## Multiple directional beaming effect of metallic subwavelength slit surrounded by periodically corrugated grooves

Optics Express, Vol. 16, Issue 7, pp. 4487-4493 (2008)

http://dx.doi.org/10.1364/OE.16.004487

Acrobat PDF (271 KB)

### Abstract

It is demonstrated that multiple directional beaming effect can be realized by a metallic subwavelength slit surrounded by finite number of grooves based on mode expansion method. Each of the directional beaming is formed by superimposing two diffraction orders of spoof surface plasmon excited on the two corrugated sides of the slit. This delivers high contrast and considerably uniform energy distribution for the beaming directions.

© 2008 Optical Society of America

## 1. Introduction

*et al.*, reported experimentally the beaming light from a subwavelength aperture surrounded by surface corrugations on a thin metal film in 2002 [1

1. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martín-Moreno, F. J. García-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science **297**, 820 (2002). [CrossRef] [PubMed]

2. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. **90**, 167401 (2003). [CrossRef] [PubMed]

*et al.*, proposed the mode expansion method to describe the connection of the excited surface mode in the grooves and the distribution of electromagnetic field in the free space [2

2. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. **90**, 167401 (2003). [CrossRef] [PubMed]

*et al.*, proposed a revised quasi perfect conductor model which can deal with the light diffraction behavior of slit-grooves structures for true metals [5

5. C. Wang, C. Du, and X. Luo, “Refining the model of light diffraction from a subwavelength slit surrounded by grooves on a metallic film,” Phys. Rev. B **74**, 245403 (2006). [CrossRef]

6. C. Wang, C. Du, Y. Lv, and X. Luo, “Surface electromagnetic wave excitation and diffraction by subwavelength slit with periodically patterned metallic grooves,” Opt. Express **14**, 5671 (2006). [CrossRef] [PubMed]

12. L.-B. Yu, D.-Z. Lin, Y.-C. Chen, Y.-C. Chang, K.-T. Huang, J.-W. Liaw, J.-T. Yeh, J.-M. Liu, C.-S. Yeh, and C.-K. Lee, “Physical origin of directional beaming emitted from a subwavelength slit,” Phys. Rev. B **71**, 041405 (2005). [CrossRef]

## 2. Beaming structure and simulation results

*N*grooves on the exit side of a metallic film. The width of the grooves and slit is

*a*, the period is

*d*, and the groove depth is

*h*. The normal incident light form left is TM polarized with magnetic field

*H*for the coordinate depicted in Fig. 1.

_{y}1. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martín-Moreno, F. J. García-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science **297**, 820 (2002). [CrossRef] [PubMed]

*λ*) and period (

*d*) of grooves mainly determine the beaming direction or the peak position in the angular spectra. This can be clearly seen from the angular spectra of diffracted light as a function of

*λ*/

*d*as plotted in Fig. 2. The geometrical parameters of slit-grooves structure are

*a*=0.110

*λ*,

*h*=0.145

*λ*,

*N*=10. The normalized angular spectrum is defined as

*I*(

*θ*)=|

*H*|

_{slit-grooves}^{2}/|

*H*

_{no-grooves}|

^{2}, where

*H*(

*θ*) denotes the diffracted angular spectra for the structure depicted in Fig. 1. On the basis of mode expansion method [2

2. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. **90**, 167401 (2003). [CrossRef] [PubMed]

*d*with fixed

*λ*.

*λ*/

*d*, the intersection of the two bright lines forms highlight spots. The spots indicate that the radiation directions of diffraction orders from both sides coincide to each other. The highlight spots have large intensity for the constructive combination of energy from both diffraction orders, while they also display small divergence angle for the doubled radiation size. For instance, the highlight spot located in the vicinity of

*λ*/

*d*=1, represents the well known one directional beaming at 0°, which is superimposed by the +1 and -1 order from the two sides of the central slit. Similarly, multiple highlight spots can be observed in the intersections of the two diffraction orders, indicating the beaming effect in multiple directions.

*d*for fixed

*λ*. The multiple directional beaming effect with coinciding diffraction orders occurs with some specific

*λ*/

*d*positions, which are listed in table 1 with beaming directions ranging from one to six. Large period yields much more beaming directions, as can be well understood from diffraction behaviors of gratings. Figure 3 and Fig. 4 present the angular spectra and space distribution of light behind the exit plane for each parameter listed in table 1. Obvious beaming phenomena in multiple directions can be observed both in the angular and space distributions of light. The beaming peaks display high contrast with the peak intensity that is about 5 to 20 times higher than that of radiation of single slit. The beaming light’s divergence is also very small (the full width at half maximum (FWHM) is usually less than 3°) due to the increased emission size. Around the diffraction peaks is the radiation background with the averaged normalized intensity of about 1 and some slight oscillations resulting from the interference effect, which is obviously caused by the directly emitted light from the central slit. Another interesting phenomenon is that the intensity of each beaming direction is nearly uniform, no matter how many beaming directions are obtained. This characteristic makes it convenient for beam shaping system.

