## Analysis of wave focusing inside a negative-index photonic-crystal slab

Optics Express, Vol. 13, Issue 8, pp. 2858-2868 (2005)

http://dx.doi.org/10.1364/OPEX.13.002858

Acrobat PDF (575 KB)

### Abstract

We analyze focusing of electromagnetic waves inside a photonic crystal slab by means of finite-difference time-domain simulations. At the frequency of the source, the photonic crystal behaves as an effective medium with an effective index of refraction of -1. Despite of the strong Bloch modulation of the field inside the slab, the presence of a well-definite internal focus is evident. The dimensions of the internal focus are similar to those of the external focus. The effect of the frequency of the wave on the focusing is also discussed.

© 2005 Optical Society of America

## 1. Introduction

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. J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. **85**, 3966–3969 (2000). [CrossRef] [PubMed]

3. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from Conductors and Enhanced Nonlinear Phenomena,” IEEE Trans. Microwave Tech. **47**, 2075–2084 (1999). [CrossRef]

5. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science **292**, 77–79 (2001). [CrossRef] [PubMed]

6. A. Grbic and G. V. Eleftheriades, “Overcoming the Diffraction Limit with a Planar Left-Handed Transmission-Line Lens,” Phys. Rev. Lett. **92**, 117403 (2004). [CrossRef] [PubMed]

7. 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]

7. 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]

*n*

_{eff}in certain spectral regions where the equifrequency contours (EFCs) become rounded. If the EFC shrinks with increasing frequency then the group velocity points inwards and a phenomenon of negative refraction can be expected at the interface between the PhC and air (or a dielectric medium) [8

8. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B **58**, 10096–10099 (1998). [CrossRef]

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

10. 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]

11. A. Martínez, H. Míguez, A. Griol, and J. Martí, “Experimental and theoretical study of the self-focusing of light by a photonic crystal lens,” Phys. Rev. B **69**, 165119 (2004). [CrossRef]

12. H.-T. Chien, H.-T. Tang, C.-H. Kuo, C.-C. Chen, and Z. Ye, “Directed diffraction without negative refraction,” Phys. Rev. B **70**, 113101 (2004). [CrossRef]

13. C.-H. Kuo and Z. Ye, “Negative-refraction-like behavior revealed by arrays of dielectric cylinders,” Phys. Rev. E **70**, 026608 (2004). [CrossRef]

14. P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, “Imaging by flat lens using negative refraction,” Nature (London) **426**, 404 (2003). [CrossRef]

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]

*n*

_{eff}=-1 does not occur in this case since the PhC slab does not behave as an effective medium [12

12. H.-T. Chien, H.-T. Tang, C.-H. Kuo, C.-C. Chen, and Z. Ye, “Directed diffraction without negative refraction,” Phys. Rev. B **70**, 113101 (2004). [CrossRef]

13. C.-H. Kuo and Z. Ye, “Negative-refraction-like behavior revealed by arrays of dielectric cylinders,” Phys. Rev. E **70**, 026608 (2004). [CrossRef]

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. 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]

## 2. Description of the scenario: a PhC slab that behaves as a NIL

*r*=0.4

*a*) on a dielectric background with

*ε*=12.96. We assume that the material has negligible absorption losses, as it would be the case of Si or GaAs at optical frequencies (

*λ*=1550 nm,

*λ*being the free-space wavelength). The lattice extends over the

*x*-

*z*plane and is infinite in the

*y*dimension. Only transverse magnetic modes with the electric field along the y direction are considered. We use a plane wave expansion method to calculate the band diagram of this PhC as shown in Fig. 1(a). Frequency is expressed in normalized units of

*a*/

*λ*. We will focus our attention on the second band. For frequencies over 0.26 the EFCs become rounded (effective refractionlike medium [7

7. 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]

*n*

_{eff}is depicted in Fig. 1(b) for the two main symmetry directions. Only propagation along ΓM is interesting because the Bloch mode along ΓK is uncoupled and cannot be excited efficiently [11

