Abstract

The spectral properties of light confined to low-index media by binary layered structures is discussed. A novel phase-based model with a simple analytical form is derived for the approximation of the center of arbitrary bandgaps of binary layered structures operating at arbitrary effective indices. An analytical approximation to the sensitivity of the bandgap center to changes in the core refractive index is thus derived. Experimentally, significant shifting of the fundamental bandgap of a hollow-core Bragg fiber with a large cladding layer refractive index contrast is demonstrated by filling the core with liquids of various refractive indices. Confirmation of these results against theory is shown, including the new analytical model, highlighting the importance of considering material dispersion. The work demonstrates the broad and sensitive tunability of Bragg structures and includes discussions on refractive index sensing.

© 2011 OSA

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  6. D. Yin, H. Schmidt, J. P. Barber, E. J. Lunt, and A. R. Hawkins, “Optical characterization of arch-shaped ARROW waveguides with liquid cores,” Opt. Express 13, 10564–10570 (2005).
    [CrossRef] [PubMed]
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  27. H. Qu and M. Skorobogatiy, “Liquid-core low-refractive-index-contrast Bragg fiber sensor,” Appl. Phys. Lett. 98, 201114 (2011).
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  28. D. Yin, H. Schmidt, J. Barber, and A. Hawkins, “Integrated ARROW waveguides with hollow cores,” Opt. Express 12, 2710–2715 (2004).
    [CrossRef] [PubMed]
  29. K. J. Rowland, S. Afshar, and T. M. Monro, “Novel low-loss bandgaps in all-silica Bragg fibers,” J. Light-wave Technol. 26, 43–51 (2008).
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2010 (2)

2009 (4)

H. T. Bookey, S. Dasgupta, N. Bezawada, B. P. Pal, A. Sysoliatin, J. E. McCarthy, M. Salganskii, V. Khopin, and A. K. Kar, “Experimental demonstration of spectral broadening in an all-silica Bragg fiber,” Opt. Express 17, 17130–17135 (2009).
[CrossRef] [PubMed]

P. Measor, S. Kühn, E. J. Lunt, B. S. Phillips, A. R. Hawkins, and H. Schmidt, “Multi-mode mitigation in an optofluidic chip for particle manipulation and sensing,” Opt. Express 17, 24342–24348 (2009).
[CrossRef]

M. Skorobogatiy, “Microstructured and Photonic Bandgap Fibers for Applications in the Resonant Bio- and Chemical Sensors,” J. Sensors 2009, 1–20 (2009).
[CrossRef]

F. Benabid, P. J. Roberts, F. Couny, and P. S. Light, “Light and gas confinement in hollow-core photonic crystal fibre based photonic microcells,” J. Eur. Opt. Soc. 4, 1–9 (2009).
[CrossRef]

2008 (5)

S. Kühn, P. Measor, E. J. Lunt, A. R. Hawkins, and H. Schmidt, “Particle manipulation with integrated optofluidic traps,” Digest of the IEEE/LEOS Summer Topical Meetings, pp. 187–188 (2008).
[CrossRef]

H. Schmidt and A. Hawkins, “Optofluidic waveguides: I. Concepts and implementations,” Microfluid. Nanofluid. 4, 3–16 (2008).
[CrossRef] [PubMed]

A. Hawkins and H. Schmidt, “Optofluidic waveguides: II. Fabrication and structures,” Microfluid. Nanofluid. 4, 17–32 (2008).
[CrossRef]

K. J. Rowland, S. Afshar, and T. M. Monro, “Novel low-loss bandgaps in all-silica Bragg fibers,” J. Light-wave Technol. 26, 43–51 (2008).
[CrossRef]

K. J. Rowland, S. Afshar, and T. M. Monro, “Bandgaps and antiresonances in integrated-ARROWs and Bragg fibers; a simple model,” Opt. Express 16, 17935–17951 (2008).
[CrossRef] [PubMed]

2006 (1)

2005 (1)

2004 (3)

2002 (4)

R. Bernini, S. Campopiano, and L. Zeni, “Design and analysis of an integrated antiresonant reflecting optical waveguide refractive-index sensor,” Appl. Opt. 41, 70–73 (2002).
[CrossRef] [PubMed]

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]

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[CrossRef] [PubMed]

D. Zhou and L. Mawst, “High-power single-mode antiresonant reflecting optical waveguide-type vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 38, 1599–1606 (2002).
[CrossRef]

1993 (1)

J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Light-wave Technol. 11, 416–423 (1993).
[CrossRef]

1986 (1)

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

1978 (1)

1977 (1)

1976 (1)

P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Abeeluck, A. K.

