Abstract

Circular Bragg resonators (CBRs) are analyzed in both the frequency domain and the time domain based on the scattering matrix method and the numerical model. The CBR with the same size as a dielectric ring can be designed to have denser resonant mode distributions in the frequency domain, and the expansion of the slow light band is imposed by the combination of multiresonant modes. Thus the expansion is independent of group velocity and is not limited by the delay-bandwidth product constraint in static photonic structures, which is deduced for a single resonant mode. Hence, the CBR can store more bits than a dielectric ring. For certain parameters, clockwise (CW) and counterclockwise (CCW) modes in the CBR are quite sensitive to dielectric perturbations, which are weak enough that they have little effect on the CW mode and CCW mode in a dielectric ring. When light propagates along a line waveguide coupled with the CBR, and if there are weak dielectric perturbations in the CBRs, extraordinary reflections could be produced and there exists strong coupling and conversion between CW and CCW modes in the CBR. The optical property indicates that extremely weak dielectric perturbations in the CBR play an important role in mode conversion. These unique properties of CBRs may find applications in the design of practical optical delay line buffers, and they also provide a new method to achieve light control by mode conversion in passive optical resonators.

© 2012 Optical Society of America

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A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E. Geluk, T. de Vries, P. Regreny, D. V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photon. 4, 182–187 (2010).
[CrossRef]

2009

2008

2007

J. B. Khurgin, “Dispersion and loss limitations on the performance of optical delay lines based on coupled resonant structures,” Opt. Lett. 32, 163–165 (2007).
[CrossRef]

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photon. 1, 65–71 (2007).
[CrossRef]

L. Chen, N. S. Droz, and M. Lipson, “Compact bandwidth-tunable microring resonators,” Opt. Lett. 32, 3361–3363 (2007).
[CrossRef]

Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys. 3, 406–410 (2007).
[CrossRef]

2006

K. Liu, X. D. Yuan, W. M. Ye, J. R. Ji, M. Zeng, and C. Zeng, “Optical filter based on omnidirectional reflectors,” Appl. Phys. B 82, 391–393 (2006).
[CrossRef]

2005

J. Scheuer, W. M. J. Green, G. DeRose, and A. Yariv, “InGaAsP annular Bragg lasers: theory, applications and modal properties,” IEEE J. Sel. Top. Quantum Electron. 11, 476–484 (2005).
[CrossRef]

2004

M. F. Yanik, W. Suh, Z. Wang, and S. H. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004).
[CrossRef]

M. T. Hill, H. J. S. Dorren1, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. Oei, H. Binsma, G. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

2003

2002

J. E. Heebner and R. W. Boyd, “ ‘Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. Mod. Opt. 49, 2629–2636 (2002).
[CrossRef]

1999

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, “Bragg reflectors for cylindrical waves,” J. Mod. Opt. 46, 875–890 (1999).
[CrossRef]

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdre, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

1990

M. Toda, “Single-mode behavior of a circular grating for potential disk-shaped DFB lasers,” IEEE J. Quantum Electron. 26, 473–481 (1990).
[CrossRef]

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

1988

D. Y. K. Ko and J. C. Inkson, “Matrix method for tunneling in heterostructures: resonant tunneling in multilayer systems,” Phys. Rev. B 38, 9945–9951 (1988).
[CrossRef]

1978

Abram, R. A.

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, “Bragg reflectors for cylindrical waves,” J. Mod. Opt. 46, 875–890 (1999).
[CrossRef]

Baets, R.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E. Geluk, T. de Vries, P. Regreny, D. V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photon. 4, 182–187 (2010).
[CrossRef]

Beausoleil, R. G.

Benisty, H.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdre, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Binsma, H.

M. T. Hill, H. J. S. Dorren1, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. Oei, H. Binsma, G. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Bolten, J.

Boyd, R. W.

J. E. Heebner and R. W. Boyd, “ ‘Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. Mod. Opt. 49, 2629–2636 (2002).
[CrossRef]

Chen, L.

Culshaw, I. S.

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

de Vries, T.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E. Geluk, T. de Vries, P. Regreny, D. V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photon. 4, 182–187 (2010).
[CrossRef]

M. T. Hill, H. J. S. Dorren1, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. Oei, H. Binsma, G. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

den Besten, J. H.

M. T. Hill, H. J. S. Dorren1, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. Oei, H. Binsma, G. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

DeRose, G.

J. Scheuer, W. M. J. Green, G. DeRose, and A. Yariv, “InGaAsP annular Bragg lasers: theory, applications and modal properties,” IEEE J. Sel. Top. Quantum Electron. 11, 476–484 (2005).
[CrossRef]

Dong, P.

Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys. 3, 406–410 (2007).
[CrossRef]

Dorren1, H. J. S.

M. T. Hill, H. J. S. Dorren1, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. Oei, H. Binsma, G. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Droz, N. S.

Erdogan, T.

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

Fan, S. H.

M. F. Yanik, W. Suh, Z. Wang, and S. H. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004).
[CrossRef]

Forst, M.

