Abstract

We rigorously compute the resonance spectrum for a deformed spherical microcavity illuminated by plane waves. Particular attention is paid to the shift of resonances due to small departures from sphericity, which includes a discussion on the behavior of individual scattering coefficients. These results are obtained for deformed microcavities that are large with respect to the wavelength of the incident wave.

© 2006 Optical Society of America

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  1. P. R. Conwell, P. W. Barber, and C. K. Rushforth, "Resonant spectra of dielectric spheres," J. Opt. Soc. Am. A 1, 62-67 (1984).
    [CrossRef]
  2. J. A. Lock, "Excitation of morphology-dependent resonances and van de Hulst's localization principle," Opt. Lett. 24, 427-429 (1999).
    [CrossRef]
  3. C. C. Lam, P. T. Leung, and K. Young, "Explicit asymptotic formulas for the positions, widths, and strengths of resonances in Mie scattering," J. Opt. Soc. Am. B 9, 1585-1592 (1992).
    [CrossRef]
  4. H. M. Lai, P. T. Leung, and K. Young, "Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets," Phys. Rev. A 41, 5187-5198 (1990).
    [CrossRef] [PubMed]
  5. H. Lai, C. Lam, P. Leung, and K. Young, "Effect of perturbations on the widths of narrow morphology-dependent resonances in Mie scattering," J. Opt. Soc. Am. B 8, 1962-1970 (1991).
    [CrossRef]
  6. G. Chern, Md. M. Mazumder, and R. K. Chang, "Laser diagnostics for droplets characterization: application of morphology dependent resonances," Prog. Energy Combust. Sci. 22, 163-188 (1996).
    [CrossRef]
  7. Y. Han, G. Gréhan, and G. Gouesbet, "Generalized Lorenz-Mie theory for a spheroidal particle with off-axis Gaussian beam illumination," Appl. Opt. 42, 6621-6629 (2003).
    [CrossRef] [PubMed]
  8. Y. Han and Z. S. Wu, "Scattering of a spheroidal particle illuminated by a Gaussian beam," Appl. Opt. 40, 2501-2509 (2001).
    [CrossRef]
  9. Y. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Gréhan, "Scattering of light by spheroids: the far field case," Opt. Commun. 210, 1-9 (2002).
    [CrossRef]
  10. M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 1999).
  11. G. Gouesbet and G. Grehan, "Generalized Lorenz-Mie theories, from past to future," Atomization Sprays 10, 277-333 (2000).
  12. G. Gouesbet, "Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres," J. Opt. Soc. Am. A 16, 1641-1650 (1999).
    [CrossRef]
  13. S. Asano and G. Yamamoto, "Light scattering by a spheroidal particle," Appl. Opt. 14, 29-49 (1975).
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  14. J. R. Probert-Jones, "Resonance component of backscattering by large dielectric spheres," J. Opt. Soc. Am. A 1, 822-830 (1984).
    [CrossRef]

2003 (1)

2002 (1)

Y. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Gréhan, "Scattering of light by spheroids: the far field case," Opt. Commun. 210, 1-9 (2002).
[CrossRef]

2001 (1)

2000 (1)

G. Gouesbet and G. Grehan, "Generalized Lorenz-Mie theories, from past to future," Atomization Sprays 10, 277-333 (2000).

1999 (2)

1996 (1)

G. Chern, Md. M. Mazumder, and R. K. Chang, "Laser diagnostics for droplets characterization: application of morphology dependent resonances," Prog. Energy Combust. Sci. 22, 163-188 (1996).
[CrossRef]

1992 (1)

1991 (1)

1990 (1)

H. M. Lai, P. T. Leung, and K. Young, "Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets," Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

1984 (2)

1975 (1)

Asano, S.

Barber, P. W.

Chang, R. K.

G. Chern, Md. M. Mazumder, and R. K. Chang, "Laser diagnostics for droplets characterization: application of morphology dependent resonances," Prog. Energy Combust. Sci. 22, 163-188 (1996).
[CrossRef]

Chern, G.

G. Chern, Md. M. Mazumder, and R. K. Chang, "Laser diagnostics for droplets characterization: application of morphology dependent resonances," Prog. Energy Combust. Sci. 22, 163-188 (1996).
[CrossRef]

Conwell, P. R.

Gouesbet, G.

Y. Han, G. Gréhan, and G. Gouesbet, "Generalized Lorenz-Mie theory for a spheroidal particle with off-axis Gaussian beam illumination," Appl. Opt. 42, 6621-6629 (2003).
[CrossRef] [PubMed]

Y. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Gréhan, "Scattering of light by spheroids: the far field case," Opt. Commun. 210, 1-9 (2002).
[CrossRef]

G. Gouesbet and G. Grehan, "Generalized Lorenz-Mie theories, from past to future," Atomization Sprays 10, 277-333 (2000).

