## Dynamic near-field calculations of surface-plasmon polariton pulses resonantly scattered at sub-micron metal defects

Optics Express, Vol. 12, Issue 5, pp. 883-894 (2004)

http://dx.doi.org/10.1364/OPEX.12.000883

Acrobat PDF (1087 KB)

### Abstract

We investigate theoretically the near-field dynamics of the scattering of a surface-plasmon polariton (SPP) pulse impinging normally on a rectangular groove on an otherwise planar metal surface. Our formulation is based on solving the reduced Rayleigh equation (derived through the use of an impedance boundary condition) for every component of the spectral decomposition of the incoming SPP pulse. Numerical calculations are carried out of the time dependence of the near-field resonant scattering effects produced at the rectangular groove. The scattering process is tracked through the (time-resolved) repartition of the incoming SPP electromagnetic energy into reflected and transmitted SPP pulses, and into pulsed scattered light. Furthermore, we directly show evidence of the excitation of single resonances, as manifested by the concentration of electric field intensity within the groove, and its subsequent leakage, over the resonance lifetime. The near-field formation of oscillations caused by the interference between two adjacent resonances simultaneously excited is also considered.

© 2004 Optical Society of America

## 1. Introduction

1. T. A. Leskova and N. I. Gapotchenko, “Fabry-Perot type interferometer for surface polaritons: resonance effects,” Solid State Commun. **53**, 351 (1985). [CrossRef]

2. B. Rothenhäusler and W. Knoll, “Surface plasmon interferometry in the visible,” Appl. Phys. Lett. **52**, 1554 (1988). [CrossRef]

3. B. Rothenhäusler and W. Knoll, “Interferometric determination of the complex wave vector of plasmon surface polaritons,” J. Opt. Soc. Am. B **5**, 1401 (1988). [CrossRef]

4. F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, “Scattering of a surface plasmon polariton by a surface defect,” Phys. Rev. B **50**, 15261 (1994). [CrossRef]

5. A. V. Shchegrov, I. V. Novikov, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a circularly symmetric surface defect,” Phys. Rev. Lett. **78**, 4269 (1997). [CrossRef]

6. J. A. Sánchez-Gil, “Surface defect scattering of surface plasmon polaritons: Mirrors and light emitters,” Appl. Phys. Lett. **73**, 3509 (1998). [CrossRef]

8. T. A. Leskova, A. A. Maradudin, and W. Zierau, “Surface plasmon polariton propagation near an index step,” Proc. SPIE **4100**, 1 (2000). [CrossRef]

2. B. Rothenhäusler and W. Knoll, “Surface plasmon interferometry in the visible,” Appl. Phys. Lett. **52**, 1554 (1988). [CrossRef]

3. B. Rothenhäusler and W. Knoll, “Interferometric determination of the complex wave vector of plasmon surface polaritons,” J. Opt. Soc. Am. B **5**, 1401 (1988). [CrossRef]

9. Z. Schlesinger and A. J. Sievers, “Infrared surface wave interferometry,” Appl. Phys. Lett. **36**, 409 (1980). [CrossRef]

10. B. Rothenhäusler and W. Knoll, “On the influence of the propagation length of plasmon surface polaritons in the visible energy range for the optical characterization of heterogeneous thin films,” Surf. Sci. **191**, 585 (1987). [CrossRef]

11. B. Rothenhäusler and W. Knoll, “Total internal diffraction of plasmon surface polaritons,” Appl. Phys. Lett. **51**, 783 (1987). [CrossRef]

12. B. Rothenhäusler and W. Knoll, “Surface plasmon microscopy,” Nature **332**, 615 (1988) [CrossRef]

13. C. E. H. Berger, R. P. H. Koioyman, and J. Greve, “Surface plasmon propagation near an index step,” Opt. Commun. **167**, 183 (1999). [CrossRef]

15. I. I. Smolyaninov, D. L. Mazzoni, J. Mait, and C. C. Davis, “Experimental study of surface plasmon scattering by individual surface defects,” Phys. Rev. B **56**, 1601 (1997). [CrossRef]

16. A. Dogariu, T. Thio, L. J. Wang, T. W. Ebbesen, and H. J. Lezec, “Delay in light transmission through small apertures,” Opt. Lett. **26**, 450 (2001). [CrossRef]

17. Y.-H. Liau, S. Egusa, and N. F. Scherer, “Ultrafast interferometric measurements of plasmonic transport in photonic crystals,” Opt. Lett. **27**, 857 (2002). [CrossRef]

