Abstract

Optical resonators are used in a variety of applications ranging from sensors to lasers and signal routing in high volume communication networks. Achieving a high quality (Q) factor is necessary for higher sensitivity in sensing applications and for narrow linewidth light emission in most lasing applications. In this work, we propose a new approach to achieve a very high Q-factor in polymeric Fabry-Perot resonators by conquering light diffraction inside the optical cavity. This can be achieved by inducing a refractive index feature inside the optical cavity that simply creates a waveguide between the two mirrors. This approach eliminates diffraction loss from the cavity and therefore the Q-factor is only limited by mirror loss and absorption. To demonstrate this claim, a device has been fabricated consisting of two dielectric Bragg reflectors with a 100 μm layer of photosensitive polymer between them. The refractive index of this polymer can be modified utilizing standard photo-lithography processes. The measured finesse of the fabricated device was 692 and the Q-factor was 55000.

© 2014 Optical Society of America

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    [Crossref]
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    [Crossref] [PubMed]
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2013 (2)

F. Ding, T. Stöferle, L. Mai, A. Knoll, and R. F. Mahrt, “Vertical microcavities with high Q and strong lateral mode confinement,” Phys. Rev. B 87(16), 161116 (2013).
[Crossref]

M. A. Tadayon and S. Ashkenazi, “Optical Micromachined Ultrasound Transducers (OMUT)-A New Approach for High-Frequency Transducers,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60(9), 2021–2030 (2013).
[Crossref]

2012 (3)

C. S. Sheaff and S. Ashkenazi, “A polyimide-etalon thin film structure for all-optical high-frequency ultrasound transduction,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59(10), 2254–2261 (2012).
[Crossref] [PubMed]

V. Govindan and S. Ashkenazi, “Bragg waveguide ultrasound detectors,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59(10), 2304–2311 (2012).
[Crossref] [PubMed]

M.-E. Baylor, B. W. Cerjan, C. R. Pfiefer, R. W. Boyne, C. L. Couch, N. B. Cramer, C. N. Bowman, and R. R. McLeod, “Monolithic integration of optical waveguide and fluidic channel structures in a thiol-ene/methacrylate photopolymer,” Opt. Mater. Express 2(11), 1548–1555 (2012).
[Crossref]

2011 (2)

2007 (1)

C. Y. Chao, S. Ashkenazi, S. W. Huang, M. O’Donnell, and L. J. Guo, “High-frequency ultrasound sensors using polymer microring resonators,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54(5), 957–965 (2007).
[Crossref] [PubMed]

2004 (2)

S. Ashkenazi, C. Y. Chao, L. J. Guo, and M. O’Donnell, “Ultrasound detection using polymer microring optical resonator,” Appl. Phys. Lett. 85(22), 5418–5420 (2004).
[Crossref]

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express 12(8), 1551–1561 (2004).
[Crossref] [PubMed]

2003 (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]

2002 (2)

2000 (3)

C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, “The atom-cavity microscope: Single atoms bound in orbit by single photons,” Science 287(5457), 1447–1453 (2000).
[Crossref] [PubMed]

P. C. Beard, A. M. Hurrell, and T. N. Mills, “Characterization of a polymer film optical fiber hydrophone for use in the range 1 to 20 MHz: A comparison with PVDF needle and membrane hydrophones,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47(1), 256–264 (2000).
[Crossref] [PubMed]

J. D. Hamilton, T. Buma, M. Spisar, and M. O’Donnell, “High frequency optoacoustic arrays using etalon detection,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47(1), 160–169 (2000).
[Crossref] [PubMed]

1998 (3)

D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. 23(4), 247–249 (1998).
[Crossref] [PubMed]

J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998).
[Crossref]

B. E. Little, H. A. Haus, J. S. Foresi, L. C. Kimerling, E. P. Ippen, and D. J. Ripin, “Wavelength switching and routing using absorption and resonance,” IEEE Photonics Technol. Lett. 10(6), 816–818 (1998).
[Crossref]

1992 (1)

D. I. Babic and S. W. Corzine, “Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28(2), 514–524 (1992).
[Crossref]

1973 (1)

J. K. Thomson, H. K. Wickramasinghe, and E. A. Ash, “A Fabry–Pérot acoustic surface vibration detector-application to acoustic holography,” J. Phys. D Appl. Phys. 6(6), 677–687 (1973).
[Crossref]

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]

Ash, E. A.

