Abstract

The construction of the multifocal Veselago lens predicted earlier [Appl. Opt. 42, 5701 (2003)] is proposed on the basis of a uniaxial photonic crystal consisting of cylindrical air holes in silicon that make a triangular lattice in a plane perpendicular to the axis of the crystal. The object and images are in air. The period of the crystal should be 0.44μm to work at the wavelength 1.5μm. The lens does not provide superlensing, but the halfwidth of the image is 0.5λ. The lens is shown to have wave-guiding properties, depending on the substrate material.

© 2006 Optical Society of America

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References

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  1. V. G. Veselago, "Properties of materials having simultaneously negative values of the dielectric (epsilon) and magnetic (μ) susceptibilities," Sov. Phys. Solid State 8, 2854-2856 (1967).
  2. A. L. Efros and A. L. Pokrovsky, "Dielectric photonic crystal as medium with negative electric permittivity and magnetic permeability," Solid State Commun. 129, 643-647 (2004).
    [CrossRef]
  3. A. L. Pokrovsky and A. L. Efros, "Sign of refractive index and group velocity in left-handed media," Solid State Commun. 124, 283-287 (2002).
    [CrossRef]
  4. P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
    [CrossRef] [PubMed]
  5. A. L. Efros, C. Y. Li, and A. L. Pokrovsky, "Evanescent waves in photonic crystals and image of Veselago lens," cond-mat/0503494.
  6. C. Y. Li, J. M. Holt, and A. L. Efros, "Far-field image of Veselago lens," J. Opt. Soc. Am. B 23, 490-497 (2006).
    [CrossRef]
  7. X. Wang, Z. F. Ren, and K. Kempa, "Unrestricted superlensing in a triangular two-dimensional photonic crystal," Opt. Express 12, 2919-2924 (2004).
    [CrossRef] [PubMed]
  8. A. L. Pokrovsky and A. L. Efros, "Lens based on the use of left-handed materials," Appl. Opt. 42, 5701-5704 (2003).
    [CrossRef] [PubMed]
  9. X. Wang, Z. F. Ren, and K. Kempa, "Improved superlensing in two-dimensional photonic crystals with a basis," Appl. Phys. Lett. 86, 061105 (2005).
    [CrossRef]
  10. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).
  11. J. B. Pendry, "Negative refraction make a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  12. F. D. M. Haldane, "Electromagnetic surface modes at interfaces with negative refractive index make a 'not-quite-perfect' lens," cond-mat/0206420.
  13. R. Ruppin, "Surface polaritons of a left-handed medium," Phys. Lett. A 277, 61-64 (2000).
    [CrossRef]

2006

2005

X. Wang, Z. F. Ren, and K. Kempa, "Improved superlensing in two-dimensional photonic crystals with a basis," Appl. Phys. Lett. 86, 061105 (2005).
[CrossRef]

2004

X. Wang, Z. F. Ren, and K. Kempa, "Unrestricted superlensing in a triangular two-dimensional photonic crystal," Opt. Express 12, 2919-2924 (2004).
[CrossRef] [PubMed]

A. L. Efros and A. L. Pokrovsky, "Dielectric photonic crystal as medium with negative electric permittivity and magnetic permeability," Solid State Commun. 129, 643-647 (2004).
[CrossRef]

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

2003

2002

A. L. Pokrovsky and A. L. Efros, "Sign of refractive index and group velocity in left-handed media," Solid State Commun. 124, 283-287 (2002).
[CrossRef]

2000

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

R. Ruppin, "Surface polaritons of a left-handed medium," Phys. Lett. A 277, 61-64 (2000).
[CrossRef]

1967

V. G. Veselago, "Properties of materials having simultaneously negative values of the dielectric (epsilon) and magnetic (μ) susceptibilities," Sov. Phys. Solid State 8, 2854-2856 (1967).

Derov, J. S.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Efros, A. L.