Geluk, E.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E. Geluk, T. de Vries, P. Regreny, D. V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photon. 4, 182–187 (2010).
[CrossRef]

Gotzinger, S.

Green, W. M. J.

J. Scheuer, W. M. J. Green, G. DeRose, and A. Yariv, “InGaAsP annular Bragg lasers: theory, applications and modal properties,” IEEE J. Sel. Top. Quantum Electron. 11, 476–484 (2005).
[CrossRef]

Hall, D. G.

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

Heebner, J. E.

J. E. Heebner and R. W. Boyd, “ ‘Slow’ and ‘fast’ light in resonator-coupled waveguides,” J. Mod. Opt. 49, 2629–2636 (2002).
[CrossRef]

Hegarty, J.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdre, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

Hill, M. T.

M. T. Hill, H. J. S. Dorren1, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. Oei, H. Binsma, G. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Houdre, R.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdre, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

Huybrechts, K.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E. Geluk, T. de Vries, P. Regreny, D. V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photon. 4, 182–187 (2010).
[CrossRef]

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Inkson, J. C.

D. Y. K. Ko and J. C. Inkson, “Matrix method for tunneling in heterostructures: resonant tunneling in multilayer systems,” Phys. Rev. B 38, 9945–9951 (1988).
[CrossRef]

Ji, J. R.

K. Liu, X. D. Yuan, W. M. Ye, J. R. Ji, M. Zeng, and C. Zeng, “Optical filter based on omnidirectional reflectors,” Appl. Phys. B 82, 391–393 (2006).
[CrossRef]

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Kaliteevski, M. A.

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, “Bragg reflectors for cylindrical waves,” J. Mod. Opt. 46, 875–890 (1999).
[CrossRef]

Khoe, G.

M. T. Hill, H. J. S. Dorren1, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. Oei, H. Binsma, G. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Khurgin, J. B.

Ko, D. Y. K.

D. Y. K. Ko and J. C. Inkson, “Matrix method for tunneling in heterostructures: resonant tunneling in multilayer systems,” Phys. Rev. B 38, 9945–9951 (1988).
[CrossRef]

Krauss, T. F.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdre, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

Kumar, R.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E. Geluk, T. de Vries, P. Regreny, D. V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photon. 4, 182–187 (2010).
[CrossRef]

Labilloy, D.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdre, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

Leijtens, X. J. M.

M. T. Hill, H. J. S. Dorren1, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. Oei, H. Binsma, G. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Li, Q.

Lipson, M.

Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys. 3, 406–410 (2007).
[CrossRef]

L. Chen, N. S. Droz, and M. Lipson, “Compact bandwidth-tunable microring resonators,” Opt. Lett. 32, 3361–3363 (2007).
[CrossRef]

Liu, F.

Liu, K.

K. Liu, X. D. Yuan, W. M. Ye, J. R. Ji, M. Zeng, and C. Zeng, “Optical filter based on omnidirectional reflectors,” Appl. Phys. B 82, 391–393 (2006).
[CrossRef]

Liu, L.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E. Geluk, T. de Vries, P. Regreny, D. V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photon. 4, 182–187 (2010).
[CrossRef]

Mahrt, R. F.

Marom, E.

Moll, N.

Morthier, G.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E. Geluk, T. de Vries, P. Regreny, D. V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photon. 4, 182–187 (2010).
[CrossRef]

Morton, P. A.

Nikolaev, V. V.

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, “Bragg reflectors for cylindrical waves,” J. Mod. Opt. 46, 875–890 (1999).
[CrossRef]

Oei, Y.

M. T. Hill, H. J. S. Dorren1, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. Oei, H. Binsma, G. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Oesterle, U.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdre, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

Offrein, B. J.

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Plotzing, T.

Qiu, M.

Regreny, P.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E. Geluk, T. de Vries, P. Regreny, D. V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photon. 4, 182–187 (2010).
[CrossRef]

Roelkens, G.

L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E. Geluk, T. de Vries, P. Regreny, D. V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photon. 4, 182–187 (2010).
[CrossRef]

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Roycroft, B.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdre, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

Sandoghdar, V.

Scheuer, J.

J. Scheuer, W. M. J. Green, G. DeRose, and A. Yariv, “InGaAsP annular Bragg lasers: theory, applications and modal properties,” IEEE J. Sel. Top. Quantum Electron. 11, 476–484 (2005).
[CrossRef]

J. Scheuer and A. Yariv, “Annular Bragg defect mode resonators,” J. Opt. Soc. Am. B 20, 2285–2291 (2003).
[CrossRef]

Schonenberger, S.

Sekaric, L.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photon. 1, 65–71 (2007).
[CrossRef]

Shaw, A.

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdre, and U. Oesterle, “Lasing properties of disk microcavity based on a circular Bragg reflector,” Appl. Phys. Lett. 75, 3051–3053 (1999).
[CrossRef]

Smalbrugge, B.

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Supplementary Material (2)

» Media 1: MPG (752 KB)     
» Media 2: MPG (752 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

Schematic optical delay line based on (a) ring resonator and (b) CBR.

