Abstract

The guided modes lying in the upper gap-edge band in the photonic band structure of photonic crystals have negative values of refractive index. This feature generates many interesting optical phenomena, and some spectacular photonic devices such as focusing slabs have been developed. We report the design of a photonic-crystal, planoconcave lens for focusing incident parallel light, and theoretically analyze the chromatic aberrations for TM and TE modes. In addition to dielectric photonic crystals, the chromatic aberration of a magnetic photonic-crystal planoconcave lens was investigated because the magnetic permeability may also contribute to the periodic index contrast in photonic crystals, especially at long wavelengths. A significant difference was found in the chromatic aberration for a TM mode propagating in a dielectric than in a magnetic photonic-crystal planoconcave lens.

© 2006 Optical Society of America

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References

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  1. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
    [CrossRef] [PubMed]
  2. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
    [CrossRef]
  3. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4186 (2000).
    [CrossRef] [PubMed]
  4. M. Notomi, "Negative refraction in photonic crystals," Opt. Quantum Electron. 34, 133-143 (2002).
    [CrossRef]
  5. C. Luo, S. G. Johnson, and J. D. Joannopoulos, "All-angle negative refraction in a three-dimensionally periodic photonic crystal," Appl. Phys. Lett. 81, 2352-2354 (2002).
    [CrossRef]
  6. K. Guven, K. Aydin, K. B. Alici, C. M. Soukoulis, and E. Ozbay, "Spectral negative refraction and focusing analysis of a two-dimensional left-handed photonic crystal lens," Phys. Rev. B 70, 205125-205129 (2004).
    [CrossRef]
  7. A. Husakou and J. Herrmann, "Superfocusing of light below the diffraction limit by photonic crystals with negative refraction," Opt. Express 12, 6491-6497 (2004).
    [CrossRef] [PubMed]
  8. I. Drikis, S. Y. Yang, H. E. Horng, C.-Y. Hong, and H. C. Yang, "Modified frequency-domain method for simulating the electromagnetics in periodic magnetoactive systems," J. Appl. Phys. 95, 5876-5881 (2004).
    [CrossRef]
  9. S. Y. Yang, C.-Y. Hong, I. Drikis, H. E. Horng, and H. C. Yang, "Photonic characterizations of triangular-arrayed rods with both dielectric and magnetic permeability functions," J. Opt. Soc. Am. B 21, 413-418 (2004).
    [CrossRef]
  10. D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, 6th ed. (Wiley, 2001), Chap. 35, pp. 841-846.
  11. S.-Y. Yang and C. T. Chang, "Birefringence prism made of photonic crystals using magnetic material," submitted to J. Opt. Soc. Am. B.

2004 (4)

K. Guven, K. Aydin, K. B. Alici, C. M. Soukoulis, and E. Ozbay, "Spectral negative refraction and focusing analysis of a two-dimensional left-handed photonic crystal lens," Phys. Rev. B 70, 205125-205129 (2004).
[CrossRef]

A. Husakou and J. Herrmann, "Superfocusing of light below the diffraction limit by photonic crystals with negative refraction," Opt. Express 12, 6491-6497 (2004).
[CrossRef] [PubMed]

I. Drikis, S. Y. Yang, H. E. Horng, C.-Y. Hong, and H. C. Yang, "Modified frequency-domain method for simulating the electromagnetics in periodic magnetoactive systems," J. Appl. Phys. 95, 5876-5881 (2004).
[CrossRef]

S. Y. Yang, C.-Y. Hong, I. Drikis, H. E. Horng, and H. C. Yang, "Photonic characterizations of triangular-arrayed rods with both dielectric and magnetic permeability functions," J. Opt. Soc. Am. B 21, 413-418 (2004).
[CrossRef]

2002 (2)

M. Notomi, "Negative refraction in photonic crystals," Opt. Quantum Electron. 34, 133-143 (2002).
[CrossRef]

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

2000 (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4186 (2000).
[CrossRef] [PubMed]

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

1996 (1)

B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Alici, K. B.

K. Guven, K. Aydin, K. B. Alici, C. M. Soukoulis, and E. Ozbay, "Spectral negative refraction and focusing analysis of a two-dimensional left-handed photonic crystal lens," Phys. Rev. B 70, 205125-205129 (2004).
[CrossRef]

Aydin, K.

