Abstract

By combining detailed imaging measurements at different tilt angles with simulations of ray emission from prolate-deformed lasing microdroplets, we conclude that the dominant contribution to the laser emission of such three-dimensional dielectric microcavities must come from modes associated with the chaotic region of the ray phase space. As a particularly striking signature, maximum emission from such chaotic lasing modes is not from tangent rays emerging from the highest curvature part of the rim. The laser emission is observed and calculated to be nontangent and displaced from the highest curvature regions owing to the presence of stable orbits. In this paper we present the first experimental evidence for this phenomenon of dynamical eclipsing.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York, 1990).
  2. J. U. Nöckel, A. D. Stone, and R. K. Chang, “Q spoiling and directionality in deformed ring cavities,” Opt. Lett. 19, 1693–1695 (1994).
    [CrossRef]
  3. A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and Q-spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
    [CrossRef] [PubMed]
  4. J. U. Nöckel, A. D. Stone, G. Chen, H. Grossman, and R. K. Chang, “Directional emission from asymmetric resonant cavities,” Opt. Lett. 21, 1609–1611 (1996).
    [CrossRef]
  5. J. U. Nöckel and A. D. Stone, “Chaotic light: a theory of asymmetric cavity resonators,” in Optical Processes in Microcavities, R. K. Chang and A. J. Campillo, eds. (World Scientific, Singapore, 1996), Chap. 11.
  6. J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant cavities,” Nature (London) 385, 45–47 (1997).
    [CrossRef]
  7. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10, 343–352 (1993).
    [CrossRef]
  8. Because the system of interest has a low refractive index, the polarization dependence of decay rates will be neglected in this work, allowing us to reduce Maxwell’s equation to the scalar wave equation.
  9. P. LeBoeuf and M. Saraceno, “Eigenfunctions of nonintegrable systems in generalized phase spaces,” J. Phys. A Math. Nucl. Gen. 23, 1745–1764 (1990).
    [CrossRef]
  10. 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]
  11. H. Waalkens, J. Wiersig, and H. Dullin, “The elliptic quantum billiard,” Ann. Phys. (N.Y.) 260, 50 (1997).
    [CrossRef]
  12. L. E. Reichl, The Transition to Chaos in Conservative Classical Systems: Quantum Manifestations (Springer-Verlag, New York, 1992).
  13. R. N. Berglund and B. Y. H. Liu, “Generation of monodis-perse aerosol standards,” Environ. Sci. Technol. 7, 147–153 (1973).
    [CrossRef]
  14. H. Lamb, Hydrodynamics (Dover, New York, 1945), pp. 473–475.
  15. S.-X. Qian, J. B. Snow, H.-M. Tzeng, and R. K. Chang, “Lasing droplets: highlighting the liquid–air interface by laser emission,” Science 231, 486 (1986).
    [CrossRef] [PubMed]
  16. The effect of gravity on the droplet trajectory is negligible owing to the relatively high speed of the droplets. The initial speed of the droplets emerging from the orifice of the droplet generator is 10 m/s, and the droplets used in the experiment are located ~1 cm below the orifice. For example, when the droplet generator shoots droplets horizontally, deviation between the true droplet trajectory and the one without gravity is only 0.028°.
  17. J. C. Swindal, D. H. Leach, R. K. Chang, and K. Young, “Precession of morphology-dependent resonances in nonspherical droplets,” Opt. Lett. 18, 191–193 (1993).
    [CrossRef]
  18. S. Chang, N. B. Rex, and R. K. Chang, “Chemical lasing in pendant droplets: lasing-spectra, emission-pattern, and cavity-lifetime measurements,” J. Opt. Soc. Am. B 16, 1224–1235 (1999).
    [CrossRef]
  19. Because the ray simulations do not include tunneling, they cannot account for the bright rims in the experimental images at θD=90°.

1999 (1)

1998 (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]

1997 (2)

H. Waalkens, J. Wiersig, and H. Dullin, “The elliptic quantum billiard,” Ann. Phys. (N.Y.) 260, 50 (1997).
[CrossRef]

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

1996 (1)

1995 (1)

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and Q-spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (2)

1990 (1)

P. LeBoeuf and M. Saraceno, “Eigenfunctions of nonintegrable systems in generalized phase spaces,” J. Phys. A Math. Nucl. Gen. 23, 1745–1764 (1990).
[CrossRef]

1986 (1)

S.-X. Qian, J. B. Snow, H.-M. Tzeng, and R. K. Chang, “Lasing droplets: highlighting the liquid–air interface by laser emission,” Science 231, 486 (1986).
[CrossRef] [PubMed]

1973 (1)

R. N. Berglund and B. Y. H. Liu, “Generation of monodis-perse aerosol standards,” Environ. Sci. Technol. 7, 147–153 (1973).
[CrossRef]

Berglund, R. N.