C. Y. Li, J. M. Holt, and A. L. Efros, "Far-field image of Veselago lens," J. Opt. Soc. Am. B 23, 490-497 (2006).
[CrossRef]

A. L. Efros and A. L. Pokrovsky, "Dielectric photonic crystal as medium with negative electric permittivity and magnetic permeability," Solid State Commun. 129, 643-647 (2004).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, "Lens based on the use of left-handed materials," Appl. Opt. 42, 5701-5704 (2003).
[CrossRef] [PubMed]

A. L. Pokrovsky and A. L. Efros, "Sign of refractive index and group velocity in left-handed media," Solid State Commun. 124, 283-287 (2002).
[CrossRef]

A. L. Efros, C. Y. Li, and A. L. Pokrovsky, "Evanescent waves in photonic crystals and image of Veselago lens," cond-mat/0503494.

Haldane, F. D. M.

F. D. M. Haldane, "Electromagnetic surface modes at interfaces with negative refractive index make a 'not-quite-perfect' lens," cond-mat/0206420.

Holt, J. M.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

Kempa, K.

X. Wang, Z. F. Ren, and K. Kempa, "Improved superlensing in two-dimensional photonic crystals with a basis," Appl. Phys. Lett. 86, 061105 (2005).
[CrossRef]

X. Wang, Z. F. Ren, and K. Kempa, "Unrestricted superlensing in a triangular two-dimensional photonic crystal," Opt. Express 12, 2919-2924 (2004).
[CrossRef] [PubMed]

Li, C. Y.

C. Y. Li, J. M. Holt, and A. L. Efros, "Far-field image of Veselago lens," J. Opt. Soc. Am. B 23, 490-497 (2006).
[CrossRef]

A. L. Efros, C. Y. Li, and A. L. Pokrovsky, "Evanescent waves in photonic crystals and image of Veselago lens," cond-mat/0503494.

Lu, W. T.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Parimi, P. V.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Pendry, J. B.

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

Pokrovsky, A. L.

A. L. Efros and A. L. Pokrovsky, "Dielectric photonic crystal as medium with negative electric permittivity and magnetic permeability," Solid State Commun. 129, 643-647 (2004).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, "Lens based on the use of left-handed materials," Appl. Opt. 42, 5701-5704 (2003).
[CrossRef] [PubMed]

A. L. Pokrovsky and A. L. Efros, "Sign of refractive index and group velocity in left-handed media," Solid State Commun. 124, 283-287 (2002).
[CrossRef]

A. L. Efros, C. Y. Li, and A. L. Pokrovsky, "Evanescent waves in photonic crystals and image of Veselago lens," cond-mat/0503494.

Ren, Z. F.

X. Wang, Z. F. Ren, and K. Kempa, "Improved superlensing in two-dimensional photonic crystals with a basis," Appl. Phys. Lett. 86, 061105 (2005).
[CrossRef]

X. Wang, Z. F. Ren, and K. Kempa, "Unrestricted superlensing in a triangular two-dimensional photonic crystal," Opt. Express 12, 2919-2924 (2004).
[CrossRef] [PubMed]

Ruppin, R.

R. Ruppin, "Surface polaritons of a left-handed medium," Phys. Lett. A 277, 61-64 (2000).
[CrossRef]

Sokoloff, J.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Sridhar, S.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Veselago, V. G.

V. G. Veselago, "Properties of materials having simultaneously negative values of the dielectric (epsilon) and magnetic (μ) susceptibilities," Sov. Phys. Solid State 8, 2854-2856 (1967).

Vodo, P.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Wang, X.

X. Wang, Z. F. Ren, and K. Kempa, "Improved superlensing in two-dimensional photonic crystals with a basis," Appl. Phys. Lett. 86, 061105 (2005).
[CrossRef]

X. Wang, Z. F. Ren, and K. Kempa, "Unrestricted superlensing in a triangular two-dimensional photonic crystal," Opt. Express 12, 2919-2924 (2004).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

X. Wang, Z. F. Ren, and K. Kempa, "Improved superlensing in two-dimensional photonic crystals with a basis," Appl. Phys. Lett. 86, 061105 (2005).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

Phys. Lett. A

R. Ruppin, "Surface polaritons of a left-handed medium," Phys. Lett. A 277, 61-64 (2000).
[CrossRef]

Phys. Rev. Lett.

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

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Solid State Commun.

