Abstract

Usual optical elements cannot focus a light beam to a spot with diameter smaller than half of the wavelength of the light; however overcoming this limit is of great importance in several applications in high-tech, such as optical lithography or magneto-optical date storage and numerous other fields of science and industry. Here we show that it is possible to focus light to spots below the diffraction limit (superfocusing) by the combination of two main elements: one which creates weak near-field evanescent components of the beam, like a wavelength-scale aperture, and an amplifier of these evanescent fields, like a slab of a photonic crystal with negative refraction.

© 2004 Optical Society of America

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References

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  1. M. A. Paesler and P. J. Moyer, Near-field optics: theory, instrumentation and applications (John Wiley and Sons, New York, 1996).
  2. J. B. Pendry, �??Negative refraction makes a perfect lens,�?? Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  3. V. G. Veselago, �??The electromagnetics of substances with simultaneously negative ε and µ,�?? Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  4. R. A. Shelby, D. R. Smith, and S. Schultz, �??Experimental verification of negative index of refraction,�?? Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  5. M. Notomi, �??Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,�?? Phys. Rev. B 62, 10696-10705 (2000).
    [CrossRef]
  6. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, �??Negative refraction by photonic crystals,�?? Nature 423, 604-605 (2003).
    [CrossRef] [PubMed]
  7. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, �??All-angle negative refraction without negative refractive index,�?? Phys. Rev. B 65, 201104 (2002).
    [CrossRef]
  8. S. Foteinopoulou and C. M. Soukoulis, �??Negative refraction and left-handed behavior in two-dimensional photonic crystals,�?? Phys. Rev. B 67, 235107 (2003).
    [CrossRef]
  9. Z.-Y. Li and L.-L. Lin, �??Evaluation of lensing in photonic crystal slabs exhibiting negative refraction,�?? Phys. Rev. B 68, 245110 (2003).
    [CrossRef]
  10. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, �??Subwavelength imaging in photonic crystals,�?? Phys. Rev. B 68, 045115 (2003).
    [CrossRef]
  11. P.V. Parimi, W. T. Lu, P. Vodo, and S. Sridar, �??Imaging by flat lens using negative refraction,�?? Nature 426, 404 (2003).
    [CrossRef] [PubMed]
  12. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, �??Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,�?? Phys. Rev. Lett. 91, 207401 (2003).
    [CrossRef] [PubMed]
  13. J. Meixner and W. Andrejewski, �??Rigorous theory of diffraction of an electromagnetic plane wave on a ideally conducting disk and on a circular opening in ideally conducting screen,�?? Ann. Phys. 7, 157-168 (1950).
    [CrossRef]
  14. D. R. Smith, D. Schurig, M. Rosenbluth, S. Shultz, S. A. Ramakrishna, and J. Pendry, �??Limitations on subdiffraction with a negative refractive slab,�?? Appl. Phys. Lett. 82, 1506-1508 (2003).
    [CrossRef]
  15. K. Sakoda, Optical properties of photonic crystals (Springer, New York, 2001).
  16. C. Luo, S. G. Johnson, J. D. Joannopoulos, �??All-angle negative refraction in a three-dimensionally periodic photonic crystal,�?? Appl. Phys. Lett. 81, 2352-2354 (2002).
    [CrossRef]

Ann. Phys. (1)

J. Meixner and W. Andrejewski, �??Rigorous theory of diffraction of an electromagnetic plane wave on a ideally conducting disk and on a circular opening in ideally conducting screen,�?? Ann. Phys. 7, 157-168 (1950).
[CrossRef]

Appl. Phys. Lett. (2)

D. R. Smith, D. Schurig, M. Rosenbluth, S. Shultz, S. A. Ramakrishna, and J. Pendry, �??Limitations on subdiffraction with a negative refractive slab,�?? Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, �??All-angle negative refraction in a three-dimensionally periodic photonic crystal,�?? Appl. Phys. Lett. 81, 2352-2354 (2002).
[CrossRef]

Nature (2)

P.V. Parimi, W. T. Lu, P. Vodo, and S. Sridar, �??Imaging by flat lens using negative refraction,�?? Nature 426, 404 (2003).
[CrossRef] [PubMed]

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, �??Negative refraction by photonic crystals,�?? Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