11. A. Martínez, H. Míguez, A. Griol, and J. Martí, “Experimental and theoretical study of the self-focusing of light by a photonic crystal lens,” Phys. Rev. B **69**, 165119 (2004). [CrossRef]

*n*

_{eff}=-1, which is the first step towards the NIL. If

*n*

_{eff}≠-1 the position of the internal and internal foci will be different depending on the angle of incidence which results in a longitudinal spreading of the focus [11

11. A. Martínez, H. Míguez, A. Griol, and J. Martí, “Experimental and theoretical study of the self-focusing of light by a photonic crystal lens,” Phys. Rev. B **69**, 165119 (2004). [CrossRef]

24. J. P. Berenger, “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,” J. Comput. Phys. **114**, 185–200 (1994). [CrossRef]

*z*direction) is along ΓM. Both interfaces are cut along ΓK. The slab thickness is

*L*. The slab is symmetric with respect to both

*x*and

*z*axes. An optical source is placed

*L*/2 below the slab (

*z*

_{source}=-

*L*). Several sets of electric field monitors (represented by dots) are placed along the dotted lines A, B, C, D, E, and F. The objective is to obtain the field (and intensity) profiles to evaluate the lensing effect. Figure 3 shows the FDTD electric field distribution obtained at different time steps of the FDTD simulation. Time is expressed in

*ct*/

*a*units,

*c*being speed of light in vacuum and

*t*absolute time. Both slab terminations are identical (the slab is symmetric with respect to the

*x*axis) and are chosen to ensure high transmission efficiency to sharpen the focusing [22

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

*L*=11.5√3-0.4)

*a*=19.52

*a*, which is much larger than the corresponding wavelength. Our FDTD simulations have proven that this kind of termination (also used in other works [19

19. 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]

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

*λ*. We have chosen this kind of source instead of a point source as in the focusing analysis of section 3 in order to better observe the shrinking and focusing of the beam inside and outside the NIL. The frequency is 0.306 in order to have

*n*

_{eff}=-1. The physical structure of the PhC slab has been removed in the snapshots to better appreciate the electromagnetic propagation inside the PhC. First we see in Fig. 3(a) the input Gaussian wave before impinging the PhC slab. At

*ct*/

*a*=20 the wave enters the slab [Fig. 3(b)] and a Bloch wave is excited. The reflection at the input interface is very low, although a perfect matching cannot be achieved, in contrast with the case of NILs based on left-handed metamaterials. After this, the beam commences to shrink and a negative curvature appears. Internal focusing is observed in Fig. 3(c) approximately at the center of the slab. It should be highlighted that such an internal focus does not occur when the focusing is at a frequency in the valence band [15

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]

*ct*/

*a*=110 we observe that the beam has reached the output interface and exits the PhC slab [see Fig. 3(d)]. However, we do not observe a beam broadening after the internal focus: this is because oblique rays need more time to reach the output interface as they travel a larger distance and still do not contribute to the beam at the time step shown in Fig. 3(d). The internal broadening for positive

*z*is clearly observed in Fig. 3(e) at a time step of

*ct*/

*a*=140, and also the formation of the external focus is evident: at the exit of the slab the electric field has a negative curvature and the radius of the phase fronts gets smaller. In Fig. 3(f) the external focus is totally formed. The beam propagation follows the geometric optics rules as predicted by Veselago [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]

25. Z. Ruan, M. Qiu, S. Xiao, S. He, and L. Thylen, “Coupling between plane waves and Bloch waves in photonic crystals with negative refraction,” Phys. Rev. B **71**, 045111 (2005). [CrossRef]

## 4. Analysis of the internal focus

*λ*, which resembles a plane wave. First we consider propagation along

*z*. Figure 6(a) shows the longitudinal profile of the field intensity along line C. The blue solid curve is the average intensity for the point source whereas the dashed black curve represents the same magnitude for the case of pseudo-plane wave. It can be easily observed that the profile is well different from that depicted in Fig. 4: the field is strongly modulated and there is a series of peaks (corresponding to the dielectric background) and valleys (corresponding to the holes) along the line C. Specially, a strong minimum of intensity is observed at

*z*=0. Despite of the modulation, the focusing is evident: for the plane wave case, the intensity maxima keep almost constant along the slab; for the case of the point source the intensity maxima are larger near the center of the slab (

*z*=0). Figure 6(b) shows the time-space diagram of the intensity along the C line. The formation of the Bloch-modulated focus is clear.

