Abstract

Selective excitation of whispering-gallery and bottle modes in a robust hollow-bottle optical microresonator, fabricated from a silica microcapillary by a pressure-compensated, “soften-and-compress” method, is demonstrated. Characteristic resonance spectra of bottle modes were obtained by using a tapered fiber coupled at different locations along the hollow bottle. The spectral characteristics (Q-factor, excitation efficiency) are shown to have high tolerance to angular misalignment of the tapered fiber. In addition, introduction of localized losses on the outer surface of the resonator results in selective clean-up of the transmission spectrum and superior performance. A theoretical analysis of modal turning points and associated resonant wavelengths is used to explain the mechanism of mode-suppression and the resultant spectral cleaning.

© 2011 OSA

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2011 (1)

2010 (4)

2009 (4)

2008 (2)

Ch. Strelow, H. Rehberg, C. M. Schultz, H. Welsch, Ch. Heyn, D. Heitmann, and T. Kipp, “Optical microcavities formed by semiconductor microtubes using a bottlelike geometry,” Phys. Rev. Lett. 101(12), 127403 (2008).
[CrossRef] [PubMed]

Y. Panitchob, G. S. Murugan, M. N. Zervas, P. Horak, S. Berneschi, S. Pelli, G. Nunzi Conti, and J. S. Wilkinson, “Whispering gallery mode spectra of channel waveguide coupled microspheres,” Opt. Express 16(15), 11066–11076 (2008).
[CrossRef] [PubMed]

2007 (2)

2006 (3)

I. M. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett. 31(9), 1319–1321 (2006).
[CrossRef] [PubMed]

V. S. Ilchenko and A. B. Matsko, “Optical Resonators with Whispering-Gallery Modes -Part II: Applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
[CrossRef]

A. B. Matsko and V. S. Ilchenko, “Optical Resonators with Whispering-Gallery Modes -Part I: Basics,” IEEE J. Sel. Top. Quantum Electron. 12(1), 3–14 (2006).
[CrossRef]

2005 (1)

A. A. Savchenkov,, A.B. Matsko, D. Strekalov, V.S. Ilchenko, and L. Maleki, “Mode filtering in optical whispering gallery resonators,” Electron. Lett. 41, 495 (2005).

2004 (1)

2003 (2)

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

1998 (1)

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

1996 (1)

1992 (1)

1991 (1)

1971 (1)

Andrés, M. V.

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

Bartlett, P. N.

G. Senthil Murugan, Y. Panitchob, E. J. Tull, P. N. Bartlett, D. W. Hewak, M. N. Zervas, and J. S. Wilkinson, “Position-dependent coupling between a channel waveguide and a distorted microsphere resonator,” J. Appl. Phys. 107(5), 053105 (2010).
[CrossRef]

Berneschi, S.

Byer, R. L.

Chu, S. T.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Díez, A.

Dulashko, Y.

Fan, X.

Foresi, J. S.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Gimeno, B.

Gloge, D.

Gorodetsky, M. L.

Greene, W.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Guo, Y.

Haus, H. A.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Heitmann, D.

Ch. Strelow, H. Rehberg, C. M. Schultz, H. Welsch, Ch. Heyn, D. Heitmann, and T. Kipp, “Optical microcavities formed by semiconductor microtubes using a bottlelike geometry,” Phys. Rev. Lett. 101(12), 127403 (2008).
[CrossRef] [PubMed]

Hewak, D. W.

G. Senthil Murugan, Y. Panitchob, E. J. Tull, P. N. Bartlett, D. W. Hewak, M. N. Zervas, and J. S. Wilkinson, “Position-dependent coupling between a channel waveguide and a distorted microsphere resonator,” J. Appl. Phys. 107(5), 053105 (2010).
[CrossRef]

Heyn, Ch.

Ch. Strelow, H. Rehberg, C. M. Schultz, H. Welsch, Ch. Heyn, D. Heitmann, and T. Kipp, “Optical microcavities formed by semiconductor microtubes using a bottlelike geometry,” Phys. Rev. Lett. 101(12), 127403 (2008).
[CrossRef] [PubMed]

Horak, P.

Ilchenko, V. S.

V. S. Ilchenko and A. B. Matsko, “Optical Resonators with Whispering-Gallery Modes -Part II: Applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
[CrossRef]

A. B. Matsko and V. S. Ilchenko, “Optical Resonators with Whispering-Gallery Modes -Part I: Basics,” IEEE J. Sel. Top. Quantum Electron. 12(1), 3–14 (2006).
[CrossRef]

M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21(7), 453–455 (1996).
[CrossRef] [PubMed]

Ilchenko, V.S.

