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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 3 — Jan. 30, 2012
  • pp: 2149–2162
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Rigorous solution for optical diffraction of a sub-wavelength real-metal slit

Yann Gravel and Yunlong Sheng  »View Author Affiliations


Optics Express, Vol. 20, Issue 3, pp. 2149-2162 (2012)
http://dx.doi.org/10.1364/OE.20.002149


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Abstract

We present a rigorous closed-form solution of the Sommerfeld integral for the optical scattering of a metal sub-wavelength slit. The two-dimensional (2D) field solution consists of the Surface Plasmon Polariton (SPP) mode at the metal surface and the 2D scattered field, which is the cylindrical harmonic of first order emitted by the electrical dipole and convolved with the 1D transient SPP along the interface. The creeping wave or quasi-cylindrical wave detected in the previous experiment is not an extra evanescent surface wave, but is the asymptotic behavior of the 2D scattered field at the proximity of the slit. Furthermore, our solution predicts a strong resonant enhancement of the scattered field at the proximity of the slit, depending on the materials and wavelength.

© 2012 OSA

1. Introduction

In fact, the radiation of an electrical dipole in the vicinity of a metallic surface has been studied for long time. The integral solution of the Helmholtz equation in the frequency domain has been investigated by Sommerfeld in 1909 [15

15. A. N. Sommerfeld, “Propagation of waves in wireless telegraphy,” Ann. Phys. (Leipzig) 28, 665–737 (1909).

] and other authors in order to study the field radiated by Hertzian dipoles over a lossy Earth, referred to as Zenneck waves and Norton trapped surface waves [16

16. J. Zenneck, “Propagation of plane EM waves along a plane conducting surface,” Ann. Phys. (Leipzig) 23, 846–866 (1907).

18

18. R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: Some early and late 20th-century controversies,” IEEE Antennas Propagat. Mag. 46(2), 64–79 (2004). [CrossRef]

]. Numerical evaluation of the Sommerfeld branch-cut integral in [13

13. P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. 2(8), 551–556 (2006). [CrossRef]

,14

14. P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep. 64(10), 453–469 (2009). [CrossRef]

] showed that the creeping wave drops along the interface at a damping rate varying from 1/x1/2 for distances from the slit less than one wavelength [13

13. P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. 2(8), 551–556 (2006). [CrossRef]

] to 1/x3/2 for distances greater than a wavelength [14

14. P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep. 64(10), 453–469 (2009). [CrossRef]

]. Asymptotic solutions of the Sommerfeld integral using a modified steepest descent method [18

18. R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: Some early and late 20th-century controversies,” IEEE Antennas Propagat. Mag. 46(2), 64–79 (2004). [CrossRef]

] also found the damping rate of 1/x1/2 at near-distances [19

19. B. Ung and Y. Sheng, “Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals revisited,” Opt. Express 16(12), 9073–9086 (2008). [CrossRef] [PubMed]

] and 1/x3/2 at far-distances [20

20. A. Y. Nikitin, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martín-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. 11(12), 123020 (2009). [CrossRef]

]. We recently obtained rigorous closed-form solution of the Sommerfeld branch-cut integral for a sub-wavelength metal slit for the field at the metal/dielectric interface. The solution includes a normal SPP mode and an extra term, which we referred to as the transient SPP and is the SPP of wave-vector ksp modulated by complex-valued envelope described by the exponential integral [21

21. Y. Gravel and Y. Sheng, “Rigorous solution for the transient Surface Plasmon Polariton launched by subwavelength slit scattering,” Opt. Express 16(26), 21903–21913 (2008). [CrossRef] [PubMed]

]. Similar result was obtained based on empirical modification of the CDEW model [22

22. G. Lévêque, O. J. F. Martin, and J. Weiner, “Transient behaviour of surface plasmon polaritons scattered at a sub-wavelength groove,” Phys. Rev. B 76(15), 155418 (2007). [CrossRef]

]. The transient SPP has a damping rate varying from –ln(x) to 1/x1/2 and to 1/x, in the distance range from the slit x < λ/20, λ/20 < x < λ and x > λ, respectively.