## 3. Discussions

### 3.1 Determination of parameters for multiple beaming

*k*=

_{xn}*k*+2

_{sp}*nπ*/

*d*, here

*k*is the wave vector of the spoof SP. In fact,

_{sp}*k*is always slightly larger than

_{sp}*k*. To make the analysis convenient,

_{0}*k*is assumed to be equal to

_{sp}*k*. So we can see that When

_{0}*λ*/

*d*>2, i.e. |

*k*|>|

_{xn}*k*|, the spoof SP displays evanescent properties in the direction normal to the grating surface and the corresponding EM field is confined to the grating surface. Therefore, we can not observe any beaming effect in this case. Otherwise, some plane wave components become propagating state and light can be radiated into the free space in the specified direction

_{0}*k*sin

_{0}*θ*=

*k*-2

_{sp}*nπ*/

*d*. For example, if 1<

*λ*/

*d*<2, only one

*k*is localized in the region [-

_{xn}*k*,

_{0}*k*]. But it can be noted that light is diffracted in two symmetrical directions due to the surface plasmon diffraction occurred at the two sides of the central slit. For

_{0}*λ*/

*d*<1, there exist more than one

*k*for light diffraction in the free space, implying that the spoof SP wave could be coupled with the plane waves in the multiple directions.

_{xn}*λ*/

*d*as shown in Fig. 2. But we can give an explicit approximation of these parameters from the fact that multiple directional beaming arises from the superimposing of the diffraction orders. Based on grating equation, the diffraction of spoof SP on the two sides of the central slit can be expressed as

*n*and

*m*are the diffraction order of the upper sides and the lower sides of the slits respectively. According to Eq. (1), when

*k*≈

_{sp}*k*, multiple directional beaming occurs with

_{0}*λ*/

*d*=1, 2/3, 2/4, 2/5, 2/6, 2/7…. Also we can obtain the approximated beaming directions and the corresponding diffraction orders for different sets of parameters.

*k*>

_{sp}*k*. Consequently, the radiation angular is slightly larger than the prediction of the grating equation. It can also be seen that the approximation works well for small

_{0}*λ*/

*d*and the error increases for larger

*d*. We believe that this point can be explained qualitatively in terms of the dispersion relation of spoof SP on 1D grooves corrugated metallic surface, which can be expressed as

*d*≪

*λ*[13]. Considering

*k*≈

_{sp}*k*, we get

_{0}*d*≪

*λ*required for Eq. (3) is not fully satisfied in our simulations.

### 3.2 Beaming intensity and efficiency analysis

*θ*=+/-36.87° and +/-11.54°. The former is generated by the +(-)1st diffraction order from the upper (lower) side and the -(+)4th diffraction order from the lower (upper) side. While the latter originates from the +(-)2nd diffraction order of the upper (lower) side and the -(+)3rd diffraction order of the lower (upper) side. It can be found that the sum of the diffraction orders |

*n*|+|

*m*| for each beaming direction is fixed. So the variation of beaming intensity at different directions can be compensated by combining higher and lower diffraction orders.

*θ*. Taking the four beaming example as well, the FWHM of beaming divergence is 1.8° and 1.44° for the beams in

*θ*=±36.87° and

*θ*=±11.54° respectively. This can be easily understood from the relationship between

*k*and

_{x}*θ*, which yields Δ

*θ*=

*λ*(2

*Nd*cos

*θ*) illustrating the FWHM increases with large

*θ*or

*λ*/

*d*. But the FWHM for beams of four beaming example derived from the equation is 1.46° and 1.20°, slightly smaller than the simulated results. This is because the derivation does not take into account that the propagation of the spoof SP decreases with the number of beams presented, thus reducing the effective

*N*values in the equation. This can also be further confirmed by the FWHM of beaming at the same value of

*θ*but with different number of beaming directions. For instance, the simulated FWHM of beaming divergence at

*θ*=0 for one, three and five directions is 3.82°, 1.78° and 1.22° respectively, and the values derived from the equation are 3.09°, 1.48° and 0.97°.

