Abstract

We investigate near-paraxial modes of high-finesse, planoconcave microresonators without using the paraxial approximation. The goal is to develop an analytical approach which is able to incorporate not only the spatial shape of the resonator boundaries, but also the dependence of reflectivities on angle of incidence. It is shown that this can be achieved using the Born-Oppenheimer method, augmented by a local Bessel wave approximation. We discuss how this approach extends standard paraxial theory. It is found that the Gouy phase of paraxial theory, which is determined purely by ray-optics, is no longer the sole parameter governing transverse mode splittings. The additional determining factor is the sensitivity with which boundary reflection phases depend on incident angle.

© 2007 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science Books, 1986).
  2. T. Klaassen, A. Hoogeboom, M. P. van Exter, and J. P. Woerdman, "Gouy phase of nonparaxial eigenmodes in a folded resonator," J. Opt. Soc. Am. A 21, 1689-1692 (2004).
    [CrossRef]
  3. G. W. Forbes, "Using rays better. IV. Theory for refraction and reflection," J. Opt. Soc. Am. A 18, 2557-2564 (2001).
    [CrossRef]
  4. D. H. Foster and J. U. Nöckel, "Bragg-induced orbital angular-momentum mixing in paraxial high-finesse cavities," Opt. Lett. 29, 2788-2790 (2004).
    [CrossRef] [PubMed]
  5. A. Fox and Y. Li, "Resonant modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-488 (1969).
  6. H. Laabs and A. T. Friberg, "Nonparaxial eigenmodes of stable resonators," IEEE J. Quantum Electron. 35, 198-207 (1999).
    [CrossRef]
  7. D. H. Foster and J. U. Nöckel, "Spatial and polarization structure in micro-domes: effects of a Bragg mirror," in Resonators and Beam Control VII (A. V. Kudryashov and A. H. Paxton, eds.), Proc. SPIE 5333, 195-203 (2004). http://arxiv.org/abs/physics/0406131
  8. D. H. Foster and J. U. Nöckel, "Methods for 3-D vector microcavity problems involving a planar dielectric mirror," Opt. Commun. 234, 351-383 (2004).
    [CrossRef]
  9. M. Aziz, J. Pfeiffer, and P. Meissner, "Modal behaviour of passive, stable microcavities," Phys. Stat. Solidi A  188, 979-982 (2001).
    [CrossRef]
  10. S. Coyle, G. V. Prakash, J. J. Baumberg, M. Abdelsalem, and P. N. Bartlett, "Spherical micromirrors from templated self-assembly: Polarization rotation on the micron scale," Appl. Phys. Lett. 83, 767-769 (2003).
    [CrossRef]
  11. G. V. Prakash, L. Besombes, T. Kelf, J. J. Baumberg, P. N. Bartlett, and M. Abdelsalem, "Tunable resonant optical microcavities by self-assembled templating," Opt. Lett. 29, 1500-1502 (2004).
    [CrossRef]
  12. G. Cui, J. M. Hannigan, R. Loeckenhoff, F. M. Matinaga, M. G. Raymer, S. Bhongale, M. Holland, S. Mosor, S. Chatterjee, H. M. Gibbs, and G. Khitrova, "A hemispherical, high-solid-angle optical micro-cavity for cavity-qed studies," Opt. Express 14, 2289-2299 (2006).
    [CrossRef] [PubMed]
  13. A. Messiah, Quantum Mechanics (North Holland, John Wiley & Sons, 1966) Vol. 2.
  14. F. Laeri and J. U. Nöckel, "Nanoporous compound materials for optical applications - Microlasers and microresonators," in Handbook of Advanced Electronic and Photonic Materials, H. S. Nalwa, ed., 6, 103-151 (Academic Press, 2001).
  15. H. E. Tureci, H. G. L. Schwefel, and A. Douglas Stone, "Gaussian-optical approach to stable periodic orbit resonances of partially chaotic dielectric micro-cavities," Opt. Express 10, 752-776 (2002)
    [PubMed]
  16. J. U. Nöckel, G. Bourdon, E. L. Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, "Mode structure and ray dynamics of a parabolic dome microcavity," Phys. Rev. E 62, 8677-8699 (2000).
    [CrossRef]
  17. S. J. M. Habraken and G. Nienhuis, "Modes of a twisted optical cavity," Phys. Rev. A 75, 033819 (2007).
    [CrossRef]
  18. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, (Feshbach Publishing, LLC, 1981) Vol. 2.
  19. O. Zaitsev, R. Narevich, and R. E. Prange, "Quasiclassical Born-Oppenheimer approximations," Found. Phys. 31, 7 (2001).
    [CrossRef]
  20. D. H. Foster, PhD thesis, http://hdl.handle.net/1794/3778 (University of Oregon, 2006).
  21. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc, 1991).
    [CrossRef]
  22. V. M. Shalaev,"Optical negative-index metamaterials," Nature Photonics 1, 41-48 (2007).
    [CrossRef]
  23. G. Nienhuis and L. Allen, "Paraxial wave optics and harmonic oscillators," Phys. Rev. A 48, 656-665 (1993).
    [CrossRef] [PubMed]
  24. O. Steuernagel, "Equivalence between focused paraxial beams and the quantum harmonic oscillator," Am. J. Phys. 73, 625-629 (2005).
    [CrossRef]
  25. F. Laeri, G. Angelow, and T. Tschudi, "Designing resonators with large mode volume and high mode discrimination," Opt. Lett. 21, 1324-1327 (1996).
    [CrossRef] [PubMed]
  26. M. Achtenhagen, A. Hardy, and E. Kapon, "Three-dimensional analysis of mode discrimination in vertical-cavity surface-emitting lasers," Appl. Opt. 44, 2832-2838 (2005).
    [CrossRef] [PubMed]
  27. A. M. Sarangan and G. M Peake, "Enhancement of lateral mode discrimination in broad-area VCSELs using curved Bragg mirrors," J. Lightwave Technol. 22, 543-549 (2004).
    [CrossRef]
  28. T. Gentsy, K. Becker, I. Fischer, W. Elsässer C. Degen, P. Debernardi, and G. P. Bava, "Enhancement of lateral mode discrimination in broad-area VCSELs using curved Bragg mirrors," Phys. Rev. Lett. 94, 233901 (2005).

