Abstract

We adapt the resonant state expansion to optical fibers such as capillary and photonic crystal fibers. As a key requirement of the resonant state expansion and any related perturbative approach, we derive the correct analytical normalization for all modes of these fiber structures, including leaky modes that radiate energy perpendicular to the direction of propagation and have fields that grow with distance from the fiber core. Based on the normalized fiber modes, an eigenvalue equation is derived that allows for calculating the influence of small and large perturbations such as structural disorder on the guiding properties. This is demonstrated for two test systems: a capillary fiber and a photonic crystal fiber.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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2018 (1)

2017 (3)

T. Weiss, M. Schäferling, H. Giessen, N. A. Gippius, S. G. Tikhodeev, W. Langbein, and E. A. Muljarov, “Analytical normalization of resonant states in photonic crystal slabs and periodic arrays of nanoantennas at oblique incidence,” Phys. Rev. B 96, 045129 (2017).
[Crossref]

G. Li, M. Zeisberger, and M. A. Schmidt, “Guiding light in a water core all-solid cladding photonic band gap fiber–an innovative platform for fiber-based optofluidics,” Opt. Express 25, 22467–22479 (2017).
[Crossref] [PubMed]

S. V. Lobanov, G. Zoriniants, W. Langbein, and E. A. Muljarov, “Resonant-state expansion of light propagation in nonuniform waveguides,” Phys. Rev. A 95, 053848 (2017).
[Crossref]

2016 (3)

T. Weiss, M. Mesch, M. Schäferling, H. Giessen, W. Langbein, and E. A. Muljarov, “From dark to bright: first-order perturbation theory with analytical mode normalization for plasmonic nanoantenna arrays applied to refractive index sensing,” Phys. Rev. Lett. 116, 237401 (2016).
[Crossref] [PubMed]

E. A. Muljarov and W. Langbein, “Resonant-state expansion of dispersive open optical systems: Creating gold from sand,” Phys. Rev. B 93, 075417 (2016).
[Crossref]

E. A. Muljarov and W. Langbein, “Exact mode volume and Purcell factor of open optical systems,” Phys. Rev. B 94, 235438 (2016).
[Crossref]

2014 (2)

L. J. Armitage, M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant-state expansion applied to planar waveguides,” Phys. Rev. A 89, 053832 (2014).
[Crossref]

M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant-state expansion applied to three-dimensional open optical systems,” Phys. Rev. A 90, 013834 (2014).
[Crossref]

2013 (4)

M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant state expansion applied to two-dimensional open optical systems,” Phys. Rev. A 87, 043827 (2013).
[Crossref]

C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013).
[Crossref] [PubMed]

P. T. Kristensen and S. Hughes, “Modes and mode volumes of leaky optical cavities and plasmonic nanoresonators,” ACS Photonics 1, 2–10 (2013).
[Crossref]

M. H. Frosz, J. Nold, T. Weiss, A. Stefani, F. Babic, S. Rammler, and P. St. J. Russell, “Five-ring hollow-core photonic crystal fiber with 1.8 dB/km loss,” Opt. Lett. 38, 2215–2217 (2013).
[Crossref] [PubMed]

2012 (1)

M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant-state expansion applied to planar open optical systems,” Phys. Rev. A 85, 023835 (2012).
[Crossref]

2011 (1)

2010 (2)

2009 (1)

2006 (2)

D. Nau, A. Schönhardt, C. Bauer, A. Christ, T. Zentgraf, J. Kuhl, and H. Giessen, “Disorder issues in metallic photonic crystals,” Phys. Status Solidi B 243, 2331–2343 (2006).
[Crossref]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

2004 (3)

2003 (1)

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

2002 (3)

2000 (1)

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photonics Technol. Lett. 12, 807–809 (2000).
[Crossref]

1997 (1)

1996 (1)

1995 (1)

S.-L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[Crossref]

1990 (1)

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187 (1990).
[Crossref] [PubMed]

1976 (2)

1964 (1)

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

Antonopoulos, G.