R. N. Berglund and B. Y. H. Liu, “Generation of monodis-perse aerosol standards,” Environ. Sci. Technol. 7, 147–153 (1973).
[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, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[CrossRef] [PubMed]

Chang, R. K.

Chang, S.

Chen, G.

J. U. Nöckel, A. D. Stone, G. Chen, H. Grossman, and R. K. Chang, “Directional emission from asymmetric resonant cavities,” Opt. Lett. 21, 1609–1611 (1996).
[CrossRef]

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and Q-spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[CrossRef] [PubMed]

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]

Dullin, H.

H. Waalkens, J. Wiersig, and H. Dullin, “The elliptic quantum billiard,” Ann. Phys. (N.Y.) 260, 50 (1997).
[CrossRef]

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]

Grossman, H.

Johnson, B. R.

Leach, D. H.

LeBoeuf, P.

P. LeBoeuf and M. Saraceno, “Eigenfunctions of nonintegrable systems in generalized phase spaces,” J. Phys. A Math. Nucl. Gen. 23, 1745–1764 (1990).
[CrossRef]

Liu, B. Y. H.

R. N. Berglund and B. Y. H. Liu, “Generation of monodis-perse aerosol standards,” Environ. Sci. Technol. 7, 147–153 (1973).
[CrossRef]

Mekis, A.

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and Q-spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[CrossRef] [PubMed]

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 cavities,” Nature (London) 385, 45–47 (1997).
[CrossRef]

J. U. Nöckel, A. D. Stone, G. Chen, H. Grossman, and R. K. Chang, “Directional emission from asymmetric resonant cavities,” Opt. Lett. 21, 1609–1611 (1996).
[CrossRef]

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and Q-spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[CrossRef] [PubMed]

J. U. Nöckel, A. D. Stone, and R. K. Chang, “Q spoiling and directionality in deformed ring cavities,” Opt. Lett. 19, 1693–1695 (1994).
[CrossRef]

Qian, S.-X.

S.-X. Qian, J. B. Snow, H.-M. Tzeng, and R. K. Chang, “Lasing droplets: highlighting the liquid–air interface by laser emission,” Science 231, 486 (1986).
[CrossRef] [PubMed]

Rex, N. B.

Saraceno, M.

P. LeBoeuf and M. Saraceno, “Eigenfunctions of nonintegrable systems in generalized phase spaces,” J. Phys. A Math. Nucl. Gen. 23, 1745–1764 (1990).
[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, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[CrossRef] [PubMed]

Snow, J. B.

S.-X. Qian, J. B. Snow, H.-M. Tzeng, and R. K. Chang, “Lasing droplets: highlighting the liquid–air interface by laser emission,” Science 231, 486 (1986).
[CrossRef] [PubMed]

Stone, A. D.

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 cavities,” Nature (London) 385, 45–47 (1997).
[CrossRef]

J. U. Nöckel, A. D. Stone, G. Chen, H. Grossman, and R. K. Chang, “Directional emission from asymmetric resonant cavities,” Opt. Lett. 21, 1609–1611 (1996).
[CrossRef]

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and Q-spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[CrossRef] [PubMed]

J. U. Nöckel, A. D. Stone, and R. K. Chang, “Q spoiling and directionality in deformed ring cavities,” Opt. Lett. 19, 1693–1695 (1994).
[CrossRef]

Swindal, J. C.

Tzeng, H.-M.

S.-X. Qian, J. B. Snow, H.-M. Tzeng, and R. K. Chang, “Lasing droplets: highlighting the liquid–air interface by laser emission,” Science 231, 486 (1986).
[CrossRef] [PubMed]

Waalkens, H.

H. Waalkens, J. Wiersig, and H. Dullin, “The elliptic quantum billiard,” Ann. Phys. (N.Y.) 260, 50 (1997).
[CrossRef]

Wiersig, J.

H. Waalkens, J. Wiersig, and H. Dullin, “The elliptic quantum billiard,” Ann. Phys. (N.Y.) 260, 50 (1997).
[CrossRef]

Young, K.