G. Gouesbet, "Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres," J. Opt. Soc. Am. A 16, 1641-1650 (1999).
[CrossRef]

Grehan, G.

G. Gouesbet and G. Grehan, "Generalized Lorenz-Mie theories, from past to future," Atomization Sprays 10, 277-333 (2000).

Gréhan, G.

Y. Han, G. Gréhan, and G. Gouesbet, "Generalized Lorenz-Mie theory for a spheroidal particle with off-axis Gaussian beam illumination," Appl. Opt. 42, 6621-6629 (2003).
[CrossRef] [PubMed]

Y. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Gréhan, "Scattering of light by spheroids: the far field case," Opt. Commun. 210, 1-9 (2002).
[CrossRef]

Han, Y.

Hovenier, J. W.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 1999).

Lai, H.

Lai, H. M.

H. M. Lai, P. T. Leung, and K. Young, "Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets," Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

Lam, C.

Lam, C. C.

Leung, P.

Leung, P. T.

C. C. Lam, P. T. Leung, and K. Young, "Explicit asymptotic formulas for the positions, widths, and strengths of resonances in Mie scattering," J. Opt. Soc. Am. B 9, 1585-1592 (1992).
[CrossRef]

H. M. Lai, P. T. Leung, and K. Young, "Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets," Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

Lock, J. A.

Mazumder, Md. M.

G. Chern, Md. M. Mazumder, and R. K. Chang, "Laser diagnostics for droplets characterization: application of morphology dependent resonances," Prog. Energy Combust. Sci. 22, 163-188 (1996).
[CrossRef]

Méès, L.

Y. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Gréhan, "Scattering of light by spheroids: the far field case," Opt. Commun. 210, 1-9 (2002).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 1999).

Probert-Jones, J. R.

Ren, K. F.

Y. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Gréhan, "Scattering of light by spheroids: the far field case," Opt. Commun. 210, 1-9 (2002).
[CrossRef]

Rushforth, C. K.

Travis, L. D.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 1999).

Wu, S. Z.

Y. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Gréhan, "Scattering of light by spheroids: the far field case," Opt. Commun. 210, 1-9 (2002).
[CrossRef]

Wu, Z. S.

Yamamoto, G.

Young, K.

Appl. Opt. (3)

Atomization Sprays (1)

G. Gouesbet and G. Grehan, "Generalized Lorenz-Mie theories, from past to future," Atomization Sprays 10, 277-333 (2000).

J. Opt. Soc. Am. A (3)

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

Opt. Commun. (1)

Y. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, and G. Gréhan, "Scattering of light by spheroids: the far field case," Opt. Commun. 210, 1-9 (2002).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

H. M. Lai, P. T. Leung, and K. Young, "Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets," Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

Prog. Energy Combust. Sci. (1)

G. Chern, Md. M. Mazumder, and R. K. Chang, "Laser diagnostics for droplets characterization: application of morphology dependent resonances," Prog. Energy Combust. Sci. 22, 163-188 (1996).
[CrossRef]

Other (1)

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 1999).

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Figures (14)

Fig. 1
Fig. 1

Definition of the configuration under study.

Fig. 2
Fig. 2

TE spectra for a microcavity ( m = 1.33 ) with a diameter d equal to 6 μ m illuminated by a spectrum of plane waves. Three independent codes are compared.

Fig. 3
Fig. 3

Individual first-order scattering coefficients computed in the framework of the Lorenz–Mie theory for d = 6 μ m and m = 1.33 .

Fig. 4
Fig. 4

TE and TM spectra with the first-order resonances named.

Fig. 5
Fig. 5

(a) FWHM Γ of the first-order resonances is evaluated from our code (triangles), Eq. (10) (diamonds), and Eq. (14) (squares) with d = 6 μ m and m = 1.33 . (b) The absolute value of errors in FWHM versus mode number.

Fig. 6
Fig. 6

Individual first-order scattering coefficient behavior in the case of a water droplet ( m = 1.33 ) with a diameter equal to 6 μ m . (a) Amplitude of scattering coefficients versus mode number. (b) Quality factor versus mode number. LMT, Lorenz–Mie theory.

Fig. 7
Fig. 7

Scattering intensity of a deformed droplet with an aspect ratio a b = 2 for TE and TM modes. (a) Angular distribution of intensity obtained from Asano and Yamamoto’s Fig. 3 in Ref. [13]. (b) Angular distribution of intensity evaluated by our code for the same conditions as in (a).

Fig. 8
Fig. 8

Angular distribution of the intensity functions for TE and TM modes obtained from our code (curve) and Mishchenko’s T-matrix code (circles), with an aspect ratio a b = 4 , m = 1.33 , and r v = 3 μ m .