However, the physical nature of the extra quasi-cylindrical wave is not fully understood. This fundamental problem is recently involved in a model of the surface wave scattering dynamic [23

23. H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452(7188), 728–731 (2008). [CrossRef] [PubMed]

] in the theory of the extraordinary optical transmission of the metallic nanohole arrays [24

24. 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(6668), 667–669 (1998). [CrossRef]

]. In fact, most studies on the Sommerfeld branch-cut integral evaluate the field only at the surface of the conductor substrate [13

13. P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. 2(8), 551–556 (2006). [CrossRef]

19

19. B. Ung and Y. Sheng, “Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals revisited,” Opt. Express 16(12), 9073–9086 (2008). [CrossRef] [PubMed]

]. This simplifies greatly the calculation, but fails to produce a complete picture of the scattered field in the dielectric space.

In this paper, we show a rigorous closed-form solution of the Sommerfeld branch-cut integral for the field scattered by a metal slit in the entire half-space over the metal substrate. The transient SPP describes the value of this field at the metal surface, while the field in the 2D space is described as a convolution between the cylindrical harmonic of first order and the transient SPP along the surface. The solution shows clearly that the so-called creeping or quasi-cylindrical field is not an extra surface wave, but is the asymptotic behavior of the scattered field in the 2D space in the region close to the slit, both at the interface and in the fee space. Moreover, the amplitude of the 2D scattered field may be enhanced significantly by surface plasmon resonance. This new theoretical understanding will be helpful in the analysis and design of nano-sized photonics devices.

2. Physical model and solution

Consider a thick planar real metal medium of dielectric function εm perforated by a sub-wavelength slit, which is infinitely long in the z-axis. Above the metal surface is a semi-infinite dielectric medium of permittivity εd. The problem is two-dimensional (2D) in the x-y space, as shown in Fig. 1
Fig. 1 Geometry of model for the diffraction of a metal sub-wavelength slit.
. The incident wave along the –y direction is monochromatic of Transverse Magnetic (TM) polarization with the magnetic field H parallel to the z-axis. Its electric components in the x-y plane induce displacements of the conduction electrons. The discontinuity of the surface at the slit prevents electrical currents from crossing the slit. Hence, the electrical charges accumulate at the two up-edges of the slit, resulting in an oscillating horizontal electric dipole, as have been observed in both experiments [25

25. 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(7), 1399–1402 (2005). [CrossRef] [PubMed]

] and numerical simulations [8

8. B. Ung and Y. Sheng, “Interference of surface waves in a metallic nanoslit,” Opt. Express 15(3), 1182–1190 (2007). [CrossRef] [PubMed]

].

The dipole acts as a source of radiation, launching the Surface Plasmons Polariton (SPP) mode along the metal/dielectric interface and radiating homogeneous waves in the dielectric half space. Thus, the problem of the diffraction by the metal slit is modeled through solving the 2D field of a horizontal electric dipole on the metallic surface. As a circulation of electric charges results in a magnetic dipole, the electric dipole can also be represented by a circulation of hypothetical magnetic charges, which takes the form of an equivalent magnetic current line along the z axis and replaces the horizontal electric dipole [13

13. P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. 2(8), 551–556 (2006). [CrossRef]

,21

21. Y. Gravel and Y. Sheng, “Rigorous solution for the transient Surface Plasmon Polariton launched by subwavelength slit scattering,” Opt. Express 16(26), 21903–21913 (2008). [CrossRef] [PubMed]

]. Moreover, for a sub-wavelength slit it is a good approximation to consider the equivalent magnetic current line source to have an infinitesimal spatial extent [13

