Abstract

We develop a numerical scheme to construct the scattering (S) matrix for optical microcavities, including the special cases with parity-time and other non-Hermitian symmetries. This scheme incorporates the explicit form of a nonlocal boundary condition, with the incident light represented by an inhomogeneous term. This approach resolves the artifact of a discontinuous normal derivative typically found in the R-matrix method. In addition, we show that, by excluding the aforementioned inhomogeneous term, the non-Hermitian Hamiltonian in our approach also determines the Periels–Kapur states, and it constitutes an alternative approach to derive the standard R-matrix result in this basis. Therefore, our scheme provides a convenient framework to explore the benefits of both approaches. We illustrate this boundary value problem using 1D and 2D scalar Helmholtz equations. The eigenvalues and poles of the S matrix calculated using our approach show good agreement with results obtained by other means.

© 2017 Chinese Laser Press

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References

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2017 (1)

L. Ge, “Symmetry-protected zero-mode laser with a tunable spatial profile,” Phys. Rev. A 95, 023812 (2017).
[Crossref]

2016 (2)

L. Ge and L. Feng, “Optical-reciprocity-induced symmetry in photonic heterostructures and its manifestation in scattering PT-symmetry breaking,” Phys. Rev. A 94, 043836 (2016).
[Crossref]

S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Controlling a microdisk laser by local refractive index perturbation,” Appl. Phys. Lett. 108, 051105 (2016).
[Crossref]

2015 (2)

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015).
[Crossref]

L. Ge, K. G. Makris, D. N. Christodoulides, and L. Feng, “Scattering in PT- and RT-symmetric multimode waveguides: Generalized conservation laws and spontaneous symmetry breaking beyond one dimension,” Phys. Rev. A 92, 062135 (2015).
[Crossref]

2014 (5)

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Singlemode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4, 031011 (2014).
[Crossref]

2013 (2)

P. Ambichl, K. G. Makris, L. Ge, Y. Chong, A. D. Stone, and S. Rotter, “Breaking of PT symmetry in bounded and unbounded scattering systems,” Phys. Rev. X 3, 041030 (2013).
[Crossref]

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

2012 (2)

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature (London) 488, 167–171 (2012).
[Crossref]

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85, 023802 (2012).
[Crossref]

2011 (2)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref]

2010 (4)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
[Crossref]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[Crossref]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref]

2009 (1)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

2008 (4)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[Crossref]

2007 (1)

2006 (1)

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[Crossref]

2004 (1)

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[Crossref]

2003 (1)

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A 5, 53–60 (2003).
[Crossref]

2002 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

2000 (1)

S. Rotter, J.-Z. Tang, L. Wirtz, J. Trost, and J. Burgdörfer, “Modular recursive Greens function method for ballistic quantum transport,” Phys. Rev. B 62, 1950–1960 (2000).
[Crossref]

1999 (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[Crossref]

1998 (3)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians naving PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref]

T. Shibata and T. Itoh, “Generalized-scattering-matrix modeling of waveguide circuits using FDTD field simulations,” IEEE Trans. Microwave Theory Tech. 46, 1742–1751 (1998).
[Crossref]

1997 (2)

C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. 69, 731–808 (1997).
[Crossref]

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature (London) 385, 45–47 (1997).
[Crossref]

1996 (2)

R. Szmytkowski and J. Hinze, “Convergence of the non-relativistic and relativistic R-matrix expansions at the reaction volume boundary,” J. Phys. B 29, 761–777 (1996).
[Crossref]

P. Yang and K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
[Crossref]

1995 (1)

1976 (1)

D. N. Pattanayak and E. Wolf, “Scattering states and bound states as solutions of the Schrodinger equation with nonlocal boundary conditions,” Phys. Rev. D 13, 913–923 (1976).
[Crossref]

1965 (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[Crossref]

1958 (1)

A. M. Lane and R. G. Thomas, “R-matrix theory of nuclear reactions,” Rev. Mod. Phys. 30, 257–353 (1958).
[Crossref]

1957 (1)

C. Bloch, “Une formulation unifièe de la théorie des réactions nucléaires,” Nucl. Phys. 4, 503–528 (1957).
[Crossref]

1947 (2)

R. H. Dicke, “A computational method applicable to microwave networks,” J. Appl. Phys. 18, 873–878 (1947).
[Crossref]

E. P. Wigner and L. Eisenbud, “Higher angular momenta and long range interaction in resonance reactions,” Phys. Rev. 72, 29–41 (1947).
[Crossref]

