## Rigorous solution for optical diffraction of a sub-wavelength real-metal slit |

Optics Express, Vol. 20, Issue 3, pp. 2149-2162 (2012)

http://dx.doi.org/10.1364/OE.20.002149

Acrobat PDF (1140 KB)

### Abstract

We present a rigorous closed-form solution of the Sommerfeld integral for the optical scattering of a metal sub-wavelength slit. The two-dimensional (2D) field solution consists of the Surface Plasmon Polariton (SPP) mode at the metal surface and the 2D scattered field, which is the cylindrical harmonic of first order emitted by the electrical dipole and convolved with the 1D transient SPP along the interface. The creeping wave or quasi-cylindrical wave detected in the previous experiment is not an extra evanescent surface wave, but is the asymptotic behavior of the 2D scattered field at the proximity of the slit. Furthermore, our solution predicts a strong resonant enhancement of the scattered field at the proximity of the slit, depending on the materials and wavelength.

© 2012 OSA

## 1. Introduction

23. H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature **452**(7188), 728–731 (2008). [CrossRef] [PubMed]

24. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature **391**(6668), 667–669 (1998). [CrossRef]

13. P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. **2**(8), 551–556 (2006). [CrossRef]

19. B. Ung and Y. Sheng, “Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals revisited,” Opt. Express **16**(12), 9073–9086 (2008). [CrossRef] [PubMed]

## 2. Physical model and solution

*ε*perforated by a sub-wavelength slit, which is infinitely long in the

_{m}*z*-axis. Above the metal surface is a semi-infinite dielectric medium of permittivity

*ε*. The problem is two-dimensional (2D) in the

_{d}*x-y*space, as shown in Fig. 1 . The incident wave along the

*–y*direction is monochromatic of Transverse Magnetic (TM) polarization with the magnetic field

*H*parallel to the

*z*-axis. Its electric components in the

*x-y*plane induce displacements of the conduction electrons. The discontinuity of the surface at the slit prevents electrical currents from crossing the slit. Hence, the electrical charges accumulate at the two up-edges of the slit, resulting in an oscillating horizontal electric dipole, as have been observed in both experiments [25

25. L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. **5**(7), 1399–1402 (2005). [CrossRef] [PubMed]

8. B. Ung and Y. Sheng, “Interference of surface waves in a metallic nanoslit,” Opt. Express **15**(3), 1182–1190 (2007). [CrossRef] [PubMed]

*z*axis and replaces the horizontal electric dipole [13

13. P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. **2**(8), 551–556 (2006). [CrossRef]

21. Y. Gravel and Y. Sheng, “Rigorous solution for the transient Surface Plasmon Polariton launched by subwavelength slit scattering,” Opt. Express **16**(26), 21903–21913 (2008). [CrossRef] [PubMed]

13. P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. **2**(8), 551–556 (2006). [CrossRef]

*-iωt*) and of TM polarization, the Maxwell’s equations in the Fourier space lead to the Helmholtz equation in the spatial frequency domain as [21

21. Y. Gravel and Y. Sheng, “Rigorous solution for the transient Surface Plasmon Polariton launched by subwavelength slit scattering,” Opt. Express **16**(26), 21903–21913 (2008). [CrossRef] [PubMed]

*x*,

*k*is the

_{x}*x*-component of the wavevector,

*γ*

^{2}=

*ε*

_{d}k_{0}

^{2}-

*k*

_{x}^{2}with

*k*

_{0}=

*ω*/

*c*

_{0}and

*c*

_{0}is the speed of light in vacuum. After solving Eq. (1) in the dielectric medium and applying the boundary conditions at the metal/dielectric interface, the inverse Fourier transform leads to a solution of the field in the

*x-y*space in the form of the Sommerfeld integral:which has two branch-cut points in the complex plane at

*k*=

_{x}*ε*

_{m}^{1/2}

*k*

_{0}and

*k*=

_{x}*ε*

_{d}^{1/2}

*k*

_{0}, and a pole at the location determined by setting the denominator of the integrant in Eq. (2) to zero, that yieldswhere

*k*is the SPP propagation constant. Sommerfeld and others have computed this type of integrals at the interface with

