Abstract

Recent experiments on similarly shaped polymer microcavity lasers show a dramatic difference in the far-field emission patterns. We show, for different deformations of the ellipse, quadrupole and hexadecapole, that the large differences in the far-field emission patterns are explained by the differing ray dynamics corresponding to each shape. Analyzing the differences in the appropriate phase space for ray motion, it is shown that the differing geometries of the unstable manifolds of periodic orbits are the decisive factors in determining the far-field pattern. Surprisingly, we find that strongly chaotic ray dynamics is compatible with highly directional emission in the far field.

© 2004 Optical Society of America

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References

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  1. R. K. Chang and A. K. Campillo, eds., Optical Processes in Microcavities (World Scientific, Singapore, 1996).
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    [CrossRef]
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    [CrossRef]
  4. J. U. Nöckel, A. D. Stone, G. Chen, H. L. Grossman, and R. K. Chang, “Directional emission from asymmetric resonant cavities,” Opt. Lett. 21, 1609–1611 (1996).
    [CrossRef]
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    [CrossRef]
  6. S. Chang, R. K. Chang, A. D. Stone, and J. U. Nöckel, “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]
  7. S. Lacey, H. Wang, D. H. Foster, and J. U. Nöckel, “Directional tunnel escape from nearly spherical optical resonators,” Phys. Rev. Lett. 91, 033902 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  12. H. E. Tureci, H. G. L. Schwefel, Ph. Jacrod, and A. Douglas Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt. (to be published).
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    [CrossRef]
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  15. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1989).
  16. 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]
  17. J. U. Nöckel, “Resonances in nonintegrable open systems,” Ph.D. dissertation (Yale University, New Haven, Conn., 1997).
  18. H. Poritsky, “The billiard ball problem on a table with convex boundary: an illustrative dynamical problem,” Ann. Math. 51, 446–470 (1950).
    [CrossRef]
  19. E. Y. Amiran, “Integrable smooth planar billiards and evolutes,” New York J. Math. 3, 32–47 (1997).
  20. L. A. Bunimovich, “On ergodic properties of nowhere dispersing billiards,” Commun. Math. Phys. 65, 295–312 (1977).
    [CrossRef]
  21. J. B. Keller, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (N.Y.) 9, 24–75 (1960).
    [CrossRef]

2003 (1)

S. Lacey, H. Wang, D. H. Foster, and J. U. Nöckel, “Directional tunnel escape from nearly spherical optical resonators,” Phys. Rev. Lett. 91, 033902 (2003).
[CrossRef]

2002 (1)

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, 094102 (2002).
[CrossRef] [PubMed]

2000 (1)

1999 (1)

S. Chang, N. B. Rex, R. K. Chang, G. B. Chong, and L. J. Guido, “Stimulated emission and lasing in whispering gallery modes of GaN microdisk cavities,” Appl. Phys. Lett. 75, 3719–3719 (1999).
[CrossRef]

1997 (2)

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

E. Y. Amiran, “Integrable smooth planar billiards and evolutes,” New York J. Math. 3, 32–47 (1997).

1996 (1)

1994 (1)

1993 (1)

Y. Yamamoto and R. E. Slusher, “Optical processes in microcavities,” Phys. Today 46, 66–73 (1993).
[CrossRef]

1981 (1)

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]

1977 (1)

L. A. Bunimovich, “On ergodic properties of nowhere dispersing billiards,” Commun. Math. Phys. 65, 295–312 (1977).
[CrossRef]

1960 (1)

J. B. Keller, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (N.Y.) 9, 24–75 (1960).
[CrossRef]

1950 (1)

H. Poritsky, “The billiard ball problem on a table with convex boundary: an illustrative dynamical problem,” Ann. Math. 51, 446–470 (1950).
[CrossRef]

1927 (1)

G. D. Birkhoff, “On the periodic motion of dynamical systems,” Acta Math. 50, 359–379 (1927).
[CrossRef]

Amiran, E. Y.

E. Y. Amiran, “Integrable smooth planar billiards and evolutes,” New York J. Math. 3, 32–47 (1997).

Berry, M. V.

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]

Birkhoff, G. D.

G. D. Birkhoff, “On the periodic motion of dynamical systems,” Acta Math. 50, 359–379 (1927).
[CrossRef]

Bunimovich, L. A.

L. A. Bunimovich, “On ergodic properties of nowhere dispersing billiards,” Commun. Math. Phys. 65, 295–312 (1977).
[CrossRef]

Chang, R. K.

