Abstract

The quasi-bound modes localized on stable periodic ray orbits of dielectric micro-cavities are constructed in the short-wavelength limit using the parabolic equation method. These modes are shown to coexist with irregularly spaced “chaotic” modes for the generic case. The wavevector quantization rule for the quasi-bound modes is derived and given a simple physical interpretation in terms of Fresnel reflection; quasi-bound modes are explictly constructed and compared to numerical results. The effect of discrete symmetries of the resonator is analyzed and shown to give rise to quasi-degenerate multiplets; the average splitting of these multiplets is calculated by methods from quantum chaos theory.

© 2002 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  29. O. A. Starykh, P. R. J. Jacquod, E. E. Narimanov, and A. D. Stone, �??Signature of dynamical localization in the resonance width distribution of wave-chaotic dielectric cavities,�?? Phys. Rev. E 62, 2078�??2084 (2000).
    [CrossRef]
  30. J. U. Nockel, �??Angular momentum localization in oval billiards,�?? Phys. Scr. T90, 263�??267 (2001).
    [CrossRef]
  31. V. M. Babi¡c and V. S. Buldyrev, Asymptotic Methods in Shortwave Di.raction Problems (Springer, New York, USA, 1991).
  32. F. Laeri and J. U. N¨ockel, �??Nanoporous compound materials for optical applications �?? Microlasers and microresonators,�?? in Handbook of Advanced Electronic and Photonic Materials, H. S. Nalwa, ed. (Academic Press, San Diego, 2001).
    [CrossRef]
  33. A. E. Siegman, Lasers (University Science Books, Mill Valley, California, 1986).
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    [CrossRef]
  35. N. A. Chernikov, �??System whose hamiltonian is a time-dependent quadratic form in x and p,�?? Sov. Phys.-Jetp Engl. Trans. 26, 603�??608 (1968).
  36. E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, T. S. S., and K. Young, �??Quasinormal-mode expansion for waves in open systems,�?? Rev. Mod. Phys. 70, 1545�??1554 (1998).
    [CrossRef]
  37. J. W. Ra, H. L. Bertoni, and L. B. Felsen, �??Reflection and transmission of beams at a dielectric interface,�?? SIAM J. Appl. Math 24, 396�??413 (1973).
    [CrossRef]
  38. J. M. Robbins, �??Discrete symmetries in periodic-orbit theory,�?? Phys. Rev. A 40, 2128�??2136 (1989).
    [CrossRef] [PubMed]
  39. M. J. Davis and E. J. Heller, �??Multidimensional wave functions from classical trajectories,�?? J. Chem. Phys. 75, 246 (1981).
    [CrossRef]
  40. O. Bohigas, S. Tomsovic, and D. Ullmo, �??Manifestations of classical phase space structures in quantum mechanics,�?? Phys. Rep. 223, 45 (1993).
    [CrossRef]
  41. S. D. Frischat and E. Doron, �??Semiclassical description of tunneling in mixed systems: case of the annular billiard,�?? Phys. Rev. Lett. 75, 3661 (1995).
    [CrossRef] [PubMed]
  42. F. Leyvraz and D. Ullmo, �??The level splitting distribution in chaos-assisted tunneling,�?? J. Phys. A 29, 2529 (1996).
    [CrossRef]
  43. E. E. Narimanov, unpublished.
  44. M. V. Berry, �??Regular and irregular semiclassical wavefunctions,�?? J. Phys. A 10, 2083 (1977).
    [CrossRef]
  45. H. E. Tureci and A. D. Stone, �??Deviation from Snell�??s law for beams transmitted near the critical angle: application to microcavity lasers,�?? Opt. Lett. 27, 7�??9 (2002).
    [CrossRef]
  46. E. E. Narimanov, G. Hackenbroich, P. Jacquod, and A. D. Stone, �??Semiclassical theory of the emission properties of wave-chaotic resonant cavities,�?? Phys. Rev. Lett. 83, 4991�??4994 (1999).
    [CrossRef]

256

H. Yokoyama, �??Physics and device applications of optical microcavities,�?? Science 256, 66�??70 (1992).
[CrossRef] [PubMed]

Ann. Phys.

