We present a study on negative refractions in the four lowest bands of two-dimensional (2D) square lattices formed by holographic lithography (HL) and compare these features with those of a lattice of the same kind but with regular dielectric columns. The plane wave calculations and FDTD simulations have shown that in some bands or for some interfaces the negative refraction can only happen in holographic structures, and generally the rightness of holographic structures and regular structures of the same kind may be different.
© 2007 Optical Society of America
The left-handed materials (LHMs) were first proposed by Veselago and have attracted much attention in recent years for their interesting properties and potential use [1–5]. Shelby  had an experimental verification of the first LHM structure based on split-ring resonators (μ<0) interconnected with a set of metallic rods (ε<0). Photonic crystals (PhCs) made of periodically modulated dielectric or metallic materials can also show effective negative refraction . Unlike the metamaterials made of conductors which have huge losses at higher frequencies, PhCs can exhibit much lower losses even at high frequencies such as in an optical band, since they are composed of insulators and the size and periodicity of their scattering elements are of the order of optical wavelength. Though to define the values of ε or μ of a PhC is impossible, many anomalous effects such as negative refraction and “super prism”  were observed, and both negative refraction and left-handedness in PhCs were experimentally demonstrated in the microwave range [8–11].
There have been many analyses of negative refractions in PhCs recently, and most of them dealt with the PhCs of regular atoms or columns with a well-defined shape [12, 13]. The holographic lithography (HL) approach for PhC fabrication has its unique advantages such as one step recording in large scale [14, 15]. Since holographic structures usually have irregular atoms or columns, and the light propagation properties of PhCs are closely related to their specific structures [16, 17], we may expect some difference in propagation behavior between regular and holographic structures. However, only a very few works have been reported by now on the negative refraction effect for the PhCs formed by HL, and they are all three-dimensional (3D) structures . Considering that two-dimensional (2D) holographic structures can be more easily made and have wide potential use, it is of interest and importance to extend the study of negative refraction to 2D HL structures. In this paper, we take a 2D square structure with circular columns connected by veins  as an example to investigate this effect. In our analyses the plane wave method (PWM)  with wave number of 729 is employed to calculate the photonic band structure and the equal frequency contours (EFCs), then the propagating direction of an electromagnetic wave (EMW) is achieved with wave vector diagram, finally the existence of negative refractions are demonstrated by finite-difference time-domain (FDTD) simulations with perfectly matched layer (PML) boundary conditions . All the calculations are performed with Matlab software.
The intensity distribution of the holographic structure we adopt in analysis is 
With a proper choice of coefficient c and a threshold intensity I t, which is a specific value the region with light intensity below it can be removed and the region above it will remain due to photopolymerization for negative photoresist, we may wash away the region of I < I t to get a normal structure. By filling this structure with a material of high dielectric constant and then removing the template, an inverse structure can be obtained . In the following discussion we assume ε =11.4 for gallium arsenide as an example and choose c = 0.16 and I t = 2.72 which lead to the largest band gap as we analyzed before . To find the difference of propagating property between the holographic structures and the regular ones, we chose several lattices shown in Fig. 1 for investigation, here (a) and (b) are the inverse and normal holographic structures at the same threshold intensity I t = 2.72 respectively, and (c) is a regular structure having the same filling ratio (FR) as (a) but circular columns with r = 0.364a, here r is the radius of column and a the lattice constant.
3. Wave vector diagrams and FDTD results
The propagation of EMWs across both ΓM and ΓX interfaces of the 2D square lattice are investigated. As examples, we took the TE mode with its magnetic field perpendicular to the 2D plane and studied its propagation behavior at four lowest bands. The dimensionless relative frequency f= ωa/2πc is used, where c is the velocity of light in vacuum.
For the first band, if we choose ΓX interface as incident plane, the gradients of EFCs are outward and negative refraction cannot happen. On the other hand, for the ΓM interface, the EFCs in air and PhCs are plotted with wave vector diagrams for the incident angle of 25° in Figs. 2 (a), 2(b) and 2(c) for the holographic inverse structure, holographic normal structure, and the regular structure shown in Figs. 1(a), 1(b) and 1(c), respectively. Here and in all the similar figures below the frequency increases when the color changes from blue to red, the black dashed line means the conservation of the parallel components of wave vectors, the blue circle represents the EFC in air, and the blue, green and red arrows denote the directions of incident wave vector K i, refractive wave vector K f and group velocity V g, respectively; and the relative frequency in air is f= 0.28 for (a) and (c), and f= 0.24 for (b). From Figs. 2(a) and 2(b) we can see that in the outside region of the vicinity of M point EFCs are convex having an inward-pointing group velocity V g, but the black dashed line intersects two points at the EFC of photonic lattices, the upper one has V g pointing towards the source, while the bottom one has V g pointing away from the source. This means that only the bottom point contributes to a propagating beam or a transmitted beam. The refractive wave vector K f pointing to the intersect point is on the same side of the normal as K i is, thus right-handed (RH) negative refractions happen with K f ∙ V g > 0 and V g pointing to the negative direction . Different from (a) and (b), the EFCs with high frequencies around M point in (c) are concave, yielding an outward-pointing V g and positive refractions. In order to get a comprehensive comparison, we have also investigated the regular structures with r/a changing form 0.1 ~ 0.5, since the columns will overlap and not be circles if r/a > 0.5. The frequencies of EFCs reduce with increasing column radius, and we find that negative refractions only happen in one situation of r/a = 0.5 when the columns are just joined together with each other. This means that negative refractions cannot occur in regular structures with separated circular dielectric columns for the first band of TE modes. To demonstrate these phenomena more clearly, we give the FDTD simulated wave patterns of TE 1 in Figs. 3(a), 3(b) and 3(c) for the cases of Figs. 2(a), 2(b) and 2(c), respectively. From Fig. 3 we can see that V g points to the negative direction in (a) and (b) and positive in (c), exactly coincident with the analysis of Fig. 2. The V g in holographic normal structure (b) is the largest, since the corresponding EFCs have the densest distribution and the largest gradient. The EFC plots are sparser for the inverse structure, and sparsest for regular lattices.
