## Investigation of the wave behaviors inside a step-modulated subwavelength metal slit |

Optics Express, Vol. 19, Issue 11, pp. 10073-10087 (2011)

http://dx.doi.org/10.1364/OE.19.010073

Acrobat PDF (2630 KB)

### Abstract

In this paper, we applied the modal expansion method (MEM) to investigate the wave behaviors inside a step-modulated subwavelength metal slit. The physical mechanism of the surface plasmon polariton (SPP) transmission is investigated in detail for slit structures with either dielectric or geometric modulation. The applicability of the effective index method is discussed. Moreover, as a special case of the geometric modulation, the evanescent-wave assisted transmission is demonstrated in a thin-modulated slit. We emphasize that a complete set is necessary in order to expand the wave functions in these kinds of structures. All the calculated results by the MEM are well retrieved by the finite-difference time-domain calculation.

© 2011 OSA

## 1. Introduction

1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) **424**, 824–830 (2003). [CrossRef]

9. C. Li, Y. S. Zhou, H. Y. Wang, and F. H. Wang, “Wavelength squeeze of surface plasmon polariton in a subwavelength metal slit,” J. Opt. Soc. Am. B **27**, 59–64 (2010). [CrossRef]

9. C. Li, Y. S. Zhou, H. Y. Wang, and F. H. Wang, “Wavelength squeeze of surface plasmon polariton in a subwavelength metal slit,” J. Opt. Soc. Am. B **27**, 59–64 (2010). [CrossRef]

22. S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B **79**, 035120 (2009). [CrossRef]

11. G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. **87**, 131102 (2005). [CrossRef]

12. H. Gao, H. Shi, C. Wang, C. Du, X. Luo, Q. Deng, Y. Lv, X. Lin, and H. Yao, “Surface plasmon polariton propagation and combination in Y-shaped metallic channels,” Opt. Express **13**, 10795–10800 (2005). [CrossRef] [PubMed]

13. T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyia, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. **85**, 5833–5835 (2004). [CrossRef]

14. Q. Zhang, X. Huang, X. Lin, J. Tao, and X. Jin, “A subwavelength coupler-type MIM optical filter,” Opt. Express **17**, 7549–7555 (2009). [CrossRef]

15. Y. S. Zhou, B. Y. Gu, and H. Y. Wang, “Band-gap structures of surface-plasmon polaritons in a subwavelength metal slit filled with periodic dielectrics,” Phys. Rev. A **81**, 015801 (2010). [CrossRef]

18. A. Houseini, H. Nejati, and Y. Massoud, “Modeling and design methodology for metal-insulator-metal plasmonic Bragg reflectors,” Opt. Express **16**, 1475–1480 (2008). [CrossRef]

24. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A **14**, 2758–2767 (1997). [CrossRef]

15. Y. S. Zhou, B. Y. Gu, and H. Y. Wang, “Band-gap structures of surface-plasmon polaritons in a subwavelength metal slit filled with periodic dielectrics,” Phys. Rev. A **81**, 015801 (2010). [CrossRef]

22. S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B **79**, 035120 (2009). [CrossRef]

15. Y. S. Zhou, B. Y. Gu, and H. Y. Wang, “Band-gap structures of surface-plasmon polaritons in a subwavelength metal slit filled with periodic dielectrics,” Phys. Rev. A **81**, 015801 (2010). [CrossRef]

21. X. S. Lin and X. G. Huang, “Numerical modeling of a teeth-shaped nanoplasmonic waveguide filter,” J. Opt. Soc. Am. B **26**, 1263–1268 (2009). [CrossRef]

19. X. S. Lin and X. G. Huang, “Tooth-shaped plasmonic waveguide filters with nanometeric sizes,” Opt. Lett. **33**, 2874–2876 (2008). [CrossRef] [PubMed]

