Abstract

Optical modes in deformed dielectric microdisk cavities often show an unexpected localization along unstable periodic ray orbits. We reveal a new mechanism for this kind of localization in weakly deformed cavities. In such systems the ray dynamics is nearly integrable and its phase space contains small island chains. When increasing the deformation the enlarging islands incorporate more and more modes. Each time a mode comes close to the border of an island chain (separatrix) the mode exhibits a strong localization near the corresponding unstable periodic orbit. Using an EBK quantization scheme taking into account the Fresnel coefficients we derive a frequency condition for the localization. Observing far field intensity patterns and tunneling distances, reveals small differences in the emission properties.

© 2017 Optical Society of America

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References

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    [Crossref]
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  8. S. B. Lee, J. H. Lee, J. S. Chang, H. J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002).
    [Crossref] [PubMed]
  9. N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, “Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators,” Phys. Rev. Lett. 88, 094102 (2002).
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  10. C. Gmachl, E. E. Narimanov, F. Capasso, J. N. Baillargeon, and A. Y. Cho, “Kolmogorov-arnold-moser transition and laser action on scar modes in semiconductor diode lasers with deformed resonators,” Opt. Lett. 27, 824–826 (2002).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  28. C.-H. Yi, J.-H. Kim, H.-H. Yu, C.-M. Lee, and J.-W. Kim, “Fermi resonance in dynamical tunneling in a chaotic billiard,” Phys. Rev. E 92, 022916 (2015).
    [Crossref]
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  33. M. Kalinski and J. H. Eberly, “New states of hydrogen in a circularly polarized electromagnetic field,” Phys. Rev. Lett. 77, 2420 (1996).
    [Crossref] [PubMed]
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  36. H. Cao and J. Wiersig, “Dielectric microcavities: Model systems for wave chaos and non-hermitian physics,” Rev. Mod. Phys. 87, 61 (2015).
    [Crossref]
  37. J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. 5, 53–60 (2003).
    [Crossref]
  38. K. Husimi, “Some formal properties of the density matrix,” Proc. Phys. Math. Soc. Jpn. 22, 264–314 (1940).
  39. M. Hentschel, H. Schomerus, and R. Schubert, “Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond,” Europhys. Lett. 62, 636–642 (2003).
    [Crossref]
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    [Crossref]
  41. J. Unterhinninghofen and J. Wiersig, “Interplay of Goos-Hänchen shift and boundary curvature in deformed microdisks,” Phys. Rev. E 82, 026202 (2010).
    [Crossref]
  42. H. E. Tureci, H. G. L. Schwefel, A. D. Stone, and E. E. Narimanov, “Gaussian-optical approach to stable periodic orbit resonances of partially chaotic dielectric micro-cavities,” Opt. Express 10, 752–776 (2002).
    [Crossref] [PubMed]
  43. J. B. Keller and S. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. 9, 24–75 (1960).
    [Crossref]
  44. V. I. Arnol’d, Mathematical Methods of Classical Mechanics, vol. 60 of Graduate Texts in Mathematics (Springer, 1978).
    [Crossref]
  45. L. J. Curtis and D. G. Ellis, “Use of the Einstein-Brillouin-Keller action quantization,” Am. J. Phys. 72, 1521 (2004).
    [Crossref]
  46. M. V. Berry, “Regularity and chaos in classical mechanics, illustrated by three deformations of a circular ’billiard’,” Eur. J. Phys. 2, 91–102 (1981).
    [Crossref]
  47. N. Mertig, J. Kullig, C. Löbner, A. Bäcker, and R. Ketzmerick, “Perturbation-free prediction of resonance-assisted tunneling in mixed regular-chaotic systems,” Phys. Rev. E 94, 062220 (2016).
    [Crossref]
  48. J. Kullig and J. Wiersig, “Q spoiling in deformed optical microdisks due to resonance-assisted tunneling,” Phys. Rev. E 94, 022202 (2016).
    [Crossref] [PubMed]
  49. O. Brodier, P. Schlagheck, and D. Ullmo, “Resonance-assisted tunneling,” Ann. Phys. (New York) 300, 88–136 (2002).
    [Crossref]
  50. J. Wiersig, “Resonance zones in action space,” Z. Naturforsch. 57a, 537–556 (2001).
  51. J. Kullig, C. Löbner, N. Mertig, A. Bäcker, and R. Ketzmerick, “Integrable approximation of regular regions with a nonlinear resonance chain,” Phys. Rev. E 90, 052906 (2014).
    [Crossref]
  52. J. Wiersig and M. Hentschel, “Unidirectional light emission from high-Q modes in optical microcavities,” Phys. Rev. A 73, 031802(R) (2006).
    [Crossref]
  53. S. C. Creagh and M. M. White, “Differences between emission patterns and internal modes of optical resonators,” Phys. Rev. E 85, 015201 (2012).
    [Crossref]
  54. M. Tomes, K. J. Vahala, and T. Carmon, “Direct imaging of tunneling from a potential well,” Opt. Express 17, 19160–19165 (2009).
    [Crossref]
  55. S.-Y. Lee and K. An, “Directional emission through dynamical tunneling in a deformed microcavity,” Phys. Rev. A 83, 023827 (2011).
    [Crossref]
  56. J.-B. Shim and J. Wiersig, “Semiclassical evaluation of frequency splittings in coupled optical microdisks,” Opt. Express 21, 24240–24253 (2013).
    [Crossref] [PubMed]
  57. J. U. Nöckel and A. D. Stone, “Chaos in optical cavities,” Nature (London) 385, 45–47 (1997).
    [Crossref]