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

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, 1972).

Armitage, L. J.

L. J. Armitage, M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant-state expansion applied to planar waveguides,” Phys. Rev. A 89, 053832 (2014).
[Crossref]

Arriaga, J.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photonics Technol. Lett. 12, 807–809 (2000).
[Crossref]

Atkin, D. M.

Babic, F.

Barber, P. W.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187 (1990).
[Crossref] [PubMed]

Bauer, C.

D. Nau, A. Schönhardt, C. Bauer, A. Christ, T. Zentgraf, J. Kuhl, and H. Giessen, “Disorder issues in metallic photonic crystals,” Phys. Status Solidi B 243, 2331–2343 (2006).
[Crossref]

Benabid, F.

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

Birks, T. A.

Botten, L. C.

Bouwmans, G.

Christ, A.

D. Nau, A. Schönhardt, C. Bauer, A. Christ, T. Zentgraf, J. Kuhl, and H. Giessen, “Disorder issues in metallic photonic crystals,” Phys. Status Solidi B 243, 2331–2343 (2006).
[Crossref]

Chung, Y.

S.-L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[Crossref]

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

Coldren, L. A.

S.-L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[Crossref]

Cucinotta, A.

F. Poli, A. Cucinotta, and S. Selleri, Photonic Crystal Fibers: Properties and Applications(Springer Science & Business Media, 2007).

Da, N.

Dagli, N.

S.-L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[Crossref]

de Sterke, C. M.

Doost, M. B.

L. J. Armitage, M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant-state expansion applied to planar waveguides,” Phys. Rev. A 89, 053832 (2014).
[Crossref]

M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant-state expansion applied to three-dimensional open optical systems,” Phys. Rev. A 90, 013834 (2014).
[Crossref]

M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant state expansion applied to two-dimensional open optical systems,” Phys. Rev. A 87, 043827 (2013).
[Crossref]

M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant-state expansion applied to planar open optical systems,” Phys. Rev. A 85, 023835 (2012).
[Crossref]

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

Fini, J. M.

Frosz, M. H.

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

Giessen, H.

T. Weiss, M. Schäferling, H. Giessen, N. A. Gippius, S. G. Tikhodeev, W. Langbein, and E. A. Muljarov, “Analytical normalization of resonant states in photonic crystal slabs and periodic arrays of nanoantennas at oblique incidence,” Phys. Rev. B 96, 045129 (2017).
[Crossref]

T. Weiss, M. Mesch, M. Schäferling, H. Giessen, W. Langbein, and E. A. Muljarov, “From dark to bright: first-order perturbation theory with analytical mode normalization for plasmonic nanoantenna arrays applied to refractive index sensing,” Phys. Rev. Lett. 116, 237401 (2016).
[Crossref] [PubMed]

D. Nau, A. Schönhardt, C. Bauer, A. Christ, T. Zentgraf, J. Kuhl, and H. Giessen, “Disorder issues in metallic photonic crystals,” Phys. Status Solidi B 243, 2331–2343 (2006).
[Crossref]

Gippius, N. A.

T. Weiss, M. Schäferling, H. Giessen, N. A. Gippius, S. G. Tikhodeev, W. Langbein, and E. A. Muljarov, “Analytical normalization of resonant states in photonic crystal slabs and periodic arrays of nanoantennas at oblique incidence,” Phys. Rev. B 96, 045129 (2017).
[Crossref]

Granzow, N.

Hansen, T. P.

Hill, S. C.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187 (1990).
[Crossref] [PubMed]

Huang, Y.

Hughes, S.

P. T. Kristensen and S. Hughes, “Modes and mode volumes of leaky optical cavities and plasmonic nanoresonators,” ACS Photonics 1, 2–10 (2013).
[Crossref]

Hugonin, J. P.

C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013).
[Crossref] [PubMed]

Humbert, G.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

Knight, J. C.

Kristensen, P. T.