J. K. Thomson, H. K. Wickramasinghe, and E. A. Ash, “A Fabry–Pérot acoustic surface vibration detector-application to acoustic holography,” J. Phys. D Appl. Phys. 6(6), 677–687 (1973).
[Crossref]

Ashkenazi, S.

M. A. Tadayon and S. Ashkenazi, “Optical Micromachined Ultrasound Transducers (OMUT)-A New Approach for High-Frequency Transducers,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60(9), 2021–2030 (2013).
[Crossref]

V. Govindan and S. Ashkenazi, “Bragg waveguide ultrasound detectors,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59(10), 2304–2311 (2012).
[Crossref] [PubMed]

C. S. Sheaff and S. Ashkenazi, “A polyimide-etalon thin film structure for all-optical high-frequency ultrasound transduction,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59(10), 2254–2261 (2012).
[Crossref] [PubMed]

C. Y. Chao, S. Ashkenazi, S. W. Huang, M. O’Donnell, and L. J. Guo, “High-frequency ultrasound sensors using polymer microring resonators,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54(5), 957–965 (2007).
[Crossref] [PubMed]

S. Ashkenazi, C. Y. Chao, L. J. Guo, and M. O’Donnell, “Ultrasound detection using polymer microring optical resonator,” Appl. Phys. Lett. 85(22), 5418–5420 (2004).
[Crossref]

Babic, D. I.

D. I. Babic and S. W. Corzine, “Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28(2), 514–524 (1992).
[Crossref]

Baylor, M.-E.

Beard, P. C.

P. C. Beard, A. M. Hurrell, and T. N. Mills, “Characterization of a polymer film optical fiber hydrophone for use in the range 1 to 20 MHz: A comparison with PVDF needle and membrane hydrophones,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47(1), 256–264 (2000).
[Crossref] [PubMed]

Bowman, C. N.

Boyne, R. W.

Buma, T.

J. D. Hamilton, T. Buma, M. Spisar, and M. O’Donnell, “High frequency optoacoustic arrays using etalon detection,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47(1), 160–169 (2000).
[Crossref] [PubMed]

Cerjan, B. W.

Chao, C. Y.

C. Y. Chao, S. Ashkenazi, S. W. Huang, M. O’Donnell, and L. J. Guo, “High-frequency ultrasound sensors using polymer microring resonators,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54(5), 957–965 (2007).
[Crossref] [PubMed]

S. Ashkenazi, C. Y. Chao, L. J. Guo, and M. O’Donnell, “Ultrasound detection using polymer microring optical resonator,” Appl. Phys. Lett. 85(22), 5418–5420 (2004).
[Crossref]

Chen, S. L.

Corzine, S. W.

D. I. Babic and S. W. Corzine, “Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors,” IEEE J. Quantum Electron. 28(2), 514–524 (1992).
[Crossref]

Costard, E.

J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998).
[Crossref]

Couch, C. L.

Cramer, N. B.

Dalton, L. R.

Ding, F.

F. Ding, T. Stöferle, L. Mai, A. Knoll, and R. F. Mahrt, “Vertical microcavities with high Q and strong lateral mode confinement,” Phys. Rev. B 87(16), 161116 (2013).
[Crossref]

Doherty, A. C.

C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, “The atom-cavity microscope: Single atoms bound in orbit by single photons,” Science 287(5457), 1447–1453 (2000).
[Crossref] [PubMed]

Foresi, J. S.

B. E. Little, H. A. Haus, J. S. Foresi, L. C. Kimerling, E. P. Ippen, and D. J. Ripin, “Wavelength switching and routing using absorption and resonance,” IEEE Photonics Technol. Lett. 10(6), 816–818 (1998).
[Crossref]

Gayral, B.

