- Email: [email protected]
Imaging properties and negative refraction of the metallic photonic crystal at near-infrared frequency
Solid State Communications 152 (2012) 929–932
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Imaging properties and negative refraction of the metallic photonic crystal at near-infrared frequency Shuai Feng n, Yuxi Li, Bo Lei, Jieying Hu, Yiquan Wang, Wenzhong Wang School of Science, Minzu University of China, Beijing 100081, China
a r t i c l e i n f o
abstract
Article history: Received 26 August 2011 Received in revised form 5 March 2012 Accepted 16 March 2012 by Z. Tang Available online 28 March 2012
Imaging properties and negative refraction of the two-dimensional metallic photonic crystal (PC) consisting of a triangular lattice of silver rods immersed in a dielectric is studied. It is found that goodquality image spots and negative refraction with an effective refractive index of 1 can be achieved for the near-infrared frequency through the adjustment of the metallic PC’s lattice constant and the permittivity of the background material together. Drude dispersion model is adopted to describe the silver rod and it is found that the influence of the metal’s absorption is trivial to debase the quality of the image spot. & 2012 Elsevier Ltd. All rights reserved.
Keywords: A. Photonic crystal D. Light imaging D. Negative refraction
1. Introduction In the past several years, the negative refractive-index material (NIM), which has negative permittivity and permeability simultaneously, has attracted a great deal of interest [1–5]. Negative refraction of electromagnetic (EM) waves is the foundation for a variety of novel phenomena and potential applications, such as negative refractive index, reversed Doppler Effect and so on. It was shown that negative refraction can also occur in photonic crystals (PCs) [6–20]. Luo showed that all-angle negative refraction (AANR) could be achieved at the lowest band of twodimensional (2D) square-lattice PCs [6,7]. Zhang showed that non-near-field images can be achieved in the second band of the PC slab consisting of a triangular lattice of coated cylinder in air [8]. Qiu reported the negative refraction and light focusing at telecommunication wavelengths by a 2D PC slab fabricated by chemically assisted ion beam etching in the InP-based contrast system [9]. In our previous work, the imaging properties of PC slabs consisting of perfect metallic rods immersed in a dielectric were systemically studied [10,11]. The metal is assumed to be perfectly conducting, so the electric field within the metal and tangential to the metal surface is equal to zero, which is suitable to the microwaves and even the lower frequencies. Owing to the strong dispersion and absorption of the metal to the incident EM waves with high frequencies (such as the near-infrared light and visible
n
Corresponding author. Tel./fax: þ 86 10 68932205. E-mail address: [email protected] (S. Feng).
0038-1098/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2012.03.008
light), the metal cannot be treated as perfect conducting. In this paper, the Drude dispersion model is adopted to describe the dielectric properties of the metal, and it can reproduce the experimental spectrum well for the near-infrared light. We will show that the near-infrared imaging and negative refraction can be obtained through a triangular-lattice PC slab consisting of metallic rods immersed in a dielectric background. The nearinfrared imaging at near-infrared wavelength around 1550 nm is achieved by the adjustment of the size of the PC’s lattice constant and the refractive index of the background dielectric. It is also shown that the absorption of the metal rods to the incident light has a trivial influence to debase the quality of the image spots.
2. Numerical simulation and discussion The structure studied in this paper is based on the 2D triangular-lattice PC consisting of square metallic rods immersed in a dielectric, the side length of the square metal is 0.3a, where a is the lattice constant of the PC (as shown in Fig. 1(a)). Fig. 1(b) shows the first Brillouin zone of the triangular lattice. pffiffiffi ! ! ! ! The elementary vectors are a 1 ¼ a e x , a 2 ¼ ða=2Þ e x þ ð 3a=2Þ ! ! e y , and the elementary reciprocal lattice vectors are b 1 ¼ pffiffiffi ! ! pffiffiffi ! ! ð2p=aÞ e x ð2p= 3aÞ e y , b 2 ¼ ð4p= 3aÞ e y . There are three high symmetric points: the ! point (0,0), the K point ðð4p=3aÞ,0Þ, and pffiffiffi the M point (ðp=a, p= 3aÞ). In Drude model, the metal’s dielectric constant is depicted as eðoÞ ¼ e0 ð1ðo2p =ðoðo þ igÞÞÞÞ, where op is the plasma frequency
930
S. Feng et al. / Solid State Communications 152 (2012) 929–932
Fig. 1. (Color online) Schematic geometry of the triangular-lattice PC consisting of square metallic rods (a), and the first Brillouin zone of the 2D triangular lattice (b).
