Abstract

We show that when a narrow beam is incident upon a dielectric interface near the critical angle for total internal reflection it will be transmitted into the far field with an angular deflection from the direction predicted by Snell’s law, because of a phenomenon that we call “Fresnel filtering.” This effect can be quite large for the parameter range that is relevant to dielectric microcavity lasers.

© 2002 Optical Society of America

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References

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  1. For examples and references, see R. K. Chang and A. J. Campillo, eds., Optical Processes in Microcavities, Vol. 3 of Advanced Series in Applied Physics (World Scientific, Singapore, 1996).
  2. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1493 (1998).
    [CrossRef]
  3. S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
    [CrossRef]
  4. S.-C. Chang, R. K. Chang, A. D. Stone, and J. U. Nöckel, J. Opt. Soc. Am. B 17, 1828 (2000).
    [CrossRef]
  5. N. B. Rex, R. K. Chang, and L. J. Guido, Proc. SPIE 3930, 163 (2000).
    [CrossRef]
  6. A. D. Stone, Phys. Scr. T 90, 248 (2001).
    [CrossRef]
  7. N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, http://xxx.lanl.gov/abs/physics/0105089 .
  8. A. W. Poon, F. Courvoisier, and R. K. Chang, Opt. Lett. 26, 632 (2001).
    [CrossRef]
  9. I. Braun, G. Ihlein, F. Laeri, J. U. Nockel, G. Schulz-Ekloff, F. Schuth, U. Vietze, O. Weiss, and D. Wohrle, Appl. Phys. B 70, 335 (2000).
    [CrossRef]
  10. J. U. Nöckel and A. D. Stone, Nature 385, 45 (1997).
    [CrossRef]
  11. T. Tamir, J. Opt. Soc. Am. A 3, 558 (1986).
    [CrossRef]
  12. J. W. Ra, H. L. Bertoni, and L. B. Felsen, SIAM J. Appl. Math. 24, 396 (1973).
    [CrossRef]
  13. Y. M. M. Antar, Can. J. Phys. 55, 2023 (1977).
    [CrossRef]
  14. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
    [CrossRef]
  15. B. Dietz and U. Smilansky, Chaos 3, 581 (1993).
    [CrossRef] [PubMed]

2001 (2)

2000 (4)

S.-C. Chang, R. K. Chang, A. D. Stone, and J. U. Nöckel, J. Opt. Soc. Am. B 17, 1828 (2000).
[CrossRef]

I. Braun, G. Ihlein, F. Laeri, J. U. Nockel, G. Schulz-Ekloff, F. Schuth, U. Vietze, O. Weiss, and D. Wohrle, Appl. Phys. B 70, 335 (2000).
[CrossRef]

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
[CrossRef]

N. B. Rex, R. K. Chang, and L. J. Guido, Proc. SPIE 3930, 163 (2000).
[CrossRef]

1998 (1)

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1493 (1998).
[CrossRef]

1997 (1)

J. U. Nöckel and A. D. Stone, Nature 385, 45 (1997).
[CrossRef]

1993 (1)

B. Dietz and U. Smilansky, Chaos 3, 581 (1993).
[CrossRef] [PubMed]

1986 (1)

1977 (1)

Y. M. M. Antar, Can. J. Phys. 55, 2023 (1977).
[CrossRef]

1973 (1)

J. W. Ra, H. L. Bertoni, and L. B. Felsen, SIAM J. Appl. Math. 24, 396 (1973).
[CrossRef]

Antar, Y. M. M.

Y. M. M. Antar, Can. J. Phys. 55, 2023 (1977).
[CrossRef]

Bertoni, H. L.

J. W. Ra, H. L. Bertoni, and L. B. Felsen, SIAM J. Appl. Math. 24, 396 (1973).
[CrossRef]

Braun, I.

I. Braun, G. Ihlein, F. Laeri, J. U. Nockel, G. Schulz-Ekloff, F. Schuth, U. Vietze, O. Weiss, and D. Wohrle, Appl. Phys. B 70, 335 (2000).
[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, Science 280, 1493 (1998).
[CrossRef]

Chang, R. K.