18. J. A. Sánchez-Gil and A. A. Maradudin, “Resonant scattering of surface-plasmon polariton pulses by nanoscale metal defects,” Opt. Lett. **28**, 2255 (2003). [CrossRef] [PubMed]

18. J. A. Sánchez-Gil and A. A. Maradudin, “Resonant scattering of surface-plasmon polariton pulses by nanoscale metal defects,” Opt. Lett. **28**, 2255 (2003). [CrossRef] [PubMed]

18. J. A. Sánchez-Gil and A. A. Maradudin, “Resonant scattering of surface-plasmon polariton pulses by nanoscale metal defects,” Opt. Lett. **28**, 2255 (2003). [CrossRef] [PubMed]

**28**, 2255 (2003). [CrossRef] [PubMed]

## 2. Pulsed SPP scattering model

*x*

_{3}=0, the semi-infinite metal occupying the lower half-space) in the positive

*x*

_{1}direction impinges from the negative

*x*

_{1}axis on a line defect characterized by its profile function

*x*

_{3}=

*f*(

*x*

_{1}) (constant along the

*x*

_{2}axis) and located at the origin. The incident

*p*-polarized SPP pulse is characterized by its only non-zero component of the magnetic field in vacuum,

*F*(

*ω*).

*k*(

*ω*) and

*β*

_{0}(

*ω*) are the SPP wave vector components [19]; since we make use of the impedance boundary condition on a plane [20] in the theoretical formulation of this problem, these components are given by:

*ω*) is the metal dielectric function.

*α*

_{0}(

*q*,

*ω*)≡(

*ω*

^{2}/

*c*

^{2}-

*q*

^{2})

^{1/2}, and

*R*(

*q*,

*ω*) the scattering amplitude for a monochromatic wave of frequency

*ω*. It is shown in Ref. [7], upon imposing the impedance boundary condition on a plane, that the scattering amplitude can be written in the form

*G*

_{0}(

*q*,

*ω*) is the Green’s function of the SPP on the unperturbed surface

*T*(

*q*,

*ω*) satisfies the equation

*V*(

*q*|

*p*) is connected to the surface impedance

*s*(

*x*

_{1}) through

*ρ*(

*ω*) and τ (

*ω*) are, respectively, the monochromatic SPP reflection and transmission amplitudes [7]:

*k*

^{R}(

*ω*) denoting the real part of

*k*(

*ω*).

*x*

_{1},

*x*

_{3})≡(

*r*cos

*θ*,

*r*sin

*θ*) (where

*θ*is the scattering angle) in the limit that

*r*≫

*λ*by means of the method of stationary phase. The resulting expression has the form of propagating pulses of outgoing cylindrical waves:

**H**Maxwell equation) from the (incident plus scattered) magnetic field, Eqs. (2) and (4). In what follows, the metal substrate is assumed to be silver, its dielectric function given by Drude’s formula ε (

*ω*)=1-

*ω*

^{2}, with

*λ*

_{p}≡2

*πc*/

*ω*

_{p}=157 nm. It is well known that for frequencies in the visible and near IR below the onset of the strong ohmic losses associated with interband transitions (at λ~400 nm for Ag), Drude’s free-electron approximation correctly describes the metal response. In addition, we neglect the absorptive contribution to Drude’s formula, since the corresponding SPP inelastic mean free path

*ℓ*

_{abs}is much larger than the defect size (e.g.

*ℓ*

_{abs}~24µm for Ag at λ=650 nm); thus no significant effects are expected in the scattering process [5

5. A. V. Shchegrov, I. V. Novikov, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a circularly symmetric surface defect,” Phys. Rev. Lett. **78**, 4269 (1997). [CrossRef]

## 3. Rectangular groove

**28**, 2255 (2003). [CrossRef] [PubMed]

*L*and depth |

*h*|,

*h*<0,

*d*(

*ω*)=

*c*[

*ε*(

*ω*)]

^{-1/2}/

*ω*is the skin depth, and Θ(

*u*) is the Heaviside unit step function. The frequency dependent factor in Eq. (15) is introduced, following Ref. [20], in order to preserve the connection between surface impedance and actual corrugation to lowest order in both the defect height and slope [21

21. Note that, strictly speaking, such a linear mapping of the surface corrugation into the surface impedance on the plane may not be correct, as discussed in Ref. [7], in the case of rectangular defects, due to the influence of higher-order terms in the slope. Nonetheless, this should not affect the results for the resonant scattering process, except for, presumably, the actual position and strength of resonances.

