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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 8 — Apr. 22, 2013
  • pp: 9734–9756
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Scattering of a surface plasmon polariton by a localized dielectric surface defect

Rodrigo E. Arias and Alexei A. Maradudin  »View Author Affiliations


Optics Express, Vol. 21, Issue 8, pp. 9734-9756 (2013)
http://dx.doi.org/10.1364/OE.21.009734


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Abstract

On the basis of a rigorous, nonperturbative, purely numerical solution of the corresponding reduced Rayleigh equation for the scattering amplitudes we have studied the scattering of a surface plasmon polariton by a two dimensional dielectric defect on a planar metal surface. The profile of the defect is assumed to be an arbitrary single-valued function of the coordinates in the plane of the metal surface, and to be differentiable with respect to those coordinates. When the defect is circularly symmetric and the dependence of the scattering amplitudes on the azimuthal angle is expressed by a rotational expansion, the reduced Rayleigh equation is transformed into a pair of one-dimensional integral equations for each value of the rotational quantum number. This approach is applied to a defect in the form of an isotropic Gaussian function. The differential cross sections for the scattering of the incident surface plasmon polariton into volume electromagnetic waves in the vacuum above the surface and into other surface plasmon polaritons are calculated, as well as the intensity of the field near the surface. These results differ significantly from the corresponding results for a metallic defect on a metallic substrate.

© 2013 OSA

A surface plasmon polariton incident on a surface defect is partly scattered into other surface plasmon polaritons and is partly converted into volume electromagnetic waves in the vacuum above the surface. The scattering out of the beam caused by the presence of surface defects decreases the propagation length of the surface plasmon polariton, and it is important for applications of these surface electromagnetic waves to be able to calculate the cross section for such scattering. At the same time surface defects of particular forms and sizes can scatter surface plasmon polaritons in desirable ways, e.g. they can act as mirrors for surface plasmon polaritons or as flashlights, and can focus them as well [1

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

10

10. G. Brucoli and L. Martín-Moreno, “Effect of defect depth on surface plasmon scattering by subwavelength surface defects,” Phys. Rev. B 83, 075433 (2011) [CrossRef] .

]. To exploit the possibilities this offers it is also necessary to be able to calculate the scattering of surface plasmon polaritons by surface defects.

Until now the great majority of such calculations have been carried out for scattering by one-dimensional defects, viz. grooves and ridges. In contrast the scattering of a surface plasmon polariton by a localized two-dimensional surface defect has been little studied. The scattering from indentations (dimples) or protuberances formed from the same metal as the substrate has been studied by a rigorous approach [11

11. 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, 4269–4272 (1997). Erratum: Phys. Rev. Lett. 79, 2597 (1997) [CrossRef] .

]. The scattering from a dielectric rectangular parallelepiped on the planar surface of metallic film in the Kretschmann attenuated total reflection geometry [12

12. E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Z. Naturforsch A23, 2135–2136 (1963).

] has been studied by a rigorous approach [13

13. M. Paulus and O. J. F. Martin, “Light propagation and scattering in stratified media: a Green’s tensor approach,” J. Opt. Soc. Am. A 18, 854–861 (2001) [CrossRef] .

]. The scattering from a dielectric defect in the shape of an anisotropic Gaussian and of an anisotropic hemiellipsoid on the planar surface of a semi-infinite metal has been calculated with the use of an effective boundary condition [14

14. B. Baumeier, F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a localized dielectric surface defect studied using an effective boundary condition,” Phys. Rev. A84, 013810 (2011).

]. The interaction of a surface plasmon polariton with a spatially localized dielectric surface defect is of experimental interest [14

14. B. Baumeier, F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a localized dielectric surface defect studied using an effective boundary condition,” Phys. Rev. A84, 013810 (2011).

], and deserves additional study. Such a study is presented in this paper.

Our treatment is based on the reduced Rayleigh equation for the electric field in the vacuum region above the metal surface and the dielectric defect on it. It is applicable to defects defined by single-valued differentiable profile functions, is rigorous for defects with small slopes, and is computationally tractable. It has worked well in a recent theoretical study of this scattering of light from a planar metal surface coated with a dielectric film whose interface with the vacuum above it is a two-dimensional randomly rough interface [15

15. I. I. Smolyaninov, J. E. Elliott, A.V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett. 94, 057401 (2005) [CrossRef] [PubMed] .

]. We expect that it will work equally well when the incident volume wave is replaced by a surface electromagnetic wave, and the rough dielectric film is replaced by a localized dielectric defect.

The physical system we consider consists of vacuum (ε1) in the region x3 > ζ (x), where x = (x1, x2, 0), a dielectric medium whose dielectric constant is ε2 in the region 0 < x3 < ζ (x), and a metal whose dielectric function is ε3, in the region x3 < 0. The surface profile function ζ (x) is a non-negative, single-valued function of x, that is differentiable with respect to x1 and x2, and is sensibly nonzero within a region of radius R on the metal surface x3 = 0. It therefore defines a dielectric defect with a finite footprint on a planar metal surface. We assume that the dielectric function ε3 is real, because the mean free path of a surface plasmon polariton on the planar portion of the vacuum-metal interface is significantly longer than the linear dimensions of the surface defect.