2007

S. J. M. Habraken and G. Nienhuis, "Modes of a twisted optical cavity," Phys. Rev. A 75, 033819 (2007).
[CrossRef]

V. M. Shalaev,"Optical negative-index metamaterials," Nature Photonics 1, 41-48 (2007).
[CrossRef]

2006

2005

M. Achtenhagen, A. Hardy, and E. Kapon, "Three-dimensional analysis of mode discrimination in vertical-cavity surface-emitting lasers," Appl. Opt. 44, 2832-2838 (2005).
[CrossRef] [PubMed]

O. Steuernagel, "Equivalence between focused paraxial beams and the quantum harmonic oscillator," Am. J. Phys. 73, 625-629 (2005).
[CrossRef]

T. Gentsy, K. Becker, I. Fischer, W. Elsässer C. Degen, P. Debernardi, and G. P. Bava, "Enhancement of lateral mode discrimination in broad-area VCSELs using curved Bragg mirrors," Phys. Rev. Lett. 94, 233901 (2005).

2004

2003

S. Coyle, G. V. Prakash, J. J. Baumberg, M. Abdelsalem, and P. N. Bartlett, "Spherical micromirrors from templated self-assembly: Polarization rotation on the micron scale," Appl. Phys. Lett. 83, 767-769 (2003).
[CrossRef]

2002

2001

G. W. Forbes, "Using rays better. IV. Theory for refraction and reflection," J. Opt. Soc. Am. A 18, 2557-2564 (2001).
[CrossRef]

O. Zaitsev, R. Narevich, and R. E. Prange, "Quasiclassical Born-Oppenheimer approximations," Found. Phys. 31, 7 (2001).
[CrossRef]

M. Aziz, J. Pfeiffer, and P. Meissner, "Modal behaviour of passive, stable microcavities," Phys. Stat. Solidi A  188, 979-982 (2001).
[CrossRef]