A. L. Efros and A. L. Pokrovsky, "Dielectric photonic crystal as medium with negative electric permittivity and magnetic permeability," Solid State Commun. 129, 643-647 (2004).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, "Sign of refractive index and group velocity in left-handed media," Solid State Commun. 124, 283-287 (2002).
[CrossRef]

Sov. Phys. Solid State

V. G. Veselago, "Properties of materials having simultaneously negative values of the dielectric (epsilon) and magnetic (μ) susceptibilities," Sov. Phys. Solid State 8, 2854-2856 (1967).

Other

F. D. M. Haldane, "Electromagnetic surface modes at interfaces with negative refractive index make a 'not-quite-perfect' lens," cond-mat/0206420.

A. L. Efros, C. Y. Li, and A. L. Pokrovsky, "Evanescent waves in photonic crystals and image of Veselago lens," cond-mat/0503494.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

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Figures (10)

Fig. 1
Fig. 1

(a) Triangular lattice of circular cylindrical holes in a dielectric matrix with ϵ m = 12 and μ m = 1 . The radius of the holes is R = 0.35 d , where d is the period. (b) The lattice with the basis and parameters as in Ref. [8]. The radii of large and small air holes are R = 0.4 d and R = 0.13 d , respectively. The dielectric matrix has ϵ m = 12.96 , μ m = 1 .

Fig. 2
Fig. 2

Six lowest bands of the photonic spectrum of the PC that has the geometry of Fig. 1a.

Fig. 3
Fig. 3

Distribution of electric energy behind the PC slab for a = 0.5 L (a) in the x - y plane and (b) along the line y = 0 . The two vertical lines show the theoretical position of the two foci. The small arrows show computational positions of the foci.

Fig. 4
Fig. 4

Same as in Fig. 3 for a = 1.5 L . In this case we get only one focus. The second maximum is interpreted as a satellite of the focus.

Fig. 5
Fig. 5

The solid curve shows the computational distribution near the focus for the case of a > L , whereas the dotted curve shows the computational distribution near the second focus for the case of a < L . There is no numerical fitting in this plot.

Fig. 6
Fig. 6

Distribution of electric energy along the lateral direction near the focus for the case of a > L . The solid curve shows the computational distribution near the focus, and the dashed curve shows the analytical result for the intensity obtained from Eq. (3). The results are normalized to have the same maximum value at the focal point.

Fig. 7
Fig. 7

Distribution of electric energy along the lateral direction near the first focus for the case of a < L . The solid curve shows the computational distribution for the PC, as shown in Fig. 1a, the dash-dotted curve shows the computational distribution for the PC with basis as shown in Fig. 1b, and the dashed curve shows the analytical result for intensity obtained from Eq. (3). The results are normalized to have the same maximum value at the focal point for the PC without basis.

Fig. 8
Fig. 8

Electric field of the evanescent wave. The cross section y = 0 is shown. The solid curve is for κ = 0.1 k 0 and k y = 1.01 k 0 , the dotted curve is for κ = 0.2 k 0 and k y = 1.04 k 0 , and the dashed curve is for κ = k 0 and k y = 2 k 0 . For other κ in the interval 0.2 k 0 and k 0 , the plots are between the dotted curve and the dashed curve. The surface cut is BH. The plot for the AH cut is indistinguishable from this plot.

Fig. 9
Fig. 9

Representation of the PC slab geometry. (a) Top view of the cross section of the PC in the x - y plane. The system is infinite in the y direction owing to periodic boundary conditions. (b) Side view. Normal light is incident through a slit on the left, whereas the remaining external boundaries are low reflecting. Normal power flux is calculated across the surface Q, indicated by the dashed line.

Fig. 10
Fig. 10

P ( x ) P ( 0 ) in percent in the PC surrounded by air (solid-curve circles) and on substrates of silicon (solid-curve triangles) and silicon dioxide (solid-curve squares). The dash-dotted curve shows the same power ratio when the PC slab is substituted by air and the “substrate” is also air. The trend line (dashed curve) indicates the least-squares linear fit of the PC in air data.

Equations (4)

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I m = ( 1 r 2 ) 2 r 4 m 4 I 0 .
r = ϵ ϵ ϵ + ϵ
E m ( x , y ) = i E m π k H k H exp i [ k y + x ( k 0 2 k 2 ) 1 2 ω t ] ( k 0 2 k 2 ) 1 2 d k .
P ( x ) = Q S d a ,

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