_{sp}18. R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: Some early and late 20^{th}-century controversies,” IEEE Antennas Propagat. Mag. **46**(2), 64–79 (2004). [CrossRef]

18. R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: Some early and late 20^{th}-century controversies,” IEEE Antennas Propagat. Mag. **46**(2), 64–79 (2004). [CrossRef]

### 2.1 Scattered field in the 2D space

21. Y. Gravel and Y. Sheng, “Rigorous solution for the transient Surface Plasmon Polariton launched by subwavelength slit scattering,” Opt. Express **16**(26), 21903–21913 (2008). [CrossRef] [PubMed]

**16**(26), 21903–21913 (2008). [CrossRef] [PubMed]

22. G. Lévêque, O. J. F. Martin, and J. Weiner, “Transient behaviour of surface plasmon polaritons scattered at a sub-wavelength groove,” Phys. Rev. B **76**(15), 155418 (2007). [CrossRef]

*ik*) with the complex-valued wavenumber

_{sp}x*k*, modulated by a complex-valued envelope described by the exponential integral of complex argument

_{sp}*E*

_{1}(

*ik*), being the case

_{sp}x*r,θ*) are the polar coordinates and

*H*(

_{b}*x,y*) represents a cylindrical harmonic of first order [27], which is a 2D solution of the scalar Helmholtz equation with a dipole source in the cylindrical coordinate system. Recall that the zero order cylindrical harmonic is the basic cylindrical wave

*h*depends on

_{sp}*k*

_{0},

*ε*and

_{d}*ε*as

_{m}*clearly represents a conventional SPP mode propagating along the interface with the wave-number*H s p ( x , y )

*k*and decaying exponentially away from

_{sp}*h*depends on

_{s}*k*

_{0},

*ε*and

_{d}*ε*as

_{m}*x*-axis of the cylindrical harmonic of first order,

**16**(26), 21903–21913 (2008). [CrossRef] [PubMed]

22. G. Lévêque, O. J. F. Martin, and J. Weiner, “Transient behaviour of surface plasmon polaritons scattered at a sub-wavelength groove,” Phys. Rev. B **76**(15), 155418 (2007). [CrossRef]

28. L. Chen, J. T. Robinson, and M. Lipson, “Role of radiation and surface plasmon polaritons in the optical interactions between a nano-slit and a nano-groove on a metal surface,” Opt. Express **14**(26), 12629–12636 (2006). [CrossRef] [PubMed]

### 2.2 Field at the interface

*y*= 0 and

*iγy*)|

_{y = 0}= 1, so that the Fourier transform

*H*(

_{b}*x,*0) is a delta function and its convolution with the transient SPP gives back the transient SPP. This is consistent with previous rigorous solution of the field at the interface [21

**16**(26), 21903–21913 (2008). [CrossRef] [PubMed]

*k*, is complex valued. Its phase, amplitude, real and imaginary parts are shown in Fig. 2 . The latters can be calculated respectively by the cosine and sine integral. Our calculation shows that the exponential integral of complex argument representing the envelope of the transient SPP has a negative phase, which cancels the positive phase of the SPP propagation, so that the phase of the transient SPP tends to a negative constant of -

_{sp}*π*/2 within the transient distance range 0 <

*x*<

*λ*from the origin.

**16**(26), 21903–21913 (2008). [CrossRef] [PubMed]

*k*, a continuous sum of all the homogeneous and evanescent plane waves propagating along the

_{sp}*x*-direction with the complex amplitude 1/(

*k*-

*k*). In this summation the evanescent components are dominant as can be seen in the spectrum of the transient SPP shown in Fig. 3 , where the evanescent components with the frequencies higher than

_{sp}*k*

_{0}have greater amplitudes than that have the propagating components with the frequencies lower than

*k*

_{0}, and the spectrum is asymmetric with respect to

*k*/

*k*

_{0}= 1, so that the scattered field at the surface is damped at a greater rate than the typically SPP mode with a propagation constant

*k*greater than

_{sp}*k*

_{0}.

*x*) and then 1/

*x*

^{1/2}, corresponding to the field naturally radiated by a punctual source. Then, the transient SPP along the surface suffers from an additional drop in amplitude due to the loss of energy through the excitation of the long-range SPP mode.