J. B. Keller and S. I. Rubinow, �??Asymptotic Solution of Eigenvalue Problems,�?? Ann. Phys. 9, 24�??75 (1960).
[CrossRef]

Appl. Phys. B

I. Braun, G. Ihlein, F. Laeri, J. U. Nockel, G. Schulz-Eklo., F. Schuth, U. Vietze, O. Weiss, and D. Wohrle, �??Hexagonal microlasers based on organic dyes in nanoporous crystals,�?? Appl. Phys. B-Lasers Opt. 70, 335�??343 (2000).
[CrossRef]

Eur. J. Phys.

M. V. Berry, �??Regularity and chaos in classical mechanics, illustrated by three deformations of a circular billiard,�?? Eur. J. Phys. 2, 91�??102 (1981).
[CrossRef]

IEEE J. Quantum Electron.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, �??Long-wavelength λ = 10μm quadrupolar-shaped GaAs-AlGaAs microlasers,�?? IEEE J. Quantum Electron. 36, 458�??464 (2000).
[CrossRef]

IEEE Photonics Technol. Lett.

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]

J. Chem. Phys.

W. H. Miller, �??Semiclassical quantization of nonseparable systems: A new look at periodic orbit theory,�?? J. Chem. Phys. 63, 996�??999 (1975).
[CrossRef]

M. J. Davis and E. J. Heller, �??Multidimensional wave functions from classical trajectories,�?? J. Chem. Phys. 75, 246 (1981).
[CrossRef]

J. Opt. Soc. Am. B

J. Opt. Soc.Am. B

S. Chang, R. K. Chang, A. D. Stone, and J. U. Nockel, �??Observation of emission from chaotic lasing modes in deformed microspheres: displacement by the stable-orbit modes,�?? J. Opt. Soc.Am. B 17, 1828�??1834 (2000).
[CrossRef]

J. Phys. A

S. D. Frischat and E. Doron, �??Quantum phase-space structures in classically mixed systems: A scattering approach,�?? J. Phys. A-Math. Gen. 30, 3613�??3634 (1997).
[CrossRef]

B. Li and M. Robnik, �??Geometry of high-lying eigenfunctions in a plane billiard system having mixed type classical dynamics,�?? J. Phys. A 28, 2799�??2818 (1995).
[CrossRef]

F. Leyvraz and D. Ullmo, �??The level splitting distribution in chaos-assisted tunneling,�?? J. Phys. A 29, 2529 (1996).
[CrossRef]

M. V. Berry, �??Regular and irregular semiclassical wavefunctions,�?? J. Phys. A 10, 2083 (1977).
[CrossRef]

Nature

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, �??Ultralow-threshold Raman laser using a spherical dielectric microcavity,�?? Nature 415, 621�??623 (2002).
[CrossRef] [PubMed]

J. U. Nockel and A. D. Stone, �??Ray and wave chaos in asymmetric resonant optical cavities,�?? Nature 385, 45�??47 (1997).
[CrossRef]

Opt. Lett.

Phys. Rep.

O. Bohigas, S. Tomsovic, and D. Ullmo, �??Manifestations of classical phase space structures in quantum mechanics,�?? Phys. Rep. 223, 45 (1993).
[CrossRef]

Phys. Rev. A

J. M. Robbins, �??Discrete symmetries in periodic-orbit theory,�?? Phys. Rev. A 40, 2128�??2136 (1989).
[CrossRef] [PubMed]

Phys. Rev. E

O. A. Starykh, P. R. J. Jacquod, E. E. Narimanov, and A. D. Stone, �??Signature of dynamical localization in the resonance width distribution of wave-chaotic dielectric cavities,�?? Phys. Rev. E 62, 2078�??2084 (2000).
[CrossRef]

Phys. Rev. Lett.