A surface plasmon polariton of frequency ω is incident on the defect from the region x1 < −R of the plane x3 = 0, where we have a planar interface between media 1 and 3 at x3 = 0. The total electric field in the region x3 > ζ (x) is the sum of the incident field and the scattered field,
E(x|ω)=cω[iβ1(k||)k^||+k||x^3]Eop(k||)exp[ik||x||β1(k||)x3]+d2q||(2π)2{cω[iβ1(q||)q^||q||x^3]ap(q||)ε3β1(q||)+ε1β3(q||)+(x^3+q^||)as(q||)β1(q||)+β3(q||)}exp[iq||x||β1(q||)x3].
(1)
In obtaining this equation a time dependence of the field of the form exp(−iωt) has been assumed but not indicated explicitly. The two-dimensional wave vector k is given by
k||=k||(cosϕ0,sinϕ0,0),
(2a)
where
k||=ωc(ε1ε3ε1+ε3)12
(2b)
is the wavenumber of the surface plasmon polariton of frequency ω at the planar interface between vacuum (ε1) and a metal (ε3). It is the solution of the equation ε3β1(k) + ε1β3(k) = 0. The angle ϕ0 is the azimuthal angle of incidence of the surface plasmon polariton, measured counterclockwise from the positive x1 axis. The functions βj(q) (j = 1, 2, 3) are defined by
βj(q||)=[q||2εj(ω/c)2]12,Reβj(q||)>0,Imβj(q||)<0.
(3)
A caret over a vector indicates that it is a unit vector. Finally, ap,s(q) are the amplitudes of the p- and s-polarized components of the scattered field with respect to the local scattering plane defined by the vectors 3 and . The amplitudes ap(q) and as(q) satisfy the pair of coupled reduced Rayleigh equations [17

17. T. A. Leskova and A. A. Maradudin (unpublished work).

]:
ap(p||)ε2ε12ε2β2(p||)d2q||(2π2){(ε2β3(p||)+ε3β2(p||))×[p||q||β2(p||)(p^||q^||)β1(q||)]J(+)(β2(p||)β1(q||)|p||q||)β2(p||)β1(q||)+(ε2β3(p||)ε3β2(p||))[p||q||+β2(p||)(p^||q^||)β1(q||)]×J()(β2(p||)+β1(q||)|p||q||)β2(p||)+β1(q||)}ap(q||)ε3β1(q||)+ε1β3(q||)ε2ε12ε2β2(p||)d2q||(2π)2{iωcβ2(p||)(p^||×q^||)3×[(ε2β3(p||)+ε3β2(p||))J(+)(β2(p||)β1(q||)|p||q||)β2(p||)β1(q||)(ε2β3(p||)ε3β2(p||))J()(β2(p||)+β1(q||)|p||q||)β2(p||)+β1(q||)]×as(q||)β3(q||)+β1(q||)=ε2ε12ε2β2(p||){(ε2β3(p||)+ε3β2(p||))[p||k||β2(p||)(p^||k^||)β1(k||)]×J(+)(β2(p||)β1(k||)|p||k||)β2(p||)β1(k||)+(ε2β3(p||)ε3β2(p||))[p||k||+β2(p||)(p^||k^||)β1(k||)]×J()(β2(p||)+β1(k||)|p||k||)β2(p||)+β1(k||)}Eop(k||)
(4a)
as(p||)(ωc)2ε2ε12β2(p||)d2q||(2π)2icω(p^||×q^||)β1(q||)×[(β3(p||)+β2(p||))J(+)(β2(p||)β1(q||)|p||q||)β2(p||)β1(q||)+(β3(p||)β2(p||))J()(β2(p||)+β1(q||)|p||q||)β2(p||)+β1(q||)]×ap(q||)ε3β1(q||)+ε1β3(q||)(ωc)2ε2ε12β2(p||)d2q||(2π)2(p^||q^||)×[(β3(p||)+β2(p||))J(+)(β2(p||)β1(q||)|p||q||)β2(p||)β1(q||)+(β3(p||)β2(p||))J()(β2(p||)+β1(q||)|p||q||)β2(p||)+β1(q||)]×as(q||)β3(q||)+β1(q||)=(ωc)2ε2ε12β2(p||)i(cω)(p^||×k^||)3β1(k||)×[(β3(p||)+β2(p||))J(+)(β2(p||)β1(k||)|p||k||)β2(p||)β1(k||)+(β3(p||)β2(p||))J()(β2(p||)+β1(k||)|p||k||)β2(p||)+β1(k||)]Eop(k||),
(4b)
where
J(±)(β2(p||)β1(q||)|p||q||)=d2x||exp[i(p||q||)x||]{exp[(±β2(p||)β1(q||))ζ(x||)]1}.
(5)

A derivation of these equations is outlined in the Appendix.

