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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 12 — Jun. 9, 2008
  • pp: 9073–9086
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Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals revisited

Bora Ung and Yunlong Sheng  »View Author Affiliations


Optics Express, Vol. 16, Issue 12, pp. 9073-9086 (2008)
http://dx.doi.org/10.1364/OE.16.009073


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Abstract

The asymptotic closed-form solution to the fundamental diffraction problem of a linear horizontal Hertzian dipole radiating over the metallo-dielectric interface is provided. For observation points just above the interface, we confirm that the total surface near-field is the sum of two components: a long-range surface plasmon polariton and a short-range radiative cylindrical wave. The relative phases, amplitudes and damping functions of each component are quantitatively elucidated through simple analytic expressions for the entire range of propagation: near and asymptotic. Validation of the analytic solution is performed by comparing the predictions of a dipolar model with recently published data.

© 2008 Optical Society of America

1. Introduction

Sparked by the discovery of enhanced optical light transmission through subwavelength holes pierced in a metal film [1

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 391, 667–669 (1998). [CrossRef]

], considerable interest has grown in the last decade for the guiding and manipulation of light in metallic nanostructures. A central role is attributed to electromagnetic surface waves (SW) that are launched by light diffraction on a structural nano-defect and subsequently propagated along the metallo-dielectric interface. While novel nano-devices operating on these surface-bound vectors are envisioned in many areas of science and engineering, fundamental-level research is still being performed in order to completely assess and control their intrinsic properties. In this respect, surface plasmon polaritons (SPP) have been identified early on to occupy a predominant role, especially in the asymptotic propagation regime. The SPP, which is a guided electromagnetic wave with fields evanescently extending on either side of the interface, represents a solution stemming from Maxwell’s equations provided the metal substrate has a finite conductivity and the magnetic field is polarized transverse (TM) to the plane of incidence. An alternate type of SW denominated “composite diffracted evanescent wave” (CDEW), and characterized by a field damping scaling as 1/x with propagation distance x, has also been proposed [2

2. 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]

]. The CDEW model is based on a number of approximations. First of all, it derives from a scalar-wave theory which ignores field polarization, and second, it assumes an opaque and infinitely thin screen as the boundary condition, thus precluding any likely involvement of the SPP mode. A key conceptual difference between the CDEW and the classical SPP model is that the former allows several propagating surface modes to be generated by diffraction while the latter only admits the SPP. Both models were applied to interpret recent experimental investigations of the near field with relative success [3–5

3. G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nanostructured surfaces and the composite diffracted evanescent wave,” Nat. Physics 2, 262–267 (2006). [CrossRef]

]: the CDEW is ostensibly better suited in the immediate vicinity of the source whereas the SPP is most accurate further away from it. The indication that each theory presents complementing pictures of the same phenomenon was made apparent by Lalanne and Hugonin [6

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

] when they suggested that the composite diffracted surface wave essentially consists of two components: a SPP and a radiative “creeping wave” characterized by a damping scaling with 1/x 1/2. In their demonstration, the authors used a rigorous Green’s function formalism to describe the field radiated by a line source over a metallo-dielectric half-space. However, no closed-form solution for the creeping wave was provided. New experimental results [7

7. L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julié, V. Mathet, and M. Mortier, “Near-field analysis of surface waves launched at nanoslit apertures,” Phys. Rev. Lett. 98, 153902 (2007). [CrossRef] [PubMed]

] taken from direct measurements of the surface near field have lent further credence to the thesis of a SW dual composition.

In this theoretical contribution, we demonstrate that the scattering process of incident TM-polarized light by a one-dimensional subwavelength nano-defect on an otherwise flat metal surface is generically modeled through the fundamental diffraction problem of a horizontal Hertzian dipole radiating over the metallo-dielectric interface. Starting directly from Maxwell’s equations and enforcing the proper boundary conditions, we write down the exact Sommerfeld-type integral for the electric field perpendicular to the interface. The integral is then rigorously solved via the modified method of steepest descents to obtain the asymptotic closed-form solution. We explicitly show that the total diffracted field along the surface is composed of two distinct contributions as predicted in [6

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

]: a short-range radiative wave and a long-range SPP. The amplitudes, phases and damping functions of both components are quantitatively revealed through simple analytic expressions. Finally, we propose a double-dipole model to characterize the near-field interactions between two nano-objects and show the excellent agreement between our model’s predictions and the experiments, thus validating the analytical solution in the same stretch.

