We have investigated for the first time the anomalous refractive effects of a photonic crystal (PhC) formed by holographic lithography (HL) with triangular rods arranged in a honeycomb lattice in air. Possibilities of left-handed negative refraction and superlens are discussed for the case of TM2 band with the index contrast n = 3.4:1. In contrast to the conventional honeycomb PhC made of regular rods in air, the HL PhCs show left-handed negative refraction over a wider and higher frequency range with high transmissivity (>90%), and the effective indices quite close to −1 for a wide range of incident angles with a larger all-angle left-handed negative refraction (AALNR) frequency range (Δω/ω ≈14.8%). Calculations and FDTD simulations demonstrate the high-performance negative refraction properties can happen in the holographic structures for a wide filling ratio and can be modulated by changing the filling ratio easily.
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Recently there has been considerable interest in realizing anomalous propagation behaviors of light such as negative refraction [1–3] in optical frequencies by means of deliberately designed artificial optical nanostructures. Shelby  had an experimental verification of the first left-handed material (LHM) structure based on split-ring resonators (μ<0) interconnected with a set of metallic rods (ε<0), such media are realized by metamaterials composed of metal/dielectric composites. However, the absorption loss in the metal limits potential optical applications. In contrast to left-handed materials, photonic crystals made of synthetic periodic dielectric materials can exhibit an extraordinarily high, nonlinear dispersion such as negative refraction and self-focusing properties that are solely determined by the characteristics of their band structures and equal frequency contours (EFC) [5–7].
The light propagation properties of PhCs are closely related to their specific structures . The properties of lattices with square and hexagonal symmetry have been widely calculated [9,10] and used in negative refraction experiments [11,12]. Derived structures have been studied together with the basic geometries. Nevertheless, traditional two-dimensional (2D) photonic crystals are synthetic periodic structures of rectangular or circle columns in air (or holes etched in dielectric slabs) with square, hexagonal or honey-comb lattices, only a few works have been reported by now on the negative refraction effect for PhCs formed by holographic lithography, and they are all three-dimensional(3D) structures . Since 2D holographic structures can be more easily fabricated and have wide potential use, it is of interest and importance to extend the study of propagation properties to 2D HL structures. In this work, we take the 2D HL honeycomb structures formed by single-exposure interference fabrication methods as examples to investigate a series of HL PhC structures in order to obtain comprehensive understanding of the anomalous refractive properties in PhCs formed by HL.
In our analyses the plane wave method (PWM)  with wave number of 729 is employed to assure the convergence of numerical calculations and the EFCs, then the propagating direction of an electromagnetic wave (EMW) is achieved with wave vector diagram, finally the existence of negative refractions and PhC superlens are demonstrated by finite-difference time-domain (FDTD) simulations with perfectly matched layer (PML) boundary conditions . Due to the desired conditions for the case under study, only the TM modes are considered here (the E field is parallel to the rods).
2. Structures and calculations
The 2D holographic structure considered in this paper is determined by the intensity distribution  of17]. The filling ratio of the HL PhC is determined by the ratio of the total exposure dose (the product of light intensity and exposure time). Since crystalline Si (c-Si) is an attractive material for its natural abundance and nearly ideal band gap, here we choose silicon as an example and assume ε = 11.56 (i.e. n = 3.4) to analyze the dispersion characteristics of the holographic PhCs. With the increase of intensity threshold I t the pattern of inverse structure changes from a noncontinuous to a continuous one as shown in Fig. 1 , which consists of honeycomb lattice with trigonal rods arranged in air. When I t = 2.0, corresponding to the filling ratio of f = 25.6%, the rods just begin to be connected and then the veins become thicker with I t increasing.
In order to find the high-performance negative refraction properties of the honeycomb structure, we first make a systematic numerical calculation and analysis of the band structures and EFCs. The frequency distribution of the HL honeycomb PhC with I t = 2.0 in the four lowest bands is plotted in Fig. 2 , in which the inset gives the corresponding irreducible Brillouin zone and three high-symmetry lattice points. The light line in a vacuum (ω = ck) separates the shadow region (ω> ck) and the shadowless region (ω< ck). As a rule, the modes in shadow region are oscillatory in air, the photonic bands above the light cone can be probed experimentally with polarized-angular dependent reflectivity measurements ; the modes below the light line are useful in engineering applications for their vertical confinement and long lifetimes. Obviously, for the HL honeycomb PhC, a large photonic band gap (PBG), from 0.395 to 0.466ωa/2πc (where c is the light velocity and a is the lattice constant), exists between the second and third bands. The frequencies are normalized as ωa/2πc (or a/λ). The symmetrical point Г is the Brillouin center and the corresponding frequencies of the second and third bands decrease with the operational reference point shifting from Г to the vicinity points M or K, which indicate that Left-handed refractive effect is possibly in the PhC around the point Г .
