Abstract

We study the output from the modes described by the superposition of Gaussian beams confined in the quasi- stadium microcavities. We experimentally observe the deviation from Snell’s law in the output when the incident angle of the Gaussian beam at the cavity interface is near the critical angle for total internal reflection, providing direct experimental evidence on the Fresnel filtering. The theory of the Fresnel filtering for a planar interface qualitatively reproduces experimental data, and a discussion is given on small deviation between the measured data and the theory.

© 2011 Optical Society of America

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References

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  1. H. E. Tureci and A. D. Stone, Opt. Lett. 27, 7 (2002).
    [CrossRef]
  2. H. E. Tureci, “Wave chaos in dielectric resonators: asymptotic and numerical approaches,” Ph.D. thesis (Yale University, 2003), downloadable from http://www.eng.yale.edu/stonegroup/publications/HakanThesis.pdf.
  3. F. Goos and H. Hänchen, Ann. Phys. 436, 333 (1947).
    [CrossRef]
  4. N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, Phys. Rev. Lett. 88, 094102 (2002).
    [CrossRef] [PubMed]
  5. S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
    [CrossRef] [PubMed]
  6. H. Schomerus and M. Hentschel, Phys. Rev. Lett. 96, 243903 (2006).
    [CrossRef] [PubMed]
  7. E. G. Altmann, G. Del Magno, and M. Hentschel, Europhys. Lett. 84, 10008 (2008).
    [CrossRef]
  8. J. Unterhinninghofen and J. Wiersig, Phys. Rev. E 82, 026202 (2010).
    [CrossRef]
  9. Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, W. Fang, and G. S. Solomon, Phys. Rev. Lett. 105, 103902 (2010).
    [CrossRef] [PubMed]
  10. H. E. Tureci, H. G. L. Schwefel, A. D. Stone, and E. E. Narimanov, Opt. Express 10, 752 (2002).
    [PubMed]
  11. T. Fukushima and T. Harayama, IEEE J. Sel. Top. Quantum Electron. 10, 1039 (2004).
    [CrossRef]

2010 (2)

J. Unterhinninghofen and J. Wiersig, Phys. Rev. E 82, 026202 (2010).
[CrossRef]

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, W. Fang, and G. S. Solomon, Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef] [PubMed]

2008 (1)

E. G. Altmann, G. Del Magno, and M. Hentschel, Europhys. Lett. 84, 10008 (2008).
[CrossRef]

2006 (1)

H. Schomerus and M. Hentschel, Phys. Rev. Lett. 96, 243903 (2006).
[CrossRef] [PubMed]

2004 (2)

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

T. Fukushima and T. Harayama, IEEE J. Sel. Top. Quantum Electron. 10, 1039 (2004).
[CrossRef]

2002 (3)

1947 (1)

F. Goos and H. Hänchen, Ann. Phys. 436, 333 (1947).
[CrossRef]

Altmann, E. G.

E. G. Altmann, G. Del Magno, and M. Hentschel, Europhys. Lett. 84, 10008 (2008).
[CrossRef]

Cao, H.

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, W. Fang, and G. S. Solomon, Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef] [PubMed]

Chang, R. K.

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, Phys. Rev. Lett. 88, 094102 (2002).
[CrossRef] [PubMed]

Choi, M.

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

Fang, W.

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, W. Fang, and G. S. Solomon, Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef] [PubMed]

Fukushima, T.

T. Fukushima and T. Harayama, IEEE J. Sel. Top. Quantum Electron. 10, 1039 (2004).
[CrossRef]

Ge, L.

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, W. Fang, and G. S. Solomon, Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef] [PubMed]

Goos, F.

F. Goos and H. Hänchen, Ann. Phys. 436, 333 (1947).
[CrossRef]

Hänchen, H.

F. Goos and H. Hänchen, Ann. Phys. 436, 333 (1947).
[CrossRef]

Harayama, T.

T. Fukushima and T. Harayama, IEEE J. Sel. Top. Quantum Electron. 10, 1039 (2004).
[CrossRef]

Hentschel, M.

E. G. Altmann, G. Del Magno, and M. Hentschel, Europhys. Lett. 84, 10008 (2008).
[CrossRef]

H. Schomerus and M. Hentschel, Phys. Rev. Lett. 96, 243903 (2006).
[CrossRef] [PubMed]

Kim, C.-M.

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

Kwon, T.-Y.

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

Lee, S.-Y.

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

Magno, G. Del

E. G. Altmann, G. Del Magno, and M. Hentschel, Europhys. Lett. 84, 10008 (2008).
[CrossRef]

Narimanov, E. E.

Rex, N. B.

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, Phys. Rev. Lett. 88, 094102 (2002).
[CrossRef] [PubMed]

Rim, S.

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

Ryu, J.-W.

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

Schomerus, H.

H. Schomerus and M. Hentschel, Phys. Rev. Lett. 96, 243903 (2006).
[CrossRef] [PubMed]

Schwefel, H. G. L.

