Abstract

We present a detailed study of the mode structure of inverse opal photonic crystal materials with an emphasis on their potential use in optical trapping and cooling. In particular, we analyze the modes corresponding to the upper and lower band edges of a high refractive index inverse opal, i.e., the so-called “air” and “dielectric” bands. In the dielectric band, we demonstrate optical intensity enhancements of two orders of magnitude which may facilitate nonlinear optical effects in the solid. In the air band, dipolar optical trapping potentials for cold atoms in the voids arise when these modes are excited by an external laser field. In addition, we discuss aspects of atom cooling through the polarization gradients provided by these modes. The results suggest that optical trapping and cooling may be achieved within a photonic crystal using a single laser source.

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References

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  1. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef] [PubMed]
  2. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  3. S. John and T. Quang "Collective Switching and Inversion without Fluctuation of Two-Level Atoms in Confined Photonic Systems," Phys. Rev. Lett. 78, 1888-1891 (1997).
    [CrossRef]
  4. S. Chu, "The manipulation of neutral particles," Rev. Mod. Phys. 70, 685-706 (1998).
    [CrossRef]
  5. C.N. Cohen-Tannoudji, "Manipulating atoms with photons," Rev. Mod. Phys. 70, 707-720 (1998).
    [CrossRef]
  6. W.D. Phillips, "Laser cooling and trapping of neutral atoms," Rev. Mod. Phys. 70, 721-742 (1998).
    [CrossRef]
  7. K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
    [CrossRef]
  8. J.D. Joannopoulos, P.R. Villeneuve, and S. Fan, "Photonic Crystals: Putting a new twist on light," Nature 386, 143 (1997).
    [CrossRef]
  9. K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
    [CrossRef] [PubMed]
  10. R. D. Meade, A. M. Rappe, K.D.Brommer, and J. D. Joannopoulos, "Accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 48, 8434-8437 (1993).
    [CrossRef]
  11. Harold J. Metcalf and Peter van der Straten, Laser Cooling and Trapping (Springer-Verlag, New York, 1999).
    [CrossRef]

Other (11)

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John and T. Quang "Collective Switching and Inversion without Fluctuation of Two-Level Atoms in Confined Photonic Systems," Phys. Rev. Lett. 78, 1888-1891 (1997).
[CrossRef]

S. Chu, "The manipulation of neutral particles," Rev. Mod. Phys. 70, 685-706 (1998).
[CrossRef]

C.N. Cohen-Tannoudji, "Manipulating atoms with photons," Rev. Mod. Phys. 70, 707-720 (1998).
[CrossRef]

W.D. Phillips, "Laser cooling and trapping of neutral atoms," Rev. Mod. Phys. 70, 721-742 (1998).
[CrossRef]

K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

J.D. Joannopoulos, P.R. Villeneuve, and S. Fan, "Photonic Crystals: Putting a new twist on light," Nature 386, 143 (1997).
[CrossRef]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

R. D. Meade, A. M. Rappe, K.D.Brommer, and J. D. Joannopoulos, "Accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 48, 8434-8437 (1993).
[CrossRef]

Harold J. Metcalf and Peter van der Straten, Laser Cooling and Trapping (Springer-Verlag, New York, 1999).
[CrossRef]

Supplementary Material (6)

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Figures (4)

Fig. 1.
Fig. 1.

Band structure for the inverse opal crystal with index 3.45. The radius of the air spheres is 0.364a and the solid dielectric coating thickness is 0.065a where a is the lattice constant of the fcc lattice. The k⃗ vector runs along a few high symmetry directions in the fcc Brillouin zone. The full photonic band gap opens between normalized frequencies [0.87–0.96].

Fig. 2.
Fig. 2.

The absolute value of the electric field in the modes corresponding to bands 8 and 9 at the W-point, respectively. |Eω (r⃗)| is color-coded according to the color bar. The three orthogonal slices are warped by the value of the dielectric function with a zero displacement in the air component and a positive displacement in the Si component. The frames in the movies correspond to different values for the z=const slice. The high-intensity regions of band 8 (1.3M) lie predominantly in the high dielectric region, thus representing a dielectric band while the high-intensity regions of band 9 (1.3M) are located within the air voids of the inverse opal, corresponding to an air band.

Fig. 3.
Fig. 3.

The absolute value of the electric field in the modes corresponding to bands 8 and 9 at the W-point, respectively. |Eω (r⃗)| is represented as a surface of constant value. The frames in the movies correspond to isosurfaces of different values and the color of the isosurface is mapped to the colorbar. The semi-transparent shape of the silicon backbone is also shown. Band 8 (2.1M) corresponds to the dielectric band, while band 9 (1.9M) represents an air band.

Fig. 4.
Fig. 4.

[Left] The ellipticity in the air mode (band 9 at W-point) is color coded according to the color bar. A value of 0 corresponds to a linearly polarized state and a value of 1 corresponds to a circularly polarized state. The frames in the movie correspond to different values for the x=const slice (1.4M). [Right] The major and minor polarization axes along a path passing entirely through the air component of the inverted opal in the same air mode. Major axis is colored blue, minor axis is green and the normal to the polarization plane is colored in red. The intersection of the path with the conventional unit cell runs from {x, y, z}={0, -1/2, -1/2} to {0, 1/2, 1/2}.(2.5M)

Equations (6)

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× ( 1 ( r ) × H ω ( r ) ) = ( ω c ) 2 H ω ( r ) .
H ω ( r ' ) = e i k · r H ω , k ( r ' ) ,
H ω , k ( r ' ) = G 1 s t B Z = 1 , 2 e , k + G u G , ( k ) e i G · r ,
E ( r , t ) = Re ( E ω ( r ) e i ω t )
E ω ( r ) = i c ω N ( r ) × H ω ( r )
U ( r ) = δ 2 ( 1 + I ( r ) I sat 1 1 + ( 2 δ γ ) 2 ) ,

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