K. Guven, K. Aydin, K. B. Alici, C. M. Soukoulis, and E. Ozbay, "Spectral negative refraction and focusing analysis of a two-dimensional left-handed photonic crystal lens," Phys. Rev. B 70, 205125-205129 (2004).
[CrossRef]

Chang, C. T.

S.-Y. Yang and C. T. Chang, "Birefringence prism made of photonic crystals using magnetic material," submitted to J. Opt. Soc. Am. B.

Drikis, I.

S. Y. Yang, C.-Y. Hong, I. Drikis, H. E. Horng, and H. C. Yang, "Photonic characterizations of triangular-arrayed rods with both dielectric and magnetic permeability functions," J. Opt. Soc. Am. B 21, 413-418 (2004).
[CrossRef]

I. Drikis, S. Y. Yang, H. E. Horng, C.-Y. Hong, and H. C. Yang, "Modified frequency-domain method for simulating the electromagnetics in periodic magnetoactive systems," J. Appl. Phys. 95, 5876-5881 (2004).
[CrossRef]

Guven, K.

K. Guven, K. Aydin, K. B. Alici, C. M. Soukoulis, and E. Ozbay, "Spectral negative refraction and focusing analysis of a two-dimensional left-handed photonic crystal lens," Phys. Rev. B 70, 205125-205129 (2004).
[CrossRef]

Halliday, D.

D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, 6th ed. (Wiley, 2001), Chap. 35, pp. 841-846.

Herrmann, J.

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Hong, C.-Y.

I. Drikis, S. Y. Yang, H. E. Horng, C.-Y. Hong, and H. C. Yang, "Modified frequency-domain method for simulating the electromagnetics in periodic magnetoactive systems," J. Appl. Phys. 95, 5876-5881 (2004).
[CrossRef]

S. Y. Yang, C.-Y. Hong, I. Drikis, H. E. Horng, and H. C. Yang, "Photonic characterizations of triangular-arrayed rods with both dielectric and magnetic permeability functions," J. Opt. Soc. Am. B 21, 413-418 (2004).
[CrossRef]

Horng, H. E.

I. Drikis, S. Y. Yang, H. E. Horng, C.-Y. Hong, and H. C. Yang, "Modified frequency-domain method for simulating the electromagnetics in periodic magnetoactive systems," J. Appl. Phys. 95, 5876-5881 (2004).
[CrossRef]

S. Y. Yang, C.-Y. Hong, I. Drikis, H. E. Horng, and H. C. Yang, "Photonic characterizations of triangular-arrayed rods with both dielectric and magnetic permeability functions," J. Opt. Soc. Am. B 21, 413-418 (2004).
[CrossRef]

Husakou, A.

Joannopoulos, J. D.

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

Johnson, S. G.

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

Luo, C.

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

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4186 (2000).
[CrossRef] [PubMed]

Notomi, M.

M. Notomi, "Negative refraction in photonic crystals," Opt. Quantum Electron. 34, 133-143 (2002).
[CrossRef]

Ozbay, E.

K. Guven, K. Aydin, K. B. Alici, C. M. Soukoulis, and E. Ozbay, "Spectral negative refraction and focusing analysis of a two-dimensional left-handed photonic crystal lens," Phys. Rev. B 70, 205125-205129 (2004).
[CrossRef]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4186 (2000).
[CrossRef] [PubMed]

Pendry, B.

B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

Resnick, R.

D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, 6th ed. (Wiley, 2001), Chap. 35, pp. 841-846.

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4186 (2000).
[CrossRef] [PubMed]

Smith, D. R.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4186 (2000).
[CrossRef] [PubMed]

Soukoulis, C. M.

K. Guven, K. Aydin, K. B. Alici, C. M. Soukoulis, and E. Ozbay, "Spectral negative refraction and focusing analysis of a two-dimensional left-handed photonic crystal lens," Phys. Rev. B 70, 205125-205129 (2004).
[CrossRef]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4186 (2000).
[CrossRef] [PubMed]

Walker, J.

D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, 6th ed. (Wiley, 2001), Chap. 35, pp. 841-846.

Yang, H. C.

