Abstract

Wave theory and geometrical optics are used to investigate the effect of small changes in size and index of refraction on the resonance wavelength of spherical microresonators. It is shown that changes in the index of refraction have two effects: These changes affect the phase jump on the surface and the optical path length in the resonator. Under certain conditions the effect of the external or internal index of refraction becomes negligible. The influence of the order number of the resonance modes is investigated. Finally, the results of the theoretical analyses are applied to calculate the effect of temperature on the resonance wavelength.

© 2006 Optical Society of America

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  1. V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, "Coupling and tunability of optical whispering-gallery modes--a basis for coordinate meter," Opt. Commun. 107, 41-48 (1994).
    [CrossRef]
  2. F. Janetta, G. Schweiger, S. Maruan, and K. Grosse, "Temperatursensor mit optischer Wirkungsweise und mikroskopischer Dimension," Photonik 3/2000, 34-38 (2000).
  3. A. T. Rosenberger and J. P. Rezac, "Whispering-gallery-mode evanescent-wave microsensor for trace-gas detection," in Biomedical Instrumentation Based on Micro- and Nanotechnology, R.P.Mariella, Jr., and D.V.Nicolau, eds., Proc. SPIE 4265, 102-112 (2001).
  4. F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
    [CrossRef]
  5. I. Teraoka, S. Arnold, and F. Vollmer, "Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium," J. Opt. Soc. Am. B 20, 1937-1946 (2003).
    [CrossRef]
  6. S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, "Shift of whispering-gallery modes in microspheres by protein adsorption," Opt. Lett. 28, 272-274 (2003).
    [CrossRef] [PubMed]
  7. S. Juodkazis, K. Fujiwara, T. Takahashi, S. Matsuo, and H. Misawa, "Morphology-dependent resonant laser emission of dye-doped ellipsoidal microcavity," J. Appl. Phys. 91, 916-921 (2002).
    [CrossRef]
  8. H. Fujiwara and K. Sasaki, "Microspherical lasing of an erbium-ion-doped glass particle," Jpn. J. Appl. Phys. Part 1 41, 46-48 (2002).
    [CrossRef]
  9. V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
    [CrossRef]
  10. T. Bilici, S. Isci, A. Kurt, and A. Serpengüzel, "GaInNAs microspheres for wavelength division multiplexing," IEE Proc.: Optoelectron. 150, 89-91 (2003).
    [CrossRef]
  11. M. Cai and K. Vahala, "Highly efficient optical power transfer to whispering-gallery modes by use of a symmetrical dual-coupling configuration," Opt. Lett. 25, 260-262 (2000).
    [CrossRef]
  12. G. Roll and G. Schweiger, "Geometrical optics model of Mie resonances," J. Opt. Soc. Am. A 17, 1301-1311 (2000).
    [CrossRef]
  13. J. Schulte and G. Schweiger, "Resonant inelastic scattering by use of geometrical optics," J. Opt. Soc. Am. A 20, 317-324 (2003).
    [CrossRef]
  14. S. Ledesma, S. N. Goyanes, and C. Duplaá, "Development of a dilatometer based on diffractometry," Rev. Sci. Instrum. 73, 3271-3274 (2002).
    [CrossRef]
  15. F. J. Duarte, A. Costela, I. Garcia-Moreno, and R. Sastre, "Measurements of ∂n/∂T in solid-state dye-laser gain media," Appl. Opt. 39, 6522-6523 (2000).

2003 (4)

2002 (4)

S. Ledesma, S. N. Goyanes, and C. Duplaá, "Development of a dilatometer based on diffractometry," Rev. Sci. Instrum. 73, 3271-3274 (2002).
[CrossRef]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

S. Juodkazis, K. Fujiwara, T. Takahashi, S. Matsuo, and H. Misawa, "Morphology-dependent resonant laser emission of dye-doped ellipsoidal microcavity," J. Appl. Phys. 91, 916-921 (2002).
[CrossRef]

H. Fujiwara and K. Sasaki, "Microspherical lasing of an erbium-ion-doped glass particle," Jpn. J. Appl. Phys. Part 1 41, 46-48 (2002).
[CrossRef]

2000 (4)

1998 (1)

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
[CrossRef]

1994 (1)

V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, "Coupling and tunability of optical whispering-gallery modes--a basis for coordinate meter," Opt. Commun. 107, 41-48 (1994).
[CrossRef]

Arnold, S.