The EFCs plot and wave vector diagram of TM2 band in the holographic PhC is shown in Fig. 3 , where the EFCs over a frequency region extend from 0.30 to 0.39, and the shapes of the EFCs around point Г are convex and shrink with increasing frequency, indicating the PhC is left-handed . For the EFCs with ω<0.348, the air EFCs are surrounded by the corresponding one in the PhC, and ω ≤ 0.5 × 2πc/a s (where a s is the ГK interface period, here a s = a). The foregoing analyses reveal that the necessary conditions for AALNR  are satisfied for the frequency range from 0.30 to 0.348 (relative bandwidth Δω /ω ≈14.8%) in the HL honeycomb PhC, which is higher and wider than the result from 0.2 to 0.218 (Δω /ω ≈8%) for the honeycomb lattice PhC composed of regular circle columns . For the TM3 band, the EFCs of a frequency region from 0.45 to 0.60 are convex in the vicinity of point K and have outward-pointing group velocities, and analyses find that left-handed refraction is present for the third band but only for limited incident angles, since the EFCs surrounding the point Г are star-like.
It has been proved that in PhCs, as for homogeneous materials, the energy velocity vector equals to the group velocity vector v gr . The propagation direction of light beam in any medium is given by the energy velocity vector. According to definition the group velocity vector v gr = ∇k ω is always oriented perpendicular to the EFC in the direction along which the frequency is increasing. For the second band of the PhC, the v gr is pointed inward from the EFC, and the phase velocity vector v ph and v gr can be antiparallel like in Veselago’s metamaterials , which denote a left-handed negative refraction. Therefore, an effective negative refractive index n eff can be obtained. The band structure predicts the behavior of the fundamental wavevector, which can be used to calculate the effective index. We define an effective index as the ratio of the (local) fundamental wavenumber and the wavenumber in free space. Figure 4 shows the effective index n eff in a wide frequency region from 0.30 to 0.39 augmenting with the increasing frequency, where the blue solid line denotes n eff = −1. The frequency of ω = 0.348, corresponding to the effective index n eff = −1, should be the optimal frequency for a 2D photonic-crystal-based superlens. In Fig. 3, the normal is along ΓM direction, the blue circle represents the EFC of ω = 0.348 in air, the dashed line means the conservation of the parallel components of wave vectors, and the blue, green and red arrows denote the directions of incident wave vector K i, refractive wave vector K r and group velocity V gr, respectively, and the blue circle is of comparable size with the corresponding PhC EFC, where K i makes an angle of 30° with the normal ΓM and K r ·V gr < 0 with V gr pointing to the negative direction with the same angle to the normal, which demonstrates the effective index are quite close to −1.
Further insight about the anomalous refractive effects of the frequency region for TM2 band can be gained by investigating the relation between the filling ratio f and the effective index n eff. Figure 5 gives the trend of frequency range varying with filling ratio for anomalous refractive effect, where the red solid line represents the optimal frequencies for PhC superlens with n eff = −1. When the filling ratio varying from 4% to 48%, the frequency scope decreases from 0.143 to the minimum of 0.086 at f = 15% (or I t = 1.85) and then increases to 0.115. By numerical calculations we know that the minimum n eff varies from −1.18 to −2.38 and the optimal frequency with n eff = −1 (the red solid line) falls from 0.506 to 0.303 with the filling ratio increasing, which imply it is easier to acquire negative refraction or PhC superlens in the high frequency range by decreasing the filling ratio of HL honeycomb PhC.
3. Numerical simulations
Considering the symmetry of Bloch modes of this PhC, the interface between PhC and free space can be arranged along ГM or ГK directions. In order to verify the previous results, different simulations have been made by using FDTD method. FDTD simulations show that the incident beam is easy to propagate through the PhC slab with the interface normal to the ГM direction, but very difficult to ГK direction. These phenomena can be explained by the symmetry mismatch between the external plane wave at normal incidence and the Bloch modes of this PhC . So the interface between PhC and free space is arranged along the ГK direction.