## 4. Conclusion

## Acknowledgments

## References and links

1. | H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martín-Moreno, F. J. García-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science |

2. | L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. |

3. | F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. |

4. | M. J. Lockyear, A. P. Hibbins, J. Roy Sambles, and C. R. Lawrence, “Surface-topography-induced enhanced transmission and directivity of microwave radiation through a subwavelength circular metal aperture.” Appl. Phys. Lett. |

5. | C. Wang, C. Du, and X. Luo, “Refining the model of light diffraction from a subwavelength slit surrounded by grooves on a metallic film,” Phys. Rev. B |

6. | C. Wang, C. Du, Y. Lv, and X. Luo, “Surface electromagnetic wave excitation and diffraction by subwavelength slit with periodically patterned metallic grooves,” Opt. Express |

7. | F. J. García-Vidal, L. Martín-Moreno, H. J. Lezec, and T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. |

8. | H. Shi, C. Du, and X. Luo, “Focal length modulation based on a metallic slit surrounded with grooves in curved depths,” Appl. Phys. Lett. |

9. | C. Huang, C. Du, and X. Luo, “A waveguide slit array antenna fabricated with subwavelength periodic grooves,” Appl. Phys. Lett. |

10. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Science |

11. | O. T. A. Janssen, H. P. Urbach, and G. W. Hooft, “Giant optical transmission of a subwavelength slit optimized using the magnetic field phase,” Phys. Rev. Lett. |

12. | L.-B. Yu, D.-Z. Lin, Y.-C. Chen, Y.-C. Chang, K.-T. Huang, J.-W. Liaw, J.-T. Yeh, J.-M. Liu, C.-S. Yeh, and C.-K. Lee, “Physical origin of directional beaming emitted from a subwavelength slit,” Phys. Rev. B |

13. | F. J. García-Vidal, L. Martín-Moreno, and J. B. Pendry, “Surface with holes in them: new plasmatic metamaterials,” J. Opt. A 7, S91 (2005). |

**OCIS Codes**

(230.7380) Optical devices : Waveguides, channeled

(240.0310) Optics at surfaces : Thin films

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: February 4, 2008

Revised Manuscript: March 13, 2008

Manuscript Accepted: March 13, 2008

Published: March 18, 2008

**Citation**

Yugang Liu, Haofei Shi, Changtao Wang, Chunlei Du, and Xiangang Luo, "Multiple directional beaming effect of metallic subwavelength slit surrounded by periodically corrugated grooves," Opt. Express **16**, 4487-4493 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-4487

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### References

- H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martín-Moreno, F. J. García-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820 (2002). [CrossRef] [PubMed]
- L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401 (2003). [CrossRef] [PubMed]
- F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, "Multiple paths to enhance optical transmission through a single subwavelength slit," Phys. Rev. Lett. 90, 213901 (2003). [CrossRef] [PubMed]
- M. J. Lockyear, A. P. Hibbins, J. Roy Sambles and C. R. Lawrence, "Surface-topography-induced enhanced transmission and directivity of microwave radiation through a subwavelength circular metal aperture." Appl. Phys. Lett. 84, 2040 (2004). [CrossRef]
- C. Wang, C. Du, and X. Luo, "Refining the model of light diffraction from a subwavelength slit surrounded by grooves on a metallic film," Phys. Rev. B 74, 245403 (2006). [CrossRef]
- C. Wang, C. Du, Y. Lv, and X. Luo, "Surface electromagnetic wave excitation and diffraction by subwavelength slit with periodically patterned metallic grooves," Opt. Express 14, 5671 (2006). [CrossRef] [PubMed]
- F. J. García-Vidal, L. Martín-Moreno, H. J. Lezec, and T. W. Ebbesen, "Focusing light with a single subwavelength aperture flanked by surface corrugations," Appl. Phys. Lett. 83, 4500 (2003). [CrossRef]
- H. Shi, C. Du, and X. Luo, "Focal length modulation based on a metallic slit surrounded with grooves in curved depths," Appl. Phys. Lett. 91, 093111 (2007). [CrossRef]
- C. Huang, C. Du, and X. Luo, "A waveguide slit array antenna fabricated with subwavelength periodic grooves," Appl. Phys. Lett. 91, 143512 (2007). [CrossRef]
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Science 424, 824 (2003).
- O. T. A. Janssen, H. P. Urbach, and G. W. Hooft, "Giant optical transmission of a subwavelength slit optimized using the magnetic field phase," Phys. Rev. Lett. 99, 043902 (2007) [CrossRef] [PubMed]
- L.-B. Yu, D.-Z. Lin, Y.-C. Chen, Y.-C. Chang, K.-T. Huang, J.-W. Liaw, J.-T. Yeh, J.-M. Liu, C.-S. Yeh, and C.-K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405 (2005). [CrossRef]
- F. J. García-Vidal, L. Martín-Moreno, and J. B. Pendry, "Surface with holes in them: new plasmatic metamaterials," J. Opt. A 7, S91 (2005).

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