## Interference of surface waves in a metallic nanoslit

Optics Express, Vol. 15, Issue 3, pp. 1182-1190 (2007)

http://dx.doi.org/10.1364/OE.15.001182

Acrobat PDF (378 KB)

### Abstract

We investigate the interference of the surface plasmon polariton (SPP) with an incident beam on a metallic slit using the FDTD. We find that the bulk waves radiated at the slit edge by scattering of the SPP leak into the slit and induce accumulated charges within the skin depth, which excite new SPPs on the slit side-walls. The SPP on the top surface of aperture is coupled into the slit and induces the 2D asymmetric field distributions, including the horizontal and vertical Fabry-Perot multi-reflection resonator modes. We show that the addition of these modes with the slit waveguide modes induced by a normally incident beam is the interference between the SPP and the incident beam, which enhances or suppresses the slit transmission, depending on their relative phase.

© 2007 Optical Society of America

## 1. Introduction

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

2. T. Thio, K.M. Pellerin, R.A. Linke, T.W. Ebbesen, and H.J. Lezec, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. **26**,1972–1974 (2001). [CrossRef]

*et al*. suggested that the enhancement of transmission could be a result of the interference of the composite diffracted evanescent waves with the normally incident light on the apertures. Their theory is within the scalar diffraction theory framework [3

3. H.J. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express **12**,3629–3651 (2004). [CrossRef] [PubMed]

*et al*. studied the optical transmission of two subwavelength slits as a function of wavelength and slit separation [4

4. 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**, 053901 (2005). [CrossRef] [PubMed]

## 2. Coupling surface plasmons polaritons into slit

*z*-axis. The surface plasmon polariton (SPP) is generated by a linearly

*p*-polarized beam, with

*E*and

_{x}*H*components, of wavelength

_{z}*λ*

_{0}=500

*nm*normally incident on a groove of width 250nm and depth 70

*nm*. As shown on Fig. 1(a), SPPs are generated by diffraction in the groove and propagate away from the groove on the top metal surface in the ±

*x*directions. The internal mechanism of the SPP generation in the groove is not our concern in this paper. Other SPP sources could also be used.

*ε*

_{r,∞}is the dielectric constant at infinite frequencies,

*ω*the plasma frequency, and

_{p}*ω*,

_{k}*f*, and Γ

_{k}*are the resonance frequency, strength and damping frequency, respectively, of*

_{k}*k*th oscillator. The Lorentz-Drude model uses

*K*damped harmonic oscillators to describe the small resonances observed in the metal’s frequency response. The values of the constants in Eq. (1) are taken from [11

11. Aleksandar D. Rakic, Aleksandra B. Djuristic, Jovan M. Elazar, and Marian L. Majewski, “Optical properties of metallic films,” Appl. Opt. **37**,5271–5283 (1998). [CrossRef]

*λ*

_{0}=500

*nm*, we obtain the complex permittivity of silver:

*ε*=-7.6321+0.7306

*i*. Perfectly-matched layer boundary conditions were applied at the limits of the computational domain [10]. The slit intensity transmission coefficient

*T*is calculated by integrating the time-averaged modulus of the Poynting vector along a plane “detector” located at the slit bottom exit (0≤

_{s}*x*≤

*a, y*=-10

*nm*), as shown on Fig. 1(a).

## 2.1 Fabry-Perot resonator of SPP between groove and slit

*H*and

_{z}*E*, which is parallel and normal to the top surface, respectively. At the slit top-left edge (

_{y}*x*=0,

*y*=

*t*), the abrupt change of permittivity of the material from metal to vacuum partially reflects the incident SPP. The remaining SPP is converted to bulk radiation by scattering. As the groove is present, the reflected SPP is again partially reflected back towards the slit from the top-right edge of the groove (

*x*=-

*L*, y=

*t*). The SPP characteristic propagation length [12]

