## Slot antenna as a bound charge oscillator |

Optics Express, Vol. 20, Issue 6, pp. 6521-6526 (2012)

http://dx.doi.org/10.1364/OE.20.006521

Acrobat PDF (1146 KB)

### Abstract

We study the scattering properties of an optical slot antenna formed from a narrow rectangular hole in a metal film. We show that slot antennas can be modeled as bound charge oscillators mediating resonant light scattering. A simple closed-form expression for the scattering spectrum of a slot antenna is obtained that reveals the nature of a bound charge oscillator and also the effect of a substrate. We find that the spectral width of scattering resonance is dominated by a radiative damping caused by the Abraham-Lorentz force acting on a bound charge. The bound charge oscillator model provides not only an intuitive physical picture for the scattering of an optical slot antenna but also reasonable numerical agreements with rigorous calculations using the finite-difference time-domain method.

© 2012 OSA

## 1. Introduction

1. H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. **66**, 163–182 (1944). [CrossRef]

3. A. Roberts, “Electromagnetic theory of diffraction by a circular aperture in a thick, perfectly conducting screen,” J. Opt. Soc. Am. A **4**, 1970–1983 (1987) [CrossRef]

4. For a review, see for example, F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength
apertures,” Rev. Mod. Phys. **82**, 729–787
(2010). [CrossRef]

5. K. G. Lee and Q-H. Park, “Coupling of surface plasmon polaritions and light in metallic nanoslits,” Phys. Rev. Lett. **95**, 103902. (2005). [CrossRef] [PubMed]

6. T.-H. Park, N. Mirin, J. B. Lassiter, C. L. Nehl, N. J. Halas, and P. Nordlander, “Optical properties of a nanosized hole in a thin metallic film,” ACS Nano **2**, 25–32. (2008). [CrossRef]

7. F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of light through a single hole,” Phys. Rev. Lett. **95**, 103901 (2005). [CrossRef] [PubMed]

7. F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of light through a single hole,” Phys. Rev. Lett. **95**, 103901 (2005). [CrossRef] [PubMed]

8. M. A. Seo, A. J. L. Adam, J. H. Kang, J. W. Lee, K. J. Ahn, Q-H. Park, P. C. M. Planken, and D. S. Kim, “Near field imaging of terahertz focusing onto rectangular apertures,” Opt. Express **16**, 20484–20489 (2008). [CrossRef] [PubMed]

11. J. H. Kang, J.-H. Choe, D. S. Kim, and Q-H. Park, “Substrate effect on aperture resonances in a thin metal film,” Opt. Express **17**, 15652–15658 (2009). [CrossRef] [PubMed]

## 2. Resonance in slot antenna

*a*×

*b*(

*b*≪

*a*) in a metal film patched on a dielectric substrate of refractive index

*n*as shown in Fig. 1(a). When light is incident upon a narrow slot with polarization normal to the long side of rectangle, transmission becomes strongly enhanced due to the capacitative coupling of light to the metallic slit structure [12

12. M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics **3**, 152–156 (2009). [CrossRef]

7. F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of light through a single hole,” Phys. Rev. Lett. **95**, 103901 (2005). [CrossRef] [PubMed]

11. J. H. Kang, J.-H. Choe, D. S. Kim, and Q-H. Park, “Substrate effect on aperture resonances in a thin metal film,” Opt. Express **17**, 15652–15658 (2009). [CrossRef] [PubMed]

11. J. H. Kang, J.-H. Choe, D. S. Kim, and Q-H. Park, “Substrate effect on aperture resonances in a thin metal film,” Opt. Express **17**, 15652–15658 (2009). [CrossRef] [PubMed]

*W*(

_{m}*m*= air, substrate and

*ε*

_{air}= 1,

*ε*

_{subs}=

*n*

^{2}) is Though

*W*has a closed-form expression, due to the complexity of integration only numerical studies have been made previously [11

_{m}**17**, 15652–15658 (2009). [CrossRef] [PubMed]

