## Image transfer by cascaded stack of photonic crystal and air layers

Optics Express, Vol. 14, Issue 2, pp. 879-886 (2006)

http://dx.doi.org/10.1364/OPEX.14.000879

Acrobat PDF (141 KB)

### Abstract

We demonstrate image transfer by a cascaded stack consisting of two and three triangular-lattice photonic crystal slabs separated by air. The quality of the image transfered by the stack is sensitive to the air/photonic crystal interface termination and the frequency. Depending on the frequency and the surface termination, the image can be transfered by the stack with very little deterioration of the resolution, that is the resolution of the final image is approximately the same as the resolution of the image formed behind one single photonic crystal slab.

© 2006 Optical Society of America

## 1. Introduction

*ε*(

**r**) and permeability

*μ*(

**r**), the electric field vector

**E**(

**r**), the magnetic field vector

**H**(

**r**) and the wave vector

**k**form a left-handed set of vectors. The Poynting vector

**S**always forms a right-handed set with the vectors

**E**and

**H**. Accordingly,

**S**and

**k**are antiparallel, that is

**S**∙

**k**< 0. In weakly dispersive media, the group velocity vg identifies the direction of energy propagation, that is

**S**·

**v**

_{g}> 0 [1

1. A. Bers, “Note on group velocity and energy propagation”, Am. J. Phys. **68**, 482–484 (2000). [CrossRef]

**v**

_{g}∙

**k**is equivalent to the sign of

**S**∙

**k**, which in this case is negative. The group velocity is called negative, that is having the opposite sign of the wave vector. The sign of the refractive index

*n*is the sign of

**S**∙

**k**and thus also of

*v*

_{g}∙

**k**. Thus, homogeneous isotropic dielectric media with simultaneously negative

*ε*and

*μ*have a negative refractive index. Therefore, these materials are often called negative index materials (NIMS) (see for example Refs. [2–4

2. J.B. Pendry and D.R. Smith, “Reversing light with negative refraction,” Phys. Today **57**, 37–43 (2004). [CrossRef]

5. V.G. Veselago, “The electrodynamics of substances with simultaneously negative values of *ε* and *μ*,” Sov. Phys. Usp. **10**, 509–514 (1968). [Translation from the original Russion version in Usp. Fiz. Nauk. **92**, 517–526 (1967). This year was mislabeled in the translation as 1964.]. [CrossRef]

*et al*. denoted these materials as double negative materials [6

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

7. R.W. Ziolkowski, “Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs,” Opt. Express **11**, 662–681 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-662. [CrossRef] [PubMed]

*et al*. suggested the name backward-wave media [8

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

9. I.V. Lindell and S. Ilvonen, “Waves in a slab of uniaxial BW medium,” J. of Electromagn. Waves and Appl. **16**, 303–318 (2002). [CrossRef]

14. A.A. Oliner and T. Tamir, “Backward waves on isotropic plasma slabs,” J. Appl. Phys. **33**, 231–233 (1962). [CrossRef]

16. H. Lamb, “On group-velocity,” Proc. London Math. Soc. **1**, 473–479 (1904). [CrossRef]

17. H.C. Pocklington, “Growth of a wave-group when the group-velocity is negative,” Nature **71**, 607–608 (1905). [CrossRef]

*D*made of a NIM with refractive index

*n̂*= - 1 and situated in vacuum (

*n*= 1) can focus radiation from a point source

*P*positioned at a distance

*L*<

*D*from one side of the plate to a point

*P*′ located at a distance

*L*′ =

*D*-

*L*from the other side of the plate [5

5. V.G. Veselago, “The electrodynamics of substances with simultaneously negative values of *ε* and *μ*,” Sov. Phys. Usp. **10**, 509–514 (1968). [Translation from the original Russion version in Usp. Fiz. Nauk. **92**, 517–526 (1967). This year was mislabeled in the translation as 1964.]. [CrossRef]

*n̂*= - 1 and surrounded by vacuum make perfect lenses or superlenses, since both propagating and evanescent waves contribute to the resolution of the image [19

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

*ε*=

*μ*= -1 that is both lossless and nondispersive [6

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

*et al*. [20

20. P.F. Loschialpo, D.L. Smith, D.W. Forester, F.J. Rachford, and J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E **67**, 025602 (2003). [CrossRef]