### 2.3 Asymptotic behavior near the source

### 2.4 Surface plasmon resonance

*h*, which depends on the permittivities of the metal and dielectric, and the incident wavelength. If Re{

_{s}*ε*}≈-

_{m}*ε*and Im{

_{d}*ε*} is small, then from Eq. (10), the amplitude of

_{m}*h*is strongly enhanced. Figure 5 shows the amplitude of

_{s}*h*as a function of the wavelength for Ag and Au slit, with the index of refraction in the dielectric medium over the metal slit

_{s}*n*= 1, 1.5, 2, 2.5 and 3. The dielectric function of the metal is computed by Lorentz-Drude dispersive model. Strong enhancements, which are more than 3000 times for Ag and 1000 times for Au are clearly shown at the resonant wavelengths for both Ag and Au slits in the visible spectrum. In the case of Ag, the strongest enhancement is observed for a dielectric refractive index of

*n*= 3 at λ = 528

*nm*, corresponding to a complex permittivity of

*ε*= −8.9594 + 0.7958i. In the case of Au, the strongest enhancement has less magnitude than for Ag, and is observed for a dielectric refractive index of

_{m}*n*= 3 at λ = 623

*nm*corresponding to a complex permittivity of

*ε*= −9.2504 + 1.9880i. Hence, the enhancement frequencies and magnitudes clearly depend on the metal considered. Furthermore, for a given metallic surface, the resonant wavelength is red shifted and the resonance is stronger, with the increase of the dielectric refractive index.

_{m}### 2.5 Field far from the source

*ik*)

_{sp}x*E*

_{1}(

*ik*) may be approximated by a delta function and the scattered field expressed in Eq. (9) is simply the first order cylindrical harmonic with the propagation constant

_{sp}x*k*:which differs from the basic cylindrical wave by the presence of the factor sin(

_{d}*θ*), corresponding to a diffraction shadow for small

*θ*. Along the

*y-*axis, the field drops as 1/

*y*

^{1/2}as the typical behavior of the Hankel function, but not as an evanescent surface wave. In the diffraction shadow region close to the interface where

*x*>>

*y*, the field takes an approximate form:

*x*

^{1/2},

*x*

^{3/2}. This result is in agreement with the work of Nikitin

*et al.*[20

20. A. Y. Nikitin, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martín-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. **11**(12), 123020 (2009). [CrossRef]

*et al.*[13

**2**(8), 551–556 (2006). [CrossRef]

*x*

^{1/2}away from the slit. When the substrate is real metal, the scattered field behaves as a flattened cylindrical wave only for the distances

*x*<

*λ*/10 as shown in Fig. 4a, and is damped more rapidly as the wave propagates along the interface. Clearly, this behavior is different from the perfect conductor case and can therefore be linked to the real metal and to the pumping of energy to the SPP mode. This result is in agreement with the numerical evaluation by Lalanne

*et al.*[13

**2**(8), 551–556 (2006). [CrossRef]

14. P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep. **64**(10), 453–469 (2009). [CrossRef]

*et al.*[19

19. B. Ung and Y. Sheng, “Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals revisited,” Opt. Express **16**(12), 9073–9086 (2008). [CrossRef] [PubMed]

## 3. Numerical solution

*μm*and time step was 0.02238

*fs*, at the wavelength λ = 500

*nm.*Results for the perfect conductor, which does not support the SPP mode, show no surface wave and the scattered field in the half space above the surface takes the form of the zero order cylindrical harmonic. As discussed in Section 2.1, this result is expected because the zero order cylindrical harmonic corresponds to the 2D Green’s function of Helmholtz equation. The scattered field behaves as the Hankel function

*y*-axis and sin

*θ*= 1 according to Eq. (6), the field behaves as

*y*

^{1/2}for

*y*larger than a wavelength, in agreement with our results presented in Section 2.5, and similarly to that in perfect conductor case, which drops as

*y*) and 1/

*y,*respectively. In the diffraction shadow, the SPP mode is dominant along the interface. The mismatch between the wavelength of the scattered field in the dielectric,

*k*

_{0}, and that of the SPP,

*k*, can be seen clearly from the distortion of the wave fronts, again in agreement with our rigorous closed solution.

