## Negative refraction can make non-diffracting beams

Optics Express, Vol. 16, Issue 19, pp. 14582-14587 (2008)

http://dx.doi.org/10.1364/OE.16.014582

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### Abstract

We report the results of simulations relating to the illumination of a structure consisting of a slab constructed from a 2-D hexagonal array of metal rods with a terahertz frequency source. As a consequence of negative refraction an essentially non-divergent beam pattern is observed. Although the results presented relate to the terahertz regime they should also be applicable at other frequencies.

© 2008 Optical Society of America

1. Kishan Dholakia, “Against the spread of the light,” Nature **451**, 413 (2008) [CrossRef] [PubMed]

*λ*and width

*w*to diverge by an angle

*δ*which satisfies the relation sin

*δ*≈

*λ*/

*w*. In an attempt to eliminate, or at least to reduce the divergence of beams, Bessel beams [2

2. J Durnin, J J Mieceli, and J H Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**, 1499–1501 (1987) [CrossRef] [PubMed]

3. G A Siviloglou, J Broky, A Dogariu, and D N Christodoulides, “Observation of accelerating Airy beams,” Phys.Rev.Lett. **99**, 213901 (2007) [CrossRef]

*A*from the right side of the slab crosses the optic axis at a distance

*D*from the other side of the slab, where

*D*depends on the angle

*α*between the ray and the optical axis. To be specific, we take the medium surrounding the slab to be vacuum when Snell’s law gives the relationship

*n*|<1, the radiation incident on the slab can experience total internal reflection, and only those rays for which the angle of incidence

*α*is smaller than a critical angle

*α*

*, given by sin*

_{tot}*α*=|

_{tot}*n*|, can enter the slab. When

*α*→0 rays cross the optic axis near

*D*

_{0}=

*L*/|

*n*|-

*A*, but when the incident angle corresponds to

*α*,

_{tot}*D*

_{0}→∞. Hence, the image of the source will not be a single point (in the 2-D plane), as would be the case if |

*n*|= 1, but will be stretched from the position

*D*

_{0}to infinity. Since

*D*

_{0}can be made arbitrarily large by reducing |

*n*|, the result suggests that it should be possible to use a slab with a negative effective refractive index of small magnitude to produce an essentially non-divergent beam of radiation from a point source in the 2-D plane. In this paper we illustrate how a non-divergent beam with a width of a few wavelengths can be produced at terahertz frequencies using a slab in the form of a metallic photonic crystal having the required properties.

*f*(

*k*), for the two-dimensional photonic crystal considered, which is in the form of gold (plasma frequency=8.9 eV) circular rods of diameter 80 µm arranged in a hexagonal lattice with period 200 µm. For radiation in the E polarization, with electric field parallel to the rods, such a structure exhibits an effective plasma frequency [11,12

12. A J Gallant, M A Kaliteevski, D Wood, M C Petty, R A Abram, S Brand, G P Swift, D A Zeze, and J M Chamberlain, “Passband filters for terahertz radiation based on dual metallic photonic structures,” Appl. Phys. Lett. **91**, 161115 (2007) [CrossRef]

9. M A Kaliteevski, S Brand, J Garvie-Cook, R A Abram, and J M Chamberlain, “Terahertz filter based on refractive properties of metallic photonic crystal,” Opt. Express **16**, 7330–7335 (2008) [CrossRef] [PubMed]

*n*=-

*k*(

*f*)/

*k*

_{0}where

*k*(

*f*) and

*k*

_{0}are the magnitudes of the wavevectors at frequency

*f*in the photonic crystal and the vacuum respectively. Also,

*k*(

*f*) decreases as the frequency is increased, and near the top of the band the effective refractive index can be very small in magnitude, making large values of

*D*

_{0}possible [13

13. Dispersion relations have been calculated using a complex photonic bandstructure method described in [11], and finite difference time domain software OMNISIM© has been employed to simulate the beam propagation. Due to minor convergence problems the frequency scales for the two methods are slightly different. In order to provide matching of the two scales, the bandstructure has been renormalized to provide matching of the frequencies at the J point which can be determined for both calculation techniques.

