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Imaging properties of triangular lattice photonic crystal at the lowest band
Physics Letters A 348 (2006) 405–409 www.elsevier.com/locate/pla
Imaging properties of triangular lattice photonic crystal at the lowest band Kun Ren ∗ , Shuai Feng, Zhi-Fang Feng, Yan Sheng, Zhi-Yuan Li ∗ , Bing-Ying Cheng, Dao-Zhong Zhang Optical Physics Laboratory, Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100080, China Received 15 August 2005; accepted 22 August 2005 Available online 30 August 2005 Communicated by V.M. Agranovich
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. We examine different surface terminations of the PC slab for getting a high-quality image. In the lowest photonic band both the TM and TE polarized EM waves can undergo negative refraction and imaging effect even at frequencies above the light line. The dependence of the image on the object distance and sample thickness is also discussed. 2005 Elsevier B.V. All rights reserved. PACS: 41.20.Jb; 42.70.Qs; 78.20.Ci
1. Introduction Recently negative refraction has been attracting a great deal of attentions. Negative refraction studies originate from left-handed metamaterials [1–4]. Now it has been shown that photonic crystals (PCs) [5–16] can exhibit negative refraction properties under certain conditions. * Corresponding authors.
E-mail addresses: [email protected] (K. Ren), [email protected] (Z.-Y. Li). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.08.053
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 [5–12], 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 equifrequency surface (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
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K. Ren et al. / Physics Letters A 348 (2006) 405–409
negatively [13–17]. 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 [12,16]. Very recently, Feng et al. [18] have demonstrated both theoretically and experimentally that negative refraction can also appear in some photonic quasicrystals (PQCs). Luo et al. [13] have shown that all angle negative refraction (AANR) could be achieved in the lowest band of two-dimensional (2D) square lattice PCs. A natural question is raised concerning whether or not negative refraction can also occur in other 2D lattices, for instance, in the triangular lattice at the lowest photonic band. In this Letter, we try to answer this question. Both the transverse magnetic (TM) polarization and transverse electric (TE) polarization modes are investigated. We have considered a series of system parameters, including different surface terminations and thicknesses of the PC slab, and the object distances. We find that negative refraction of EM waves can occur in the triangular lattice at the lowest band. In this frequency window we have S · k > 0, where S represents the Poynting vector.
2. Dispersion characteristics analysis We consider PC structures made from a triangular lattice of air holes embedded in a dielectric background of permittivity ε = 12.56. The air hole radius is r = 0.15a, 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. Planewave expansion method (PWEM) is used to obtain the band diagram and EFS, in which Bloch waves are expanded by approximately 1000 plane waves. The frequency is normalized as ωa/2πc (c is the light speed in vacuum). Fig. 1(a) and (b) display the band structure and EFS at several relevant frequencies for the TM-polarized Bloch modes within the first Brillouin zone (BZ). The results for the TE polarized modes are shown in Fig. 2(a) and (b), respectively.
Fig. 1. (a) TM polarized band structure of the 2D triangular lattice PC with air hole radius r = 0.15a and background dielectric constant ε = 12.56. The light line shift to K is shown in dashed line. (b) Several EFS contours of the first band at the TM mode in the first Brillouin zone.
In Fig. 1(b) and Fig. 2(b), the EFS contours of some frequencies such as 0.05, 0.1, 0.15 (in the unit of normalized frequency 2πc/a) are close to a perfect circle, indicating that the crystal behaves like an effective homogeneous medium at these long wavelengths. But the 0.175 contour is significantly distorted from a circle. And besides, the EFS contours of the two polarization modes within the first band are convex at certain frequency windows, and this is similar to the square lattice. We follow the analysis of Ref. [13] to study the negative refraction behavior. To realize negative refraction for superlensing, the required conditions are that only one single mode of Bloch waves can be excited and the EFS contour of this mode is convex everywhere. To determine what Bloch modes can be excited, the conservation of frequency and surface-
K. Ren et al. / Physics Letters A 348 (2006) 405–409
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Fig. 3. The geometry of a point source against a triangular-lattice photonic crystal slab. δ determines the surface termination of the PC slab.
Fig. 2. (a) TE polarized band structure of the 2D triangular lattice PC with air hole radius r = 0.15a and background dielectric constant ε = 12.56. The light line shift to K is shown in dashed line. (b) Several equal frequency contours of TE mode of the first band drawn in the first Brillouin zone.
parallel component of the wave vector are followed. Under these conditions, a light beam incident on the surface will couple to a single Bloch mode whose propagation direction is on the negative side of the surface normal and thus negative refraction occurs at that particular frequency.