1940 (1)

G. Breit, “Scattering matrix of radioactive states,” Phys. Rev. 58, 1068–1074 (1940).
[Crossref]

1938 (1)

P. L. Kapur and R. Peierls, “The dispersion formula for nuclear reactions,” Proc. R. Soc. A 166, 277–295 (1938).
[Crossref]

1937 (1)

J. A. Wheeler, “On the mathematical description of light nuclei by the method of resonating group structure,” Phys. Rev. 52, 1107–1122 (1937).
[Crossref]

1928 (1)

G. Gamow, “Zur quantentheorie des atomkernes,” Z. Phys. 51, 204–212 (1928).
[Crossref]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Almeida, V. R.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Ambichl, P.

P. Ambichl, K. G. Makris, L. Ge, Y. Chong, A. D. Stone, and S. Rotter, “Breaking of PT symmetry in bounded and unbounded scattering systems,” Phys. Rev. X 3, 041030 (2013).
[Crossref]

Beenakker, C. W. J.

C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. 69, 731–808 (1997).
[Crossref]

Bender, C. M.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians naving PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Bersch, C.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature (London) 488, 167–171 (2012).
[Crossref]

Bloch, C.

C. Bloch, “Une formulation unifièe de la théorie des réactions nucléaires,” Nucl. Phys. 4, 503–528 (1957).
[Crossref]

Boettcher, S.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians naving PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Breit, G.

G. Breit, “Scattering matrix of radioactive states,” Phys. Rev. 58, 1068–1074 (1940).
[Crossref]

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

Burgdörfer, J.

S. Rotter, J.-Z. Tang, L. Wirtz, J. Trost, and J. Burgdörfer, “Modular recursive Greens function method for ballistic quantum transport,” Phys. Rev. B 62, 1950–1960 (2000).
[Crossref]

Campillo, A. J.

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996).

Cao, H.

S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Controlling a microdisk laser by local refractive index perturbation,” Appl. Phys. Lett. 108, 051105 (2016).
[Crossref]

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref]

Capasso, F.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref]

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C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
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L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
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M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Westview, 1995).

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C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
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S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Controlling a microdisk laser by local refractive index perturbation,” Appl. Phys. Lett. 108, 051105 (2016).
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L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4, 031011 (2014).
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[Crossref]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref]

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W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
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S. Rotter, J.-Z. Tang, L. Wirtz, J. Trost, and J. Burgdörfer, “Modular recursive Greens function method for ballistic quantum transport,” Phys. Rev. B 62, 1950–1960 (2000).
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H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
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L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Singlemode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
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L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
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L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
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S. F. Liew, L. Ge, B. Redding, G. S. Solomon, and H. Cao, “Controlling a microdisk laser by local refractive index perturbation,” Appl. Phys. Lett. 108, 051105 (2016).
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W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
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L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
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L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
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Figures (5)

Fig. 1.
Fig. 1.

(a) Total wave function and (b) its flux depicted by black thick lines for a half-gain-half-loss microcavity with light incident from the left. The wave vector k=12/L and refractive indices n1=n2*=20.2i are used. The expansion in Eq. (14) with 50 CF states is plotted by the red thin lines as a comparison, which can barely be distinguished from the black line in (a) but shows a significant deviation near the left boundary in (b). The black dot in (b) shows the analytical result at x=L/2 given by Eq. (18).

Fig. 2.
Fig. 2.

Schematic of an optical microcavity (shaded area) and the circular LSS (solid line) in 2D. The finite-difference grid is indicated by the dots and dashed lines.

Fig. 3.
Fig. 3.

Resonances of a microdisk cavity with a uniform index n=1.5. The crosses are the analytical results given by Eq. (52), and the circles are the poles of the S matrix constructed using Eq. (28).

Fig. 4.
Fig. 4.

(a) Spontaneous symmetry breaking of S-matrix eigenvalues sn in a microdisk cavity with PT and RT symmetries. Its refractive index is given by n(x)=1.5+0.4sinθ, the imaginary part of which is shown schematically by the inset in (b). (b) A real-valued eigenvector of S at kR=4 in the PT- and RT-symmetric phase. The blue (pink) bars show symmetric and antisymmetric components with opposite m’s. The corresponding wave function is shown in (d), where the cavity boundary is marked by the white circle. The wave function of a scattering eigenstate in the broken-symmetry phase at kR=4 is shown in (c) as a comparison.