Bilici, T.

T. Bilici, S. Isci, A. Kurt, and A. Serpengüzel, "GaInNAs microspheres for wavelength division multiplexing," IEE Proc.: Optoelectron. 150, 89-91 (2003).
[CrossRef]

Braun, D.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Cai, M.

Costela, A.

Duarte, F. J.

Duplaá, C.

S. Ledesma, S. N. Goyanes, and C. Duplaá, "Development of a dilatometer based on diffractometry," Rev. Sci. Instrum. 73, 3271-3274 (2002).
[CrossRef]

Fujiwara, H.

H. Fujiwara and K. Sasaki, "Microspherical lasing of an erbium-ion-doped glass particle," Jpn. J. Appl. Phys. Part 1 41, 46-48 (2002).
[CrossRef]

Fujiwara, K.

S. Juodkazis, K. Fujiwara, T. Takahashi, S. Matsuo, and H. Misawa, "Morphology-dependent resonant laser emission of dye-doped ellipsoidal microcavity," J. Appl. Phys. 91, 916-921 (2002).
[CrossRef]

Garcia-Moreno, I.

Gorodetsky, M. L.

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
[CrossRef]

V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, "Coupling and tunability of optical whispering-gallery modes--a basis for coordinate meter," Opt. Commun. 107, 41-48 (1994).
[CrossRef]

Goyanes, S. N.

S. Ledesma, S. N. Goyanes, and C. Duplaá, "Development of a dilatometer based on diffractometry," Rev. Sci. Instrum. 73, 3271-3274 (2002).
[CrossRef]

Grosse, K.

F. Janetta, G. Schweiger, S. Maruan, and K. Grosse, "Temperatursensor mit optischer Wirkungsweise und mikroskopischer Dimension," Photonik 3/2000, 34-38 (2000).

Hollberg, L.

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
[CrossRef]

Holler, S.

Ilchenko, V. S.

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
[CrossRef]

V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, "Coupling and tunability of optical whispering-gallery modes--a basis for coordinate meter," Opt. Commun. 107, 41-48 (1994).
[CrossRef]

Isci, S.

T. Bilici, S. Isci, A. Kurt, and A. Serpengüzel, "GaInNAs microspheres for wavelength division multiplexing," IEE Proc.: Optoelectron. 150, 89-91 (2003).
[CrossRef]

Janetta, F.

F. Janetta, G. Schweiger, S. Maruan, and K. Grosse, "Temperatursensor mit optischer Wirkungsweise und mikroskopischer Dimension," Photonik 3/2000, 34-38 (2000).

Juodkazis, S.

S. Juodkazis, K. Fujiwara, T. Takahashi, S. Matsuo, and H. Misawa, "Morphology-dependent resonant laser emission of dye-doped ellipsoidal microcavity," J. Appl. Phys. 91, 916-921 (2002).
[CrossRef]

Khoshsima, M.

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, "Shift of whispering-gallery modes in microspheres by protein adsorption," Opt. Lett. 28, 272-274 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Kurt, A.

T. Bilici, S. Isci, A. Kurt, and A. Serpengüzel, "GaInNAs microspheres for wavelength division multiplexing," IEE Proc.: Optoelectron. 150, 89-91 (2003).
[CrossRef]

Ledesma, S.

S. Ledesma, S. N. Goyanes, and C. Duplaá, "Development of a dilatometer based on diffractometry," Rev. Sci. Instrum. 73, 3271-3274 (2002).
[CrossRef]

Libchaber, A.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Maruan, S.

F. Janetta, G. Schweiger, S. Maruan, and K. Grosse, "Temperatursensor mit optischer Wirkungsweise und mikroskopischer Dimension," Photonik 3/2000, 34-38 (2000).

Matsuo, S.

S. Juodkazis, K. Fujiwara, T. Takahashi, S. Matsuo, and H. Misawa, "Morphology-dependent resonant laser emission of dye-doped ellipsoidal microcavity," J. Appl. Phys. 91, 916-921 (2002).
[CrossRef]

Misawa, H.