2000

J. U. Nöckel, G. Bourdon, E. L. Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, "Mode structure and ray dynamics of a parabolic dome microcavity," Phys. Rev. E 62, 8677-8699 (2000).
[CrossRef]

1999

H. Laabs and A. T. Friberg, "Nonparaxial eigenmodes of stable resonators," IEEE J. Quantum Electron. 35, 198-207 (1999).
[CrossRef]

1996

1993

G. Nienhuis and L. Allen, "Paraxial wave optics and harmonic oscillators," Phys. Rev. A 48, 656-665 (1993).
[CrossRef] [PubMed]

1969

A. Fox and Y. Li, "Resonant modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-488 (1969).

Abdelsalem, M.

G. V. Prakash, L. Besombes, T. Kelf, J. J. Baumberg, P. N. Bartlett, and M. Abdelsalem, "Tunable resonant optical microcavities by self-assembled templating," Opt. Lett. 29, 1500-1502 (2004).
[CrossRef]

S. Coyle, G. V. Prakash, J. J. Baumberg, M. Abdelsalem, and P. N. Bartlett, "Spherical micromirrors from templated self-assembly: Polarization rotation on the micron scale," Appl. Phys. Lett. 83, 767-769 (2003).
[CrossRef]

Abram, I.

J. U. Nöckel, G. Bourdon, E. L. Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, "Mode structure and ray dynamics of a parabolic dome microcavity," Phys. Rev. E 62, 8677-8699 (2000).
[CrossRef]

Achtenhagen, M.

Adams, R.

J. U. Nöckel, G. Bourdon, E. L. Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, "Mode structure and ray dynamics of a parabolic dome microcavity," Phys. Rev. E 62, 8677-8699 (2000).
[CrossRef]

Allen, L.

G. Nienhuis and L. Allen, "Paraxial wave optics and harmonic oscillators," Phys. Rev. A 48, 656-665 (1993).
[CrossRef] [PubMed]

Angelow, G.

Aziz, M.

M. Aziz, J. Pfeiffer, and P. Meissner, "Modal behaviour of passive, stable microcavities," Phys. Stat. Solidi A  188, 979-982 (2001).
[CrossRef]

Bartlett, P. N.

G. V. Prakash, L. Besombes, T. Kelf, J. J. Baumberg, P. N. Bartlett, and M. Abdelsalem, "Tunable resonant optical microcavities by self-assembled templating," Opt. Lett. 29, 1500-1502 (2004).
[CrossRef]

S. Coyle, G. V. Prakash, J. J. Baumberg, M. Abdelsalem, and P. N. Bartlett, "Spherical micromirrors from templated self-assembly: Polarization rotation on the micron scale," Appl. Phys. Lett. 83, 767-769 (2003).
[CrossRef]

Baumberg, J. J.

G. V. Prakash, L. Besombes, T. Kelf, J. J. Baumberg, P. N. Bartlett, and M. Abdelsalem, "Tunable resonant optical microcavities by self-assembled templating," Opt. Lett. 29, 1500-1502 (2004).
[CrossRef]

S. Coyle, G. V. Prakash, J. J. Baumberg, M. Abdelsalem, and P. N. Bartlett, "Spherical micromirrors from templated self-assembly: Polarization rotation on the micron scale," Appl. Phys. Lett. 83, 767-769 (2003).
[CrossRef]

Becker, K.

T. Gentsy, K. Becker, I. Fischer, W. Elsässer C. Degen, P. Debernardi, and G. P. Bava, "Enhancement of lateral mode discrimination in broad-area VCSELs using curved Bragg mirrors," Phys. Rev. Lett. 94, 233901 (2005).

Besombes, L.

Bhongale, S.

Bourdon, G.

J. U. Nöckel, G. Bourdon, E. L. Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, "Mode structure and ray dynamics of a parabolic dome microcavity," Phys. Rev. E 62, 8677-8699 (2000).
[CrossRef]

Chatterjee, S.

Coyle, S.

S. Coyle, G. V. Prakash, J. J. Baumberg, M. Abdelsalem, and P. N. Bartlett, "Spherical micromirrors from templated self-assembly: Polarization rotation on the micron scale," Appl. Phys. Lett. 83, 767-769 (2003).
[CrossRef]

Cui, G.