Fig. 9
Fig. 9

Behavior of first-order scattering coefficients versus wavelength for m = 1.33 , r v = 3 μ m .

Fig. 10
Fig. 10

Locations of the resonance of first-order scattering coefficients b n , 1 ( n = 29 35 ) obtained numerically from our theory (triangles) and Lai’s Eq. (17) (curves) for m = 1.33 , r v = 3 μ m .

Fig. 11
Fig. 11

Width Δ λ n , 1 ’s are obtained numerically from our theory (triangles) and Eq. (18) (curves) as a function of ellipticity a b for r v = 3 μ m . (a) m = 1.33 , (b) m = 1.38 .

Fig. 12
Fig. 12

Behavior of the quality factor Q versus the ellipticity for r v = 3 μ m with m = 1.33 , 1.38 .

Fig. 13
Fig. 13

Values of Q versus a b for m = 1.33 , r v = 10 μ m . Triangles, numerical results from our theory; curves, from Eq. (19).

Fig. 14
Fig. 14

Strength of the resonance of first-order scattering coefficients versus a b for (a) b 26 , 1 and b 27 , 1 with r v = 2 μ m , m = 1.33 , (b) b 32 , 1 and b 33 , 1 with r v = 2 μ m , m = 1.33 , 1.38 , respectively. Curves, from Eq. (20); triangles, from our theory.

Tables (1)

Tables Icon

Table 1 Location and Quality Factor for Some Individual Scattering Coefficients a

Equations (36)