2. Derivation of the closed-form solution

It was demonstrated that the scattering of incident TM-polarized light on a subwavelength metallic slit induces oscillating electric charges around the sharp slit edges that emulate a horizontal electric dipole [8

8. 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]

, 9

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

]. It can be argued that this distinctive feature equally applies to the subwavelength groove if it is considered as a partially filled slit. Hence, the physical problem of light scattering on a nano-slit, or nano-groove, can be represented by the basic diffraction problem of an infinitely small horizontal electric dipole radiating over the interface. That assumption is corroborated by other analytical and experimental investigations [10

10. G. Lévêque, O. J. F. Martin, and J. Weiner, “Transient behavior of surface plasmon polaritons scattered at a subwavelength groove,” Phys. Rev. B. 76, 155418 (2007). [CrossRef]

,11

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

]. From our calculations, this approximation is fairly accurate if the width w of the given nano-object is smaller than a half-wavelength, w<λ/2, and if observation points are located at a distance x from the source larger than x>λ/π. The remaining discussion in this Section follows the contemporary mathematical treatment derived by R. E. Collin for the distinct case of a 3D vertical point dipole radiating over the earth’s surface [12

12. R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: some early and late 20th-century controversies,” IEEE Antennas Propag. Mag. 46, 64–79 (2004). [CrossRef]

]; which was the original diffraction problem famously addressed by Arnold Sommerfeld one century ago.

2.1 Expression of the Sommerfeld integral for a linear horizontal Hertzian dipole

Fig. 1. Radiating horizontal electric dipole located in region 1 (air).

We consider the 2D linear electric dipole oriented along the x-axis and located in the half-space z≥0 at a height h from the interface separating two semi-infinite nonmagnetic and isotropic dielectrics (Fig. 1). The infinitesimal line dipole is described by its surface current density Jx=δ(xδ(z-h) normalized to unit electric moment. We will not cover here the details of the straightforward though lengthy derivation of Eq. (1), which can be found in textbooks [13

13. R. W. P. King, M. Owens, and T. T. Wu, Lateral Electromagnetic Waves (Springer-Verlag, New York, 1992). [CrossRef]

], but rather briefly summarize the procedures therein. After enforcing the boundary conditions at the interface for the Ex, Ez and Hy fields (TM-polarization) in Maxwell’s curl equations then taking the inverse Fourier transform of the fields and choosing the appropriate Green’s function, one derives the Sommerfeld-type integral for the normal electric field in region 1 (z≥0):

E1z=ωμ02πk02+γ2eiγ1hε2γ1+ε1γ2+sin(γ1h)iε1eiγ1zkeikxdk
(1)

2.2 Asymptotic solution to the integral via the modified method of steepest descents

The integral in Eq. (1) is separated in two parts, E 1z=K(I 1+I 2), with K=ωμ 0/2π, and I 1 and I 2 respectively corresponding to the first and second term inside the brackets. We first consider the integral I 1 and impose γ 1 and γ 2 to have positive imaginary parts in order for the field to be bounded at infinity. In the corresponding first quadrant of the complex k-plane, we discard the branch cut running from the point k=k 2 since its contribution - characterized by an attenuation factor exp(-Im{k 2x) that drops to nearly zero within a quarter-wavelength distance x - is negligible for the optical frequencies of interest compared to that of the branch cut running from k=k 1. There is also a pole singularity in the denominator of I 1 located at ±kp=±k0εp where εp=ε 1 ε 2/(ε 1+ε 2) defines the SPP’s “effective permittivity”. The proper pole (+kp) lies very close to the branch point k 1 and we will see later on that this aspect entails particular considerations. We make the simplification ε 1=1 such that k 1=k 0 and perform the successive transformations k=k 0 sin α, x=R 2 sin θ and (z+h)=R 2 cos θ, where R 2 is the distance from the mirror image of the dipole in region 2 to the observation point. These procedures enable one to express I 1 as the integral over an angular spectrum of plane waves:

I1=Csinα·cosα·ε2sin2α·eik0R2cos(αθ)(ε2cosα+ε2sin2α)·dα
(2)

The complex variable α=σ+ defines the angle between the direction of propagation and the x-axis. The saddle-point is found by setting the derivative of the argument in exponential equal to zero, thus yielding α=θ where θ is real. The location of the pole in the α -plane is given by kp=k 0 sin αp. As a numerical example, at the excitation wavelength λ=852nm and ε 2=-33.22+1.17i (silver metal) we have αp=π-Arcsin (kp/k 0)=1.574-i0.175 where Arcsin (v)=0vdy1y2 . As shown on Fig. 2, the original integration contour C is then deformed into the steepest-descent contour (SDC) whose path, cos(σ-θ)cosh η=1, is shifted to pass through the saddle-point at θ=π/2 where the highest accuracy is assigned to observation points along the surface. Since the pole αp has been crossed by the path in the process, and is positioned below the SDC and very close to the saddle-point (see Fig. 2), the pole is captured and its contribution must be accounted for with a residue [12

12. R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: some early and late 20th-century controversies,” IEEE Antennas Propag. Mag. 46, 64–79 (2004). [CrossRef]

,14

14. P. C. Clemmow, “A note on the diffraction of a cylindrical wave by a perfectly conducting half-plane,” Q. J. Mech. Appl. Math. 3, 377–384 (1950). [CrossRef]

].

Fig. 2. Integration contours and location of the single pole in the α-plane.

To carry out the steepest-descent integration in Eq. (2), we first perform the change of variable τ=2e /4 sin((α-θ)2), which yields the relations dαdτ=eiπ4cos((αθ)2)=2iτ24i , such that the integral I 1 is rewritten:

I1=eik0R2+f(τ)·exp(k0R2τ22)·dτ
(3)

where f(τ)=g(τ)/h(τ) and:

g(τ)=sinα·cosαε2sin2α·dαdτ
(4)
h(τ)=ε2cosα+ε2sin2α
(5)

Because a pole singularity lies very close to the saddle-point θ=π/2, the classical steepest-descent method cannot be applied directly. To circumvent this problem, the procedure that we will employ here is a modified steepest-descent technique described in [12

12. R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: some early and late 20th-century controversies,” IEEE Antennas Propag. Mag. 46, 64–79 (2004). [CrossRef]

]. In this approach we separate the pole and analytical contributions, respectively fP(τ) and fA(τ), by subtracting out the pole term from the main integrand:

f(τ)=Aττp+f(τ)·(ττp)Aττp
 =fp(τ)+fA(τ)
(6)

A=g(τ)τh(τ)τp=g(τp)αh(α).αττp=εp2ε21
(7)

We separate the integral I 1 in two parts, I1=IP+IA, where IP and IA respectively denote the pole and analytical contributions from fP(τ) and fA(τ). We first consider the analytical part:

IA=eik0R2+fA(0)·exp(k0R2τ22)·dτ
(8)

Since the function fA(0) has no dependence in τ, it can be removed from the integrand. Upon evaluating the remaining kernel, +exp(k0R2τ22)·dτ=2πk0R2 , we obtain:

IA=[eiπ4sinθcosθ·ε2sin2θ(ε2cosθ+ε2sin2θ)+Aeiπ42·sin((αpθ)2)]2πk0R2eik0R2
(9)

IP=eik0R2+fP(τ)·exp(k0R2τ22)·dτ=Aeik0R2P(τ)
(10)

where P(τ)=+dτ·eχ2τ2(ττp) and χ=k0R22 . Special care must be taken to evaluate P(τ) since it can be defined by different functions depending on whether the pole is located in the lower-half (Im{τp}<0) or upper-half (Im{τp}>0) τ -plane. We can find a solution valid over the entire τ -plane by first evaluating P(τ) for Im{τp}>0 and then assuming Im{τp}<0 in the resulting integral expression. In which case we obtain P(τ)=iπeχ2τp2[22πtet2dt] with t=-iχτp, such that:

IP=2πi·A·U(R2,θ)·exp[ik0R2cos(αpθ)]
(11)

where:

U(R2,θ)=112erfc[(1+i)k0R22(cos(αpθ)1)]
(12)

II=sinθcosθ·eiπ422πk0R2exp(ik0R2)
(13)
ID=+sinθcosθ·eiπ422πk0R1exp(ik0R1)
(14)

The complete air-side normal E-field is written E 1z=K(IP+IA+II+ID). Upon adding the terms IA and II together and performing some algebraic manipulations, one obtains the following general closed-form expression:

E1z=ωμ04πsinθcosθ·eiπ42πk0R1eik0R1Directdipolefield
(15)
+ωμ04πsinθcosθ·rTM·eiπ42πk0R2eik0R2Reflectedfield
(16)
+ωμ04πA·eiπ44πk0R2(cos(αpθ)1)eik0R2Boundarywavefield
(17)
+ωμ0A·eiπ2·U(R2,θ)·exp[ik0R2cos(αpθ)] SPPfield
(18)

Equations (15)–(16) define the geometrical-optics field while Eqs. (17)–(18) describe the diffracted field. The term rTM=(ε2sin2θε2cosθ)(ε2sin2θ+ε2cosθ) present in Eq. (16), is the Fresnel reflection coefficient for TM-polarized light incident on the air-metal interface. The function U(R 2,θ), found in Eq. (18) and determined in Eq. (12), defines a complex envelope multiplying the SPP phasor.

3. Analysis of the surface near field

3.1 Reduction of the general solution for observation points along the surface

For the critical case of observation points just above and parallel the interface at θ=π2, we get x=R 2 sin θ=R 2 and the geometrical-optics field components [Eqs. (15)–(16)] vanish while the diffracted field components [Eqs. (17)–(18)] remain. The total near field along the interface then describes a composite surface wave (SW) created by two co-propagating vectors: a surface plasmon polariton (SPP) evanescent wave and a “boundary wave” (BW) having essentially a free-space cylindrical nature. The boundary wave lies in the geometrical shadow so as to compensate for the discontinuity in the geometrical-optics field across the planar interface [15

15. M. Born and E. Wolf, Principles of optics 7th ed, (Cambridge University Press, Oxford, 1999), Chap. 8.9.

]. We refrain from using the denomination “creeping wave”, which was previously chosen in [6

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

], to identify this radiative wave since it generally refers to a surface mode propagating in the geometrical shadow around convex surfaces. The contributions of the SPP and BW respectively correspond to the first and second terms in Eq. (19). In that expression, the “mismatch parameter” Δk=(kp-k 0) defines the difference between the evanescent and free-space wavevectors. We normalize the total surface wave field, which we denote Esw, by taking out the factor ωμ 0 Ae from Eqs. (17)–(18) and moving the √x root from the denominator to the numerator in order to eliminate any sign ambiguity when x<0:

Esw+(x)=U(x)·exp[i(kpxπ2)]+14π·Δkxxexp[i(k0x3π4)]
=Espp+(x)+  Ebw+(x)
(19)

The + superscript in Eq. (19) indicates that the SW originates on the right of the source dipole and propagates in the +x direction. An identical SW is excited in the -x direction whose field is the complex conjugate: E - sw (x)=Ē + sw(x). We note that the SPP and BW are initially phase-shifted by π/4. One may also notice that the expression of the BW involves the asymptotic form of the first-order Hankel function of the first kind, H1(1)(k0x)2(πk0x)exp[i(k0x3π4)] , which is accurate for k 0 x>2. This outcome is not incidental because the steepest-descent method - which is asymptotically exact - was used to obtain the solution. Thus in principle, the BW could be written in “exact” form with the Hankel function, Ebw+(x)=k0(8·Δk)·H1(1)(k0x) , which highlights the cylindrical character of the BW. From the preceding remarks we expect Eq. (19) to be likewise accurate for k 0 x>2.