21. X. S. Lin and X. G. Huang, “Numerical modeling of a teeth-shaped nanoplasmonic waveguide filter,” J. Opt. Soc. Am. B **26**, 1263–1268 (2009). [CrossRef]

28. Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. **88**, 057403 (2002). [CrossRef] [PubMed]

**81**, 015801 (2010). [CrossRef]

22. S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B **79**, 035120 (2009). [CrossRef]

**81**, 015801 (2010). [CrossRef]

21. X. S. Lin and X. G. Huang, “Numerical modeling of a teeth-shaped nanoplasmonic waveguide filter,” J. Opt. Soc. Am. B **26**, 1263–1268 (2009). [CrossRef]

20. Y. Matsuzaki, T. Okamoto, M. Haraguchi, M. Fukui, and M. Nakagaki, “Characteristics of gap plasmon waveguide with stub structures,” Opt. Express **16**, 16314–16325 (2008). [CrossRef] [PubMed]

17. J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express **17**, 20134–20139 (2009). [CrossRef] [PubMed]

20. Y. Matsuzaki, T. Okamoto, M. Haraguchi, M. Fukui, and M. Nakagaki, “Characteristics of gap plasmon waveguide with stub structures,” Opt. Express **16**, 16314–16325 (2008). [CrossRef] [PubMed]

*et al*. discussed the eigen problems of a metal slit structure and find a more general and more precise way to investigate the transmission [22

**79**, 035120 (2009). [CrossRef]

**79**, 035120 (2009). [CrossRef]

29. The commercially available software developed by Rsoft Design Group http://www.rsoftdesign.com is used for the numerical simulations.

## 2. The modal expansion method for metal slit structures

24. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A **14**, 2758–2767 (1997). [CrossRef]

30. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta **28**, 1087–1102 (1981). [CrossRef]

32. L. C. Botten and R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta **32**, 1479–1488 (1985). [CrossRef]

*λ*

_{0}is evaluated as

34. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

*λ*

_{0}= 1

*μ*m, thus,

*ε*= −50.76 + 0.083

_{Ag}*i*.

24. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A **14**, 2758–2767 (1997). [CrossRef]

30. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta **28**, 1087–1102 (1981). [CrossRef]

32. L. C. Botten and R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta **32**, 1479–1488 (1985). [CrossRef]

**14**, 2758–2767 (1997). [CrossRef]

35. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A **3**, 1780–1787 (1986). [CrossRef]

36. J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. **72**, 839–846 (1982). [CrossRef]

35. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A **3**, 1780–1787 (1986). [CrossRef]

36. J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. **72**, 839–846 (1982). [CrossRef]

*x*direction. The eigenfunctions and their derivatives should be continuous at the boundaries of the metal, and must be zero at the walls. Under these boundary conditions, the eigenequation can be solved. In the

*l*th layer of Fig. 1, the eigenfunctions along the

*x*direction are expressed as

*means the complex conjugate of*φ ¯

*φ*and

*δ*is the Kronecker delta. The equation satisfied by

_{mn}*φ*

^{+}is the adjoint of that by

*φ*. The proof of Eq. (3) is straightforward and similar to that given in Ref. [30

30. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta **28**, 1087–1102 (1981). [CrossRef]

*y*direction in Fig. 1 are that the magnetic field and its derivative at

*y*=

*Q*

^{(1)}and

*Q*

^{(2)}must be continuous. Applying these conditions, we obtain the coupled-wave equations as follows:

*q*

^{(1)}=

*Q*

^{(1)}−

*Q*

^{(0)}and

*q*

^{(2)}=

*Q*

^{(2)}−

*Q*

^{(1)}. The coefficients

*R*,

_{n}*T*,

_{n}*E*and

_{n}*F*can be numerically calculated by means of Eq. (6) and then are used to compute all the physical quantities.