A. A. Savchenkov,, A.B. Matsko, D. Strekalov, V.S. Ilchenko, and L. Maleki, “Mode filtering in optical whispering gallery resonators,” Electron. Lett. 41, 495 (2005).

Ippen, E. P.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Kimerling, L. C.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Kipp, T.

Ch. Strelow, H. Rehberg, C. M. Schultz, H. Welsch, Ch. Heyn, D. Heitmann, and T. Kipp, “Optical microcavities formed by semiconductor microtubes using a bottlelike geometry,” Phys. Rev. Lett. 101(12), 127403 (2008).
[CrossRef] [PubMed]

Kippenberg, T. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

Lam, C. C.

Leung, P. T.

Li, F.

Li, H.

Little, B. E.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Maleki, L.

A. A. Savchenkov,, A.B. Matsko, D. Strekalov, V.S. Ilchenko, and L. Maleki, “Mode filtering in optical whispering gallery resonators,” Electron. Lett. 41, 495 (2005).

Matsko, A. B.

V. S. Ilchenko and A. B. Matsko, “Optical Resonators with Whispering-Gallery Modes -Part II: Applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
[CrossRef]

A. B. Matsko and V. S. Ilchenko, “Optical Resonators with Whispering-Gallery Modes -Part I: Basics,” IEEE J. Sel. Top. Quantum Electron. 12(1), 3–14 (2006).
[CrossRef]

Matsko, A.B.

A. A. Savchenkov,, A.B. Matsko, D. Strekalov, V.S. Ilchenko, and L. Maleki, “Mode filtering in optical whispering gallery resonators,” Electron. Lett. 41, 495 (2005).

Mi, Z.

Murugan, G. S.

Nic Chormaic, S.

Nunzi Conti, G.

O’Shea, D.

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q Tunable Whispering-Gallery-Mode Microresonator,” Phys. Rev. Lett. 103(5), 053901 (2009).
[CrossRef] [PubMed]

Oveys, H.

Panitchob, Y.

G. Senthil Murugan, Y. Panitchob, E. J. Tull, P. N. Bartlett, D. W. Hewak, M. N. Zervas, and J. S. Wilkinson, “Position-dependent coupling between a channel waveguide and a distorted microsphere resonator,” J. Appl. Phys. 107(5), 053105 (2010).
[CrossRef]

Y. Panitchob, G. S. Murugan, M. N. Zervas, P. Horak, S. Berneschi, S. Pelli, G. Nunzi Conti, and J. S. Wilkinson, “Whispering gallery mode spectra of channel waveguide coupled microspheres,” Opt. Express 16(15), 11066–11076 (2008).
[CrossRef] [PubMed]

Pelli, S.

Pöllinger, M.

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q Tunable Whispering-Gallery-Mode Microresonator,” Phys. Rev. Lett. 103(5), 053901 (2009).
[CrossRef] [PubMed]

Rauschenbeutel, A.

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q Tunable Whispering-Gallery-Mode Microresonator,” Phys. Rev. Lett. 103(5), 053901 (2009).
[CrossRef] [PubMed]

Reddy, K.

Rehberg, H.

Ch. Strelow, H. Rehberg, C. M. Schultz, H. Welsch, Ch. Heyn, D. Heitmann, and T. Kipp, “Optical microcavities formed by semiconductor microtubes using a bottlelike geometry,” Phys. Rev. Lett. 101(12), 127403 (2008).
[CrossRef] [PubMed]

Savchenkov, A. A.

Savchenkov,, A. A.

A. A. Savchenkov,, A.B. Matsko, D. Strekalov, V.S. Ilchenko, and L. Maleki, “Mode filtering in optical whispering gallery resonators,” Electron. Lett. 41, 495 (2005).

Schiller, S.

Schultz, C. M.

Ch. Strelow, H. Rehberg, C. M. Schultz, H. Welsch, Ch. Heyn, D. Heitmann, and T. Kipp, “Optical microcavities formed by semiconductor microtubes using a bottlelike geometry,” Phys. Rev. Lett. 101(12), 127403 (2008).
[CrossRef] [PubMed]

Senthil Murugan, G.

G. Senthil Murugan, Y. Panitchob, E. J. Tull, P. N. Bartlett, D. W. Hewak, M. N. Zervas, and J. S. Wilkinson, “Position-dependent coupling between a channel waveguide and a distorted microsphere resonator,” J. Appl. Phys. 107(5), 053105 (2010).
[CrossRef]

G. Senthil Murugan, J. S. Wilkinson, and M. N. Zervas, “Selective excitation of whispering gallery modes in a novel bottle microresonator,” Opt. Express 17(14), 11916–11925 (2009).
[CrossRef] [PubMed]

Spillane, S. M.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

Steinmeyer, G.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Strekalov, D.