21. P.F. Loschialpo, D.W. Forester, D.L. Smith, F.J. Rachford, and C. Monzon, “Optical properties of an ideal homogeneous causal left-handed material slab,” Phys. Rev. E **70**, 036605 (2004). [CrossRef]

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

*ε*> 0 and a magnetic permeability

*μ*= 1, near the bandgap frequency behave as if they have an effective refractive index [22

22. 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]

22. 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]

22. 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]

23. 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]

**62**, 10696–10705 (2000). [CrossRef]

24. A. Martinez, H. Miguez, A. Griol, and J. Marti, “Experimental and theoretical analysis of the self-focusing of light by a photonic crystal lens,” Phys. Rev. B **69**, 165119 (2004). [CrossRef]

**62**, 10696–10705 (2000). [CrossRef]

34. 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]

**62**, 10696–10705 (2000). [CrossRef]

27. M. Notomi, “Negative refraction in photonic crystals,” Opt. Quant. Electr. **34**, 133–143 (2002). [CrossRef]

27. M. Notomi, “Negative refraction in photonic crystals,” Opt. Quant. Electr. **34**, 133–143 (2002). [CrossRef]

35. S. He and Z. Ruan, “A completely open cavity realized with photonic crystal wedges,” J. Zhejiang Univ. SCI **6A**, 355–357 (2005). [CrossRef]

## 2. Numerical method

**r**=

**r**

_{0}and modeled by

**n**defines the direction of the electric current (TM-mode) and

*ω*is the angular frequency of the source. The point source is placed in front of the stack at a distance

*L*from the most left air/PhC interface. The location of the source is chosen such that it is located at one of the

*E*-points of the two-dimensional Yee grid (TM-mode). As indicated by the step function in Eq. (1), the source is turned on at

_{z}*t*= 0. In order to study the imaging properties of the cascaded stack we solve the time-dependent Maxwell equations (TDME) by means of a finite-difference time-domain method that, in the absence of external currents, conserves the energy exactly [38

38. J.S. Kole, M.T. Figge, and H. De Raedt, “Unconditionally stable algorithms to solve the time-dependent Maxwell equations,” Phys. Rev. E **64**, 066705 (2001). [CrossRef]

*π*/

*ω*. We study the image quality from the electric field intensity distribution, plotted along the image center perpendicular (along the

*x*-axis) and parallel (along the

*y*-axis) to the air/PhC slab interfaces. The corresponding longitudinal and transversal spatial resolutions, defined as the ratio of the full width at half maximum (FWHM) to the wavelength

*λ*of the light, are

*R*and

_{x}*R*, respectively. For reference, and because it is most frequently shown in the current literature, we also show the electric field amplitude.

_{y}*λ*of light in vacuum. Time and frequency are then expressed in units of

*λ*/

*c*and

*c*/

*λ*, respectively, where

*c*denotes the velocity of light in vacuum. For PhCs with lattice constant

*a*,

*f*=

*ωa*/(2

*πc*) is the dimensionless frequency. We employ a square grid with lattice spacing

*δ*= 0.01

*λ*. We solve the TDME using a fourth-order unconditionally stable algorithm [38

38. J.S. Kole, M.T. Figge, and H. De Raedt, “Unconditionally stable algorithms to solve the time-dependent Maxwell equations,” Phys. Rev. E **64**, 066705 (2001). [CrossRef]

*δt*= 0.001

*λ*/

*c*. Smaller values for the lattice spacing

*δ*and the time step

*δt*give similar results as the ones presented in this paper.

## 3. Parameters of the photonic crystal slabs

*ε*= 12.96,

*μ*= 1. The holes have a radius

*r*= 0.4

*a*. We use the MIT Photonic-Bands (MPB) package [40

40. S.G. Johnson and J.D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173. [CrossRef] [PubMed]

*f*=

*ωa*/2

*πc*= 0.26 - 0.33, corresponding to frequencies in band two, the EFS plot for the TM mode is depicted in the left panel of Fig. 1. For some frequencies, the shape of the EFS is circular. In this case, we can extract an effective refractive index

*n̂*from the radius of the EFS using Snell’s law [22

**62**, 10696–10705 (2000). [CrossRef]

*n̂*is determined by the behavior of the EFSs as a function of

*f*. From Fig. 1 it can be seen that the EFSs move inwards with increasing frequency. Hence,

*n̂*< 0 (

**v**

_{g}∙

**k**< 0) [22

**62**, 10696–10705 (2000). [CrossRef]

*n̂*as a function of the angle

*θ*of the incoming wave vector

**k**is shown in the right panel of Fig. 1. For

*f*= 03,

*n̂*is almost independent of

*θ*and is almost equal to -1. Small changes in

*f*lead to large changes in

*n̂*.

