Abstract

This work presents a comprehensive analysis of electromagnetic wave propagation inside a two-dimensional photonic crystal in a spectral region in which the crystal behaves as an effective medium to which a negative effective index of refraction can be associated. It is obtained that the main plane wave component of the Bloch mode that propagates inside the photonic crystal has its wave vector k out of the first Brillouin zone and it is parallel to the Poynting vector (S⃗·k>0), so light propagation in these composites is different from that reported for left-handed materials despite the fact that negative refraction can take place at the interface between air and both kinds of composites. However, wave coupling at the interfaces is well explained using the reduced wave vector (k⃗) in the first Brillouin zone, which is opposed to the energy flow, and agrees well with previous works dealing with negative refraction in photonic crystals.

© 2005 Optical Society of America

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References

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    [CrossRef] [PubMed]
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IEEE J. Sel. Top. Quantum Electron. (1)

M. Qiu, L. Thyl�n, M. Swillo, and B. Jaskorzynska, �Wave propagation through a photonic crystal in a negative phase refractive-index region,� IEEE J. Sel. Top. Quantum Electron. 9, 106-110 (2003).
[CrossRef]

IEEE Trans. Microwave 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 Tech. 47, 2075-2084 (1999).
[CrossRef]

J. Comput. Phys. (1)

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

J. Opt. Soc. Amer. A (1)

B. Gralak, S. Enoch, and G. Tayeb, �Anomalous refractive properties of photonic crystals,� J. Opt. Soc. Amer. A 17, 1012-1020 (2000).
[CrossRef]

Nature (2)

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

J. D. Joannopoulos, P. Villeneuve, and S. Fan, �Photonic crystals: putting a new twist on light,� Nature (London) 386, 143-149 (1997).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (6)

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]

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]

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

A. Mart�nez, H. M�guez, A. Griol, and J. Mart�, �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]

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]

Phys. Rev. E (2)

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, �Electromagnetic waves focused by a negative-index planar lens,� Phys. Rev. E 67, 025602 (2003).
[CrossRef]

R.W. Ziolkowski and E. Heyman, �Wave propagation in media having negative permittivity and permeability,� Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Phys. Rev. Lett. (3)

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]

C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Koltenbah, and M. Tanielian, �Experimental Verification and Simulation of Negative Index of Refraction Using Snell�s Law,� Phys. Rev. Lett. 90, 107401 (2003)
[CrossRef] [PubMed]

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]

Science (1)

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. (1)

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

Other (3)

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

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

A. Yariv, P. Yeh, Optical Waves in Crystals : Propagation and Control of Laser Radiation (New York, Wiley, 1984).

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Figures (9)

Fig. 1.
Fig. 1.

(a) TM-polarization photonic band structure of the 2D PhC under study: a triangular lattice (the inset shows its first BZ) of dielectric rods with ε=10.3 and radius r=0.4a. Effective refractive index n eff of (b) the first and (c) the second photonic bands for the ΓM and ΓK directions of propagation.

Fig. 2.
Fig. 2.

FDTD simulation of the refraction of a monochromatic wave (fa/c=0.3153) that propagates along the ΓM direction in a 2D PhC wedge. The output interface is along (a) ΓM and (b) ΓK directions. The electric field parallel to the rods’ axis is shown. The output PhC-air interface is highlighted with a bold solid line. The normal to the interface is highlighted with a bold dashed line. Arrows show the propagation direction of the waves inside the PhC and in air.

Fig. 3.
Fig. 3.

Diagrams showing the evolution of the electric field with time (vertical axis) and space (horizontal axis). 101 field monitors are employed in a 2D FDTD simulation. In all cases the wave is monochromatic and TM-polarized. (a) Propagation in air, fa/c=0.3153; (b) Propagation inside the PhC described in Fig. 1, fa/c=0.15, (c) Propagation inside the PhC described in Fig. 1, fa/c=0.3153. The phase fronts are the lines of the same color and the inverse of their slope gives the phase velocity. The slope of the arriving impulse stands for the group velocity, which can be obtained from the angle α in Fig. 3(c). The dashed lines in (b) and (c) highlight the slope of the main phase front.

Fig. 4.
Fig. 4.

Detected electric field at the field monitors 50 and 51 (spaced √3a/20 along the ΓM direction). A TM-polarized monochromatic wave propagating along ΓM and with frequencies fa/c=0.15 [(a) and (b)] and 0.3153 [(c) and (d)] is injected in the 2D PhC. The diagrams (a) and (c) corresponds to the arrival of the leading edge (related to the group velocity). The diagrams (b) and (d) correspond to a time step for which the steady state has been reached and the signal can be considered almost monochromatic (related to the phase velocity).

Fig. 5.
Fig. 5.

Study of the 2D PhC in Ref. 11: a triangular lattice of dielectric rods (r=0.182 and ε=11), fa/c=0.8648. (a) FDTD simulation (electric field) of the refraction of a monochromatic wave that propagates along the ΓM direction in a 2D PhC wedge. Output interface along the ΓM direction. (b) Diagram showing the evolution of the electric field with time (vertical axis) and space (horizontal axis) inside the PhC. 101 field monitors are employed. (c) Detected electric field at the field monitors 50 and 51 (spaced √3a/20 along the ΓM direction).

Fig. 6.
Fig. 6.

Wave vector diagram of the 2D PhC under study (Fig. 1). The red circular contours correspond to the EFSs at frequency fa/c=0.3153. The fundamental vectors of the reciprocal lattice are G1 and G2. First and second BZs are highlighted with different gray tones.

Fig. 7.
Fig. 7.

Plane-wave decomposition of the space sampling of the electric field that propagates inside the PhC under study along the ΓM direction. The field correspond to a simulation time step for which the wave has reached its steady state. (a) Transverse sampling (a/20 spacing); (b) longitudinal sampling (√3a/40 spacing); (c) longitudinal sampling: √3a/20 spacing (solid curve) and √3a/5 spacing (dashed curve). The peaks corresponds to the amplitude of a certain plane wave component of the whole Bloch wave.

Fig. 8.
Fig. 8.

Plane-wave decomposition of the longitudinal sampling (√3a/20 spacing) of the electric field that propagates inside the PhC under study along the ΓM direction. The field correspond to a simulation time step for which the wave has reached its steady state. (a) first band: fa/c=0.1 (solid curve); fa/c=0.15 (dotted curve); (b) second band: fa/c=0.3 (solid curve); fa/c=0.33 (dashed curve), fa/c=0.36 (dotted curve). Only positive wave vectors are shown.

Fig. 9.
Fig. 9.

Schematic explanation of the refraction at the output PhC-air interface using a wave vector diagram. The circle corresponds to the EFS of air. The hexagon is the first BZ of the PhC. The bold solid gray lines show the interface: (a) along ΓK; (b) along ΓM. The dashed lines represent the condition of conservation of the wave vector components parallel to the interface. The arrows stand for the energy flow of the wave before and after the interface.

Equations (2)

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E z ( r , k ) = m = n = E m , n exp [ j ( k + m G 1 + n G 2 ) ]
k PhC = k i + 2 π l a

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