P. T. Kristensen and S. Hughes, “Modes and mode volumes of leaky optical cavities and plasmonic nanoresonators,” ACS Photonics 1, 2–10 (2013).
[Crossref]

Kuhl, J.

D. Nau, A. Schönhardt, C. Bauer, A. Christ, T. Zentgraf, J. Kuhl, and H. Giessen, “Disorder issues in metallic photonic crystals,” Phys. Status Solidi B 243, 2331–2343 (2006).
[Crossref]

Kuhlmey, B. T.

Lai, H. M.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187 (1990).
[Crossref] [PubMed]

Lalanne, P.

C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013).
[Crossref] [PubMed]

Langbein, W.

S. V. Lobanov, G. Zoriniants, W. Langbein, and E. A. Muljarov, “Resonant-state expansion of light propagation in nonuniform waveguides,” Phys. Rev. A 95, 053848 (2017).
[Crossref]

T. Weiss, M. Schäferling, H. Giessen, N. A. Gippius, S. G. Tikhodeev, W. Langbein, and E. A. Muljarov, “Analytical normalization of resonant states in photonic crystal slabs and periodic arrays of nanoantennas at oblique incidence,” Phys. Rev. B 96, 045129 (2017).
[Crossref]

E. A. Muljarov and W. Langbein, “Resonant-state expansion of dispersive open optical systems: Creating gold from sand,” Phys. Rev. B 93, 075417 (2016).
[Crossref]

E. A. Muljarov and W. Langbein, “Exact mode volume and Purcell factor of open optical systems,” Phys. Rev. B 94, 235438 (2016).
[Crossref]

T. Weiss, M. Mesch, M. Schäferling, H. Giessen, W. Langbein, and E. A. Muljarov, “From dark to bright: first-order perturbation theory with analytical mode normalization for plasmonic nanoantenna arrays applied to refractive index sensing,” Phys. Rev. Lett. 116, 237401 (2016).
[Crossref] [PubMed]

L. J. Armitage, M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant-state expansion applied to planar waveguides,” Phys. Rev. A 89, 053832 (2014).
[Crossref]

M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant-state expansion applied to three-dimensional open optical systems,” Phys. Rev. A 90, 013834 (2014).
[Crossref]

M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant state expansion applied to two-dimensional open optical systems,” Phys. Rev. A 87, 043827 (2013).
[Crossref]

M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant-state expansion applied to planar open optical systems,” Phys. Rev. A 85, 023835 (2012).
[Crossref]

E. A. Muljarov, W. Langbein, and R. Zimmermann, “Brillouin-Wigner perturbation theory in open electromagnetic systems,” Europhys. Lett. 92, 50010 (2010).
[Crossref]

Lee, S.-L.

S.-L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[Crossref]

Leung, P. T.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187 (1990).
[Crossref] [PubMed]

Li, G.

Liu, X.

Lobanov, S. V.

S. V. Lobanov, G. Zoriniants, W. Langbein, and E. A. Muljarov, “Resonant-state expansion of light propagation in nonuniform waveguides,” Phys. Rev. A 95, 053848 (2017).
[Crossref]

Love, J. D.

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

Ludvigsen, H.

Maksymov, I. S.

C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013).
[Crossref] [PubMed]

Mangan, B. J.

Marcatili, E. A.

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

Marcuse, D.

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

Maystre, D.

McPhedran, R. C.

Mesch, M.

T. Weiss, M. Mesch, M. Schäferling, H. Giessen, W. Langbein, and E. A. Muljarov, “From dark to bright: first-order perturbation theory with analytical mode normalization for plasmonic nanoantenna arrays applied to refractive index sensing,” Phys. Rev. Lett. 116, 237401 (2016).
[Crossref] [PubMed]

Muljarov, E. A.