*L*=1/(2∙Im(

_{spp}*k*))=52

_{spp}*μm*is significantly longer than the propagation distance

*L*in our experiments. Thus, the multiple reflections on the section -

*L*<

*x*<0 of the metal top surface create a Fabry-Perot (F-P) type interference of the SPP. In Fig. 1(b) we show the slit energy transmission

*T*as a function of the cavity length

_{s}*L*, when the sole SPP is incident to the slit from the left ridge. The main features of the curve are its initial rapid decay for 0<

*L*<

*μm*and the oscillation period of

*λ*/2, where the SPP wavelength

_{spp}*λ*=

_{spp}*λ*

_{0}/Re(

*n*)=470

_{spp}*nm*. The resonance peaks of

*T*have low magnitudes corresponding to small SPP reflectivities

_{s}*R*at the slit and groove edges. At the left-end ridge of the metal film the SPP propagating in the −

*x*direction is also partially reflected back. However, the propagation distance of the SPP in the −

*x*direction,

*L*’, is chosen to be 1.5

*λ*such that their interference is destructive and their perturbation to the +

_{spp}*x*propagating SPP on the right-side of the groove is minimized.

## 2.2 Bulk waves Fabry-Perot resonator modes between slit walls

*H*and

_{z}*E*components, as for the incident SPP, hit and are reflected from the right-side-wall of the slit, and are then reflected back by the left-side-wall of the slit. The multiple reflections between the two slit walls form F-P resonator modes with slightly inclined bulk wave input from the top of the cavity. When the slit width is close to integer multiples of

_{y}*λ*

_{0}/2, the cavity resonance creates strong standing wave fringes of

*E*and

_{y}*H*with a fringe period of

_{z}*λ*

_{0}/2 as shown in Fig. 2(a) and 2(c). The fringes of

*H*are interlaced with the fringes of

_{z}*E*. The maxima of

_{y}*H*are at the slit walls, while the maxima of

_{z}*E*are shifted by

_{y}*λ*

_{0}/4 from the slit walls [5

5. A.R. Zakharian, M. Mansuripur, and J.V. Moloney, “Transmission of light through small elliptical apertures,” Opt. Express **12**,2631–2648 (2004). [CrossRef] [PubMed]

*T*are maximal. Out of the resonance, both

_{s}*E*and

_{y}*H*in the slit decrease to low intensities and

_{z}*T*is minimal, as pictured in Fig. 2(b).

_{s}*x*>

*a*,

*y*=0 and

*y*=

*t*). On the top surface, the SPP on the right-side of the slit is referred to as the transmitted SPP over the slit. This result is consistent with previous studies [8

8. J. A. Sanchez-Gil and A. A. Maradudin, “Near-field and far-field scattering of surface plasmon polaritons by one-dimensional surface defects,” Phys. Rev. B **60**,8359–8367 (1999). [CrossRef]

9. J. A. Sanchez-Gil and A. A. Maradudin, “Dynamic near-field calculations of surface-plasmon polariton pulses resonantly scattered at sub-micron metal defects,” Opt. Express **12**,883–894 (2004). [CrossRef] [PubMed]

13. T. A. Leskova, “Theory of a Fabry-Perot Type Interferometer for Surface Polaritons,” Solid State Commun. **50**,869–873 (1984). [CrossRef]

*ε*

_{2}, or when the slit is replaced by a surface defect, such as a groove or bump of any shape, one can write down the analytic expressions for

*H*in the free space above the top surface (

_{z}*y*≥

*t*), which should satisfy the boundary conditions at the surface

*y*=

*t*[8

8. J. A. Sanchez-Gil and A. A. Maradudin, “Near-field and far-field scattering of surface plasmon polaritons by one-dimensional surface defects,” Phys. Rev. B **60**,8359–8367 (1999). [CrossRef]

9. J. A. Sanchez-Gil and A. A. Maradudin, “Dynamic near-field calculations of surface-plasmon polariton pulses resonantly scattered at sub-micron metal defects,” Opt. Express **12**,883–894 (2004). [CrossRef] [PubMed]

