## Imaging properties of photonic crystals

Optics Express, Vol. 15, Issue 12, pp. 7786-7801 (2007)

http://dx.doi.org/10.1364/OE.15.007786

Acrobat PDF (987 KB)

### Abstract

We observe, by means of finite element calculations, that some photonic crystals produce negative refraction with almost circular isofrequency lines of their band diagram, so that a slab of this structure presents a large degree of isoplanatism and thus can behave like an imaging system. However, it has aberrations on comparison with a model of ideal lossless left-handed material within an effective medium theory. Further, we see that it does not produce subwavelength focusing. We discuss the limitations and requirements for such photonic crystal slabs to yield superresolved images of extended objects.

© 2007 Optical Society of America

## 1. Introduction

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

15. 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 (2000) [CrossRef]

16. 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]

17. D.N. Chigrin, S. Enoch, C.M. Sotomayor-Torres, and G. Tayeb, “Self-guiding in two-dimensional photonic crystals,” Opt. Express **11**, 1203 (2003). http://www.opticsexpress.org/abstract.cfm?uri=OE-11-10-1203 [CrossRef] [PubMed]

18. R. Iliew, C. Etrich, and F. Lederer, “Self-collimation of light in three-dimensional photonic crystals,” Opt. Express **13**, 7076 (2005). http://www.opticsexpress.org/abstract.cfm?uri=OE-13-18-7076 [CrossRef] [PubMed]

19. J.L. Garcia-Pomar and M. Nieto-Vesperinas, “Waveguiding, collimation and subwavelength concentration in photonic crystals,” Opt. Express **13**, 7997–8007 (2005). http://www.opticsexpress.org/abstract.cfm?uri=OE-13-20-7997 [CrossRef] [PubMed]

26. X. Wang, Z.F. Ren, and K. Kempa, “Improved superlensing in two-dimensional photonic crystals with a basis,” Appl. Phys. Lett. **86**, 061105 (2004). [CrossRef]

27. R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, and C. M. Soukoulis “Negative refraction and superlens behavior in a two-dimensional photonic crystal” Phys. Rev. B **71**, 085106 (2005) [CrossRef]

28. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J.B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B **68**, 045115 (2003) [CrossRef]

28. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J.B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B **68**, 045115 (2003) [CrossRef]

16. 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]

28. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J.B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B **68**, 045115 (2003) [CrossRef]

29. T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic Crystal lens: From negative refraction and negative index to negative permittivity and permeability”, Phys. Rev. Lett. **97**073905 (2006). [CrossRef] [PubMed]

26. X. Wang, Z.F. Ren, and K. Kempa, “Improved superlensing in two-dimensional photonic crystals with a basis,” Appl. Phys. Lett. **86**, 061105 (2004). [CrossRef]

27. R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, and C. M. Soukoulis “Negative refraction and superlens behavior in a two-dimensional photonic crystal” Phys. Rev. B **71**, 085106 (2005) [CrossRef]

## 2. Propagation of extended wavefronts in photonic crystal slabs

*ε*= 12.96, lattice constant

*a*and radius

*r*= 0.4

*a*. The geometry is shown in Figs. 1(a) and 1(b) both for a slab of this crystal and its Brillouin zone. This structure was proposed in [26

26. X. Wang, Z.F. Ren, and K. Kempa, “Improved superlensing in two-dimensional photonic crystals with a basis,” Appl. Phys. Lett. **86**, 061105 (2004). [CrossRef]

*k*in this regime of frequencies, which are in the second band, for the directions Γ

*K*and Γ

*M*. We observe that the the magnitude of the permitted wavevectors is different for the Γ

*K*and Γ

*M*directions in this range of frequencies (bottom inset of Fig. 2 (b)), and therefore the effective refractive index is not the same for both directions, which causes the deformation of the isofrequency from a circle one; this deformation being the cause of an aberration in the imaging process by a slab of this crystal, as shown next. We shall denote as the

*input plane*that at which we specify the limiting value of the wavefield illuminating a slab of thickness

*d*of this array. This plane is taken at distance

*z*

_{0}from the entrance surface of the crystal slab. Let us now consider a point source, in the input plane, at distance

*z*

_{0}= 3.442

*a*, which emits linearly polarized harmonic waves around a normalized frequency

*ω*= 0.305 × 2π

*c*/

*a*,

*λ*being the wavelength, with the electric vector

*E*parallel to the cylinder axis.

