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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 23 — Nov. 13, 2006
  • pp: 11178–11183
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Layered superlensing in two-dimensional photonic crystals

Haifei Zhang, Linfang Shen, Lixin Ran, Yu Yuan, and Jin Au Kong  »View Author Affiliations


Optics Express, Vol. 14, Issue 23, pp. 11178-11183 (2006)
http://dx.doi.org/10.1364/OE.14.011178


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Abstract

We demonstrate layered superlensing in two-dimensional photonic crystals structured by both square and triangular lattices. In virtue of equifrequency contour analysis and FDTD calculation, both near field and far field imaging are displayed. Layered superlensing consisting of only triangular lattice photonic crystal is also studied and it exhibits more flexibility than the single layer counterpart. That is, the objective distance can be changed freely while keeping the image distance constant and vice versa. Hence, farther field imaging is achieved.

© 2006 Optical Society of America

1. Introduction

Materials with negative refractive index predicted by Veselago in 1968 [1

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

] attracted lots of attention in recent years. These materials, generally referred to as left-handed materials (LHM) by Veseloago, double negative materials by Ziolkowski [2

2. R. W. Ziolkowski, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001). [CrossRef]

], and backward-wave media by Lindell et al. [3

3. I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media- media with negative parameters, capable of supporting backward waves,” Microwave Opt. Technol. Lett. 31, 129–133 (2001). [CrossRef]

], was first realized in 2000 by Smith et al. [4

4. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [CrossRef] [PubMed]

] as metamaterials composed of wires [5

5. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996). [CrossRef] [PubMed]

] and split ring resonators [6

6. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Technol. 47, 2075–2084 (1999). [CrossRef]

]. Later, variations of such structure were designed and experiments were performed [7

7. L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Cheng, T. M. Grzegorczyk, and J. A. Kong, “Experimental study on several left-handed metamaterials,” PIER 51, 249–279 (2005). [CrossRef]

]. One of the most interesting properties of such LHM is the superlensing effect suggested by Pendry [8

8. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]

]. Meanwhile, an effective negative refractive index in the vicinity of photonic band gap in photonic crystals was shown theoretically by Notomi [9

9. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000). [CrossRef]

]. Later, negative refraction of electromagnetic waves in such photonic crystals was demonstrated numerically and experimentally by Soukoulis et al. [10–12

10. S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a Negative Refractive Index,” Phys. Rev. Lett. 90, 107402 (2003). [CrossRef] [PubMed]

].

To our knowledge, negative refraction in 2D PCs generally occur under two conditions. One is that, as Notomi suggested [9

9. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000). [CrossRef]

], the EFC (equifrequency contour) of the PC is rounded and its radius shrinks as the frequency approaches to the band gap, which is common for frequencies above the first band near the Brillouin-zone center (Γ), so the wave vector and the group velocity are antiparallel. The other condition lies in frequencies where the EFC is hyperbolic like and the normal components of wave vector and group velocity are parallel, which usually occurs near a Brillouin-zone corner farthest from the center (Γ) [13

13. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002). [CrossRef]

]. Yet, there is a striking difference between the two cases, i.e. the refractive index is negative for the first case and positive for the second case. It is anisotropy that restricted far field image formation in the second case [13

13. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002). [CrossRef]

] and imaging of the first case was first demonstrated by Wang et al. [14

14. X. Wang, Z. F. Ren, and K. Kempa, “Unrestricted superlensing in a triangular two dimensional photonic crystal,” Opt. Express 12, 2919–2924 (2004). [CrossRef] [PubMed]

]. Recently, image quality, transmission efficiency, surface termination and lattice orientation were well studied [15–20

15. Z.-Y. Li and L.-L. Lin, “Evaluation of lensing in photonic crystal slabs exhibiting negative refraction,” Phys. Rev. B 68, 245110 (2003). [CrossRef]

]. However, a clear and explicit distinction between the concepts of negative refraction, negative refractive index, and superlensing was not drawn until the work of Li et al. [21

21. C. Li, J. M. Holt, and A. L. Efros, “Far-field imaging by the Veselago lens made of a photonic crystal,” J. Opt. Soc. Am. B 23, 490–497 (2006). [CrossRef]