Equations (4a) and (4b) are valid for any localized surface defect whose profile function ζ (x) satisfies the assumptions about it stated above. However, they simplify significantly when ζ (x) is a function of x only through its magnitude x. In this case we introduce the expansions
ap,s(p||)=k=ak(p,s)(p||)exp(ikϕp),
(6)
where ϕp is the azimuthal angle of the vector p, measured counterclockwise from the positive x1 axis. We also have the expansions
J(+)(β2(p||)β1(q||)|p||q||)=k=ck(+)(p|||q||)exp[ik(ϕpϕq)]
(7a)
J()(β2(p||)+β1(q||)|p||q||)=k=ck()(p|||q||)exp[ik(ϕpϕq)]
(7b)
where
ck(+)(p|||q||)=2πn=1(β2(p||)β1(q||))nn!×0dx||x||ζn(x||)Jk(p||x||)Jk(q||x||)
(8a)
ck()(p|||q||)=2πn=1(1)n(β2(p||)+β1(q||))nn!×0dx||x||ζn(x||)Jk(p||x||)Jk(q||x||),
(8b)
and Jk(z) is a Bessel function of the first kind and order k. The equations satisfied by the amplitudes { ak(p,s)(p||)} are
ak(p)(p||)ε2ε12ε2β2(p||)0dq||2πq||{ε2β3(p||)+ε3β2(p||)β2(p||)β1(q||)×[p||q||ck(+)(p|||q||)12β2(p||)β1(q||)(ck1(+)(p|||q||)+ck+1(+)(p|||q||))+ε2β3(p||)ε3β2(p||)β2(p||)+β1(q||)×[p||q||ck()(p|||q||)+12β2(p||)β1(q||)(ck1()(p|||q||)+ck+1()(p|||q||))]}ak(p)(q||)ε3β1(q||)+ε1β3(q||)ε2ε14ε2ωc0dq||2πq||{ε2β3(p||)+ε3β3(p||)β2(p||)β1(q||)×[ck1(+)(p|||q||)ck+1(+)(p|||q||)]ε2β3(p||)ε3β2(p||))β2(p||)+β1(q||)[ck1()(p|||q||)ck+1()(p|||q||)]}ak(s)(q||)β3(q||)+β1(q||)=(ε2ε1)2ε2β2(p||){ε2β3(p||)+ε3β2(p||)β2(p||)β1(k||)[p||k||ck(+)(p|||k||)12β2(p||)β1(k||)(ck1(+)(p|||k||)+ck+1(+)(p|||k||))]+ε2β3(p||)ε3β2(p||)β2(p||)+β1(k||)[p||k||ck()(p|||k||)+12β2(p||)β1(k||)(ck1()(p|||k||)+ck+1()(p|||k||))]}×exp(ikϕ0)Eop(k||).
(9a)
ak(s)(p||)+ωc(ε2ε1)4β2(p||)0dq||2πq||{β3(p||)+β2(p||)β2(p||)β1(q||)×[ck1(+)(p|||q||)ck+1(+)(p|||q||)]+β3(p||)β2(p||)β2(p||)+β1(q||)×[ck1()(p|||q||)ck+1()(p|||q||)]}β1(q||)ak(p)(q||)ε3β1(q||)+ε1β3(q||)ω2c2ε2ε14β2(p||)0dq||(2π)q||{β3(p||)+β2(p||)β2(p||)β1(q||)×[ck1(+)(p|||q||)+ck+1(+)(p|||q||)]+β3(p||)β2(p||)β2(p||)+β1(q||)×[ck1()(p|||q||)+ck+1()(p|||q||)]}ak(s)(q||)β3(q||)+β1(q||)=ωcε2c14β2(p||){β3(p||)+β2(p||)β2(p||)β1(k||)[ck1(+)(p|||k||)ck+1(+)(p|||k||)]+β3(p||)β2(p||)β2(p||)+β1(k||)[ck1()(p|||k||)ck+1()(p|||k||)]}×β1(k||)exp[ikϕ0]Eop(k||).
(9b)
Thus, for a circularly symmetric defect the equations for different k values decouple, reducing the problem to one of solving one-dimensional integral equations.

Equations (9) for each value of the azimuthal integer k were solved by transforming them into linear matrix equations following the spirit of the method used in Ref. [18

18. A. A. Maradudin and W. M. Visscher, “Electrostatic and electromagnetic surface shape resonances,” Z. Phys. B60, 215–230 (1985) [CrossRef] .

]. The integration over the wavenumber q was carried out over a dimensionless variable y defined by q||=2y/R, where R is the effective radius of the defect, and a dimensionless frequency Ω was introduced through (ω/c)=2Ω/R. The integration over the wavenumber variable y was split into six intervals: (0, Ω), (Ω, ypδ), (yp + δ, 2yp − Ω), (2yp − Ω, ε2Ω), (ε2Ω, KΩ), (KΩ, ∞), where K is an integer (chosen to be 7) and ε2 is the dielectric constant of the defect, plus the contribution from the pole corresponding to the wavenumber of the surface plasmon polariton at y = yp. This separation of the region of integration was based on the behavior of β1(q) and β3(q) as functions of the wave numbers q, namely on the values of the latter variable at which they change from purely imaginary to purely real, on the location of the surface plasmon polariton pole, as well as on how the integration to infinite wave number was to be carried out. The integral over the range (KΩ, ∞) was calculated by Gauss-Laguerre integration, while the integrals over the preceding five intervals were calculated by Gauss-Legendre integration. The number of discretization points in each interval was determined by evaluating the integrals over q on the left-hand sides of Eqs. (9a) and (9b), with the expressions for akp,s(q||) (with k = 1) appearing in the integrands given by the right-hand sides of these equations, by two different methods. In one method the integral over each interval was evaluated by an integration software package (Mathematica, that uses a method with iteration until it attains a specified accuracy). In the second method a Gaussian integration scheme was used, with the number of abscissas being increased until the same accuracy was achieved. In both of these methods the integrand is known analytically, but the discretization used by them is different. The total number of abscissas used in carrying out the q integrations in the region (0, ∞) produced by this approach was 180. The contribution from the surface plasmon polariton pole, i.e. from the interval (ypδ, yp + δ), was calculated by the use of the prescription presented in Ref. [11

11. 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, 4269–4272 (1997). Erratum: Phys. Rev. Lett. 79, 2597 (1997) [CrossRef] .

], namely
1fp(q||)=P(1fp(q||))+iπε1β3(k||)(ε12ε32)k||δ(q||k||),
(10a)
fp(q||)ε3β1(q||)+ε1β3(q||),
(10b)
P denotes the Cauchy principal part, ε(ω) = ε3 is the dielectric function of the metal, and k is the zero of fp(q). Finally, in carrying out these calculations we set the amplitude Eop(k) = 1/(2πR).