*z*=0: maxima are slightly larger for positive values of

*z*. Moreover, a second focus seems to appear around

*z*=6

*a*, which is also observed in the external focus (see Fig. 5(a)). This can be explained by considering the amplification of evanescent waves along the slab, which would explain the subwavelength focusing observed in Fig. 5(b) [6

6. A. Grbic and G. V. Eleftheriades, “Overcoming the Diffraction Limit with a Planar Left-Handed Transmission-Line Lens,” Phys. Rev. Lett. **92**, 117403 (2004). [CrossRef] [PubMed]

*x*

_{1}are refracted and, since

*n*

_{eff}is slightly lower than −1 for incident angles >0°, they exit the PhC slab at

*x*

_{2}<0 that verifies |

*x*

_{2}|<|

*x*

_{1}|. Results shown below in Fig. 8 contribute to this explanation. We think that both phenomena (growing of evanescent waves and residual collimating behavior) contribute to the observed effect.

*x*=0 is observed, in agreement with the results shown in Fig. 6(a). This is, the input transversal intensity is rearranged as the field propagates through the NIL so the power concentration is higher near

*x*=0 as the wave reaches the output interface. As explained before, both the growing of evanescent waves and the residual collimating behavior of the PhC are probably the origin of this phenomenon.

*a*/2 rightwards. The obtained longitudinal and transversal intensity profiles are shown as black curves in Fig. 9(a) and Fig. 9(b) respectively. For the sake of comparison, the intensity profiles obtained previously (hole in the center of the slab) are also shown with blue curves. It is interesting to note that in this case an intensity maximum occurs at the center of the slab whereas intensity minima are observed corresponding to the presence of holes. Despite of the well different intensity profiles inside the slab for each slab configuration, it must be noted that the external focus is exactly the same for both of them. We measured the same longitudinal and transversal profiles represented in Fig. 5 for the slab with dielectric at its center, which confirms that the PhC slab behaves as an isotropic NIL. We also include the intensity profile of the external foci as red dashed curves. The longitudinal external focus in Fig. 9(a) has been represented inverted with respect to

*z*=

*L*/2. It can be observed the resemblance of the internal and external focus if the Bloch-modulation is not taken into account. The intensity level is higher in the internal focus, which would mean that approximately half the intensity is lost when the wave travels from the NIL to the external medium. However, taking into account that inside the PhC there is a series of intensity maxima (dielectric) and minima (holes), we can state that the mean intensity is similar for both the internal and external focus, which agrees well with the low reflection observed at the air-PhC interfaces. From Fig. 9 it can also be deduced that the values of FWHM are similar for both the internal and external foci. We must also note that the presence of the second maximum in the longitudinal external focus at

*z*=13.5

*a*in Fig. 5(a) is also observed in the internal focus so it can be concluded that the second interface produces a high-quality image of the filed inside the slab and the first interface is the main origin of distortion of the final image. This is consistent with the fact that evanescent waves are strongly attenuated before entering the NIL and can only be restored in part, so the loss of information takes part mainly in this region.

*n*

_{eff}varies as in Fig. 1(b). Assuming that the NIL under study follows the geometric optics rules, it can be stated if the frequency is chosen so that the index is different from −1, not all the rays converge in the same point both inside and outside the slab and the position of the internal and external foci is modified [11

**69**, 165119 (2004). [CrossRef]

*n*

_{eff}| decreases (increases) so the internal focus position (proportional to |

*n*

_{eff}|) comes to a lower (larger) value of

*z*and the external focus position (proportional to 1/|

*n*

_{eff}|) is displaced to larger (smaller) values of

*z*. We measured the characteristics of the internal and external foci for the slab with

*L*=19.52

*a*at frequencies

*a*/

*λ*=0.295, 0.306 and 0.32. The corresponding field intensity along the

*z*axis is represented in Fig. 10, in which it can be observed that the internal and the external foci are displaced as expected for each frequency. The broadening of the external focus is more evident for

*a*/

*λ*=0.32. We think that in the case of

*a*/

*λ*=0.296 the external focus is not broadened because of the proximity of the PhC surface, which gives rise to a near-field focusing. Anyway, we can conclude that both the internal and external foci follow the rules of geometric optics when the frequency of the source is varied.