*L*=1200 µm irradiated with line sources (perpendicular to the plane shown) of frequencies 1.621 THz, 1.667 THz and 1.715 THz placed at a distance

*A*=1.8 mm to the right of the structure. For a frequency of 1.621 THz (wavelength

*λ*=185 µm) the beam has a width of about 3

*λ*and it propagates for about 10 mm (50

*λ*) before there is significant divergence. The divergence of the beam is illustrated in Fig. 4a by the black solid line, which shows the intensity profile at a distance of 7.5 mm to the left of the slab (approximately in the middle of Fig. 3), and by the black dashed line which shows the profile at 16 mm (at the left edge of Fig. 3). With increasing frequency, |

*n*| reduces, as shown in Fig. 2, and

*D*

_{0}increases, as is implicit in Figs. 3(b) and 3(c).

*λ*and shows no sign of divergence at a propagation distance corresponding to 100

*λ*.

*D*

_{0}and waist widths

*w*are shown in Table 1.

14. H Kosaka, T Kawashima, A Tomita, M Notomi, T Tamamura, T Sato, and S Kawakami, Appl. Phys. Lett. **74**, 1212–1214 (1999) [CrossRef]

15. D N Chigrin, S Enoch, C M Sotomayor Torres, and G Tayeb, Opt. Express **11**, 1203–1211 (2003) [CrossRef] [PubMed]

*A*=4.5 mm and

*A*=5.3 mm respectively. It is apparent that the width of the beam is not increased in proportion to

*A*, as would be expected on purely geometrical grounds, and in fact for all three source distances considered, the parameters of the non-diffracting beam differ only slightly (although when the distance is smaller than 1 mm, results not shown here demonstrate that the divergence of the beam becomes significantly more pronounced). However, the data in Table 1 do show a clear increase in the waist of the beam with decreasing angle of total reflection, which is the opposite to what would be expected on geometrical grounds. These results can be explained, at least intuitively, by a Fourier argument where the width of the beam is determined by the spread of its transverse wavevectors within the slab, which in turn is determined principally by sin

*α*. Then, as

_{tot}*α*is decreased, the wavevector spread is decreased and, through the usual Fourier relationship, there is an increase in the beam width in real space.

_{tot}*α*is only

_{tot}*α*/180, and of this, somewhat less than 10% is transferred into the collimated beam. However, more efficient illumination schemes could be employed to increase the fraction of emitted power that is transmitted.

_{tot}## Acknowledgement

## References and links

1. | Kishan Dholakia, “Against the spread of the light,” Nature |

2. | J Durnin, J J Mieceli, and J H Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

3. | G A Siviloglou, J Broky, A Dogariu, and D N Christodoulides, “Observation of accelerating Airy beams,” Phys.Rev.Lett. |

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

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

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

7. | V. Patanjali, V. Parimi, Wentao T. Lu, Plarenta Vodo, and Srinivas Sridhar, “Photonic crystals: Imaging by flat lens using negative refraction,” Nature |

8. | E Cubukcu, K Aydin, E Ozbay, S Foteinopoulou, and C M Soukoulis, “Electromagnetic waves: Negative refraction by photonic crystals,” Nature |

9. | M A Kaliteevski, S Brand, J Garvie-Cook, R A Abram, and J M Chamberlain, “Terahertz filter based on refractive properties of metallic photonic crystal,” Opt. Express |

10. | D O S Melville, R J Blaikie, and C R Wolf, “Submicron imaging with a planar silver lens,” Appl. Phys. Lett. |

11. | S Brand, R A Abram, and M A Kaliteevski, “Complex photonic bandstructure and effective plasma frequency of a two-dimensional array of metal rods,” Phys. Rev. |