*W*. The real part of

_{m}*W*results from the integration over the domain

_{m}*b/λ*. This approximation is valid for the narrow slot case (

*b*≪

*λ*). The imaginary part of

*W*, with the integral domain

_{m}*k*-integration while the remaining d

_{y}*k*-integration can be carried out through the contour integral with a contour as in Fig. 1(b) (detailed derivations are suppressed). The final result is

_{x}*G*are cosine-integral, sine-integral and the Meijer G function respectively. To help understand the spectral behavior of

*W*, we also keep only the leading order term in

_{m}*b/λ*for the imaginary part of

*W*. For the real part of

_{m}*W*, we make an additional approximation by keeping the leading order term in

_{m}*a/λ*which is valid since

*b*≪

*a*<

*λ*. This brings

*W*into a simple form where

_{m}*γ*(≈ 0.577) is the Euler-Gamma constant. It was noted that resonance occurs around the zero of the sum of imaginary part of

*W*such that [11

**17**, 15652–15658 (2009). [CrossRef] [PubMed]

*W*in Eq. (4) is that now we can find the resonance condition explicitly in terms of geometrical parameters. Solving Eq. (5) for

_{m}*λ*

_{res}, we obtain the resonance condition, Figure 2(a) shows that the resonance condition in Eq. (6) agrees nicely with rigorous numerical results of the coupled-mode theory. In particular, resonance wavelength is proportional to the long side of the slot and depends on the refractive index of the substrate. If the substrate is absent (

*n*= 1), resonant wavelength becomes two times of the long side of the slot consistent with the resonance condition in rectangular waveguide [9, 10].

*n*of substrate while the dependence is much weaker for the quality factor. As expected, the quality factor increases as the slot becomes narrower. In Fig. 2(c), Lorentzian shape spectrums calculated for four distinct cases of

*n*using three different methods are compared. Dashed lines representing results of the coupled-mode theory show that resonance peaks are slightly blue-shifted compared to the rigorous numerical results using the Finite Difference Time Domain(FDTD) method. This is caused by the single-mode approximation adopted in the coupled-mode theory calculation which does not apply well for thin layer holes. Solid lines representing closed form of

## 3. Bound Charge Oscillator

*n*= 1). In terms of angular frequency

*ω*(= 2

*πc/λ*), the total scattering cross section of free standing slot antenna is obtained from Eq. (1) and Eq. (4) such that where The logarithmic factor Δ(

*ω*

^{2}) is a slowly varying function of

*ω*if

*ω*≠

*ω*

_{0}. The quasi-Lorentzian total cross section in Eq. (7) may be compared to the case of light scattering by a bound charge oscillator. If radiative effects are small, the total cross section of scattering of radiation by a bound charge oscillator can be written [9] where Γ and Γ

*are the radiative decay and total decay constants. Note that the total cross section in Eq. (7) has a similar Lorentzian shape as in Eq. (9) but with a different line broadening factor. The*

_{t}*ω*

^{6}-dependence of the line broadening term

*T*

^{2}

*ω*

^{6}comes from the fact that the energy loss of a slot antenna is due to the radiation damping. If radiation is the dominant mechanism in the energy loss of the bound charge oscillator, the motion of bound charge driven by external light field is governed by the Abraham-Lorentz equation of motion [9], where the first two terms on the r.h.s. represent the restoring and external forces respectively. The last term is a radiative reaction force responsible for the radiative energy loss. From the steady-state solution of the Abraham-Lorentz equation we may calculate the radiation field and subsequently the total scattering cross section, where

*τ*(=

*μ*

_{0}

*e*

^{2}/

*6πmc*) is the characteristic time of Abraham-Lorentz force. Comparing Eqs. (7) and (12) and associating

*τ*with

*T*/Δ, we find that the total scattering cross section of a slot antenna is indeed that of a bound charge oscillator with a factor 1/2 because we observe antenna radiation only from one side. The characteristic time

*τ*of electron is of the order of time taken for light to travel across nucleus (10

^{−24}second), which is too small to have any significant change on non-relativistic motions. On the contrary, radiation is the dominant damping mechanism of a slot antenna so that a slot antenna is an efficient low

*Q*radiator as in Fig. 2(b). The characteristic time (

*T*/Δ) of a slot antenna depends on

*ω*but only weakly away from resonance. It is of the order of time taken for light to travel the short side of rectangle and significant radiative loss arises during time

*T*/Δ.