3. 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 Γ K direction. We employ the finite difference time domain (FDTD) method with the use of perfectly-matched layer (PML) boundary condition.
The operating frequency is selected as ω = 0.179 × (2πc/a), which is above the light line of the air background. Recently a few works [18–21] have shown that with an appropriate termination of the PC slab surface, the incident wave can efficiently excite the surface waves and enable the PC slab to improve the quality of the image. In order to obtain a high quality image, we examine different surface terminations. The schematic diagram of the PC slab is displayed in Fig. 3. The distance between the left boundary of the slab and the center of the first column of holes and the distance between the right boundary and the center of the final column of holes are both δ. We first consider a point source placed a away from the left surface of a nine-layer thick PC slab. Fig. 4 shows the field distribution across the PC slab for TM and TE polarizations. Clearly high-quality images are formed against the PC slab for both polarization modes. A careful look at the data of the light intensity across the image, we find that the transverse size, as measured by full width at half maximum, is about 0.58λ for the TM mode and 0.5λ for the TE mode. It can also be found that the TM mode exhibits a better imaging performance than the TE mode. The radiation field is more dispersed within the PC slab in the TE polarization. Note that here negative refraction and imaging are realized in the first photonic band that consists of forward propagating waves (S · k > 0), but not backward propagating waves as in left-handed material. We have also considered different object distances. We place the source at a distance 0.5a away
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K. Ren et al. / Physics Letters A 348 (2006) 405–409
Fig. 4. Simulated field distribution of a point source ω = 0.179 × (2π c/a) located at a away from the 9-layer air-hole slab with surface termination denoted by (a) δ = 0.6a for Ez at the TM mode and (b) δ = a for Hz at the TE mode.
from the left surface of the 9-layer air-hole slab and a focusing spot is also observed on the right side of the slab with appropriate termination. In order to see the dependence of the focusing on slab thickness, we perform the numerical simulation under various thickness circumstances. Fig. 5 shows the imaging behavior of a PC slab that consists of 15layer air holes. The image spots are both located in the near-field region of the opposite side of the slab for the two polarized modes. The image distance has little dependence on the slab thickness. The imaging behavior clearly do not obey the well-known wave beam refraction law as expected, instead, it is dominantly governed by the self-collimation effect [14] that
Fig. 5. Simulated field distribution of a point source ω = 0.179 × (2π c/a) located at a away from the PC slab consisting of 15-layer air holes. The surface termination of the slab is represented by (a) δ = 0.5a for Ez at the TM mode and (b) δ = 0.1a for Hz at the TE mode.
is found for the square-lattice photonic crystal in the lowest photonic band. We also check the lateral intensity profiles at image plane and find that the transverse size of the image spot is about 0.34λ for the TM mode and 0.51λ for the TE mode. We can obtain an optimum-performance image by tuning parameters such as thickness of slab, source distance and surface termination. In contrast to the conventional lens which are made of positive refraction index materials, a lens based on negative refraction can be used as superlens to achieve superresolution, which overcomes the diffraction limit of the conventional far-field imaging systems.
K. Ren et al. / Physics Letters A 348 (2006) 405–409
4. Conclusion In conclusion, we have presented imaging property analysis of 2D triangular photonic crystal with air holes in a dielectric background. Self-collimation effect is found in the lowest photonic band at frequencies that are located even above the light line. A superlens has been designed based on this effect. An image spot can be formed against the PC slab without employing a negative index or a back wave effect for both the TE and TM polarization modes. The advantages of the lowest valence band are the single-beam propagation and high transmission efficiency, and these should be valuable for realistic negative-refraction superlens design.
Acknowledgements This work was supported by the National Key Basic Research Special Foundation of China at No. 2004CB719804 and No. 2001CB610402, and the National Natural Science Foundation of China at No. 10404036. The supports from the Supercomputing Center, CNIC, CAS are acknowledged.
References [1] R.A. Shelby, D.R. Smith, S. Schultz, Science 292 (2001) 779.