When ap,s(q) have been determined we can use the results to calculate the scattered field in the vacuum region in the far zone by the use of the method described in Ref. [19

19. D. L. Mills, “Attenuation of surface polaritons by surface roughness,” Phys. Rev. B 12, 4036–4046 (1975). Erratum: Phys. Rev. B14, 5539 (1976) [CrossRef] .

]. The result has the form of an outgoing spherical wave
Evac(sc)(x|ω)==iε1ωcosθx2πceiε1(ω/c)xx×[ε1e^pAp(ε1x^||(ωc)sinθx)+e^sAs(e1x^||(ωc)sinθx)],ε1(ω/c)x1.
(11)
In this expression êp and ês are unit vectors defined by
e^p=(cosθxcosϕx,cosθxsinϕx,sinθx)
(12a)
e^s=(sinϕx,cosϕx,0),
(12b)
with
cosθx=x3x,sinθx=x||x
(13a)
cosϕx=x1x||,sinϕx=x2x||,
(13b)
where (θx, ϕx) are the polar and azimuthal angles of the vector , while
Ap(q||)=ap(q||)ε3β1(q||)+ε1β3(q||)
(14a)
As(q||)=as(q||)β1(q||)+β3(q||).
(14b)

The surface plasmon polariton contribution to the scattered field is given by the residue at the pole of the integrand in Eq. (1) at q = k. It is given by an outgoing cylindrical wave
Espp(sc)(x|ω)=ieik||x||β1(k||)x3iπ4(2πk||x||)12×cε1β3(k||)ωix^||β1(k||)x^3k||ε32ε12ap(x^||k||),k||x||1.
(15)

We introduce the differential cross sections, measured in units of length, for scattering into the vacuum and into other surface waves by
σvac(θx,ϕx)=Pvac(θx,ϕx)Pinc
(16)
σspp(ϕx)=Pspp(ϕx)Pinc,
(17)
where Pvac(θx, ϕx) is the power scattered into the vacuum away from the surface in the direction , Pspp(ϕx) is the power scattered into other surface waves in the direction , and Pinc is the incident power per unit length in the x2 direction. These powers are given by
Pvac(θx,ϕx)=c8πε1(ω2πc)2cos2θx×{ε1|Ap(ε1x^||(ωc)sinθx)|2+|As(ε1x^||(ωc)sinθx)|2}
(18)
Pspp(ϕx)=12ε13(c4π)2β32(k||)ωβ1(k||)|ap(x^||k||)|2(ε32ε12)2
(19)
Pinc=ε1c28πωk||2β1(k||)(11ε32(ω))|Eop(k||)|2.
(20)
In what follows we will neglect the second term on the right-hand side of Eq. (20) due to its smallness relative to the first term.

We present numerical results for a dielectric defect of Gaussian form defined by the surface profile function
ζ(x||)=Aexp(x||2/R2),
(21)
In this case the coefficients ck(±)(p|||q||) are given by
ck(±)(p|||q||)=πR2n=1[±A(β2(p||)β1(q||))]nnn!×Ik(R2p||q||2n)exp[R2(p||2+q||2)4n],
(22)
where Ik(z) is a modified Bessel function of the first kind of order k. The Gaussian defect is characterized by a 1/e half width R = 0.25μm. It is assumed to be deposited on a planar aluminum surface. It is illuminated by a surface plasmon polariton propagating in the positive x1 direction (ϕ0 = 0°). The wavelength of the incident surface plasmon polariton is λ =632.8 nm, and the dielectric function of aluminum at this wavelength is ε3=ε3r(ω)+iε3i(ω)=57.19+i11.19[20

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

] of which we used only the real part. The energy mean free path of the surface plasmon polariton at a planar vacuum-aluminum interface at this wavelength is spp(ω)=(λ/2π)|ε3r(ω)|12(|ε3r(ω)|ε1)3/2/[ε13/2ε3i(ω)]=28.67μm. This is more than two orders of magnitude longer than the 0.25μm radius of the surface defect, and justifies our treating ε3 as real. In these calculations the rotational quantum k ranged in integer setps from −30 to +30.

In Fig. 1 we present a contour plot of σvac(θx, ϕx) for a value of the dielectric constant of the defect ε2 = 2.69 and (a) A/R = 0.1, (b) A/R = 0.2, (c) A/R = 0.3. For each of these values of A/R the maximum of the intensity of the field radiated into the vacuum occurs at θx = 70°, ϕx = 0°. More radiation into smaller values of θx with increasing values of A/R is seen, especially in Fig. 1(c). The total cross sections for the waves radiated into the vacuum are (a) 0.0021μm, (b) 0.0057μm, (c) 0.0099 μm respectively. Thus they increase as the amplitude of the defect increases, with the values of the other parameters kept fixed.

Fig. 1 A contour plot of σvac(θx, ϕx). The concentric dashed circles are the lines of constant θx, with θx = 0° at the center, and θx = 90° at the boundary. The azimuthal angle ϕx varies from 0° to 360°. ε2 = 2.69;A/R = 0.1 (a), 0.2 (b), 0.3 (c).

When the dielectric constant of the defect is increased to ε2 = 5.0, with the remaining parameters maintaining the values they have in Fig. 1 (Fig. 2), the maximum of the intensity of the field scattered into the vacuum remains at θx = 70°, ϕx = 0°, but the strength of the scattering is nearly doubled in comparison with the corresponding results presented in connection with Fig. 1. The total cross sections for the waves radiated into the vacuum are (a) 0.0040μm, (b) 0.0125μm, (c) 0.0290μm. Some radiation into the backward directions is seen in Fig. 2(c).

