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Waveguiding, collimation and subwavelength concentration in photonic crystals

J. L. Garcia-Pomar and M. Nieto-Vesperinas

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Abstract

By means of both finite elements and FDTD calculations, we demonstrate that a structure of photonic crystal, constituted by two dimensional arrays of dielectric cylinders in air, or viceversa, previously proposed as capable of producing negative refraction with superlensing properties and subsequently proved to lack this characteristic, do possesses however the property of giving rise to effects of total internal reflection that allow both waveguiding, bending and collimation with high intensity subwavelength concentration of wavefronts. This is a consequence of both the dominant propagation along the ΓM direction due to diffraction, and of intensity localization in the cylinder regions as a result of the operating frequency being in the lower part of the bandgap, namely, in the so-called dielectric band.

©2005 Optical Society of America

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Figures (9)
Fig. 1.
Fig. 1. (a) Band diagram for the photonic crystal structure for TM polarization (Ez parallel to the cylinders) The illumination frequency is marked by the red line. (b) Isofrequency curves of air (bellow) and PC(above) at this frequency.vg is the group velocity.

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Fig. 2.
Fig. 2. (a) Maps of the modulus |Ez | of the electric field E for two point sources separated a distance s from each other. (White spots in the space of simulation are out of the color scale). (b) Intensity of the point sources (bottom) and the intensity of the “image” (top), at distance 0.7mm from the tangent plane to the top cylinders, for different values of s. The peak abscissas coincide with those of illuminated cylinders

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Fig. 5.
Fig. 5. Maps of the modulus |Ez | of the electric field E for (a) a Gaussian beam entering in a prism in the interface parallel to XM, (b) and (c) a Gaussian beam entering in a prism in the interface parallel to ΓM and reflection in the tilted interfaces. (d) Isofrequency lines in air and in the PC, and illustration at the entrance (left) and at the reflection interface (right). Conservation of the parallel component of k at the entrance interface gives a broad range of k-vectors in the PC giving rise to group velocity vectors normal to the flat portion of the isofrequency line and thus propagating in the ΓM direction. On the other hand, conservation of the parallel component of k at the reflection interface gives rise to almost all k-vectors in air being evanescent (i.e. outside the k = ω/c circle).

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Fig. 3.
Fig. 3. (a) Same as Fig. 2(a) but with the upper row removed. (b) Normalized intensity of the image in the case of Fig. 2(a), i.e. 16 rows (dashed red line) and of Fig 3(a), namely, 17 rows (solid black line), and with 18 rows (dash-dot blue line) at the plane shown in (a) at distance 0.7mm from the tangent plane to the cylinders. In the three cases the peak abscissas fit with the position of the illuminated cylinders in the exit interface, and it is independent of the position of the two point sources.

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Fig. 6.
Fig. 6. (a) Map of the electric field Ez for a prism with a tilt angle ϕ = 63.43 o . (b) Conservation of the parallel components in the Bragg law for the tilted interface for refraction, where Γ2 is the origin of the second Brillouin zone. Notice that the pink broken line is normal to the tilted interface direction. The point at which this line crosses the air isofrequency circle marks the orientation of the refraction k-vector. (c) Conservation of the parallel components of the Bragg law for the tilted interface in the reflection process. (d) conservation of the component of k parallel to the right side interface in the refraction process.

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Fig. 4.
Fig. 4. Normalized intensity in a plane away 0.7 mm from the exit row of the response of the PC to a Gaussian beam of FWHM = 2 27 m m (solid black line) and a Gaussian beam with FWHM = 30mm (dashed red line).

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Fig. 7.
Fig. 7. (a)(Movie of the electric field with FDTD 560KB) Map of the modulus |Ez | of E for the tip structure with the upper cylinder tapered to sharper shape. (b) Intensity along the yellow line of Fig.11(a) showing the exponential decay typical of an evanescent wave. (c) Variation of the intensity at 0.7mm from the cylinder of the apex as a function of the number of rows of the tip, showing a resonator behavior. (d) Intensity along a line parallel to x-direction at 0.7mm from the apex with a circular top cylinder, i.e. not tapered to a sharper shape,(shaded curves) and with a sharpener cylinder vertex (solid lines) for two Gaussian beams with FWHM 12.65mm (red curves) and 30mm (black curves), respectively.

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Fig. 8.
Fig. 8. (a) Maps of the modulus |Ez | of the electric field E for a two apex structure in front of a point source and (b) linear response of the exit apex in the top to different intensities of the point source.

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Fig. 9.
Fig. 9. (a)Maps of the modulus |Ez | of the electric field E of a two-apex crystal.(b)Intensity of the wavefront emerging from an extended object, (solid line), compared with the output signal, (open circles), close to the upper apex when the lower apex and an aperture close to it raster scan the object wavefront. Both signals are normalized to unity. The peak of the output is the same as that of the object.

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