Abouraddy, A. F.

Afshar, S.

K. J. Rowland, S. Afshar, and T. M. Monro, “Bandgaps and antiresonances in integrated-ARROWs and Bragg fibers; a simple model,” Opt. Express 16, 17935–17951 (2008).
[CrossRef] [PubMed]

K. J. Rowland, S. Afshar, and T. M. Monro, “Novel low-loss bandgaps in all-silica Bragg fibers,” J. Light-wave Technol. 26, 43–51 (2008).
[CrossRef]

K. J. Rowland, S. Afshar, A. Stolyarov, Y. Fink, and T. M. Monro, “Spectral properties of liquid-core Bragg fibers”, Conference on Lasers and Electro-Optics (CLEO), Baltimore, Maryland, US, June 2–4 2009.

Archambault, J. L.

J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Light-wave Technol. 11, 416–423 (1993).
[CrossRef]

Barber, J.

Barber, J. P.

Bayindir, M.

Benabid, F.

F. Benabid, P. J. Roberts, F. Couny, and P. S. Light, “Light and gas confinement in hollow-core photonic crystal fibre based photonic microcells,” J. Eur. Opt. Soc. 4, 1–9 (2009).
[CrossRef]

Benoit, G.

Bernini, R.

Bezawada, N.

Black, R. J.

J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Light-wave Technol. 11, 416–423 (1993).
[CrossRef]

Bookey, H. T.

Brewster, M. M.

Bures, J.

J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Light-wave Technol. 11, 416–423 (1993).
[CrossRef]

Campopiano, S.

Couny, F.

F. Benabid, P. J. Roberts, F. Couny, and P. S. Light, “Light and gas confinement in hollow-core photonic crystal fibre based photonic microcells,” J. Eur. Opt. Soc. 4, 1–9 (2009).
[CrossRef]

Dasgupta, S.

Duguay, M. A.

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

Eggleton, B. J.

Fink, Y.

O. Shapira, K. Kuriki, N. D. Orf, A. F. Abouraddy, G. Benoit, J. F. Viens, A. Rodriguez, M. Ibanescu, J. D. Joannopoulos, Y. Fink, and M. M. Brewster, “Surface-emitting fiber lasers,” Opt. Express 14, 3929–3935 (2006).
[CrossRef] [PubMed]

K. Kuriki, O. Shapira, S. Hart, G. Benoit, Y. Kuriki, J. Viens, M. Bayindir, J. Joannopoulos, and Y. Fink, “Hollow multilayer photonic bandgap fibers for NIR applications,” Opt. Express 12, 1510–1517 (2004).
[CrossRef] [PubMed]

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[CrossRef] [PubMed]

K. J. Rowland, S. Afshar, A. Stolyarov, Y. Fink, and T. M. Monro, “Spectral properties of liquid-core Bragg fibers”, Conference on Lasers and Electro-Optics (CLEO), Baltimore, Maryland, US, June 2–4 2009.

Hart, S.

Hart, S. D.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[CrossRef] [PubMed]

Hawkins, A.

H. Schmidt and A. Hawkins, “Optofluidic waveguides: I. Concepts and implementations,” Microfluid. Nanofluid. 4, 3–16 (2008).
[CrossRef] [PubMed]

A. Hawkins and H. Schmidt, “Optofluidic waveguides: II. Fabrication and structures,” Microfluid. Nanofluid. 4, 17–32 (2008).
[CrossRef]

D. Yin, H. Schmidt, J. Barber, and A. Hawkins, “Integrated ARROW waveguides with hollow cores,” Opt. Express 12, 2710–2715 (2004).
[CrossRef] [PubMed]

Hawkins, A. R.

Headley, C.

Hong, C. S.

Hsueh, W. J.

Huang, Y.

Ibanescu, M.

Joannopoulos, J.

Joannopoulos, J. D.

O. Shapira, K. Kuriki, N. D. Orf, A. F. Abouraddy, G. Benoit, J. F. Viens, A. Rodriguez, M. Ibanescu, J. D. Joannopoulos, Y. Fink, and M. M. Brewster, “Surface-emitting fiber lasers,” Opt. Express 14, 3929–3935 (2006).
[CrossRef] [PubMed]

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[CrossRef] [PubMed]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

Kar, A. K.