*y*-polarized for slot antenna and

*x*-polarized for bound charge oscillator), thus the direction of charge oscillations are also orthogonal. They result in nearly the same radiation pattern despite the opposite polarization of incident light according to the Babinet’s principle. This strengthens our interpretation of slot antenna as a bound charge oscillator.

## 4. Discussion

## Acknowledgments

## References and links

1. | H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. |

2. | C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Philips Res. Rep. |

3. | A. Roberts, “Electromagnetic theory of diffraction by a circular aperture in a thick, perfectly conducting screen,” J. Opt. Soc. Am. A |

4. | For a review, see for example, F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength
apertures,” Rev. Mod. Phys. |

5. | K. G. Lee and Q-H. Park, “Coupling of surface plasmon polaritions and light in metallic nanoslits,” Phys. Rev. Lett. |

6. | T.-H. Park, N. Mirin, J. B. Lassiter, C. L. Nehl, N. J. Halas, and P. Nordlander, “Optical properties of a nanosized hole in a thin metallic film,” ACS Nano |

7. | F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of light through a single hole,” Phys. Rev. Lett. |

8. | M. A. Seo, A. J. L. Adam, J. H. Kang, J. W. Lee, K. J. Ahn, Q-H. Park, P. C. M. Planken, and D. S. Kim, “Near field imaging of terahertz focusing onto rectangular apertures,” Opt. Express |

9. | J. D. Jackson, |

10. | J. D. Kraus and R. J. Marhefka, |

11. | J. H. Kang, J.-H. Choe, D. S. Kim, and Q-H. Park, “Substrate effect on aperture resonances in a thin metal film,” Opt. Express |

12. | M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics |

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

**OCIS Codes**

(050.1220) Diffraction and gratings : Apertures

(050.1960) Diffraction and gratings : Diffraction theory

(050.5745) Diffraction and gratings : Resonance domain

(290.5825) Scattering : Scattering theory

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: August 22, 2011

Revised Manuscript: October 17, 2011

Manuscript Accepted: February 27, 2012

Published: March 6, 2012

**Citation**

Jong-Ho Choe, Ji-Hun Kang, Dai-Sik Kim, and Q-Han Park, "Slot antenna as a bound charge oscillator," Opt. Express **20**, 6521-6526 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-6-6521

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### References

- H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev.66, 163–182 (1944). [CrossRef]
- C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Philips Res. Rep.5, 321–332 (1950).
- A. Roberts, “Electromagnetic theory of diffraction by a circular aperture in a thick, perfectly conducting screen,” J. Opt. Soc. Am. A4, 1970–1983 (1987) [CrossRef]
- For a review, see for example, F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys.82, 729–787 (2010). [CrossRef]
- K. G. Lee and Q-H. Park, “Coupling of surface plasmon polaritions and light in metallic nanoslits,” Phys. Rev. Lett.95, 103902. (2005). [CrossRef] [PubMed]
- T.-H. Park, N. Mirin, J. B. Lassiter, C. L. Nehl, N. J. Halas, and P. Nordlander, “Optical properties of a nanosized hole in a thin metallic film,” ACS Nano2, 25–32. (2008). [CrossRef]
- F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of light through a single hole,” Phys. Rev. Lett.95, 103901 (2005). [CrossRef] [PubMed]
- M. A. Seo, A. J. L. Adam, J. H. Kang, J. W. Lee, K. J. Ahn, Q-H. Park, P. C. M. Planken, and D. S. Kim, “Near field imaging of terahertz focusing onto rectangular apertures,” Opt. Express16, 20484–20489 (2008). [CrossRef] [PubMed]
- J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2001).
- J. D. Kraus and R. J. Marhefka, Antennas For All Applications, 3rd ed. (McGraw-Hill, 2002).
- J. H. Kang, J.-H. Choe, D. S. Kim, and Q-H. Park, “Substrate effect on aperture resonances in a thin metal film,” Opt. Express17, 15652–15658 (2009). [CrossRef] [PubMed]
- M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics3, 152–156 (2009). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

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