A. A. Savchenkov,, A.B. Matsko, D. Strekalov, V.S. Ilchenko, and L. Maleki, “Mode filtering in optical whispering gallery resonators,” Electron. Lett. 41, 495 (2005).

Strelow, Ch.

Ch. Strelow, H. Rehberg, C. M. Schultz, H. Welsch, Ch. Heyn, D. Heitmann, and T. Kipp, “Optical microcavities formed by semiconductor microtubes using a bottlelike geometry,” Phys. Rev. Lett. 101(12), 127403 (2008).
[CrossRef] [PubMed]

Sumetsky, M.

Sun, Y.

Thoen, E. R.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Tull, E. J.

G. Senthil Murugan, Y. Panitchob, E. J. Tull, P. N. Bartlett, D. W. Hewak, M. N. Zervas, and J. S. Wilkinson, “Position-dependent coupling between a channel waveguide and a distorted microsphere resonator,” J. Appl. Phys. 107(5), 053105 (2010).
[CrossRef]

Vahala, K. J.

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

Vicknesh, S.

Ward, J.

Warken, F.

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q Tunable Whispering-Gallery-Mode Microresonator,” Phys. Rev. Lett. 103(5), 053901 (2009).
[CrossRef] [PubMed]

Watkins, A.

Welsch, H.

Ch. Strelow, H. Rehberg, C. M. Schultz, H. Welsch, Ch. Heyn, D. Heitmann, and T. Kipp, “Optical microcavities formed by semiconductor microtubes using a bottlelike geometry,” Phys. Rev. Lett. 101(12), 127403 (2008).
[CrossRef] [PubMed]

White, I. M.

Wilkinson, J. S.

Windeler, R. S.

Wu, Y.

Young, K.

Zamora, V.

Zervas, M. N.

Appl. Opt. (1)

Electron. Lett. (1)

A. A. Savchenkov,, A.B. Matsko, D. Strekalov, V.S. Ilchenko, and L. Maleki, “Mode filtering in optical whispering gallery resonators,” Electron. Lett. 41, 495 (2005).

IEEE J. Sel. Top. Quantum Electron. (2)

V. S. Ilchenko and A. B. Matsko, “Optical Resonators with Whispering-Gallery Modes -Part II: Applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
[CrossRef]

A. B. Matsko and V. S. Ilchenko, “Optical Resonators with Whispering-Gallery Modes -Part I: Basics,” IEEE J. Sel. Top. Quantum Electron. 12(1), 3–14 (2006).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

J. Appl. Phys. (1)

G. Senthil Murugan, Y. Panitchob, E. J. Tull, P. N. Bartlett, D. W. Hewak, M. N. Zervas, and J. S. Wilkinson, “Position-dependent coupling between a channel waveguide and a distorted microsphere resonator,” J. Appl. Phys. 107(5), 053105 (2010).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature (2)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[CrossRef] [PubMed]

K. J. Vahala, “Optical microcavities,” Nature 424(6950), 839–846 (2003).
[CrossRef] [PubMed]

Opt. Express (5)

Opt. Lett. (9)

V. Zamora, A. Díez, M. V. Andrés, and B. Gimeno, “Interrogation of whispering-gallery modes resonances in cylindrical microcavities by backreflection detection,” Opt. Lett. 34(7), 1039–1041 (2009).
[CrossRef] [PubMed]

S. Schiller and R. L. Byer, “High-resolution spectroscopy of whispering gallery modes in large dielectric spheres,” Opt. Lett. 16(15), 1138–1140 (1991).
[CrossRef] [PubMed]

M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21(7), 453–455 (1996).
[CrossRef] [PubMed]

F. Li, Z. Mi, and S. Vicknesh, “Coherent emission from ultrathin-walled spiral InGaAs/GaAs quantum dot microtubes,” Opt. Lett. 34(19), 2915–2917 (2009).
[CrossRef] [PubMed]

M. Sumetsky, Y. Dulashko, and R. S. Windeler, “Optical microbubble resonator,” Opt. Lett. 35(7), 898–900 (2010).
[CrossRef] [PubMed]

G. S. Murugan, J. S. Wilkinson, and M. N. Zervas, “Optical excitation and probing of whispering gallery modes in bottle microresonators: potential for all-fiber add-drop filters,” Opt. Lett. 35(11), 1893–1895 (2010).
[CrossRef] [PubMed]

A. Watkins, J. Ward, Y. Wu, and S. Nic Chormaic, “Single-input spherical microbubble resonator,” Opt. Lett. 36(11), 2113–2115 (2011).
[CrossRef] [PubMed]

M. Sumetsky, “Whispering-gallery-bottle microcavities: the three-dimensional etalon,” Opt. Lett. 29(1), 8–10 (2004).
[CrossRef] [PubMed]