and g is the damping coefficient. The metal studied in paper is Ag, the corresponding values are op ¼ 1:37 1016 Hz, g ¼ 2:73 1013 Hz, which can reproduce the experimental spectrum well at the nearinfrared range. For the 2D PC, the electromagnetic waves can be separated into two independent polarized modes, TM-polarized mode and TE-polarized mode. In this paper, we only consider the TM polarization modes, where the electric field Ez is kept parallel to the extension axis of the metallic rods (assumed to be the z axis). In all of the simulations throughout this paper, we employ the finite-difference time-domain (FDTD) method with periodic boundary conditions to calculate the band structures and EFS contours in the reciprocal space, while Berenger’s perfectly matched layer absorbing boundary condition is employed to calculating the wave propagating in the finite real space [21,22]. In all the simulations, the size of the lattice constant is divided into 30 lattice grids, and good accuracy is ensured. At first, we set the length of the PC’s lattice constant is 430 nm, and the refractive index of the background dielectric is 3.45 (the permittivity of silicon at the near-infrared wavelength). Ignoring the absorption of the metal to the EM waves, the calculated band structure for the several lowest photonic bands along the highsymmetric lines in the first Brillouin zone is shown in Fig. 2(a). It can be seen that the metallic rods lift the first TM-polarized band and cause a stop band at the lowest frequencies. The cutoff frequency of this metallic PC is 91.6 THz (1 THz ¼1012 Hz). For such a perfect PC, the lowest photonic band extending from 91.6 THz to 144.9 THz, and the second band spans from 144.9 THz to 232.9 THz. Fig. 2(b) shows the several calculated equal-frequency surface (EFS) contours in the second band. It can be seen that some EFS contours such as 1.8, 1.935, and 2.0 (in unit of 1014 Hz) are very
Fig. 2. The TM-polarized photonic band structure (a) and the EFS contour in the second band (b) of the metallic PC (in a unit of 100 THz). The PC is composed of a triangular lattice of square metallic rods immersed in the silicon background, whose refractive index is set to be 3.45. The lattice constant of the PC is a¼ 430 nm, and the side length of the square metal is 0.3a.
close to a perfect circle, indicating that the PC can be regarded as an effective homogeneous medium at these frequencies. These frequencies increase inwards, so the group velocities are opposite to the phase velocities, meaning that the transmitting features of the EM waves in the PC are negative refraction behavior. The alteration of the metal’s shape from a circle to a square does not change the EFS’s shape. the Now we visualize the propagation of EM waves through this kind of metallic PC slab by employing the FDTD method with a boundary treatment of perfectly matched layers, and the actual absorption of the metal to the incident EM wave is also considered. The rectangular metallic PC slabs are 11, 21 and 41 layers thick, whose surface normal is along the !M direction. The whole PC slab is embedded in the air environment. The distances from the four slab edges to the center of the adjacent metallic rods are 0.3a. Thus the exact thicknesses of the PC slabs are 9.3a, 17.9a, 35.3a in Fig. 3(a), (b), and (c), respectively. A point source of continuous wave resonating at 193.5 THz (the corresponding wavelength in vacuum is 1550 nm) is placed at the left side of the PC slab, the calculated Ez field patterns are shown in Fig. 3(a), (b), and (c), respectively. It can be
S. Feng et al. / Solid State Communications 152 (2012) 929–932
931
1.0 0.9 Relative Light Intensity
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -4
-3
-2
-1 0 1 Position (µm)
2
3
4
-4
-3
-2
-1 0 1 Position (µm)
2
3
4
1.0 0.9 Relative Light Intensity
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
Fig. 4. (Color online) Lateral light intensity distributions centered at the image spot formed by the light propagation of a point source across the 21-layer (a) and 41-layer (b) 2D PC slabs. The solid line corresponding to the metallic PC with no absorption, while the blue dotted line corresponding to the metal with absorption. The full width at half maximum of the image spot are all about 0.5l.