13. P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. 2(8), 551–556 (2006). [CrossRef]

] and to describe it as a 2D delta function of the form: Jm*=Jm*δ(x)δ(y)z^, where Jm* is the complex amplitude of the magnetic current density. The Maxwell’s equations in such formalism lead to a single differential equation to solve for the magnetic component instead of two coupled equations for two electric components, resulting in a simplification of the problem. For time harmonic fields described by exp(-iωt) and of TM polarization, the Maxwell’s equations in the Fourier space lead to the Helmholtz equation in the spatial frequency domain as [21

21. Y. Gravel and Y. Sheng, “Rigorous solution for the transient Surface Plasmon Polariton launched by subwavelength slit scattering,” Opt. Express 16(26), 21903–21913 (2008). [CrossRef] [PubMed]

]
(y2+γ2)H˜(kx,y)=iωεdJm*δ(y)
(1)
where H˜(kx,y) is the Fourier spectrum of the magnetic field H(x,y) with respect to x, kx is the x-component of the wavevector, γ2 = εdk02-kx2 with k0 = ω/c0 and c0 is the speed of light in vacuum. After solving Eq. (1) in the dielectric medium and applying the boundary conditions at the metal/dielectric interface, the inverse Fourier transform leads to a solution of the field in the x-y space in the form of the Sommerfeld integral:
H(x,y)=ωJm*2πeikxxeiγy1εdεdk02kx2+1εmεmk02kx2dkx
(2)
which has two branch-cut points in the complex plane at kx = εm1/2k0 and kx = εd1/2k0, and a pole at the location determined by setting the denominator of the integrant in Eq. (2) to zero, that yields
kx=k0εdεmεd+εmksp
(3)
where ksp is the SPP propagation constant. Sommerfeld and others have computed this type of integrals at the interface with y=0 using the steepest descent approach in investigation of the Zenneck and Norton surface waves [15

15. A. N. Sommerfeld, “Propagation of waves in wireless telegraphy,” Ann. Phys. (Leipzig) 28, 665–737 (1909).

18

18. R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: Some early and late 20th-century controversies,” IEEE Antennas Propagat. Mag. 46(2), 64–79 (2004). [CrossRef]

]. In general, the solutions for the field of an electric dipole close to or on a metal surface take complex forms [18

18. R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: Some early and late 20th-century controversies,” IEEE Antennas Propagat. Mag. 46(2), 64–79 (2004). [CrossRef]

, 26

26. R. W. P. King, M. Owens, and T. T. Wu, Lateral Electromagnetic Waves: Theory and Applications to Communications, Geophysical Exploration and Remote Sensing (Springer-Verlag, New York, 1992).

].

2.1 Scattered field in the 2D space

To find a closed-form solution of the field in the entire half space with y0 we consider. the integrant in Eq. (2) as a product of the term H˜b(kx,y)=exp(iγy) withy0and the remaining term, denoted by H˜a(k), such that the field H(x,y) is a Fourier transform of the product of H˜a(k) and H˜b(k,y) which can solved by the 1D convolution of the inverse Fourier transforms of H˜a(k)and H˜b(k,y), as H(x,y)=Ha(x)Hb(x,y). In fact, Ha(x) corresponds to the field at the interface y=0, which has been solved by two consequent branch-cut integrals with parameterization of the branch-cut paths in Ref. [21

21. Y. Gravel and Y. Sheng, “Rigorous solution for the transient Surface Plasmon Polariton launched by subwavelength slit scattering,” Opt. Express 16(26), 21903–21913 (2008). [CrossRef] [PubMed]

] as
Ha(x)=H(x,y=0)=iωJm*εdεm[(εdεm)12εd2+εm2eikspx+12π(εd+εm)eikspxE1(ikspx)]
(4)
where the first term is the contribution of the pole to the residue in the Cauchy integral and represents the SPP mode at the interface. The second term is the contribution of the two branch points to the residue, referred to as the transient SPP [21