_{y}*z*

_{0}= 3.442

*a*. The thickness of the PC slab is

*d*= 2

*z*

_{0}. The calculation gives the distribution of the field propagated throughout the crystal and out of it. In particular, its response in the image plane at the Veselago distance

*z*

_{0}from the exit surface is evaluated. This is shown for

*λ*= 3.28

*a*in Fig. 3.

*isoplanatic*condition, according to which the image field distribution

*i*(

*x*) is given in terms of the object wavefront distribution

*o*(

*x*) by the convolution [30], [31]:

*P*(

*x*) is the point spread function of the system, or response to a point source, (i.e.,

*P*(

*x*) constitutes the wavefunction at the focus), and it allows, by Fourier transformation, to define a transfer function of the lens or optical system. This also involves that in the region of isoplanacity the response

*o*(

*x*)

_{i}*P*(

*x*-

*x*) in the detection plane to each

_{i}*ith*sampling point

*o*(

*x*)

_{i}*δ*(

*x*-

*x*) of the object wavefield distribution in the input plane, only depends on the difference of coordinates

_{i}*x*-

*x*and not on the position

_{i}*x*.

_{i}*P*(

*x*) is performed, then this result which is the transfer function

*T*(

*u*), is multiplied by the Fourier spectrum

*O*(

*u*) of any extended object

*o*(

*x*) placed at the input plane

*z*

_{0}. Subsequently, an inverse Fourier transform of this product should yield a field distribution

*i*(

*x*) in the image plane according to Eq.(1), and this result ought to be very similar to that obtained by direct FE calculation of the propagation of this extended object wavefront through the crystal slab, in the image plane.

*λ*= 3.24

*a*, 3.28

*a*, 3.32

*a*and 3.36

*a*. On the other hand, Fig. 5 depicts the transfer function at different distances

*z*

_{0}of the source from the slab. We observe that the widths of these transfer functions do not exceed the Rayleigh limit of resolution 1/

*λ*. The comparison of the images of an extended object that has no subwavelength details, obtained by the above mentioned procedures is displayed in Fig. 6 for

*λ*= 3.28

*a*measured at a distance

*z*

_{0}from the slab. They indicate that in the range of wavelengths and distances of operation studied here, there is good degree of isoplanatism in the crystal, however due to a diffraction effects and since the isofrequency lines are not perfect circles, the image possesses aberrations.

*x*) and longitudinal (∆

*y*) aberration distances, defining the vertices of the resulting ray tracing caustic, (cf. Figs. 7(a) and 7(b)), are shown in Fig. 7(d). The distances ∆

*x*and ∆

*y*of the aberration due only to deformations of the isofrequency from a circle, increase with the wavelength, (this is clear by looking at the bottom inset of Fig. 2(b), where details of the separation between the Γ

*K*and Γ

*M*curves are appreciated). On the other hand, the distances ∆

*x*and ∆

*y*of the aberration in the real case, namely that in which we have also had into account that the refractive index of the surrounding air, being different to that of the PC, gives rise to transmission into the PC that varies with the angle of incidence, decrease with the wavelength.

*Z*= √

*ε*/

*μ*, which has been demonstrated to be different from

*Z*= 1 in these PC slabs [29

29. T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic Crystal lens: From negative refraction and negative index to negative permittivity and permeability”, Phys. Rev. Lett. **97**073905 (2006). [CrossRef] [PubMed]

32. J. L. Garcia-Pomar and M. Nieto-Vesperinas, “Transmission study of prisms and slabs of lossy negative index media,” Opt. Express **12**, 2081 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2081 [CrossRef] [PubMed]

## 3. LHM slabs

*ε*̂ =

*μ*̂ = -1 +

*i*0.001, which involves that the complex refractive index will be

*n*̂ = - 1 +

*i*0.001. A time harmonic dependence exp(

*iωt*) for the wave is assumed. Figs. 8(a)–(c) show the spatial distribution of the electric field modulus when a point source, made by a narrow slit (Fig. 8(d)), is placed at distances

*z*

_{0}=

*λ*,

*λ*/3 and

*λ*/6 from the entrance surface of the slab.