2016 (3)

C.-H. Yi, H.-H. Yu, and C.-M. Kim, “Resonant torus-assisted tunneling,” Phys. Rev. E 93, 012201 (2016).
[Crossref] [PubMed]

N. Mertig, J. Kullig, C. Löbner, A. Bäcker, and R. Ketzmerick, “Perturbation-free prediction of resonance-assisted tunneling in mixed regular-chaotic systems,” Phys. Rev. E 94, 062220 (2016).
[Crossref]

J. Kullig and J. Wiersig, “Q spoiling in deformed optical microdisks due to resonance-assisted tunneling,” Phys. Rev. E 94, 022202 (2016).
[Crossref] [PubMed]

2015 (4)

D. A. Wisniacki and P. Schlagheck, “Quantum manifestations of classical nonlinear resonances,” Phys. Rev. E 92, 062923 (2015).
[Crossref]

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

C.-H. Yi, H.-H. Yu, J.-W. Lee, and C.-M. Kim, “Fermi resonance in optical microcavities,” Phys. Rev. E 91, 042903 (2015).
[Crossref]

C.-H. Yi, J.-H. Kim, H.-H. Yu, C.-M. Lee, and J.-W. Kim, “Fermi resonance in dynamical tunneling in a chaotic billiard,” Phys. Rev. E 92, 022916 (2015).
[Crossref]

2014 (1)

J. Kullig, C. Löbner, N. Mertig, A. Bäcker, and R. Ketzmerick, “Integrable approximation of regular regions with a nonlinear resonance chain,” Phys. Rev. E 90, 052906 (2014).
[Crossref]

2013 (1)

2012 (1)

S. C. Creagh and M. M. White, “Differences between emission patterns and internal modes of optical resonators,” Phys. Rev. E 85, 015201 (2012).
[Crossref]

2011 (3)

S.-Y. Lee and K. An, “Directional emission through dynamical tunneling in a deformed microcavity,” Phys. Rev. A 83, 023827 (2011).
[Crossref]

D. Wisniacki, M. Saraceno, F. J. Arranz, R. M. Benito, and F. Borondo, “Poincaré-birkhoff theorem in quantum mechanics,” Phys. Rev. E 84, 026206 (2011).
[Crossref]

C.-H. Yi, S. H. Lee, M.-W. Kim, J. Cho, J. Lee, S.-Y. Lee, J. Wiersig, and C.-M. Kim, “Light emission of a scarlike mode with assistance of quasiperiodicity,” Phys. Rev. A 84, 041803 (2011).
[Crossref]

2010 (2)

J. Wiersig, J. Unterhinninghofen, H. Schomerus, U. Peschel, and M. Hentschel, “Electromagnetic modes in cavities made of negative-index metamaterials,” Phys. Rev. A 81, 023809 (2010).
[Crossref]

J. Unterhinninghofen and J. Wiersig, “Interplay of Goos-Hänchen shift and boundary curvature in deformed microdisks,” Phys. Rev. E 82, 026202 (2010).
[Crossref]

2009 (2)

M. Tomes, K. J. Vahala, and T. Carmon, “Direct imaging of tunneling from a potential well,” Opt. Express 17, 19160–19165 (2009).
[Crossref]