3.2 Fresnel diffraction effects and the asymptotic propagation regime

The inspection of Eq. (19) reveals that the SPP excited through diffraction at the interface is not pure but rather modulated by a slowly oscillating envelope U(x) owing to Fresnel diffraction effects [Eq. (12) evaluated at θ=π/2]:

U(x)=112erfc[(1+i)Δk·x2]
(20)

The first term on the right-hand side of Eq. (20) defines the classical SPP mode whereas the second erfc term describes the fringe pattern generated by the process of coupling incident homogeneous light into the evanescent mode via scattering. As a note, Eq. (20) can equivalently be expressed using the conventional complex Fresnel integral, F(ν)=∫ν 0exp(iπy 2/2)·dy, instead of the erfc: U(x)=12+12(1i)·F(iΔk·2xπ) . As shown on Fig. 3, the modulation function Re{U(x)} evolves from U(0)=0.5 at the origin before reaching the first peak at some distance x 1 and then oscillates onward with a large distinctive period ΔL=2π/|Δk| to lower amplitudes before asymptotically growing towards ever higher amplitudes due to the positive imaginary component inside the argument of erfc.

Fig. 3. Amplitude moduli of the total field (solid line), SPP field (dotted line), boundary wave field (dashed line) and the envelope function U(x) (dashed-dotted line) for ε 1=1, ε 2=-33.22+i1.17 and λ=852nm. The successive peaks of U(x) are numbered x 1, x 2,…, xm.

3.3 Characterization of the field at the origin and in the near-zone propagation regime

Fig. 4. Real components of the total field (solid line), boundary wave field(dashed line), SPP field (dotted line) and their moduli for λ/8≤x≤10λ

4. Dipolar model of the near-field interactions between nano-objects

As indicated earlier, when light is incident on a nano-object (nano-slit or nano-groove) with dimensions smaller than the wavelength of light, the scattered far-field radiation from the nano-object can be considered as originating from an infinitesimal horizontal electric dipole (HED) located on the structure. This assumption has been rigorously validated in Green’s tensor numerical calculations by Lévêque, et al., [10

10. G. Lévêque, O. J. F. Martin, and J. Weiner, “Transient behavior of surface plasmon polaritons scattered at a subwavelength groove,” Phys. Rev. B. 76, 155418 (2007). [CrossRef]

]. Hence in principle, many diffraction problems involving nanostructured surfaces can be modeled by replacing the individual nano-objects with point-sized dipoles. Indeed we demonstrate in this Section that recently reported experiments using groove-slit and double-slit configurations are accurately modeled by placing radiating linear HEDs at the corresponding sites of the surface nano-defects.

4.1 Groove-slit transmission

We first consider the groove-slit experiment performed by Gay, et al., [3

3. G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nanostructured surfaces and the composite diffracted evanescent wave,” Nat. Physics 2, 262–267 (2006). [CrossRef]

] and theoretically investigated by Lalanne and Hugonin [6

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

]. The main interactions in this setup are described by the simple model illustrated in the diagram below (Fig. 5) where the groove-slit center-tocenter separation distance is controlled by the variable d.

Fig. 5. Schematic of the near-field interactions in the groove-slit setup

Fig. 6. Normalized slit transmission as a function of groove-slit distance d. Comparison between experimental (circles) [3], numerical (dashed line) [16], and the analytical (solid line) results with λ=852nm and ε 2=-33.22+1.17i.

4.2 Double-slit near field

The field just above the surface between two subwavelength slits milled in a gold film, separated by 2d=10.44µm and illuminated with λ=974.3nm normal incident TM-polarized light, was recently measured through a scanning near-field optical microscope (SNOM) by Aigouy and co-workers [7

7. L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julié, V. Mathet, and M. Mortier, “Near-field analysis of surface waves launched at nanoslit apertures,” Phys. Rev. Lett. 98, 153902 (2007). [CrossRef] [PubMed]

]. Neglecting the tangential component of the E-field at the interface, the patterns recorded from fluorescence emission are expected to scale with |Ez|4. Modeling once more each nano-object (i.e. the slits) as a linear HED, the resulting near-field intensity pattern arises from the interference of two counter-propagating SWs, Ez(x)=E + sw(x+d)+E - sw(x-d), where the origin of the x-axis is located at the half slit-to-slit separation distance. In Fig. 7 the near field between the slits, a|Ez(x)|4+b, is plotted in the vertical axis, where b=0.33 is the background illumination offset taken from the original data and a=0.031 is the best-fit gain factor. The decay, phase and pattern periodicity predicted by the analytical model, with ε 1=1 and ε 2=-44.05+3.24i, closely match the experimental data. Clearly, the dipolar model is again fully consistent with the near-field structure depicted in real-world experiments.