Khopin, V.

Koch, T. L.

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

Kokubun, Y.

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

Kühn, S.

P. Measor, S. Kühn, E. J. Lunt, B. S. Phillips, A. R. Hawkins, and H. Schmidt, “Multi-mode mitigation in an optofluidic chip for particle manipulation and sensing,” Opt. Express 17, 24342–24348 (2009).
[CrossRef]

S. Kühn, P. Measor, E. J. Lunt, A. R. Hawkins, and H. Schmidt, “Particle manipulation with integrated optofluidic traps,” Digest of the IEEE/LEOS Summer Topical Meetings, pp. 187–188 (2008).
[CrossRef]

Kuriki, K.

Kuriki, Y.

Lacroix, S.

J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Light-wave Technol. 11, 416–423 (1993).
[CrossRef]

Light, P. S.

F. Benabid, P. J. Roberts, F. Couny, and P. S. Light, “Light and gas confinement in hollow-core photonic crystal fibre based photonic microcells,” J. Eur. Opt. Soc. 4, 1–9 (2009).
[CrossRef]

Litchinitser, N. M.

Lunt, E. J.

Marom, E.

Mawst, L.

D. Zhou and L. Mawst, “High-power single-mode antiresonant reflecting optical waveguide-type vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 38, 1599–1606 (2002).
[CrossRef]

McCarthy, J. E.

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

Measor, P.

P. Measor, S. Kühn, E. J. Lunt, B. S. Phillips, A. R. Hawkins, and H. Schmidt, “Multi-mode mitigation in an optofluidic chip for particle manipulation and sensing,” Opt. Express 17, 24342–24348 (2009).
[CrossRef]

S. Kühn, P. Measor, E. J. Lunt, A. R. Hawkins, and H. Schmidt, “Particle manipulation with integrated optofluidic traps,” Digest of the IEEE/LEOS Summer Topical Meetings, pp. 187–188 (2008).
[CrossRef]

Monro, T. M.

K. J. Rowland, S. Afshar, and T. M. Monro, “Bandgaps and antiresonances in integrated-ARROWs and Bragg fibers; a simple model,” Opt. Express 16, 17935–17951 (2008).
[CrossRef] [PubMed]

K. J. Rowland, S. Afshar, and T. M. Monro, “Novel low-loss bandgaps in all-silica Bragg fibers,” J. Light-wave Technol. 26, 43–51 (2008).
[CrossRef]

K. J. Rowland, S. Afshar, A. Stolyarov, Y. Fink, and T. M. Monro, “Spectral properties of liquid-core Bragg fibers”, Conference on Lasers and Electro-Optics (CLEO), Baltimore, Maryland, US, June 2–4 2009.

Orf, N. D.

Pal, B. P.

Pfeiffer, L.

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

Phillips, B. S.

Qu, H.

H. Qu and M. Skorobogatiy, “Liquid-core low-refractive-index-contrast Bragg fiber sensor,” Appl. Phys. Lett. 98, 201114 (2011).
[CrossRef]

Roberts, P. J.

F. Benabid, P. J. Roberts, F. Couny, and P. S. Light, “Light and gas confinement in hollow-core photonic crystal fibre based photonic microcells,” J. Eur. Opt. Soc. 4, 1–9 (2009).
[CrossRef]

Rodriguez, A.

Rowland, K. J.

K. J. Rowland, S. Afshar, and T. M. Monro, “Novel low-loss bandgaps in all-silica Bragg fibers,” J. Light-wave Technol. 26, 43–51 (2008).
[CrossRef]

K. J. Rowland, S. Afshar, and T. M. Monro, “Bandgaps and antiresonances in integrated-ARROWs and Bragg fibers; a simple model,” Opt. Express 16, 17935–17951 (2008).
[CrossRef] [PubMed]

K. J. Rowland, S. Afshar, A. Stolyarov, Y. Fink, and T. M. Monro, “Spectral properties of liquid-core Bragg fibers”, Conference on Lasers and Electro-Optics (CLEO), Baltimore, Maryland, US, June 2–4 2009.

Salganskii, M.

Sarro, P. M.

Scheuer, J.

J. Scheuer and X. Sun, “Radial Bragg resonators,” in Photonic Microresonator Research and Applications, I. Chremmos, O. Schwelb, and N. Uzunoglu, eds. (Springer Series in Optical Sciences, 2010), Chap. 15.
[CrossRef]

Schmidt, H.