_{n}*x*direction is removed, i.e.,

1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) **424**, 824–830 (2003). [CrossRef]

9. C. Li, Y. S. Zhou, H. Y. Wang, and F. H. Wang, “Wavelength squeeze of surface plasmon polariton in a subwavelength metal slit,” J. Opt. Soc. Am. B **27**, 59–64 (2010). [CrossRef]

*x*direction.

*y*direction, they influence the scattering but have little contribution to the transmission. Since

*μ*m, among the lowest six modes, the first four are guided modes and the next two are radiation modes, as shown in Fig. 2(c).

*et al*. provided a powerful program, based on Cauchy principle, to solve this problem [38

38. L. C. Botten, M. S. Craig, and R. C. McPhedran, “Complex zeros of analytic functions,” Computer Phys. Commun. **29**, 245–259 (1983). [CrossRef]

32. L. C. Botten and R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta **32**, 1479–1488 (1985). [CrossRef]

*et al*. could characterize the slit by two thin metal layers on its both sides [28

28. Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. **88**, 057403 (2002). [CrossRef] [PubMed]

*L*−

_{r}*L*= 2

_{l}*μ*m, the convergence is less than 1% for all the calculations in this paper.

## 3. Numerical results and analysis

**27**, 59–64 (2010). [CrossRef]

**81**, 015801 (2010). [CrossRef]

**26**, 1263–1268 (2009). [CrossRef]

*I*=

_{m}*δ*

_{m}_{1}in Eq. (6).

### 3.1. Dielectric modulation

*H*amplitude distribution, for [1 – 6 – 1]. Along the central axis of the slit, a good fit is obtained between the MEM and FDTD. In the slit, as pointed out already, only the SPP mode propagates and all the other modes attenuate. In Layer 3, the steady distribution is a straight line because only the upward propagating SPP mode survives. In Layers 1 and 2, due to the interference of the upward and downward propagating SPP modes, the oscillating period of the steady field distribution is half of the wavelength of corresponding SPP modes; for instance, in Layer 1, the oscillation period is 0.450

_{z}*μ*m, and the corresponding SPP wavelength is 0.900

*μ*m.

**81**, 015801 (2010). [CrossRef]

*q*

^{(2)}, for three structures [1 – 3 – 1], [1 – 6 – 1], and [3 – 1 – 6]. The results are plotted in Fig. 5. For comparison, we use the FDTD method and EIM-based triple-layer Fresnel equation to compute the same quantity, and the results are also plotted in Fig. 5. These three kinds of curves fit quite well, even for a very small

*q*

^{(2)}, namely, the thin modulation. Note that the EIM-based Fresnel equation only takes into account the non-phase-shift SPP modes. It is known that only the SPP mode propagates and multi-reflects in Layer 2. The phase shift of SPP reflection coefficients is too small (note that the

*y*-axis in Figs. 4 (b) and (d) is less than 1°) to cause an observable variation in the position of the transmission peaks in Fig. 5. These facts manifest the weak scattering. We conclude that the OMA is applicable in describing the transmission behavior in the dielectric-modulated case. However, neglecting the small contribution of the higher modes will cause the small difference in the height of the peaks between the results of EIM and MEM.

### 3.2. Geometric modulation

*w*

^{(1)}∼

*w*

^{(2)}∼

*w*

^{(3)}]. For example, [0.2 ∼ 0.4 ∼ 0.2] represents the structure shown in Fig. 6.

*q*

^{(2)}, are calculated for three different structures, [0.2 ∼ 0.4 ∼ 0.2], [0.4 ∼ 0.2 ∼ 0.4] and [0.4 ∼ 0.2 ∼ 0.6].