, 22

22. C. Y. Li, J. M. Holt, and A. L. Efros, “Imaging by the Veselago lens based upon a two-dimensional photonic crystal with a triangular lattice,” J. Opt. Soc. Am. B 23, 963–968 (2006). [CrossRef]

]. Superlensing in the sense of Pendry is due to interface modes which can be of at least two types: plasmons in metal layers, or defect interface layer in photonic crystals. Only when the surface is carefully terminated, exciting the surface modes and an improvement of the image is achieved and hence the superlensing effect, not owing to negative permittivity and negative permeability, or inward directed equifrequency contours.

However, these lenses were all composed of single PC slab and layered superlens haven’t yet been demonstrated. Lately, image transfer with very little deterioration of the resolution by a cascaded stack consisting of two or three two-dimensional PC slabs separated by air is demonstrated [23

23. C. Shen, K. Michielsen, and H. De Raedt, “Image transfer by cascaded stack of photonic crystal and air layers,” Opt. Express 14, 879–886 (2006). [CrossRef] [PubMed]

].

In this paper, we propose a layered superlens composed of both square lattice and triangular lattice PC as well as a layered superlens composed of air sandwiched by triangular lattice PC.

2. Numerical method

Plane wave expansion method [24

24. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

, 25

25. M. Plihal, A. Shambrook, and A. A. Maradudin, “Two-dimensional photonic band structures,” Optics Commun. 80, 3–4 (1991). [CrossRef]

] was employed to calculate photonic band diagrams and equifrequency contours, in which Bloch waves are expanded by 625 plane waves. We examine mainly TM modes (electric field lies perpendicular to the 2D plane). To solve the Maxwell equations, the finite-difference time-domain (FDTD) method [26

26. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966). [CrossRef]

, 27

27. A. Taflove and S. C. Hagness, Computational Electrodynamics—the Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2000).

] was used with uniaxial perfectly matched layer (UPML) boundary conditions [28

28. S. D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices,” IEEE Trans. Antennas Propagat. 44, 1630–1639 (1996). [CrossRef]

]. Distances were normalized by units of the lattice constant a. Time and frequency are then expressed in units of a/c and c/a, respectively, where c denotes the velocity of light in vacuum. f=ωa/2πc is the dimensionless frequency. Square meshes with mesh size δ=0.02a and time step Δt=0.95δ/√2c were employed. More than 10000 time steps were run to reach the steady state.

3. Layered superlens structured by triangular lattice PC

Consider photonic crystal slabs made of a triangular lattice of air holes with radius r=0.4a (a is the lattice constant) embedded in dielectric matrix with permittivity 12.96. Its photonic bands (TM modes) are illustrated in Fig. 1(a) showing negative refraction of the first case at frequency 0.306 (normalized by ωa/2πc) with effective refractive index n=-1 [9

9. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000). [CrossRef]

]. In addition, both slab terminations are identical and are chosen to ensure high transmission efficiency and interface modes which are critical to the superlensing effect. To this end, the distance between the centers of the outmost holes and the surface is (√3a/4-0.2)a.

Fig. 1. Calculated photonic band structures for (a) triangular 2D PC with air holes of radius r=0.4atri in dielectric matrix with ε=12.96 and (b) square 2D PC with dielectric rods in air, radius r=0.3asqua, ε=14. Frequencies for negative refraction are marked for (a) ω1=0.306 and (b) ω2=0.192.

As we know, a single triangular PC slab could act as a superlens and the relation Do+Di=T holds [14

14. X. Wang, Z. F. Ren, and K. Kempa, “Unrestricted superlensing in a triangular two dimensional photonic crystal,” Opt. Express 12, 2919–2924 (2004). [CrossRef] [PubMed]

], where Do is the distance from the object to one side of the PC slab (we called it objective distance hereafter), Di is the distance from the other side of the PC slab to the image (we called it image distance hereafter), and T is the thickness of the slab. If the objective distance Do is greater than the thickness of the PC slab T, the image won’t appear.

