stress anisotropy and geometric deformation,” IEEE J. Selected Topics Quantum Elec. 6, 227–232 (2000).
[Crossref]

1999 (3)

M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. 163, 86–94 (1999).
[Crossref]

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

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Tech. 17, 2039–2041 (1999).
[Crossref]

1998 (1)

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[Crossref] [PubMed]

1996 (2)

R. S. Grant, “Effective non-linear coefficients in optical waveguides,” Optical and Quantum Elec. 28, 1161–1173 (1996).
[Crossref]

L. A. Yudin, S. P. Efimov, M. I. Kapchinsky, and I. L. Korenev, “Electrodynamics as a problem of eigenvalues,” Phys. Plasmas 3, 42–58 (1996).
[Crossref]

1995 (3)

1994 (1)

C. M. de Sterke and I. M. Bassett, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680–688 (1994).
[Crossref]

1992 (2)

1990 (1)

1988 (1)

A. N. Lazarchik, “Bragg fiber lightguides,” Radiotekhnika i electronika 1, 36–43 (1988).

1986 (1)

Z. Pantic and R. Mittra, “Quasi-TEM analysis of microwave transmission lines by the finite-element method,” IEEE Trans. Microwave Theory Tech. MTT-34, 1096–1103 (1986).
[Crossref]

1984 (3)

M. Miyagi and S. Kawakami, “Design theory of dielectric-coated circular metallic waveguides for infrared transmission,” J. Lightwave Tech. 2, 116–126 (1984).
[Crossref]

M. Miyagi, K. Harada, and S. Kawakami, “Wave propagation and attenuation in the general class of circular hollow waveguides with uniform curvature,” IEEE Trans. Microwave Theory Tech. MTT-32, 513–521 (1984).
[Crossref]

V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence of a three-layer elliptic single-mode fiber waveguide,” Sov. J. Quantum Elec. 14, 427–430 (1984).
[Crossref]

1983 (3)

V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence in a single-mode elliptic optical fiber,” Sov. J. Quantum Elec. 13, 109–111 (1983).
[Crossref]

N. J. Doran and K. J. Bulow, “Cylindrical Bragg fibers: a design and feasibility study for optical communications,” J. Lightwave Tech. 1, 588–590 (1983).
[Crossref]

M. Miyagi, A. Hongo, and S. Kawakami, “Transmission characteristics of dielectric-coated metallic waveguides for infrared transmission: slab waveguide model,” IEEE J. Quantum Elec. QE-19, 136–145 (1983).
[Crossref]

1981 (2)

A. Kumar, S. I. Hosain, and A. K. Ghatak, “Propagation characteristics of weakly guiding lossy fibers: an exact and perturbation analysis,” Optica Acta 28, 559–566 (1981).
[Crossref]

D. Sarid and G. I. Stegeman, “Optimization of the effects of power dependent refractive indices in optical waveguides,” J. Appl. Phys. 52, 5439–5441 (1981).
[Crossref]

1978 (1)

1977 (1)

W. D. Warters, “WT4 millimeter waveguide system: introduction,” Bell Syst. Tech. J. 56, 1825–1827 (1977), the introduction to a special issue with many useful articles.

1974 (1)

1970 (1)

A. W. Snyder, “Radiation losses due to variations of radius on dielectric or optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-18, 608–615 (1970).
[Crossref]

1964 (1)

E. A. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt Saunders, Philadelphia, 1976).

Bahlmann, N.

M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. 163, 86–94 (1999).
[Crossref]

Bassett, I. M.

C. M. de Sterke and I. M. Bassett, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680–688 (1994).
[Crossref]

Birks, T. A.

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

Brechet, F.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Elec. Lett. 36, 514–515 (2000).
[Crossref]

F. Brechet, P. Leproux, P. Roy, J. Marcou, and D. Pagnoux, “Analysis of bandpass filtering behavior of singlemode depressed-core-index photonic bandgap fibre,” Elec. Lett. 36, 870–872 (2000).
[Crossref]

Bulow, K. J.

N. J. Doran and K. J. Bulow, “Cylindrical Bragg fibers: a design and feasibility study for optical communications,” J. Lightwave Tech. 1, 588–590 (1983).
[Crossref]

Chang, D. C.

L. Lewin, D. C. Chang, and F. Kuester, Electromagnetic Waves and Curved Structures (P. Peregrinus, England, 1977).

Chen, C.

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Tech. 17, 2039–2041 (1999).
[Crossref]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[Crossref] [PubMed]

Chowdhury, D. Q.