C.-M. Kim, S. H. Lee, K. R. Oh, and J. H. Kim, “Experimental verification of quasiscarred resonance mode,” Appl. Phys. Lett. 94, 231120 (2009).
[Crossref]

2008 (4)

E. G. Altmann, G. Del Magno, and M. Hentschel, “Non-hamiltonian dynamics in optical microcavities resulting from wave-inspired corrections to geometric optics,” Euro. Phys. Lett. 84, 10008 (2008).
[Crossref]

J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
[Crossref] [PubMed]

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, “Ray and wave dynamical properties of a spiral-shaped dielectric microcavity,” J. Phys. A: Math. Theor. 41, 275102 (2008).
[Crossref]

J. Unterhinninghofen, J. Wiersig, and M. Hentschel, “Goos-Hänchen shift and localization of optical modes in deformed microcavities,” Phys. Rev. E 78, 016201 (2008).
[Crossref]

2007 (3)

W. Fang, H. Cao, and G. S. Solomon, “Control of lasing in fully chaotic open microcavities by tailoring the shape factor,” Appl. Phys. Lett. 90, 081108 (2007).
[Crossref]

W. Fang and H. Cao, “Wave interference effect on polymer microstadium laser,” Appl. Phys. Lett. 91, 041108 (2007).
[Crossref]

S.-B. Lee, J. Yang, S. Moon, J.-H. Lee, K. An, J.-B. Shim, H.-W. Lee, and S. W. Kim, “Universal output directionality of single modes in a deformed microcavity,” Phys. Rev. A 75, 011802 (2007).
[Crossref]

2006 (2)

J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901 (2006).
[Crossref]

J. Wiersig and M. Hentschel, “Unidirectional light emission from high-Q modes in optical microcavities,” Phys. Rev. A 73, 031802(R) (2006).
[Crossref]

2005 (2)

W. Fang, A. Yamilov, and H. Cao, “Analysis of high-quality modes in open chaotic microcavities,” Phys. Rev. A 72, 023815 (2005).
[Crossref]

S.-Y. Lee, J.-W. Ryu, T.-Y. Kwon, S. Rim, and C.-M. Kim, “Scarred resonances and steady probability distribution in a chaotic microcavity,” Phys. Rev. A 72, 061801 (2005).
[Crossref]

2004 (2)

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, “Quasiscarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[Crossref] [PubMed]

L. J. Curtis and D. G. Ellis, “Use of the Einstein-Brillouin-Keller action quantization,” Am. J. Phys. 72, 1521 (2004).
[Crossref]

2003 (3)

M. Hentschel, H. Schomerus, and R. Schubert, “Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond,” Europhys. Lett. 62, 636–642 (2003).
[Crossref]

T. Harayama, T. Fukushima, P. Davis, P. Vaccaro, T. Miyasaka, T. Nishimura, and T. Aida, “Lasing on scar modes in fully chaotic microcavities,” Phys. Rev. E 67, 015207 (2003).
[Crossref]

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

2002 (6)

H. E. Tureci and A. D. Stone, “Deviation from Snell’s law for beams transmitted near the critical angle: application to microcavity lasers,” Opt. Lett. 27, 7–9 (2002).
[Crossref]

S. B. Lee, J. H. Lee, J. S. Chang, H. J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002).
[Crossref] [PubMed]

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, “Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators,” Phys. Rev. Lett. 88, 094102 (2002).
[Crossref] [PubMed]

C. Gmachl, E. E. Narimanov, F. Capasso, J. N. Baillargeon, and A. Y. Cho, “Kolmogorov-arnold-moser transition and laser action on scar modes in semiconductor diode lasers with deformed resonators,” Opt. Lett. 27, 824–826 (2002).
[Crossref]

H. E. Tureci, H. G. L. Schwefel, A. D. Stone, and E. E. Narimanov, “Gaussian-optical approach to stable periodic orbit resonances of partially chaotic dielectric micro-cavities,” Opt. Express 10, 752–776 (2002).
[Crossref] [PubMed]

O. Brodier, P. Schlagheck, and D. Ullmo, “Resonance-assisted tunneling,” Ann. Phys. (New York) 300, 88–136 (2002).
[Crossref]

2001 (2)

J. Wiersig, “Resonance zones in action space,” Z. Naturforsch. 57a, 537–556 (2001).

M. Sieber and K. Richter, “Correlations between periodic orbits and their role in spectral statistics,” Physica Scripta T90, 128–133 (2001).
[Crossref]