Ann. Phys. (N.Y.) (1)

H. Waalkens, J. Wiersig, and H. Dullin, “The elliptic quantum billiard,” Ann. Phys. (N.Y.) 260, 50 (1997).
[CrossRef]

Environ. Sci. Technol. (1)

R. N. Berglund and B. Y. H. Liu, “Generation of monodis-perse aerosol standards,” Environ. Sci. Technol. 7, 147–153 (1973).
[CrossRef]

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

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

J. Phys. A Math. Nucl. Gen. (1)

P. LeBoeuf and M. Saraceno, “Eigenfunctions of nonintegrable systems in generalized phase spaces,” J. Phys. A Math. Nucl. Gen. 23, 1745–1764 (1990).
[CrossRef]

Nature (London) (1)

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

Opt. Lett. (3)

Phys. Rev. Lett. (1)

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and Q-spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[CrossRef] [PubMed]

Science (2)

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]

S.-X. Qian, J. B. Snow, H.-M. Tzeng, and R. K. Chang, “Lasing droplets: highlighting the liquid–air interface by laser emission,” Science 231, 486 (1986).
[CrossRef] [PubMed]

Other (7)

The effect of gravity on the droplet trajectory is negligible owing to the relatively high speed of the droplets. The initial speed of the droplets emerging from the orifice of the droplet generator is 10 m/s, and the droplets used in the experiment are located ~1 cm below the orifice. For example, when the droplet generator shoots droplets horizontally, deviation between the true droplet trajectory and the one without gravity is only 0.028°.

H. Lamb, Hydrodynamics (Dover, New York, 1945), pp. 473–475.

L. E. Reichl, The Transition to Chaos in Conservative Classical Systems: Quantum Manifestations (Springer-Verlag, New York, 1992).

J. U. Nöckel and A. D. Stone, “Chaotic light: a theory of asymmetric cavity resonators,” in Optical Processes in Microcavities, R. K. Chang and A. J. Campillo, eds. (World Scientific, Singapore, 1996), Chap. 11.

M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York, 1990).

Because the system of interest has a low refractive index, the polarization dependence of decay rates will be neglected in this work, allowing us to reduce Maxwell’s equation to the scalar wave equation.

Because the ray simulations do not include tunneling, they cannot account for the bright rims in the experimental images at θD=90°.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Husimi plots (red, high intensity; white, no weight) superimposed onto classical surfaces of section for the ray dynamics inside 2D deformed dielectric cavities with refractive index n=1.36: (a) CWGM at ka=52.85 (k is the wave number) in a quadrupolar cavity parametrized by r(θ)=1+ cos 2θ with =0.113. (b) WGM at ka=52.04 in an ellipse with b/a=1/27. Insets: the corresponding wave functions in real space. (c) Sketch of quasi-periodic four-bounce orbits (green) and a chaotic orbit (red) inside the quadrupolar cavity of (a).

Fig. 2
Fig. 2

Schematic of experimental setup for taking images of lasing microdroplets at various inclination angles (θD). (L1, L2, camera lenses; F1, F2, color filters; M1, M2, mirrors; BS, beam splitter). Color filters were used to block scattered pump laser light. Focused pumping with <5-µm beam diameter was used for the experiment. Images (I-1 and I-2) of the prolate droplet (b/a=1.3) at θD=140° are shown for three different locations of the focused pump beam: (i) equatorial rim, (ii) 45° above the equator, and (iii) north pole.

Fig. 3
Fig. 3

(a) Experimental lasing images (I-1 and I-2) of broadly pumped (beam diameter>100 µm) prolate droplets (b/a=1.29) are shown for θD=90°, 122°, and 146° (pump laser was vertically polarized, and an f/16 lens was used). Corresponding calculated lasing images (in the I-2 direction) at θD=90°, 120°, and 150° are shown for (b) quadrupole and (c) elliptic deformations. In the simulation, equally mixed TE and TM polarizations were used to calculate Fresnel coefficients for the emerging laser rays.

Fig. 4
Fig. 4

(a) Images of broadly pumped lasing prolate droplets (b/a=1.29) at θD=142° with various f-numbers (f/#) (f/16,f/8,f/4, and f/2). Because the solid angle of light acceptance is proportional to 1/(f/#)2, the color intensity scale for each f/# is adjusted accordingly. The depth of field [(f/#)2] for each f/# is shown relative to the droplet size. The imaging direction is on the right of the droplet. (b) Lasing images of broadly pumped prolate droplets with various aspect ratios. Dynamical eclipsing is not present for small deformation (b/a=1.07).

Metrics