D. Q. Chowdhury, “Comparison between optical fiber birefringence induced by stress anisotropy and geometric deformation,” IEEE J. Selected Topics Quantum Elec. 6, 227–232 (2000).
[Crossref]

D. Q. Chowdhury and D. A. Nolan, “Perturbation model for computing optical fiber birefringence from a two-dimensional refractive-index profile,” Opt. Lett. 20, 1973–1975 (1995).
[Crossref] [PubMed]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Din, and F. Laloë, Quantum Mechanics (Hermann, Paris, 1977), Vol. One, ch. 2; and Vol. Two, ch. 11 and 13.

Cregan, R. F.

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

Damsgaard, H.

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, and H. Damsgaard, “Dispersion compensating fibers,” Optical Fiber Tech. 6, 164–180 (2000).
[Crossref]

de Sterke, C. M.

C. M. de Sterke and I. M. Bassett, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680–688 (1994).
[Crossref]

Din, B.

C. Cohen-Tannoudji, B. Din, and F. Laloë, Quantum Mechanics (Hermann, Paris, 1977), Vol. One, ch. 2; and Vol. Two, ch. 11 and 13.

Doran, N. J.

N. J. Doran and K. J. Bulow, “Cylindrical Bragg fibers: a design and feasibility study for optical communications,” J. Lightwave Tech. 1, 588–590 (1983).
[Crossref]

Edvold, B.

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, and H. Damsgaard, “Dispersion compensating fibers,” Optical Fiber Tech. 6, 164–180 (2000).
[Crossref]

Efimov, S. P.

L. A. Yudin, S. P. Efimov, M. I. Kapchinsky, and I. L. Korenev, “Electrodynamics as a problem of eigenvalues,” Phys. Plasmas 3, 42–58 (1996).
[Crossref]

Fan, S.

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]

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Tech. 17, 2039–2041 (1999).
[Crossref]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[Crossref] [PubMed]

Fink, Y.

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]

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Tech. 17, 2039–2041 (1999).
[Crossref]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[Crossref] [PubMed]

Fontaine, M.

Ford, G. W.

Ghatak, A. K.

A. Kumar, S. I. Hosain, and A. K. Ghatak, “Propagation characteristics of weakly guiding lossy fibers: an exact and perturbation analysis,” Optica Acta 28, 559–566 (1981).
[Crossref]

Godbout, N.

Grant, R. S.

R. S. Grant, “Effective non-linear coefficients in optical waveguides,” Optical and Quantum Elec. 28, 1161–1173 (1996).
[Crossref]

Grattan, K. T. V.

C. Themistos, B. M. A. Rahman, A. Hadjicharalambous, and K. T. V. Grattan, “Loss/gain characterization of optical waveguides,” J. Lightwave Tech. 13, 1760–1765 (1995).
[Crossref]

Grüner-Nielsen, L.

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, and H. Damsgaard, “Dispersion compensating fibers,” Optical Fiber Tech. 6, 164–180 (2000).
[Crossref]

Gupta, V. L.

Hadjicharalambous, A.

C. Themistos, B. M. A. Rahman, A. Hadjicharalambous, and K. T. V. Grattan, “Loss/gain characterization of optical waveguides,” J. Lightwave Tech. 13, 1760–1765 (1995).
[Crossref]

Harada, K.

M. Miyagi, K. Harada, and S. Kawakami, “Wave propagation and attenuation in the general class of circular hollow waveguides with uniform curvature,” IEEE Trans. Microwave Theory Tech. MTT-32, 513–521 (1984).
[Crossref]

Harrington, J. A.

J. A. Harrington, “A review of IR transmitting, hollow waveguides,” Fiber Integr. Opt. 19, 211–227 (2000).
[Crossref]

Hertel, P.

M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. 163, 86–94 (1999).
[Crossref]

Hongo, A.

M. Miyagi, A. Hongo, and S. Kawakami, “Transmission characteristics of dielectric-coated metallic waveguides for infrared transmission: slab waveguide model,” IEEE J. Quantum Elec. QE-19, 136–145 (1983).
[Crossref]

Hosain, S. I.

A. Kumar, S. I. Hosain, and A. K. Ghatak, “Propagation characteristics of weakly guiding lossy fibers: an exact and perturbation analysis,” Optica Acta 28, 559–566 (1981).
[Crossref]

Ibanescu, M.

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]

M. Ibanescuet al., to be published in 2002.

Izutsu, M.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).

Jacobs, S. A.

S. A. Jacobset al., to be published in 2002.

Joannopoulos, J. D.

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]

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Tech. 17, 2039–2041 (1999).
[Crossref]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[Crossref] [PubMed]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton, 1995).

Johnson, S. G.

S. G. Johnsonet al., to be published in 2002.

Kalosha, V. P.