*x*-

*z*plane since the lens is invariant along the

*y*axis. This is, the source is actually a line source, not a point source. To achieve point focusing from a true point source, a three-dimensional PhC slab is required. In this case, we should work in a frequency region where the EFCs are spheres (instead of circles as in the two-dimensional case) whose radii decrease with increasing frequency. This has been recently demonstrated to be possible using a bicontinuous PhC of a

*bcc*lattice, in which subwavelength resolution was shown to occur along both transversal directions [26

26. X. Ao and S. He, “Three-dimensional photonic crystal of negative refraction achieved by interference lithography,” Opt. Lett. **29**, 2542–2544 (2004). [CrossRef] [PubMed]

*y*and the

*z*axes.

## 5. Conclusion

## Acknowledgments

## References and Links

1. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. |

2. | J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. |

3. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from Conductors and Enhanced Nonlinear Phenomena,” IEEE Trans. Microwave Tech. |

4. | J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. |

5. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

6. | A. Grbic and G. V. Eleftheriades, “Overcoming the Diffraction Limit with a Planar Left-Handed Transmission-Line Lens,” Phys. Rev. Lett. |

7. | M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B |

8. | H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B |

9. | S. Foteinopoulou and C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B |

10. | C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B |

11. | A. Martínez, H. Míguez, A. Griol, and J. Martí, “Experimental and theoretical study of the self-focusing of light by a photonic crystal lens,” Phys. Rev. B |

12. | H.-T. Chien, H.-T. Tang, C.-H. Kuo, C.-C. Chen, and Z. Ye, “Directed diffraction without negative refraction,” Phys. Rev. B |

13. | C.-H. Kuo and Z. Ye, “Negative-refraction-like behavior revealed by arrays of dielectric cylinders,” Phys. Rev. E |

14. | P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, “Imaging by flat lens using negative refraction,” Nature (London) |

15. | Z.-Y. Li and L.-L. Lin, “Evaluation of lensing in photonic crystal slabs exhibiting negative refraction,” Phys. Rev. B |

16. | X. Wang, Z. F. Ren, and K. Kempa, “Unrestricted superlensing in a triangular two-dimensional photonic crystal,” Opt. Express |

17. | A. Martínez and J. Martí, submitted to Phys. Rev. B. |

18. | A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. |

19. | 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. |

20. | K. Guven, K. Aydin, K. B. Alici, C. M. Soukoulis, and E. Ozbay, “Spectral negative refraction and focusing analysis of a two-dimensional left-handed photonic crystal lens,” Phys Rev. B |

21. | X. Zhang, “Image resolution depending on slab thickness and object distance in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. B |

22. | X. Wang and K. Kempa, “Effects of disorder on subwavelength lensing in two-dimensional photonic crystal slabs,” Phys. Rev. B |

23. | A. Taflove, |

24. | J. P. Berenger, “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,” J. Comput. Phys. |

25. | Z. Ruan, M. Qiu, S. Xiao, S. He, and L. Thylen, “Coupling between plane waves and Bloch waves in photonic crystals with negative refraction,” Phys. Rev. B |

26. | X. Ao and S. He, “Three-dimensional photonic crystal of negative refraction achieved by interference lithography,” Opt. Lett. |

**OCIS Codes**

(160.4760) Materials : Optical properties

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 4, 2005

Revised Manuscript: March 30, 2005

Published: April 18, 2005

**Citation**

Alejandro Martinez and Javier Marti, "Analysis of wave focusing inside a negative-index photonic-crystal slab," Opt. Express **13**, 2858-2868 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-8-2858