Fig. 2. Electric field distribution with frequency 0.306 *ωa/2πc for layered superlens structured by triangular lattice PC, varying the position of the two slabs. Geometric optics analysis are presented in the right column, wherein the PC is considered as an effective medium with n=-1.

Yet, a closer inspection shows that a virtual image is formed and we can use a second PC slab to reconverge it to form a real image as pictured in Fig. 2(d). Actually the function of the objective lens (with thickness To) is to draw the object nearer by a distance of 2*To and a virtual image is formed, hence we call the first lens objective lens. The image lens sees the virtual image and a real one is formed, hence we call the second lens image lens. So, even the objective distance is larger than the thickness of objective lens a real image is still formed which farther free the object from near field imaging. Likewise, even the image distance is greater than the thickness of the image lens a image is still formed as pictured in Fig. 2(a).

It is true that a thicker slab could form a image far from the slab, but with the increase of the slab thickness, resolution dropped. This is also the reason why image transfer rather than a thicker slab was used to increase the distance between the image and the object as addressed in Ref. [23

23. C. Shen, K. Michielsen, and H. De Raedt, “Image transfer by cascaded stack of photonic crystal and air layers,” Opt. Express 14, 879–886 (2006). [CrossRef] [PubMed]

]. Moreover, layered superlens provides more flexibility than the single layer counterpart since we could easily change, holding the distance from the object to the image constant though, the objective distance or image distance freely while keeping the other constant, which is rather useful in optical imaging system and particularly optical lithography. As illustrated in Fig. 2, since the distance between the object and the image are two times the total thickness of objective lens and image lens, it is free to change the objective distance and image distance respectively without change in image position and loss in resolution which is always about 0.44λ (full width at half maximum) in the transversal direction.

4. Layered superlens structured by both square and triangular lattice PC

The triangular lattice photonic crystal consist of air holes with radius 0.4 r=atri (atri is the lattice constant) embedded in dielectric matrix with permittivity 12.96 while the square lattice photonic crystal consists of a periodic array of infinitely long, cylindrical dielectric rods with radius r=0.3asqua (asqua is the lattice constant) and permittivity 14 embedded in air. Their photonic bands (TM modes) are shown in Fig. 1, respectively. the triangular lattice PC show negative refractive of the first case at frequency 0.306 (normalized by ωatri/2πc) with effective refraction index n=-1 [9

9. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000). [CrossRef]

], while the square lattice PC exhibit a narrow frequency range around 0.192 (normalized by ωasqua/2πc) of negative refraction of the second case discussed above [13

13. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002). [CrossRef]

]. What if we bring them together by scaling atri, pegging 0.306*ωatri/2πc to 0.192*ωasqua/2πc. Specifically, the triangular lattice constant atri is 0.306/0.192 that of the square lattice constant asqua.

Fig. 3. Left: Equifrequency contour analysis of the refraction at the interface between the square lattice PC and triangular lattice PC whose normal are along the ΓΜ and ΓΚ directions. The incident (ki) and refracted (kf) wave vectors as well as the group velocity of incident wave (vgi) and the refracted wave (vgf) are shown. The dashed line represents the conservation of parallel component of wave vector. Right: Schematics representation of the refraction at the square-triangular lattice PC interface.
Fig. 4. Electric field distribution of (a) square-triangular lattice layered PC slab with near field source, (b) square-triangular lattice layered PC slab with far field source, (c) triangular-square lattice layered PC slab with near field source, and (d) triangular-square lattice layered PC slab with far field source.

5. Discussions and conclusions

When the electromagnetic wave transports through the layered samples, the scattering loss by the interface is inevitable. Although the transmission efficiency is rather low, there is no pronounced decrease in the resolution for such two layered superlenses since the surface termination and lattice orientation are carefully chosen to ensure high transmission efficiency and interface modes. As for superlenses consisting of more layers, the image quality requires the solution of an optimization problem involving the frequency of the light, the distance between the slabs, the surface termination and lattice orientation, and the thickness of the photonic crystal slabs. In conclusion, we present the study of layered superlening effect in two dimensional photonic crystals including square-triangular layered, triangular-square layered and triangular-air-triangular sandwiched superlenses. Both the square-triangular layered and triangular-square layered superlenses are able to form either near field or far field image. And the triangular sandwiched superlenses provide more flexibility than the single layer counterpart in changing the objective and image distances which is useful in the potential subwavelength optical lithography.