21. Y. Gravel and Y. Sheng, “Rigorous solution for the transient Surface Plasmon Polariton launched by subwavelength slit scattering,” Opt. Express 16(26), 21903–21913 (2008). [CrossRef] [PubMed]

,22

22. G. Lévêque, O. J. F. Martin, and J. Weiner, “Transient behaviour of surface plasmon polaritons scattered at a sub-wavelength groove,” Phys. Rev. B 76(15), 155418 (2007). [CrossRef]

], which is the SPP, exp(ikspx) with the complex-valued wavenumber ksp, modulated by a complex-valued envelope described by the exponential integral of complex argument E1(ikspx), being the case n=1of a general set of the functions defined by

En(z)=1ezttndt
(5)

The inverse Fourier transform of H˜b(kx,y)=exp(iγy) is performed analytically (see Appendix A) as
Hb(x,y)=i2kdyx2+y2H1(1)(kdx2+y2)=ikd2sin(θ)H1(1)(kdr)
(6)
where kdεdk0, (r,θ) are the polar coordinates and H1(1) is the Hankel function of first kind and first order. Thus, Hb(x,y) represents a cylindrical harmonic of first order [27

27. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

], which is a 2D solution of the scalar Helmholtz equation with a dipole source in the cylindrical coordinate system. Recall that the zero order cylindrical harmonic is the basic cylindrical wave H0(1)(kdr) representing the outward propagating wave emitted by a line source.

Finally, according to Eqs. (4) and (6), the convolution H(x,y)=Ha(x)Hb(x,y) for the total magnetic field in the 2D dielectric media above the metallic surface is expressed as a sum of two convolutions: H(x,y)=Hsp(x,y)+Hs(x,y) (see Appendix B). The first convolution Hsp(x,y) is easily solved analytically (see Appendix C):
Hsp(x,y)=2hspikdeikspxikd2ksp2y
(7)
where the constant hsp depends on Jm*, k0, εd and εm as

hsp=ωkdJm*(εdεm)3/22(εd2+εm2)
(8)

Hsp(x,y) clearly represents a conventional SPP mode propagating along the interface with the wave-number ksp and decaying exponentially away from y=0, with its evanescence in the direction normal to the interface as exp(ikd2ksp2y).

The term of interest is Hs(x,y) and can be expressed as (see Appendix B)
Hs(x,y)hsy(xx')2+y2H1(1)(kd(xx')2+y2)eikspx'E1(ikspx')dx'
(9)
where the constant hs depends on Jm*, k0, εd and εm as

hs=ωkdJm*εdεm4π(εd+εm)
(10)

Equation (9) shows clearly that the field scattered into the 2D space above the metallic surface is a 1D convolution along the x-axis of the cylindrical harmonic of first order, sin(θ)H1(1)(kdr) with the transient SPP [21

21. Y. Gravel and Y. Sheng, “Rigorous solution for the transient Surface Plasmon Polariton launched by subwavelength slit scattering,” Opt. Express 16(26), 21903–21913 (2008). [CrossRef] [PubMed]

,22

22. G. Lévêque, O. J. F. Martin, and J. Weiner, “Transient behaviour of surface plasmon polaritons scattered at a sub-wavelength groove,” Phys. Rev. B 76(15), 155418 (2007). [CrossRef]

], exp(ikspx)E1(ikspx). As a comparison, recall that in the case of a perfect conductor slit, which does not support the SPP mode, the diffracted field would be a summation of cylindrical wavelets, or the cylindrical harmonics of zero order, emitted from hypothetical sources inside the slit, according to the Huygens principle. As the width of the slit becomes negligible with respect to the wavelength, the scattered field takes the form of a cylindrical harmonic of order zero, or a typical cylindrical wave, as shown by the simulation in Ref. [28

28. L. Chen, J. T. Robinson, and M. Lipson, “Role of radiation and surface plasmon polaritons in the optical interactions between a nano-slit and a nano-groove on a metal surface,” Opt. Express 14(26), 12629–12636 (2006). [CrossRef] [PubMed]

]. However, in the case of a real metal slit, according to Eq. (9) the scattered field is a continuous sum of cylindrical harmonics of first order, emitted from a set of hypothetic dipolar sources with their amplitude and phase following a spatial distribution as the transient SPP function, which extends from the inside to the outside of the slit along the metal/dielectric interface.