J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998).
[Crossref]

Gérard, J. M.

J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998).
[Crossref]

Govindan, V.

V. Govindan and S. Ashkenazi, “Bragg waveguide ultrasound detectors,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59(10), 2304–2311 (2012).
[Crossref] [PubMed]

Guo, L. J.

T. Ling, S. L. Chen, and L. J. Guo, “Fabrication and characterization of high Q polymer micro-ring resonator and its application as a sensitive ultrasonic detector,” Opt. Express 19(2), 861–869 (2011).
[Crossref] [PubMed]

C. Y. Chao, S. Ashkenazi, S. W. Huang, M. O’Donnell, and L. J. Guo, “High-frequency ultrasound sensors using polymer microring resonators,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54(5), 957–965 (2007).
[Crossref] [PubMed]

S. Ashkenazi, C. Y. Chao, L. J. Guo, and M. O’Donnell, “Ultrasound detection using polymer microring optical resonator,” Appl. Phys. Lett. 85(22), 5418–5420 (2004).
[Crossref]

Hamilton, J. D.

J. D. Hamilton, T. Buma, M. Spisar, and M. O’Donnell, “High frequency optoacoustic arrays using etalon detection,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47(1), 160–169 (2000).
[Crossref] [PubMed]

Haus, H. A.

B. E. Little, H. A. Haus, J. S. Foresi, L. C. Kimerling, E. P. Ippen, and D. J. Ripin, “Wavelength switching and routing using absorption and resonance,” IEEE Photonics Technol. Lett. 10(6), 816–818 (1998).
[Crossref]

Hood, C. J.

C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, “The atom-cavity microscope: Single atoms bound in orbit by single photons,” Science 287(5457), 1447–1453 (2000).
[Crossref] [PubMed]

Huang, S. W.

C. Y. Chao, S. Ashkenazi, S. W. Huang, M. O’Donnell, and L. J. Guo, “High-frequency ultrasound sensors using polymer microring resonators,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54(5), 957–965 (2007).
[Crossref] [PubMed]

Hurrell, A. M.

P. C. Beard, A. M. Hurrell, and T. N. Mills, “Characterization of a polymer film optical fiber hydrophone for use in the range 1 to 20 MHz: A comparison with PVDF needle and membrane hydrophones,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47(1), 256–264 (2000).
[Crossref] [PubMed]

Ilchenko, V. S.

Ippen, E. P.

B. E. Little, H. A. Haus, J. S. Foresi, L. C. Kimerling, E. P. Ippen, and D. J. Ripin, “Wavelength switching and routing using absorption and resonance,” IEEE Photonics Technol. Lett. 10(6), 816–818 (1998).
[Crossref]

Khurgin, J. B.

Kimble, H. J.

C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, “The atom-cavity microscope: Single atoms bound in orbit by single photons,” Science 287(5457), 1447–1453 (2000).
[Crossref] [PubMed]

D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. 23(4), 247–249 (1998).
[Crossref] [PubMed]

Kimerling, L. C.

B. E. Little, H. A. Haus, J. S. Foresi, L. C. Kimerling, E. P. Ippen, and D. J. Ripin, “Wavelength switching and routing using absorption and resonance,” IEEE Photonics Technol. Lett. 10(6), 816–818 (1998).
[Crossref]

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]

Knoll, A.

F. Ding, T. Stöferle, L. Mai, A. Knoll, and R. F. Mahrt, “Vertical microcavities with high Q and strong lateral mode confinement,” Phys. Rev. B 87(16), 161116 (2013).
[Crossref]

Kuramochi, E.

Legrand, B.

J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998).
[Crossref]

Ling, T.

Little, B. E.

B. E. Little, H. A. Haus, J. S. Foresi, L. C. Kimerling, E. P. Ippen, and D. J. Ripin, “Wavelength switching and routing using absorption and resonance,” IEEE Photonics Technol. Lett. 10(6), 816–818 (1998).
[Crossref]

Lynn, T. W.