_{sp}## 4. Conclusion

23. H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature **452**(7188), 728–731 (2008). [CrossRef] [PubMed]

## Appendix A

*k*, so that

*K*

_{1}is the modified Bessel function of the second kind. Hence, Eq. (A3) becomes

*K*

_{1}function with an imaginary argument can be expressed in term of the Hankel function:where we replaced

*k*

_{0}by

*k*. The negative sign has been chosen in order to retrieve the Hankel function of the first kind representing waves out-coming from the source. We then obtain the result presented in Eq. (9).

_{d}## Appendix B

*H*and the following constants have been defined:

_{a}## Appendix C

*x’*in the integrant are even, except sin(

*k*), which is odd. We also separate the Hankel function in its Bessel components:

_{sp}x’23. H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature **452**(7188), 728–731 (2008). [CrossRef] [PubMed]

## Appendix D

*H*(

_{s}*x,y*) in the form shown in Eq. (B8):

*z*, the Hankel function can be approximated aswhich is valid for 0<

*z*<<2

^{1/2}and

*z*=

*k*[(

_{d}*x-x’*)

^{2}+

*y*

^{2}]

^{1/2}where

*x’*is smaller than a wavelength for

*E*

_{1}(

*ik*)≠0. The exponential integral can be expressed aswhere

_{sp}x*γ*is the Euler constant. Hence, in the region with the distances to the slit smaller than the wavelength, Eq. (D1) can be approximated as

*x’*=±

*iy*. Closing the contour with a half-circular path in the upper half space encloses the positive pole. The integral is then computed with Cauchy’s theorem trough evaluation of the residue around the pole

*x’*=

*iy*, resulting in the following solution satisfying the radiation condition:which according to Eq. (D3) can be expressed in terms of a new exponential integral function:

## References and links

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

2. | S. A. Maier, |

3. | T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature |

4. | L. Matrin-Moreno, F. J. Gracia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through sub-wavelength hole arrays,” Phys. Rev. Lett. |

5. | J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science |

6. | Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. |

7. | H. J. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express |

8. | B. Ung and Y. Sheng, “Interference of surface waves in a metallic nanoslit,” Opt. Express |

9. | M. W. Kowarz, “Homogeneous and evanescent contributions in scalar near-field diffraction,” Appl. Opt. |

10. | H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. Hooft, D. Lenstra, and E. R. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. |

11. | G. Gay, O. Alloschery, B. Viaris De Lesegno, C. O'Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nanostructured surfaces and the composite diffracted evanescent wave model,” Nat. Phys. |

12. | G. Gay, O. Alloschery, J. Weiner, H. J. Lezec, C. O’Dwyer, M. Sukharev, and T. Seideman, “Surface quality and surface waves on subwavelength-structured silver films,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

13. | P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. |

14. | P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep. |

15. | A. N. Sommerfeld, “Propagation of waves in wireless telegraphy,” Ann. Phys. (Leipzig) |

16. | J. Zenneck, “Propagation of plane EM waves along a plane conducting surface,” Ann. Phys. (Leipzig) |

17. | K. A. Norton, “The propagation of radio waves over the surface of the earth and in the upper atmosphere,” Proc. IRE |

18. | R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: Some early and late 20 |

19. | B. Ung and Y. Sheng, “Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals revisited,” Opt. Express |

20. | A. Y. Nikitin, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martín-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. |

21. | Y. Gravel and Y. Sheng, “Rigorous solution for the transient Surface Plasmon Polariton launched by subwavelength slit scattering,” Opt. Express |

22. | G. Lévêque, O. J. F. Martin, and J. Weiner, “Transient behaviour of surface plasmon polaritons scattered at a sub-wavelength groove,” Phys. Rev. B |

23. | H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature |

24. | T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature |

25. | L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. |

26. | R. W. P. King, M. Owens, and T. T. Wu, |

27. | R. F. Harrington, |

28. | L. Chen, J. T. Robinson, and M. Lipson, “Role of radiation and surface plasmon polaritons in the optical interactions between a nano-slit and a nano-groove on a metal surface,” Opt. Express |

29. | I. S. Gradshteyn and I. m. Ryzhik, |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(260.2110) Physical optics : Electromagnetic optics