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, 094102 (2002).
[CrossRef] [PubMed]

S. Chang, R. K. Chang, A. D. Stone, and J. U. Nöckel, “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]

S. Chang, N. B. Rex, R. K. Chang, G. B. Chong, and L. J. Guido, “Stimulated emission and lasing in whispering gallery modes of GaN microdisk cavities,” Appl. Phys. Lett. 75, 3719–3719 (1999).
[CrossRef]

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

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]

Chang, S.

S. Chang, R. K. Chang, A. D. Stone, and J. U. Nöckel, “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]

S. Chang, N. B. Rex, R. K. Chang, G. B. Chong, and L. J. Guido, “Stimulated emission and lasing in whispering gallery modes of GaN microdisk cavities,” Appl. Phys. Lett. 75, 3719–3719 (1999).
[CrossRef]

Chen, G.

Chong, G. B.

S. Chang, N. B. Rex, R. K. Chang, G. B. Chong, and L. J. Guido, “Stimulated emission and lasing in whispering gallery modes of GaN microdisk cavities,” Appl. Phys. Lett. 75, 3719–3719 (1999).
[CrossRef]

Foster, D. H.

S. Lacey, H. Wang, D. H. Foster, and J. U. Nöckel, “Directional tunnel escape from nearly spherical optical resonators,” Phys. Rev. Lett. 91, 033902 (2003).
[CrossRef]

Grossman, H. L.

Guido, L. J.

S. Chang, N. B. Rex, R. K. Chang, G. B. Chong, and L. J. Guido, “Stimulated emission and lasing in whispering gallery modes of GaN microdisk cavities,” Appl. Phys. Lett. 75, 3719–3719 (1999).
[CrossRef]

Keller, J. B.

J. B. Keller, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (N.Y.) 9, 24–75 (1960).
[CrossRef]

Lacey, S.

S. Lacey, H. Wang, D. H. Foster, and J. U. Nöckel, “Directional tunnel escape from nearly spherical optical resonators,” Phys. Rev. Lett. 91, 033902 (2003).
[CrossRef]

Nöckel, J. U.

Poritsky, H.

H. Poritsky, “The billiard ball problem on a table with convex boundary: an illustrative dynamical problem,” Ann. Math. 51, 446–470 (1950).
[CrossRef]

Rex, N. B.

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, 094102 (2002).
[CrossRef] [PubMed]

S. Chang, N. B. Rex, R. K. Chang, G. B. Chong, and L. J. Guido, “Stimulated emission and lasing in whispering gallery modes of GaN microdisk cavities,” Appl. Phys. Lett. 75, 3719–3719 (1999).
[CrossRef]

Schwefel, H. G. L.

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, 094102 (2002).
[CrossRef] [PubMed]

Slusher, R. E.

Y. Yamamoto and R. E. Slusher, “Optical processes in microcavities,” Phys. Today 46, 66–73 (1993).
[CrossRef]

Stone, A. D.

Tureci, H. E.

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, 094102 (2002).
[CrossRef] [PubMed]

Wang, H.

S. Lacey, H. Wang, D. H. Foster, and J. U. Nöckel, “Directional tunnel escape from nearly spherical optical resonators,” Phys. Rev. Lett. 91, 033902 (2003).
[CrossRef]

Yamamoto, Y.

Y. Yamamoto and R. E. Slusher, “Optical processes in microcavities,” Phys. Today 46, 66–73 (1993).
[CrossRef]

Acta Math. (1)

G. D. Birkhoff, “On the periodic motion of dynamical systems,” Acta Math. 50, 359–379 (1927).
[CrossRef]

Ann. Math. (1)

H. Poritsky, “The billiard ball problem on a table with convex boundary: an illustrative dynamical problem,” Ann. Math. 51, 446–470 (1950).
[CrossRef]

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

J. B. Keller, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (N.Y.) 9, 24–75 (1960).
[CrossRef]

Appl. Phys. Lett. (1)

S. Chang, N. B. Rex, R. K. Chang, G. B. Chong, and L. J. Guido, “Stimulated emission and lasing in whispering gallery modes of GaN microdisk cavities,” Appl. Phys. Lett. 75, 3719–3719 (1999).
[CrossRef]

Commun. Math. Phys. (1)

L. A. Bunimovich, “On ergodic properties of nowhere dispersing billiards,” Commun. Math. Phys. 65, 295–312 (1977).
[CrossRef]

Eur. J. Phys. (1)

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]

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

Nature (1)

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

New York J. Math. (1)

E. Y. Amiran, “Integrable smooth planar billiards and evolutes,” New York J. Math. 3, 32–47 (1997).

Opt. Lett. (2)

Phys. Rev. Lett. (2)

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, 094102 (2002).
[CrossRef] [PubMed]

S. Lacey, H. Wang, D. H. Foster, and J. U. Nöckel, “Directional tunnel escape from nearly spherical optical resonators,” Phys. Rev. Lett. 91, 033902 (2003).
[CrossRef]

Phys. Today (1)

Y. Yamamoto and R. E. Slusher, “Optical processes in microcavities,” Phys. Today 46, 66–73 (1993).
[CrossRef]

Other (7)

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

N. B. Rex, “Regular and chaotic orbit gallium nitride microcavity lasers,” Ph.D. dissertation (Yale University, New Haven, Conn., 2001).