13. T. A. Leskova, “Theory of a Fabry-Perot Type Interferometer for Surface Polaritons,” Solid State Commun. **50**,869–873 (1984). [CrossRef]

*ε*

_{2}has a width

*a*>>

*λ*the structure is referred to as a SPP Fabry-Perot interferometer. Its SPP energy transmission coefficient

_{spp}*T*oscillates with period Δ

*a*=

*λ*/2 [13

_{spp}13. T. A. Leskova, “Theory of a Fabry-Perot Type Interferometer for Surface Polaritons,” Solid State Commun. **50**,869–873 (1984). [CrossRef]

*et al*. found that the SPP intensity transmission coefficient

*T*across the defect exhibits a cavity-like oscillation with a period Δ

*a*∼1.4(

*λ*/2) [8

_{spp}8. J. A. Sanchez-Gil and A. A. Maradudin, “Near-field and far-field scattering of surface plasmon polaritons by one-dimensional surface defects,” Phys. Rev. B **60**,8359–8367 (1999). [CrossRef]

*et al*. showed in the case of a simple surface defect (groove or bump) that the scattering efficiency

*S*of the waves into the free space above the top surface depends on the SPP transmission

*T*(Fig. 1). When the surface defect is replaced by a slit, we observed that the scattering

*S*and its direction of maximum radiation both depend on the F-P resonator modes in the slit, which also determine the SPP transmission

*T*on the top surface and the slit transmission Ts coefficients. The latter is the energy leaked from the slit’s bottom exit in the form of the transmitted bulk waves and of the SPPs excited on the bottom plane of the metal film (

*x*<0 and

*x*>

*a*,

*y*=0).

## 2.3 Excitation of new SPPs on slit walls

*E*component. The scattered bulk waves entering the slit contains

_{x}*E*which is responsible for the downward component of the bulk wave vector

_{x}*k*⃗

_{0}. In addition, new

*E*fields are excited on the slit walls. Indeed, the

_{x}*H*of the incident SPP induces surface currents parallel to the top metal surface (-

_{z}*L*≤

*x*≤0,

*y*=

*t*), while the bulk

*H*leaked into the slit induces surface currents parallel to the slit left-side wall (

_{z}*x*=0, 0≤

*y*≤

*t*). The surface currents associated with the incoming SPP must turn around the top-left corner (

*x*=0,

*y*=

*t*) as shown in Fig. 3. There is a corresponding sharp discontinuity in the amplitude of

*H*on the top surface SPP and that of the leaked bulk

_{z}*H*, in the vicinity of the slit top-left edge, as shown in Fig. 2. According to the charge conservation equation, ∁ ∙

_{z}*J*⃑ = -

*∂ρ*/

*∂t*, the corresponding current discontinuity creates the accumulation of electric charges at the slit corner as an oscillating monopole, which can be observed by a high concentration of

*E*on the top surface (

_{y}*x*≤0,

*y*=

*t*) and of

*E*on the slit left-side wall (

_{x}*x*=0, 0≤

*y*≤

*t*), as shown in Fig. 2. The oscillating monopole radiates waves which excites the SPP on the slit left-side wall (

*x*=0, 0≤

*y*≤

*t*).

*x*=

*a*, 0≤

*y*≤

*t*), there is no significant

*E*when the slit width

_{x}*a*>>

*λ*

_{0}. However, the scattered bulk

*H*induces surface currents and accumulated charges on the slit right-side wall within the skin depth. The FDTD calculation shows that the phase of

_{z}*H*is shifted by

_{z}*π*from the left-side wall to the right-side wall of the slit, as shown in Fig. 2(d), so that the surface currents are in the same direction along both walls. As a consequence, the accumulated charges on the slit right-side wall are of the opposite sign to that at the slit top-left edge, and the two poles are in-phase in their oscillation with time resulting in an oblique dipole across the slit as shown on Fig. 3. This dipole is the source of excitation of new SPPs on both slit walls, although the two poles can have different amount of charges, and the rightside pole is not necessarily located at the slit bottom-right corner but near the middle of the right-side wall. The distributions of field components

*E*and

_{x}*H*and of the accumulated charges on the two walls vary with the slit geometry and are asymmetric with respect to the slit axis, see Fig. 2(c).