(260.3910) Physical optics : Metal optics

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: October 28, 2011

Revised Manuscript: December 14, 2011

Manuscript Accepted: December 18, 2011

Published: January 17, 2012

**Citation**

Yann Gravel and Yunlong Sheng, "Rigorous solution for optical diffraction of a sub-wavelength real-metal slit," Opt. Express **20**, 2149-2162 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2149

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### References

- W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424(6950), 824–830 (2003). [CrossRef] [PubMed]
- S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, Berlin, 2007).
- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998). [CrossRef]
- L. Matrin-Moreno, F. J. Gracia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through sub-wavelength hole arrays,” Phys. Rev. Lett.86, 1112–1117 (2001).
- J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305(5685), 847–848 (2004). [CrossRef] [PubMed]
- Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett.88(5), 057403–057407 (2002). [CrossRef] [PubMed]
- H. J. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express12(16), 3629–3651 (2004). [CrossRef] [PubMed]
- B. Ung and Y. Sheng, “Interference of surface waves in a metallic nanoslit,” Opt. Express15(3), 1182–1190 (2007). [CrossRef] [PubMed]
- M. W. Kowarz, “Homogeneous and evanescent contributions in scalar near-field diffraction,” Appl. Opt.34(17), 3055–3063 (1995). [CrossRef] [PubMed]
- H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. Hooft, D. Lenstra, and E. R. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett.94(5), 053901 (2005). [CrossRef] [PubMed]
- G. Gay, O. Alloschery, B. Viaris De Lesegno, C. O'Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nanostructured surfaces and the composite diffracted evanescent wave model,” Nat. Phys.2(4), 262–267 (2006). [CrossRef]
- G. Gay, O. Alloschery, J. Weiner, H. J. Lezec, C. O’Dwyer, M. Sukharev, and T. Seideman, “Surface quality and surface waves on subwavelength-structured silver films,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.75(1), 016612 (2007). [CrossRef] [PubMed]
- P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys.2(8), 551–556 (2006). [CrossRef]
- P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep.64(10), 453–469 (2009). [CrossRef]
- A. N. Sommerfeld, “Propagation of waves in wireless telegraphy,” Ann. Phys. (Leipzig)28, 665–737 (1909).
- J. Zenneck, “Propagation of plane EM waves along a plane conducting surface,” Ann. Phys. (Leipzig)23, 846–866 (1907).
- K. A. Norton, “The propagation of radio waves over the surface of the earth and in the upper atmosphere,” Proc. IRE24(10), 1367–1387 (1936). [CrossRef]
- R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: Some early and late 20th-century controversies,” IEEE Antennas Propagat. Mag.46(2), 64–79 (2004). [CrossRef]
- B. Ung and Y. Sheng, “Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals revisited,” Opt. Express16(12), 9073–9086 (2008). [CrossRef] [PubMed]
- A. Y. Nikitin, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martín-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys.11(12), 123020 (2009). [CrossRef]
- Y. Gravel and Y. Sheng, “Rigorous solution for the transient Surface Plasmon Polariton launched by subwavelength slit scattering,” Opt. Express16(26), 21903–21913 (2008). [CrossRef] [PubMed]
- G. Lévêque, O. J. F. Martin, and J. Weiner, “Transient behaviour of surface plasmon polaritons scattered at a sub-wavelength groove,” Phys. Rev. B76(15), 155418 (2007). [CrossRef]
- H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature452(7188), 728–731 (2008). [CrossRef] [PubMed]
- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998). [CrossRef]
- L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett.5(7), 1399–1402 (2005). [CrossRef] [PubMed]
- R. W. P. King, M. Owens, and T. T. Wu, Lateral Electromagnetic Waves: Theory and Applications to Communications, Geophysical Exploration and Remote Sensing (Springer-Verlag, New York, 1992).
- R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).
- L. Chen, J. T. Robinson, and M. Lipson, “Role of radiation and surface plasmon polaritons in the optical interactions between a nano-slit and a nano-groove on a metal surface,” Opt. Express14(26), 12629–12636 (2006). [CrossRef] [PubMed]
- I. S. Gradshteyn and I. m. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

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