Fig. 3. Simulated Ez field distributions of a point source and its image across a metallic PC slab. The rectangular PC slabs are 11 (a), 21 (b), and 41 (c) layers thick. The lattice constant of the PC is a¼ 430 nm, and the refractive index of the background material is 3.45. The slab’s surface normal is along the !M direction. The distances from the left and right slab edges to the center of the adjacent rods are all 0.3a. Dark and bright regions correspond to negative and positive Ez, respectively.
seen from Fig. 3 that the EM waves emitted from the point sources transmit through the PC slabs and good-quality image spots are formed in the right side of the PC slabs. The distances
from the point source to the center of the image spots are 18.8a, 36.0a, 70.8a in Fig. 3(a), (b), and (c), respectively. Because the shape of the EFS of the frequency 193.5 THz is a perfect circle, the rectangular PC slab can be looked as an isotropic material for this frequency. The light beams, which are radiated from a point source placed on the left side of the slab, transmit the PC slab and the refractive light beams inside the PC slab propagate on the same side of the surface normal owing to negative refraction. We know that the effective refractive index of the PC is determined by the EFS of the PC and the EFS of the source in the background. If the EFS of the PC is circular and inward at some fixed frequency, at the same time the radius of that EFS is equal to the radius of the EFS of the source in the background. Thus, the effective refractive index of the PC is 1. Light focusing is observed inside the PC above slabs, and the distance from the light sources to the image spots is twice the thickness of the PC slab. The rectangular PC slab can be looked as an isotropic material with an effective refractive index of 1. The size of the image spots in the lateral direction is about 1.8a (where a¼430 nm), corresponding to 0.5l (l ¼1550 nm). This value is much smaller than the traditional image spot restricted by the diffraction limit inherent in conventional lens.
932
S. Feng et al. / Solid State Communications 152 (2012) 929–932
square metallic rods immersed in a dielectric background. The light imaging and negative refraction at near-infrared wavelength around 1550 nm can be obtained by adjusting the lattice constant of PC and the permittivity of the background material together. Drude dispersion model is adopted to describe the metal’s permittivity, and it is found that the absorption of the metal rod is trivial to debase the image spot’s intensity, especially the image’s quality. This offers a powerful way to engineer the negative refraction and focusing properties of PC slab lens at the telecommunication wavelengths.
900
Lattice constance (nm)
800 700 600 500 400 300
Acknowledgments
200 1
2
3 4 5 6 7 Refractive Index of the background
8
9
Fig. 5. The size of the PC’s lattice constant varies with the refractive index of the background material, where an effective refractive index of 1 occurs for nearinfrared wavelength around 1550 nm. The side length of the metal is 0.3a.
Fig. 4 shows the lateral light intensity distributions centered at the image spot formed by the light propagation of a point source across the 21-layer and 41-layer PC slabs. The solid red lines corresponding to the metallic PC with no absorption of the metal, while the dotted blue line taking into account of the metal’s absorption to the incident light wave. It is found that the relative intensity of the image spot reduces to be 0.89 through the 21-layer PC slab, while the value reduces to be about 0.8 through the 41-layer PC slab owing to the metal’s absorption. The influence of the metal’s absorption is severe to debase the quality of the image spot. And the full-widths at half-maximum of the image spots are all about 0.5l, so high-quality images are obtained. According to Pendry’s theory [6], the single-beam behavior is only possible under the condition that o r0:5 2pc=as (where as is the surface-parallel period) in order to avoid diffraction. So the lattice constant of the metallic PC must less than 775 nm to ensure the good-quality imaging at the near-infrared wavelength 1550 nm. Based on the systematical simulation, keeping the side length of the square metal equal to 0.3a, the advisable values for the PC’s lattice constant and the background’s refractive index are shown in Fig. 5 to ensuring the high-quality imaging and negative refraction with an effective refractive index of 1 at the nearinfrared wavelength around 1550 nm.