Acknowledgments

This work is supported by the Key Project of National Science Foundation of China (NSFC) under Contract No. 60531020, and in part by NSFC60671003 and ZJNSF R105253.

References and links

1.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

2.

R. W. Ziolkowski, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001). [CrossRef]

3.

I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media- media with negative parameters, capable of supporting backward waves,” Microwave Opt. Technol. Lett. 31, 129–133 (2001). [CrossRef]

4.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [CrossRef] [PubMed]

5.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996). [CrossRef] [PubMed]

6.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Technol. 47, 2075–2084 (1999). [CrossRef]

7.

L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Cheng, T. M. Grzegorczyk, and J. A. Kong, “Experimental study on several left-handed metamaterials,” PIER 51, 249–279 (2005). [CrossRef]

8.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]

9.

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000). [CrossRef]

10.

S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a Negative Refractive Index,” Phys. Rev. Lett. 90, 107402 (2003). [CrossRef] [PubMed]

11.

S. Foteinopoulou and C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2003). [CrossRef]

12.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Negative refraction by photonic crystals,” Nature (London) 423, 604–605 (2003). [CrossRef] [PubMed]

13.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002). [CrossRef]

14.

X. Wang, Z. F. Ren, and K. Kempa, “Unrestricted superlensing in a triangular two dimensional photonic crystal,” Opt. Express 12, 2919–2924 (2004). [CrossRef] [PubMed]

15.

Z.-Y. Li and L.-L. Lin, “Evaluation of lensing in photonic crystal slabs exhibiting negative refraction,” Phys. Rev. B 68, 245110 (2003). [CrossRef]

16.

S. Xiao, M. Qiu, Z. Ruan, and S. He, “Influence of the surface termination to the point imaging by a photonic crystal slab with negative refraction,” Appl. Phys. Lett. 85, 4269–4271 (2004). [CrossRef]

17.

X. Zhang, “Image resolution depending on slab thickness and object distance in a two-dimensional photoniccrystal-based superlens,” Phys. Rev. B 70, 195110 (2004). [CrossRef]

18.

X. Wang and K. Kempa, “Effects of disorder on subwavelength lensing in two-dimensional photonic crystal slabs,” Phys. Rev. B 71, 085101 (2005). [CrossRef]

19.

A. Martinez and J. Marti, “Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination,” Phys. Rev. B 71, 235115 (2005). [CrossRef]

20.

A. Martinez and J. Marti, “Analysis of wave focusing inside a negative-index photonic-crystal slab,” Opt. Express 13, 2858–2868 (2005). [CrossRef] [PubMed]

21.

C. Li, J. M. Holt, and A. L. Efros, “Far-field imaging by the Veselago lens made of a photonic crystal,” J. Opt. Soc. Am. B 23, 490–497 (2006). [CrossRef]

22.

C. Y. Li, J. M. Holt, and A. L. Efros, “Imaging by the Veselago lens based upon a two-dimensional photonic crystal with a triangular lattice,” J. Opt. Soc. Am. B 23, 963–968 (2006). [CrossRef]

23.

C. Shen, K. Michielsen, and H. De Raedt, “Image transfer by cascaded stack of photonic crystal and air layers,” Opt. Express 14, 879–886 (2006). [CrossRef] [PubMed]

24.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

25.

M. Plihal, A. Shambrook, and A. A. Maradudin, “Two-dimensional photonic band structures,” Optics Commun. 80, 3–4 (1991). [CrossRef]

26.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966). [CrossRef]

27.

A. Taflove and S. C. Hagness, Computational Electrodynamics—the Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2000).

28.