V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence of a three-layer elliptic single-mode fiber waveguide,” Sov. J. Quantum Elec. 14, 427–430 (1984).
[Crossref]

V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence in a single-mode elliptic optical fiber,” Sov. J. Quantum Elec. 13, 109–111 (1983).
[Crossref]

Kapchinsky, M. I.

L. A. Yudin, S. P. Efimov, M. I. Kapchinsky, and I. L. Korenev, “Electrodynamics as a problem of eigenvalues,” Phys. Plasmas 3, 42–58 (1996).
[Crossref]

Katsenelenbaum, B. Z.

B. Z. Katsenelenbaum, L. Mercader del Río, M. Pereyaslavets, M. Sorolla Ayza, and M. Thumm, Theory of Nonuniform Waveguides: The Cross-Section Method (Inst. of Electrical Engineers, London, 1998).
[Crossref]

Kawakami, S.

M. Miyagi, K. Harada, and S. Kawakami, “Wave propagation and attenuation in the general class of circular hollow waveguides with uniform curvature,” IEEE Trans. Microwave Theory Tech. MTT-32, 513–521 (1984).
[Crossref]

M. Miyagi and S. Kawakami, “Design theory of dielectric-coated circular metallic waveguides for infrared transmission,” J. Lightwave Tech. 2, 116–126 (1984).
[Crossref]

M. Miyagi, A. Hongo, and S. Kawakami, “Transmission characteristics of dielectric-coated metallic waveguides for infrared transmission: slab waveguide model,” IEEE J. Quantum Elec. QE-19, 136–145 (1983).
[Crossref]

Kawanishi, T.

Khapalyuk, A. P.

V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence of a three-layer elliptic single-mode fiber waveguide,” Sov. J. Quantum Elec. 14, 427–430 (1984).
[Crossref]

V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence in a single-mode elliptic optical fiber,” Sov. J. Quantum Elec. 13, 109–111 (1983).
[Crossref]

Knight, J. C.

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

Knudsen, S. N.

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, and H. Damsgaard, “Dispersion compensating fibers,” Optical Fiber Tech. 6, 164–180 (2000).
[Crossref]

Korenev, I. L.

L. A. Yudin, S. P. Efimov, M. I. Kapchinsky, and I. L. Korenev, “Electrodynamics as a problem of eigenvalues,” Phys. Plasmas 3, 42–58 (1996).
[Crossref]

Kuester, F.

L. Lewin, D. C. Chang, and F. Kuester, Electromagnetic Waves and Curved Structures (P. Peregrinus, England, 1977).

Kumar, A.

A. Kumar, S. I. Hosain, and A. K. Ghatak, “Propagation characteristics of weakly guiding lossy fibers: an exact and perturbation analysis,” Optica Acta 28, 559–566 (1981).
[Crossref]

Lacroix, S.

Laloë, F.

C. Cohen-Tannoudji, B. Din, and F. Laloë, Quantum Mechanics (Hermann, Paris, 1977), Vol. One, ch. 2; and Vol. Two, ch. 11 and 13.

Larsen, C. C.

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, and H. Damsgaard, “Dispersion compensating fibers,” Optical Fiber Tech. 6, 164–180 (2000).
[Crossref]

Lazarchik, A. N.

A. N. Lazarchik, “Bragg fiber lightguides,” Radiotekhnika i electronika 1, 36–43 (1988).

Lee, R. K.

Leproux, P.

F. Brechet, P. Leproux, P. Roy, J. Marcou, and D. Pagnoux, “Analysis of bandpass filtering behavior of singlemode depressed-core-index photonic bandgap fibre,” Elec. Lett. 36, 870–872 (2000).
[Crossref]

Lewin, L.

L. Lewin, D. C. Chang, and F. Kuester, Electromagnetic Waves and Curved Structures (P. Peregrinus, England, 1977).

Lohmeyer, M.

M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. 163, 86–94 (1999).
[Crossref]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

Magnussen, D.

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, and H. Damsgaard, “Dispersion compensating fibers,” Optical Fiber Tech. 6, 164–180 (2000).
[Crossref]

Mangan, B. J.

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

Marcatili, E. A.

E. A. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).

Marcou, J.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Elec. Lett. 36, 514–515 (2000).
[Crossref]

F. Brechet, P. Leproux, P. Roy, J. Marcou, and D. Pagnoux, “Analysis of bandpass filtering behavior of singlemode depressed-core-index photonic bandgap fibre,” Elec. Lett. 36, 870–872 (2000).
[Crossref]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Marom, E.

McCarthy, S. L.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton, 1995).

Mercader del Río, L.

B. Z. Katsenelenbaum, L. Mercader del Río, M. Pereyaslavets, M. Sorolla Ayza, and M. Thumm, Theory of Nonuniform Waveguides: The Cross-Section Method (Inst. of Electrical Engineers, London, 1998).
[Crossref]

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt Saunders, Philadelphia, 1976).