S. B. Lee, J. H. Lee, J. S. Chang, H. J. Moon, S. W. Kim, and K. An, �??Observation of scarred modes in asymmetrically deformed microcylinder lasers,�?? Phys. Rev. Lett. 88, art. no.033903 (2002).
[CrossRef] [PubMed]

E. J. Heller, �??Bound-state eigenfunctions of classically chaotic hamiltonian-systems - Scars of periodic orbits,�?? Phys. Rev. Lett. 53, 1515�??1518 (1984).
[CrossRef]

A. Mekis, J. U. Nockel, 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]

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, �??Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators,�?? Phys. Rev. Lett. 88, art. no.094 102 (2002).
[CrossRef]

A. J. Campillo, J. D. Eversole, and H. B. Lin, �??Cavity quantum electrodynamic enhancement of stimulated-emission in microdroplets,�?? Phys. Rev. Lett. 67, 437�??440 (1991).
[CrossRef] [PubMed]

E. E. Narimanov, G. Hackenbroich, P. Jacquod, and A. D. Stone, �??Semiclassical theory of the emission properties of wave-chaotic resonant cavities,�?? Phys. Rev. Lett. 83, 4991�??4994 (1999).
[CrossRef]

S. D. Frischat and E. Doron, �??Semiclassical description of tunneling in mixed systems: case of the annular billiard,�?? Phys. Rev. Lett. 75, 3661 (1995).
[CrossRef] [PubMed]

Phys. Scr.

A. D. Stone, �??Wave-chaotic optical resonators and lasers,�?? Phys. Scr. T90, 248�??262 (2001).
[CrossRef]

J. U. Nockel, �??Angular momentum localization in oval billiards,�?? Phys. Scr. T90, 263�??267 (2001).
[CrossRef]

QELS Technical Digest

H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, and A. D. Stone, �??Dramatic shape sensitivity of emission patterns for similarly deformed cylindrical polymer lasers,�?? in QELS Technical Digest (Optical Society of America, Washington, D.C., 2002), pp. 24�??25.

Rev. Mod. Phys.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, T. S. S., and K. Young, �??Quasinormal-mode expansion for waves in open systems,�?? Rev. Mod. Phys. 70, 1545�??1554 (1998).
[CrossRef]

Science

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, 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�??488 (1986).
[CrossRef] [PubMed]

SIAM J. Appl. Math

J. W. Ra, H. L. Bertoni, and L. B. Felsen, �??Reflection and transmission of beams at a dielectric interface,�?? SIAM J. Appl. Math 24, 396�??413 (1973).
[CrossRef]

Sov. Phys.-Jetp Engl. Trans.

N. A. Chernikov, �??System whose hamiltonian is a time-dependent quadratic form in x and p,�?? Sov. Phys.-Jetp Engl. Trans. 26, 603�??608 (1968).

Other

E. E. Narimanov, unpublished.

N. B. Rex, Regular and chaotic orbit Gallium Nitride microcavity lasers, Ph.D. thesis, Yale University (2001).

R. K. Chang, A. K. Campillo, eds., Optical Processes in Microcavities (World Scienti.c, Singapore, 1996).

M. C. Gutzwiller, Chaos in classical and quantum mechanics (Springer, New York, USA, 1990).

V. M. Babi¡c and V. S. Buldyrev, Asymptotic Methods in Shortwave Di.raction Problems (Springer, New York, USA, 1991).

F. Laeri and J. U. N¨ockel, �??Nanoporous compound materials for optical applications �?? Microlasers and microresonators,�?? in Handbook of Advanced Electronic and Photonic Materials, H. S. Nalwa, ed. (Academic Press, San Diego, 2001).
[CrossRef]

