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Positive phase evolution of waves propagating along a photonic crystal with negative index of refraction

Alejandro Martínez and Javier Martí

Open Access Open Access

Abstract

We analyze propagation of electromagnetic waves in a photonic crystal at frequencies at which it behaves as an effective medium with a negative index in terms of refraction at its interface with free space. We show that the phase evolution along the propagation direction is positive, despite the fact that the photonic crystal displays negative refraction following Snell’s law, and explain it in terms of the Fourier components of the Bloch wave. Two distinct behaviors are found at frequencies far and close to the band edge of the negative-index photonic band. These findings contrast with the negative phase evolution that occurs in left-handed materials, so care has to be taken when applying the term left-handed to photonic crystals.

©2006 Optical Society of America

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Figures (6)
Fig. 1.
Fig. 1. (a) Schematic EFCs of the PhC under study for frequencies between 0.29 and 0.34. The direction of the input external wave, as well as the direction of k 0,0 and v g inside the PhC, are depicted; (b) vg (normalized to c) vs. k 0,0 (normalized to Gy ) for the frequency range under study: results obtained by derivation of the photonic band (solid line) and by weighted summation of the group velocity of the different Fourier components (open circles). Inset: detailed view of the group velocity near the band edge.

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Fig. 2.
Fig. 2. (a), (b) Amplitudes of the coefficients e m,n and Δm,n obtained from the 2D Fourier transformation as a function of frequency. The inset in (b) shows in detail the region close to the band edge.

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Fig. 3.
Fig. 3. (a)–(d) Distribution of the real part of E z in the computational domain at a normalized frequency of 0.3 at different time steps (see the figure) of the FDTD simulations. (e) Left-hand side: detail of the computation domain at the free space-PhC interface; Right-hand side: detail of E z inside the dashed rectangle in (d).

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Fig. 4.
Fig. 4. Phase evolution of the Bloch wave with normalized frequency of 0.3 along a period in the y direction for different values of the x coordinate in a period. (b) is a the projection of (a) on a plane perpendicular to the x-axis. Values in (b) stand for the values of the x coordinate for which the phase evolution is obtained. The phase shifts that would occur in an ideal LH medium with negative wave vector k 0,0 are also shown as a black line in (b).

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Fig. 5.
Fig. 5. (a)–(d) As in Fig. 3, but for a frequency of 0.346. (e) Detail of Ez inside the dashed rectangle in (d).

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Fig. 6.
Fig. 6. As Fig. 4, but for a normalized frequency of 0.346. In (b) the values indicate the interval of the x axis for which the phase path are obtained.

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Equations (3)

Equations on this page are rendered with MathJax. Learn more.

E z ( x , y ) = E 0 ∑ m , n ∞ e m , n e i [ m G x x ] e i [ ( k 0,0 + n G y ) y ]
E z ( x , y ) = E 0 e ik 0,0 y ( ( e 0,0 + 2 e 2,0 cos ( 2 G x x ) + … ) + ( Δ 0,2 e i 2 G y y + 2 Δ 1,1 cos ( G x x ) e i G y y + … ) + ( 2 e 0 , − 2 cos ( 2 G y y ) + 4 e 1 , − 1 cos ( G x x ) cos ( G y y ) + … ) )
v → g = c 2 ω ∑ m , n ∞ ( k → 0,0 + m G → x + n G → y ) e m , n 2
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