*w*

^{(2)}= 0.2

*μ*m and 0.4

*μ*m,

*μ*m, respectively, which indicates only the SPP mode propagates and resonates in Layer 2. The wave length and the position of the first peak can be used to calculate phase shift of the SPP mode. For [0.4 ∼ 0.2 ∼ 0.4], the phase shift is ▵

*θ*= 2 * (0.386/0.9006) * 360° ≈ 308.59°, where the factor 2 is introduced by the absolute value of the fields. To confirm the result, we read the corresponding phase shift of SPP reflection coefficients from Fig. 7(b) to be ▵

*θ*= |2 *(–154.30°)| = 308.60°. For [0.4 ∼ 0.2 ∼ 0.6], the phase shifts calculates from Fig. 8(a) and read from Fig. 7(b) are ▵

_{SPP}*θ*= 2 *(0.356/0.9006) *360° ≈ 284.60°, and ▵

*θ*= |–154.30° – 129.93°| = 284.23°, respectively. For [0.2 ∼ 0.4 ∼ 0.2], the position of the first peak is 0.484

_{SPP}*μ*m, larger than the half of resonant SPP wavelength, which means that the first peak should locate at (0.484 – 0.9449/2) = 0.01155

*μ*m; thus the phase shift ▵

*θ*= 2 *(0.01155/0.9449) *360° ≈ 8.80°; this time the phase shift of SPP reflection coefficient should be read form Fig.7 (d), and it is ▵

*θ*= |2 *(–4.41°)| = 8.82°. These three cases clearly show that it is the phase shift of the SPP wave that causes the displacement of transmission peaks. The effect of the phase shift of SPP transmission coefficients is canceled by the field absolute value.

_{SPP}*q*

^{(2)}< 0.2

*μ*m, the curve calculated by the multi-reflection model deviate from that by MEM for a [0.2 ∼ 0.4 ∼ 0.2] structure. This is because higher even eigenmodes in Layer 2 also make contribution to the transmission in such a thin modulation, as illustrated in Fig. 8(b). Although these modes are evanescent wave, it does not attenuate to a negligible value in this short distance, so that the single mode multi-reflection model cannot well describe the transmission. By introducing more modes into the multi-reflection process, the results converge to that of the MEM (the reflection time is set as 50 in Fig. 8(b)). This evanescent-wave assisted transmission in a structure beyond the skin-depth limit is wildly applied in optics [39

39. P. Yeh, “A new optical model for wire grid polarizers,” Opt. Commun. **26**, 289–292 (1978). [CrossRef]

42. W. Stork, N. Sreibl, H. Haidner, and P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. **16**, 1921–1923 (1991). [CrossRef] [PubMed]

43. 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]

## 4. Conclusion

## Acknowledgments

## References and links

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

2. | J. Pendry, “Playing tricks with light,” Science |

3. | H. Raether, |

4. | F. Villa, T. Lopez-Rios, and L. E. Regalado, “Electromagnetic modes in metal-insulator-metal structures,” Phys. Rev. B |

5. | J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B |

6. | Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B |

7. | B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B |

8. | A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Surface plasmon polaritons on metallic surfaces,” Opt. Express |

9. | C. Li, Y. S. Zhou, H. Y. Wang, and F. H. Wang, “Wavelength squeeze of surface plasmon polariton in a subwavelength metal slit,” J. Opt. Soc. Am. B |

10. | E. Feigenbaum and M. Orenstein, “Modeling of complementary (void) plasmon waveguiding,” J. Lightwave Technol. |

11. | G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. |

12. | H. Gao, H. Shi, C. Wang, C. Du, X. Luo, Q. Deng, Y. Lv, X. Lin, and H. Yao, “Surface plasmon polariton propagation and combination in Y-shaped metallic channels,” Opt. Express |

13. | T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyia, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. |

14. | Q. Zhang, X. Huang, X. Lin, J. Tao, and X. Jin, “A subwavelength coupler-type MIM optical filter,” Opt. Express |

15. | Y. S. Zhou, B. Y. Gu, and H. Y. Wang, “Band-gap structures of surface-plasmon polaritons in a subwavelength metal slit filled with periodic dielectrics,” Phys. Rev. A |

16. | A. Hosseini and Y. Massoud, “A low-loss metal-insulator-metal plasmonic bragg reflector,” Opt. Express |