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

H. E. Tureci, H. G. L. Schwefel, Ph. Jacrod, and A. Douglas Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt. (to be published).

H. Poincaré, Les Méthodes Nouvelles de la Méchanique Céleste (Gauthier-Villars, Paris, France, 1892).

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1989).

J. U. Nöckel, “Resonances in nonintegrable open systems,” Ph.D. dissertation (Yale University, New Haven, Conn., 1997).

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Figures (14)

Fig. 1
Fig. 1

Cross-sectional shapes of micropillar resonators studied: (a) the quadrupole, defined in polar coordinates by R=R0(1+ cos 2ϕ); (b) the ellipse, defined by R=R0{1+[(1+)4-1]sin2 ϕ}-1/2; and (c) the hexadecapole, defined by R=R0[1+(cos2 ϕ+3/2 cos4 ϕ)], all at a deformation of =0.12. Note that all shapes have horizontal and vertical reflection symmetry and have been defined so that the same value of corresponds to approximately the same major-to-minor axis ratio. In (b), χ is the angle of incidence of a ray with respect to the local normal. In (a) and (c), we show short periodic orbits (diamond, rectangle, triangle) relevant to the discussion in text.

Fig. 2
Fig. 2

Two-dimensional display of the experimental data showing in false-color scale the emission intensity as a function of sidewall angle ϕ (converted from ICCD images) and of the far-field angle θ (camera angle). Columns from left to right represent the quadrupole, ellipse, and hexadecapole, respectively. Insets show the cross-sectional shapes of the pillars in each case (for definitions, see Fig. 1). The graphs at the bottom show the far-field patterns obtained by integration over ϕ for each θ, normalized to unity in the direction of maximal intensity. The deformations are =0.12, 0.16, 0.18, 0.20 (red, blue, black, and green, respectively).

Fig. 3
Fig. 3

Far-field intensity for the quadrupole with (a) =0.12 and (b) 0.18. The dash–dotted curve is the experimental result, dashed is the ray simulation, and solid is a numerical solution of the wave equation. The ray simulation was performed starting with 6000 random initial conditions above the critical line and then propagated into the far field in the manner described in the text. The numerical solutions selected have kR0=49.0847-0.0379i with a Q=2593.05 and kR0=49.5927-0.0679i with Q=1460.72 for =0.12 and 0.18, respectively.

Fig. 4
Fig. 4

Far-field intensity for the ellipse with (a) =0.12 and (b) 0.18. Dash–dotted, dashed, and solid curves are experiment, ray simulation, and wave solution. The ray simulation was performed starting with 6000 initial conditions spread over seven KAM curves separated by Δ sin χ=0.02 below the critical KAM curve (that just touches the critical line). The numerical wave solutions shown correspond to kR0=49.1787-0.0028i with Q=17481.38 and kR0=49.2491-0.0110i with Q=4488.20 for =0.12 and 0.18, respectively.

Fig. 5
Fig. 5

Far-field intensity for the hexadecapole with (a) =0.12 and (b) 0.18. Dash–dotted, dashed, and solid curves are experiment, ray simulation, and wave solution, as described in the caption to Fig. 3. The numerical wave solutions shown correspond to kR0=50.5761-0.0024i with Q=42573.77 and kR0=49.5642-0.0092i with Q=10741.93 for =0.12 and 0.18, respectively.

Fig. 6
Fig. 6

Poincaré surface of section for (A) the quadrupole and (B) the ellipse with =0.072. The schematics (A) (a–c) on the right show three classes of orbits for the quadrupole: (A)(a) a quasi-periodic orbit on a KAM curve, (A)(b) a stable period-four orbit, (the “diamond”), and (A)(c) a chaotic orbit. Schematics (B)(a, b) show the two types of orbits that exist in the ellipse, (B)(a) the whispering-gallery type, with an elliptical and (B)(b), the bouncing ball type, with a hyperbolic.