A. E. Siegman, Lasers (University Science Books, Mill Valley, California, 1986).

V. P. Maslov and M. V. Fedoriuk, Semiclassical Approximations in Quantum Mechanics (Reidel, Boston, USA, 1981).
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

Surface of section illustrating the different regions of phase space for a closed quadrupole billiard with boundary given by r(ϕ) = R(1 + cos 2ϕ) for = 0.072. Real-space ray trajectories corresponding to each region are indicated at right: a) A quasi-periodic, marginally stable orbit. b) A stable four-bounce “diamond” periodic orbit (surrounded by stability “islands” in the SOS) c) A chaotic ray trajectory. Orbits of type (b) have associated with them regular gaussian solutions as we will show below.

Fig. 2.
Fig. 2.

Black background gives the surface of section for the quadrupole at = 0.17 for which the four small islands correspond to a stable bow-tie shaped orbit (inset). A numerical solution of the Helmholtz equation for this resonator can be projected onto this surface of section via the Husimi transform[28] and is found to have high intensity (in false color scale) precisely on these islands, indicating that this is a mode associated with the bow-tie orbit.

Fig. 3.
Fig. 3.

(a) Vertical lines indicate wavevectors of bound states of the closed quadrupole resonator for = 0.17; no regular spacings are visible. (b) Spectrum weighted by overlap of the Husimi function of the solution with the bow-tie island as in Fig. 2. Note the emergence of regularly spaced levels with two main spacings Δklong and Δktrans . These spacings, indicated by the arrows, are calculated from the length of the bow-tie orbit and the associated Floquet phase (see Section 2.4 below). The color coding corresponds to the four possible symmetry types of the solutions (see Section 4 below). In the inset is a magnified view showing the splitting of quasi-degenerate doublets as discussed in Section 4.3. Note the pairing of the (+, +) and (+, -) symmetry types as predicted in section 4.2. The different symmetry pairs alternate every free spectral range (Δklong ).

Fig. 4.
Fig. 4.

Illustration of the reduction of the Maxwell equation for an infinite dielectric cylinder to the 2D Helmholtz equation for the TM case (E field parallel to axis) and k = 0.

Fig. 5.
Fig. 5.

Coordinate system and variables used in the text displayed for the case of a quadrupolar boundary ∂D and the diamond four-bounce PO. A fixed coordinate system (X, Z) is attached to the origin. The “mobile” coordinate systems (xm ,zm ) are fixed on segments of the periodic orbit so that their respective z-axes are parallel to the segment, while their origins are set back a distance Lm (or nLm for transmitted beam axes), so as to account for zeroth order phase accumulation between successive bounce-points. ξ 1, ξ 2 are the common local coordinates at each bounce (index m suppressed). Scaled coordinates are denoted by tildes, e.g. m = √kxm . The coordinate transformations at each bounce m are given by zi = Lm + ξ 1 sin χi + ξ 2 cos χi , zr = Lm + ξ 1 sin χi - ξ 2 cos χi , zt = nLm + ξ 1 sin χt + ξ 2 cos χt and xi = ξ 1 cos χi - ξ 2 sin χi , xr = ξ 1 cos χi + ξ 2 sin χi , xt = ξ 1 cos χt - ξ 2 sin χt , where i, r, t refer to the incident, transmitted and reflected solutions.

Fig. 6.
Fig. 6.

Intensity of the TM solution for a bow-tie mode plotted in a false color scale, (a) calculated numerically and (b) from the gaussian optical theory with parameters m = 100, φ = 2.11391, Nμ = 1 and N = 4. Note the excellent agreement of the quantized values for kR (R is the average radius of the quadrupole).

Fig. 7.
Fig. 7.

Schematic indicating a direct tunneling process (black arrow) and a chaos-assisted tunneling process (yellow arrow) which would contribute to splitting of bow-tie doublets.

Fig. 8.
Fig. 8.