E. A. Muljarov and T. Weiss, “Resonant-state expansion for open optical systems: generalization to magnetic, chiral, and bi-anisotropic materials,” Opt. Lett. 43, 1978–1981 (2018).
[Crossref] [PubMed]

S. V. Lobanov, G. Zoriniants, W. Langbein, and E. A. Muljarov, “Resonant-state expansion of light propagation in nonuniform waveguides,” Phys. Rev. A 95, 053848 (2017).
[Crossref]

T. Weiss, M. Schäferling, H. Giessen, N. A. Gippius, S. G. Tikhodeev, W. Langbein, and E. A. Muljarov, “Analytical normalization of resonant states in photonic crystal slabs and periodic arrays of nanoantennas at oblique incidence,” Phys. Rev. B 96, 045129 (2017).
[Crossref]

T. Weiss, M. Mesch, M. Schäferling, H. Giessen, W. Langbein, and E. A. Muljarov, “From dark to bright: first-order perturbation theory with analytical mode normalization for plasmonic nanoantenna arrays applied to refractive index sensing,” Phys. Rev. Lett. 116, 237401 (2016).
[Crossref] [PubMed]

E. A. Muljarov and W. Langbein, “Resonant-state expansion of dispersive open optical systems: Creating gold from sand,” Phys. Rev. B 93, 075417 (2016).
[Crossref]

E. A. Muljarov and W. Langbein, “Exact mode volume and Purcell factor of open optical systems,” Phys. Rev. B 94, 235438 (2016).
[Crossref]

M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant-state expansion applied to three-dimensional open optical systems,” Phys. Rev. A 90, 013834 (2014).
[Crossref]

L. J. Armitage, M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant-state expansion applied to planar waveguides,” Phys. Rev. A 89, 053832 (2014).
[Crossref]

M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant state expansion applied to two-dimensional open optical systems,” Phys. Rev. A 87, 043827 (2013).
[Crossref]

M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant-state expansion applied to planar open optical systems,” Phys. Rev. A 85, 023835 (2012).
[Crossref]

E. A. Muljarov, W. Langbein, and R. Zimmermann, “Brillouin-Wigner perturbation theory in open electromagnetic systems,” Europhys. Lett. 92, 50010 (2010).
[Crossref]

Nau, D.

D. Nau, A. Schönhardt, C. Bauer, A. Christ, T. Zentgraf, J. Kuhl, and H. Giessen, “Disorder issues in metallic photonic crystals,” Phys. Status Solidi B 243, 2331–2343 (2006).
[Crossref]

Nold, J.

Ortigosa-Blanch, A.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photonics Technol. Lett. 12, 807–809 (2000).
[Crossref]

Peng, M.

Petersen, J. C.

Poli, F.

F. Poli, A. Cucinotta, and S. Selleri, Photonic Crystal Fibers: Properties and Applications(Springer Science & Business Media, 2007).

Rammler, S.

Ren, X.

Renversez, G.

Ritari, T.

Roberts, P. J.

Russell, P. St. J.

M. H. Frosz, J. Nold, T. Weiss, A. Stefani, F. Babic, S. Rammler, and P. St. J. Russell, “Five-ring hollow-core photonic crystal fiber with 1.8 dB/km loss,” Opt. Lett. 38, 2215–2217 (2013).
[Crossref] [PubMed]

N. Granzow, P. Uebel, M. A. Schmidt, A. S. Tverjanovich, L. Wondraczek, and P. St. J. Russell, “Bandgap guidance in hybrid chalcogenide–silica photonic crystal fibers,” Opt. Lett. 36, 2432–2434 (2011).
[Crossref] [PubMed]

M. A. Schmidt, N. Granzow, N. Da, M. Peng, L. Wondraczek, and P. St. J. Russell, “All-solid bandgap guiding in tellurite-filled silica photonic crystal fibers,” Opt. Lett. 34, 1946–1948 (2009).
[Crossref] [PubMed]

G. Humbert, J. C. Knight, G. Bouwmans, P. St. J. Russell, D. P. Williams, P. J. Roberts, and B. J. Mangan, “Hollow core photonic crystal fibers for beam delivery,” Opt. Express 12, 1477–1484 (2004).
[Crossref] [PubMed]

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

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

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photonics Technol. Lett. 12, 807–809 (2000).
[Crossref]

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[Crossref] [PubMed]

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996).
[Crossref] [PubMed]

Sammut, R.