Chang, S.-C.

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, Science 280, 1493 (1998).
[CrossRef]

Courvoisier, F.

Dietz, B.

B. Dietz and U. Smilansky, Chaos 3, 581 (1993).
[CrossRef] [PubMed]

Faist, J.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1493 (1998).
[CrossRef]

Felsen, L. B.

J. W. Ra, H. L. Bertoni, and L. B. Felsen, SIAM J. Appl. Math. 24, 396 (1973).
[CrossRef]

Gianordoli, S.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
[CrossRef]

Gmachl, C.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1493 (1998).
[CrossRef]

Gornik, E.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
[CrossRef]

Guido, L. J.

N. B. Rex, R. K. Chang, and L. J. Guido, Proc. SPIE 3930, 163 (2000).
[CrossRef]

Hvozdara, L.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
[CrossRef]

Ihlein, G.

I. Braun, G. Ihlein, F. Laeri, J. U. Nockel, G. Schulz-Ekloff, F. Schuth, U. Vietze, O. Weiss, and D. Wohrle, Appl. Phys. B 70, 335 (2000).
[CrossRef]

Laeri, F.

I. Braun, G. Ihlein, F. Laeri, J. U. Nockel, G. Schulz-Ekloff, F. Schuth, U. Vietze, O. Weiss, and D. Wohrle, Appl. Phys. B 70, 335 (2000).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
[CrossRef]

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, Science 280, 1493 (1998).
[CrossRef]

Nockel, J. U.

I. Braun, G. Ihlein, F. Laeri, J. U. Nockel, G. Schulz-Ekloff, F. Schuth, U. Vietze, O. Weiss, and D. Wohrle, Appl. Phys. B 70, 335 (2000).
[CrossRef]

Nöckel, J. U.

S.-C. Chang, R. K. Chang, A. D. Stone, and J. U. Nöckel, J. Opt. Soc. Am. B 17, 1828 (2000).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1493 (1998).
[CrossRef]

J. U. Nöckel and A. D. Stone, Nature 385, 45 (1997).
[CrossRef]

Poon, A. W.

Ra, J. W.

J. W. Ra, H. L. Bertoni, and L. B. Felsen, SIAM J. Appl. Math. 24, 396 (1973).
[CrossRef]

Rex, N. B.

N. B. Rex, R. K. Chang, and L. J. Guido, Proc. SPIE 3930, 163 (2000).
[CrossRef]

Schrenk, W.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
[CrossRef]

Schulz-Ekloff, G.

I. Braun, G. Ihlein, F. Laeri, J. U. Nockel, G. Schulz-Ekloff, F. Schuth, U. Vietze, O. Weiss, and D. Wohrle, Appl. Phys. B 70, 335 (2000).
[CrossRef]

Schuth, F.

I. Braun, G. Ihlein, F. Laeri, J. U. Nockel, G. Schulz-Ekloff, F. Schuth, U. Vietze, O. Weiss, and D. Wohrle, Appl. Phys. B 70, 335 (2000).
[CrossRef]

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, Science 280, 1493 (1998).
[CrossRef]

Smilansky, U.

B. Dietz and U. Smilansky, Chaos 3, 581 (1993).
[CrossRef] [PubMed]

Stone, A. D.

A. D. Stone, Phys. Scr. T 90, 248 (2001).
[CrossRef]

S.-C. Chang, R. K. Chang, A. D. Stone, and J. U. Nöckel, J. Opt. Soc. Am. B 17, 1828 (2000).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1493 (1998).
[CrossRef]

J. U. Nöckel and A. D. Stone, Nature 385, 45 (1997).
[CrossRef]

Strasser, G.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
[CrossRef]

Tamir, T.

Vietze, U.

I. Braun, G. Ihlein, F. Laeri, J. U. Nockel, G. Schulz-Ekloff, F. Schuth, U. Vietze, O. Weiss, and D. Wohrle, Appl. Phys. B 70, 335 (2000).
[CrossRef]

Weiss, O.