C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, “The atom-cavity microscope: Single atoms bound in orbit by single photons,” Science 287(5457), 1447–1453 (2000).
[Crossref] [PubMed]

Mabuchi, H.

Mahrt, R. F.

F. Ding, T. Stöferle, L. Mai, A. Knoll, and R. F. Mahrt, “Vertical microcavities with high Q and strong lateral mode confinement,” Phys. Rev. B 87(16), 161116 (2013).
[Crossref]

Mai, L.

F. Ding, T. Stöferle, L. Mai, A. Knoll, and R. F. Mahrt, “Vertical microcavities with high Q and strong lateral mode confinement,” Phys. Rev. B 87(16), 161116 (2013).
[Crossref]

McLeod, R. R.

Mills, T. N.

P. C. Beard, A. M. Hurrell, and T. N. Mills, “Characterization of a polymer film optical fiber hydrophone for use in the range 1 to 20 MHz: A comparison with PVDF needle and membrane hydrophones,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47(1), 256–264 (2000).
[Crossref] [PubMed]

Mitsugi, S.

Notomi, M.

O’Donnell, M.

C. Y. Chao, S. Ashkenazi, S. W. Huang, M. O’Donnell, and L. J. Guo, “High-frequency ultrasound sensors using polymer microring resonators,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54(5), 957–965 (2007).
[Crossref] [PubMed]

S. Ashkenazi, C. Y. Chao, L. J. Guo, and M. O’Donnell, “Ultrasound detection using polymer microring optical resonator,” Appl. Phys. Lett. 85(22), 5418–5420 (2004).
[Crossref]

J. D. Hamilton, T. Buma, M. Spisar, and M. O’Donnell, “High frequency optoacoustic arrays using etalon detection,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47(1), 160–169 (2000).
[Crossref] [PubMed]

Painter, O.

Parkins, A. S.

C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, “The atom-cavity microscope: Single atoms bound in orbit by single photons,” Science 287(5457), 1447–1453 (2000).
[Crossref] [PubMed]

Pfiefer, C. R.

Pruessner, M. W.

Rabiei, P.

Rabinovich, W. S.

Ripin, D. J.

B. E. Little, H. A. Haus, J. S. Foresi, L. C. Kimerling, E. P. Ippen, and D. J. Ripin, “Wavelength switching and routing using absorption and resonance,” IEEE Photonics Technol. Lett. 10(6), 816–818 (1998).
[Crossref]

Ryu, H.

Sermage, B.

J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998).
[Crossref]

Sheaff, C. S.

C. S. Sheaff and S. Ashkenazi, “A polyimide-etalon thin film structure for all-optical high-frequency ultrasound transduction,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59(10), 2254–2261 (2012).
[Crossref] [PubMed]

Shinya, A.

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]

Spisar, M.

J. D. Hamilton, T. Buma, M. Spisar, and M. O’Donnell, “High frequency optoacoustic arrays using etalon detection,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 47(1), 160–169 (2000).
[Crossref] [PubMed]

Srinivasan, K.

Steier, W. H.

Stievater, T. H.

Stöferle, T.

F. Ding, T. Stöferle, L. Mai, A. Knoll, and R. F. Mahrt, “Vertical microcavities with high Q and strong lateral mode confinement,” Phys. Rev. B 87(16), 161116 (2013).
[Crossref]

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Tadayon, M. A.

M. A. Tadayon and S. Ashkenazi, “Optical Micromachined Ultrasound Transducers (OMUT)-A New Approach for High-Frequency Transducers,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60(9), 2021–2030 (2013).
[Crossref]

Thierry-Mieg, V.

J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998).
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Wickramasinghe, H. K.