Fig. 2.
Fig. 2.

log10(1/e0) versus normalized frequency. a is equal to one unit length, n represents azimuthal mode numbers. (a) Single ring resonator suspended in air, with refractive index of the ring 3.4, normalized ring width w=0.8, and normalized outer radius r=8.6. (b) CBR, with one disk and five rings, refraction index of disk and rings is 3.4, with normalized ring width w=0.6, and normalized outmost radius r=8.6. The normalized distance d between each ring is d=0.08.

Fig. 3.
Fig. 3.

Transmitted power in different parts of CBR. Distance between the bus line waveguide and CBR is 0.1. (a) Normalized distance between disk and innermost ring is d=0.08, and normalized distance between each ring is d=0.08. (b) Distance between disk and innermost ring is d=0.065, and distance between each ring is d=0.065.

Fig. 4.
Fig. 4.

Snapshots of the magnetic field distributions with different continuous incident waves in a steady state. (a) Normalized frequency, f=0.01760, (b) f=0.17775, and (c) f=0.17975.

Fig. 5.
Fig. 5.

Temporal evolution of the Gaussian pulse. The intensities of the input and the output pulse are normalized with the maximum intensity of the input pulse with center frequency.

Fig. 6.
Fig. 6.

Schematic representation of SCISSOR composed of CBRs.

Fig. 7.
Fig. 7.

Temporal evolution of the Gaussian pulse, with two CBRs.

Fig. 8.
Fig. 8.

Dielectric distribution of the CBR coupled with line waveguides. Inset shows a magnification of the dielectric thin line with normalized width 0.05 as dielectric perturbation on the rings.

Fig. 9.
Fig. 9.

Transmission spectra and transmitted power of TE polarized light through CBR (a) Transmission spectra. (b) Normalized transmitted power through five rings from the outmost layer to the inner layer of the CBR with dielectric perturbations (shown in Fig. 8). (c) Normalized transmitted power through five rings the outmost layer to the inner layer of CBR without dielectric perturbations. Positive transmitted power means light travels through the rings in CCW direction, and negative transmitted power in CW direction.

Fig. 10.
Fig. 10.

Snapshots of the magnetic field distributions with different continuous incident waves in a steady state with normalized frequency, 0.2123. (a) CBR without dielectric perturbation (Media 1) and (b) CBR with dielectric perturbations as shown in Fig. 8 (Media 2).

Equations (24)

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Nres3Nst3/2.
[1ρ1ρ(ρ1ρ)+1ρ22φ2+k02n2(ρ)]Hz=0,
(μ0ε0Hz(ρ)Eφ(ρ))=A(ρ)(albl).
(aN+1b1)=S(1,N+1)(a1bN+1)=(S11S12S21S22)(a1bN+1).
(a1b1)=Ta0=(T1T2)a0.
aN+1=[S11T1+S12S221(T2S21T1)]a0,
bN+1=S221(T2S21T1)]a0.
detU(ω)=det[S11T1+S12S221(T2S21T1)]=0.
ΔTd×ΔB2/π.
Hz(ρ,φ)=Hz(ρ)exp(inφ)Eφ(ρ,φ)=Eφ(ρ)exp(inφ),
(μ0ε0Hz(ρ)Eφ(ρ))=(Jn(kρlρ)Yn(kρlρ)ik0urlkρlJn(kρlρ)ik0urlkρlYn(kρlρ))(albl).
(μ0ε0Hz(ρ)Eφ(ρ))=(Jn(kρ0ρ)ik0ur0kρJn(kρ0ρ))(a0).
(μ0ε0Hz(ρ)Eφ(ρ))=(Jn(kρN+1ρ)Hn(kρN+1ρ)ik0ur(N+1)kρN+1Jn(kρN+1ρ)ik0ur(N+1)kρN+1Hn(kρN+1ρ))(aN+1bN+1),
(aN+1b1)=S(1,N+1)(a1bN+1)=(S11S12S21S22)(a1bN+1).
(al+1bl)=C(l,l+1)(albl+1)=(C11C12C21C22)(albl+1).
S11(l,l+1)=C11(l,l+1)Vaa(l,l)S11(l,l),
S12(l,l+1)=C12(l,l+1)+C11(l,l+1)Vaa(l,l)S12(l,l)C22(l,l+1),
S21(l,l+1)=S21(l,l)+S22(l,l)C21(l,l+1)Vaa(l,l)S11(l,l),
S22(l,l+1)=S22(l,l)[C21(l,l+1)Vaa(l,l)S12(l,l)+I]C22(l,l+1),
Vaa(l,l)=[IS12(l,l)C21(l,l+1)]1.
(a1b1)=(Jn(kρlR)Yn(kρlR)ik0urlkρlJn(kρlR)ik0urlkρlYn(kρlR))1(Jn(kρ0R)ik0ur0kρJn(kρ0R))a0=(T1T2)a0.
aN+1=[S11T1+S12S221(T2S21T1)]a0,
bN+1=[S221(T2S21T1)]a0.
detU(ω)=det[S11T1+S12S221(T2S21T1)]=0.

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