Abstract

Numerical calculation of vector electromagnetic modes of plano–concave microcavities reveals that the polarization-dependent reflectivity of a flat Bragg mirror can lead to unexpected cavity field distribution for nominally paraxial modes. Even in a rotationally symmetric resonator, certain pairs of orbital angular momenta are necessarily mixed in an excitation-independent way to form doublets. A characteristic mixing angle is identified, which even in the paraxial limit can be designed to have large values. This correction to Gaussian theory is of zeroth order in deviations from paraxiality. We discuss the resultant nonuniform polarization fields. Observation will require small cavities with sufficiently high Q. Possible applications are proposed.

© 2004 Optical Society of America

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References

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  1. M. Padgett, J. Courtial, and L. Allen, Phys. Today 57(5), 35 (2004).
    [CrossRef]
  2. D. G. Grier, Nature 424, 810 (2003).
    [CrossRef] [PubMed]
  3. T. M. Stace, G. J. Milburn, and C. H. W. Barnes, Phys. Rev. B 67, 085317 (2003).
    [CrossRef]
  4. D. H. Foster and J. U. Nöckel, Opt. Commun. 234, 351 (2004).
    [CrossRef]
  5. D. H. Foster and J. U. Nöckel, Proc. SPIE 5333, 195 (2004).
    [CrossRef]
  6. V. M. Babic and V. S. Buldyrev, Short-Wavelength Diffraction Theory (Springer-Verlag, Berlin, 1972).
  7. J. U. Nöckel and D. H. Foster are preparing a manuscript entitled “Coupling of optical Born–Oppenheimer modes in near-paraxial microcavities.”
  8. E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901 (2003).
    [CrossRef]

2004 (3)

M. Padgett, J. Courtial, and L. Allen, Phys. Today 57(5), 35 (2004).
[CrossRef]

D. H. Foster and J. U. Nöckel, Opt. Commun. 234, 351 (2004).
[CrossRef]

D. H. Foster and J. U. Nöckel, Proc. SPIE 5333, 195 (2004).
[CrossRef]

2003 (3)

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef]

D. G. Grier, Nature 424, 810 (2003).
[CrossRef] [PubMed]

T. M. Stace, G. J. Milburn, and C. H. W. Barnes, Phys. Rev. B 67, 085317 (2003).
[CrossRef]

Allen, L.

M. Padgett, J. Courtial, and L. Allen, Phys. Today 57(5), 35 (2004).
[CrossRef]

Babic, V. M.

V. M. Babic and V. S. Buldyrev, Short-Wavelength Diffraction Theory (Springer-Verlag, Berlin, 1972).

Barnes, C. H. W.

T. M. Stace, G. J. Milburn, and C. H. W. Barnes, Phys. Rev. B 67, 085317 (2003).
[CrossRef]

Buldyrev, V. S.

V. M. Babic and V. S. Buldyrev, Short-Wavelength Diffraction Theory (Springer-Verlag, Berlin, 1972).

Courtial, J.

M. Padgett, J. Courtial, and L. Allen, Phys. Today 57(5), 35 (2004).
[CrossRef]

Crawford, P. R.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef]

Foster, D. H.

D. H. Foster and J. U. Nöckel, Opt. Commun. 234, 351 (2004).
[CrossRef]

D. H. Foster and J. U. Nöckel, Proc. SPIE 5333, 195 (2004).
[CrossRef]

J. U. Nöckel and D. H. Foster are preparing a manuscript entitled “Coupling of optical Born–Oppenheimer modes in near-paraxial microcavities.”

Galvez, E. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef]

Grier, D. G.

D. G. Grier, Nature 424, 810 (2003).
[CrossRef] [PubMed]

Haglin, P. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef]

Milburn, G. J.

T. M. Stace, G. J. Milburn, and C. H. W. Barnes, Phys. Rev. B 67, 085317 (2003).
[CrossRef]

Nöckel, J. U.

D. H. Foster and J. U. Nöckel, Proc. SPIE 5333, 195 (2004).
[CrossRef]

D. H. Foster and J. U. Nöckel, Opt. Commun. 234, 351 (2004).
[CrossRef]

J. U. Nöckel and D. H. Foster are preparing a manuscript entitled “Coupling of optical Born–Oppenheimer modes in near-paraxial microcavities.”

Padgett, M.

M. Padgett, J. Courtial, and L. Allen, Phys. Today 57(5), 35 (2004).
[CrossRef]

Pysher, M. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef]

Stace, T. M.

T. M. Stace, G. J. Milburn, and C. H. W. Barnes, Phys. Rev. B 67, 085317 (2003).
[CrossRef]

Sztul, H. I.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef]

Williams, R. E.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef]

Nature (1)

D. G. Grier, Nature 424, 810 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

D. H. Foster and J. U. Nöckel, Opt. Commun. 234, 351 (2004).
[CrossRef]

Phys. Rev. B (1)

T. M. Stace, G. J. Milburn, and C. H. W. Barnes, Phys. Rev. B 67, 085317 (2003).
[CrossRef]

Phys. Rev. Lett. (1)

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef]

Phys. Today (1)

M. Padgett, J. Courtial, and L. Allen, Phys. Today 57(5), 35 (2004).
[CrossRef]

Proc. SPIE (1)

D. H. Foster and J. U. Nöckel, Proc. SPIE 5333, 195 (2004).
[CrossRef]

Other (2)

V. M. Babic and V. S. Buldyrev, Short-Wavelength Diffraction Theory (Springer-Verlag, Berlin, 1972).

J. U. Nöckel and D. H. Foster are preparing a manuscript entitled “Coupling of optical Born–Oppenheimer modes in near-paraxial microcavities.”

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic cross section of the cavity. The dome is metal; the base is either metal or a Bragg mirror. Radius of curvature R is ten times larger than length L. (b) Polarization-dependent reflectivity phases ϕs/pargrs/p of the mirrors used in the calculation, versus angle of incidence θ. (c) Index profiles of the quarter-wave Bragg mirrors used in Figs. 2 and 3 below. Each has 36 pairs of layers of indices 3.0 and 3.5.

Fig. 2
Fig. 2

Mixing angle α for numerically calculated vector cavity modes versus (a) relative detuning between mean wavelength λ of doublet C, D and Bragg design wavelength λd and (b) ϕs-ϕp. Line fits use the function given in the text. Cavity dimensions are L=10 µm and R=100 µm.

Fig. 3
Fig. 3

Cross-sectional electric field of C and D for the point marked by the arrow in Fig. 2(a) (mirror design 2; λ=401.1 nm; divergence angle, θ3.6°). The ellipses show the xy component of the field; white or black filling indicates the sense of rotation. Each polarization ellipse is decorated with a line pointing along the instantaneous field vector to indicate the phase. Here the mixing angle is α=39°, and ωD-ωC=2π×3.4 GHz. Inset, Ex in the xz plane.

Equations (3)

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LGplρ,ϕ2ρwlLpl2ρ2w2exp-ρ2/w2expilϕ,
ExEy=p,lLGplρ,ϕAp,lσˆ++Bp,lσˆ-,
CD=cos α-sin αsin αcos αAB,

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