Sauvan, C.

C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013).
[Crossref] [PubMed]

Schäferling, M.

T. Weiss, M. Schäferling, H. Giessen, N. A. Gippius, S. G. Tikhodeev, W. Langbein, and E. A. Muljarov, “Analytical normalization of resonant states in photonic crystal slabs and periodic arrays of nanoantennas at oblique incidence,” Phys. Rev. B 96, 045129 (2017).
[Crossref]

T. Weiss, M. Mesch, M. Schäferling, H. Giessen, W. Langbein, and E. A. Muljarov, “From dark to bright: first-order perturbation theory with analytical mode normalization for plasmonic nanoantenna arrays applied to refractive index sensing,” Phys. Rev. Lett. 116, 237401 (2016).
[Crossref] [PubMed]

Schmeltzer, R. A.

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

Schmidt, M. A.

Schönhardt, A.

D. Nau, A. Schönhardt, C. Bauer, A. Christ, T. Zentgraf, J. Kuhl, and H. Giessen, “Disorder issues in metallic photonic crystals,” Phys. Status Solidi B 243, 2331–2343 (2006).
[Crossref]

Selleri, S.

F. Poli, A. Cucinotta, and S. Selleri, Photonic Crystal Fibers: Properties and Applications(Springer Science & Business Media, 2007).

Simonsen, H. R.

Snyder, A. W.

Sørensen, T.

Stefani, A.

Tai, C.-T.

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994).

Tikhodeev, S. G.

T. Weiss, M. Schäferling, H. Giessen, N. A. Gippius, S. G. Tikhodeev, W. Langbein, and E. A. Muljarov, “Analytical normalization of resonant states in photonic crystal slabs and periodic arrays of nanoantennas at oblique incidence,” Phys. Rev. B 96, 045129 (2017).
[Crossref]

Tuominen, J.

Tverjanovich, A. S.

Uebel, P.

Wadsworth, W. J.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photonics Technol. Lett. 12, 807–809 (2000).
[Crossref]

Wang, Y.

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, 1972).

Weiss, T.

E. A. Muljarov and T. Weiss, “Resonant-state expansion for open optical systems: generalization to magnetic, chiral, and bi-anisotropic materials,” Opt. Lett. 43, 1978–1981 (2018).
[Crossref] [PubMed]

T. Weiss, M. Schäferling, H. Giessen, N. A. Gippius, S. G. Tikhodeev, W. Langbein, and E. A. Muljarov, “Analytical normalization of resonant states in photonic crystal slabs and periodic arrays of nanoantennas at oblique incidence,” Phys. Rev. B 96, 045129 (2017).
[Crossref]

T. Weiss, M. Mesch, M. Schäferling, H. Giessen, W. Langbein, and E. A. Muljarov, “From dark to bright: first-order perturbation theory with analytical mode normalization for plasmonic nanoantenna arrays applied to refractive index sensing,” Phys. Rev. Lett. 116, 237401 (2016).
[Crossref] [PubMed]

M. H. Frosz, J. Nold, T. Weiss, A. Stefani, F. Babic, S. Rammler, and P. St. J. Russell, “Five-ring hollow-core photonic crystal fiber with 1.8 dB/km loss,” Opt. Lett. 38, 2215–2217 (2013).
[Crossref] [PubMed]

White, T. P.

Williams, D. P.

Wondraczek, L.

Young, K.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187 (1990).
[Crossref] [PubMed]

Zeisberger, M.

Zentgraf, T.

D. Nau, A. Schönhardt, C. Bauer, A. Christ, T. Zentgraf, J. Kuhl, and H. Giessen, “Disorder issues in metallic photonic crystals,” Phys. Status Solidi B 243, 2331–2343 (2006).
[Crossref]

Zhang, X.

Zheng, L.

Zimmermann, R.