*R*

_{SPP}≡|

*ρ*|

^{2},

*T*

_{SPP}≡|τ|

^{2}), and of S, the total, normalized power carried away from the surface in the form of volume electromagnetic waves (obtained by integrating over

*θ*the differential reflection coefficient [7]), presented in Fig. 2 for

*h*=-

*λ*

_{p}=-157.1 nm and

*L*=5

*λ*

_{p}=785 nm. (Incidentally, the results corresponding to a rectangular ridge with identical parameters (

*h*>0), also shown here, reveal no evidence of resonances, with structureless, monotonic spectral variations qualitatively similar to those of Gaussian ridges [18

**28**, 2255 (2003). [CrossRef] [PubMed]

*ω*) and the complex zeroes of the corresponding homogeneous reduced Rayleigh equation [Eq. (7) without the independent term], exhibiting a complicated pattern that cannot be readily fitted to a sum of well defined Lorentzians τ(

*ω*)∝Γ/[

*ω*-(

*ω*

_{R}-

*i*Γ)], but is still roughly attributable to the transmission peaks in Fig. 2.

## 3.1. Single resonance

*ω*

_{0}/

*ω*

_{p}=0.275 (

*λ*

_{0}=571 nm) and spectral width Δ

*ω*/

*ω*

_{0}=0.02 (FWHM~60 fs) as shown in Fig. 2, where it is clearly seen that the pulse spectral envelope covers (only) the resonance at

*ω*

_{0}/

*ω*

_{p}=0.275. Dynamic near-field maps [22] are presented in the movie of Fig. 3 in a logarithmic scale. At negative times, only the evanescent field of the approaching SPP pulse located to the left of the defect is observed. At

*t*=0 (the peak of the incoming SPP pulse right at the center of the defect), the signatures of the scattering process appear in several ways: a reflected SPP that interferes with the incoming pulse and yields the observed fringes to the left of the defect; propagating waves that are scattered into the vacuum with a characteristic angular pattern (discussed below); and the evanescent field of the transmitted SPP seen to the right of the defect. These signatures differ from one defect (or SPP central frequency) to another, revealing the peculiarities of the scattering process for each case. At a certain positive time, the central parts of the reflected and transmitted SPP, and also of the scattered light, are displaced towards their corresponding propagation directions. Finally, all of them should be out of the range of the maps at

*t*>

*x*

_{max}/

*v*

_{SPP}[where the SPP group velocity is given from Eq. (3) by

*v*

_{SPP}/

*c*=(

*ω*/(

*ck*(

*ω*)))

^{3}]. However, as a consequence of the excitation of the resonance in the rectangular groove (see Fig. 2), EM energy is stored in the defect that leaks out, even long after the SPP incoming pulse is gone, in the form of reflected and transmitted SPP (the reflected SPP is weak), and scattered light, with typical decay times given by the resonance lifetime; as discussed below, the latter must be longer than the SPP pulse width for this resonant leakage to be observable.

*h*>0). Although the scattering process at times

*t*≤(Δ

*ω*)

^{-1}is qualitatively similar to that in Fig. 3 (of course, with a different energy balance for the outgoing channels), no resonance is excited at the defect, however. Thus no particular features are observed in the near electric field intensity at times after the central lobes of the reflected and transmitted SPP and of the scattered light leave the scene.

*d*=400 (2

*πc*/

*ω*

_{0}) from the defect are shown in Fig. 5 for both the rectangular groove and ridge; Also included is the time-dependence of the scattered light amplitude

*r*

^{1/2}|

*H*

_{s}| at the same distance and given scattering angle

*θ*=

*θ*

_{max}(at which the corresponding angular distribution of scattered power is maximum). In the case of the rectangular groove at resonance, Fig. 5(a), the negative exponential tails of all the pulses quantitatively confirm the resonant scattering process described in the movie of Fig. 3. By fitting the tails to exp(-

*t*Γ), with linewidth Γ/

*ω*

_{0}=8.7·10

^{-3}±10

^{-5}, the resonance lifetime is obtained (time decay Γ

^{-1}≈35 fs). This value is in fairly good agreement with the resonance HWHM as inferred from

*T*

_{SPP}(

*ω*). The resonance life time can be also probed through the delay time of the transmitted SPP with respect to the freely propagating SPP, more accurately determined by calculating the cross-correlation with longer pulses [18