12. | A J Gallant, M A Kaliteevski, D Wood, M C Petty, R A Abram, S Brand, G P Swift, D A Zeze, and J M Chamberlain, “Passband filters for terahertz radiation based on dual metallic photonic structures,” Appl. Phys. Lett. |

13. | Dispersion relations have been calculated using a complex photonic bandstructure method described in [11], and finite difference time domain software OMNISIM© has been employed to simulate the beam propagation. Due to minor convergence problems the frequency scales for the two methods are slightly different. In order to provide matching of the two scales, the bandstructure has been renormalized to provide matching of the frequencies at the J point which can be determined for both calculation techniques. |

14. | H Kosaka, T Kawashima, A Tomita, M Notomi, T Tamamura, T Sato, and S Kawakami, Appl. Phys. Lett. |

15. | D N Chigrin, S Enoch, C M Sotomayor Torres, and G Tayeb, Opt. Express |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(160.5298) Materials : Photonic crystals

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 11, 2008

Revised Manuscript: August 29, 2008

Manuscript Accepted: August 29, 2008

Published: September 2, 2008

**Citation**

M. Kaliteevski, S. Brand, R. A. Abram, A. J. Gallant, and J. M. Chamberlain, "Negative refraction can make non-diffracting
beams," Opt. Express **16**, 14582-14587 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-14582

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### References

- K. Dholakia, "Against the spread of the light," Nature 451, 413 (2008) [CrossRef] [PubMed]
- J .Durnin, J. J. Mieceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys.Rev.Lett. 99, 213901 (2007) [CrossRef]
- V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and ?," 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]
- R. A. Shelby, D. R. Smith, and S. Shultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2002) [CrossRef]
- V. Patanjali V. Parimi, W. T. Lu, Plarenta Vodo and Srinivas Sridhar, "Photonic crystals: Imaging by flat lens using negative refraction," Nature 426, 404 (2003) [CrossRef]
- E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou and C. M. Soukoulis, "Electromagnetic waves: Negative refraction by photonic crystals," Nature 423, 604-605 (2003) [CrossRef] [PubMed]
- M. A. Kaliteevski, S. Brand, J. Garvie-Cook, R. A. Abram, and J. M. Chamberlain, "Terahertz filter based on refractive properties of metallic photonic crystal," Opt. Express 16, 7330-7335 (2008) [CrossRef] [PubMed]
- D. O. S. Melville, R. J .Blaikie, and C. R. Wolf, "Submicron imaging with a planar silver lens," Appl. Phys. Lett. 84, 4403-4405 (2004) [CrossRef]
- S. Brand, R. A. Abram and M. A. Kaliteevski, "Complex photonic bandstructure and effective plasma frequency of a two-dimensional array of metal rods," Phys. Rev. B 75, 035102, (2007)
- A. J. Gallant, M. A. Kaliteevski, D. Wood, M. C. Petty, .R A. Abram, S. Brand, G. P. Swift, D. A. Zeze and J. M. Chamberlain, "Passband filters for terahertz radiation based on dual metallic photonic structures," Appl. Phys. Lett. 91, 161115 (2007) [CrossRef]
- Dispersion relations have been calculated using a complex photonic bandstructure method described in [11], and finite difference time domain software OMNISIM© has been employed to simulate the beam propagation. Due to minor convergence problems the frequency scales for the two methods are slightly different. In order to provide matching of the two scales, the bandstructure has been renormalized to provide matching of the frequencies at the J point which can be determined for both calculation techniques.
- H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Appl. Phys. Lett. 74, 1212-1214 (1999) [CrossRef]
- D. N. Chigrin, S. Enoch, C. M. Sotomayor Torres and G. Tayeb, "Self-guiding in two-dimensional photonic crystals," Opt. Express 11, 1203-1211 (2003). [CrossRef] [PubMed]

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