Fig. 7. Near-field fringe pattern in the double-slit setup. Comparison between the experimental (circles) [7] and analytical (solid line) results with 2d=10.44µm, λ=974.3nm and ε 2=-44.05+3.24i

It is worth to emphasize that the two studies [3

3. G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nanostructured surfaces and the composite diffracted evanescent wave,” Nat. Physics 2, 262–267 (2006). [CrossRef]

,7

7. L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julié, V. Mathet, and M. Mortier, “Near-field analysis of surface waves launched at nanoslit apertures,” Phys. Rev. Lett. 98, 153902 (2007). [CrossRef] [PubMed]

] reported in this Section both assess the SW field in the near-zone propagation regime (0<x<2Lbw). The corresponding results yielded for either investigation by the analytical dipolar model indicate that the asymptotic solution is fairly accurate inside this transient near-zone as expected from the accuracy range k 0 x>2, or stated alternatively, x>λ/π (see Subsection 3.1).

5. Conclusion

In summary, we provide a rigorous closed-form description of the surface wave generated via the diffraction of a horizontal Hertzian dipole over the air-metal interface by solving the corresponding Sommerfeld integral with a modified method of steepest descents. The asymptotic solution - accurate for distances x>λ/π from the origin - demonstrates that the total surface near field is composed of two distinct components as previously evidenced in [6

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

]: a long-range surface plasmon polariton (SPP) evanescent wave and a short-range boundary wave (BW) with free-space cylindrical properties. The dynamics of both constituent waves are fully revealed through simple analytic expressions, and appropriate parameters are defined to quantitatively and continuously describe their properties across the entire propagation range: near and asymptotic. Moreover, our calculations predict that the wavevector mismatch of the two co-propagating surface modes creates a weak periodic beating of the total field implitude, which is noticeable at relatively large distances from the source and high optical frequencies. The closed-form solution is further validated via comparison between a dipolar model and recent experimental data, which demonstrates excellent quantitative agreement. In the process we show that the generation of surface waves by diffraction - and their ensuing interactions between one-dimensional subwavelength-sized nano-defects along the metallo-dielectric interface - are conveniently and accurately modeled with linear Hertzian dipoles. Hence, the theoretical formalism presented in this contribution can be used for the characterization and engineering of the near-field interactions in plasmonic and nano-optical devices while alleviating the reliance on time-consuming computer simulations.

Acknowledgments

The authors cordially thank Robert E. Collin from Case Western Reserve University for his valuable remarks and suggestions. Helpful discussions with J. Weiner of IRSAMC/LCAR and P. Lalanne of the Institut d’Optique are also gratefully acknowledged.

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 391, 667–669 (1998). [CrossRef]

2.

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]

3.

G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nanostructured surfaces and the composite diffracted evanescent wave,” Nat. Physics 2, 262–267 (2006). [CrossRef]

4.

G. Gay, O. Alloschery, B. Viaris de Lesegno, J. Weiner, and H. J. Lezec, “Surface wave generation and propagation on metallic subwavelength structures measured by far-field interferometry,” Phys. Rev. Lett. 96, 213901 (2006). [CrossRef] [PubMed]

5.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W.’t Hooft, D. Lenstra, and E. R. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005). [CrossRef] [PubMed]

6.

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

7.

L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julié, V. Mathet, and M. Mortier, “Near-field analysis of surface waves launched at nanoslit apertures,” Phys. Rev. Lett. 98, 153902 (2007). [CrossRef] [PubMed]

8.

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]

9.

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

10.

G. Lévêque, O. J. F. Martin, and J. Weiner, “Transient behavior of surface plasmon polaritons scattered at a subwavelength groove,” Phys. Rev. B. 76, 155418 (2007). [CrossRef]

11.

