Abstract

Dielectric microcavities based on cylindrical and deformed cylindrical shapes have been employed as resonators for microlasers. Such systems support spiral resonances with finite momentum along the cylinder axis. For such modes the boundary conditions do not separate, and simple TM and TE polarization states do not exist. We formulate a theory for the dispersion relations and polarization properties of such resonances for an infinite dielectric rod of arbitrary cross section and then solve for these quantities for the case of a circular cross section (cylinder). Useful analytic formulas are obtained using the eikonal (Einstein–Brillouin–Keller) method, which are shown to be excellent approximations to the exact results from the wave equation. The major finding is that the polarization of the radiation emitted into the far field is linear up to a polarization critical angle (PCA) at which it changes to elliptical. The PCA always lies between the Brewster’s and total-internal-reflection angles for the dielectric, as is shown by an analysis based on the Jones matrices of the spiraling rays.

© 2005 Optical Society of America

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References

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  1. 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] [PubMed]
  2. J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature  385, 45–47 (1997).
    [CrossRef]
  3. 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]
  4. G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett.  83, 1710–1712 (2003).
    [CrossRef]
  5. H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, “Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers,” J. Opt. Soc. Am. B  21, 923–934 (2004), physics/0308001.
    [CrossRef]
  6. A. W. Poon, R. K. Chang, and J. A. Lock, “Spiral morphology-dependent resonances in an optical fiber: effects of fiber tilt and focused Gaussian beam illumination,” Opt. Lett.  23, 1105–1107 (1998).
    [CrossRef]
  7. J. A. Lock, “Morphology-dependent resonances of an infinitely long circular cylinder illuminated by a diagonally incident plane wave or a focused Gaussian beam,” J. Opt. Soc. Am. A  14, 653–661 (1997).
    [CrossRef]
  8. G. Roll and G. Schweiger, “Resonance shift of obliquely illuminated dielectric cylinders: geometrical-optics estimates,” Appl. Opt.  37, 5628–5630 (1998).
    [CrossRef]
  9. H. G. L. Schwefel and A. D. Stone, “Vector resonances in chaotic dielectric rods” (in preparation; available from the authors at the address on the title page).
  10. H. E. Türeci, “Wave chaos in dielectric resonators: asymptotic and numerical approaches,” Ph.D. thesis (Yale University, New Haven, Connecticut, 2003).
  11. H. G. L. Schwefel, “Directionality and vector resonances of regular and chaotic dielectric microcavities,” Ph.D. thesis (Yale University, New Haven, Connecticut, 2004).
  12. T. Harayama, P. Davis, and K. S. Ikeda, “Stable oscillations of a spatially chaotic wave function in a microstadium laser,” Phys. Rev. Lett.  90, 063901 (2003).
    [CrossRef] [PubMed]
  13. H. E. Tureci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt.  47 (2005).
    [CrossRef]
  14. SLATEC, “SLATEC Common Mathematical Library (Version 4.1),” July 1993, http://www.netlib.org/slatec/.
  15. J. B. Keller, “Corrected Bohr–Sommerfeld quantum conditions for nonseparable systems,” Ann. Phys. (N.Y.)  4, 180–188 (1958).
    [CrossRef]
  16. J. B. Keller and S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (N.Y.)  9, 24–75 (1960).
    [CrossRef]
  17. A. Einstein, “Zum Quantensatz von Sommerfeld und Epstein,” Verh. Dtsch. Phys. Ges.  19, 82–92 (1917).
  18. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).
  19. R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am.  31, 488–493 (1941).
    [CrossRef]

2005 (1)

H. E. Tureci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt.  47 (2005).
[CrossRef]

2004 (1)

2003 (2)

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett.  83, 1710–1712 (2003).
[CrossRef]

T. Harayama, P. Davis, and K. S. Ikeda, “Stable oscillations of a spatially chaotic wave function in a microstadium laser,” Phys. Rev. Lett.  90, 063901 (2003).
[CrossRef] [PubMed]

2002 (1)

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]

1998 (3)

1997 (2)

1960 (1)

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

1958 (1)

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

1941 (1)

1917 (1)

A. Einstein, “Zum Quantensatz von Sommerfeld und Epstein,” Verh. Dtsch. Phys. Ges.  19, 82–92 (1917).

Ben-Messaoud, T.

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] [PubMed]

Chang, R. K.

H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, “Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers,” J. Opt. Soc. Am. B  21, 923–934 (2004), physics/0308001.
[CrossRef]

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett.  83, 1710–1712 (2003).
[CrossRef]

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]

A. W. Poon, R. K. Chang, and J. A. Lock, “Spiral morphology-dependent resonances in an optical fiber: effects of fiber tilt and focused Gaussian beam illumination,” Opt. Lett.  23, 1105–1107 (1998).
[CrossRef]

Chern, G. D.