Messiah, A.

A. Messiah, Quantum Mechanics: Vol. II (Wiley, New York, 1976), ch. 17.

Michel, J.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[Crossref] [PubMed]

Mittra, R.

Z. Pantic and R. Mittra, “Quasi-TEM analysis of microwave transmission lines by the finite-element method,” IEEE Trans. Microwave Theory Tech. MTT-34, 1096–1103 (1986).
[Crossref]

Miyagi, M.

M. Miyagi, K. Harada, and S. Kawakami, “Wave propagation and attenuation in the general class of circular hollow waveguides with uniform curvature,” IEEE Trans. Microwave Theory Tech. MTT-32, 513–521 (1984).
[Crossref]

M. Miyagi and S. Kawakami, “Design theory of dielectric-coated circular metallic waveguides for infrared transmission,” J. Lightwave Tech. 2, 116–126 (1984).
[Crossref]

M. Miyagi, A. Hongo, and S. Kawakami, “Transmission characteristics of dielectric-coated metallic waveguides for infrared transmission: slab waveguide model,” IEEE J. Quantum Elec. QE-19, 136–145 (1983).
[Crossref]

Nolan, D. A.

Pagnoux, D.

F. Brechet, P. Leproux, P. Roy, J. Marcou, and D. Pagnoux, “Analysis of bandpass filtering behavior of singlemode depressed-core-index photonic bandgap fibre,” Elec. Lett. 36, 870–872 (2000).
[Crossref]

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Elec. Lett. 36, 514–515 (2000).
[Crossref]

Pantic, Z.

Z. Pantic and R. Mittra, “Quasi-TEM analysis of microwave transmission lines by the finite-element method,” IEEE Trans. Microwave Theory Tech. MTT-34, 1096–1103 (1986).
[Crossref]

Pereyaslavets, M.

B. Z. Katsenelenbaum, L. Mercader del Río, M. Pereyaslavets, M. Sorolla Ayza, and M. Thumm, Theory of Nonuniform Waveguides: The Cross-Section Method (Inst. of Electrical Engineers, London, 1998).
[Crossref]

Rahman, B. M. A.

C. Themistos, B. M. A. Rahman, A. Hadjicharalambous, and K. T. V. Grattan, “Loss/gain characterization of optical waveguides,” J. Lightwave Tech. 13, 1760–1765 (1995).
[Crossref]

Ramaswami, R.

R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Academic Press, London, London, 1998).

Ripin, D. J.

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Tech. 17, 2039–2041 (1999).
[Crossref]

Roberts, P. J.

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

Roy, P.

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Elec. Lett. 36, 514–515 (2000).
[Crossref]

F. Brechet, P. Leproux, P. Roy, J. Marcou, and D. Pagnoux, “Analysis of bandpass filtering behavior of singlemode depressed-core-index photonic bandgap fibre,” Elec. Lett. 36, 870–872 (2000).
[Crossref]

Russell, P. S.-J.

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

Sarid, D.

D. Sarid and G. I. Stegeman, “Optimization of the effects of power dependent refractive indices in optical waveguides,” J. Appl. Phys. 52, 5439–5441 (1981).
[Crossref]

Schmeltzer, R. A.

E. A. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).

Sharma, E. K.

She, S. X.

Sivarajan, K. N.

R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Academic Press, London, London, 1998).

Skorobogatiy, M.

M. Skorobogatiyet al., to be published in 2002.

Snyder, A. W.

A. W. Snyder, “Radiation losses due to variations of radius on dielectric or optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-18, 608–615 (1970).
[Crossref]

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

Song, G. H.

Sorolla Ayza, M.

B. Z. Katsenelenbaum, L. Mercader del Río, M. Pereyaslavets, M. Sorolla Ayza, and M. Thumm, Theory of Nonuniform Waveguides: The Cross-Section Method (Inst. of Electrical Engineers, London, 1998).
[Crossref]

Stegeman, G. I.

D. Sarid and G. I. Stegeman, “Optimization of the effects of power dependent refractive indices in optical waveguides,” J. Appl. Phys. 52, 5439–5441 (1981).
[Crossref]

Themistos, C.

C. Themistos, B. M. A. Rahman, A. Hadjicharalambous, and K. T. V. Grattan, “Loss/gain characterization of optical waveguides,” J. Lightwave Tech. 13, 1760–1765 (1995).
[Crossref]

Thomas, E. L.

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]

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Tech. 17, 2039–2041 (1999).
[Crossref]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[Crossref] [PubMed]

Thumm, M.