17. | J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express |

18. | A. Houseini, H. Nejati, and Y. Massoud, “Modeling and design methodology for metal-insulator-metal plasmonic Bragg reflectors,” Opt. Express |

19. | X. S. Lin and X. G. Huang, “Tooth-shaped plasmonic waveguide filters with nanometeric sizes,” Opt. Lett. |

20. | Y. Matsuzaki, T. Okamoto, M. Haraguchi, M. Fukui, and M. Nakagaki, “Characteristics of gap plasmon waveguide with stub structures,” Opt. Express |

21. | X. S. Lin and X. G. Huang, “Numerical modeling of a teeth-shaped nanoplasmonic waveguide filter,” J. Opt. Soc. Am. B |

22. | S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B |

23. | R. E. Collin, |

24. | L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A |

25. | T. Itoh, |

26. | Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. |

27. | J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. |

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

29. | The commercially available software developed by Rsoft Design Group http://www.rsoftdesign.com is used for the numerical simulations. |

30. | L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta |

31. | L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta |

32. | L. C. Botten and R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta |

33. | Since the slit is an infinitely long one, the wave source has to be located in Layer 1. This can be easily done in FDTD by setting the coordinates of the source to |

34. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

35. | M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A |

36. | J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. |

37. | E. A. Coddington and N. Levinson, |

38. | L. C. Botten, M. S. Craig, and R. C. McPhedran, “Complex zeros of analytic functions,” Computer Phys. Commun. |

39. | P. Yeh, “A new optical model for wire grid polarizers,” Opt. Commun. |

40. | M. E. Motamedi, W. H. Southwell, and W. J. Gunning, “Antireflection surfaces in silicon using binary optics technology,” Appl. Opt. |

41. | W. M. Farn, “Binary gratings with increased efficiency,” Appl. Opt. |

42. | W. Stork, N. Sreibl, H. Haidner, and P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. |

43. | 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 |

**OCIS Codes**

(230.7380) Optical devices : Waveguides, channeled

(240.6680) Optics at surfaces : Surface plasmons

(260.3910) Physical optics : Metal optics

(290.5825) Scattering : Scattering theory

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: February 23, 2011

Revised Manuscript: April 9, 2011

Manuscript Accepted: April 22, 2011

Published: May 9, 2011

**Citation**

Chao Li, Yun-Song Zhou, Huai-Yu Wang, and Fu-He Wang, "Investigation of the wave behaviors inside a step-modulated subwavelength metal slit," Opt. Express **19**, 10073-10087 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-11-10073