E. A. Muljarov, W. Langbein, and R. Zimmermann, “Brillouin-Wigner perturbation theory in open electromagnetic systems,” Europhys. Lett. 92, 50010 (2010).
[Crossref]

Zoriniants, G.

S. V. Lobanov, G. Zoriniants, W. Langbein, and E. A. Muljarov, “Resonant-state expansion of light propagation in nonuniform waveguides,” Phys. Rev. A 95, 053848 (2017).
[Crossref]

ACS Photonics (1)

P. T. Kristensen and S. Hughes, “Modes and mode volumes of leaky optical cavities and plasmonic nanoresonators,” ACS Photonics 1, 2–10 (2013).
[Crossref]

Appl. Opt. (3)

Bell Labs Tech. J. (1)

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

Europhys. Lett. (1)

E. A. Muljarov, W. Langbein, and R. Zimmermann, “Brillouin-Wigner perturbation theory in open electromagnetic systems,” Europhys. Lett. 92, 50010 (2010).
[Crossref]

IEEE J. Quantum Electron. (1)

S.-L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31, 1790–1802 (1995).
[Crossref]

IEEE Photonics Technol. Lett. (1)

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photonics Technol. Lett. 12, 807–809 (2000).
[Crossref]

J. Opt. Soc. Am. B (3)

Opt. Express (3)

Opt. Lett. (6)

Phys. Rev. A (6)

M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant-state expansion applied to planar open optical systems,” Phys. Rev. A 85, 023835 (2012).
[Crossref]

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187 (1990).
[Crossref] [PubMed]

L. J. Armitage, M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant-state expansion applied to planar waveguides,” Phys. Rev. A 89, 053832 (2014).
[Crossref]

S. V. Lobanov, G. Zoriniants, W. Langbein, and E. A. Muljarov, “Resonant-state expansion of light propagation in nonuniform waveguides,” Phys. Rev. A 95, 053848 (2017).
[Crossref]

M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant state expansion applied to two-dimensional open optical systems,” Phys. Rev. A 87, 043827 (2013).
[Crossref]

M. B. Doost, W. Langbein, and E. A. Muljarov, “Resonant-state expansion applied to three-dimensional open optical systems,” Phys. Rev. A 90, 013834 (2014).
[Crossref]

Phys. Rev. B (3)

E. A. Muljarov and W. Langbein, “Exact mode volume and Purcell factor of open optical systems,” Phys. Rev. B 94, 235438 (2016).
[Crossref]

E. A. Muljarov and W. Langbein, “Resonant-state expansion of dispersive open optical systems: Creating gold from sand,” Phys. Rev. B 93, 075417 (2016).
[Crossref]

T. Weiss, M. Schäferling, H. Giessen, N. A. Gippius, S. G. Tikhodeev, W. Langbein, and E. A. Muljarov, “Analytical normalization of resonant states in photonic crystal slabs and periodic arrays of nanoantennas at oblique incidence,” Phys. Rev. B 96, 045129 (2017).
[Crossref]

Phys. Rev. Lett. (2)

T. Weiss, M. Mesch, M. Schäferling, H. Giessen, W. Langbein, and E. A. Muljarov, “From dark to bright: first-order perturbation theory with analytical mode normalization for plasmonic nanoantenna arrays applied to refractive index sensing,” Phys. Rev. Lett. 116, 237401 (2016).
[Crossref] [PubMed]

C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013).
[Crossref] [PubMed]

Phys. Status Solidi B (1)

D. Nau, A. Schönhardt, C. Bauer, A. Christ, T. Zentgraf, J. Kuhl, and H. Giessen, “Disorder issues in metallic photonic crystals,” Phys. Status Solidi B 243, 2331–2343 (2006).
[Crossref]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

Science (2)

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

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

Other (7)

F. Poli, A. Cucinotta, and S. Selleri, Photonic Crystal Fibers: Properties and Applications(Springer Science & Business Media, 2007).

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

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

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, 1972).

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994).