I. Braun, G. Ihlein, F. Laeri, J. U. Nockel, G. Schulz-Ekloff, F. Schuth, U. Vietze, O. Weiss, and D. Wohrle, Appl. Phys. B 70, 335 (2000).
[CrossRef]

Wohrle, D.

I. Braun, G. Ihlein, F. Laeri, J. U. Nockel, G. Schulz-Ekloff, F. Schuth, U. Vietze, O. Weiss, and D. Wohrle, Appl. Phys. B 70, 335 (2000).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
[CrossRef]

Appl. Phys. B (1)

I. Braun, G. Ihlein, F. Laeri, J. U. Nockel, G. Schulz-Ekloff, F. Schuth, U. Vietze, O. Weiss, and D. Wohrle, Appl. Phys. B 70, 335 (2000).
[CrossRef]

Can. J. Phys. (1)

Y. M. M. Antar, Can. J. Phys. 55, 2023 (1977).
[CrossRef]

Chaos (1)

B. Dietz and U. Smilansky, Chaos 3, 581 (1993).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
[CrossRef]

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

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

Nature (1)

J. U. Nöckel and A. D. Stone, Nature 385, 45 (1997).
[CrossRef]

Opt. Lett. (1)

Phys. Scr. T (1)

A. D. Stone, Phys. Scr. T 90, 248 (2001).
[CrossRef]

Proc. SPIE (1)

N. B. Rex, R. K. Chang, and L. J. Guido, Proc. SPIE 3930, 163 (2000).
[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, Science 280, 1493 (1998).
[CrossRef]

SIAM J. Appl. Math. (1)

J. W. Ra, H. L. Bertoni, and L. B. Felsen, SIAM J. Appl. Math. 24, 396 (1973).
[CrossRef]

Other (3)

For examples and references, see R. K. Chang and A. J. Campillo, eds., Optical Processes in Microcavities, Vol. 3 of Advanced Series in Applied Physics (World Scientific, Singapore, 1996).

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, http://xxx.lanl.gov/abs/physics/0105089 .

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Angular far-field intensity distributions Iϕ=Eeϕ2 for (dotted curve) critical incidence on a planar interface with n=1.56, Δ=8.82, z0=5, with the Gaussian model [(GM) Eq. (4)]. Solid curve, exact quasi-normal mode with diamond geometry at nk0r090, for a quadrupolar ARC with =0.1 and n=1.56. Dotted–dashed curve, chiral version of diamond resonance (see text). Inset, coordinates and variables for the GM calculation.

Fig. 2
Fig. 2

(a) Field-intensity plot (gray scale) for a diamond resonance of the quadrupole at critical incidence for the points at ϕw=0, π, calculated numerically at nk0r090, n=1.56, and =0.1. Note that there is negligible emission from the upper and lower bounce points at ϕw=±90° because they are above the critical angle. (b) Chiral counterpart of this exact resonance.

Fig. 3
Fig. 3

Comparison of peak angular far-field values ϕmax for varying critical angle θc=sin-11/n. Diamonds, exact resonances at nk0r090; solid curve, GM calculation with Δ8.82; dashed curve, Snell’s law prediction: sin ϕmax=n sin θi, where θi39°. ΔθFFc, deviation from Snell’s law at θc=θi.

Equations (6)

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

Eiαxi,zi=E0Δ2π-ds exp-Δ22s2+iΔsxi+czi,
Eeαρ,ϕ=E0Δ2π-dsTαsGs×expiΔnρ cosϕ-θe-Δθe.
Gs=exp-Δ22s2+iΔ1-s2z0.
Eeαϕ=E0Δ2iΔnρ1-s02cos ϕn2-sin2ϕ×Tαs0Gs0expiΔnρ,
Tαs0ϕ=2nn2-sin2ϕμn2-sin2ϕ+n21-sin2ϕ.
ΔθFFc2/tan θc1/2Δ-1/2,

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