The angular dependence of σspp(ϕx) is presented in Fig. 3. The dielectric constant of the defect is ε2 = 2.69, and (a) A/R = 0.1, (b) A/R = 0.2, (c) A/R = 0.3. We see that there is no scattering of the surface plasmon polariton into the backward direction for all three values of A/R: all of the scattering is into the forward direction. These results are due presumably to the transparency of the dielectric defect. The shapes of these scattering patterns do not change with increasing values of A/R, but the strength of the scattering does. The total cross sections for the scattering of the incident surface plasmon polariton into other surface plasmon polaritons are (a) 0.0019 μm, (b) 0.0053 μm, (c) 0.0087 μm. These values are nearly the same as the values of σvac(θx, ϕx) for the corresponding parameter values in Fig. 3.

Fig. 2 The same as Fig. 1, but with ε2 = 5.0.
Fig. 3 Plots of σspp(ϕx). ε2 = 2.69;A/R = 0.1 (a), 0.2 (b), 0.3 (c).

The same qualitative behavior of σspp(ϕx) is observed when ε2 is increased to ε2 = 5.0. This is seen from the results presented in Fig. 4. The values of A/R for which these results are calculated are (a) A/R = 0.1, (b) A/R = 0.2, (c) A/R = 0.3. All of the scattering is in the forward direction. The shapes of the scattering patterns do not change as the value of A/R increases, while the strength of the scattering increases with increasing values of A/R. The total cross sections for the scattering of the incident surface plasmon polariton into other surface plasmon polaritons are (a) 0.0035μm, (b) 0.0100μm, (c) 0.0176μm. For each value of A/R increasing ε2 from 2.69 to 5.0 nearly doubles the total scattering cross section from the value it has for the data used in obtaining the corresponding results in connection with Fig. 3.

Fig. 4 The same as Fig. 3, but with ε2 = 5.0.

The forward scattering of the surface plasmon polariton is clearly seen in the results presented in Figs. 5 and 6 which show the field intensity |E(x|ω)|2 as a function of x at 5 nm above the surface profile (x3 = ζ (x) + 5nm), for ε2 = 2.69 and 5.0, respectively, while (a) A/R = 0.1, (b) A/R = 0.2, (c) A/R = 0.3. Both figures show a weak diffractive spreading of the intensity of the field after its interaction with the defect. There is very little indication of the scattering of the surface plasmon polariton in backward directions, in agreement with the results presented in Figs. 12 and 34.

Fig. 5 The field intensity |E(x|ω)|2 as a function of x at x3 = ζ (x)+5 nm. ε2 = 2.69; A/R = 0.1 (a), 0.2 (b), 0.3 (c).
Fig. 6 The same as Fig. 5, but with ε2 = 5.0.

The present results qualitatively resemble analogous results obtained in Ref. [14

14. B. Baumeier, F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a localized dielectric surface defect studied using an effective boundary condition,” Phys. Rev. A84, 013810 (2011).

], bearing in mind that in that reference the surface profile functions of the dielectric surface defects were an anisotropic Gaussian and an anisotropic hemiellipsoid.

The preceding results can be understood if in Eqs. (18) and (19) we substitutee for ap(q) and as(q) the expressions for them obtained in the small roughness limit of the first Born approximation, namely
ap(q||)=ε2ε1ε2ζ^(q||k||)[ε3q||k||ε2β3(q||)(q^||k^||)β1(k||)]E0p(k||)
(23a)
as(q||)=i(ε2ε1)ζ^(q||k||)ωc(q^||×k^||)3β1(k||)E0p(k||),
(23b)
where
ζ^(Q||)=d2x||ζ(x||)exp(iQ||x||)=πAR2exp(R2Q||2/4).
(24)
The second of Eqs. (24) gives the expression for ζ̂(Q) corresponding to the Gaussian surface profile function (21).

On substituting Eqs. (23) into Eq. (18) and making use of Eqs. (16) and (20) we obtain σvac(θx, ϕx) in the form
σvac(θx,ϕx)=12R(AR)2(ωRc)5ε15/2ε22|ε2|12(ε2ε1)2(ε32ε12)cos2θx×exp[R22(k||22ε1ωck||sinθxcosϕx+ε1ω2c2sin2θx)]×a0(θx)+a1(θx)cosϕx+a2(θx)cos2ϕx[|ε3|(|ε3|ε1)sin2θx],
(25)
where
a0(θx)=ε22|ε3|cos2θx+(|ε3|3+ε22ε1)sin2θx
(26a)
a1(θx)=2ε2|ε3|3/2sinθx(|ε3|+ε1sin2θx)12
(26b)
a2(θx)=ε22|ε3|sin2θx.
(26c)
This cross section is proportional to the square of the aspect ratio (A/R) of the defect. It is an increasing function of ε2, and has a maximum at (θx, ϕx) = (70.75°, 0°) for the values of the parameters assumed in obtaining Fig. 1, and at (70.5, 0°) for the values assumed in obtaining Fig. 2. The insensitivity of these angles to changes of ε2, and their closeness to the angular position of the maximum in these figures, namely (70°, 0°), is gratifying, given the simplicity of this approximate calculation.

When Eq. (23a) is substituted in Eq. (19), and Eqs. (17) and (20) are used, we obtain σspp(ϕx) in the form
σspp(θx)=π2R(AR)(ωRc)5|ε3|7/2ε17/2ε22(ε2ε1)2(|ε2|ε1)9/2(|ε2|+ε1)2×exp(2R2k||2sin212ϕx)(|ε3|+ε2cosϕx)2.
(27)
Although the factor (|ε3| + ε2 cosϕx)2 favors scattering into the forward direction by a small amount, because of the large value of |ε3|, the suppression of scattering into backward directions is due entirely to the factor exp(2R2k||2sin212ϕx), which is a consequence of the Gaussian form of ζ̂(Q), Eq. (24). The cross section (27) is proportional to the square of the defects aspect ratio, and is an increasing function of ε2.

