Elsevier

Optik

Volume 124, Issue 4, February 2013, Pages 361-365
Optik

Intensity depending on object distance in a two-dimensional photonic crystal

https://doi.org/10.1016/j.ijleo.2011.12.008Get rights and content

Abstract

The propagation of electromagnetic (EM) waves in two-dimensional triangular-lattice photonic crystals (PCs) is investigated through dispersion characteristics analysis and numerical simulation of field pattern. The designed PC structure can exhibit all angle negative refraction in the second band. A flat superlens formed from such a PC has been designed and its imaging properties have been investigated systematically. Good-quality images and focusing, with relative refractive index of −1, have been observed in this system for TE mode waves. In contrast to the images in near-field region for the lowest valence band, non-near-field images, explicitly following the well-known wave-beam negative refraction law, have been demonstrated. The relation between the intensity of the image and the object distance has also been discussed in this paper. The extensive applications of such a phenomenon to optical devices are anticipated.

Introduction

Recently, there has been a great deal of interest in studying class of media that are known as the left-handed materials (LHMs)[1], [2], [3], [4], [5], [6]. These materials are characterized negative permittivity ɛ and negative permeability μ simultaneously. Properties of such materials were analyzed theoretically by Veselago over 40 years ago [1]. Due to the absence of naturally occurring materials having both negative ɛ and negative permeability μ, Veselago's prediction did not receive much attention until recently, when a system consisting of an array of split-ring resonators and metallic wires was prepared and demonstrated to have negative refractive index experimentally [3], [4].

It was shown that negative refraction could also occur in photonic crystals (PCs) [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. The physical principles that allow negative refraction in them arise from the dispersion characteristics of wave propagation in a periodic medium, which can be well described by analyzing the equifrequency surface (EFS) of the band structures [7], [8], [9], [10], [11], [12], [13], [14], [15].

Theoretical studies indicate that the underlying mechanism for negative refraction of electromagnetic (EM) waves in PCs is not unique. The first is the lefthanded behavior [7], [8], [18], [19], [20], [21], [22], where the group velocity and the phase velocity derived from the band dispersion are antiparallel to each other for all the values of wave vector k, leading to neff < 0 for the PC. In another mechanism, negative refraction may occur when the incident field couples to a band with convex EFS contours in k-space, where the conservation of the surface parallel component of the wave vector k, combined with the “negative” curvature of the band causes the incident beam to bend negatively [12], [15], [17], [23], [24], [25]. In this case, neither the group velocity nor the effective refractive index is negative and the PC is essentially a positive index medium that however exhibits negative refraction. Both mechanisms have been well confirmed by the experimental observations [10], [12].

Pendry [5] predicted that the LHM slab can amplify the evanescent waves and the flat lens constructed from such a material with ɛ = μ = −1 which could in principle work as “perfect” lens (superlens). Recently, such image behaviors have been observed by some numerical simulations [26], [27], [28] and experimental measurements [28], [29], [30], [31]. However, only near-field images were demonstrated and extensive applications of such a phenomenon were limited [26], [27], [28], [29], [30], [31].

How to realize a good-quality non-near-field image becomes an important issue. The prerequisite condition to realize such a phenomenon is the negative refraction which possesses the single-mode and high transmission. The investigations [8] have shown that air-hole-type 2D PCs possess good lefthanded behavior for TM modes, and a pillar type 2D PC prefers TE modes. In this paper, the propagation of electromagnetic (EM) waves in 2D triangular-lattice PCs is investigated through dispersion characteristics analysis and numerical simulation of field pattern. A flat superlens formed from such a PC has been designed and its imaging properties have been investigated systematically. The image resolution, the slab thickness and the object distance in the 2D PC superlens also have been discussed.

Section snippets

Dispersion characteristics analysis

We consider PC structures made from a triangular lattice of dielectric rods of permittivity ɛ = 12 embedded in air background. The radius of dielectric rods is r = 0.4a, where a is the lattice constant. To study EM wave beam propagation in the PC, we must consider the group velocity vector. We thus need to investigate the EFS contours of the PC because the group velocities of the excited photonic modes are given by the gradient vectors of the EFS in k-space. Plane wave expansion method (PWEM) is

Numerical simulation of field patterns

To test our analysis, we perform numerical simulations to investigate the focusing ability of the present triangular-lattice PC structures with the surface normal along the ΓK direction. We employ the finite difference time domain (FDTD) method with the use of perfectly-matched layer (PML) boundary condition. Fig. 2 shows the effective phase index neff versus frequency f for the second band along the ΓK direction. In 2D PCs, neff had been discussed in detail in Ref. [9]. From Fig. 2, we can see

Effect of absorption and gain

The earlier investigations have shown that the PC slab consisting of triangular lattice of dielectric rods is actually considered as a good superlens for TE mode waves. However, the absorption for the systems is inevitable. Lou et al. [16] have pointed that the central image peak disappears and the image degrades gradually with the increase in absorption. Fortunately, the loss can be overcome by introducing the optical gain in the systems.

In this paper, we also attempt to investigate the

Conclusion

Through the exact numerical simulation and physical analysis, we have demonstrated all-angle single-beam lefthanded behavior for TE mode in the 2D triangular photonic crystal with dielectric rods in air background. More interestingly, the relative refractive index of −1 has been found. The ideal LHM system can serve as a perfect superlens. Good-quality images and focusing, with relative refractive index of −1 and explicitly following the well-known wave-beam negative refraction law, have been

Acknowledgments

This work was supported by the National Key Basic Research Special Foundation of China at no. 2005CB724304, and the National Natural Science Foundation of China at no. 60777045, Shanghai Education committee innovation project.

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