B. Z. Katsenelenbaum, L. Mercader del Río, M. Pereyaslavets, M. Sorolla Ayza, and M. Thumm, Theory of Nonuniform Waveguides: The Cross-Section Method (Inst. of Electrical Engineers, London, 1998).
[Crossref]

Tomlinson, W. J.

Tzolov, V. P.

Veng, T.

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, and H. Damsgaard, “Dispersion compensating fibers,” Optical Fiber Tech. 6, 164–180 (2000).
[Crossref]

Warters, W. D.

W. D. Warters, “WT4 millimeter waveguide system: introduction,” Bell Syst. Tech. J. 56, 1825–1827 (1977), the introduction to a special issue with many useful articles.

Weber, W. H.

Winn, J. N.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[Crossref] [PubMed]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton, 1995).

Xu, Y.

Y. Xu and A. Yariv, “Asymptotic analysis of Bragg fibers and dielectric coaxial fibers,” In Proc. SPIE, A. Dutta, A. A. S. Awwal, N. K. Dutta, and K. Okamoto, eds., 4532, 191–205 (2001).
[Crossref]

Y. Xu, R. K. Lee, and A. Yariv, “Asymptotic analysis of Bragg fibers,” Opt. Lett. 25, 1756–1758 (2000).
[Crossref]

Yariv, A.

Y. Xu and A. Yariv, “Asymptotic analysis of Bragg fibers and dielectric coaxial fibers,” In Proc. SPIE, A. Dutta, A. A. S. Awwal, N. K. Dutta, and K. Okamoto, eds., 4532, 191–205 (2001).
[Crossref]

Y. Xu, R. K. Lee, and A. Yariv, “Asymptotic analysis of Bragg fibers,” Opt. Lett. 25, 1756–1758 (2000).
[Crossref]

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

Yeh, P.

Yudin, L. A.

L. A. Yudin, S. P. Efimov, M. I. Kapchinsky, and I. L. Korenev, “Electrodynamics as a problem of eigenvalues,” Phys. Plasmas 3, 42–58 (1996).
[Crossref]

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

E. A. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).

W. D. Warters, “WT4 millimeter waveguide system: introduction,” Bell Syst. Tech. J. 56, 1825–1827 (1977), the introduction to a special issue with many useful articles.

Elec. Lett. (2)

F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, “Singlemode propagation into depressed-core-index photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths,” Elec. Lett. 36, 514–515 (2000).
[Crossref]

F. Brechet, P. Leproux, P. Roy, J. Marcou, and D. Pagnoux, “Analysis of bandpass filtering behavior of singlemode depressed-core-index photonic bandgap fibre,” Elec. Lett. 36, 870–872 (2000).
[Crossref]

Fiber Integr. Opt. (1)

J. A. Harrington, “A review of IR transmitting, hollow waveguides,” Fiber Integr. Opt. 19, 211–227 (2000).
[Crossref]

IEEE J. Quantum Elec. (1)

M. Miyagi, A. Hongo, and S. Kawakami, “Transmission characteristics of dielectric-coated metallic waveguides for infrared transmission: slab waveguide model,” IEEE J. Quantum Elec. QE-19, 136–145 (1983).
[Crossref]

IEEE J. Selected Topics Quantum Elec. (1)

D. Q. Chowdhury, “Comparison between optical fiber birefringence induced by stress anisotropy and geometric deformation,” IEEE J. Selected Topics Quantum Elec. 6, 227–232 (2000).
[Crossref]

IEEE Trans. Microwave Theory Tech. (3)

M. Miyagi, K. Harada, and S. Kawakami, “Wave propagation and attenuation in the general class of circular hollow waveguides with uniform curvature,” IEEE Trans. Microwave Theory Tech. MTT-32, 513–521 (1984).
[Crossref]

A. W. Snyder, “Radiation losses due to variations of radius on dielectric or optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-18, 608–615 (1970).
[Crossref]

Z. Pantic and R. Mittra, “Quasi-TEM analysis of microwave transmission lines by the finite-element method,” IEEE Trans. Microwave Theory Tech. MTT-34, 1096–1103 (1986).
[Crossref]

In Proc. SPIE (1)

Y. Xu and A. Yariv, “Asymptotic analysis of Bragg fibers and dielectric coaxial fibers,” In Proc. SPIE, A. Dutta, A. A. S. Awwal, N. K. Dutta, and K. Okamoto, eds., 4532, 191–205 (2001).
[Crossref]

J. Appl. Phys. (2)

C. M. de Sterke and I. M. Bassett, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680–688 (1994).
[Crossref]

D. Sarid and G. I. Stegeman, “Optimization of the effects of power dependent refractive indices in optical waveguides,” J. Appl. Phys. 52, 5439–5441 (1981).
[Crossref]