Douglas Stone, A.

Fischer, I.

T. Gentsy, K. Becker, I. Fischer, W. Elsässer C. Degen, P. Debernardi, and G. P. Bava, "Enhancement of lateral mode discrimination in broad-area VCSELs using curved Bragg mirrors," Phys. Rev. Lett. 94, 233901 (2005).

Forbes, G. W.

Foster, D. H.

D. H. Foster and J. U. Nöckel, "Bragg-induced orbital angular-momentum mixing in paraxial high-finesse cavities," Opt. Lett. 29, 2788-2790 (2004).
[CrossRef] [PubMed]

D. H. Foster and J. U. Nöckel, "Methods for 3-D vector microcavity problems involving a planar dielectric mirror," Opt. Commun. 234, 351-383 (2004).
[CrossRef]

Fox, A.

A. Fox and Y. Li, "Resonant modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-488 (1969).

Friberg, A. T.

H. Laabs and A. T. Friberg, "Nonparaxial eigenmodes of stable resonators," IEEE J. Quantum Electron. 35, 198-207 (1999).
[CrossRef]

Gentsy, T.

T. Gentsy, K. Becker, I. Fischer, W. Elsässer C. Degen, P. Debernardi, and G. P. Bava, "Enhancement of lateral mode discrimination in broad-area VCSELs using curved Bragg mirrors," Phys. Rev. Lett. 94, 233901 (2005).

Gibbs, H. M.

Habraken, S. J. M.

S. J. M. Habraken and G. Nienhuis, "Modes of a twisted optical cavity," Phys. Rev. A 75, 033819 (2007).
[CrossRef]

Hannigan, J. M.

Hardy, A.

Holland, M.

Hoogeboom, A.

Kapon, E.

Kelf, T.

Khitrova, G.

Klaassen, T.

Laabs, H.

H. Laabs and A. T. Friberg, "Nonparaxial eigenmodes of stable resonators," IEEE J. Quantum Electron. 35, 198-207 (1999).
[CrossRef]

Laeri, F.

Li, Y.

A. Fox and Y. Li, "Resonant modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-488 (1969).

Loeckenhoff, R.

Matinaga, F. M.

Meissner, P.

M. Aziz, J. Pfeiffer, and P. Meissner, "Modal behaviour of passive, stable microcavities," Phys. Stat. Solidi A  188, 979-982 (2001).
[CrossRef]

Moison, J.-M.

J. U. Nöckel, G. Bourdon, E. L. Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, "Mode structure and ray dynamics of a parabolic dome microcavity," Phys. Rev. E 62, 8677-8699 (2000).
[CrossRef]

Mosor, S.

Narevich, R.

O. Zaitsev, R. Narevich, and R. E. Prange, "Quasiclassical Born-Oppenheimer approximations," Found. Phys. 31, 7 (2001).
[CrossRef]

Nienhuis, G.

S. J. M. Habraken and G. Nienhuis, "Modes of a twisted optical cavity," Phys. Rev. A 75, 033819 (2007).
[CrossRef]

G. Nienhuis and L. Allen, "Paraxial wave optics and harmonic oscillators," Phys. Rev. A 48, 656-665 (1993).
[CrossRef] [PubMed]

Nöckel, J. U.

D. H. Foster and J. U. Nöckel, "Methods for 3-D vector microcavity problems involving a planar dielectric mirror," Opt. Commun. 234, 351-383 (2004).
[CrossRef]

D. H. Foster and J. U. Nöckel, "Bragg-induced orbital angular-momentum mixing in paraxial high-finesse cavities," Opt. Lett. 29, 2788-2790 (2004).
[CrossRef] [PubMed]

J. U. Nöckel, G. Bourdon, E. L. Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, "Mode structure and ray dynamics of a parabolic dome microcavity," Phys. Rev. E 62, 8677-8699 (2000).
[CrossRef]

Peake, G. M

Pfeiffer, J.