409
[2] D.R. Smith, W.J. Padilla, D.C. Nemat-Nasser, S. Schultz, Phys. Rev. Lett. 84 (2000) 4184. [3] C.G. Parazzoli, R.B. Greegor, K. Li, B.E.C. Koltenbah, M. Tanielian, Phys. Rev. Lett. 90 (2003) 107401. [4] P. Markos, C.M. Soukoulis, Phys. Rev. B 65 (2001) 033401. [5] H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, Phys. Rev. B 58 (1998) R10096. [6] M. Notomi, Phys. Rev. B 62 (2000) 10696. [7] B. Gralak, S. Enoch, G. Tayeb, J. Opt. Soc. Am. A 17 (2000) 1012. [8] C. Luo, M. Ibanescu, S.G. Johnson, J.D. Joannopoulos, Science 299 (2003) 368. [9] E.J. Read, M. Soljacic, J.D. Joannopoulos, Phys. Rev. Lett. 91 (2003) 133901. [10] S. Foteinopoulou, E.N. Economou, C.M. Soukoulis, Phys. Rev. Lett. 90 (2003) 107402. [11] X. Zhang, Phys. Rev. B 70 (2004) 195110. [12] P.V. Parimi, W.T. Lu, P. Vodo, J. Sokoloff, J.S. Derov, S. Sridhar, Phys. Rev. Lett. 92 (2004) 127401. [13] C. Luo, S.G. Johnson, J.D. Joannopoulos, J.B. Pendry, Phys. Rev. B 65 (2002) 201104(R). [14] Z.Y. Li, L.L. Lin, Phys. Rev. B 68 (2003) 245110. [15] X. Zhang, Phys. Rev. B 70 (2004) 205102. [16] E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, C.M. Soukoulis, Nature (London) 423 (2003) 604. [17] S. Feng, Z.Y. Li, Z.F. Feng, B.Y. Cheng, D.Z. Zhang, Phys. Rev. B 72 (2005) 075101. [18] Z.F. Feng, X.D. Zhang, Y.Q. Wang, Z.Y. Li, B.Y. Cheng, D.Z. Zhang, Phys. Rev. Lett. 94 (2005) 247402. [19] S. Xiao, M. Qiu, Z. Ruan, S. He, Appl. Phys. Lett. 85 (2004) 4269. [20] R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, C.M. Soukoulis, Phys. Rev. B 71 (2005) 085106. [21] X. Zhang, Phys. Rev. B 71 (2005) 165116.
Imaging properties of triangular lattice photonic crystal at the lowest band Kun Ren ∗ , Shuai Feng, Zhi-Fang Feng, Yan Sheng, Zhi-Yuan Li ∗ , Bing-Ying Cheng, Dao-Zhong Zhang Optical Physics Laboratory, Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100080, China Received 15 August 2005; accepted 22 August 2005 Available online 30 August 2005 Communicated by V.M. Agranovich
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. We examine different surface terminations of the PC slab for getting a high-quality image. In the lowest photonic band both the TM and TE polarized EM waves can undergo negative refraction and imaging effect even at frequencies above the light line. The dependence of the image on the object distance and sample thickness is also discussed. 2005 Elsevier B.V. All rights reserved. PACS: 41.20.Jb; 42.70.Qs; 78.20.Ci
1. Introduction Recently negative refraction has been attracting a great deal of attentions. Negative refraction studies originate from left-handed metamaterials [1–4]. Now it has been shown that photonic crystals (PCs) [5–16] can exhibit negative refraction properties under certain conditions. * Corresponding authors.
E-mail addresses: [email protected] (K. Ren), [email protected] (Z.-Y. Li). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.08.053
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 [5–12], 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 equifrequency surface (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
406
K. Ren et al. / Physics Letters A 348 (2006) 405–409
negatively [13–17]. 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 [12,16]. Very recently, Feng et al. [18] have demonstrated both theoretically and experimentally that negative refraction can also appear in some photonic quasicrystals (PQCs). Luo et al. [13] have shown that all angle negative refraction (AANR) could be achieved in the lowest band of two-dimensional (2D) square lattice PCs. A natural question is raised concerning whether or not negative refraction can also occur in other 2D lattices, for instance, in the triangular lattice at the lowest photonic band. In this Letter, we try to answer this question. Both the transverse magnetic (TM) polarization and transverse electric (TE) polarization modes are investigated. We have considered a series of system parameters, including different surface terminations and thicknesses of the PC slab, and the object distances. We find that negative refraction of EM waves can occur in the triangular lattice at the lowest band. In this frequency window we have S · k > 0, where S represents the Poynting vector.