P. Measor, S. Kühn, E. J. Lunt, B. S. Phillips, A. R. Hawkins, and H. Schmidt, “Multi-mode mitigation in an optofluidic chip for particle manipulation and sensing,” Opt. Express 17, 24342–24348 (2009).
[CrossRef]

S. Kühn, P. Measor, E. J. Lunt, A. R. Hawkins, and H. Schmidt, “Particle manipulation with integrated optofluidic traps,” Digest of the IEEE/LEOS Summer Topical Meetings, pp. 187–188 (2008).
[CrossRef]

A. Hawkins and H. Schmidt, “Optofluidic waveguides: II. Fabrication and structures,” Microfluid. Nanofluid. 4, 17–32 (2008).
[CrossRef]

H. Schmidt and A. Hawkins, “Optofluidic waveguides: I. Concepts and implementations,” Microfluid. Nanofluid. 4, 3–16 (2008).
[CrossRef] [PubMed]

D. Yin, H. Schmidt, J. P. Barber, E. J. Lunt, and A. R. Hawkins, “Optical characterization of arch-shaped ARROW waveguides with liquid cores,” Opt. Express 13, 10564–10570 (2005).
[CrossRef] [PubMed]

D. Yin, H. Schmidt, J. Barber, and A. Hawkins, “Integrated ARROW waveguides with hollow cores,” Opt. Express 12, 2710–2715 (2004).
[CrossRef] [PubMed]

Shapira, O.

Skorobogatiy, M.

H. Qu and M. Skorobogatiy, “Liquid-core low-refractive-index-contrast Bragg fiber sensor,” Appl. Phys. Lett. 98, 201114 (2011).
[CrossRef]

M. Skorobogatiy, “Microstructured and Photonic Bandgap Fibers for Applications in the Resonant Bio- and Chemical Sensors,” J. Sensors 2009, 1–20 (2009).
[CrossRef]

Stolyarov, A.

K. J. Rowland, S. Afshar, A. Stolyarov, Y. Fink, and T. M. Monro, “Spectral properties of liquid-core Bragg fibers”, Conference on Lasers and Electro-Optics (CLEO), Baltimore, Maryland, US, June 2–4 2009.

Sun, X.

J. Scheuer and X. Sun, “Radial Bragg resonators,” in Photonic Microresonator Research and Applications, I. Chremmos, O. Schwelb, and N. Uzunoglu, eds. (Springer Series in Optical Sciences, 2010), Chap. 15.
[CrossRef]

Sysoliatin, A.

Temelkuran, B.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[CrossRef] [PubMed]

Testa, G.

Viens, J.

Viens, J. F.

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

Wun, S. J.

Yariv, A.

Yeh, P.

Yin, D.

Yu, T. H.

Zeni, L.

Zhou, D.

D. Zhou and L. Mawst, “High-power single-mode antiresonant reflecting optical waveguide-type vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 38, 1599–1606 (2002).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

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

H. Qu and M. Skorobogatiy, “Liquid-core low-refractive-index-contrast Bragg fiber sensor,” Appl. Phys. Lett. 98, 201114 (2011).
[CrossRef]

Digest of the IEEE/LEOS Summer Topical Meetings (1)

S. Kühn, P. Measor, E. J. Lunt, A. R. Hawkins, and H. Schmidt, “Particle manipulation with integrated optofluidic traps,” Digest of the IEEE/LEOS Summer Topical Meetings, pp. 187–188 (2008).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Zhou and L. Mawst, “High-power single-mode antiresonant reflecting optical waveguide-type vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 38, 1599–1606 (2002).
[CrossRef]

J. Eur. Opt. Soc. (1)

F. Benabid, P. J. Roberts, F. Couny, and P. S. Light, “Light and gas confinement in hollow-core photonic crystal fibre based photonic microcells,” J. Eur. Opt. Soc. 4, 1–9 (2009).
[CrossRef]

J. Light-wave Technol. (2)

J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Light-wave Technol. 11, 416–423 (1993).
[CrossRef]

K. J. Rowland, S. Afshar, and T. M. Monro, “Novel low-loss bandgaps in all-silica Bragg fibers,” J. Light-wave Technol. 26, 43–51 (2008).
[CrossRef]

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Nature (1)

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Figures (6)

Fig. 1
Fig. 1

A schematic of a Bragg fiber with a filled core such that ncoren0 < n1. The diagram to the right can also represent an arbitrary planar low-index core Bragg waveguide.