The PhC slab used for negative refraction simulations has eight layers in the propagation direction (ΓM) with f = 0.256. The high index contrast honeycomb structure can achieve good optical confinement, but it also induces serious impedance mismatch that leads to strong reflection and scattering at the interface between the input free space and the surface of PhC slab. To conquer this obstacle, we dispose the surface of the PhC slab with the trigonal dielectric flange (cut 0.4a) as shown in Fig. 6(a) to effectively reduce the reflection and scattering losses [20,21], for the trigonal dielectric flange has an effective index gradually varying to match the input free space. A source emits a continuous monochromatic TM polarization plane wave of desired frequency ω = 0.348, which is incident upon the PhC slab with angles of θ = 30° and 60° to the normal of the interface. The simulated wave patterns are shown in Fig. 6(b) and 6(c) respectively. In both cases the incident beams are refracted in the opposite directions of the reflected beams, namely, the effective refractive index of this PhC is n eff = −1, and the negative refraction in this holographic PhC is an absolutely left-handed behavior with K r ·V gr<0, which is in good accordance with the calculation results in section II.
We further calculated the field patterns of flat superlens made by this PhC based on its absolutely left-handed behavior. A continuous-wave point source of ω = 0.348 is located on the upper side of the PhC slab. In Fig. 7(a) , the distance from the point source to the upper interface (i.e. the object distance) is d o1 = 8.0a, and the image approximately locates at the edge of the lower interface (i.e. the image distance d i1 = 0). In Fig. 7(b), the object distance is d o2 = 3.5a and the relevant image distance becomes d i2≈4.6a. It is obvious that the sum of d o and d i is mostly a constant for this PhC slab, which satisfies the Snell’s law for a flat lens with n eff = −1. Moreover, it is worth emphasizing that in Fig. 7(b) there is an internal focus inside the PhC slab, which is a clear evidence of LHMs following the rules of geometric optics. The similar simulations have been made for other PhC slabs with different filling ratios to prove that superlens properties can be acquired in the HL honeycomb PhC slabs easily. For the fixed source and PhC slab, the focus distance can be modulated by changing the frequency of incident light freely. If the same point source is located in front of the PhC slab with the interface normal to ГK direction, the field pattern is shown in Fig. 7(c). It is clear that the wave is difficult to propagate through the PhC slab perpendicularly due to the symmetry mismatch, yet, some beams leak out with ± 60° to the normal, which can be explained for the excited Bloch wave is nearly in the ΓM direction.
The normalized transmission and reflection spectra of the HL honeycomb PhC are given in Fig. 8 . A Gaussian wave with high center frequency ω 0 = 0.424ωa/2πc incidents normally upon the PhC slab with f = 10.1% (or I t = 1.75). The transmission spectrum is measured at the back of the PhC slab. Figure 8 shows the transmission (red line) and reflection (blue line) spectra, the black arrow indicates the frequency with n eff = −1. In accord with the predicted results by the frequency scope of Fig. 5, the Gaussian wave over a wide range of frequencies from 0.38 to 0.46 can propagate through the HL honeycomb PhC slab and the other part is forbidden by the side PBGs. Particularly, the Gaussian wave with frequency below the light cone can propagate through the PhC slabs with a high transmission (>90%), which is not affected by undesired diffraction. For the part of waves with the frequency above light cone, the reflection and diffraction losses lead to the decrease of transmissivity. Obviously, comparing with the conventional honeycomb lattices made of regular rods in air, the left-handed negative refraction with high transmissivity (>90%) can be more easily acquired in a wider and higher frequency scope in the HL honeycomb PhCs.
In conclusion, we have theoretically investigated the absolute left-handed behaviors in a 2D HL PhC made of triangular rods arranged in a honeycomb lattice in air. The high-performance negative refraction properties are found by analyzing the photonic band structures and EFCs for the contrast n = 3.4:1. We studied the relation between the filling ratio f and the effective refractive index n eff, and demonstrate that the left-handed negative refraction exists over a wider and higher range of frequency with high transmissivity (>90%) compared with the conventional honeycomb PhC made of regular rods in air, and the frequency range can be upgraded by decreasing the filling ratio. Moreover, it is found that the effective index is quite close to −1 for a wide range of incident angles with a larger AALNR frequency range (Δω /ω ≈14.8%). Typical left-handed behaviors such as negative refraction, flat superlens are simulated by FDTD method, which accord well with the results of our numerical calculations. We believe that these results are important and useful for understanding and designing the anomalous optical behavior of PhC applications fabricated by HL.
This work is supported by the National Natural Science Foundation (60777008, 10804063 and 60907005), the Doctoral Research Foundation of Shandong Jianzhu University (XNBS0903) and the Project-sponsored by SRF for ROCS, SEM, China.
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