S. Juodkazis, K. Fujiwara, T. Takahashi, S. Matsuo, and H. Misawa, "Morphology-dependent resonant laser emission of dye-doped ellipsoidal microcavity," J. Appl. Phys. 91, 916-921 (2002).
[CrossRef]

Rezac, J. P.

A. T. Rosenberger and J. P. Rezac, "Whispering-gallery-mode evanescent-wave microsensor for trace-gas detection," in Biomedical Instrumentation Based on Micro- and Nanotechnology, R.P.Mariella, Jr., and D.V.Nicolau, eds., Proc. SPIE 4265, 102-112 (2001).

Roll, G.

Rosenberger, A. T.

A. T. Rosenberger and J. P. Rezac, "Whispering-gallery-mode evanescent-wave microsensor for trace-gas detection," in Biomedical Instrumentation Based on Micro- and Nanotechnology, R.P.Mariella, Jr., and D.V.Nicolau, eds., Proc. SPIE 4265, 102-112 (2001).

Sasaki, K.

H. Fujiwara and K. Sasaki, "Microspherical lasing of an erbium-ion-doped glass particle," Jpn. J. Appl. Phys. Part 1 41, 46-48 (2002).
[CrossRef]

Sastre, R.

Schulte, J.

Schweiger, G.

Serpengüzel, A.

T. Bilici, S. Isci, A. Kurt, and A. Serpengüzel, "GaInNAs microspheres for wavelength division multiplexing," IEE Proc.: Optoelectron. 150, 89-91 (2003).
[CrossRef]

Takahashi, T.

S. Juodkazis, K. Fujiwara, T. Takahashi, S. Matsuo, and H. Misawa, "Morphology-dependent resonant laser emission of dye-doped ellipsoidal microcavity," J. Appl. Phys. 91, 916-921 (2002).
[CrossRef]

Teraoka, I.

Vahala, K.

Vassiliev, V. V.

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
[CrossRef]

Velichansky, V. L.

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
[CrossRef]

Vollmer, F.

Vyatchanin, S. P.

V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, "Coupling and tunability of optical whispering-gallery modes--a basis for coordinate meter," Opt. Commun. 107, 41-48 (1994).
[CrossRef]

Yarovitsky, A. V.

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

IEE Proc.: Optoelectron. (1)

T. Bilici, S. Isci, A. Kurt, and A. Serpengüzel, "GaInNAs microspheres for wavelength division multiplexing," IEE Proc.: Optoelectron. 150, 89-91 (2003).
[CrossRef]

J. Appl. Phys. (1)

S. Juodkazis, K. Fujiwara, T. Takahashi, S. Matsuo, and H. Misawa, "Morphology-dependent resonant laser emission of dye-doped ellipsoidal microcavity," J. Appl. Phys. 91, 916-921 (2002).
[CrossRef]

J. Opt. Soc. Am. A (2)

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

Jpn. J. Appl. Phys. Part 1 (1)

H. Fujiwara and K. Sasaki, "Microspherical lasing of an erbium-ion-doped glass particle," Jpn. J. Appl. Phys. Part 1 41, 46-48 (2002).
[CrossRef]

Opt. Commun. (2)

V. V. Vassiliev, V. L. Velichansky, V. S. Ilchenko, M. L. Gorodetsky, L. Hollberg, and A. V. Yarovitsky, "Narrow-line-width diode laser with a high-Q microsphere resonator," Opt. Commun. 158, 305-312 (1998).
[CrossRef]

V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, "Coupling and tunability of optical whispering-gallery modes--a basis for coordinate meter," Opt. Commun. 107, 41-48 (1994).
[CrossRef]

Opt. Lett. (2)

Photonik (1)

F. Janetta, G. Schweiger, S. Maruan, and K. Grosse, "Temperatursensor mit optischer Wirkungsweise und mikroskopischer Dimension," Photonik 3/2000, 34-38 (2000).