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E i = E 0 l = 0 n = l i n [ G n l , TE M e l n r ( 1 ) ( c ( I ) , ζ , η , ϕ ) + i G n l , TM N o l n r ( 1 ) ( c ( I ) , ζ , η , ϕ ) ] .
E TE s = l = 0 n = l i n [ b n l , TE M e l n r ( 3 ) ( c ( I ) , ζ , η , ϕ ) + i a n l , TE N o l n r ( 3 ) ( c ( I ) , ζ , η , ϕ ) ] ,
H TE s = k 1 ω μ 1 l = 0 n = l i n [ a n l , TE M o l n r ( 3 ) ( c ( I ) , ζ , η , ϕ ) i b n l , TE N e l n r ( 3 ) ( c ( I ) , ζ , η , ϕ ) ] ,
E TE w = l = 0 n = l i n [ d n l , TE M e l n r ( 1 ) ( c ( I I ) , ζ , η , ϕ ) + i c n l , TE N o l n r ( 1 ) ( c ( I I ) , ζ , η , ϕ ) ] ,
H TE w = k 2 ω μ 2 l = 0 n = l i n [ c n l , TE M o l n r ( 1 ) ( c ( I I ) , ζ , η , ϕ ) i d n l , TE N e l n r ( 1 ) ( c ( I I ) , ζ , η , ϕ ) ] .
E η i + E η s = E η w ,
E ϕ i + E ϕ s = E ϕ w ,
H η i + H η s = H η w ,
H ϕ i + H ϕ s = H ϕ w ,
n = l i n [ V n l ( 3 ) , t U n l ( 3 ) , t V n l ( 1 ) , t U n l ( 1 ) , t Y n l ( 3 ) , t X n l ( 3 ) , t Y n l ( 1 ) , t X n l ( 1 ) , t U n l ( 3 ) , t V n l ( 3 ) , t L U n l ( 1 ) , t L V n l ( 1 ) , t X n l ( 3 ) , t Y n l ( 3 ) , t L X n l ( 1 ) , t L Y n l ( 1 ) , t ] ( a n l b n l c n l d n l ) = n = l i n ( G n l , TE U n l ( 1 ) , t + G n l , TM V n l ( 1 ) , t G n l , TE X n l ( 1 ) , t + G n l , TM Y n l ( 1 ) , t G n l , TM U n l ( 1 ) , t + G n l , TE V n l ( 1 ) , t G n l , TM X n l ( 1 ) , t + G n l , TE Y n l ( 1 ) , t ) ( l = 0 , 1 , 2 , ; t = 0 , 1 , 2 , ) ,
Q = λ n , l Δ λ n , l ,
m x n , l = ν + 2 1 3 τ i ν 1 3 P ( m 2 1 ) 1 2 + ( 3 10 2 2 3 ) τ i 2 ν 1 3 2 1 3 P [ m 2 ( 2 P 2 3 ) ] ( m 2 1 ) 3 2 τ i ν 2 3 + O ( ν 1 )
Γ n , l Lam = 2 ( N x n , l Y n 2 2 ) 1
ν = n + 1 2 ,
P = { m , for TE modes 1 m , for TM modes } ,
N = { m 2 1 for TE mode ( m 2 1 ) [ ( n + 0.5 x n , l ) 2 ( 1 + 1 m 2 ) 1 ] for TM mode ) .
Γ n , l Prober = 2 m P ( m 2 1 ) [ ( 2 u 0 ν ) 1 2 1 4 u 0 ] exp [ 4 u 0 3 ( 2 u 0 ν ) 1 2 ] ,
r v = ( a b 2 ) 1 3 ,
r v = ( a 2 b ) 1 3 ,
λ n , l = λ n , l ( 0 ) { 1 + e 6 [ 1 3 α 2 n ( n + 1 ) ] } ,
Δ λ n , 1 = Δ λ n , 1 0 { 1 + ( 0.02 n 0.37 ) e + [ 0.113 n 2 ( m 2 1 ) 3.5 n + 45.6 ] e 2 } ,
Q n = λ n , 1 Δ λ n , 1 = Q 0 ( 1 e 6 ) { 1 + ( 0.02 n 0.37 ) e + [ 0.113 n 2 ( m 2 1 ) 3.5 n + 45.6 ] e 2 } ,
P = P n , 1 0 { 0.97 + 0.003 n 0.00005 n 2 + [ 3 ( 1.16 m 2 1.26 ) n ] e } ,
U n l ( j ) , t ( c ( h ) ) = l ζ 0 R n l ( j ) [ ( ζ 0 2 1 ) 2 B t n l + 2 ( ζ 0 2 1 ) A t n l + E t n l ] ,
V n l ( j ) , t = i c { l 2 R n l ( j ) ζ 0 2 1 [ ( ζ 0 2 1 ) 2 D t n l + 2 ( ζ 0 2 1 ) C t n l + F t n l ] R n l ( j ) [ λ n l ( c ζ 0 ) 2 + l 2 ζ 0 2 1 ] ] [ ( ζ 0 2 1 ) C t n l + F t n l ] + ζ 0 ( ζ 0 2 1 ) d d ζ R n l ( j ) ζ 0 [ 2 C t n l ζ 0 + ( ζ 0 2 1 ) G t n l + I t n l ] { + R n l ( j ) [ ( ζ 0 2 1 ) 2 G t n l + ( 3 ζ 0 2 1 ) I t n l ] } ,
X n l ( j ) , t = ζ 0 R n l ( j ) G t n l d d ζ R n l ( j ) ζ 0 C t n l ,
Y n l ( j ) , t = i l c { ( ζ 0 2 1 ) 1 R n l ( j ) ( A t n l + H t n l ) + B t n l [ R n l ( j ) + ζ 0 d d ζ R n l ( j ) ζ 0 ] } .
A t n l = N l 1 , l 1 t 1 r = 0 , 1 d r n l 1 + 1 ( 1 η 2 ) 1 2 P l + r l ( η ) P l 1 + t l 1 ( η ) d ( η ) ,
B t n l = N l t , l 1 + t 1 r = 0 , 1 d r n l 1 + 1 ( 1 η 2 ) 1 2 P 1 + r l ( η ) P l 1 + t l 1 d ( η ) ,
C t n l = N l t , l 1 + t 1 r = 0 , 1 d r n l 1 + 1 η ( 1 η 2 ) 1 2 P l + r l ( η ) P l 1 + t l 1 d ( η ) ,
D t n l = N l t , l 1 + t 1 r = 0 , 1 d r n l 1 + 1 η ( 1 η 2 ) 1 2 P l + r l ( η ) P l 1 + t l 1 d ( η ) ,
E t n l = N l t , l 1 + t 1 r = 0 , 1 d r n l 1 + 1 ( 1 η 2 ) 3 2 P l + r l ( η ) P l 1 + t l 1 d ( η ) ,
F t n l = N l 1 , l 1 + t 1 r = 0 , 1 d r n l 1 + 1 η ( 1 η 2 ) 3 2 P l + r 1 ( η ) P t 0 ( η ) d ( η ) ,
G t n l = N l 1 , l 1 + t 1 r = 0 , 1 d r n l 1 + 1 ( 1 η 2 ) 1 2 d P l + r l ( η ) d η P l 1 + t l 1 ( η ) d ( η ) ,
H t n l = N l 1 , l 1 + t 1 r = 0 , 1 d r n l 1 + 1 η ( 1 η 2 ) 1 2 d P l + r l ( η ) d η P l 1 + t l 1 d ( η ) ,
I t n l = N l 1 , l 1 + t 1 r = 0 , 1 d r n l 1 + 1 ( 1 η 2 ) 3 2 d P l + r l ( η ) d η P l 1 + t l 1 d ( η ) .

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