I. Drikis, S. Y. Yang, H. E. Horng, C.-Y. Hong, and H. C. Yang, "Modified frequency-domain method for simulating the electromagnetics in periodic magnetoactive systems," J. Appl. Phys. 95, 5876-5881 (2004).
[CrossRef]

S. Y. Yang, C.-Y. Hong, I. Drikis, H. E. Horng, and H. C. Yang, "Photonic characterizations of triangular-arrayed rods with both dielectric and magnetic permeability functions," J. Opt. Soc. Am. B 21, 413-418 (2004).
[CrossRef]

Yang, S. Y.

S. Y. Yang, C.-Y. Hong, I. Drikis, H. E. Horng, and H. C. Yang, "Photonic characterizations of triangular-arrayed rods with both dielectric and magnetic permeability functions," J. Opt. Soc. Am. B 21, 413-418 (2004).
[CrossRef]

I. Drikis, S. Y. Yang, H. E. Horng, C.-Y. Hong, and H. C. Yang, "Modified frequency-domain method for simulating the electromagnetics in periodic magnetoactive systems," J. Appl. Phys. 95, 5876-5881 (2004).
[CrossRef]

Yang, S.-Y.

S.-Y. Yang and C. T. Chang, "Birefringence prism made of photonic crystals using magnetic material," submitted to J. Opt. Soc. Am. B.

Youngs, I.

B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

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

IEEE Trans. Microwave Theory Tech. (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

J. Appl. Phys. (1)

I. Drikis, S. Y. Yang, H. E. Horng, C.-Y. Hong, and H. C. Yang, "Modified frequency-domain method for simulating the electromagnetics in periodic magnetoactive systems," J. Appl. Phys. 95, 5876-5881 (2004).
[CrossRef]

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

Opt. Express (1)

Opt. Quantum Electron. (1)

M. Notomi, "Negative refraction in photonic crystals," Opt. Quantum Electron. 34, 133-143 (2002).
[CrossRef]

Phys. Rev. B (1)

K. Guven, K. Aydin, K. B. Alici, C. M. Soukoulis, and E. Ozbay, "Spectral negative refraction and focusing analysis of a two-dimensional left-handed photonic crystal lens," Phys. Rev. B 70, 205125-205129 (2004).
[CrossRef]

Phys. Rev. Lett. (2)

B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4186 (2000).
[CrossRef] [PubMed]

Other (2)

D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, 6th ed. (Wiley, 2001), Chap. 35, pp. 841-846.

S.-Y. Yang and C. T. Chang, "Birefringence prism made of photonic crystals using magnetic material," submitted to J. Opt. Soc. Am. B.

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Figures (4)

Fig. 1
Fig. 1

Photonic band structure of the photonic crystal made of triangularly arrayed magnetic (solid curves) or dielectric (dashed curves) rods for (a) TM, (b) TE modes. The magnetic rods have a ratio of (10:1.5) for ( ϵ rod , μ rod ) , and the dielectric rods have (15:1). The ratio of the rod radius to the rod spacing a 2 d is 0.2.

Fig. 2
Fig. 2

Scheme of a photonic-crystal planoconcave lens. The rods are infinitely long along the direction outward from the page. The ratio of the rod radius to the rod spacing a 2 d is 0.2. The radius of curvature of the lens is denoted by R. The gray shadow represents the incoming plane wave from the flat side of the lens and focused out of the concave side. The focal length is f.

Fig. 3
Fig. 3

Frequency-dependent phase index n p for (a) TM, (b) TE modes propagating along the Γ M direction of the photonic crystal shown in Fig. 2. The dashed/solid curves correspond to the cases where the photonic crystal is composed of dielectric/magnetic rods with the ( ϵ rod , μ ro ) ratio being (15:1) and (10:1.5).

Fig. 4
Fig. 4

Normalized focal length as a function of frequency for (a) TM, (b) TE modes propagating through the photonic-crystal planoconcave lens shown in Fig. 2. The dashed/solid curves correspond to the cases where the photonic crystal is composed of dielectric/magnetic rods with the ( ϵ rod , μ ro ) ratio being ( 15 : 1 ) ( 10 : 1.5 ) .

Tables (1)

Tables Icon

Table 1 Architectural Parameters for the Photonic-Crystal Planoconcave Lens a

Equations (5)

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1 μ x 1 ε x 1 μ B = ( ω c ) 2 1 μ B
n p = ± k N ω N ,
f = R ( 1 n p ) ,
TM TE
2.38 1.69

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