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett.  83, 1710–1712 (2003).
[CrossRef]

Cho, A. Y.

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] [PubMed]

Davis, P.

T. Harayama, P. Davis, and K. S. Ikeda, “Stable oscillations of a spatially chaotic wave function in a microstadium laser,” Phys. Rev. Lett.  90, 063901 (2003).
[CrossRef] [PubMed]

Einstein, A.

A. Einstein, “Zum Quantensatz von Sommerfeld und Epstein,” Verh. Dtsch. Phys. Ges.  19, 82–92 (1917).

Faist, J.

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] [PubMed]

Gmachl, C.

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] [PubMed]

Harayama, T.

T. Harayama, P. Davis, and K. S. Ikeda, “Stable oscillations of a spatially chaotic wave function in a microstadium laser,” Phys. Rev. Lett.  90, 063901 (2003).
[CrossRef] [PubMed]

Ikeda, K. S.

T. Harayama, P. Davis, and K. S. Ikeda, “Stable oscillations of a spatially chaotic wave function in a microstadium laser,” Phys. Rev. Lett.  90, 063901 (2003).
[CrossRef] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

Jacquod, P.

H. E. Tureci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt.  47 (2005).
[CrossRef]

Johnson, N. M.

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett.  83, 1710–1712 (2003).
[CrossRef]

Jones, R. C.

Keller, J. B.

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

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

Kneissl, M.

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett.  83, 1710–1712 (2003).
[CrossRef]

Lock, J. A.

Narimanov, E. E.

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] [PubMed]

Nöckel, J. U.

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] [PubMed]

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

Poon, A. W.

Rex, N. B.

H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, “Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers,” J. Opt. Soc. Am. B  21, 923–934 (2004), physics/0308001.
[CrossRef]

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]

Roll, G.

Rubinow, S. I.

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

Schwefel, H. G.

H. E. Tureci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt.  47 (2005).
[CrossRef]

H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, “Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers,” J. Opt. Soc. Am. B  21, 923–934 (2004), physics/0308001.
[CrossRef]

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]

H. G. L. Schwefel, “Directionality and vector resonances of regular and chaotic dielectric microcavities,” Ph.D. thesis (Yale University, New Haven, Connecticut, 2004).

H. G. L. Schwefel and A. D. Stone, “Vector resonances in chaotic dielectric rods” (in preparation; available from the authors at the address on the title page).

Schweiger, G.

Sivco, D. L.

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] [PubMed]

Stone, A. D.

H. E. Tureci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt.  47 (2005).
[CrossRef]

H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, “Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers,” J. Opt. Soc. Am. B  21, 923–934 (2004), physics/0308001.
[CrossRef]

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett.  83, 1710–1712 (2003).
[CrossRef]

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, 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] [PubMed]

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

H. G. L. Schwefel and A. D. Stone, “Vector resonances in chaotic dielectric rods” (in preparation; available from the authors at the address on the title page).

Tureci, H. E.

H. E. Tureci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt.  47 (2005).
[CrossRef]

H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, “Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers,” J. Opt. Soc. Am. B  21, 923–934 (2004), physics/0308001.
[CrossRef]

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett.  83, 1710–1712 (2003).
[CrossRef]

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]

Türeci, H. E.

H. E. Türeci, “Wave chaos in dielectric resonators: asymptotic and numerical approaches,” Ph.D. thesis (Yale University, New Haven, Connecticut, 2003).

Zyss, J.

Ann. Phys. (N.Y.) (2)

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

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

Appl. Opt. (1)

Appl. Phys. Lett. (1)

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, “Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars,” Appl. Phys. Lett.  83, 1710–1712 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

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

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

Nature (1)

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

Opt. Lett. (1)

Phys. Rev. Lett. (2)

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]

T. Harayama, P. Davis, and K. S. Ikeda, “Stable oscillations of a spatially chaotic wave function in a microstadium laser,” Phys. Rev. Lett.  90, 063901 (2003).
[CrossRef] [PubMed]

Prog. Opt. (1)

H. E. Tureci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt.  47 (2005).
[CrossRef]

Science (1)

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] [PubMed]

Verh. Dtsch. Phys. Ges. (1)

A. Einstein, “Zum Quantensatz von Sommerfeld und Epstein,” Verh. Dtsch. Phys. Ges.  19, 82–92 (1917).

Other (5)

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

SLATEC, “SLATEC Common Mathematical Library (Version 4.1),” July 1993, http://www.netlib.org/slatec/.