*n̂*= - 1 embedded in air is not reflected at the air/slab interfaces. For a slab made of a PhC with an effective refractive index of

*n̂*= - 1 the transmission of light is not 100% and strongly depends on the orientation of the PhC and on the terminations of the air/slab interfaces. Transmission is maximal if the surface normal of the slabs is along the Γ

_{M}direction [41

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

29. 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]

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

*N*layers of air spheres with cutted air/PhC interfaces normal to the Γ

*M*direction. Simulation results for PhC slabs with

*N*= 9 indicate that images with the best spatial resolutions

*R*and

_{x}*R*are obtained if the cutted area at both air/PhC interfaces has a width

_{y}*C*= 0.5

*r*(results not shown). Concerning the maximum of the transmitted intensity (maximum of the intensity behind the slab),

*C*= 0.3

*r*is a slightly better choice than

*C*= 0. 5

*r*. Because the maximum of the transmitted intensity is low anyway (5.2% for

*C*= 0.3

*r*compared to 3.8% for

*C*= 0.5

*r*, if we normalize the maximum of the source intensity to 100%), we only consider PhC slabs with

*C*= 0.5

*r*. Hence, the thickness of the slabs is given by

*D*= (

*N*- 1)√3

*a*/2 + 0.4

*a*.

## 4. Image transfer by a cascaded stack

*D*made of NIM with

*n̂*= - 1 and situated in vacuum can focus radiation from a point source

*P*positioned at a distance

*L*<

*D*from one side of the plate to a point

*P*′ located at a distance

*L*′ =

*D*-

*L*from the other side of the plate [5

5. V.G. Veselago, “The electrodynamics of substances with simultaneously negative values of *ε* and *μ*,” Sov. Phys. Usp. **10**, 509–514 (1968). [Translation from the original Russion version in Usp. Fiz. Nauk. **92**, 517–526 (1967). This year was mislabeled in the translation as 1964.]. [CrossRef]

*L*+

*L*′ =

*D*, the distance between

*P*and

*P*′ is always equal to 2

*D*. Hence, to increase the distance between

*P*and

*P*′ there are two possibilities: Increasing

*D*or applying the principle of image transfer by a cascaded stack. If the flat plates are made of a PhC with an effective refractive index

*n̂*= - 1, only one option is feasible: Taking one PhC slab and increasing its thickness leads to worsening spatial resolutions of the image that is formed behind the slab.

*n̂*≈ - 1, we consider a source, emitting TM waves with a frequency

*f*= 0.299. We first consider a cascaded stack composed of two PhC slabs with thickness

*D*= 6.33

*λ*(

*N*= 25). The slabs are separated by an air layer of thickness 6.43

*λ*. The source is placed at a distance

*L*= 3.29

*λ*from the left surface of the first PhC slab. Figure 2 depicts the electric field intensity and amplitude. The material structure of the PhC slab is not shown in order to give a better image of the light focusing inside the slabs. The lines indicate the direction of propagation according to Snell’s law for the case of a homogeneous isotropic slab with

*n̂*= - 1. From Fig. 2 it can be seen that there is a relatively good agreement between our simulation results and the results obtained from the rules of geometric optics. Light focusing occurs inside the slabs, in between the two slabs and behind the two slabs. The focus behind the two slabs is not circular, but elongated in the direction of propagation. This has also been observed in previous works for single slab imaging [25

25. 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]

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

33. A. Martinez and J. Marti, “Analysis of wave focusing inside a negative-index photonic-crystal slab,” Opt. Express **13**, 2858–2868 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-8-2858. [CrossRef] [PubMed]

*n̂*from -1 for different angles of incidence. Hence, we do not observe point focusing and in a strict sense, there is no image formation. However, we can say we observe image formation with some aberrations. The spatial resolutions of the image behind the second slab are