The authors gratefully acknowledge the National Natural Science Foundation of China (Grant nos. 10904176 and 11004169), the Research Foundation of the State Ethnic Affairs Commission of People’s Republic of China through Grant no. 10ZY05, the NMOE Project of China through Grant no. 2010110009, the ‘‘985 Project’’ and ‘‘211 Project’’ of the Ministry of Education of China.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
3. Conclusion In summary, we have studied the light imaging and negative refraction through the 2D triangular-lattice PC slabs consisting of
[20] [21] [22]
J.B. Pendry, Phys. Rev. Lett. 85 (2000) 3966. R.A. Shelby, D.R. Smith, S. Schultz, Science 292 (2001) 77–79. A.A. Houck, J.B. Brock, I.L. Chuang, Phys. Rev. Lett. 90 (2003) 137401. D.R. Smith, D. Schurig, Phys. Rev. Lett. 90 (2003) 077405. J. Pacheco, T.M. Grzegorczyk, B.I. Wu, Y. Zhang, J.A. Kong, Phys. Rev. Lett. 89 (2002) 257401. C.Y. Luo, S.G. Johnson, J.D. Joannopoulos, J.B. Pendry, Phys. Rev. B 65 (2002) 201104. (R). C.Y. Luo, S.G. Johnson, J.D. Joannopoulos, J.B. Pendry, Phys. Rev. B 68 (2003) 045115. X.D. Zhang, Phys. Rev. B 70 (2004) 195110. A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thyle´n, A. Talneau, S. Anand, Phys. Rev. Lett. 93 (2004) 073902. S. Feng, Z.Y. Li, Z.F. Feng, B.Y. Cheng, D.Z. Zhang, Appl. Phys. Lett. 88 (2006) 031104. S. Feng, Y.Q. Wang, Z.Y. Li, B.Y. Cheng, D.Z. Zhang, Chin. Phys. Lett. 24 (2007) 229–232. S. Feng, Y.Q. Wang, Z.Y. Li, B.Y. Cheng, D.Z. Zhang, Chin. Phys. 16 (2007) 1689–1693. Z.Y. Li, L.L. Lin, Phys. Rev. B 68 (2003) 245110. X.D. Zhang, L.M. Li, Appl. Phys. Lett. 86 (2005) 121103. C.Y. Qiu, X.D. Zhang, Z.Y. Liu, Phys. Rev. B 71 (2005) 054302. L. Gan, Y.Z. Liu, J.Y. Li, Z.B. Zhang, D.Z. Zhang, Z.Y. Li, Opt. Express 17 (2009) 9962–9970. J. Tian, M. Yan, M. Qiu, C.G. Ribbing, Y.Z. Liu, D.Z. Zhang, Z.Y. Li, Appl. Phys. Lett. 93 (2008) 191114. G.Y. Dong, J. Zhou, X.L. Yang, L.Z. Cai, Opt. Express 19 (2011) 12119–12124. G.P. Swift, A.J. Gallant, N. Kaliteevskaya, M.A. Kaliteevski, S. Brand, D. Dai, A.J. Baragwanath, I. Iorsh, R.A. Abram, J.M. Chamberlain, Opt. Lett. 36 (2011) 1641–1643. S.L. He, Z.C. Ruan, L. Chen, J.Q. Shen, Phys. Rev. B 70 (2004) 115113. K.S. Yee, IEEE Trans. Antennas Propag. 14 (1966) 302–307. J.P. Berenger, J. Comput. Phys. 127 (1996) 363–379.