Fig. 7
Fig. 7

Comparison of the Poincaré surface of section for (a) the quadrupole and (b) the ellipse with =0.12 showing mostly chaotic behavior in the former case and completely regular motion in the latter. The dash–dotted line denotes sin χc=1/n, the critical value for total internal reflection; rays above that line are trapped, and those below escape rapidly by refraction. The quadrupole still exhibits stable islands at ϕ=0, π and sin χ=sin χc, which prevent escape at the points of highest curvature in the tangent direction. In (a) we show an adiabatic curve (solid black curve) that has the minimum on the critical line.

Fig. 8
Fig. 8

Poincaré surface of section for the quadrupole with =0.18. The black line indicates the critical angle of incidence. The diamonds indicate the location of the fixed points of the (now) unstable “diamond” orbit, and the squares indicate the fixed points of the unstable rectangular orbit. In the inset, we show the trace of the monodromy (stability) matrix (see Section 5) for the diamond orbit versus deformation. When |Tr[M]|>2, its eigenvalues become real, the periodic motion becomes unstable, and the associated islands vanish. For the diamond, this happens at =0.1369 (see dashed vertical line in the inset), and the simple dynamical eclipsing picture of Fig. 7 does not apply at larger deformations.

Fig. 9
Fig. 9

Ray simulations of the far-field emission patterns for the quadrupole with (a) =0.12 and (b) =0.18 with different types of initial conditions. The solid curve is the result of choosing random initial conditions above the critical line sin χ=1/n; the dashed curve is for initial conditions on the adiabatic curve with a minimum value at the critical line. The dotted curve results for initial conditions localized around the unstable fixed point of the rectangle periodic orbit. In each of the ray simulations, 6000 rays were started with unit amplitude, and the amplitude was reduced according to Fresnel’s law upon each reflection, with the refracted amplitude “collected” in the far field. The dash–dotted curve is the experimental result; clearly all three choices give similar results, in good agreement with experiment.

Fig. 10
Fig. 10

(a) Ray simulations of short-term dynamics for random initial conditions above the critical line, propagated for 10 iterations, plotted on the surface of section for the quadrupole with =0.18. The areas of the SOS covered are delineated very accurately by the unstable manifolds of the short periodic orbits, which are indicated in the schematics at right. These manifolds are overlaid in the figure with appropriate color coding. (b) Flow of phase-space volume in the surface of section of the quadrupole with =0.18. A localized but arbitrary cloud of initial conditions (red) is iterated six times to illustrate the flow. The initial volume is the circle at the far left; successive iterations are increasingly stretched by the chaotic map. The stretching clearly follows closely the unstable manifold of the rectangle orbit, which we have plotted in blue.

Fig. 11
Fig. 11

(a) Ray simulation of emission: emitted-ray amplitude (color scale) overlaid on the surface of section for the quadrupole with =0.18. (b) Far-field intensity from experimental image data (Fig. 2) projected in false-color scale onto the surface of section for the quadrupole with =0.18. The blue curve is the unstable manifold of the periodic rectangle orbit. In green, we have the curve of constant 34° far field (see the discussion in Section 6). Absence of projected intensity near ϕ=±π in (b) is due to collection of experimental data only in the first quadrant.

Fig. 12
Fig. 12

Ray-emission amplitude (color scale) overlaid on the surface of section for the hexadecapole with (a) =0.12 and (b) =0.20. Solid blue and magenta curves are the unstable manifold of the diamond orbit (a) and of the unstable rectangular orbit (b). In green and turquoise, we plot the curves of constant emission in the 75° and 30° directions in the far field.

Fig. 13
Fig. 13

Far-field emission patterns for the stadium with (a) =0.12 and (b) =0.18. The dash–dotted curve is the ray simulation, and the solid is a numerical solution of the wave equation; no experimental data were taken for this shape. The ray simulation was performed with random initial conditions exactly as in Fig. 3. The numerical solutions were for resonances with kR=50.5401-0.0431i with Q=2342.71 and kR=48.7988-0.1192i with Q=818.83 for =0.12 and 0.18, respectively. The inset shows the shape of the stadium; it is defined by two half circles with radius one and a straight line segment of length 2.

Fig. 14
Fig. 14

Ray-emission amplitude (color scale) overlaid on the surface of section for the stadium with (a) =0.12 and (b) =0.18. The solid blue curve is the unstable manifold of the periodic rectangle orbit. The green curve is (a) the line of constant 55° and (b) the 48° emission direction into the far field. The thick black lines mark the end of the circular segments of the boundary and coincide with discontinuities in the manifolds.

Equations (1)

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sin χ(ϕ)=1+(S2-1)κ2/3(ϕ, ),

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