I. M. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett. 31(9), 1319–1321 (2006).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q Tunable Whispering-Gallery-Mode Microresonator,” Phys. Rev. Lett. 103(5), 053901 (2009).
[CrossRef] [PubMed]

Ch. Strelow, H. Rehberg, C. M. Schultz, H. Welsch, Ch. Heyn, D. Heitmann, and T. Kipp, “Optical microcavities formed by semiconductor microtubes using a bottlelike geometry,” Phys. Rev. Lett. 101(12), 127403 (2008).
[CrossRef] [PubMed]

Other (4)

M. N. Zervas, G. S. Murugan, M. N. Petrovich, and J. S. Wilkinson, “Hollow-Bottle Optical Microresonator”, in Conference on Lasers and Electro-Optics 2011, OSA Technical Digest Series (Optical Society of America, 2011), paper JTuI14.

X. Fan, I. W. White, H. Zhu, J. D. Suter, and H. Oveys, “Overview of novel integrated optical ring resonator bio/chemical sensors”, Proc. SPIE vol. 6452, 64520M (2007).

M. N. Zervas, G. S. Murugan, and J. S. Wilkinson, “Demonstration of novel high-Q fiber WGM “bottle” microresonators”, Proc. 10th anniversary International Conference on Transparent Optical Networks 4, 58 (2008).

G. S. Murugan, J. S. Wilkinson, and M. N. Zervas, “Experimental demonstration of a bottle microresonator,” in Conference on Lasers and Electro-Optics 2009, OSA Technical Digest Series (Optical Society of America, 2009), paper JTuD87.

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Figures (10)

Fig. 1
Fig. 1

Hollow-bottle microresonator with fitted truncated harmonic-oscillator profile (red line). Capillary diameter Dc = 218μm, bottle outer diameter Db = 265 μm, bottle length Lb≈550 μm, wall thickness Tb = 13.5 μm (for the fitted harmonic oscillator profile: Δk = 0.0027μm−1).

Fig. 2
Fig. 2

Transmission spectra of the tapered fiber-coupled hollow-bottle microresonator excited at the center and at several positions along the HMBR. The resonantor length is 550μm.

Fig. 3
Fig. 3

Transmission spectra of the fiber-coupled hollow-bottle microresonator excited at the center, with the excitation tapered fiber (a) perpendicular to the bottle axis and (b) misaligned by 3.25°.

Fig. 4
Fig. 4

(a) Normal transmission spectrum of the fiber-coupled hollow-bottle microresonator excited at the center, and (b) cleaned-up spectrum with localized loss element placed on the bottle outer surface, (c) detailed comparison of an individual resonance before and after “cleaning-up”. Cleaning-up liquid refractive index = 1.64.

Fig. 5
Fig. 5

Intensity distribution (a) along the length and (b) over transverse cross-section of the microbottle for mode (733,3,4). The resonant wavelength and turning point are 1554.61nm and 24.8µm, respectively.

Fig. 6
Fig. 6

Power distribution of: (a) an un-attenuated mode (742,2,10) with resonant wavelength 1554.13nm and turning point 36µm, and (b) a mode (738,1,58), which would experience a strong diffractive loss from a high-index liquid drop placed at the position of the red dot (resonant wavelength is 1555.65nm and turning point is 90.5µm).

Fig. 7
Fig. 7

(a) Azimuthal mode number versus turning point and (b) azimuthal mode number versus resonant wavelength.

Fig. 8
Fig. 8

(a) Azimuthal mode number versus turning point and (b) azimuthal mode number versus resonant wavelengths for longer turning point range.

Fig. 9
Fig. 9

Experimental “cleaned-up” spectrum. Superimposed are the theoretical resonant wavelengths and the corresponding azimuthal mode numbers (right axis). The corresponding radial mode numbers are p = 1(○), 2( + ) and 3(Δ).

Fig. 10
Fig. 10

Experimental normal spectrum. Superimposed are the theoretical resonant wavelengths and the corresponding azimuthal mode numbers (right axis).

Equations (5)

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Φ(r,z)={ A m J m ( U mp r/R (z) ) J m ( U mp ) rR(z) A m K m ( U mp r/R (z) ) K m ( U mp ) r>R(z)
U mp =m+ α p ( m 2 ) 1/3 +( 3 20 ) α p 2 ( m 2 ) 1 /3
Ψ(z)= C mq H q ( Δ E m 2 z )exp( Δ E m 4 z 2 )
z c =± [ 4 Δ E m ( q+ 1 2 ) ] 1 2
λ mpq =2π n 0 [ ( U mp R b ) 2 +( q+ 1 2 )Δ E m ] 1 2

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