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### References

- W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) 424, 824–830 (2003). [CrossRef]
- J. Pendry, “Playing tricks with light,” Science 285, 1687–1688 (1999). [CrossRef]
- H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).
- F. Villa, T. Lopez-Rios, and L. E. Regalado, “Electromagnetic modes in metal-insulator-metal structures,” Phys. Rev. B 63, 165103 (2001). [CrossRef]
- J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006). [CrossRef]
- Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B 75, 035411 (2007). [CrossRef]
- B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007). [CrossRef]
- A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Surface plasmon polaritons on metallic surfaces,” Opt. Express 15, 183–197 (2007). [CrossRef] [PubMed]
- C. Li, Y. S. Zhou, H. Y. Wang, and F. H. Wang, “Wavelength squeeze of surface plasmon polariton in a subwavelength metal slit,” J. Opt. Soc. Am. B 27, 59–64 (2010). [CrossRef]
- E. Feigenbaum and M. Orenstein, “Modeling of complementary (void) plasmon waveguiding,” J. Lightwave Technol. 25, 2547–2562 (2007). [CrossRef]
- G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87, 131102 (2005). [CrossRef]
- H. Gao, H. Shi, C. Wang, C. Du, X. Luo, Q. Deng, Y. Lv, X. Lin, and H. Yao, “Surface plasmon polariton propagation and combination in Y-shaped metallic channels,” Opt. Express 13, 10795–10800 (2005). [CrossRef] [PubMed]
- T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyia, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85, 5833–5835 (2004). [CrossRef]
- Q. Zhang, X. Huang, X. Lin, J. Tao, and X. Jin, “A subwavelength coupler-type MIM optical filter,” Opt. Express 17, 7549–7555 (2009). [CrossRef]
- Y. S. Zhou, B. Y. Gu, and H. Y. Wang, “Band-gap structures of surface-plasmon polaritons in a subwavelength metal slit filled with periodic dielectrics,” Phys. Rev. A 81, 015801 (2010). [CrossRef]
- A. Hosseini and Y. Massoud, “A low-loss metal-insulator-metal plasmonic bragg reflector,” Opt. Express 14, 11318–11323 (2006). [CrossRef]
- J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express 17, 20134–20139 (2009). [CrossRef] [PubMed]
- A. Houseini, H. Nejati, and Y. Massoud, “Modeling and design methodology for metal-insulator-metal plasmonic Bragg reflectors,” Opt. Express 16, 1475–1480 (2008). [CrossRef]
- X. S. Lin and X. G. Huang, “Tooth-shaped plasmonic waveguide filters with nanometeric sizes,” Opt. Lett. 33, 2874–2876 (2008). [CrossRef] [PubMed]
- Y. Matsuzaki, T. Okamoto, M. Haraguchi, M. Fukui, and M. Nakagaki, “Characteristics of gap plasmon waveguide with stub structures,” Opt. Express 16, 16314–16325 (2008). [CrossRef] [PubMed]
- X. S. Lin and X. G. Huang, “Numerical modeling of a teeth-shaped nanoplasmonic waveguide filter,” J. Opt. Soc. Am. B 26, 1263–1268 (2009). [CrossRef]
- S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009). [CrossRef]
- R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, 1966).
- L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997). [CrossRef]
- T. Itoh, Numerical Techniques for Microwave and Millimeter Wave Passive Structures (Wiley, 1989).
- Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86, 5601–5603 (2001). [CrossRef] [PubMed]
- J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999). [CrossRef]
- Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 057403 (2002). [CrossRef] [PubMed]
- The commercially available software developed by Rsoft Design Group http://www.rsoftdesign.com is used for the numerical simulations.
- L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981). [CrossRef]
- L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981). [CrossRef]
- L. C. Botten and R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta 32, 1479–1488 (1985). [CrossRef]
- Since the slit is an infinitely long one, the wave source has to be located in Layer 1. This can be easily done in FDTD by setting the coordinates of the source to y = H(0). A “slab mode” (a normal mode of a slab waveguide with characteristics matching the input waveguide, a feature provieded by the RSOFT Fullwave software) is used to excite the SPP mode. However, for the amplitude of the source in FDTD, it should be adjusted to the same value of the normalized SPP mode in Layer 1 as pointed out later. The grid size of the simulation is set as 2.5 nm × 2.5 nm.
- P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]
- M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986). [CrossRef]
- J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982). [CrossRef]
- E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, 1955).
- L. C. Botten, M. S. Craig, and R. C. McPhedran, “Complex zeros of analytic functions,” Computer Phys. Commun. 29, 245–259 (1983). [CrossRef]
- P. Yeh, “A new optical model for wire grid polarizers,” Opt. Commun. 26, 289–292 (1978). [CrossRef]
- M. E. Motamedi, W. H. Southwell, and W. J. Gunning, “Antireflection surfaces in silicon using binary optics technology,” Appl. Opt. 31, 4371–4376 (1992) [CrossRef] [PubMed]
- W. M. Farn, “Binary gratings with increased efficiency,” Appl. Opt. 31, 4453–4458 (1992). [CrossRef] [PubMed]
- W. Stork, N. Sreibl, H. Haidner, and P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 16, 1921–1923 (1991). [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. Photonics 3, 152–156 (2009). [CrossRef]

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