M. Aziz, J. Pfeiffer, and P. Meissner, "Modal behaviour of passive, stable microcavities," Phys. Stat. Solidi A  188, 979-982 (2001).
[CrossRef]

Prakash, G. V.

G. V. Prakash, L. Besombes, T. Kelf, J. J. Baumberg, P. N. Bartlett, and M. Abdelsalem, "Tunable resonant optical microcavities by self-assembled templating," Opt. Lett. 29, 1500-1502 (2004).
[CrossRef]

S. Coyle, G. V. Prakash, J. J. Baumberg, M. Abdelsalem, and P. N. Bartlett, "Spherical micromirrors from templated self-assembly: Polarization rotation on the micron scale," Appl. Phys. Lett. 83, 767-769 (2003).
[CrossRef]

Prange, R. E.

O. Zaitsev, R. Narevich, and R. E. Prange, "Quasiclassical Born-Oppenheimer approximations," Found. Phys. 31, 7 (2001).
[CrossRef]

Raymer, M. G.

Robert, I.

J. U. Nöckel, G. Bourdon, E. L. Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, "Mode structure and ray dynamics of a parabolic dome microcavity," Phys. Rev. E 62, 8677-8699 (2000).
[CrossRef]

Ru, E. L.

J. U. Nöckel, G. Bourdon, E. L. Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, "Mode structure and ray dynamics of a parabolic dome microcavity," Phys. Rev. E 62, 8677-8699 (2000).
[CrossRef]

Sarangan, A. M.

Schwefel, H. G. L.

Shalaev, V. M.

V. M. Shalaev,"Optical negative-index metamaterials," Nature Photonics 1, 41-48 (2007).
[CrossRef]

Steuernagel, O.

O. Steuernagel, "Equivalence between focused paraxial beams and the quantum harmonic oscillator," Am. J. Phys. 73, 625-629 (2005).
[CrossRef]

Tschudi, T.

Tureci, H. E.

van Exter, M. P.

Woerdman, J. P.

Zaitsev, O.

O. Zaitsev, R. Narevich, and R. E. Prange, "Quasiclassical Born-Oppenheimer approximations," Found. Phys. 31, 7 (2001).
[CrossRef]

Am. J. Phys.

O. Steuernagel, "Equivalence between focused paraxial beams and the quantum harmonic oscillator," Am. J. Phys. 73, 625-629 (2005).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

S. Coyle, G. V. Prakash, J. J. Baumberg, M. Abdelsalem, and P. N. Bartlett, "Spherical micromirrors from templated self-assembly: Polarization rotation on the micron scale," Appl. Phys. Lett. 83, 767-769 (2003).
[CrossRef]

Bell Syst. Tech. J.

A. Fox and Y. Li, "Resonant modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-488 (1969).

Found. Phys.

O. Zaitsev, R. Narevich, and R. E. Prange, "Quasiclassical Born-Oppenheimer approximations," Found. Phys. 31, 7 (2001).
[CrossRef]

IEEE J. Quantum Electron.

H. Laabs and A. T. Friberg, "Nonparaxial eigenmodes of stable resonators," IEEE J. Quantum Electron. 35, 198-207 (1999).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. A

Nature Photonics

V. M. Shalaev,"Optical negative-index metamaterials," Nature Photonics 1, 41-48 (2007).
[CrossRef]

Opt. Commun.

D. H. Foster and J. U. Nöckel, "Methods for 3-D vector microcavity problems involving a planar dielectric mirror," Opt. Commun. 234, 351-383 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

S. J. M. Habraken and G. Nienhuis, "Modes of a twisted optical cavity," Phys. Rev. A 75, 033819 (2007).
[CrossRef]

G. Nienhuis and L. Allen, "Paraxial wave optics and harmonic oscillators," Phys. Rev. A 48, 656-665 (1993).
[CrossRef] [PubMed]

Phys. Rev. E

J. U. Nöckel, G. Bourdon, E. L. Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, "Mode structure and ray dynamics of a parabolic dome microcavity," Phys. Rev. E 62, 8677-8699 (2000).
[CrossRef]

Phys. Rev. Lett.