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, Phys. Rev. Lett. 88, 094102 (2002).
[CrossRef] [PubMed]

H. E. Tureci, H. G. L. Schwefel, A. D. Stone, and E. E. Narimanov, Opt. Express 10, 752 (2002).
[PubMed]

Shim, J.-B.

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, W. Fang, and G. S. Solomon, Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef] [PubMed]

Solomon, G. S.

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, W. Fang, and G. S. Solomon, Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef] [PubMed]

Song, Q. H.

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, W. Fang, and G. S. Solomon, Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef] [PubMed]

Stone, A. D.

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, W. Fang, and G. S. Solomon, Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef] [PubMed]

H. E. Tureci, H. G. L. Schwefel, A. D. Stone, and E. E. Narimanov, Opt. Express 10, 752 (2002).
[PubMed]

H. E. Tureci and A. D. Stone, Opt. Lett. 27, 7 (2002).
[CrossRef]

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, Phys. Rev. Lett. 88, 094102 (2002).
[CrossRef] [PubMed]

Tureci, H. E.

H. E. Tureci and A. D. Stone, Opt. Lett. 27, 7 (2002).
[CrossRef]

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, Phys. Rev. Lett. 88, 094102 (2002).
[CrossRef] [PubMed]

H. E. Tureci, H. G. L. Schwefel, A. D. Stone, and E. E. Narimanov, Opt. Express 10, 752 (2002).
[PubMed]

H. E. Tureci, “Wave chaos in dielectric resonators: asymptotic and numerical approaches,” Ph.D. thesis (Yale University, 2003), downloadable from http://www.eng.yale.edu/stonegroup/publications/HakanThesis.pdf.

Unterhinninghofen, J.

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, W. Fang, and G. S. Solomon, Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef] [PubMed]

J. Unterhinninghofen and J. Wiersig, Phys. Rev. E 82, 026202 (2010).
[CrossRef]

Wiersig, J.

J. Unterhinninghofen and J. Wiersig, Phys. Rev. E 82, 026202 (2010).
[CrossRef]

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, W. Fang, and G. S. Solomon, Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef] [PubMed]

Ann. Phys. (1)

F. Goos and H. Hänchen, Ann. Phys. 436, 333 (1947).
[CrossRef]

Europhys. Lett. (1)

E. G. Altmann, G. Del Magno, and M. Hentschel, Europhys. Lett. 84, 10008 (2008).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

T. Fukushima and T. Harayama, IEEE J. Sel. Top. Quantum Electron. 10, 1039 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. E (1)

J. Unterhinninghofen and J. Wiersig, Phys. Rev. E 82, 026202 (2010).
[CrossRef]

Phys. Rev. Lett. (4)

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, W. Fang, and G. S. Solomon, Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef] [PubMed]

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, Phys. Rev. Lett. 88, 094102 (2002).
[CrossRef] [PubMed]

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

H. Schomerus and M. Hentschel, Phys. Rev. Lett. 96, 243903 (2006).
[CrossRef] [PubMed]

Other (1)

H. E. Tureci, “Wave chaos in dielectric resonators: asymptotic and numerical approaches,” Ph.D. thesis (Yale University, 2003), downloadable from http://www.eng.yale.edu/stonegroup/publications/HakanThesis.pdf.

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Figures (2)

Fig. 1
Fig. 1

(a) Geometry of the quasi-stadium cavity with L = 600 μm and W = 150 μm . The radius R of curvature of both curved edges is 600 μm . Inside the cavity, the ring orbit reflected at the boundary points A, B, C, and D is shown. (b) Resonant mode associated with the ring orbit with the wavelength λ 0.856 μm calculated by the Gaussian-optic theory.

Fig. 2
Fig. 2

Far-field emission patterns of the quasi-stadium microlasers with (a)  W = 100 μm , (b)  W = 150 μm , (c)  W = 190 μm , and (d)  W = 200 μm . The definition of the far-field angle ϕ is given in the inset of (d). Experimental data are plotted with solid curves, while the predictions by the FF theory in dotted curves. The predictions by Snell’s law are indicated by vertical lines.

Equations (6)

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

θ i = sin 1 ( W L 2 + W 2 ) .
E i ( x i , z i ) = E 0 w 0 w ( z i ) exp [ ( x i w ( z i ) ) 2 + i n k z i ] ,
E e ( ρ , ϕ ) n k w 0 E 0 2 i k ρ t ( s 0 ) G ( s 0 ) J ( ϕ , s 0 ) e i k ρ ,
t ( s ) = 2 n cos ( Θ i ( s ) ) cos ( Θ i ( s ) ) + n 2 sin 2 θ c sin 2 ( Θ i ( s ) ) ,
G ( s ) = exp [ ( n k w 0 2 ) 2 s 2 + i n k z 0 1 s 2 ] ,
J ( ϕ , s ) = cos ϕ 1 s 2 n 2 sin 2 ϕ ,

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