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Imaging properties and negative refraction of the metallic photonic crystal at near-infrared frequency Shuai Feng n, Yuxi Li, Bo Lei, Jieying Hu, Yiquan Wang, Wenzhong Wang School of Science, Minzu University of China, Beijing 100081, China
a r t i c l e i n f o
abstract
Article history: Received 26 August 2011 Received in revised form 5 March 2012 Accepted 16 March 2012 by Z. Tang Available online 28 March 2012
Imaging properties and negative refraction of the two-dimensional metallic photonic crystal (PC) consisting of a triangular lattice of silver rods immersed in a dielectric is studied. It is found that goodquality image spots and negative refraction with an effective refractive index of 1 can be achieved for the near-infrared frequency through the adjustment of the metallic PC’s lattice constant and the permittivity of the background material together. Drude dispersion model is adopted to describe the silver rod and it is found that the influence of the metal’s absorption is trivial to debase the quality of the image spot. & 2012 Elsevier Ltd. All rights reserved.
Keywords: A. Photonic crystal D. Light imaging D. Negative refraction
1. Introduction In the past several years, the negative refractive-index material (NIM), which has negative permittivity and permeability simultaneously, has attracted a great deal of interest [1–5]. Negative refraction of electromagnetic (EM) waves is the foundation for a variety of novel phenomena and potential applications, such as negative refractive index, reversed Doppler Effect and so on. It was shown that negative refraction can also occur in photonic crystals (PCs) [6–20]. Luo showed that all-angle negative refraction (AANR) could be achieved at the lowest band of twodimensional (2D) square-lattice PCs [6,7]. Zhang showed that non-near-field images can be achieved in the second band of the PC slab consisting of a triangular lattice of coated cylinder in air [8]. Qiu reported the negative refraction and light focusing at telecommunication wavelengths by a 2D PC slab fabricated by chemically assisted ion beam etching in the InP-based contrast system [9]. In our previous work, the imaging properties of PC slabs consisting of perfect metallic rods immersed in a dielectric were systemically studied [10,11]. The metal is assumed to be perfectly conducting, so the electric field within the metal and tangential to the metal surface is equal to zero, which is suitable to the microwaves and even the lower frequencies. Owing to the strong dispersion and absorption of the metal to the incident EM waves with high frequencies (such as the near-infrared light and visible
n
Corresponding author. Tel./fax: þ 86 10 68932205. E-mail address: [email protected] (S. Feng).
0038-1098/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2012.03.008
light), the metal cannot be treated as perfect conducting. In this paper, the Drude dispersion model is adopted to describe the dielectric properties of the metal, and it can reproduce the experimental spectrum well for the near-infrared light. We will show that the near-infrared imaging and negative refraction can be obtained through a triangular-lattice PC slab consisting of metallic rods immersed in a dielectric background. The nearinfrared imaging at near-infrared wavelength around 1550 nm is achieved by the adjustment of the size of the PC’s lattice constant and the refractive index of the background dielectric. It is also shown that the absorption of the metal rods to the incident light has a trivial influence to debase the quality of the image spots.
2. Numerical simulation and discussion The structure studied in this paper is based on the 2D triangular-lattice PC consisting of square metallic rods immersed in a dielectric, the side length of the square metal is 0.3a, where a is the lattice constant of the PC (as shown in Fig. 1(a)). Fig. 1(b) shows the first Brillouin zone of the triangular lattice. pffiffiffi ! ! ! ! The elementary vectors are a 1 ¼ a e x , a 2 ¼ ða=2Þ e x þ ð 3a=2Þ ! ! e y , and the elementary reciprocal lattice vectors are b 1 ¼ pffiffiffi ! ! pffiffiffi ! ! ð2p=aÞ e x ð2p= 3aÞ e y , b 2 ¼ ð4p= 3aÞ e y . There are three high symmetric points: the ! point (0,0), the K point ðð4p=3aÞ,0Þ, and pffiffiffi the M point (ðp=a, p= 3aÞ). In Drude model, the metal’s dielectric constant is depicted as eðoÞ ¼ e0 ð1ðo2p =ðoðo þ igÞÞÞÞ, where op is the plasma frequency
930
S. Feng et al. / Solid State Communications 152 (2012) 929–932
Fig. 1. (Color online) Schematic geometry of the triangular-lattice PC consisting of square metallic rods (a), and the first Brillouin zone of the 2D triangular lattice (b).