T. Gentsy, K. Becker, I. Fischer, W. Elsässer C. Degen, P. Debernardi, and G. P. Bava, "Enhancement of lateral mode discrimination in broad-area VCSELs using curved Bragg mirrors," Phys. Rev. Lett. 94, 233901 (2005).

Phys. Stat. Solidi

M. Aziz, J. Pfeiffer, and P. Meissner, "Modal behaviour of passive, stable microcavities," Phys. Stat. Solidi A  188, 979-982 (2001).
[CrossRef]

Other

D. H. Foster and J. U. Nöckel, "Spatial and polarization structure in micro-domes: effects of a Bragg mirror," in Resonators and Beam Control VII (A. V. Kudryashov and A. H. Paxton, eds.), Proc. SPIE 5333, 195-203 (2004). http://arxiv.org/abs/physics/0406131

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, (Feshbach Publishing, LLC, 1981) Vol. 2.

D. H. Foster, PhD thesis, http://hdl.handle.net/1794/3778 (University of Oregon, 2006).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc, 1991).
[CrossRef]

A. E. Siegman, Lasers (University Science Books, 1986).

A. Messiah, Quantum Mechanics (North Holland, John Wiley & Sons, 1966) Vol. 2.

F. Laeri and J. U. Nöckel, "Nanoporous compound materials for optical applications - Microlasers and microresonators," in Handbook of Advanced Electronic and Photonic Materials, H. S. Nalwa, ed., 6, 103-151 (Academic Press, 2001).

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Figures (5)

Fig. 1.
Fig. 1.

(a) Sketch of the cavity geometry. The vertical axis is z, and the radial distance from the z axis is r. (b) Example for a paraxial mode (gray scale: intensity). The height function of the top mirror is z M, and its maximum is at z = h. The Born-Oppenheimer potential V BO, superimposed as a sketch, is responsible for the transverse confinement of the mode. Here, h = 3μm and kh = 37.5; the field is calculated using the method described in this paper.

Fig. 2.
Fig. 2.

Wave number k for four longitudinal modes, each with their lowest three transverse levels, as a function of cavity height h, as given by Eq. (10). In varying h, the longitudinal phase, +π/2 -φ/2, is adjusted such that a fundamental transverse mode exists at k = 12.5(μm)-1; this fixes the lowest mode in the plots to be a horizontal line. The grouping of curves into triplets corresponds to clustering of modes with the same longitudinal quantum number v, i.e., same phase +π/2 -φ/2. The transverse levels in each cluster are enumerated by the principal quantum number, N = 0, 1, 2. For mirror radius of curvature R = 45μm, the transverse mode spacings are seen to be much smaller than the longitudinal spacing, so that the displayed clusters do not intersect. For smaller radius of curvature, R = 10μm, the spacing of k 2 between transverse and longitudinal modes becomes comparable near h ≈ 17μm when R = 10μm. Note that for R = 45μm, the Born-Oppenheimer method is valid over the entire range of h shown.

Fig. 3.
Fig. 3.

Effective radial potential V eff(r) for the Laguerre-Gauss modes (solid lines, labeled by orbital angular momentum = 0, 1, 2). Shown as horizontal lines are three values of the effective energy K 2 for mode orders N = 0, 1, 2. Laguerre-Gaussians are superimposed with the same line style and at the same ordinate as the K 2 to which they correspond, and labeled by their -value.

Fig. 4.
Fig. 4.

Wave number spectra: (a), same as in Fig. 2(a), but for angle dependent reflection phase, with ε = -5. The Born-Oppenheimer separation of transverse and longitudinal splittings is still valid, but deviations from Fig. 2(a) are visible at small h. In (b), the phase coefficient ε is varied while keeping the resonator geometry fixed at h = 2μm and R = 45μm. In both plots, we adjust the constant part of the longitudinal phase following the same prescription as in Fig. 2. The three lowest transverse modes from part (a) shift and become more degenerate as ε increases.

Fig. 5.
Fig. 5.

Competition between (a) oscillator frequency, Ωv, and (b) the effective “zero of energy” of the harmonic Born-Oppenheimer potential, k z 2(0), as a function of the quadratic reflection phase coefficient ε. The cavity parameters are the same as in Fig. 4.