Fig. 5.
Fig. 5.

Resonant modes corresponding to the scattering eigenstates in Figs. 4(c) and 4(d).

Equations (54)

Equations on this page are rendered with MathJax. Learn more.

[x2+ϵ(x)k2]Ψ(x,k)=0,
Ψ(x,k)={ΨL+rLΨL+,(x<L/2)tLΨR+,(x>L/2),
xΨ|L=ik(2Ψ|L),
xΨ|R=ikΨ|R,
1Δ2[Ψi+12Ψi+Ψi1]+ϵik2Ψi=0,
Ψ0=2+iq2iqΨ1+η,
ΨN+1=2+iq2iqΨN,
Hψm=qm2ϵψm.
Hij=[2+2+iq2iq(δi,1+δi,N)]δij+(δi+1,j+δi1,j),
HΨ+F=q2ϵΨ,
Ψ=[H+q2ϵ]1F.
rL=2Ψ1(2+iq)2iq,tL=2ΨN2iq,
S=(rLtLtRrR),
Ψ=ΔLmψmψmTq2qm2F,
ψmTϵψn=LΔδm,n.
ψ(x)=ηΔLmψm(x)ψm(0)k2km2=2ikLmψm(x)ψm(0)k2km2.
am=1L0LϵψmΨdx=1L[ΨxψmψmxΨ]0Lk2km2.
Im[Ψ*xΨ]L=k(1|rL|2),rL=G+iFD,
Ψm=Hm(nekr)Hm(nekR)eimθ,Ψm+=Hm+(nek*r)Hm+(nek*R)eimθ,
Ψ>=Ψm0+mSm,m0Ψm+,
Ψ>r=Vm0(r)eim0θ+mSm,m0Vm+*(r)eimθ,
ΨNr,ν=mbmeimθν,
ΨNr+1,ν=mbm(1+cmΔr)eimθν,
ΨNr+1,ν+ΨNr,ν2=mbm(1+cmΔr2)eimθν,
ΨNr+1,νΨNr,νΔr=mbmcmeimθν,
Sm,m0=bmcmVm0δm,m0Vm+*,
Sm,m0=bm(1+cmΔr2)δm,m0,
Sm,m0=bm(1Vm0Δr2)δm,m01Vm+*Δr2.
bm=νΔθ2πeimθνΨNr,ν.
ΨNr+1,ν=νOν,νΨNr,ν+fν(m0),
Oν,νΔθ2πmHm+(nek*R+)Hm+(nek*R)eim(θνθν),
fν(m0)=(Vm0Vm0+*)Δr1Vm0+*Δr2eim0θν,
(H+k2ϵ)ψ+F(m0)=0.
ψ=(ψ1,1ψ1,Nθψ2,1ψNr,1ψNr,Nθ)T
Hψn=kn2ϵψn,
{H(μ1)Nθ+ν,(μ1)Nθ+ν=2(Δr)22(rμΔθ)2H(μ1)Nθ+ν,(μ1)Nθ+ν+1=1(rμΔθ)2H(μ1)Nθ+ν,μNθ+ν=rμ+12(Δr)2rμrμ+1,
Hν,ν=1(Δr)2RROν,ν.
ψ=(H+k2ϵ)1F(m0),
Sm,m0Δr0bmδm,m0.
ψ=ΔrΔθπR2nψnψnTk2kn2F(m0).
ψnTϵψn=πR2ΔrΔθδn,n,
systemϵΨnΨnrdrdϕ=πR2δn,n
Ψn|r=R=mzm(n)Ψm+|r=R=mzm(n)eimθ,
ψnTF(m0)=2RΔrΔθ(Vm0Vm0+*)zm0(n),
Sm,m0Δr02R(Vm0+*Vm0)Rm,m0δm,m0,
Rm,m0=nzm(n)zm0(n)k2kn2
an=1πR2systemϵΨnΨdx=1πRLSS[ΨrΨnΨnrΨ]Rdθk2kn2.
[2+ne2k2]X=0,X=Ψ,Ψn.
an=1πRLSS[Ψm0rΨnΨnrΨm0]Rdθk2kn2
rΨn|r=R=mzm(n)Vm+*eimθ,
Ψ(x)=2R(Vm0+*Vm0)nzm0(n)Ψn(x)k2kn2.
Hm+(kR)Jm+(nkR)Hm+(kR)Jm(nkR)=n.
vm=±()mvm(vmR),
vm=±()mvm(vmC).