S. D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices,” IEEE Trans. Antennas Propagat. 44, 1630–1639 (1996). [CrossRef]

OCIS Codes
(110.2960) Imaging systems : Image analysis
(260.2110) Physical optics : Electromagnetic optics

ToC Category:
Metamaterials

History
Original Manuscript: August 23, 2006
Revised Manuscript: October 21, 2006
Manuscript Accepted: October 22, 2006
Published: November 13, 2006

Citation
Haifei Zhang, Linfang Shen, Lixin Ran, Yu Yuan, and Jin Au Kong, "Layered superlensing in two-dimensional photonic crystals," Opt. Express 14, 11178-11183 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11178


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References

  1. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of and," Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
  2. R. W. Ziolkowski, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 056625 (2001). [CrossRef]
  3. I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, "BW media- media with negative parameters, capable of supporting backward waves," Microwave Opt. Technol. Lett. 31, 129-133 (2001). [CrossRef]
  4. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000). [CrossRef] [PubMed]
  5. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996). [CrossRef] [PubMed]
  6. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Technol. 47, 2075-2084 (1999). [CrossRef]
  7. L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Cheng, T. M. Grzegorczyk, and J. A. Kong, "Experimental study on several left-handed metamaterials," PIER 51, 249-279 (2005). [CrossRef]
  8. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
  9. M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000). [CrossRef]
  10. S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, "Refraction in media with a Negative Refractive Index," Phys. Rev. Lett. 90, 107402 (2003). [CrossRef] [PubMed]
  11. S. Foteinopoulou and C. M. Soukoulis, "Negative refraction and left-handed behavior in two-dimensional photonic crystals," Phys. Rev. B 67, 235107 (2003). [CrossRef]
  12. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Negative refraction by photonic crystals," Nature (London) 423, 604-605 (2003). [CrossRef] [PubMed]
  13. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002). [CrossRef]
  14. X. Wang, Z. F. Ren, and K. Kempa, "Unrestricted superlensing in a triangular two dimensional photonic crystal," Opt. Express 12,2919-2924 (2004). [CrossRef] [PubMed]
  15. Z.-Y. Li and L.-L. Lin, "Evaluation of lensing in photonic crystal slabs exhibiting negative refraction," Phys. Rev. B 68, 245110 (2003). [CrossRef]
  16. S. Xiao, M. Qiu, Z. Ruan, and S. He, "Influence of the surface termination to the point imaging by a photonic crystal slab with negative refraction," Appl. Phys. Lett. 85, 4269-4271 (2004). [CrossRef]
  17. X. Zhang, "Image resolution depending on slab thickness and object distance in a two-dimensional photoniccrystal-based superlens," Phys. Rev. B 70, 195110 (2004). [CrossRef]
  18. X. Wang and K. Kempa, "Effects of disorder on subwavelength lensing in two-dimensional photonic crystal slabs," Phys. Rev. B 71, 085101 (2005). [CrossRef]
  19. A. Martinez and J. Marti, "Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination," Phys. Rev. B 71, 235115 (2005). [CrossRef]
  20. A. Martinez and J. Marti, "Analysis of wave focusing inside a negative-index photonic-crystal slab," Opt. Express 13, 2858-2868 (2005). [CrossRef] [PubMed]
  21. C. Li, J. M. Holt, and A. L. Efros, "Far-field imaging by the Veselago lens made of a photonic crystal," J. Opt. Soc. Am. B 23, 490-497 (2006). [CrossRef]
  22. C. Y. Li, J. M. Holt, and A. L. Efros, "Imaging by the Veselago lens based upon a two-dimensional photonic crystal with a triangular lattice," J. Opt. Soc. Am. B 23, 963-968 (2006). [CrossRef]
  23. C. Shen, K. Michielsen, and H. De Raedt, "Image transfer by cascaded stack of photonic crystal and air layers," Opt. Express 14, 879-886 (2006). [CrossRef] [PubMed]
  24. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
  25. M. Plihal, A. Shambrook, and A. A. Maradudin, "Two-dimensional photonic band structures," Optics Commun. 80, 3-4 (1991). [CrossRef]
  26. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966). [CrossRef]
  27. A. Taflove and S. C. Hagness, Computational Electrodynamics—the Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2000).
  28. S. D. Gedney, "An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices," IEEE Trans. Antennas Propagat. 44, 1630-1639 (1996). [CrossRef]

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