**28**, 2255 (2003). [CrossRef] [PubMed]

*t*

_{SPP}=

*d*/

*v*

_{SPP}≈

*d*/(0.89

*c*)≈450(2

*π*/

*ω*

_{0}). The reflected SPP and light scattered pulses are highly distorted, and present two lobes (advanced and retarded) stemming from the dip of

*R*

_{SPP}(

*ω*) and

*S*(

*ω*) at

*ω*/

*ω*

_{p}=0.275 [see Fig. 2(b)]. On the other hand, the response of the rectangular ridge, see Fig. 5(b), is much simpler, as expected from the fairly dispersion-less

*R*

_{SPP}(

*ω*),

*T*

_{SPP}(

*ω*), and

*S*(

*ω*) in Fig. 2, and in agreement with the movie of Fig. 3. All the outgoing pulses preserve the Gaussian shape of the incoming SPP pulse, rescaled by the corresponding coefficients, and with central lobes occurring at the expected times given either by the SPP group velocity (for the reflected and transmitted SPP) or by the speed of light (for the scattered light).

*I*(

*θ*,

*t*). For the rectangular groove at resonance [see Fig. 6(a)], the entire angular distribution, which consists of a diffraction-like fringe pattern, is distorted along its time evolution; moreover, such distortion differs slightly from one scattering angle to another, thus leading to qualitative changes in the angular distribution. In contrast, the fringe pattern associated with the rectangular ridge [see Fig. 6(b)] evolves uniformly in time according to the Gaussian pulse envelope, as expected.

## 3.2. Two resonances

*ω*

_{0}/

*ω*

_{p}=0.247 (

*λ*

_{0}=636 nm) and Δ

*ω*/

*ω*

_{0}=0.035 (FWHM~39 fs), which ensure that the pulse spectral envelope covers the two resonances at, respectively,

*ω*

_{1}/

*ω*

_{p}=0.253 and

*ω*

_{2}/

*ω*

_{p}=0.241 (see Fig. 2). The near field maps showing the approach of the incoming SPP and subsequent scattering, before the reflected and transmitted SPP and the scattered light move away from the rectangular defect, are similar to what is shown above for a single resonance. Nevertheless, at longer time steps, the electric field intensity on the defect bounces back and forth from one end of the defect to the other, simultaneously leaking out predominantly from the end with a higher concentration of EM energy density. This is a manifestation of the interference between the two resonances being excited [23].

*θ*

_{max}, are neatly observed superimposed on negative exponential tails at longer times. Such tails provide information on the resonance linewidths and shifts; assuming that both resonances have similar lifetimes, the exponentially decaying envelope yields Γ

_{1}/ω0≈Γ

_{2}/

*ω*

_{0}≈0.012±10

^{-3}, whereas the frequency of the oscillations leads to

*ω*

_{1}-

*ω*

_{2}≈0.05

*ω*

_{0}≈0.012

*ω*

_{p}, in good agreement with the values inferred from Fig. 2.

## 4. Gaussian groove

**28**, 2255 (2003). [CrossRef] [PubMed]

**28**, 2255 (2003). [CrossRef] [PubMed]

*Q*value of the excited mode of the Gaussian groove as compared to that of the rectangular groove (at close frequencies), despite being deeper and narrower, revealing the critical role played by the shape of the groove.

## 5. Conclusions

*k*-space formulation, based on the reduced Rayleigh equation, obtained by means of an impedance boundary condition, for every spectral component of the incoming pulse, which enables us to study the dynamics of the scattering of a pulsed-SPP by a surface profile line defect, both in the far field and in the near field. This formulation has been exploited to investigate the time dependence of the near-field, resonant scattering effects produced at a rectangular groove. The scattering process is tracked through the (time-resolved) repartition of the incoming SPP EM energy into reflected and transmitted SPP pulses, and into pulsed, scattered light. Furthermore, we directly show evidence of the excitation of single resonances, as manifested by the concentration of electric field intensity within the groove during the resonance lifetime and long after the incoming SPP pulse is gone (provided that the pulse width is sufficiently shorter than this lifetime); and then by the subsequent leakage of electric field intensity from the groove, on a time scale also controlled by the resonance lifetime, leading in the far field to exponential tails of the reflected and transmitted SPP pulses, and also of the time dependence of the angular distribution of scattered light. Simultaneous excitation of two resonances is also considered, exhibiting the near-field formation of Rabi-like oscillations. Interestingly, our formulation can be straightforwardly employed to study the near-field (and far-field) dynamics for a finite number of line defects, arbitrarily distributed, which is of much interest in SPP NanoOptics [24