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

B. T. Kuhlmey, “Computer code CUDOS MOF Utilities,” available at http://www.physics.usyd.edu.au/cudos/mofsoftware/index.html .

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Figures (5)

Fig. 1
Fig. 1 (a) Axial component of the time-averaged Poynting vector of the fundamental core mode of a step index fiber with refractive indices of 1 and 1.44 in the core and cladding region, respectively, and a core radius of 1 µm (core region indicated by the green solid line) at a wavelength of 1 µm. (b) Axial component of the time-averaged Poynting vector for a higher-order core mode of a silica-air photonic crystal fiber with four rings of air holes of radius 0.25 µm and pitch 2.3 µm around a single-defect core. The refractive index of silica is taken as 1.44. The considered wavelength is 1 µm. Both modes in (a) and (b) exhibit fields that grow in the exterior with distance from the core. The bottom panels depict the real (c) and imaginary (d) parts of the surface term (blue solid line) and line term (red dotted line) of the normalization Eq. (8) as a function of the radius of normalization. Evidently, the divergence of the fields is manifested in the surface and line terms, while it is countervailed in their sum as the normalization constant.
Fig. 2
Fig. 2 Effective refractive indices of modes in a capillary fiber with a homogeneous perturbation in the core region of (a) Δn = 0.07 and (b) Δn = 0.17. The results from the resonant state expansion (red crosses) are compared with the exact analytical solution (blue circles) for the perturbed system at a wavelength of 1 µm. The unperturbed system has a core index of 1, cladding index of 1.44, and a radius of 8 µm, with its effective refractive indices denoted by black squares. The number of modes used is 154. The black arrow indicates the fundamental core mode.
Fig. 3
Fig. 3 (a) Spatial distribution of the time-averaged Poynting vector of a higher-order core mode supported by a capillary fiber with parameters as used in Fig. 2. The fiber core is indicated by the green solid line. The effective index of the unperturbed mode is 0.03139 + 1.0103i. (b) Relative error of the effective index of the higher-order mode with respect to the number of modes used in Eq. (16). Two refractive index differences have been considered as perturbations (dashed blue line: Δn = 0.07, solid red line: Δn = 0.17).
Fig. 4
Fig. 4 Axial component of the time-averaged Poynting vector of the fundamental core mode of a silica-air photonic crystal fiber with diameter disorder for disorder parameter (a) Δ = 0 µm and (b) Δ = 0.1 µm. The disorder parameter provides the range of radii in the disordered fiber as r0 ± Δ, with r0 being the radius of the air holes in the ordered fiber. The geometrical parameters of the fiber are the same as in Fig. 1(b). Panels (c) and (d) show the comparison of the real and imaginary parts of the effective indices from the resonant state expansion (red crosses) with the exact numerical solution of the perturbed system (black circles) for 20 realizations of disorder at a wavelength of 1.55 µm. The number of modes used for the resonant state expansion is 190. The blue dotted line indicates the effective index for an unperturbed cladding.
Fig. 5
Fig. 5 Real (a) and imaginary (b) part of the effective index of the fundamental core mode as a function of the disorder parameter Δ averaged over 200 realizations of diameter disorder at a wavelength of 1.55 µm. The averaged real part grows almost linearly with increasing Δ, while the imaginary part is growing quadratically. The standard deviation is indicated by the errorbars. The blue dotted line indicates the effective index of the unperturbed cladding.

Equations (29)