Equations (34)

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

2 ψ ( x ) + k 2 ψ = 2 ψ z 2 + 2 ψ ( x ) + k 2 ψ = 0 ,
0 = η ( z M ; x , y ) = e ik z z M + e e i k z z M k z = 1 z M x y ( + π 2 φ 2 )
( 2 x 2 + 2 y 2 + k 2 ) χ x y + V BO ( v ) x y χ x y = K 2 χ x y
K 2 k 2 k z 2 0 0 = k 2 1 h 2 ( + π 2 φ 2 ) 2
V BO ( v ) x y k z 2 x y k z 2 0 0 = ( 1 z M 2 x y 1 h 2 ) ( + π 2 φ 2 ) 2 ,
V BO ( v ) ( r ) = 1 2 Ω v 2 r 2 , Ω v = 2 h 3 R ( + π 2 φ 2 ) .
ψ p , ; v r ϕ = e iℓϕ R p , ; v ( r ) e iℓϕ 2 p ! π w v 2 ( p + ) ! exp ( r 2 w v 2 ) ( 2 r 2 w v 2 ) 2 L p ( 2 r 2 w v 2 )
w v = 2 3 4 Ω v = 2 h 3 R + π 2 φ 2
K N ; v 2 = 2 Ω v ( N + 1 )
θ p arctan λ π w v λ π w v = 2 k w v = Ω v k 2 1 4
k N ; v 2 = K N ; v 2 + hR 2 Ω v 2 = K N ; v 2 + k z 2 ( r = 0 ) .
k N ; v 1 h ( + π 2 φ 2 ) + 1 hR ( N + 1 ) = hR 2 Ω v + k G ( N + 1 )
k G = 1 hR
R p , ; v ( ρ w v ) = 0 ug p , ; v ( u ) J ( ) du
R p , ; v ( r ) exp ( r 2 w v 2 ) ( 2 r 2 w v 2 ) 2 L p ( 2 r 2 w v 2 ) ,
S p , ; v ( u ) = C exp ( u 2 ) ( 2 u 2 ) 2 L p ( 2 u 2 )
R p , , v ( r ) e i k z ( r ) z e ilϕ = 0 π 2 sin θg ( θ ) J ( kr sin θ ) e ikz cos θ e ilϕ
g p , , v ( θ ) = CR p , , v ( w v θ θ p )
z M ( r ) R p , , v ( r ) = 0 π 2 sin θ g ( θ ) J ( kr sin θ ) × exp ( i 2 z M ( r ) ( k cos θ k z ( r ) ) ) sin [ 1 2 ( k cos θ k z ( r ) ) z M ( r ) ] 1 2 ( k cos θ k z ( r ) )
k cos θ k z ( r ) ,
R p , , v ( r ) 0 θ g ( θ ) J ( kr θ )
cos θ ( r ) = k z ( r ) k .
θ ( r ) = arcsin k 2 k z 2 ( r ) k 2 θ p N + 1 r 2 w v 2
r outer = w v N + 1 = 2 r rms
r max min = w v N + 1 2 1 ± 1 ( N + 1 ) 2
r min r rms r max r outer ,
θ max min = ( θ p w v ) r max r min .
φ ( θ ) φ 0 + 2 ε θ 2 + O ( θ 4 ) .
α + π 2 φ 0 2 = const
k z 2 ( r ) 1 h 2 ( 1 + r 2 hR ) ( α 2 ε ( 1 k z 2 ( r ) k 2 ) ) 2
k z 2 ( r ) = ( α ε ) 2 h 2 cos 4 γ 2 { 1 + r 2 hR cos γ }
cos γ h 2 k 2 4 ε ( α ε ) hk .
Ω v , ε 2 ( k ) = [ 2 r 2 k z 2 r k ] r = 0 = 2 ( α ε ) 2 h 3 R cos γ cos 4 γ 2 .
2 Ω v , ε 2 ( k ) ( N + 1 ) = k 2 k z 2 ( 0 ) = k 2 ( α ε ) 2 h 2 cos 4 γ 2 ,

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