H. G. L. Schwefel and A. D. Stone, “Vector resonances in chaotic dielectric rods” (in preparation; available from the authors at the address on the title page).

H. E. Türeci, “Wave chaos in dielectric resonators: asymptotic and numerical approaches,” Ph.D. thesis (Yale University, New Haven, Connecticut, 2003).

H. G. L. Schwefel, “Directionality and vector resonances of regular and chaotic dielectric microcavities,” Ph.D. thesis (Yale University, New Haven, Connecticut, 2004).

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Figures (15)

Fig. 1
Fig. 1

Schematics of a spiraling ray in a rod with arbitrary cross section. The ray can refract out, and the polarization can be defined in the far field with respect to a local coordinate system. For a general deformation, the polarization will change with the azimuthal angle ϕ. Here, we will focus only on the circular cross section, where the polarization is constant (with respect to ϕ) when referred to the local coordinate system oriented along and perpendicular to a ray in the far field.

Fig. 2
Fig. 2

(a) Coordinates used for ray dynamics in a rod with arbitrary cross section. The ray can refract out, and the polarization can be defined in the far field with respect to a local Cartesian coordinate system tilted so that one of the axes is along the propagation direction. We define η as the angle of incidence in the plane of incidence; χ, the angle of incidence projected into the transverse plane; σ, the projected far-field angle; θ, the tilt angle measured from the cross-sectional x y plane, given by tan θ = k z γ 1 ; α, the external tilt angle tan α = k z γ 2 ; (b), (c) Schematics highlighting the trigonometric relations among the inside and outside quantities. Note that α can be found through the application of Snell’s law, n sin θ = sin α .

Fig. 3
Fig. 3

Comparison of scattering and emission pictures for quasi-bound modes. The complex quasi-bound mode frequencies are plotted on the Re [ k R ] Im [ k R ] plane. On the back panel we plot the real k S-matrix, scattering cross section at 170° with respect to the incoming wave direction. Notice that the most prominent peaks in scattering intensity are found at the values of k where a quasi-bound mode frequency is closest to the real axis. These are the long-lived resonances of the cavity. Also visible is the contribution of resonances with shorter lifetimes (higher values of Im [ k R ] ) to broader peaks and the scattering background. Calculations are for a dielectric cylinder with n = 1.5 and for k z = 0 .

Fig. 4
Fig. 4

TE-like resonances (gray circles) and TM-like resonances (black circles) for a cylinder with n = 2 , θ = 0.2 ; dashed line corresponds to the critical angle defined by γ 1 = m n , and the solid line corresponds to the Brewster’s condition γ 1 = m n 2 + 1 . Note that the TE or TM association does not work around the Brewster’s angle.

Fig. 5
Fig. 5

Exact calculation of the coefficient for the resonant shift (circle) TM-like and (cross) TE-like. The coefficient is expected to be unity for small sin χ .

Fig. 6
Fig. 6

(a) Schematic of wave front-matching argument. The internal spiral wave of a tilted optical fiber with respect to the incoming wave. The phase-matching condition between the spiral mode and the external incident wave reduces the effective cavity length for the spiral wave by a distance d n , where n is the refractive index. (b) The spiral quadratic blueshift can be interpreted by one’s unwrapping the circular fiber. The dashed lines indicate the wavefront. The wavefront-matched path is only 2 π a cos θ ; therefore the resonances are quadratic blueshifted with the tilt angle. The figures are adapted from Poon et al.[6]

Fig. 7
Fig. 7

Shift of the resonance condition with the quantum numbers j = 21 , m = 20 ( sin χ 0.33 ) , and n = 2 with respect to the internal tilt angle θ. Crosses are the numerical solutions following Eq. (32), and circles show the small θ expansion, following Eq. (29). Note the discontinuity in the exact numerical values around θ = 0.33 ; this is due to the onset of the PCA, which will be discussed in Section 7. This discontinuity occurs beyond the regime of the small θ expansion.

Fig. 8
Fig. 8

(a) Path of the first curve Γ 1 . (b) Second path of Γ 2 , of length L.

Fig. 9
Fig. 9

Schematics of scattering on the projected plane. We will expand the wave solution inside into an incoming component Ψ i and a reflected Ψ r . The outside transmitted component is given by Ψ t .

Fig. 10
Fig. 10

Functional dependence of f ( θ ) for n = 2 . Note that f ( θ ) diverges at the critical angle when sin θ = 1 n .