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( k 0 ε × × k 0 μ ) M 0 ( E i H ) F = ( J E i J H ) J ,
f ^ ( r ; β ) = 1 2 π d z f ( r ; z ) e i β z ,
( k 0 ε ^ β × ^ β × k 0 μ ) ( E ^ i H ^ ) ( J ^ E i J ^ H ) , with ^ β ( x y i β ) .
M ^ 0 ( r ; β ) G ^ ( r , r ; β ) = δ ( r r ) ,
F ^ ( r ) = d r G ^ ( r , r ; β ) = J ^ ( r ) .
M ^ 0 ( r ; β n ) F ^ n = 0 .
G ^ ( r , r ; β ) = n F ^ n ( r ) F ^ n R ( r ) 2 N n ( β β n ) + Δ G ^ cuts ,
N n = S n + L n ,
S n = 0 R ρ d ρ 0 2 π d ϕ ( E ^ n , ρ H ^ n , ϕ E ^ n , ϕ H ^ n , ρ ) ,
L n = ε μ k 0 2 + β n 2 2 ϰ n 4 0 2 π d ϕ ( E ^ n , z H ^ n , z ϕ H ^ n , z E ^ n , z ϕ ) R + k 0 β n R 2 2 ϰ n 4 0 2 π d ϕ { μ [ ( H ^ n , z ρ ) 2 ρ H ^ n , z ρ ( 1 ρ H ^ n , z ρ ) ] + ε [ ( E ^ n , z ρ ) 2 ρ E ^ n , z ( 1 ρ E ^ n , z ρ ) ] } R ,
ϰ n 2 = ε μ k 0 2 β n 2 .
Δ M ^ ( r ) = ( k 0 Δ ε ( r ) 0 0 k 0 Δ μ ( r ) ) .
M ^ 0 ( r ; β ν ) F ^ ν ( r ) = Δ M ^ ( r ) F ^ ν ( r ) ,
F ^ ν ( r ) = d r G ^ ( r , r ; β ν ) Δ M ^ ( r ) F ^ ν ( r ) .
F ^ ν ( r ) = n b n ( ν ) F ^ n ( r ) ,
β ν b n ( ν ) = β n b n ( ν ) + 1 2 n V n n b n ( ν ) ,
V n n = d r F ^ n R ( r ) Δ M ( r ) F ^ n ( r ) .
f ( r ) = { 1 2 Δ for r 0 Δ r r 0 + Δ 0 for r < r 0 Δ or r > r 0 + Δ
M ^ 0 ( r ; β ) F ^ = ( β β n ) σ n ( r ) .
F ^ n ( r ) = lim β β n n 1 2 N n β β n β β n F ^ n ( r ) d r F ^ n R ( r ) σ n ( r ) .
d r F ^ n R ( r ) σ n ( r ) = 2 N n .
0 = F ^ ( r ; β ) M ^ 0 ( r ; β n ) F ^ n R ( r ) ,
F ^ n R ( r ) M ^ 0 ( r ; β n ) F ^ ( r ; β ) F ^ ( r ; β ) M ^ 0 ( r ; β n ) F ^ n R ( r ) = ( β β n ) F ^ n R ( r ) σ n ( r ) .
2 N n = lim β β n d r i β β n [ E ^ ( r ; β ) × H ^ n R ( r ) E ^ n R ( r ) × H ^ ( r ; β ) ] + d r [ E ^ n ( r ) × H ^ n R ( r ) E ^ n R ( r ) × H ^ ( r ) ] z .
F ^ ( r ; β ) = F ^ n ( r ) + ( β β n ) F ^ ( r ; β ) β | β n + ( β β n ) 2 2 2 F ^ ( r ; β ) β 2 | β n + ,
N n = β n R 2 i ϰ n 0 2 π d ϕ ( E ^ n , ϕ ϰ H ^ n , z + E ^ n , z ϰ H ^ n , ϕ H ^ n , ϕ ϰ E ^ n , z H ^ n , z ϰ E ^ n , ϕ ) + d r ( E ^ n , ρ H ^ n , ϕ E ^ n , ϕ H ^ n , ρ ) .
E ^ z ϰ = ρ ϰ E ^ z ρ , H ^ z ϰ = ρ ϰ H ^ z ρ .
E ^ ϕ = i β ϰ 2 ρ E ^ z ϕ i k 0 μ ϰ 2 H ^ z ρ , H ^ ϕ = i β ϰ 2 ρ H ^ z ϕ + i k 0 ε ϰ 2 E ^ z ρ ,
0 2 π d ϕ f ϕ g = 0 2 π d ϕ f g ϕ ,

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