_{z}*E*and

_{x}*H*on the slit side walls. In contrast with the

_{z}*E*, whose maximum is shifted from the reflecting wall surfaces by

_{y}*λ*

_{0}/4, the

*E*is stuck on the walls. Apparently, the multiple reflection resonances of the bulk waves between the two slit walls have little effect to the

_{x}*E*on the walls. When the bulk wave is out of resonance,

_{x}*E*is minimum inside the slit while the

_{y}*E*and

_{x}*H*components are non null on the slit walls, as shown in Fig. 2(b), where they sustain the SPPs on the slit side walls and contribute to coupling the incoming SPP on the top surface into the slit cavity.

_{z}## 2.4 Coupling the incident SPP into the slit

*S*⃑ associated with a single SPP incident on the slit. Since the magnetic field has only

*H*component, the direction of

_{z}*S*⃑ is determined by the local ratio of

*E*and

_{x}*E*. On the top surface, the

_{y}*S*⃑ of the incident SPP is in +

*x*direction. The

*S*⃑ then enters inside the slit by turning around the positively charged pole at the slit top-left edge due to the strong presence of accumulated charges at the edge. The

*S*⃑ then flows downward supported by the excited SPP (

*E*) on the slit left-side wall. After propagating through a small depth in the slit, the magnitude of Ex diminishes, and

_{x}*S*⃑ changes the direction right conducted by the rising of

*E*in the middle of the slit, as shown in Fig. 2(c), to meet the right-side wall of the slit where the negatively charged pole and the excited SPP, sustained by

_{y}*E*and

_{x}*H*, leads the

_{z}*S*⃑ flux towards the bottom-right edge and to the slit bottom exit. In addition, new SPPs are excited on the bottom-right metal plane (

*x*≥

*a*,

*t*=0).

## 2.5 Fabry-Perot resonator modes along the slit axis

14. Y. Takakura, “Optical Resonance in a Narrow Slit in a Thick Metallic Screen,” Phys. Rev. Lett. **86**,5601–5603 (2001). [CrossRef] [PubMed]

*a*≤

*λ*

_{0}/4, the

*E*distribution is oblique only at the top of the slit inside the first period, (

_{x}*t-λ*)/2<

_{spp}*y*<

*t*, of the standing-wave fringes. As shown in Fig. 4(b), the standing-wave fringes of

*E*become uniform across the slit width for the remaining slit length, 0<

_{x}*y*<(

*t-λ*/2), as in the conventional slit F-P waveguide modes for the slit illuminated by a normal incident beam.

_{spp}## 2.6 Field distributions inside the slit

*a*≥

*λ*

_{0}and

*a*≥

*λ*0/4 respectively. In general, a single incident SPP on the top surface induces 2D field distributions in the slit, which are asymmetric with respect to the slit axis.

*T*normalized by the slit width

_{s}*a*as a function of slit width

*a*and thickness

*t*, when a single SPP on the top surface impinges on the slit. Peak transmission occurs when both the vertical and horizontal F-P resonator modes are at resonance. Our computations show that the values of

*a*and

*t*for the resonances are shifted from those for the classical cavity resonances,

*a*=

*n*(

*λ*

_{0}/2) and

*t*=

*m*(

*λ*/2), where

_{eff}*m*and

*n*are integers and

*λ*=

_{eff}*λ*/

_{spp}*n*is the effective wavelength with slit waveguide mode effective index

_{eff}*n*[14

_{eff}14. Y. Takakura, “Optical Resonance in a Narrow Slit in a Thick Metallic Screen,” Phys. Rev. Lett. **86**,5601–5603 (2001). [CrossRef] [PubMed]