The numerically determined splittings of bow-tie doublets for a closed quadrupole resonator with ε = 0.14 (black dots) vs kR; the red line denotes the prediction of Eq. (60) for the average splitting, the blue line an estimate of the splitting based on the “direct” coupling. Note the large enhancement due to chaos-assisted tunneling and the large fluctuations around the mean splitting.

Tables (2)

Tables Icon

Table 1. Comparison of the gaussian optical predictions

Tables Icon

Table 2. Illustration of the symmetry rules for the quadrupole

Equations (72)

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

( 2 + k 2 ) E = 0
E ( X , Z ) = m = 1 N E m ( x m ( X , Z ) , z m ( X , Z ) )
E m ( x m , z m ) = u m ( x m , z m ) e ik z m
( m 2 + k 2 ) E m = 0
E m + E m + 1 | D = 0
u xx + u zz + 2 ik u z = 0
u x ˜ x ˜ + 1 k u zz + 2 i u z = 0 .
u ( x ˜ , z ) = 0
u ( x , z ) = c A ( z ) exp [ i 2 Ω ( z ) x ˜ 2 ]
Ω 2 + Ω = 0
A Ω + 2 A = 0
Ω = Q ( z ) Q ( z )
Q = 0
Q Q + 2 A A = 0
c m Q m ( z m ) exp ( ik z m + i 2 Ω ( z m ) x ˜ m 2 )
+ c m + 1 Q m + 1 ( z m + 1 ) exp ( ik z m + 1 + i 2 Ω ( z m + 1 ) x ˜ m + 1 2 ) | D = 0
k ( l m + 1 k ξ ˜ 1 sin χ m 1 k ξ ˜ 1 2 2 ρ m cos χ m ) + 1 2 Q m Q m ( ξ ˜ 1 cos χ m + 1 k ξ ˜ 1 2 2 ρ m sin χ m ) 2 =
k ( l m + 1 k ξ ˜ 1 sin χ m + 1 k ξ ˜ 1 2 2 ρ m cos χ m ) + 1 2 Q m + 1 Q m + 1 ( ξ ˜ 1 cos χ m + 1 k ξ ˜ 1 2 2 ρ m sin χ m ) 2
c m Q m ( L m ) + c m + 1 Q m + 1 ( L m ) = 0
( Q m + 1 Q m + 1 ) = ( 1 0 2 ρ m cos χ m 1 ) ( Q m Q m ) m ( Q m Q m )
C m + 1 = e i π C m
= ( Q 1 Q 2 P 1 P 2 )
d dz = H
( Q ( z ) P ( z ) ) = ( z ) h m
( Q m ( z + l ) P m ( z + l ) ) = ( l ) ( Q m ( z ) P m ( z ) ) .
( Q m + 1 ( z ) P m + 1 ( z ) ) = ( z L m ) m ( L m z ) ( Q m ( z ) P m ( z ) )
𝚳 ( z ) = ( z L m 1 ) m 1 ( l m 1 ) ( l m + 1 ) m ( L m z )
E ( x , z + L ) = E ( x , z ) .
u ( x , z + L ) e ikL = u ( x , z ) .
p q * q p * = i .
u ( x ˜ , z + L ) = e i φ 2 ( N μ + N ) u ( x ˜ , z )
kL = 1 2 φ + 2 π m + mod 2 π [ ( N + N μ ) π ]
N μ = [ 1 2 π i 0 L d ( ln q ( z ) ) ]
Λ ( z ) = iq ( z ) x ˜ + p ( z ) x ˜