*z*

_{0}/

*λ*decreases. Also, while at

*z*

_{0}=

*λ*there is focusing inside the slab (cf. Fig. 5(a)), which manifests the dominant contribution of the propagating components of the wavefield, at subwavelength distances

*z*

_{0}this focusing is almost imperceptible, showing the predominance of the evanescent components of the wavefield angular spectrum. In addition in Figs. 8(a)–(c) one sees how both the object and the image fields are spatially coupled with intensity enhancements in the entrance and exit surfaces of the slab, respectively. Both enhancements being due to the excitation of surface plasmon polaritons (SPPs) by the evanescent components of the wave, and thus being increasingly prominent as

*z*

_{0}/

*λ*decreases; this is shown in Fig. 10. These SPPs are present for both TE and TM polarization cf.[33

33. R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter **13**, 1811 (2001). [CrossRef]

*z*

_{0}/

*λ*diminishes, is what broadens the bandwidth of the transfer function. Figs. 8(a)–(c) and 9 confirm it.

*z*

_{0}/

*λ*decreases beyond 1/2. Since the image field is constituted by the distribution expelled out from the standing wave pattern of the SPP at the exit interface. This progressive lack of a focus plane as one approaches the quasi-electrostatic limit is in agreement with a similar observation of lack of intensity maximum in

*z*> 0 in [28

**68**, 045115 (2003) [CrossRef]

*λ*. The same can be said with respect to the field of the incident wave in front of the entrance surface of the slab.

*z*

_{0}from the exit surface of the slab for an incident extended object wavefront, on the input plane at distance

*z*

_{0}from the entrance surface. The superresolution attained as

*z*

_{0}decreases, in association with that of the corresponding transfer functions (cf Fig. 9) is evident. Also, in these Figs. 11(a)–(c) we have performed the same operation as in the PC slab, namely, a comparison between the image obtained by FE simulation of the propagation process throughout the slab and that obtained by Fourier transformation via the transfer function. The agreement shown, even in those cases demanding superresolution, demonstrates the isoplanatism of this system in the range of distances and wavelengths employed.

## 4. Discussion on the conditions for superresolution

*ε*′ =

*μ*′ =

*n*′ = - 1. These values are not strictly physical, since some absorption should be present [9

9. N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. **88**, 207403 (2002). [CrossRef] [PubMed]

**k**= (

*k*,

_{x}*k*) and

_{z}**k**′ = (

*k′*,

_{x}*k′*) be the wavevectors of one evanescent plane wave component, incident and transmitted into the slab, respectively. Namely,

_{z}*ae*

^{ikxx-kzz}is refracted in the first interface of the LHM slab into

*ae*

^{ik′x-k′zZ}, where

*s*and

*s*′ be the directional sines of

**k**and

**k**′ for

*k*and

_{x}*k′*, respectively, namely,

_{x}*k*=

_{x}*k*

_{0}

*s*,

*k′*=

_{x}*k*

_{0}

*n′s*, then the Snell law derived from the continuity equation:

*k′*=

_{x}*k*implies that

_{x}*s*′ = -

*s*and thus

*k′*= -

_{z}*k*which is usually interpreted as the amplification of the incident evanescent wave component by the LHM slab.

_{z}*ε*′ = -1,

*μ*= 1 and thus

*n*′ =

*i*, then the Snell law yields

*is*′ =

*s*, i.e.:

*s*′

^{2}= -

*s*

^{2}; thus the z-component of the wavevector transmitted into the slab is in this case

*s*>> 1 which yields exactly

*k′*= -

_{z}*k*. Thus the silver slab acts in a similar way as the LHM slab in this limit which involves scales of the slab thickness, and distances outside it, much smaller than

_{z}*λ*.

33. R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter **13**, 1811 (2001). [CrossRef]

*ω*being the plasma frequency in the bulk. If we look into the dispersion relation:

_{P}*ε*≈ - 1 then from Eq. (6) one sees that the surface plasmon satisfies that

*k*(

_{x}*ω*) → ∞, this is the essence of the electrostatic limit:

*k*(

_{x}*ω*) >>

*k*

_{0}at subwavelength scales, and thus from Eq. (5) we have that its excitation frequency

*ω*is:

_{sp}*k*,

_{x}*k*)-complex plane in the evanescent zone, given by:

_{z}*ω*. However, The SPP plasmon dispersion relation shown in Fig. 10 shows that this is only possible in the flat region of this curve which corresponds to large values of

_{sp}*k*at which Eq. (8) takes on the asymptotic form given by the straight line:

_{x}*k*in the flat zone of the dispersion relation, all of them satisfying it at the plasmon frequency

_{x}*ω*.This could not happen, however, in the curved portion of the dispersion relation, since there only one

_{sp}*k*corresponds to each

_{x}*ω*, and thus only one evanescent component of the incident wavefront would excite it.