*R*= 1.76 and

_{x}*R*= 0.46. As can be clearly seen from the left panel of Fig. 2, this image is different from the image in between the two slabs. However, if we remove the second slab and repeat the same calculation, then an image with the same spatial resolutions is formed (results not shown). Hence, the first image (behind the first slab) is transfered through the second slab, so that the second image (behind the second slab) is a copy of the first image. The first image gets distorted by reflections from the second slab. This would not happen in the ideal case, that is in the case of two slabs made of a homogeneous isotropic material with

_{y}*n̂*= -1. Note that the picture of the electric field amplitude (see right panel of Fig. 2) suggests a more circular image that is transfered by the cascaded stack and thus better imaging properties of the slabs. However, the picture of the electric field amplitude displays only a snapshot. Moreover, it is not sufficient to study field amplitudes in order to investigate the imaging properties of a PhC system since they cannot be measured directly.

*f*= 0.300. For this frequency images are also formed inside the slabs, in between the two slabs and behind the two slabs, but the deviations from propagation through a homogeneous isotropic slab with

*n̂*= -1 is much larger than for

*f*= 0.299, as indicated by the propagation lines obtained from Snell’s law. Moreover, the image formed behind the two slabs is no longer an exact copy of the image formed behind the first slab. The spatial resolution changes from

*R*= 1.60,

_{x}*R*= 0.42 for the first image to

_{y}*R*= 1.70,

_{x}*R*= 0.42 for the second image. Hence, for this particular case, additional longitudinal aberrations are induced by the second slab. Note however, that the spatial resolutions of the image formed behind the second slab in Fig. 3 (left panel) are better than the ones of the image formed behind the second slab in Fig. 2 (left panel). The right panel of Fig. 3 shows the electric field intensity for the same setup as the one that was used to obtain the results displayed in Fig. 2 but the air/PhC interfaces of the slabs are not cut. It is clearly seen that in this case, no image transfer by the cascaded stack can be observed.

_{y}*N*= 15, show that the intensity maximum in the first slab corresponds to 7.0%, in between the two slabs to 3.8%, inside the second slab to 2.6% and behind the second slab to 1.3%. Hence, even for cascaded stacks with thin PhC slabs the maximum of the transmitted intensity is rather low. This observation is in agreement with the observation that the maximum coupling coefficient between a plane wave in air and the Bloch wave in the PhC is only about 65% and rapidly decreases for angles of incidence larger than 30°, for an air/PhC interface normal to the Γ

*M*direction [41

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

*D*= 3.74

*λ*(

*N*= 15). The PhC slabs are separated by air layers of the same thickness as the slabs. The source is placed at a distance

*L*= 1.87

*λ*from the left surface of the first PhC slab. Figure 4 depicts the electric field intensity. The intensity maximum in the first slab corresponds to 6.6%, in between slab 1 and slab 2 to 3.6%, in the second slab to 3.3%, in between slab 2 and 3 to 1.5%, in the third slab to 1.6% and behind the third slab to 0.7%. Hence, for cascaded stacks with three PhC layers with

*N*= 15, our simulations show that the maximum of the intensity behind the last slab decreases, compared to the case of a cascaded stack with only two PhC slabs: For three slabs, the maximum of the intensity behind the last slab corresponds to 0.7%, while for the two slab case it corresponds to 1.3%, as shown before. Similar simulations with one single slab show that for a single slab the maximum of the transmitted intensity corresponds to 3.4%. These results indicate that each slab that is added to the stack reduces the intensity by a factor of two. In this particular case, additional longitudinal aberrations are induced by each slab that is added to the stack. The transversal spatial resolution of the image remains approximately the same upon transfer of the image through the stack.

## 5. Summary and conclusions

## Acknowledgments

## References and links

1. | A. Bers, “Note on group velocity and energy propagation”, Am. J. Phys. |

2. | J.B. Pendry and D.R. Smith, “Reversing light with negative refraction,” Phys. Today |

3. | S. Foteinopoulou, E.N. Economou, and C.M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. |

4. | J.B. Brock, A.A. Houck, and I.L. Chuang, “Focusing inside negative index materials,” Appl. Phys. Lett. |

5. | V.G. Veselago, “The electrodynamics of substances with simultaneously negative values of |

6. | R.W. Ziolkowski, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E |

7. | R.W. Ziolkowski, “Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs,” Opt. Express |

8. | I.V. Lindell, S.A. Tretyakov, K.I. Nikoskinen, and S. Ilvonen, “BW media- media with negative parameters, capable of supporting backward waves,” Microw. Opt. Tech. Lett. |