Λ ( z ) = i q * ( z ) x ˜ + p * ( z ) x ˜ .
u ( l ) ( x ˜ , z ) = ( Λ ) l u ( 0 )
kL = ( l + 1 2 ) φ + 2 π m + mod 2 π [ ( N + N μ ) π ]
u ( l ) ( x ˜ , z ) = ( i q * ( z ) 2 q ( z ) ) l H l ( I m [ p ( z ) q ( z ) ] x ˜ ) u ( 0 ) ( x ˜ , z )
( 2 + n ( r ) 2 k 2 ) Ψ = 0
E i + E r | D = E t | D +
n E i + n E r | D = n E t | D +
Φ i = nk ( L m + 1 nk ξ ˜ 1 sin χ 1 nk ξ ˜ 1 2 2 ρ cos χ ) + 1 2 Q i Q i ( ξ ˜ 1 cos χ + 1 nk ξ ˜ 1 2 2 ρ sin χ ) 2
Φ t = k ( n L m + 1 nk ξ ˜ 1 sin χ t 1 nk ξ ˜ 1 2 2 ρ cos χ t ) + 1 2 Q t Q t ( ξ ˜ 1 cos χ t + 1 nk ξ ˜ 1 2 2 ρ sin χ t ) 2
( Q t Q t ) = ( 1 μ 0 2 ( 1 μ ) ρ cos χ n μ ) ( Q i Q i )
c i Q i + c r Q r = c t Q t
c i + c r = μ c t
c i n Φ i + c r n Φ r = μ c t n Φ t .
n μ ( c i c r ) = c t
c r = 1 + 1 c i .
c r 2 = n cos χ i cos χ t 2 n cos χ i + cos χ t 2 c i 2
E ( x , z + L ) = E ( x , z )
u ( x , z + L ) e inkL = u ( x , z )
nkL = 1 2 φ + 2 πm + mod 2 π [ ( N + N μ ) π ] i b = 1 N log [ n μ b 1 n μ n + 1 ] .
Re [ nkL ] = 2 πm + mod 2 π [ ( N + N μ ) π ] + φ 2 + φ f
Im [ nkL ] = γ f .
P m E ( x ) = d m G g G χ m ( g ) E ( g x )
E ( + + ) = 1 4 ( e 1 + e 2 + e 3 + e 4 ) , E ( + ) = 1 4 ( e 1 + e 2 e 3 e 4 )
E ( + ) = 1 4 ( e 1 e 2 + e 3 e 4 ) , E ( ) = 1 4 ( e 1 e 2 e 3 + e 4 )
e 1 = E ( X , Z ) , e 2 = E ( X , Z ) , e 3 = E ( X , Z ) , e 4 = E ( X , Z ) .
e 1 = E ( g x ) = 1 q ( z ) exp [ ikz + i 2 Ω ( z ) x 2 ]
e 2 = E ( g x ) = e 1 2 i π 1 e i φ 2 q ( z ) exp [ ik ( z + L 2 + i 2 Ω ( z ) x 2 ) ] e i ζ e 1
e 3 = E ( g x ) = 1 q ( 1 z ) exp [ ik ( 1 z ) + i 2 Ω ( 1 z ) x 2 ]
e 4 = E ( g x ) = e 1 2 i π 1 e i φ 2 q ( 1 z ) exp [ ik ( L 2 + 1 z ) + i 2 Ω ( 1 z ) x 2 ] e i ζ e 3
E ( + + ) = 1 2 ( e 1 + e 3 ) e i ζ 2 cos ζ 2
E ( + ) = 1 2 ( e 1 e 3 ) e i ζ 2 cos ζ 2
E ( + ) = 1 2 i ( e 1 + e 3 ) e i ζ 2 sin ζ 2
E ( ) = 1 2 i ( e 1 + e 3 ) e i ζ 2 sin ζ 2
E ( + + ) , E ( + ) cos ζ 2 = 0 m = 1,3,5 , . . .
E ( + ) , E ( ) sin ζ 2 = 0 m = 0,2,4 , . . .
Δ E CAT V RC 2 E R E C
V RC 2 d ϕ d sin χ W C ( ϕ , sin χ ) W R ( ϕ , sin χ )
Δ E CAT exp ( Ak R 0 )

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