_{sp}*a*× 0.8

*a*with permittivity

*ε*= 9.61 in air, like those used in reference [27

27. R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, and C. M. Soukoulis “Negative refraction and superlens behavior in a two-dimensional photonic crystal” Phys. Rev. B **71**, 085106 (2005) [CrossRef]

*a*the first and last rows of prisms, and we have calculated it by the supercell method [35–37

35. F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: Effect of the position of the surface plane,” Phys. Rev. B **59**, 15112 (1999) [CrossRef]

*k*are very close to the light cone

_{x}*ω*/

*c*and thus they are not sufficiently high to reach the hyperbolic isofrequency curve posed in our Eq. (8) in the evanescent zone, or conversely the corresponding dispersion relation line, (green curve in Fig. 13). Furthermore we have a limited spectrum of permitted

*k*, with a maximum

_{x}*k*that does not allow this subwavelength detail.

_{M}*k*) of Fig. 13, one sees that this range lies inside the zone in which the hyperbola of Eq. (8) does not take the asymptotic form Eq. (9); namely, that range lies in the curved region of the plasmon dispersion relation which is between the light line (value

_{x}*ω*/

*c*) and the flat zone region

*ω*=

*constant*of the dispersion curve. Therefore, the excited surface waves do not follow the required isofrequency line in the evanescent zone.

## 5. Conclusions

## Acknowledgments

## References and links

1. | V.G. Veselago,“The electrodynamics of substances with simultanenous negative values of |

2. | J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Magnetism from conductors and enhanced non-linear phenomena,” IEEE Trans. Microwave Theory Tech. |

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

4. | A.A. Houck, J.B. Brock, and I.L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. |

5. | C.G. Parazzoli, R.B. Greegor, K. Li, B.E.C. Koltenbach, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. |

6. | N. Garcia and M. Nieto-Vesperinas, “Is there an experimental verification of a negative index of refraction yet?,” Opt. Lett. |

7. | N.-C. Panoiu and R.M. Osgood, “Influence of the dispersive properties of metals on the transmission characteristics of left-handed materials,” Phys. Rev. E |

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

9. | N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. |

10. | D.R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S.A. Ramakrishna, and J.B. Pendry, “Limitations on sub-diffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. |

11. | M. Nieto-Vesperinas,“Problem of image superresolution with a negative-refractive-index slab,” J. Opt. Soc. Am. A |

12. | R. Merlin,“Analytical solution of the almost-perfect-lens problem” Appl. Phys. Lett. |

13. | V.A. Podolskiy and E.E. Narimanov, “Near-sighted superlens,” Opt. Lett. |

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

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

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

17. | D.N. Chigrin, S. Enoch, C.M. Sotomayor-Torres, and G. Tayeb, “Self-guiding in two-dimensional photonic crystals,” Opt. Express |

18. | R. Iliew, C. Etrich, and F. Lederer, “Self-collimation of light in three-dimensional photonic crystals,” Opt. Express |

19. | J.L. Garcia-Pomar and M. Nieto-Vesperinas, “Waveguiding, collimation and subwavelength concentration in photonic crystals,” Opt. Express |

20. | P.V. Parimi, W.T. Lu, P. Vodo, J. Sokoloff, J.S. Derov, and S. Sridhar, “Negative Refraction and Left-Handed Electromagnetism in Microwave Photonic Crystals,” Phys. Rev. Lett. |

21. | B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am. A |

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

23. | E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C.M. Soukoulis, “Subwavelength Resolution in a Two-Dimensional Photonic-Crystal-Based Superlens,” Phys. Rev. Lett. |

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

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

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

27. | R. Moussa, S. Foteinopoulou, L. Zhang, G. Tuttle, K. Guven, E. Ozbay, and C. M. Soukoulis “Negative refraction and superlens behavior in a two-dimensional photonic crystal” Phys. Rev. B |

