Abstract

We demonstrate lasing in liquid pendant droplets through a chemiluminescence process, which uniquely provides spatially uniform pumping throughout the sample. Pendant droplets of 2-mm equatorial radius are formed at the tip of a capillary tube through which the chemiluminescence material is injected. The chemiluminescence spectra along the highlighted rim of the droplet show redshifted intensity enhancement in the wavelength region where the absorption is low. The lasing threshold is found by addition of different amounts of absorbers. The observed nonuniform laser-emission intensity distribution along the droplet rim is caused by a spatially varying rate of diffractive-light leakage related to the droplet surface curvature. Using WKB approximation, we express the diffractive-light leakage rate on a curved surface as an exponentially decreasing function of angle of incidence. The standard laser rate equation with distributed leakage loss is employed to express the laser-emission output intensity from the pendant droplet. The light leakage from the surface was further investigated by localized perturbations formed by poking of the surface with a sharply tipped fiber. The Q of cavity modes as high as 3.5×108 in the pendant droplet was determined from the cavity-lifetime measurement.

© 1999 Optical Society of America

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B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

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. M. Hartings, J. L. Cheung, and R. K. Chang, “Temporal beating of nondegenerate azimuthal modes in nonspherical microdroplets: technique for determining the distortion amplitude,” Appl. Opt. 37, 3306–3310 (1998).
[CrossRef]

1997

J. M. Hartings, A. Poon, X. Pu, R. K. Chang, and T. M. Leslie, “Second harmonic generation and fluorescence images from surfactants on hanging droplets,” Chem. Phys. Lett. 281, 389–393 (1997).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (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

1995

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

1993

L. Collot, V. Lefevre-Seguine, M. Brune, J. M. Raimonde, and S. Haroche, “Very high-Q whispering-gallery resonances observed on fused-silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310–1312 (1993).
[CrossRef]

B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10, 343–352 (1993).
[CrossRef]

J. C. Swindal, D. H. Leach, R. K. Chang, and K. Young, “Precession of morphology-dependent resonances in non-spherical droplets,” Opt. Lett. 18, 191–193 (1993).
[CrossRef]

1992

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

1991

1990

1989

H. M. Nussenzveig, “Tunneling effects in diffractive scattering and resonances,” Comments At. Mol. Phys. 23, 175–187 (1989).

1986

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, 486–488 (1986).
[CrossRef] [PubMed]

1974

A. G. Mohan and N. J. Turro, “A facile and effective chemiluminescence demonstration experiment,” J. Chem. Educ. 51, 528–529 (1974).
[CrossRef]

Acker, W. P.

Brune, M.

L. Collot, V. Lefevre-Seguine, M. Brune, J. M. Raimonde, and S. Haroche, “Very high-Q whispering-gallery resonances observed on fused-silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

Campillo, A. J.

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.

J. M. Hartings, J. L. Cheung, and R. K. Chang, “Temporal beating of nondegenerate azimuthal modes in nonspherical microdroplets: technique for determining the distortion amplitude,” Appl. Opt. 37, 3306–3310 (1998).
[CrossRef]

J. M. Hartings, A. Poon, X. Pu, R. K. Chang, and T. M. Leslie, “Second harmonic generation and fluorescence images from surfactants on hanging droplets,” Chem. Phys. Lett. 281, 389–393 (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. C. Swindal, D. H. Leach, R. K. Chang, and K. Young, “Precession of morphology-dependent resonances in non-spherical droplets,” Opt. Lett. 18, 191–193 (1993).
[CrossRef]

G. Chen, W. P. Acker, R. K. Chang, and S. C. Hill, “Fine structures in the angular distribution of stimulated Raman scattering from single droplets,” Opt. Lett. 16, 177–179 (1991).
[CrossRef]

J.-Z. Zhang, G. Chen, and R. K. Chang, “Pumping of stimulated Raman scattering by stimulated Brillouin scattering within a single droplet: input laser linewidth effects,” J. Opt. Soc. Am. B 7, 108–115 (1990).
[CrossRef]

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, 486–488 (1986).
[CrossRef] [PubMed]

Chen, G.