2. Dispersion characteristics analysis We consider PC structures made from a triangular lattice of air holes embedded in a dielectric background of permittivity ε = 12.56. The air hole radius is r = 0.15a, 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. Planewave expansion method (PWEM) is used to obtain the band diagram and EFS, in which Bloch waves are expanded by approximately 1000 plane waves. The frequency is normalized as ωa/2πc (c is the light speed in vacuum). Fig. 1(a) and (b) display the band structure and EFS at several relevant frequencies for the TM-polarized Bloch modes within the first Brillouin zone (BZ). The results for the TE polarized modes are shown in Fig. 2(a) and (b), respectively.
Fig. 1. (a) TM polarized band structure of the 2D triangular lattice PC with air hole radius r = 0.15a and background dielectric constant ε = 12.56. The light line shift to K is shown in dashed line. (b) Several EFS contours of the first band at the TM mode in the first Brillouin zone.
In Fig. 1(b) and Fig. 2(b), the EFS contours of some frequencies such as 0.05, 0.1, 0.15 (in the unit of normalized frequency 2πc/a) are close to a perfect circle, indicating that the crystal behaves like an effective homogeneous medium at these long wavelengths. But the 0.175 contour is significantly distorted from a circle. And besides, the EFS contours of the two polarization modes within the first band are convex at certain frequency windows, and this is similar to the square lattice. We follow the analysis of Ref. [13] to study the negative refraction behavior. To realize negative refraction for superlensing, the required conditions are that only one single mode of Bloch waves can be excited and the EFS contour of this mode is convex everywhere. To determine what Bloch modes can be excited, the conservation of frequency and surface-
K. Ren et al. / Physics Letters A 348 (2006) 405–409
407
Fig. 3. The geometry of a point source against a triangular-lattice photonic crystal slab. δ determines the surface termination of the PC slab.
Fig. 2. (a) TE polarized band structure of the 2D triangular lattice PC with air hole radius r = 0.15a and background dielectric constant ε = 12.56. The light line shift to K is shown in dashed line. (b) Several equal frequency contours of TE mode of the first band drawn in the first Brillouin zone.
parallel component of the wave vector are followed. Under these conditions, a light beam incident on the surface will couple to a single Bloch mode whose propagation direction is on the negative side of the surface normal and thus negative refraction occurs at that particular frequency.
3. 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 Γ K direction. We employ the finite difference time domain (FDTD) method with the use of perfectly-matched layer (PML) boundary condition.
The operating frequency is selected as ω = 0.179 × (2πc/a), which is above the light line of the air background. Recently a few works [18–21] have shown that with an appropriate termination of the PC slab surface, the incident wave can efficiently excite the surface waves and enable the PC slab to improve the quality of the image. In order to obtain a high quality image, we examine different surface terminations. The schematic diagram of the PC slab is displayed in Fig. 3. The distance between the left boundary of the slab and the center of the first column of holes and the distance between the right boundary and the center of the final column of holes are both δ. We first consider a point source placed a away from the left surface of a nine-layer thick PC slab. Fig. 4 shows the field distribution across the PC slab for TM and TE polarizations. Clearly high-quality images are formed against the PC slab for both polarization modes. A careful look at the data of the light intensity across the image, we find that the transverse size, as measured by full width at half maximum, is about 0.58λ for the TM mode and 0.5λ for the TE mode. It can also be found that the TM mode exhibits a better imaging performance than the TE mode. The radiation field is more dispersed within the PC slab in the TE polarization. Note that here negative refraction and imaging are realized in the first photonic band that consists of forward propagating waves (S · k > 0), but not backward propagating waves as in left-handed material. We have also considered different object distances. We place the source at a distance 0.5a away
408
K. Ren et al. / Physics Letters A 348 (2006) 405–409
Fig. 4. Simulated field distribution of a point source ω = 0.179 × (2π c/a) located at a away from the 9-layer air-hole slab with surface termination denoted by (a) δ = 0.6a for Ez at the TM mode and (b) δ = a for Hz at the TE mode.