28. | C. Luo, S. G. Johnson, J. D. Joannopoulos, and J.B. Pendry, “Subwavelength imaging in photonic crystals,” Phys. Rev. B |

29. | T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic Crystal lens: From negative refraction and negative index to negative permittivity and permeability”, Phys. Rev. Lett. |

30. | M. Born and E. Wolf, |

31. | J.W. Goodman, |

32. | J. L. Garcia-Pomar and M. Nieto-Vesperinas, “Transmission study of prisms and slabs of lossy negative index media,” Opt. Express |

33. | R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter |

34. | A.L. Efros, C. Y. Li, and A. L. Pokrovsky, “Evanescent waves in photonic crystals and image of Veselago lens,” cond-mat/0503494(2005) |

35. | F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: Effect of the position of the surface plane,” Phys. Rev. B |

36. | W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. |

37. | R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B |

**OCIS Codes**

(100.6640) Image processing : Superresolution

(110.2990) Imaging systems : Image formation theory

(110.4850) Imaging systems : Optical transfer functions

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: February 6, 2007

Revised Manuscript: April 27, 2007

Manuscript Accepted: May 16, 2007

Published: June 8, 2007

**Citation**

J. L. Garcia-Pomar and M. Nieto-Vesperinas, "Imaging properties of photonic crystals," Opt. Express **15**, 7786-7801 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-12-7786

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### References

- V. G. Veselago,"The electrodynamics of substances with simultanenous negative values of ∑ and μ," Sov. Phys. Usp. 10, 509 (1968). [CrossRef]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced non-linear phenomena," IEEE Trans. Microwave Theory Tech. MTT-47, 195 (1999).
- R. A. Shelby, D. R. Smith, S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77 (2001). [CrossRef] [PubMed]
- A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell’s law," Phys. Rev. Lett. 90, 137401 (2003). [CrossRef] [PubMed]
- C. G. Parazzoli, R. B. Greegor, and K. Li, B. E. C. Koltenbach, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell’s law," Phys. Rev. Lett. 90, 107401 (2003). [CrossRef] [PubMed]
- N. Garcia and M. Nieto-Vesperinas, "Is there an experimental verification of a negative index of refraction yet?," Opt. Lett. 27, 885 (2002). [CrossRef]
- N.-C. Panoiu and R. M. Osgood, "Influence of the dispersive properties of metals on the transmission characteristics of left-handed materials," Phys. Rev. E 68, 016611 (2003). [CrossRef]
- J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- N. Garcia and M. Nieto-Vesperinas, "Left-handed materials do not make a perfect lens," Phys. Rev. Lett. 88, 207403 (2002). [CrossRef] [PubMed]
- D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, "Limitations on subdiffraction imaging with a negative refractive index slab," Appl. Phys. Lett. 82, 1506 (2003). [CrossRef]
- M. Nieto-Vesperinas, "Problem of image superresolution with a negative-refractive-index slab," J. Opt. Soc. Am. A 21, 491 (2004). [CrossRef]
- R. Merlin, "Analytical solution of the almost-perfect-lens problem," Appl. Phys. Lett. 84, 1290 (2004). [CrossRef]
- V. A. Podolskiy and E. E. Narimanov, "Near-sighted superlens," Opt. Lett. 30, 75 (2005). [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 (2000) [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]
- D. N. Chigrin, S. Enoch C. M. Sotomayor-Torres and G. Tayeb, "Self-guiding in two-dimensional photonic crystals," Opt. Express 11, 1203 (2003). http://www.opticsexpress.org/abstract.cfm?uri=OE-11-10-1203> [CrossRef] [PubMed]
- R. Iliew, C. Etrich, and F. Lederer, "Self-collimation of light in three-dimensional photonic crystals," Opt. Express 13, 7076 (2005). http://www.opticsexpress.org/abstract.cfm?uri=OE-13-18-7076> [CrossRef] [PubMed]
- J. L. Garcia-Pomar and M. Nieto-Vesperinas, "Waveguiding, collimation and subwavelength concentration in photonic crystals," Opt. Express 13, 7997-8007 (2005). http://www.opticsexpress.org/abstract.cfm?uri=OE-13-20-7997> [CrossRef] [PubMed]
- P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004). [CrossRef] [PubMed]
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