Cheung, J. L.

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]

Chu, S. T.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Chýlek, P.

Collot, L.

L. Collot, V. Lefevre-Seguine, M. Brune, J. M. Raimonde, and S. Haroche, “Very high-Q whispering-gallery resonances observed on fused-silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

Eversole, J. D.

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]

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Foresi, J. S.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[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, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[CrossRef] [PubMed]

Greene, W.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Grossman, H.

Haroche, S.

L. Collot, V. Lefevre-Seguine, M. Brune, J. M. Raimonde, and S. Haroche, “Very high-Q whispering-gallery resonances observed on fused-silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

Hartings, J. M.

J. M. Hartings, J. L. Cheung, and R. K. Chang, “Temporal beating of nondegenerate azimuthal modes in nonspherical microdroplets: technique for determining the distortion amplitude,” Appl. Opt. 37, 3306–3310 (1998).
[CrossRef]

J. M. Hartings, A. Poon, X. Pu, R. K. Chang, and T. M. Leslie, “Second harmonic generation and fluorescence images from surfactants on hanging droplets,” Chem. Phys. Lett. 281, 389–393 (1997).
[CrossRef]

Haus, H. A.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Hill, S. C.

Ippen, E. P.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Johnson, B. R.

Kimble, H. J.

Kimerling, L. C.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Knotts, M. E.

M. E. Knotts, “Fun with lightsticks,” Opt. Photon. News 7(1), 40 (1996).
[CrossRef]

Laine, J.-P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Leach, D. H.

Lefevre-Seguine, V.

L. Collot, V. Lefevre-Seguine, M. Brune, J. M. Raimonde, and S. Haroche, “Very high-Q whispering-gallery resonances observed on fused-silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

Leslie, T. M.

J. M. Hartings, A. Poon, X. Pu, R. K. Chang, and T. M. Leslie, “Second harmonic generation and fluorescence images from surfactants on hanging droplets,” Chem. Phys. Lett. 281, 389–393 (1997).
[CrossRef]

Levi, A. F. J.

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310–1312 (1993).
[CrossRef]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Lin, H.-B.

Little, B. E.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Logan, R. A.

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310–1312 (1993).
[CrossRef]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Mabuchi, H.

McCall, S. L.

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310–1312 (1993).
[CrossRef]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[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]

Mohan, A. G.

A. G. Mohan and N. J. Turro, “A facile and effective chemiluminescence demonstration experiment,” J. Chem. Educ. 51, 528–529 (1974).
[CrossRef]

Mohideen, U.

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310–1312 (1993).
[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, “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]

Nussenzveig, H. M.

H. M. Nussenzveig, “Tunneling effects in diffractive scattering and resonances,” Comments At. Mol. Phys. 23, 175–187 (1989).

Pearton, S. J.

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310–1312 (1993).
[CrossRef]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Poon, A.

J. M. Hartings, A. Poon, X. Pu, R. K. Chang, and T. M. Leslie, “Second harmonic generation and fluorescence images from surfactants on hanging droplets,” Chem. Phys. Lett. 281, 389–393 (1997).
[CrossRef]

Pu, X.

J. M. Hartings, A. Poon, X. Pu, R. K. Chang, and T. M. Leslie, “Second harmonic generation and fluorescence images from surfactants on hanging droplets,” Chem. Phys. Lett. 281, 389–393 (1997).
[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, 486–488 (1986).
[CrossRef] [PubMed]

Raimonde, J. M.

L. Collot, V. Lefevre-Seguine, M. Brune, J. M. Raimonde, and S. Haroche, “Very high-Q whispering-gallery resonances observed on fused-silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[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]

Slusher, R. E.