and g is the damping coefficient. The metal studied in paper is Ag, the corresponding values are op ¼ 1:37 1016 Hz, g ¼ 2:73 1013 Hz, which can reproduce the experimental spectrum well at the nearinfrared range. For the 2D PC, the electromagnetic waves can be separated into two independent polarized modes, TM-polarized mode and TE-polarized mode. In this paper, we only consider the TM polarization modes, where the electric field Ez is kept parallel to the extension axis of the metallic rods (assumed to be the z axis). In all of the simulations throughout this paper, we employ the finite-difference time-domain (FDTD) method with periodic boundary conditions to calculate the band structures and EFS contours in the reciprocal space, while Berenger’s perfectly matched layer absorbing boundary condition is employed to calculating the wave propagating in the finite real space [21,22]. In all the simulations, the size of the lattice constant is divided into 30 lattice grids, and good accuracy is ensured. At first, we set the length of the PC’s lattice constant is 430 nm, and the refractive index of the background dielectric is 3.45 (the permittivity of silicon at the near-infrared wavelength). Ignoring the absorption of the metal to the EM waves, the calculated band structure for the several lowest photonic bands along the highsymmetric lines in the first Brillouin zone is shown in Fig. 2(a). It can be seen that the metallic rods lift the first TM-polarized band and cause a stop band at the lowest frequencies. The cutoff frequency of this metallic PC is 91.6 THz (1 THz ¼1012 Hz). For such a perfect PC, the lowest photonic band extending from 91.6 THz to 144.9 THz, and the second band spans from 144.9 THz to 232.9 THz. Fig. 2(b) shows the several calculated equal-frequency surface (EFS) contours in the second band. It can be seen that some EFS contours such as 1.8, 1.935, and 2.0 (in unit of 1014 Hz) are very
Fig. 2. The TM-polarized photonic band structure (a) and the EFS contour in the second band (b) of the metallic PC (in a unit of 100 THz). The PC is composed of a triangular lattice of square metallic rods immersed in the silicon background, whose refractive index is set to be 3.45. The lattice constant of the PC is a¼ 430 nm, and the side length of the square metal is 0.3a.
close to a perfect circle, indicating that the PC can be regarded as an effective homogeneous medium at these frequencies. These frequencies increase inwards, so the group velocities are opposite to the phase velocities, meaning that the transmitting features of the EM waves in the PC are negative refraction behavior. The alteration of the metal’s shape from a circle to a square does not change the EFS’s shape. the Now we visualize the propagation of EM waves through this kind of metallic PC slab by employing the FDTD method with a boundary treatment of perfectly matched layers, and the actual absorption of the metal to the incident EM wave is also considered. The rectangular metallic PC slabs are 11, 21 and 41 layers thick, whose surface normal is along the !M direction. The whole PC slab is embedded in the air environment. The distances from the four slab edges to the center of the adjacent metallic rods are 0.3a. Thus the exact thicknesses of the PC slabs are 9.3a, 17.9a, 35.3a in Fig. 3(a), (b), and (c), respectively. A point source of continuous wave resonating at 193.5 THz (the corresponding wavelength in vacuum is 1550 nm) is placed at the left side of the PC slab, the calculated Ez field patterns are shown in Fig. 3(a), (b), and (c), respectively. It can be
S. Feng et al. / Solid State Communications 152 (2012) 929–932
931
1.0 0.9 Relative Light Intensity
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -4
-3
-2
-1 0 1 Position (µm)
2
3
4
-4
-3
-2
-1 0 1 Position (µm)
2
3
4
1.0 0.9 Relative Light Intensity
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
Fig. 4. (Color online) Lateral light intensity distributions centered at the image spot formed by the light propagation of a point source across the 21-layer (a) and 41-layer (b) 2D PC slabs. The solid line corresponding to the metallic PC with no absorption, while the blue dotted line corresponding to the metal with absorption. The full width at half maximum of the image spot are all about 0.5l.