1997 (4)

H. Waalkens, J. Wiersig, and H. R. Dullin, “The elliptic quantum billiard,” Ann. Phys. 260, 50–90 (1997).
[Crossref]

D. Delande, J. Zakrzewski, and A. Buchleitner, “Comment on new states of hydrogen in a circularly polarized electromagnetic field,” Phys. Rev. Lett. 79, 3541 (1997).
[Crossref]

H. J. Korsch, C. Müller, and H. Wiescher, “On the zeros of the husimi distribution,” J. Phys. A 30, L677 (1997).
[Crossref]

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

1996 (1)

M. Kalinski and J. H. Eberly, “New states of hydrogen in a circularly polarized electromagnetic field,” Phys. Rev. Lett. 77, 2420 (1996).
[Crossref] [PubMed]

1984 (1)

E. J. Heller, “Bound-states eigenfunctions of classically chaotic Hamiltionian systems: Scars of periodic orbits,” Phys. Rev. Lett. 53, 1515–1518 (1984).
[Crossref]

1981 (1)

M. V. Berry, “Regularity and chaos in classical mechanics, illustrated by three deformations of a circular ’billiard’,” Eur. J. Phys. 2, 91–102 (1981).
[Crossref]

1971 (1)

M. C. Gutzwiller, “Periodic orbits and classical quantization conditions,” J. Math. Phys. 12, 343–358 (1971).
[Crossref]

1970 (1)

M. C. Gutzwiller, “Energy spectrum according to classical mechanics,” J. Math. Phys. 11, 1791–1806 (1970).
[Crossref]

1967 (1)

M. C. Gutzwiller, “Phase-integral approximation in momentum space and the bound states of an atom,” J. Math. Phys. 8, 1979–2001 (1967).
[Crossref]

1960 (1)

J. B. Keller and S. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. 9, 24–75 (1960).
[Crossref]

1958 (1)

J. B. Keller, “Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems,” Ann. Phys. 4, 180–188 (1958).
[Crossref]

1940 (1)

K. Husimi, “Some formal properties of the density matrix,” Proc. Phys. Math. Soc. Jpn. 22, 264–314 (1940).

Aida, T.

T. Harayama, T. Fukushima, P. Davis, P. Vaccaro, T. Miyasaka, T. Nishimura, and T. Aida, “Lasing on scar modes in fully chaotic microcavities,” Phys. Rev. E 67, 015207 (2003).
[Crossref]

Altmann, E. G.

E. G. Altmann, G. Del Magno, and M. Hentschel, “Non-hamiltonian dynamics in optical microcavities resulting from wave-inspired corrections to geometric optics,” Euro. Phys. Lett. 84, 10008 (2008).
[Crossref]

An, K.

S.-Y. Lee and K. An, “Directional emission through dynamical tunneling in a deformed microcavity,” Phys. Rev. A 83, 023827 (2011).
[Crossref]

S.-B. Lee, J. Yang, S. Moon, J.-H. Lee, K. An, J.-B. Shim, H.-W. Lee, and S. W. Kim, “Universal output directionality of single modes in a deformed microcavity,” Phys. Rev. A 75, 011802 (2007).
[Crossref]

S. B. Lee, J. H. Lee, J. S. Chang, H. J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002).
[Crossref] [PubMed]

Arnol’d, V. I.

V. I. Arnol’d, Mathematical Methods of Classical Mechanics, vol. 60 of Graduate Texts in Mathematics (Springer, 1978).
[Crossref]

Arranz, F. J.

D. Wisniacki, M. Saraceno, F. J. Arranz, R. M. Benito, and F. Borondo, “Poincaré-birkhoff theorem in quantum mechanics,” Phys. Rev. E 84, 026206 (2011).
[Crossref]

Bäcker, A.

N. Mertig, J. Kullig, C. Löbner, A. Bäcker, and R. Ketzmerick, “Perturbation-free prediction of resonance-assisted tunneling in mixed regular-chaotic systems,” Phys. Rev. E 94, 062220 (2016).
[Crossref]

J. Kullig, C. Löbner, N. Mertig, A. Bäcker, and R. Ketzmerick, “Integrable approximation of regular regions with a nonlinear resonance chain,” Phys. Rev. E 90, 052906 (2014).
[Crossref]

Baillargeon, J. N.

Benito, R. M.