J. Lightwave Tech. (4)

C. Themistos, B. M. A. Rahman, A. Hadjicharalambous, and K. T. V. Grattan, “Loss/gain characterization of optical waveguides,” J. Lightwave Tech. 13, 1760–1765 (1995).
[Crossref]

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Tech. 17, 2039–2041 (1999).
[Crossref]

N. J. Doran and K. J. Bulow, “Cylindrical Bragg fibers: a design and feasibility study for optical communications,” J. Lightwave Tech. 1, 588–590 (1983).
[Crossref]

M. Miyagi and S. Kawakami, “Design theory of dielectric-coated circular metallic waveguides for infrared transmission,” J. Lightwave Tech. 2, 116–126 (1984).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. 163, 86–94 (1999).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Optica Acta (1)

A. Kumar, S. I. Hosain, and A. K. Ghatak, “Propagation characteristics of weakly guiding lossy fibers: an exact and perturbation analysis,” Optica Acta 28, 559–566 (1981).
[Crossref]

Optical and Quantum Elec. (1)

R. S. Grant, “Effective non-linear coefficients in optical waveguides,” Optical and Quantum Elec. 28, 1161–1173 (1996).
[Crossref]

Optical Fiber Tech. (1)

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, and H. Damsgaard, “Dispersion compensating fibers,” Optical Fiber Tech. 6, 164–180 (2000).
[Crossref]

Phys. Plasmas (1)

L. A. Yudin, S. P. Efimov, M. I. Kapchinsky, and I. L. Korenev, “Electrodynamics as a problem of eigenvalues,” Phys. Plasmas 3, 42–58 (1996).
[Crossref]

Radiotekhnika i electronika (1)

A. N. Lazarchik, “Bragg fiber lightguides,” Radiotekhnika i electronika 1, 36–43 (1988).

Science (3)

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[Crossref] [PubMed]

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]

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

Sov. J. Quantum Elec. (2)

V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence in a single-mode elliptic optical fiber,” Sov. J. Quantum Elec. 13, 109–111 (1983).
[Crossref]

V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence of a three-layer elliptic single-mode fiber waveguide,” Sov. J. Quantum Elec. 14, 427–430 (1984).
[Crossref]

Other (16)

M. Skorobogatiyet al., to be published in 2002.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Characteristics of a single-mode optical fibre cable (Intl. Telecom. Union, 2000), No. G.652.

S. G. Johnsonet al., to be published in 2002.

C. Cohen-Tannoudji, B. Din, and F. Laloë, Quantum Mechanics (Hermann, Paris, 1977), Vol. One, ch. 2; and Vol. Two, ch. 11 and 13.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton, 1995).

B. Z. Katsenelenbaum, L. Mercader del Río, M. Pereyaslavets, M. Sorolla Ayza, and M. Thumm, Theory of Nonuniform Waveguides: The Cross-Section Method (Inst. of Electrical Engineers, London, 1998).
[Crossref]

L. Lewin, D. C. Chang, and F. Kuester, Electromagnetic Waves and Curved Structures (P. Peregrinus, England, 1977).

R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Academic Press, London, London, 1998).

M. Ibanescuet al., to be published in 2002.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

S. A. Jacobset al., to be published in 2002.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt Saunders, Philadelphia, 1976).

A. Messiah, Quantum Mechanics: Vol. II (Wiley, New York, 1976), ch. 17.

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Figures (16)

Fig. 1.
Fig. 1.

(a) Hollow dielectric waveguide of radius R. Light is confined in the hollow core by a multilayer dielectric mirror made of alternating layers with high (blue) and low (green) indices of refraction. (b) Hollow metallic waveguide of radius R. Light is confined in the hollow core by a metallic cylinder.

Fig. 2.
Fig. 2.

(Left) Projected band structure associated with the planar dielectric mirror. The blue regions correspond to (β, ω) pairs for which light can propagate within the mirror. White and gray regions correspond to situations where light cannot propagate within the mirror. The thick black line represents the light line (ω=). Shown in gray are the two omnidirectional frequency ranges of the mirror. (Right) Dispersion relations ω(β) of the lowest 7 modes supported by a hollow metallic waveguide of radius R=2a are plotted. TE/TM-polarized modes are shown in red/blue, and the modes have angular dependence eimφ . Note the degeneracy of the TE01 and the TM11 modes.

Fig. 3.
Fig. 3.

Guided modes supported by a hollow OmniGuide fiber of radius R=2a: red lines are for TE and HE modes, while blue is for TM and EH modes. In black is the light line (ω=), and the solid blue regions represent the continuum of modes that propagate within the multilayer cladding. Only the first three modes in each band gap are labeled.

Fig. 4.
Fig. 4.