The proportionality of the cross sections (25) and (27) to ω5 can be understood in the following way. The cross section for the Rayleigh scattering of a volume wave from a d-dimensional object is proportional to ωd+1. Since the scatterer in the present case is defined by x3ζ(x) = 0, it is a three-dimensional scatterer as far as Rayleigh scattering is concerned. The additional factor of ω arises because the decay length of the surface plasmon polariton into the vacuum is proportional to its wavelength parallel to the surface, which reduces the volume within which its interaction with the defect occurs [21

21. A. A. Maradudin and D. L. Mills, “Attenuation of Rayleigh surface waves by surface roughness,” Ann. Phys. (N.Y.) 100262–309 (1976), Ref. 9 [CrossRef] .

].

In this paper we have derived a pair of coupled two-dimensional integral equations – reduced Rayleigh equations – for the amplitudes of the p- and s-polarized components of the scattered elsctric field produced when a surface plasmon polariton is incident on a localized dielectric surface defect on a planar metal surface. When the defect is circularly symmetric with respect to the normal to the surface a rotational expansion of these scattering amplitudes transforms the reduced Rayleigh equations into an infinite set of uncoupled one-dimensional integral equations that have been solved numerically. We have applied this approach to calculate the near-field and far-field angular distributions of the intensity of the field scattered by a dielectric defect in the form of an isotropic Gaussian on a metal surface for two values of its dielectric constant, and three values of its aspect ratio. It is found that the intensity of the volume electromagnetic field scattered into the vacuum above the surface and defect is a maximum for in-plane scattering at a polar scattering angle of θx = 70°. This value is independent of the aspect ratios and dielectric constants of the defect assumed in our work. It is a significantly larger angle than the angle θx = 28° at which the corresponding maxima occurs in the scattering of a surface plasmon polariton from a circularly symmetric Gaussian indentation (dimple) in a planar metal surface [11

11. 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, 4269–4272 (1997). Erratum: Phys. Rev. Lett. 79, 2597 (1997) [CrossRef] .

]. This result suggests that the excitation of a surface plasma polariton by illuminating this dielectric defect by a p-polarized volume electromagnetic wave will be most efficient for a polar angle of incidence θ0 = 70° [22

22. M. Kretschmann, T. A. Leskova, and A.A. Maradudin, “Excitation of surface plasmon polaritons by the scattering of a volume electromagnetic beam from a circularly symmetric defect on a planar metal surface,” Proc. SPIE 4447, 24–33 (2001) [CrossRef] .

]. It is also found that the surface plasmon polariton is scattered into other surface plasmon polaritons primarily in the forward direction. There is no shadow behind the defect as in the case of the scattering of a surface plasmon polariton from a circularly symmetric indentation in a metal surface [11

11. 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, 4269–4272 (1997). Erratum: Phys. Rev. Lett. 79, 2597 (1997) [CrossRef] .

]. This is due presumably to the transparency of the dielectric defect. Thus there are significant differences between the cross section for the scattering of a surface plasmon polariton from a metallic defect on a metallic surface and for its scattering from a dielectric defect on a metallic surface.

The approach developed here can also be applied to dielectric surface defects that are not circularly symmetric. In this case a rotational expansion of the scattering amplitudes produces a set of one-dimensional integral equations that, unlike Eqs. (9), are coupled for different values of the rotational quantum number k[14

14. B. Baumeier, F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a localized dielectric surface defect studied using an effective boundary condition,” Phys. Rev. A84, 013810 (2011).

].