D. Wisniacki, M. Saraceno, F. J. Arranz, R. M. Benito, and F. Borondo, “Poincaré-birkhoff theorem in quantum mechanics,” Phys. Rev. E 84, 026206 (2011).
[Crossref]

Berry, M. V.

M. V. Berry, “Regularity and chaos in classical mechanics, illustrated by three deformations of a circular ’billiard’,” Eur. J. Phys. 2, 91–102 (1981).
[Crossref]

Borondo, F.

D. Wisniacki, M. Saraceno, F. J. Arranz, R. M. Benito, and F. Borondo, “Poincaré-birkhoff theorem in quantum mechanics,” Phys. Rev. E 84, 026206 (2011).
[Crossref]

Brodier, O.

O. Brodier, P. Schlagheck, and D. Ullmo, “Resonance-assisted tunneling,” Ann. Phys. (New York) 300, 88–136 (2002).
[Crossref]

Buchleitner, A.

D. Delande, J. Zakrzewski, and A. Buchleitner, “Comment on new states of hydrogen in a circularly polarized electromagnetic field,” Phys. Rev. Lett. 79, 3541 (1997).
[Crossref]

Cao, H.

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

W. Fang and H. Cao, “Wave interference effect on polymer microstadium laser,” Appl. Phys. Lett. 91, 041108 (2007).
[Crossref]

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Figures (11)

Fig. 1
Fig. 1 Sketch of an island chain in the SOS of a near-integrable microcavity. The coordinate s is the arc length along the boundary of the cavity and p = sin χ is its canonical conjugate momentum; χ is the angle of incidence. The thin curves are sections through invariant tori. The thick curve marks the separatrix which separates the interior from the exterior of the island chain.
Fig. 2
Fig. 2 (a) and (b) Intensity of modes for ε = 0.0025 with dimensionless frequencies Re(ne kR) = 95.726 and 95.692, respectively. The solid lines mark the stable (a) and the unstable PO (b). (c) and (d) show the corresponding Husimi functions with ray dynamics superimposed. Only the counter-clockwise direction (p ≥ 0) is shown by virtue of the spatial mirror-symmetry. The localization on a stable and unstable period-3 orbit can be clearly seen. The yellow dashed line at p = 0.5 is a guide to the eye.
Fig. 3
Fig. 3 Dimensionless frequencies (a) and Husimi functions on unstable points (b) related to period-3 POs. The solid, the dashed, the dot-dashed, and the dotted curves originate from the (l, m) = (11, 48), the (10, 51), the (9, 54), and the (8, 57) mode in the circular cavity, respectively. (a) The squares (triangles) are quantized stable (unstable) POs with N = 76, see Eq. (7). Vertical lines at ε1 = 0.002, ε2 = 0.0175, and ε3 = 0.049 mark the maximal localization on the unstable PO. There are two kinds of avoided crossings in (a). The first is a broad one along the quantized unstable PO and the second is a sharp one at ε ≈ 0.0415. For the sharp one, we focus on the diabatic continuation of the modes by jumping at this point to the other branch.
Fig. 4
Fig. 4 (a), (b), and (c) Intensity of modes at ε = 0.0175 for the solid, dashed and dot-dashed curves in Fig. 3, respectively. The solid lines mark the stable PO (a)–(b) and the unstable PO (c). (d)–(f) are the Husimi functions of the modes in (a)–(c) superimposed on the SOS (red dots). The yellow dashed line at p = 0.5 is a guide to the eye.
Fig. 5
Fig. 5 Husimi functions at p = pt:r for the selected parameters. Solid, dashed, dot-dashed, and dotted curves are for (11,48), (10,51), (9,54), and (8,57) modes of the circular cavity, respectively. Husimi functions are normalized by total intensity in phase space and recorded in an arbitrary unit. The values of ε1, ε2, and ε3 are same as those in Fig. 3.
Fig. 6
Fig. 6 (a) Ratio of the Husimi functions of the modes from Fig. 3 inside the separatrix normalized by the total Husimi functions on the entire phase space. The shaded region marks values below 1/2. (b) Expectation values of the distances measured from pt:r to p having probabilities hu(s,p). The shaded region indicates values smaller than the width of the island chain from Eq. (11). Vertical lines labeled by ε1, ε2, and ε3 are same as those in Fig. 3.
Fig. 7
Fig. 7 Angular distributed far-field intensities. (a), (b), and (c) are overlapped far-fields of the island ground and separatrix modes at ε1 = 0.002, ε2 = 0.0175, and ε3 = 0.049. Black curves in all panels are for the island ground modes (l, m) = (11, 48). Red curves in (a), (b), and (c) correspond to the separatrix modes (10, 51), (9, 54), and (8, 57). (d), (e), and (f) are expanded plots of the first prominent lobes of (a), (b), and (c), respectively.
Fig. 8
Fig. 8 Logarithmic plot of Husimi functions of the island ground and separatrix modes. (a), (b), and (c) are for the island modes (l, m) = (11, 48) at ε1 = 0.002, ε2 = 0.0175, and ε3 = 0.049, respectively. (c), (d), and (e) are, respectively, for the separatrix modes (10, 51), (9, 54), and (8, 57) at the same deformation in the same column. Dashed lines correspond to the critical line pc = 1/ne ≈ 0.303. Rightmost panels illustrate the tunneling paths of the modes in the same rows. pI(S) and ΓI(S) correspond to the tunneling starting point and distance of the island ground (separatrix) mode. The maximum and minimum values of the Husimi functions are shown in yellow and black.
Fig. 9
Fig. 9 Logarithmic plot of the intensity. (a) and (b) are the island ground mode (l, m)=(11,48) and the separatrix mode (8,57) at ε3, respectively. In (a), a horizontal solid line at y=1.5 is a plane for the emission Husimi function. The vertical line at x = xb is a guide for the left end of the cavity boundary. A line with arrow heads at both ends in (a) visualizes the departure point, remote from xb, and the radiation directions. All quantities are in units of R.
Fig. 10
Fig. 10 Emission Husimi function τ(x) with the setup described in Fig. 9(a). Solid and dashed curves in (a), (b), and (c) correspond to the island ground mode (l, m) =(11,48) and the separatrix mode (8,57) at ε = ε1 = 0.002, ε2 = 0.0175, and ε3 = 0.049, respectively. Horizontal and vertical axes are scaled in R and arbitrary unit, respectively.
Fig. 11
Fig. 11 Weight function P(m) = | (m)|2 versus angular momentum number m of the far field intensity. Open circle with solid lines and filled square with dashed lines are for the island ground and separatrix modes. (a), (b), and (c) are at ε1 = 0.002 ε2 = 0.0175, and ε3 = 0.049, respectively.