An OmniGuide fiber with core radius R=30a, the parameters that we employ in the remainder of this paper. The omnidirectional mirror here comprises 17 layers, starting with a high-index layer, with indices 4.6/1.6 and thicknesses 0.22a/0.78a, respectively. (The omnidirectional mirror is surrounded by some coating for mechanical support; this layer is not shown to scale.) We choose a=0.434µm, so that the lowest dissipation losses occur roughly at λ=1.55µm.

Fig. 5.
Fig. 5.

Transverse electric-field distributions in the OmniGuide fiber of Fig. 4 for the TE01 mode (left) and the EH11 mode (right), which have β=0.27926 · 2π/a and β=0.27955·2π/a, respectively, at ω=0.28·2πc/a.

Fig. 6.
Fig. 6.

The (unnormalized) electric field Eφ for the TE01 mode in the OmniGuide fiber of Fig. 4. The lower plot displays the same field, but with the vertical scale exaggerated in order to show the field amplitude in the cladding. The field has a node near the core interface at R, and so the field amplitude in the cladding is determined by the slope at that point.

Fig. 7.
Fig. 7.

Radiation leakage through a finite number (17) of cladding layers in the OmniGuide fiber of Fig. 4. The lowest-loss mode is TE01 (solid blue) and the next-lowest is TE02 (red dots), while the linearly-polarized EH11 mode (black circles) typifies the higher losses for mixed-polarization modes due to the smaller TM band gap.

Fig. 8.
Fig. 8.

Group-velocity (chromatic) dispersion of the TE01 mode in both the OmniGuide fiber of Fig. 4 (solid blue) and a hollow metallic waveguide with the same core radius (green circles).

Fig. 9.
Fig. 9.

Absorption losses due to the cladding materials the OmniGuide fiber (with core radius 30a), as a fraction of the bulk cladding losses. The lowest-loss mode is TE01 (solid blue) and the next-lowest is TE02 (red dots), while the linearly-polarized EH11 mode (black circles) typifies the higher losses for mixed-polarization modes due to the smaller TM band gap.

Fig. 10.
Fig. 10.

The TE01 mode’s suppression factor for cladding nonlinearities in the OmniGuide fiber of Fig. 4, relative to nonlinearities that include the core.

Fig. 11.
Fig. 11.

Scaling of the cladding absorption and nonlinearity suppression factors for a core radius R varying from 7a to 30a (taking the minimum over the TE01 band at each radius). Hollow squares/circles show the computed values, and the solid lines display the values predicted by starting from the 30a value and applying the scaling laws.

Fig. 12.
Fig. 12.

Minimum bending radius R 0.1% to achieve 0.1% worst-case scattering losses for the TE01 mode in the OmniGuide fiber of Fig. 4. Conversely, the losses for a given bending radius Rb are 0.1% · (R 0.1%/Rb )2. The sharp peak (actually a divergence) in R 0.1% is due to the point of degeneracy between TE01 and EH11.

Fig. 13.
Fig. 13.

Scaling of the ellipticity-induced phase shift ℜ[Δ β ˜ (2)] and loss Δα(2) for the OmniGuide-fiber TE01 mode at λ=1.55µm as a function of the core radius R, varying from 20–80a. The abscissa is the expected scaling form of 1/R 2 and 1/R, respectively. The amount of ellipticity is δ=1% and the cladding has 17 layers.

Fig. 14.
Fig. 14.

The estimated radiative loss α+Δα(2) of the TE01 mode at λ=1.55µm for the OmniGuide fiber of Fig. 4 with an elliptical perturbation, plotted versus the number of cladding layers for three ellipticities: δ=0.5%,1%,2%. Red circles indictate the losses for the perturbed fibers, while solid blue lines above and below are the losses of the unperturbed HE21 and TE01 modes, respectively. As the number of layers increases, the losses become dominated by coupling to HE21, due to the weaker band gap for TM polarizations.

Fig. 15.
Fig. 15.

Enhanced loss of the TE01 mode at λ=1.55µm from randomly-varying ellipticity with rms δ0 =1%, as a function of the ellipticity correlation length Lc , due to coupling with the HE21 and EH21 modes. The inset shows the fractional scattered power in the parasitic HE21/EH21 modes as a function of Lc . If the correlation length exceeds ~1.6mm, the induced losses become smaller than 0.01 dB/km and the fractional parasitic power is less than 10-4.

Fig. 16.
Fig. 16.

Roughness radiation efficiency ρs , as defined by Eq. (24) for the TE01 mode of the OmniGuide fiber of Fig. 4, based on the | E |2 at the core/cladding interface (blue dots) or averaged over a 10nm interval (red circles). Approximate roughness losses are computed by multiplying ρs with a dimensionful quantity s that is dependent on the scatterer quantity; e.g. s=0.03 dB/km for one 10nm scatterer every 10µm2.