24. W. L. Barnes, A. Dereux, and T.W. Ebbesen, “Surface plasmon sub-wavelength optics,” Nature **424**, 824 (2003). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | T. A. Leskova and N. I. Gapotchenko, “Fabry-Perot type interferometer for surface polaritons: resonance effects,” Solid State Commun. |

2. | B. Rothenhäusler and W. Knoll, “Surface plasmon interferometry in the visible,” Appl. Phys. Lett. |

3. | B. Rothenhäusler and W. Knoll, “Interferometric determination of the complex wave vector of plasmon surface polaritons,” J. Opt. Soc. Am. B |

4. | F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, “Scattering of a surface plasmon polariton by a surface defect,” Phys. Rev. B |

5. | A. V. Shchegrov, I. V. Novikov, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a circularly symmetric surface defect,” Phys. Rev. Lett. |

6. | J. A. Sánchez-Gil, “Surface defect scattering of surface plasmon polaritons: Mirrors and light emitters,” Appl. Phys. Lett. |

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

8. | T. A. Leskova, A. A. Maradudin, and W. Zierau, “Surface plasmon polariton propagation near an index step,” Proc. SPIE |

9. | Z. Schlesinger and A. J. Sievers, “Infrared surface wave interferometry,” Appl. Phys. Lett. |

10. | B. Rothenhäusler and W. Knoll, “On the influence of the propagation length of plasmon surface polaritons in the visible energy range for the optical characterization of heterogeneous thin films,” Surf. Sci. |

11. | B. Rothenhäusler and W. Knoll, “Total internal diffraction of plasmon surface polaritons,” Appl. Phys. Lett. |

12. | B. Rothenhäusler and W. Knoll, “Surface plasmon microscopy,” Nature |

13. | C. E. H. Berger, R. P. H. Koioyman, and J. Greve, “Surface plasmon propagation near an index step,” Opt. Commun. |

14. | I. Smolyaninov, D. L. Mazzoni, and C. C. Davis, “Imaging of surface plasmon scattering by lithographically created individual surface defects,” Phys. Rev. Lett. |

15. | I. I. Smolyaninov, D. L. Mazzoni, J. Mait, and C. C. Davis, “Experimental study of surface plasmon scattering by individual surface defects,” Phys. Rev. B |

16. | A. Dogariu, T. Thio, L. J. Wang, T. W. Ebbesen, and H. J. Lezec, “Delay in light transmission through small apertures,” Opt. Lett. |

17. | Y.-H. Liau, S. Egusa, and N. F. Scherer, “Ultrafast interferometric measurements of plasmonic transport in photonic crystals,” Opt. Lett. |

18. | J. A. Sánchez-Gil and A. A. Maradudin, “Resonant scattering of surface-plasmon polariton pulses by nanoscale metal defects,” Opt. Lett. |

19. | H. Raether, |

20. | A. A. Maradudin, “An impedance boundary condition for a rough surface,” in |

21. | Note that, strictly speaking, such a linear mapping of the surface corrugation into the surface impedance on the plane may not be correct, as discussed in Ref. [7], in the case of rectangular defects, due to the influence of higher-order terms in the slope. Nonetheless, this should not affect the results for the resonant scattering process, except for, presumably, the actual position and strength of resonances. |

22. | In our two-dimensional geometry, a near-field area at constant height converts into a single line; we actually merge in a single map all such line scans, from the vacuum-metal interface up to a certain, maximum height. |

23. | We have verified that both resonances are decoupled by separately probing each with surface plasmon polariton pulses appropriately tuned. |

24. | W. L. Barnes, A. Dereux, and T.W. Ebbesen, “Surface plasmon sub-wavelength optics,” Nature |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(260.5740) Physical optics : Resonance

(290.5880) Scattering : Scattering, rough surfaces

(320.2250) Ultrafast optics : Femtosecond phenomena

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 16, 2003

Revised Manuscript: February 27, 2004

Published: March 8, 2004

**Citation**

Jose Sanchez-Gil and Alexei Maradudin, "Dynamic near-field calculations of surface-plasmon polariton pulses resonantly scattered at sub-micron metal defects," Opt. Express **12**, 883-894 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-5-883