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

12.

R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: some early and late 20th-century controversies,” IEEE Antennas Propag. Mag. 46, 64–79 (2004). [CrossRef]

13.

R. W. P. King, M. Owens, and T. T. Wu, Lateral Electromagnetic Waves (Springer-Verlag, New York, 1992). [CrossRef]

14.

P. C. Clemmow, “A note on the diffraction of a cylindrical wave by a perfectly conducting half-plane,” Q. J. Mech. Appl. Math. 3, 377–384 (1950). [CrossRef]

15.

M. Born and E. Wolf, Principles of optics 7th ed, (Cambridge University Press, Oxford, 1999), Chap. 8.9.

16.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bientsman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. 2, 07022 (2007). [CrossRef]

17.

P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett. 95, 263902 (2005). [CrossRef]

OCIS Codes
(050.1970) Diffraction and gratings : Diffractive optics
(240.6680) Optics at surfaces : Surface plasmons
(240.6690) Optics at surfaces : Surface waves
(260.3910) Physical optics : Metal optics
(310.2790) Thin films : Guided waves
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Optics at Surfaces

History
Original Manuscript: April 8, 2008
Revised Manuscript: May 27, 2008
Manuscript Accepted: May 30, 2008
Published: June 4, 2008

Citation
Bora Ung and Yunlong Sheng, "Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals revisited," Opt. Express 16, 9073-9086 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-9073


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References

  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 391, 667-669 (1998). [CrossRef]
  2. 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]
  3. G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O???Dwyer, J. Weiner, and H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave," Nat. Physics 2, 262-267 (2006). [CrossRef]
  4. G. Gay, O. Alloschery, B. Viaris de Lesegno, J. Weiner, and H. J. Lezec, "Surface wave generation and propagation on metallic subwavelength structures measured by far-field interferometry," Phys. Rev. Lett. 96, 213901 (2006). [CrossRef] [PubMed]
  5. H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. ???t Hooft, D. Lenstra and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young???s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005). [CrossRef] [PubMed]
  6. P. Lalanne and J. P. Hugonin, "Interaction between optical nano-objects at metallo-dielectric interfaces," Nat. Physics 2, 551-556 (2006). [CrossRef]
  7. L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julié, V. Mathet, and M. Mortier, "Near-field analysis of surface waves launched at nanoslit apertures," Phys. Rev. Lett. 98, 153902 (2007). [CrossRef] [PubMed]
  8. 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]
  9. B. Ung and Y. Sheng, "Interference of surface waves in a metallic nanoslit," Opt. Express 15, 1182-1190 (2007). [CrossRef] [PubMed]
  10. G. Lévêque, O. J. F. Martin, and J. Weiner, "Transient behavior of surface plasmon polaritons scattered at a subwavelength groove," Phys. Rev. B. 76, 155418 (2007). [CrossRef]
  11. L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, "Subwavelength focusing and guiding of surface plasmons," Nano. Lett. 5, 1399-1402 (2005). [CrossRef]
  12. R. E. Collin, "Hertzian dipole radiating over a lossy earth or sea: some early and late 20th-century controversies," IEEE Antennas Propag. Mag. 46, 64-79 (2004). [CrossRef]
  13. R. W. P. King, M. Owens, and T. T. Wu, Lateral Electromagnetic Waves (Springer-Verlag, New York, 1992). [CrossRef]
  14. P. C. Clemmow, "A note on the diffraction of a cylindrical wave by a perfectly conducting half-plane," Q. J. Mech. Appl. Math. 3, 377-384 (1950). [CrossRef]
  15. M. Born and E. Wolf, Principles of optics 7th ed, (Cambridge University Press, Oxford, 1999), Chap. 8.9.
  16. M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bientsman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, "Numerical analysis of a slit-groove diffraction problem," J. Eur. Opt. Soc. 2, 07022 (2007). [CrossRef]
  17. P. Lalanne, J. P. Hugonin, and J. C. Rodier, "Theory of surface plasmon generation at nanoslit apertures," Phys. Rev. Lett. 95, 263902 (2005). [CrossRef]

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