Rev. Sci. Instrum. (1)

S. Ledesma, S. N. Goyanes, and C. Duplaá, "Development of a dilatometer based on diffractometry," Rev. Sci. Instrum. 73, 3271-3274 (2002).
[CrossRef]

Other (1)

A. T. Rosenberger and J. P. Rezac, "Whispering-gallery-mode evanescent-wave microsensor for trace-gas detection," in Biomedical Instrumentation Based on Micro- and Nanotechnology, R.P.Mariella, Jr., and D.V.Nicolau, eds., Proc. SPIE 4265, 102-112 (2001).

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Figures (6)

Fig. 1
Fig. 1

Logarithmic presentation of the sensitivity functions for TE modes (solid curves) and TM modes (dashed curves) plotted versus the relative index of refraction for three values z = ( n + 1 2 ) y .

Fig. 2
Fig. 2

Logarithmic sensitivity functions for TE modes (solid curves) and TM modes (dashed curves) multiplied by the internal size parameter for three indices of refraction. The functions are plotted over the ratio z = ( n + 1 2 ) y .

Fig. 3
Fig. 3

Comparison of sensitivity functions calculated from geometrical optics (solid curves) and wave theory (dashed curves) for two relative indices of refraction.

Fig. 4
Fig. 4

Calculated shift of the resonance wavelength with temperature for TE modes; a = 140 μ m , m = 1.4959 , λ = 632.8 nm , l = 800 . Solid curves; geometrical optics; dashed curves; wave theory; dotted curves; approximation for F = 0 .

Fig. 5
Fig. 5

Experimental shift of resonances with temperature in PMMA particles; a = 140 μ m and λ = 632.8 nm .

Fig. 6
Fig. 6

Light emitted by a microsphere ( d = 140 μ m ) in dependence of the wavelength.

Equations (45)