J. K. Thomson, H. K. Wickramasinghe, and E. A. Ash, “A Fabry–Pérot acoustic surface vibration detector-application to acoustic holography,” J. Phys. D Appl. Phys. 6(6), 677–687 (1973).
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Appl. Phys. Lett. (1)

S. Ashkenazi, C. Y. Chao, L. J. Guo, and M. O’Donnell, “Ultrasound detection using polymer microring optical resonator,” Appl. Phys. Lett. 85(22), 5418–5420 (2004).
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J. Lightwave Technol. (1)

J. Phys. D Appl. Phys. (1)

J. K. Thomson, H. K. Wickramasinghe, and E. A. Ash, “A Fabry–Pérot acoustic surface vibration detector-application to acoustic holography,” J. Phys. D Appl. Phys. 6(6), 677–687 (1973).
[Crossref]

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]

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[Crossref] [PubMed]

Opt. Express (4)

Opt. Lett. (1)

Opt. Mater. Express (1)

Phys. Rev. B (1)

F. Ding, T. Stöferle, L. Mai, A. Knoll, and R. F. Mahrt, “Vertical microcavities with high Q and strong lateral mode confinement,” Phys. Rev. B 87(16), 161116 (2013).
[Crossref]

Phys. Rev. Lett. (1)

J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998).
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R. Paschotta, Encyclopedia of Laser Physics and Technology (Wiley-VCH, 2008), Vol. 1.

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Figures (6)

Fig. 1
Fig. 1 Schematic of the Fabry-Perot Resonator with a waveguide having a core refractive index of n1 and a cladding refractive index of n2. The layers of dielecteric mirrors are presented by high (nh) and low nl refractive index.

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Fig. 2
Fig. 2 Gaussian beam propagation in the cavity (a) without and (b) with waveguide embedded in the Fabry-Perot etalon layer (t:transmission, r:reflection).

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Fig. 3
Fig. 3 (a) Resonance reflection spectrum of a 100 μm Fabry-Perot optical resonator excited with a Gaussian wave and a plane wave. (b) Finesse variation versus reflectivity for different cavity length with 10 μm steps .

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Fig. 4
Fig. 4 (a) Comparison of plane-wave resonance in an un-guided cavity with multi-mode resonance in the waveguide cavity, (b) cross-sectional plot of the modes inside the waveguide cavity.

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Fig. 5
Fig. 5 (a) Fabrication steps of waveguide-Fabry-Perot device by permanent refractive index modification in a photopolymer, (b) microscope image of the fabricated array of the devices.

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Fig. 6
Fig. 6 (a) The experimental set-up for testing reflection spectrum of the device (b) characteristic reflection spectrum curve of the tested device.

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Equations (6)

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E 0 ( r , z ) = A 0 w 0 w ( z ) exp [ − i   ( k z − η ( z ) ) − r 2 ( 1 w ( z ) 2 + i k 2 R ( z ) ) ] ,
B 0 ( r ) = − R 1 E 0 ( r , 0 ) , B s ( r ) =   T 1 R 2 (   R 1 R 2 ) s − 1   E 0 ( r , 2 s L ) ,     f o r   s > 0 ,
B t ( r ) = ∑ s = 0 ∞ B s ( r ) = − R 1 E 0 ( r , 0 ) + T 1 R 2 ∑ s = 1 ∞ (   R 1 R 2 ) s − 1   E 0 ( r , 2 s L ) ,
P t = 2 π ∫ 0 r domain I t ( r ) r d r ,
E x l m ( r , ϕ , z ) = 0 ;   E y l m ( r < a , ϕ , z ) = A l m J l ( h l m r ) e i l ϕ e − i β l m z , E y l m ( r > a , ϕ , z ) = B l m K l ( q l m r ) e i l ϕ e − i β l m z ; E z l m (   r < a , ϕ , z ) = h l m β l m A l m 2 [   J l + 1 ( h l m r ) e i ( l + 1 ) ϕ + J l − 1 ( h l m r ) e i ( l − 1 ) ϕ ] e − i β l m z ,  E z l m (   r > a , ϕ , z ) = q l m β l m B l m 2 [   K l + 1 ( q l m r ) e i ( l + 1 ) ϕ − K l − 1 ( q l m r ) e i ( l − 1 ) ϕ ] e − i β l m z ;
B t l m ( r , ϕ ) = E l m ( r , ϕ ) [ − R 1 + T 1 R 2 ∑ s = 1 ∞ (   R 1 R 2 ) s − 1   e − i β l m 2 s L ] ,

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