Phys. Rev. B (5)

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, �??All-angle negative refraction without negative refractive index,�?? Phys. Rev. B 65, 201104 (2002).
[CrossRef]

S. Foteinopoulou and C. M. Soukoulis, �??Negative refraction and left-handed behavior in two-dimensional photonic crystals,�?? Phys. Rev. B 67, 235107 (2003).
[CrossRef]

Z.-Y. Li and L.-L. Lin, �??Evaluation of lensing in photonic crystal slabs exhibiting negative refraction,�?? Phys. Rev. B 68, 245110 (2003).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, �??Subwavelength imaging in photonic crystals,�?? Phys. Rev. B 68, 045115 (2003).
[CrossRef]

M. Notomi, �??Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,�?? Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

Phys. Rev. Lett. (2)

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, �??Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,�?? Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef] [PubMed]

J. B. Pendry, �??Negative refraction makes a perfect lens,�?? Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Science (1)

R. A. Shelby, D. R. Smith, and S. Schultz, �??Experimental verification of negative index of refraction,�?? Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, �??The electromagnetics of substances with simultaneously negative ε and µ,�?? Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Other (2)

M. A. Paesler and P. J. Moyer, Near-field optics: theory, instrumentation and applications (John Wiley and Sons, New York, 1996).

K. Sakoda, Optical properties of photonic crystals (Springer, New York, 2001).

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Figures (4)

Fig. 1.
Fig. 1.

Superfocusing in the effective-medium theory with a negative refractive index. The spatial spectrum |E(kx ,ky )| is shown in (a) after the aperture |E(kx ,ky ,0)| (red, surface I) and after the slab |E(kx ,ky ,L)| (green, surface II). The spatial distribution of the energy density after the slab is shown in (b) and the dependence of the FWHM diameters along x (triangles) and y (circles) directions in (c). The parameters for (a),(b) are L=0.4λ and ε=µ=-1-0.0001(i+1). In (c), ε=µ=-1-0.0001(i+1) and ε=µ=-1-0.01(i+1) for solid (green) and open (magenta) points, correspondingly. Curves in (c) are guide for the eyes; the aperture diameter is 2λ.

Fig. 2.
Fig. 2.

Superfocusing in a hexagonal rods-in-air lattice PC with an effective negative refraction index. Transfer into the 0th Bragg order is shown in (a), the spatial spectrum (solid, red) and phase (dashed, green) of the field Hz are given in (b). The spatial distribution of the field is shown in (c), and the dependence of the FWHM of the beam at the output (solid, red) and the square of the maximum field |Hz (0)|2 (dashed, green) on the aperture width in given in (d). The input beam has a Gaussian shape with FWHM of 3.2λ, the photonic crystal slab consists of 4 layers of circular rods with radius 0.35a with lattice constant a. The frequency of the field is 0.57×2πc/a, the dielectric permittivity of the rods ε=12.96+0.01i. The aperture width in cases (b) and (c) is 0.55λ.

Fig. 3.
Fig. 3.

Superfocusing in a square air-holes lattice PC with an all-angle negative refraction. Transfer into the 0th Bragg order is shown in (a), the spatial spectrum (solid, red) and phase (dashed, green) of the field Hz is given in (b). The spatial distribution of the field is shown in (c), and the dependence of the FWHM of the beam at the output (solid, red) and the square of the maximum field |Hz (0)|2 (dashed, green) on the aperture width is given in (d). The input beam has a Gaussian shape with FWHM of 3.2λ, the photonic crystal slab consists of 4 layers of ‘+’-shaped air holes with parameters as given in Fig. 4(b). The frequency of the field is 0.27×2πc/a, the dielectric permittivity of the bulk is ε=12+0.01i. The aperture width in cases (b) and (c) is 1.15λ.

Fig. 4.
Fig. 4.

Distribution of the field of the beam (a) and the structure of the unit cell of the photonic crystal for the same system as in Fig. 3. The value of |Hz | is illustrated by shadows of blue in (a), and the aperture is represented by green; the reflected beam is omitted for the sake of clarity.

Equations (1)

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T s , p = 4 [ ( 2 + κ ) exp ( i q z L ) + ( 2 κ ) exp ( i q z L ) ] 1 .

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