*a*≤

*λ*

_{0}/4, the F-P resonances along the slit thickness

*t*are clearly visible in Fig. 4(a), which occur when

*t*is

*m*(

*λ*/2). The peak regions of

_{eff}*T*have an inclined shape because

_{s}*λ*and

_{eff}*n*are functions of the slit width

_{eff}*a*. With increasing of the slit thickness

*t*,

*T*decreases due to attenuation losses of the SPPs’ propagation along the slit walls. When the slit width

_{s}*a*is around

*λ*

_{0}/2, the F-P resonances along the slit thickness

*t*are clearly visible in Fig. 4(a), which occur when

*t*is

*m*(

*λ*/2). The peak regions of

_{eff}*T*have an inclined shape because

_{s}*λ*and

_{eff}*n*are functions of the slit width

_{eff}*a*. With increasing of the slit thickness

*t*,

*T*decreases due to attenuation losses of the SPPs’ propagation along the slit walls. When the slit width a is around

_{s}*λ*

_{0}/2, the F-P resonances as a function of

*t*are still present, but with different shapes and locations of the

*T*peaks in the

_{s}*a*-

*t*map due to the onset of bulk wave horizontal F-P modes. When the slit width

*a*is around

*λ*

_{0}and 3

*λ*

_{0}/4, the peaks of

*T*with

_{s}*t*dependence can still be seen in Fig. 4(a). However, the appearance of the resonances with variation of

*t*are not as regular as for small widths,

*a*≤

*λ*

_{0}/4, because of the 2D asymmetric nature of the field distributions in the slit region and the weak confinement ability of the slit cavities for large widths

*a*and small thicknesses

*t*.

*t*∼

*λ*/2=235

_{spp}*nm*, the peaks of

*T*for the slit widths

_{s}*a*around

*λ*

_{0}/2,

*λ*

_{0}and 3

*λ*

_{0}/2, are due to the horizontal bulk wave F-P resonances. The peaks at

*a*∼

*λ*

_{0}/2 and ⃑∼λ

*λ*

_{0}are spaced with an interval slightly larger than

*λ*

_{0}/2.

## 3. Enhancement and suppression of transmission through field interference

*E*,

_{x}*E*,

_{y}*H*and electrical charge distributions when a single

_{z}*p*-polarized narrow beam is normally incident on the groove. As described in Section 2, this beam generates the SPP propagating on the top metal surface and impinging on the slit from the left side.

*p*-polarized narrow beam is normally incident on the slit. For this latter configuration, the FDTD results have been well described in [5

5. A.R. Zakharian, M. Mansuripur, and J.V. Moloney, “Transmission of light through small elliptical apertures,” Opt. Express **12**,2631–2648 (2004). [CrossRef] [PubMed]

6. Y. Xie, A.R. Zakharian, J.V. Moloney, and M. Mansuripur, “Transmission of light through slit apertures in metallic films,” Opt. Express **12**,6106–6121 (2004). [CrossRef] [PubMed]

*x*directions on both the top and bottom metal film surfaces.

*a*, thickness

*t*and the incident SPP propagation distance

*L*, which controls the relative phase of the SPP-induced distribution with respect that induced by the normally incident beam. The slit transmission is enhanced by the incident SPP, when the two distributions are superposed constructively. It is reduced when the two distributions are superposed destructively.

*T*with the two incident beams, on the groove and on the slit, normalized by that of a single beam normally incident on the slit, as a function of the SPP propagation distance

_{s}*L*. For constructive interference, the bare slit transmittance can be enhanced by more than 1.3 by the incident SPP on the slit. When the two distributions are superposed destructively, the transmittance can drop as low as 0.72 compared to the bare slit transmittance.