APPENDIX

To transform them into equations describing the scattering of a surface plasmon polariton from a localized dielectric defect on a planar metallic surface we first set d = 0 and restrict ζ (x) to be a non-negative function of x. We also set E0s(k) ≡ 0 because the incident field is p polarized, and then rewrite I(γ|Q) as
I(γ|Q||)=(2π)2δ(Q||)+J(γ|Q||),
(A.30a)
where
J(γ|Q||)=d2x||eiQ||x||(eiγζ(x||)1).
(A.30b)
The relevant elements of the matrices M(p|q) and N(p|k) then become
Mpp(p|||q||)=(2π)2δ(p||q||)2ε2α2(p||)ε2ε1[ε1α3(p||)+ε3α1(p||)]+M˜pp(p|||q||)
(A.31a)
Mps(p|||q||)=M˜ps(p|||q||)
(A.31b)
Msp(p|||q||)=M˜sp(p|||q||)
(A.31c)
Mss(p|||q||)=(2π)2δ(p|||q||)c2ω22α2(p||)ε2ε1[α3(p||)+α1(p||)]+M˜ss(p|||q||)
(A.31d)
Npp(p|||k||)=(2π)2δ(p|||k||)2ε2α2(k||)ε2ε1[ε3α1(k||)ε1α3(k||)]+N˜pp(p|||k||)
(A.32a)
Nsp(p|||k||)=N˜sp(p|||k||),
(A.32b)
where
M˜pp(p|||q||)=dp(p||)[α2(p||)(p^||q^||)α1(q||)+p||q||]×J(α2(p||)α1(q||)|p||q||)α2(p||)α1(q||)Δp(p||)[α2(p||)(p^||q^||)α1(q||)p||q||]×J((α2(p||)+α1(q||))|p||k||)α2(p||)+α1(q||)
(A.33a)
M˜ps(p|||q||)=ωcα2(p||)(p^||×q^||)3×{dp(p||)J(α2(p||)α1(q||)|p||q||)α2(p||)α1(q||)Δp(p||)J((α2(p||)+α1(q||))|p||q||)α2(p||)+α1(q||)}
(A.33b)
M˜sp(p|||q||)=cω(p^||×q^||)3α1(q||)×{ds(p||)J(α2(p||)α1(q||)|p||q||)α2(p||)α1(q||)+Δs(p||)J((α2(p||)+α1(q||))|p||q||)α2(p||)+α1(q||)}
(A.33c)
M˜ss(p|||q||)=(p^||q^||){ds(p||)J(α2(p||)α1(q||)|p||q||)α2(p||)α1(q||)+Δs(p||)J((α2(p||)+α1(q||))|p||q||)α2(p||)+α1(q||)}
(A.33d)
N˜pp(p|||k||)=dp(p||)[α2(p||)(p^||k^||)α1(k||)p||k||]×J(α2(p||)+α1(k||)|p||k||)α2(p||)+α1(k||)Δp(p||)[α2(p||)(p^||k^||)α1(k||)+p||k||]×J((α2(p||)α1(k||))|p||k||)α2(p||)α1(k||)
(A.34a)
N˜sp(p|||k||)=cω(p^||×k^||)3α1(k||)×{ds(p||)J(α2(p||)+α1(k||)|p||k||)α2(p||)+α1(k||)+Δs(p||)J((α2(p||)α1(k||))|p||k||)α2(p||)α1(k||)}.
(A.34b)
Equations (A.26) and (A.27) now become
[ε1α3(p||)+ε3α1(p||)]Ap(p||)+ε2ε12ε2α2(p||)d2q||(2π)2[M˜pp(p|||q||)Ap(q||)+M˜ps(p|||q||)As(q||)]={(2π)2δ(p||k||)[ε3α1(k||)ε1α3(k||)]+ε2ε12ε2α2(p||)N˜pp(p|||k||)}E0p(k||)
(A.35)
[α3(p||)+α1(p||)]As(p||)+ω2c2ε2ε12α2(p||)d2q||(2π)2[M˜sp(p|||q||)Ap(q||)+M˜ss(p|||q||)As(q||)]=ω2c2ε2ε12α2(p||)N˜sp(p|||k||)E0p(k||).
(A.36)
We now make use of the analytic continuations
α1(k||)=iβ1(k||)
(A.37a)
α1(q||)=iβ1(q||)
(A.37b)
α2(q||)=iβ2(q||)
(A.37c)
α3(q||)=iβ3(q||),
(A.37d)
where
βj(q||)=[q||2εj(ω/c)2]12,Reβj(q||)>0,Imβj(q||)<0.
(A.38)
With the definitions
J(i(β2(p||)β1(q||))|p||q||)=d2x||ei(p||q||)x||(e(β2(p||)β1(q||))ζ(x||)1)=J(+)(β2(p||)β1(q||)|p||q||)
(A.39a)
J(i(β2(p||)+β1(q||))|p||q||)=d2x||ei(p||q||)x||(e(β2(p||)+β1(q||))ζ(x||)1)=J()(β2(p||)+β1(q||)|p||q||)
(A.39b)
and
Ap(q||)=ap(q||)ε3β1(q||)+ε1β3(q||)
(A.40a)
As(q||)=as(q||)β1(q||)+β3(q||),
(A.40b)
the substitution of Eqs. (A.37)(A.40) into Eqs. (A.35) and (A.36) yields Eqs. (4). We have used the result that ε3β1(k) + ε1β3(k) = 0. This is the dispersion relation for surface plasmon polaritons at a planar vacuum (ε1)-metal (ε3) interface.

Acknowledgments

The research of R.E.A. was supported by DOE grant DE-FG02-84ER45083, and partially by “ Financiamiento Basal para Centros científicos y tecnológicos de excelencia,” project FB0807 (Chile). The research of A.A.M. was supported in part by AFRL contract FA9453-08-C-0230.

References and links

1.

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

2.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett. 5, 1726–1729 (2005) [CrossRef] [PubMed] .

3.

I. I. Smolyaninov, C. C. Davis, and A. V. Zayats, “Image formation in surface plasmon polariton mirrors: applications in high resolution optical microscopy,” New. J. Phys. 7, 175 (2005) [CrossRef] .

4.

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006) [CrossRef] [PubMed] .

5.

I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Imaging and focusing properties of plasmonic metamaterial devices,” Phys. Rev. B 76, 205424, (2007) [CrossRef] .

6.

H. Kim, J. Hahn, and B. Lee, “Focusing properties of surface plasmon polariton floating dielectric lenses,” Opt. Express 16, 3049–3057 (2008) [CrossRef] [PubMed] .

7.

J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature 460, 498–501 (2009) [CrossRef] .

8.

I. Chremmos, “Magnetic field integral equation analysis of surface plasmon scattering by rectangular dielectric channel discontinuities,” J. Opt. Soc. Am. A 27, 85–94 (2010) [CrossRef] .

9.

G. Brucoli and L. Martín-Moreno, “Comparative study of surface plasmon polariton scattering by shallow ridges and grooves,” Phys. Rev. B 83, 045422 (2011) [CrossRef] .

10.

G. Brucoli and L. Martín-Moreno, “Effect of defect depth on surface plasmon scattering by subwavelength surface defects,” Phys. Rev. B 83, 075433 (2011) [CrossRef] .

11.

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, 4269–4272 (1997). Erratum: Phys. Rev. Lett. 79, 2597 (1997) [CrossRef] .

12.

E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Z. Naturforsch A23, 2135–2136 (1963).

13.

M. Paulus and O. J. F. Martin, “Light propagation and scattering in stratified media: a Green’s tensor approach,” J. Opt. Soc. Am. A 18, 854–861 (2001) [CrossRef] .

14.