9. | I.V. Lindell and S. Ilvonen, “Waves in a slab of uniaxial BW medium,” J. of Electromagn. Waves and Appl. |

10. | R.G.E. Hutter, |

11. | P.E. Mayes, G.A. Deschamps, and W.T. Patton, “Backward-wave radiation from periodic structures and application to the design of frequency-independent antennas,” Proc. IRE |

12. | J.L. Altman, |

13. | R.E. Collin, |

14. | A.A. Oliner and T. Tamir, “Backward waves on isotropic plasma slabs,” J. Appl. Phys. |

15. | |

16. | H. Lamb, “On group-velocity,” Proc. London Math. Soc. |

17. | H.C. Pocklington, “Growth of a wave-group when the group-velocity is negative,” Nature |

18. | R.A. Silin, “Possibility of creating plane-parallel lenses,” Opt. Spectrosc. (USSR) |

19. | J.B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

20. | P.F. Loschialpo, D.L. Smith, D.W. Forester, F.J. Rachford, and J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E |

21. | P.F. Loschialpo, D.W. Forester, D.L. Smith, F.J. Rachford, and C. Monzon, “Optical properties of an ideal homogeneous causal left-handed material slab,” Phys. Rev. E |

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

23. | 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. |

24. | A. Martinez, H. Miguez, A. Griol, and J. Marti, “Experimental and theoretical analysis of the self-focusing of light by a photonic crystal lens,” Phys. Rev. B |

25. | 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 |

26. | E. Ozbay, I. Bulu, K. Aydin, H. Caglayan, K.B. Alici, and K. Guven, “Highly directive radiation and negative refraction using photonic crystals,” Laser Phys. |

27. | M. Notomi, “Negative refraction in photonic crystals,” Opt. Quant. Electr. |

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

29. | 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. |

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

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

32. | X. Wang, Z.F. Ren, and K. Kempa, “Improved superlensing in two-dimensional photonic crystals with a basis,” Appl. Phys. Lett. |

33. | A. Martinez and J. Marti, “Analysis of wave focusing inside a negative-index photonic-crystal slab,” Opt. Express |

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

35. | S. He and Z. Ruan, “A completely open cavity realized with photonic crystal wedges,” J. Zhejiang Univ. SCI |

36. | Z. Ruan and S. He, “Open cavity formed by a photonic crystal with negative effective index of refraction,” Opt. Lett. |

37. | S. He, Y. Jin, Z. Ruan, and J. Kuang, “On subwavelength and open resonators involving metamaterials of negative refraction index,” New Journal of Physics |

38. | J.S. Kole, M.T. Figge, and H. De Raedt, “Unconditionally stable algorithms to solve the time-dependent Maxwell equations,” Phys. Rev. E |