from the left surface of the 9-layer air-hole slab and a focusing spot is also observed on the right side of the slab with appropriate termination. In order to see the dependence of the focusing on slab thickness, we perform the numerical simulation under various thickness circumstances. Fig. 5 shows the imaging behavior of a PC slab that consists of 15layer air holes. The image spots are both located in the near-field region of the opposite side of the slab for the two polarized modes. The image distance has little dependence on the slab thickness. The imaging behavior clearly do not obey the well-known wave beam refraction law as expected, instead, it is dominantly governed by the self-collimation effect [14] that
Fig. 5. Simulated field distribution of a point source ω = 0.179 × (2π c/a) located at a away from the PC slab consisting of 15-layer air holes. The surface termination of the slab is represented by (a) δ = 0.5a for Ez at the TM mode and (b) δ = 0.1a for Hz at the TE mode.
is found for the square-lattice photonic crystal in the lowest photonic band. We also check the lateral intensity profiles at image plane and find that the transverse size of the image spot is about 0.34λ for the TM mode and 0.51λ for the TE mode. We can obtain an optimum-performance image by tuning parameters such as thickness of slab, source distance and surface termination. In contrast to the conventional lens which are made of positive refraction index materials, a lens based on negative refraction can be used as superlens to achieve superresolution, which overcomes the diffraction limit of the conventional far-field imaging systems.
K. Ren et al. / Physics Letters A 348 (2006) 405–409
4. Conclusion In conclusion, we have presented imaging property analysis of 2D triangular photonic crystal with air holes in a dielectric background. Self-collimation effect is found in the lowest photonic band at frequencies that are located even above the light line. A superlens has been designed based on this effect. An image spot can be formed against the PC slab without employing a negative index or a back wave effect for both the TE and TM polarization modes. The advantages of the lowest valence band are the single-beam propagation and high transmission efficiency, and these should be valuable for realistic negative-refraction superlens design.
Acknowledgements This work was supported by the National Key Basic Research Special Foundation of China at No. 2004CB719804 and No. 2001CB610402, and the National Natural Science Foundation of China at No. 10404036. The supports from the Supercomputing Center, CNIC, CAS are acknowledged.
References [1] R.A. Shelby, D.R. Smith, S. Schultz, Science 292 (2001) 779.
409
[2] D.R. Smith, W.J. Padilla, D.C. Nemat-Nasser, S. Schultz, Phys. Rev. Lett. 84 (2000) 4184. [3] C.G. Parazzoli, R.B. Greegor, K. Li, B.E.C. Koltenbah, M. Tanielian, Phys. Rev. Lett. 90 (2003) 107401. [4] P. Markos, C.M. Soukoulis, Phys. Rev. B 65 (2001) 033401. [5] H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, Phys. Rev. B 58 (1998) R10096. [6] M. Notomi, Phys. Rev. B 62 (2000) 10696. [7] B. Gralak, S. Enoch, G. Tayeb, J. Opt. Soc. Am. A 17 (2000) 1012. [8] C. Luo, M. Ibanescu, S.G. Johnson, J.D. Joannopoulos, Science 299 (2003) 368. [9] E.J. Read, M. Soljacic, J.D. Joannopoulos, Phys. Rev. Lett. 91 (2003) 133901. [10] S. Foteinopoulou, E.N. Economou, C.M. Soukoulis, Phys. Rev. Lett. 90 (2003) 107402. [11] X. Zhang, Phys. Rev. B 70 (2004) 195110. [12] P.V. Parimi, W.T. Lu, P. Vodo, J. Sokoloff, J.S. Derov, S. Sridhar, Phys. Rev. Lett. 92 (2004) 127401. [13] C. Luo, S.G. Johnson, J.D. Joannopoulos, J.B. Pendry, Phys. Rev. B 65 (2002) 201104(R). [14] Z.Y. Li, L.L. Lin, Phys. Rev. B 68 (2003) 245110. [15] X. Zhang, Phys. Rev. B 70 (2004) 205102. [16] E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, C.M. Soukoulis, Nature (London) 423 (2003) 604. [17] S. Feng, Z.Y. Li, Z.F. Feng, B.Y. Cheng, D.Z. Zhang, Phys. Rev. B 72 (2005) 075101. [18] Z.F. Feng, X.D. Zhang, Y.Q. Wang, Z.Y. Li, B.Y. Cheng, D.Z. Zhang, Phys. Rev. Lett. 94 (2005) 247402. [19] S. Xiao, M. Qiu, Z. Ruan, S. He, Appl. Phys. Lett. 85 (2004) 4269. [20] R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, C.M. Soukoulis, Phys. Rev. B 71 (2005) 085106. [21] X. Zhang, Phys. Rev. B 71 (2005) 165116.