Fig. 3. Simulated Ez field distributions of a point source and its image across a metallic PC slab. The rectangular PC slabs are 11 (a), 21 (b), and 41 (c) layers thick. The lattice constant of the PC is a¼ 430 nm, and the refractive index of the background material is 3.45. The slab’s surface normal is along the !M direction. The distances from the left and right slab edges to the center of the adjacent rods are all 0.3a. Dark and bright regions correspond to negative and positive Ez, respectively.
seen from Fig. 3 that the EM waves emitted from the point sources transmit through the PC slabs and good-quality image spots are formed in the right side of the PC slabs. The distances
from the point source to the center of the image spots are 18.8a, 36.0a, 70.8a in Fig. 3(a), (b), and (c), respectively. Because the shape of the EFS of the frequency 193.5 THz is a perfect circle, the rectangular PC slab can be looked as an isotropic material for this frequency. The light beams, which are radiated from a point source placed on the left side of the slab, transmit the PC slab and the refractive light beams inside the PC slab propagate on the same side of the surface normal owing to negative refraction. We know that the effective refractive index of the PC is determined by the EFS of the PC and the EFS of the source in the background. If the EFS of the PC is circular and inward at some fixed frequency, at the same time the radius of that EFS is equal to the radius of the EFS of the source in the background. Thus, the effective refractive index of the PC is 1. Light focusing is observed inside the PC above slabs, and the distance from the light sources to the image spots is twice the thickness of the PC slab. The rectangular PC slab can be looked as an isotropic material with an effective refractive index of 1. The size of the image spots in the lateral direction is about 1.8a (where a¼430 nm), corresponding to 0.5l (l ¼1550 nm). This value is much smaller than the traditional image spot restricted by the diffraction limit inherent in conventional lens.
932
S. Feng et al. / Solid State Communications 152 (2012) 929–932
square metallic rods immersed in a dielectric background. The light imaging and negative refraction at near-infrared wavelength around 1550 nm can be obtained by adjusting the lattice constant of PC and the permittivity of the background material together. Drude dispersion model is adopted to describe the metal’s permittivity, and it is found that the absorption of the metal rod is trivial to debase the image spot’s intensity, especially the image’s quality. This offers a powerful way to engineer the negative refraction and focusing properties of PC slab lens at the telecommunication wavelengths.
900
Lattice constance (nm)
800 700 600 500 400 300
Acknowledgments
200 1
2
3 4 5 6 7 Refractive Index of the background
8
9
Fig. 5. The size of the PC’s lattice constant varies with the refractive index of the background material, where an effective refractive index of 1 occurs for nearinfrared wavelength around 1550 nm. The side length of the metal is 0.3a.
Fig. 4 shows the lateral light intensity distributions centered at the image spot formed by the light propagation of a point source across the 21-layer and 41-layer PC slabs. The solid red lines corresponding to the metallic PC with no absorption of the metal, while the dotted blue line taking into account of the metal’s absorption to the incident light wave. It is found that the relative intensity of the image spot reduces to be 0.89 through the 21-layer PC slab, while the value reduces to be about 0.8 through the 41-layer PC slab owing to the metal’s absorption. The influence of the metal’s absorption is severe to debase the quality of the image spot. And the full-widths at half-maximum of the image spots are all about 0.5l, so high-quality images are obtained. According to Pendry’s theory [6], the single-beam behavior is only possible under the condition that o r0:5 2pc=as (where as is the surface-parallel period) in order to avoid diffraction. So the lattice constant of the metallic PC must less than 775 nm to ensure the good-quality imaging at the near-infrared wavelength 1550 nm. Based on the systematical simulation, keeping the side length of the square metal equal to 0.3a, the advisable values for the PC’s lattice constant and the background’s refractive index are shown in Fig. 5 to ensuring the high-quality imaging and negative refraction with an effective refractive index of 1 at the nearinfrared wavelength around 1550 nm.