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310–1312 (1993).
[CrossRef]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

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, 486–488 (1986).
[CrossRef] [PubMed]

Steinmeyer, G.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[CrossRef]

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]

Swindal, J. C.

Thoen, E. R.

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Figures (10)

Fig. 1
Fig. 1

Chemically initiated electron-exchange luminescence is a photoemission process associated with chemiluminescent pendant droplets formed from ultrahigh-intensity light sticks. The diagram shows the chemical reactions upon mixing of two solutions from the light stick. Dioxetandione is produced from a reaction of oxalate ester and hydrogen peroxide (H2O2). Dioxetandione (peroxide) and rubrene (fluor) generate a charge-transfer complex. The charge complex breaks into two CO2 molecules and an excited dye molecule, which subsequently emits a photon.

Fig. 2
Fig. 2

Color photographs of chemiluminescent pendant droplets (10× magnification) with and without absorber. The pendant droplets are generated at the tip of a capillary tube of 700-µm outer diameter. The equatorial diameter of the droplets is ∼2 mm. (a) High-intensity light stick [fluor: 9,10-bis (phenyl-ethynyl) anthracene] without an absorber. Camera exposure time (Texp)=1/4 s. (b) Ultrahigh-intensity light stick (fluor: rubrene) without an absorber. Texp=1/60 s. (c) High-intensity light stick with an absorber (0.4 g/l of Nigrosin). Texp=16 s. (d) Ultrahigh-intensity light stick with an absorber (1.0 g/l of Nigrosin). Texp=16 s.

Fig. 3
Fig. 3

False-color presentation of chemiluminescent pendant droplets with their surfaces perturbed by a pulled-fiber tip: (a) The pendant droplet is formed from the original solution extracted from an ultrahigh-intensity light stick without perturbation. The pulled-fiber tip touches (b) the equator, (c) below the equator from the left, and (d) above the equator from the left. All the perturbed locations appear brighter. Note that the droplet edge diagonally opposite the perturbation location also brightens.

Fig. 4
Fig. 4

(a) Emission and absorption curves of rubrene (taken from Ref. 17). The absorption curve is presented in units of molar-extinction coefficient for the rubrene dissolved in benzene with a concentration of 0.1 g/l. The molar-extinction coefficient (mole-1 cm-1) is defined as =A/c=(αabs/c)log10 e, where A (cm-1) is the absorbance, c (mole) is the concentration of the molecule, and αabs (cm-1) is the absorption coefficient. (b) The absorption coefficient of 0.1-g/l nigrosin dissolved in ethanol, measured by a spectrophotometer (Hitachi U-2001).

Fig. 5
Fig. 5

Emission spectra at selected locations along the rim of various pendant droplets. The inset shows the image of the pendant droplet on the slit (along the z axis) of the spectrometer. The emission spectra are taken along the rim at z=0 mm (the equator) and from z=0.1 mm to z=0.4 mm of the upper hemisphere. (a) Droplet A is made of the original solution extracted from an ultrahigh-intensity light stick. (b) Droplet B is made of the original solution plus 0.1-g/l of Nigrosin. (c) Droplet C is made of the original solution plus 0.4-g/l of Nigrosin.

Fig. 6
Fig. 6

Intensity profiles (along z) of the chemiluminescent pendant droplets at various selective wavelengths. We obtained each intensity profile curve by integrating counts from the CCD camera over 2 nm, centered at the indicated wavelength. We used the same data as in Fig. 5 for (a) droplet A, (b) droplet B, and (c) droplet C.