Tables (1)

Tables Icon

Table 1. Scaling relations with core radius R for TE01 in OmniGuide fibers.

Equations (50)

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d hi d lo = n lo 2 1 n hi 2 1
normalized T E 0 cladding E 1 R 2 .
fraction of E 2 in cladding for TE 0 1 R 3 ,
mode separation Δ β 1 R 2 ;
TE 01 and EH 11 mode separation Δ β 1 R .
f m ( ω , β ˜ ) det [ M m ( ω , β ˜ ) ] ,
TE 0 radiation leakage α 1 R 3 ,
D ω 2 2 π c d 2 β d ω 2 ,
Δ ε = 2 n n 2 E 2 .
TE 0 cladding absorption 1 R 3 .
TE 0 cladding nonlinearity γ γ 0 1 R 5 .
n Δ A ̂ n = E x E y E z H z H y H z ω x cR b ( ε ε ε μ μ μ ) E x E y E z H z H y H z
n Δ A ̂ n = β n x R b ( E t * × H t + E t × H t * )
i R b ( E y * H z H y * E z ) .
n ; m Δ A ̂ n ; m = ( r integral ) · 0 2 π e i Δ m φ ( e i φ + e i φ ) d φ 4 π
= ( r integral ) · δ Δ m , ± 1 2 ,
minimum bend R b for 0.1 % losses , R 0.1 % R 2 .
Δ α n ( 2 ) n n n Δ A ̂ n 2 Δ β n n 2 ( α n α n ) ,
n ; m Δ A ̂ n ; m = E r E φ H r H φ ω c b η ( ε ± ε ± i ε ε μ ± i μ ± i μ μ ) E r E φ H r H φ δ m , m 2 , ,
TE 0 Δ A ̂ m = ± 2 δ R 2 .
[ Δ β ˜ ( 2 ) ] δ 2 R 2 .
Δ α ( 2 ) = 2 [ Δ β ˜ ( 2 ) ] δ 2 R .
P n z = α n P n + m M n m ( P m P n ) ,
P s = c 2 μ 0 ε 0 12 π ( ω c ) 4 p 2 = ω 4 12 π Δ ε E V s 2 ,
α = P s · 2 π R σ s P = ( ω 4 12 π Δ ε E 2 · 2 π R · 4 ) · σ s V s 2 ,
ρ s 2 ω 4 R 3 Δ ε E 2 ,
s σ s V s 2 a 5 · 10 ln 10 ,
κ te n ˜ lo n ˜ hi ,
κ tm n lo 2 n ˜ hi n hi 2 n ˜ lo > κ te .
fraction of E 2 in cladding for TE 0 f hi 2 ( 1 κ te 2 ) ( a R ) 3 .
TE 0 cladding absorption n ̅ f hi 2 ( 1 κ te 2 ) ( a R ) 3 .
TE 0 cladding nonlinearity γ γ 0 n ̅ f hi 4 ( 1 κ te 4 ) ( a R ) 5 .
Δ ω tm ω 0 = 4 π sin 1 ( n hi 2 n ˜ lo n lo 2 n ˜ hi n hi 2 n ˜ lo + n lo 2 n ˜ hi ) ,
ω 0 = n ˜ lo + n ˜ hi 4 n ˜ lo n ˜ hi · 2 π c a .
H z = i c ω μ t × E t ,
E z = i c ω ε t × H t
A ̂ ψ = i z B ̂ ψ ,
ψ ( E t ( z ) H t ( z ) ) ,
ψ ψ E t * · E t + H t * · H t ,
A ̂ ( ω ε c c ω t × 1 μ t × 0 0 ω μ c c ω t × 1 ε t × ) ,
B ̂ ( 0 z ̂ × z ̂ × 0 ) = ( 1 1 1 1 ) = B ̂ 1 .
e i ( β z ω t ) ψ .
A ̂ ψ = β B ̂ ψ .
ψ B ̂ ψ = z ̂ · E t * × H t + E t × H t * ,
Δ β n ( 1 ) = n Δ A ̂ n ,
Δ β n ( 2 ) = n n n Δ A ̂ n 2 Δ β n n ,
dc n dz = i β n c n + i n n Δ A ̂ n c n .
c n ( z ) 2 c n ( 0 ) 2 4 n Δ A ̂ n 2 Δ β n n 2 sin 2 ( Δ β n n z 2 ) .
dc n dz = i β n c n n n n d A ̂ dz n Δ β n n c n .
n Δ A ̂ n = E x E y E z ω c ( Δ ε Δ ε Δ ε ) E x E y E z + O ( Δ ε 2 ) .

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