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### References

- T. A. Leskova and N. I. Gapotchenko, �??Fabry-Perot type interferometer for surface polaritons: resonance effects,�?? Solid State Commun. 53, 351 (1985). [CrossRef]
- B. Rothenhausler and W. Knoll, �??Surface plasmon interferometry in the visible,�?? Appl. Phys. Lett. 52, 1554 (1988). [CrossRef]
- B. Rothenhausler and W. Knoll, �??Interferometric determination of the complex wave vector of plasmon surface polaritons,�?? J. Opt. Soc. Am. B 5, 1401 (1988). [CrossRef]
- F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, �??Scattering of a surface plasmon polariton by a surface defect,�?? Phys. Rev. B 50, 15261 (1994). [CrossRef]
- A. V. Shchegrov, I. V. Novikov, and A. A. Maradudin, �??Scattering of surface plasmon polaritons by a circularly symmetric surface defect,�?? Phys. Rev. Lett. 78, 4269 (1997). [CrossRef]
- J. A. Sanchez-Gil, �??Surface defect scattering of surface plasmon polaritons: Mirrors and light emitters,�?? Appl. Phys. Lett. 73, 3509 (1998). [CrossRef]
- 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 (1999).
- T. A. Leskova, A. A. Maradudin, and W. Zierau, �??Surface plasmon polariton propagation near an index step,�?? Proc. SPIE 4100, 1 (2000). [CrossRef]
- Z. Schlesinger and A. J. Sievers, �??Infrared surface wave interferometry,�?? Appl. Phys. Lett. 36, 409 (1980). [CrossRef]
- B. Rothenh¨ausler and W. Knoll, �??On the influence of the propagation length of plasmon surface polaritons in the visible energy range for the optical characterization of heterogeneous thin films,�?? Surf. Sci. 191, 585 (1987). [CrossRef]
- B. Rothenhausler and W. Knoll, �??Total internal diffraction of plasmon surface polaritons,�?? Appl. Phys. Lett. 51, 783 (1987). [CrossRef]
- B. Rothenhausler and W. Knoll, �??Surface plasmon microscopy,�?? Nature 332, 615 (1988) [CrossRef]
- C. E. H. Berger, R. P. H. Koioyman, and J. Greve, �??Surface plasmon propagation near an index step,�?? Opt. Commun. 167, 183 (1999). [CrossRef]
- I. Smolyaninov, D. L. Mazzoni, and C. C. Davis, �??Imaging of surface plasmon scattering by lithographically created individual surface defects,�?? Phys. Rev. Lett. 78, 2823 (1997).
- I. I. Smolyaninov, D. L. Mazzoni, J. Mait, and C. C. Davis, �??Experimental study of surface plasmon scattering by individual surface defects,�?? Phys. Rev. B 56, 1601 (1997). [CrossRef]
- A. Dogariu, T. Thio, L. J. Wang, T. W. Ebbesen, and H. J. Lezec, �??Delay in light transmission through small apertures,�?? Opt. Lett. 26, 450 (2001). [CrossRef]
- Y.-H. Liau, S. Egusa, and N. F. Scherer, �??Ultrafast interferometric measurements of plasmonic transport in photonic crystals,�?? Opt. Lett. 27, 857 (2002). [CrossRef]
- J. A. Sanchez-Gil and A. A. Maradudin, �??Resonant scattering of surface-plasmon polariton pulses by nanoscale metal defects,�?? Opt. Lett. 28, 2255 (2003). [CrossRef] [PubMed]
- H. Raether, Surface Polaritons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988).
- A. A. Maradudin, �??An impedance boundary condition for a rough surface,�?? in Topics in Condensed Matter Physics, ed. M. P. Das (Nova, New York, 1994), p. 33.
- Note that, strictly speaking, such a linear mapping of the surface corrugation into the surface impedance on the plane may not be correct, as discussed in Ref. [7], in the case of rectangular defects, due to the influence of higher-order terms in the slope. Nonetheless, this should not affect the results for the resonant scattering process, except for, presumably, the actual position and strength of resonances.
- In our two-dimensional geometry, a near-field area at constant height converts into a single line; we actually merge in a single map all such line scans, from the vacuum-metal interface up to a certain, maximum height.
- We have verified that both resonances are decoupled by separately probing each with surface plasmon polariton pulses appropriately tuned.
- W. L. Barnes, A. Dereux, and T.W. Ebbesen, �??Surface plasmon sub-wavelength optics,�?? Nature 424, 824 (2003). [CrossRef] [PubMed]

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