Fig. 7
Fig. 7

(a) Effective radial potential well/barrier Veff(r)=k2[1-n2(r)]+l(l+1)/r2 in the quantum-mechanical analogy for WGM’s in a spherical cavity with radius a, where k=2π/λ, n(r) is the refractive index as a function of r, and l is the angular mode number of an WGM, usually labeled with (ν, l), where ν is the radial order. Then, (ν, l) WGM’s correspond to the quasi-bound states (labeled as ν1, ν2, ) in an l-potential well. A WGM with wave number k sees the effective work function Weff and effective thickness deff of the potential barrier. The transmittance T in relation (3) can also be expressed in terms of Weff and deff, Texp(-2/3αdeff Weff). 11 (b) Diffractive or tunneling leakage (in the effective potential picture) occurs even when χ>χc in the spherical cavity. The tunneling probability is given by relation (3). Using semiclassical approximation,7 each (ν, l) WGM can be related to the angle of incidence χ by the relation sin χν, l=l/nkν, la, where a is the radius of the sphere. For a given l potential, lower ν WGM’s have higher χ, and consequently they are more tightly bound to the rim of the sphere, resulting in less tunneling loss and a higher Q value.

Fig. 8
Fig. 8

Precessing tilted orbits in the pendant droplet. The pendant droplet is axisymmetric (axis of symmetry: z axis), and hence Lz is conserved. As a result, the angular-momentum vector L, which is the normal of the plane of motion, rotates around the z axis. The Ω denotes the precession frequency. For each precessing tilted orbit the highest curvature occurs at the highest latitude of the orbit because of the shape in the upper half of the pendant droplet. The bottom half is more like a hemisphere.

Fig. 9
Fig. 9

(a) Emission spectra at selected locations along the rim of pendant droplet A, made of the original solution extracted from an ultrahigh-intensity light stick, with surface perturbation applied at the equator (z=0) by a sharp pulled-fiber tip. The enhanced light emission at z=0 is due to increased output coupling of lasing light at the perturbed equatorial location. (b) Spatial intensity profile along z at selected wavelength regions (each intensity curve is obtained by integration of the photon counts over 2 nm) of the same droplet. The largest enhancement occurs at λ=620 nm, where the net unsaturated gain of lasing is the highest.

Fig. 10
Fig. 10

Cavity decay time τ0 measurements from the droplet D, pumped by a 30-ns laser pulse at 532 nm. (a) The longest decay times measured from SRS: τ0=60 ns (Q=1.8×108) when pumped at the equator and detected at the equator (equatorial orbit); τ0=86 ns (Q=2.6×108) when pumped at θ=+45° and detected at θ=135° (precessing orbit). (b) The longest decay time measured at the green wavelength (532 nm) when pumped at the equator by a green beam and detected at the equator (equatorial orbit): τ0=31 ns (Q=1.1×108). (c) Various decay times measured at the green wavelength (532 nm) when pumped by a green beam at θ=+45° and detected at θ=135° (precessing tilted orbit): (i) τ0=37 ns (Q=1.3×108), (ii) τ0=68 ns (Q=2.4×108), and (iii) τ0=40 ns (Q=1.4×108) and τ0=98 ns (Q=3.5×108).

Equations (17)

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αleak¯L=0Lαleakdl,
1Q=1Qabs+1Qleak=1ωcn(αabs+αleak¯)=1ωcnαabs+1L0Lαleakdl,
Texp-2nka sin χ31-sin χcsin χ23/2,
sin χν, l=lnkν, l a.
dN2dt=Rp-σ(ν)Φν(N2-N1)-Γ21N2,
dN1dt=σ(ν)Φν(N2-N1)+Γ21N2-Γ10N1,
dΦνdt=cn[σ(ν)(N2-N1)-αleak-αabs]Φν,
ΔNSS=Rpσ(ν)Φν+Γ21,
gSSσ(ν)ΔNSS=αtot,
Icirc=hνΦν2=Isat2g0αtot-1,
dIout=αleakdlIcirc=αleakdlg0αabs+αleak-1 Isat2.
αleak-opt=-αabs+αabsg0.
dIoutαleakαabs+αleakIsat2g0dl,
dIabsαabsαabs+αleakIsat2g0dl,
αleakΔl>αabsΔl,
χ¯=χ0,
χ¯>χ0.

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