Fig. 2
Fig. 2

Material dispersion data for the materials constituting the Bragg fiber layers: As2S3 (top) and PEI (bottom).

Fig. 3
Fig. 3

A schematic of the Bragg fiber filling and spectral measurement configuration. The light beam exiting the fiber is (arbitrarily) colored to represent the spectral filtering effect upon the white light due to the cladding structure. Each cleaved end of the Bragg fiber is hermetically sealed within its own liquid-filled windowed cell, as shown in the zoom-in region in the bottom right of the figure.

Fig. 4
Fig. 4

Left: measured transmission spectra for the filled and unfilled Bragg fiber (as per the schematic of Fig. 3). The peaks are labelled and color coded to indicate an empty fiber (ncore ≈ 1) and the fiber filled with liquids of refractive indices nliquid ≈ 1.4018, 1.4720 and 1.5780. Right: spectral positions of the peaks’ maxima vs. the core refractive index. The color matched horizontal rectangular regions correspond to the TM bandgap for the given core index (cf. Fig. 5, bottom). Horizontal lines, right: the position of Pμ (an approximation of the bandgap central frequency, § 4) at the given ncore.

Fig. 5
Fig. 5

Top: the fundamental bandgap 〈m1, m0〉 = 〈1,0〉 of the used Bragg fiber, neglecting material dispersion of the layers (with material indices assumed to be those at λ = 700 nm). Bottom: the same spectrum with layer material dispersion (Fig. 2). Solid black curve: center of the TE bandgap ( λ c Bloch, mean of the edge points in k for a given ñ). Dashed black curve: center of the TM gap. Green curve: parametric central gap point Pμ ; note the very close overlap with the TE and, especially, TM gap centers. Horizontal lines: ñ = ncore (colors as per Fig. 4). Dashed lines: ñ = nB (white, Brewster) and n0 (cyan, low layer index).

Fig. 6
Fig. 6

Sensitivity of the exact and approximate center of the fundamental bandgap to changes in ñ for the layer properties described in § 2. Top: without material dispersion [ni = ni(λ = 700 nm)]. Bottom: with material dispersion [ni = ni(λ)]. Black: numerically calculated sensitivity of the Bloch bandgap center [from k c Bloch, Eq. (2)]; solid: TE; dashed: TM. Green: the wavelength sensitivity of the Pμ point in to changes in ñ; solid: analytic approximation to derivative of Pμ neglecting material dispersion derivatives [Eq. (11) –setting ∂ni/∂λ = 0 but allowing ni = ni(λ)]; dashed: numerical derivative of Pμ including material dispersion [∂ni/∂λ ≠ 0].

Tables (1)

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Table 1 Summary of the Filled Bragg Fiber Transmission Peaks

Equations (11)

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cos ϕ 1 cos ϕ 0 Λ sin ϕ 1 sin ϕ 0 = ζ ,
k c Bloch = k + Bloch + k Bloch 2 ,
k m i = m i π t i n i n ˜ m i .
k c res . = k m i ( n ˜ ) + k m j ( n ˜ ) 2 ,
P μ = ( k μ , n ˜ μ ) = ( π ρ 1 2 ρ 0 2 n 1 2 n 0 2 , n 1 2 n 0 2 η 2 1 η 2 ) ,
n ˜ μ k μ = n ˜ μ η η μ μ k μ = n ˜ μ η η μ / k μ μ ,
n ˜ μ η = η 1 η 2 n 0 2 n ˜ μ 2 n ˜ μ ,
η μ = t 0 + t 1 η ( m 0 + μ ) t 1 .
k μ μ = π 2 ( n 1 2 n 0 2 ) k μ ( m 1 μ t 1 2 + m 0 + μ t 0 2 ) .
k μ n ˜ μ = π 2 1 η 2 ( n 1 2 n 0 2 ) η ( m 0 + μ ) t 1 t 0 + t 1 η ( m 1 μ t 1 2 + m 0 + μ t 0 2 ) n ˜ μ ( n 0 2 n ˜ μ 2 ) k μ .
λ μ n ˜ μ = 2 π k μ 2 k μ n ˜ μ = 2 π 3 1 η 2 ( n 1 2 n 0 2 ) η ( m 0 + μ ) t 1 t 0 + t 1 η ( m 1 μ t 1 2 + m 0 + μ t 0 2 ) n ˜ μ ( n 0 2 n ˜ μ 2 ) k μ 3 .

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