Fig. 11
Fig. 11

Absolute value (solid gray and black curves) and the phase divided scaled by π (dashed gray and black curves) of the eigenvalues of R for n = 2 and tan θ = 0.2 versus sin χ . The dotted black vertical line is the Brewster’s angle, and the black dashed line is the critical angle. Gray indicates the TE-like component, and black indicates the TM-like component.

Fig. 12
Fig. 12

Comparison of the eigenvalues of the rotated Jones matrix, absolute value (solid gray and black curves) and the phase (divided by π)— (dashed gray and black curves) with the eigenvalues of R (gray and black circles). Parameters of the calculation are tan θ = 0.2 and n = 2 .

Fig. 13
Fig. 13

(a) Absolute value of the two eigenvalues ν 1 , 2 of the rotated Jones matrix (solid gray and black curves). The phase of the eigenvalues (divided by π) is plotted in dashes. The eigenvalues become complex at the point where the two curves meet and join. This point lies between the Brewster’s angle (dashed vertical black line) and the effective critical angle (dotted vertical black line). Calculated for tan θ = 0.2 , n = 2 . (b) Black curve, the sine of the PCA at which the eigenvalue of R gets complex. Dotted curve, sine of the Brewster’s angle. Dashed curve, sine of the critical angle of TIR.

Fig. 14
Fig. 14

(a) Phase differences between the E z and the B z fields, for TE- and TM-like modes. (b) Absolute values of the ratio of the eigenvector components of R. For normal incidence sin χ = 0 , the modes are clearly either TE or TM; right at the PCA they completely mix. Calculations are done for n = 2 , tan θ = 0.2 . The solid vertical black line is the effective critical angle, and the dashed line is the effective Brewster’s angle.

Fig. 15
Fig. 15

Respective (a) phase differences and (b) absolute values of the ratio of the eigenvector components of R (gray solid curves) in the far field. Black circles are the exact numerical solutions following Eq. (22) ( m [ 0 , 50 ] and γ 1 < 50 ). For n = 2 , tan θ = 0.2 . The solid vertical black line is the effective critical angle, and the dashed line is the effective Brewster’s angle.

Tables (1)

Tables Icon

Table 1 Spiral Resonances of the Cylinder with n = 2 , θ = 0.1 , 0.2 a

Equations (62)