The EFC plots for the second band are shown in Fig. 4; here the incident angle is 20°. Figures 4(a) and 4(b) represent the HL inverse and normal lattices respectively. For these two structures, EFCs become smaller and convex when frequency increases, so negative refractions can be achieved easily when waves of inner higher frequencies are incident upon both ΓX and ΓM interfaces, and in the both cases we can get left-handed (LH) negative refractions with K f∙V g < 0 with V g pointing to the negative direction. The corresponding FDTD simulations of refractions for these two structures are given in GIF image format with increasing incident angles from 5° to 60° in Figs. 5(a) and 5(b), where the incident frequency is taken as 0.35 and 0.32 respectively. From these images we can see clearly that in almost the whole angular range negative refraction exists in the two structures. On the contrary, for lattices with regular circular columns in Fig. 4(c), there is only one convex EFC of f= 0.36 around the origin, thus negative refraction happens only for the waves incident across ΓM interface; and different from the former two lattices, the negative refraction here is similar as in Figs. 2 (a) and 2(b) with K f∙V g > 0 and V g is negative. The FDTD simulations given in Fig. 6 are the wave propagation patterns of TE waves at an incidence of 20° on the ΓM interface, (a), (b) and (c) are for the three structures respectively. The consistency of pointing direction of V g in these figures demonstrated our analysis again.
The EFC plots for the third band are four similar parts distributed symmetrically around the origin, and the shape of each part just seems like concave squares with their centers at M point. Different from other bands, the EFC plots and correspondingly the refractive properties here for the three different structures are very similar, so we won’t say more about this band.
For the forth band the EFC plots are shown in Fig. 7 for the same three structures as in Figs. 2 and 4. Now what appear in the four corners are almost circles centered at M with the frequencies increasing towards to the origin. And from calculations we find that there are overlapped band areas between band 3 and band 4 in holographic structures but not in regular lattices, which means that holographic structures have larger band ranges than regular lattices for this band, so the distributions of EFCs around M point are denser in (a) and (b) than in (c). If a wave is incident on ΓX interface at a small angle, there is no intersection point between the dashed conservation line and EFCs. When incident angle increases, intersection points may appear in (a) and (b) but still not in (c), for in the case of (c) there are few EFCs around M point due to their low density. For an incident angle large enough, the coupling efficiency would be strongly affected. Therefore negative refraction happens only in holographic structures. This fact is verified in Fig. 8 using FDTD simulations of TE 4 wave incident on ΓX interface at 45°, the incident relative frequency is 0.50 for (a) and (c), and 0.43 for (b).
At last the propagating properties of the second and forth band for regular structures with changing column radius have also been studied for comparison, and we found that when r/a is large enough the EFCs of regular structures become similar to those of holographic structures, and negative refraction may happen in these situations similarly as in HL structures, but the column radius is restricted within a narrow range.
From the analyses above we can conclude that there are some differences in light propagation behavior between the holographic lattices and the lattice of regular columns. For TE1 mode, right-handed refraction happens in the former for ΓM interface, but not in the latter due to its concave EFCs near M point except the case of r/a = 0.5. And for TE2 mode, holographic structures can yield left-handed negative refractions when EMWs are incident either on ΓM or ΓX interface, for EFCs around Γ point are convex and have inward gradients; on the other hand, the regular lattice can only achieve right-handed negative refractions for ΓM interface incident when r/a is not large enough. The most similar properties for these three structures occur in the third band, where LH negative refraction happens in all lattices for a small wave band as the shapes of their EFCs are quite alike. On the contrary, the propagation properties in holographic and regular structures with a small ratio of r/a in band 4 are considerably different. The overlap of bands 3 and 4 in holographic structures leads to dense EFCs and negative refraction in band 4 lacked in regular structure. The negative refraction may appear in regular structures only for a large enough r/a. In table 1 we give all frequency bands that could intricate negative refractions for each structure, from it one can see that negative refraction seems more likely to happen in holographic structures, at least they cover a larger wave band.
We have investigated the negative refraction in 2D PhC structures made by HL and in the lattices of regular circular dielectric columns and compared the propagation properties of the three different structures. The results show that in some bands or for some interfaces the negative refraction can only happen in holographic structures, and generally the rightness of holographic structures and regular structures of the same kind may be different. Considering that HL has its unique advantages such as one-step recording, large scale and low cost fabrication, holographic structures may have a promising potential when negative refractions are concerned in practice. We may also combine the two kinds of structures together so that they would complement each other and find more applications. The more comprehensive study in this field will be our next task.
This work is supported by the National Science Foundation of China (60477005, 60677026) and the National Science Foundation of Shandong Province (Y2006A09), China.
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