## 4. Conclusion

## References and links

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

2. | T. Thio, K.M. Pellerin, R.A. Linke, T.W. Ebbesen, and H.J. Lezec, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. |

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

4. | 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. |

5. | A.R. Zakharian, M. Mansuripur, and J.V. Moloney, “Transmission of light through small elliptical apertures,” Opt. Express |

6. | Y. Xie, A.R. Zakharian, J.V. Moloney, and M. Mansuripur, “Transmission of light through slit apertures in metallic films,” Opt. Express |

7. | Y. Xie, A.R. Zakharian, J.V. Moloney, and M. Mansuripur, “Transmission of light through a periodic array of slits in a thick metallic film,” Opt. Express |

8. | J. A. Sanchez-Gil and A. A. Maradudin, “Near-field and far-field scattering of surface plasmon polaritons by one-dimensional surface defects,” Phys. Rev. B |

9. | J. A. Sanchez-Gil and A. A. Maradudin, “Dynamic near-field calculations of surface-plasmon polariton pulses resonantly scattered at sub-micron metal defects,” Opt. Express |

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

11. | Aleksandar D. Rakic, Aleksandra B. Djuristic, Jovan M. Elazar, and Marian L. Majewski, “Optical properties of metallic films,” Appl. Opt. |

12. | H. Raether, |

13. | T. A. Leskova, “Theory of a Fabry-Perot Type Interferometer for Surface Polaritons,” Solid State Commun. |

14. | Y. Takakura, “Optical Resonance in a Narrow Slit in a Thick Metallic Screen,” Phys. Rev. Lett. |

**OCIS Codes**

(050.1220) Diffraction and gratings : Apertures

(240.6680) Optics at surfaces : Surface plasmons

(240.6690) Optics at surfaces : Surface waves

(260.3160) Physical optics : Interference

(260.3910) Physical optics : Metal optics

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: November 14, 2006

Revised Manuscript: January 19, 2007

Manuscript Accepted: January 22, 2007

Published: February 5, 2007

**Citation**

Bora Ung and Yunlong Sheng, "Interference of surface waves in a metallic nanoslit," Opt. Express **15**, 1182-1190 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-3-1182

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### References

- T.W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature (London) 391, 667-669 (1998). [CrossRef]
- T. Thio, K. M. Pellerin, R. A. Linke, T. W. Ebbesen, and H. J. Lezec, "Enhanced light transmission through a single subwavelength aperture," Opt. Lett. 26, 1972-1974 (2001). [CrossRef]
- H. J. Lezec and T. Thio, "Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays," Opt. Express 12, 3629-3651 (2004). [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, 053901 (2005). [CrossRef] [PubMed]
- A. R. Zakharian, M. Mansuripur, and J. V. Moloney, "Transmission of light through small elliptical apertures," Opt. Express 12, 2631-2648 (2004). [CrossRef] [PubMed]
- Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, "Transmission of light through slit apertures in metallic films," Opt. Express 12, 6106-6121 (2004). [CrossRef] [PubMed]
- Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, "Transmission of light through a periodic array of slits in a thick metallic film," Opt. Express 13, 4485-4491 (2005). [CrossRef] [PubMed]
- J. A. Sanchez-Gil and A. A. Maradudin, "Near-field and far-field scattering of surface plasmon polaritons by one-dimensional surface defects," Phys. Rev. B 60, 8359-8367 (1999). [CrossRef]
- J. A. Sanchez-Gil and A. A. Maradudin, "Dynamic near-field calculations of surface-plasmon polariton pulses resonantly scattered at sub-micron metal defects," Opt. Express 12, 883-894 (2004). [CrossRef] [PubMed]
- A. Taflove and S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method, 2nd edition, (Artech House, Boston, 2000).
- A. D. Rakic, A. B. Djuristic, J. M. Elazar, and M. L. Majewski, "Optical properties of metallic films," Appl. Opt. 37, 5271-5283 (1998). [CrossRef]
- H. Raether, Surface Polaritons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988).
- T. A. Leskova, "Theory of a Fabry-Perot Type Interferometer for Surface Polaritons," Solid State Commun. 50, 869-873 (1984). [CrossRef]
- Y. Takakura, "Optical resonance in a narrow slit in a thick metallic screen," Phys. Rev. Lett. 86, 5601-5603 (2001). [CrossRef] [PubMed]

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