39. | A. Taflove and S.C. Hagness, |

40. | S.G. Johnson and J.D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

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

**OCIS Codes**

(110.2990) Imaging systems : Image formation theory

(120.5710) Instrumentation, measurement, and metrology : Refraction

(220.3630) Optical design and fabrication : Lenses

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Photonic Crystals

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-2-879

Sort: Journal | Reset

### References

- A. Bers, "Note on group velocity and energy propagation", Am. J. Phys. 68, 482 - 484 (2000). [CrossRef]
- J.B. Pendry and D.R. Smith, "Reversing light with negative refraction," Phys. Today 57, 37 - 43 (2004). [CrossRef]
- 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]
- J.B. Brock, A.A. Houck, and I.L. Chuang, "Focusing inside negative index materials," Appl. Phys. Lett. 85, 2472 - 2474 (2004). [CrossRef]
- V.G. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and µ," Sov. Phys. Usp. 10, 509 - 514 (1968). [Translation from the original Russion version in Usp. Fiz. Nauk. 92, 517 - 526 (1967). This year was mislabeled in the translation as 1964.]. [CrossRef]
- R.W. Ziolkowski, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 056625 (2001). [CrossRef]
- R.W. Ziolkowski, "Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs," Opt. Express 11, 662 - 681 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-662">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-662</a>. [CrossRef] [PubMed]
- I.V. Lindell, S.A. Tretyakov, K.I. Nikoskinen, and S. Ilvonen, "BW media- media with negative parameters, capable of supporting backward waves," Microw. Opt. Tech. Lett. 31, 129 - 133 (2001). [CrossRef]
- I.V. Lindell and S. Ilvonen, "Waves in a slab of uniaxial BW medium," J. of Electromagn. Waves and Appl. 16, 303 - 318 (2002). [CrossRef]
- R.G.E. Hutter, Beam and wave electronics in microwave tubes, (Van Nostrand, Princeton, NJ, 1960), p.220.
- P.E. Mayes, G.A. Deschamps, and W.T. Patton, "Backward-wave radiation from periodic structures and application to the design of frequency-independent antennas," Proc. IRE 49, 962 - 963 (1961).
- J.L. Altman, Microwave circuits, (Van Nostrand, Princeton, NJ, 1964), chap.7, p.304.
- R.E. Collin, Foundations for vmicrowave engineering, (McGraw-Hill, New York, 1966).
- A.A. Oliner and T. Tamir, "Backward waves on isotropic plasma slabs," J. Appl. Phys. 33, 231 - 233 (1962). [CrossRef]
- H. Lamb, "On group-velocity," Proc. London Math. Soc. 1, 473 - 479(1904). [CrossRef]
- H.C. Pocklington, "Growth of a wave-group when the group-velocity is negative," Nature 71, 607 - 608 (1905). [CrossRef]
- R.A. Silin, "Possibility of creating plane-parallel lenses," Opt. Spectrosc. (USSR) 44, 109 - (1978). [Translation from the original Russian version in Opt. Spektrosk. 44, 189 - 191 (1978).]
- J.B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966 - 3969 (2000). [CrossRef] [PubMed]
- P.F. Loschialpo, D.L. Smith, D.W. Forester, F.J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003). [CrossRef]
- P.F. Loschialpo, D.W. Forester, D.L. Smith, F.J. Rachford, and C. Monzon, "Optical properties of an ideal homogeneous causal left-handed material slab," Phys. Rev. E 70, 036605 (2004). [CrossRef]
- 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]
- 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]
- A. Martinez, H. Miguez, A. Griol, and J. Marti, "Experimental and theoretical analysis of the self-focusing of light by a photonic crystal lens," Phys. Rev. B 69, 165119 (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]
- E. Ozbay, I. Bulu, K. Aydin, H. Caglayan, K.B. Alici, and K. Guven, "Highly directive radiation and negative refraction using photonic crystals," Laser Phys. 15, 217 - 224 (2005).
- M. Notomi, "Negative refraction in photonic crystals," Opt. Quant. Electr. 34, 133 - 143 (2002). [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]
- S. Xiao, M. Qiu, Z. Ruan, 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]
- X. Zhang, "Image resolution depending on slab thickness and object distance in a two-dimensional photonic-crystal-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]
- X. Wang, Z.F. Ren, and K. Kempa, "Improved superlensing in two-dimensional photonic crystals with a basis," Appl. Phys. Lett. 86, 061105 (2005). [CrossRef]
- A. Martinez and J. Marti, "Analysis of wave focusing inside a negative-index photonic-crystal slab," Opt. Express 13, 2858 -2868 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-8-2858">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-8-2858</a>. [CrossRef] [PubMed]
- C. Luo, S.G. Johnson, J.D. Joannopoulos, J.B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002). [CrossRef]
- S. He and Z. Ruan, "A completely open cavity realized with photonic crystal wedges," J. Zhejiang Univ. SCI 6A, 355 - 357 (2005). [CrossRef]
- Z. Ruan and S. He, "Open cavity formed by a photonic crystal with negative effective index of refraction," Opt. Lett. 30, 2308 - 2310 (2005). [CrossRef] [PubMed]
- S. He, Y. Jin, Z. Ruan, and J. Kuang, "On subwavelength and open resonators involving metamaterials of negative refraction index," New Journal of Physics 7, 210 (2005). [CrossRef]
- J.S. Kole, M.T. Figge, and H. De Raedt, "Unconditionally stable algorithms to solve the time-dependent Maxwell equations," Phys. Rev. E 64, 066705 (2001). [CrossRef]
- A. Taflove and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd edition, (Artech House, MA USA, 2005).
- S.G. Johnson and J.D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173 - 190 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173</a>. [CrossRef] [PubMed]
- Z. Ruan, M. Qiu, S. Xiao, S. He, and L. Thylén, "Coupling between plane waves and Bloch waves in photonic crystals with negative refraction," Phys. Rev. B 71, 045111 (2005). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.