B. Baumeier, F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a localized dielectric surface defect studied using an effective boundary condition,” Phys. Rev. A84, 013810 (2011).

15.

I. I. Smolyaninov, J. E. Elliott, A.V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett. 94, 057401 (2005) [CrossRef] [PubMed] .

16.

T. Nordam, P. A. Letnes, I. Simonsen, and A. A. Maradudin, “Satellite peaks in the scattering of light from the two-dimensional randomly rough surface of a dielectric film on a planar metal surface,” Opt. Express 20, 11336– 11350 (2012) [CrossRef] [PubMed] .

17.

T. A. Leskova and A. A. Maradudin (unpublished work).

18.

A. A. Maradudin and W. M. Visscher, “Electrostatic and electromagnetic surface shape resonances,” Z. Phys. B60, 215–230 (1985) [CrossRef] .

19.

D. L. Mills, “Attenuation of surface polaritons by surface roughness,” Phys. Rev. B 12, 4036–4046 (1975). Erratum: Phys. Rev. B14, 5539 (1976) [CrossRef] .

20.

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

21.

A. A. Maradudin and D. L. Mills, “Attenuation of Rayleigh surface waves by surface roughness,” Ann. Phys. (N.Y.) 100262–309 (1976), Ref. 9 [CrossRef] .

22.

M. Kretschmann, T. A. Leskova, and A.A. Maradudin, “Excitation of surface plasmon polaritons by the scattering of a volume electromagnetic beam from a circularly symmetric defect on a planar metal surface,” Proc. SPIE 4447, 24–33 (2001) [CrossRef] .

OCIS Codes
(240.5420) Optics at surfaces : Polaritons
(240.6680) Optics at surfaces : Surface plasmons
(250.5403) Optoelectronics : Plasmonics
(290.5825) Scattering : Scattering theory

ToC Category:
Optics at Surfaces

History
Original Manuscript: February 11, 2013
Revised Manuscript: March 26, 2013
Manuscript Accepted: April 2, 2013
Published: April 12, 2013

Citation
Rodrigo E. Arias and Alexei A. Maradudin, "Scattering of a surface plasmon polariton by a localized dielectric surface defect," Opt. Express 21, 9734-9756 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-8-9734


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References

  1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London)424, 824–830 (2003). [CrossRef]
  2. Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5, 1726–1729 (2005). [CrossRef] [PubMed]
  3. I. I. Smolyaninov, C. C. Davis, and A. V. Zayats, “Image formation in surface plasmon polariton mirrors: applications in high resolution optical microscopy,” New. J. Phys.7, 175 (2005). [CrossRef]
  4. E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science311, 189–193 (2006). [CrossRef] [PubMed]
  5. I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Imaging and focusing properties of plasmonic metamaterial devices,” Phys. Rev. B76, 205424, (2007). [CrossRef]
  6. H. Kim, J. Hahn, and B. Lee, “Focusing properties of surface plasmon polariton floating dielectric lenses,” Opt. Express16, 3049–3057 (2008). [CrossRef] [PubMed]
  7. J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature460, 498–501 (2009). [CrossRef]
  8. I. Chremmos, “Magnetic field integral equation analysis of surface plasmon scattering by rectangular dielectric channel discontinuities,” J. Opt. Soc. Am. A27, 85–94 (2010). [CrossRef]
  9. G. Brucoli and L. Martín-Moreno, “Comparative study of surface plasmon polariton scattering by shallow ridges and grooves,” Phys. Rev. B83, 045422 (2011). [CrossRef]
  10. G. Brucoli and L. Martín-Moreno, “Effect of defect depth on surface plasmon scattering by subwavelength surface defects,” Phys. Rev. B83, 075433 (2011). [CrossRef]
  11. 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, 4269–4272 (1997). Erratum: Phys. Rev. Lett. 79, 2597 (1997). [CrossRef]
  12. E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Z. NaturforschA23, 2135–2136 (1963).
  13. M. Paulus and O. J. F. Martin, “Light propagation and scattering in stratified media: a Green’s tensor approach,” J. Opt. Soc. Am. A18, 854–861 (2001). [CrossRef]
  14. B. Baumeier, F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a localized dielectric surface defect studied using an effective boundary condition,” Phys. Rev.A84, 013810 (2011).
  15. I. I. Smolyaninov, J. E. Elliott, A.V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett.94, 057401 (2005). [CrossRef] [PubMed]
  16. T. Nordam, P. A. Letnes, I. Simonsen, and A. A. Maradudin, “Satellite peaks in the scattering of light from the two-dimensional randomly rough surface of a dielectric film on a planar metal surface,” Opt. Express20, 11336– 11350 (2012). [CrossRef] [PubMed]
  17. T. A. Leskova and A. A. Maradudin (unpublished work).
  18. A. A. Maradudin and W. M. Visscher, “Electrostatic and electromagnetic surface shape resonances,” Z. Phys.B60, 215–230 (1985). [CrossRef]
  19. D. L. Mills, “Attenuation of surface polaritons by surface roughness,” Phys. Rev. B12, 4036–4046 (1975). Erratum: Phys. Rev. B14, 5539 (1976). [CrossRef]
  20. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972). [CrossRef]
  21. A. A. Maradudin and D. L. Mills, “Attenuation of Rayleigh surface waves by surface roughness,” Ann. Phys. (N.Y.)100262–309 (1976), Ref. 9. [CrossRef]
  22. M. Kretschmann, T. A. Leskova, and A.A. Maradudin, “Excitation of surface plasmon polaritons by the scattering of a volume electromagnetic beam from a circularly symmetric defect on a planar metal surface,” Proc. SPIE4447, 24–33 (2001). [CrossRef]

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