Tables (1)

Tables Icon

Table 1 Tunneling distance (measured in units of R) depending on deformation ε. δI, δS, and Δδ are the tunneling distance of the island ground mode, of the separatrix modes and the difference of them, respectively. The second and third columns are obtained from the peak positions of Fig. 10 and Eq. (15).

Equations (15)

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x 2 + y 2 ( 1 + ε x ) = R 2 ,
ω l ω m = | Δ n m Δ n l | = r t ,
l n e k d l = 2 π ( N + ψ )
ψ = 1 2 + μ 4 + ϕ 4 π + α 2 π .
( Δ s 1 Δ p 1 ) = ( s 1 s 0 s 1 p 0 p 1 s 0 p 1 p 0 ) ( Δ s 0 Δ p 0 ) = M ( Δ s 0 Δ p 0 ) ,
α = Re [ i j = 1 3 ln ( n e β ( χ j ) 1 n e β ( χ j ) + 1 ) ] ,
n e k R = 2 π ( N + 1 2 ) + π μ 2 + ϕ 2 + α l p o / R ,
H t : r ( s , p ) = H 0 ( p ) + 2 V t : r ( ε ) cos ( 2 π r s s max )
H 0 ( p ) = 2 ( p cos 1 ( p ) 1 p 2 + 1 p t : r 2 ) ω ( p t : r ) p
2 V t : r ( ε ) = 1 p t : r 2 2 r 2 ( cos 1 [ Tr { M t : r ( ε ) } 2 ] ) 2 .
p sep ( s ) = p t : r ± ( 2 V t : r 1 p t : r 2 [ 1 cos ( 2 π r s s max ) ] ) 1 2 = p t : r ± Δ p ( s ) ,
δ = R ( n e sin χ 1 ) = n e R ( p p c ) = n e R Γ p ,
τ ( x ) = | d x Φ ( x ) G ( x , x ) | 2 ,
( 2 m ) = 1 𝒵 d ϕ I ( ϕ ) e i m ϕ 𝒵 = m i m f | ( 2 m ) | 2 .
δ f = 1 k [ m i m f m P ( m ) k R ] .

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