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### References

- V. G. Veselago, �??The electrodynamics of substances with simultaneously negative values of ε and μ,�?? Sov. Phys. Usp. 10, 509-514 (1968) [CrossRef]
- J. B. Pendry, �??Negative Refraction Makes a Perfect Lens,�?? Phys. Rev. Lett. 85, 3966-3969 (2000) [CrossRef] [PubMed]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, �??Magnetism from Conductors and Enhanced Nonlinear Phenomena,�?? IEEE Trans. Microwave Tech. 47, 2075-2084 (1999) [CrossRef]
- 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]
- R. A. Shelby, D. R. Smith, and S. Schultz, �??Experimental verification of a negative index of refraction,�?? Science 292, 77-79 (2001) [CrossRef] [PubMed]
- A. Grbic and G. V. Eleftheriades, �??Overcoming the Diffraction Limit with a Planar Left-Handed Transmission-Line Lens,�?? Phys. Rev. Lett. 92, 117403 (2004) [CrossRef] [PubMed]
- 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]
- H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, �??Superprism phenomena in photonic crystals,�?? Phys. Rev. B 58, 10096-10099 (1998) [CrossRef]
- S. Foteinopoulou and C. M. Soukoulis, �??Negative refraction and left-handed behavior in two-dimensional photonic crystals,�?? Phys. Rev. B 67, 235107 (2003) [CrossRef]
- 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]
- A. Martínez, H. Míguez, A. Griol, and J. Martí, �??Experimental and theoretical study of the self-focusing of light by a photonic crystal lens,�?? Phys. Rev. B 69, 165119 (2004) [CrossRef]
- H.-T. Chien, H.-T. Tang, C.-H. Kuo, C.-C. Chen, and Z. Ye, �??Directed diffraction without negative refraction,�?? Phys. Rev. B 70, 113101 (2004) [CrossRef]
- C.-H. Kuo and Z. Ye, �??Negative-refraction-like behavior revealed by arrays of dielectric cylinders,�?? Phys. Rev. E 70, 026608 (2004) [CrossRef]
- P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, �??Imaging by flat lens using negative refraction,�?? Nature (London) 426, 404 (2003). [CrossRef]
- Z.-Y. Li and L.-L. Lin, �??Evaluation of lensing in photonic crystal slabs exhibiting negative refraction,�?? Phys. Rev. B 68, 245110 (2003) [CrossRef]
- X. Wang, Z. F. Ren, and K. Kempa, �??Unrestricted superlensing in a triangular two-dimensional photonic crystal,�?? Opt. Express 12, 2919-2924 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2919">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2919</a>. [CrossRef] [PubMed]
- A. Martínez and J. Martí, submitted to Phys. Rev. B.
- A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylén, A. Talneau and S. Anand, �??Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,�?? Phys. Rev. Lett. 93, 073902 (2004) [CrossRef] [PubMed]
- 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]
- K. Guven, K. Aydin, K. B. Alici, C. M. Soukoulis, and E. Ozbay, �??Spectral negative refraction and focusing analysis of a two-dimensional left-handed photonic crystal lens,�?? Phys Rev. B 70, 205125 (2004) [CrossRef]
- 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]
- X. Wang and K. Kempa, �??Effects of disorder on subwavelength lensing in two-dimensional photonic crystal slabs,�?? Phys. Rev. B 71, 085101 (2005) [CrossRef]
- A. Taflove, Computational Electrodynamics�??The Finite Difference Time-Domain Method (Artech House, Boston, 1995)
- J. P. Berenger, �??A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,�?? J. Comput. Phys. 114, 185-200 (1994) [CrossRef]
- Z. Ruan, M. Qiu, S. Xiao, S. He, and L. Thylen, �??Coupling between plane waves and Bloch waves in photonic crystals with negative refraction,�?? Phys. Rev. B 71, 045111 (2005) [CrossRef]
- X. Ao and S. He, �??Three-dimensional photonic crystal of negative refraction achieved by interference lithography,�?? Opt. Lett. 29, 2542-2544 (2004) [CrossRef] [PubMed]

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