Abstract

We analyze focusing of electromagnetic waves inside a photonic crystal slab by means of finite-difference time-domain simulations. At the frequency of the source, the photonic crystal behaves as an effective medium with an effective index of refraction of -1. Despite of the strong Bloch modulation of the field inside the slab, the presence of a well-definite internal focus is evident. The dimensions of the internal focus are similar to those of the external focus. The effect of the frequency of the wave on the focusing is also discussed.

© 2005 Optical Society of America

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Appl. Phys. Lett.

S. Xiao, M. Qiu, Z. Ruan, and S. He, �??Influence of the surface termination to the point imaging by a photonic crystal slab with negative refraction,�?? Appl. Phys. Lett. 85, 4269-4271 (2004)
[CrossRef]

IEEE Trans. Microwave Tech.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, �??Magnetism from Conductors and Enhanced Nonlinear Phenomena,�?? IEEE Trans. Microwave Tech. 47, 2075-2084 (1999)
[CrossRef]

J. Comput. Phys.

J. P. Berenger, �??A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,�?? J. Comput. Phys. 114, 185-200 (1994)
[CrossRef]

Nature

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

Opt. Express

Opt. Lett.

Phys Rev. 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 (2004)
[CrossRef]

Phys. Rev. B

X. Zhang, �??Image resolution depending on slab thickness and object distance in a two-dimensional photoniccrystal- based superlens,�?? Phys. Rev. B 70, 195110 (2004)
[CrossRef]

X. Wang and K. Kempa, �??Effects of disorder on subwavelength lensing in two-dimensional photonic crystal slabs,�?? Phys. Rev. B 71, 085101 (2005)
[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]

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]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, �??Superprism phenomena in photonic crystals,�?? Phys. Rev. B 58, 10096-10099 (1998)
[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]

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

A. Martínez, H. Míguez, A. Griol, and J. Martí, �??Experimental and theoretical study of the self-focusing of light by a photonic crystal lens,�?? Phys. Rev. B 69, 165119 (2004)
[CrossRef]

H.-T. Chien, H.-T. Tang, C.-H. Kuo, C.-C. Chen, and Z. Ye, �??Directed diffraction without negative refraction,�?? Phys. Rev. B 70, 113101 (2004)
[CrossRef]

Z. Ruan, M. Qiu, S. Xiao, S. He, and L. Thylen, �??Coupling between plane waves and Bloch waves in photonic crystals with negative refraction,�?? Phys. Rev. B 71, 045111 (2005)
[CrossRef]

Phys. Rev. E

C.-H. Kuo and Z. Ye, �??Negative-refraction-like behavior revealed by arrays of dielectric cylinders,�?? Phys. Rev. E 70, 026608 (2004)
[CrossRef]

Phys. Rev. Lett.

J. 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]

J. B. Pendry, �??Negative Refraction Makes a Perfect Lens,�?? Phys. Rev. Lett. 85, 3966-3969 (2000)
[CrossRef] [PubMed]

A. Grbic and G. V. Eleftheriades, �??Overcoming the Diffraction Limit with a Planar Left-Handed Transmission-Line Lens,�?? Phys. Rev. Lett. 92, 117403 (2004)
[CrossRef] [PubMed]

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylén, A. Talneau and S. Anand, �??Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,�?? Phys. Rev. Lett. 93, 073902 (2004)
[CrossRef] [PubMed]

Science

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

Sov. Phys. Usp.

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

Other

A. Martínez and J. Martí, submitted to Phys. Rev. B.

A. Taflove, Computational Electrodynamics�??The Finite Difference Time-Domain Method (Artech House, Boston, 1995)

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Figures (10)

Fig. 1.
Fig. 1.

(a) Photonic band diagram (transverse magnetic modes) of the proposed PhC; (b) effective index of refraction of the second photonic band.

Fig. 2.
Fig. 2.

Scenario under study: a PhC slab with thickness L oriented so that the interfaces are along the ΓK direction. Slab terminations are carefully chosen to ensure the excitation of a surface mode. Several sets of electric field monitors are placed in the structure under analysis: lines A, B, C, D, E and F. The optical source is placed at z=-L and radiates upwards.

Fig. 3.
Fig. 3.

Electric field distribution for the scenario described in Fig. 2. Gaussian source with initial width 0.4λ and frequency a/λ=0.306. The PhC slab thickness is L=(11.5 3-0.4)a=19.52a. The snapshots show different time steps of an FDTD simulation.

Fig. 4.
Fig. 4.

Time-space evolution of the field intensity at the NIL output. (a) Longitudinal direction; (b) transversal direction.

Fig. 5.
Fig. 5.

(a) Longitudinal and (b) transversal profiles of the field intensity in the external focus.

Fig. 6.
Fig. 6.

(a) Longitudinal profile of the field intensity inside the NIL along line C. The intensity produced by a point source (blue solid curve) and a plane-like wave (black dashed curve) are compared. (b) Time-space evolution of the field intensity inside the NIL along the longitudinal direction.

Fig. 7.
Fig. 7.

(a) Transversal profile of the field intensity inside the NIL along line E. The intensity produced by a point source (blue solid curve) and a plane-like wave (black dashed curve) are compared. (b) Time-space evolution of the field intensity inside the NIL along the transversal direction.

Fig. 8.
Fig. 8.

Transversal profiles of the field intensity inside the NIL along lines D (black), E (blue) and F (red) for L=19.52a.

Fig. 9.
Fig. 9.

(a) Longitudinal and (b) transversal intensity patterns for the internal focus in the cases of a slab with dielectric (black curve) or a hole (blue curve) at its center. For the sake of comparison, the red curve shows the profiles of the external focus with the intensity double In order to get a better visual comparison. the longitudinal external pattern in (a) is represented inverted with respect to z=L/2.

Fig. 10.
Fig. 10.

Longitudinal intensity patterns for three different frequencies of the source: a/λ=0.296 (red), 0.306 (blue) and 0.32 (black). The vertical dotted line stands for the PhC-air output interface.

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