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ϕ R = ( m 1 R , m 2 R , a R , λ R ) = 0 ,
d ϕ ( a , λ 0 , m 1 , m 2 ) = ϕ x ( x a d a + x λ d λ + x m 2 d m 2 ) + ϕ y ( y a d a + y λ d λ + y m 1 d m 1 ) = 0 ,
x = m 2 2 π a λ 0 , y = m 1 2 π a λ 0 .
( y R ϕ y + x R ϕ x ) ( d a a R d λ λ 0 R ) + y R ϕ y d m 1 m 1 R + x R ϕ x d m 2 m 2 R = 0 .
d λ λ = d a a + d m 1 m 1 ( 1 x ϕ x y ϕ y + x ϕ x ) + d m 2 m 2 x ϕ x y ϕ y + x ϕ x .
F ( x , y ) = x ϕ x x ϕ x + y ϕ y .
d λ λ = d a a + d m 1 m 1 [ 1 F ( x , y ) ] + d m 2 m 2 F ( x , y ) .
ϕ n TE ( x , y ) = χ n ( x ) ψ n ( x ) [ x G n ( x ) y D n ( y ) x D n ( x ) y D n ( y ) ] = 0 ,
ϕ n TM ( x , y ) = χ n ( x ) ψ n ( x ) [ y G n ( x ) x D n ( y ) y D n ( x ) x D n ( y ) ] = 0 ,
ϕ n TE ( x , y ) x = χ n ( x ) ψ n ( x ) [ G n ( x ) + x G n ( x ) x D n ( x ) y D n ( y ) ] ,
ϕ n TE ( x , y ) y = χ n ( x ) ψ n ( x ) [ D n ( y ) y D n ( y ) x D n ( x ) y D n ( y ) ] ,
ϕ n TM ( x , y ) x = χ n ( x ) ψ n ( x ) [ y G n ( x ) D n ( y ) y D n ( x ) x D n ( y ) ] ,
ϕ n TM ( x , y ) y = χ n ( x ) ψ n ( x ) [ G n ( x ) x D n ( y ) y D n ( x ) x D n ( y ) ] .
F n TE ( m , x ) = 1 m 2 1 [ G n ( x ) + 1 x G n ( x ) ] ,
G n TM ( m , x , y ) = 1 m 2 1 G n ( x ) 1 x G n ( x ) n ( n + 1 ) y 2 + G 2 ( x ) .
d λ λ = d m 2 m 2 1 m 2 1 [ G n ( x ) + 1 x G n ( x ) ] , m = m 1 m 2 .
d λ λ = m 2 d m 2 m 1 2 m 2 2 { n ( n + 1 ) x 2 1 + 1 x χ n ( x ) χ n ( x ) [ χ n ( x ) χ n ( x ) ] 2 } .
F n TM = m 2 2 m 1 2 m 2 2 n ( n + 1 ) x 2 1 1 x χ n ( x ) χ n ( x ) [ χ n ( x ) χ n ( x ) ] 2 n ( n + 1 ) y 2 + [ χ n ( x ) χ n ( x ) ] 2 .
ϕ ( a , λ R , m 1 , m 2 ) = y 2 Λ 2 Λ arccos ( Λ y ) β π 4 ( ν 1 ) π = 0 ,
β = arctan [ α 2 ( Λ 2 x 2 y 2 Λ 2 ) 1 2 ] ,
TE mode : α = 1 , TM mode : α = m .
a = Λ 2 x 2 , b = y 2 Λ 2
ϕ ( x , y ) y = ( a 2 + b 2 ) b 2 + y 2 a y b ( a 2 + b 2 ) , ϕ ( x , y ) x = x b a ( a 2 + b 2 )
ϕ ( x , y ) y = ( y 4 a 2 + x 4 b 2 ) b 2 x 2 y 2 a ( 2 b 2 y 2 ) y b ( y 4 a 2 + x 4 b 2 ) ,
ϕ ( x , y ) x = x b y 2 ( x 2 + 2 a 2 ) a ( y 4 a 2 + x 4 b 2 ) .
F TE ( x , y , m ) = ( y 2 Λ 2 ) ( m 2 1 ) [ Λ 2 x 2 ( y 2 Λ 2 ) + Λ 2 ] ,
F TM ( x , y , m ) = m 2 m 2 1 ( y 2 Λ 2 ) ( 2 Λ 2 x 2 ) Λ 2 x 2 ( y 2 Λ 2 ) [ ( 1 + m 2 ) Λ 2 y 2 ] + y 2 Λ 2 .
F TE ( y , z , m ) = 1 m 2 1 1 z 2 y ( 1 z 2 ) z 2 m 2 + z 2 ,
F TM ( y , z , m ) = m 2 m 2 1 ( 1 z 2 ) ( 2 z 2 m 2 ) y ( 1 z 2 ) z 2 m 2 [ z 2 ( 1 + m 2 ) 1 ] + z 2 ,
z = Λ y = sin φ , 1 m z 1 .
F TE ( y , m ) = m ( m 2 1 ) 3 2 1 y ,
F TM ( y , m ) = m ( m 2 1 ) 3 2 2 m 2 y .
Δ λ λ = [ ( 1 a d a d T ) + ( 1 m 1 d m 1 d T ) ] Δ T .
ψ ( m k r ) = m k r j n ( m k r ) , χ ( m k r ) = m k r n n ( m k r )
z 2 w ( z ) + [ z 2 n ( n + 1 ) ] w ( z ) = 0 ,
w ( x ) w ( x ) = m 2 w ( y ) w ( y ) + m 2 1 ;
x = m 2 z , y = m 1 z , m = m 1 m 2 .
G n = χ n ( x ) χ n ( x ) , D n = ψ n ( x ) ψ n ( x ) ,
G n = χ n ( x ) χ n ( x ) G n 2 = G n ( x ) = n ( n + 1 ) x 2 1 G n 2 ,
D n = ψ n ( x ) ψ n ( x ) D n 2 = D n ( x ) = n ( n + 1 ) x 2 1 D n 2 ,
G n ( x ) D n ( x ) = [ ψ n ( x ) χ n ( x ) ] 1 ;
TE mode : G ( x ) m D ( m x ) = 0 G n ( x ) m 2 D n ( m x ) = m 2 1 ,
G n ( x ) D n ( y ) = ( m 2 1 ) [ 1 + D n ( y ) ] ;
TM mode : m G ( x ) D ( m x ) = 0 ,
G n ( x ) D n ( y ) = ( m 2 1 ) [ n ( n + 1 ) y 2 + G 2 ( x ) ] .

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