The authors gratefully acknowledge the National Natural Science Foundation of China (Grant nos. 10904176 and 11004169), the Research Foundation of the State Ethnic Affairs Commission of People’s Republic of China through Grant no. 10ZY05, the NMOE Project of China through Grant no. 2010110009, the ‘‘985 Project’’ and ‘‘211 Project’’ of the Ministry of Education of China.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
3. Conclusion In summary, we have studied the light imaging and negative refraction through the 2D triangular-lattice PC slabs consisting of
[20] [21] [22]
J.B. Pendry, Phys. Rev. Lett. 85 (2000) 3966. R.A. Shelby, D.R. Smith, S. Schultz, Science 292 (2001) 77–79. A.A. Houck, J.B. Brock, I.L. Chuang, Phys. Rev. Lett. 90 (2003) 137401. D.R. Smith, D. Schurig, Phys. Rev. Lett. 90 (2003) 077405. J. Pacheco, T.M. Grzegorczyk, B.I. Wu, Y. Zhang, J.A. Kong, Phys. Rev. Lett. 89 (2002) 257401. C.Y. Luo, S.G. Johnson, J.D. Joannopoulos, J.B. Pendry, Phys. Rev. B 65 (2002) 201104. (R). C.Y. Luo, S.G. Johnson, J.D. Joannopoulos, J.B. Pendry, Phys. Rev. B 68 (2003) 045115. X.D. Zhang, Phys. Rev. B 70 (2004) 195110. A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thyle´n, A. Talneau, S. Anand, Phys. Rev. Lett. 93 (2004) 073902. S. Feng, Z.Y. Li, Z.F. Feng, B.Y. Cheng, D.Z. Zhang, Appl. Phys. Lett. 88 (2006) 031104. S. Feng, Y.Q. Wang, Z.Y. Li, B.Y. Cheng, D.Z. Zhang, Chin. Phys. Lett. 24 (2007) 229–232. S. Feng, Y.Q. Wang, Z.Y. Li, B.Y. Cheng, D.Z. Zhang, Chin. Phys. 16 (2007) 1689–1693. Z.Y. Li, L.L. Lin, Phys. Rev. B 68 (2003) 245110. X.D. Zhang, L.M. Li, Appl. Phys. Lett. 86 (2005) 121103. C.Y. Qiu, X.D. Zhang, Z.Y. Liu, Phys. Rev. B 71 (2005) 054302. L. Gan, Y.Z. Liu, J.Y. Li, Z.B. Zhang, D.Z. Zhang, Z.Y. Li, Opt. Express 17 (2009) 9962–9970. J. Tian, M. Yan, M. Qiu, C.G. Ribbing, Y.Z. Liu, D.Z. Zhang, Z.Y. Li, Appl. Phys. Lett. 93 (2008) 191114. G.Y. Dong, J. Zhou, X.L. Yang, L.Z. Cai, Opt. Express 19 (2011) 12119–12124. G.P. Swift, A.J. Gallant, N. Kaliteevskaya, M.A. Kaliteevski, S. Brand, D. Dai, A.J. Baragwanath, I. Iorsh, R.A. Abram, J.M. Chamberlain, Opt. Lett. 36 (2011) 1641–1643. S.L. He, Z.C. Ruan, L. Chen, J.Q. Shen, Phys. Rev. B 70 (2004) 115113. K.S. Yee, IEEE Trans. Antennas Propag. 14 (1966) 302–307. J.P. Berenger, J. Comput. Phys. 127 (1996) 363–379.