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

( 2 + n 2 k 2 ) { E ( x , y , z ) B ( x , y , z ) } = 0 ,
E ( x ) = E ( x , y ) exp ( i k z z ) , B ( x ) = B ( x , y ) exp ( i k z z ) ;
( 2 + γ 2 ) { E z ( x , y ) B z ( x , y ) } = 0 ,
E z 1 = E z 2 or t E z 1 = t E z 2 ,
B z 1 = B z 2 or t B z 1 = t B z 2 ,
k γ 1 2 n B z 1 k γ 2 2 n B z 2 = ( k z γ 1 2 k z γ 2 2 ) t E z 1 ,
n 1 2 k γ 1 2 n E z 1 n 2 2 k γ 2 2 n E z 2 = + ( k z γ 1 2 k z γ 2 2 ) t B z 1 .
B = i γ 2 n 2 k ( y E z x E z ) ,
E = i γ 2 k ( y B z x B z ) .
D = R ( z , ϕ ) z R , ϕ [ 0 , 2 π ] .
( E z < B z < ) = m = [ ( α m ξ m ) H m + ( γ 1 r ) + ( β m η m ) H m ( γ 1 r ) ] exp ( i m ϕ ) ,
( E z > B z > ) = m = [ ( v m ζ m ) H m + ( γ 2 r ) + ( δ m ϑ m ) H m ( γ 2 r ) ] exp ( i m ϕ ) ,
( E ρ E φ E z ) 2 ( k z γ 2 v m exp [ i m ( φ π 2 ) ] k γ 2 ζ m exp [ i m ( φ π 2 ) ] v m exp [ i m ( φ π 2 ) ] ) 2 = ( k z γ 2 E z k γ 2 B z E z ) 2 .
[ cos α 0 sin α 0 1 0 sin α 0 cos α ] ( tan α E z sec α B z E z ) = ( 0 sec α B z sec α E z ) .
J m J m ( γ 1 R ) , H m H m + ( γ 2 R ) ,
E z < ( r ; m , j ) = α m J m ( γ 1 m , j r ) exp ( i m φ ) r < R ,
B z < ( r ; m , j ) = ξ m J m ( γ 1 m , j r ) exp ( i m φ ) r < R ,
E z > ( r ; m , j ) = v m H m + ( γ 2 m , j r ) exp ( i m φ ) r > R ,
B z > ( r ; m , j ) = ζ m H m + ( γ 2 m , j r ) exp ( i m φ ) r > R .
v m = J m H m α m , ζ m = J m H m ξ m .
[ i m ( n n 3 ) sin θ J m H m cos 2 α H m ρ J m n 2 cos 2 θ J m ρ H m n 2 cos 2 α H m ρ J m n 2 cos 2 θ J m ρ H m i m ( n 3 n ) sin θ J m H m ] ( α m ξ m ) = 0 ,
( 1 n 2 ) 2 m 2 sin 2 θ = 1 J m H m [ cos 2 α H m ρ J m cos 2 θ J m ρ H m ] × 1 J m H m [ cos 2 α H m ρ J m n 2 cos 2 θ J m ρ H m ] G TM G TE ,
G TE = 1 J m H m [ cos 2 α H m ρ J m n 2 cos 2 θ J m ρ H m ] ,
G TM = 1 J m H m [ cos 2 α H m ρ J m cos 2 θ J m ρ H m ] .
0 = G TE G TM = [ H m ρ J m n 2 J m ρ H m ]
× [ H m ρ J m J m ρ H m ] .
ξ m = i m ( n n 3 ) sin θ n 2 G TE α m ,
α m = i m ( n 3 n ) sin θ n 2 G TM ξ m .
P = B z > E z > = ξ m α m = i m ( n n 3 ) sin θ n 2 G TE .
Δ k 0 k 0 = 1 2 α θ 2 ,
( E z B z ) = Ψ ( r ) = A 1 exp [ i γ S 1 ( r ) ] + A 2 exp [ i γ S 2 ( r ) ] ,
γ Γ i dq S = 2 π l i + Φ i i = 1 , 2 .
sin χ = m γ R ,
γ L = [ 2 cos χ 2 ( π 2 χ ) sin χ ] R = 2 π j + Φ 2 ,
Ψ m = ( E z , m B z , m ) exp ( i γ S m ) ,
n S i = i γ 1 cos χ , n S r = i γ 1 cos χ , n S t = i γ 2 cos σ ,
t S i = i γ 1 sin χ , t S r = i γ 1 sin χ , t S t = i γ 2 sin σ .
( E z B z ) i exp ( i γ 1 S i ) + ( E z B z ) r exp ( i γ 1 S r ) = ( E z B z ) t exp ( i γ 2 S t ) .
γ 1 S i = γ 1 S r = γ 2 S t .
γ 1 sin χ = γ 2 sin σ sin χ = γ 2 γ 1 sin σ .
n sin χ = sin σ .
sin σ = γ 1 γ 2 sin χ = sin α cos θ cos α sin θ sin χ = f ( θ ) n sin χ ,
f ( θ ) = cos θ ( 1 n 2 sin 2 θ ) 1 2 = ( 1 sin 2 θ 1 n 2 sin 2 θ ) 1 2 1 .
B i ( E z B z ) i exp ( i γ 1 S i ) + B r ( E z B z ) r exp ( i γ 1 S r ) = B t ( E z B z ) t exp ( i γ 2 S t ) ,
B ( i , r ) = [ ( n n 3 ) sin θ t cos 2 α n n 2 cos 2 α n ( n 3 n ) sin θ t ] ,
B t = [ 0 n 2 cos 2 θ n n 2 cos 2 θ n 0 ] .
Ψ r = R Ψ i ,
Ψ t = T Ψ i ,
R = ( B r B t ) 1 ( B t B i )
T = ( B t B r ) 1 ( B i B r ) .
R = [ n cos χ cos σ n cos χ + cos σ 0 0 cos χ n cos σ cos χ + n cos σ ] = ̂ R = [ r s 0 0 r p ] ,
T = [ 2 n cos χ n cos χ + cos σ 0 0 2 cos χ cos χ + n cos σ ] = ̂ T = [ t s 0 0 t p ] .
R a = Λ a .
2 γ ( 1 m 2 γ 2 ) 1 2 + 2 m arcsin ( m γ ) = 2 π j + π 2 + m π + ζ + i ln r ,
γ = π [ j + m 2 + 1 4 ] + f ( χ , θ ) ,
n k = n k o ( 1 + 1 2 θ 2 ) ,
E = ( E p E s ) = ( E 0 p exp ( i ϕ p ) E 0 s exp ( i ϕ s ) ) ,
J r = [ r p 0 0 r s ]
J t = [ t p 0 0 t s ]
I ( E p E s ) 1 , 2 = ν 1 , 2 ( E p E s ) 1 , 2 ,
ν 1 , 2 = 1 2 ( r s r p ) cos ξ ± 1 2 cos 2 ξ ( r s r p ) 2 + 4 r p r s .
R ( ϴ ) J t a ,

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