2.2 Field at the interface

The exponential integral of complex argument, ksp, is complex valued. Its phase, amplitude, real and imaginary parts are shown in Fig. 2
Fig. 2 Exponential integral function E1(ikspx): (a): Real, imaginary parts and amplitude;(b): Phase. Constitutive parameters have been taken for an air-silver interface at λ=1 μm.
. The latters can be calculated respectively by the cosine and sine integral. Our calculation shows that the exponential integral of complex argument representing the envelope of the transient SPP has a negative phase, which cancels the positive phase of the SPP propagation, so that the phase of the transient SPP tends to a negative constant of -π/2 within the transient distance range 0 < x < λ from the origin.

In fact, as shown in Ref. [21

21. Y. Gravel and Y. Sheng, “Rigorous solution for the transient Surface Plasmon Polariton launched by subwavelength slit scattering,” Opt. Express 16(26), 21903–21913 (2008). [CrossRef] [PubMed]

], the transient SPP is given by
eikspxE1(ikspx)=0eikxkkspdk
(11)
which is the inverse Fourier transform of a simple pole centered at ksp, a continuous sum of all the homogeneous and evanescent plane waves propagating along the x-direction with the complex amplitude 1/(k-ksp). In this summation the evanescent components are dominant as can be seen in the spectrum of the transient SPP shown in Fig. 3
Fig. 3 Spectrum of the transient SPP for an air-silver interface at λ=1 μm with the pole at ksp slightly higher than k0.
, where the evanescent components with the frequencies higher than k0 have greater amplitudes than that have the propagating components with the frequencies lower than k0, and the spectrum is asymmetric with respect to k/k0 = 1, so that the scattered field at the surface is damped at a greater rate than the typically SPP mode with a propagation constant ksp greater than k0.

The transient SPP is first launched from the slit as a typical cylindrical wave with its amplitude dropping as characterized by –ln(x) and then 1/x1/2, corresponding to the field naturally radiated by a punctual source. Then, the transient SPP along the surface suffers from an additional drop in amplitude due to the loss of energy through the excitation of the long-range SPP mode.

2.3 Asymptotic behavior near the source

2.4 Surface plasmon resonance

The scattered field described in Eq. (9) can have strong resonant enhancement by several orders of magnitude due to the coefficient hs, which depends on the permittivities of the metal and dielectric, and the incident wavelength. If Re{εm}≈-εd and Im{εm} is small, then from Eq. (10), the amplitude of hs is strongly enhanced. Figure 5
Fig. 5 Amplitude of the coefficient hs of the scattered field in Eq. (9) in arbitrary units as a function of the incident wavelength for (a): Ag; and (b): Au. The index of refraction in the dielectric medium over the metal slit is n = 1 (black), 1.5 (blue), 2 (red), 2.5 (green), 3 (orange).
shows the amplitude of hs as a function of the wavelength for Ag and Au slit, with the index of refraction in the dielectric medium over the metal slit n = 1, 1.5, 2, 2.5 and 3. The dielectric function of the metal is computed by Lorentz-Drude dispersive model. Strong enhancements, which are more than 3000 times for Ag and 1000 times for Au are clearly shown at the resonant wavelengths for both Ag and Au slits in the visible spectrum. In the case of Ag, the strongest enhancement is observed for a dielectric refractive index of n = 3 at λ = 528 nm, corresponding to a complex permittivity of εm = −8.9594 + 0.7958i. In the case of Au, the strongest enhancement has less magnitude than for Ag, and is observed for a dielectric refractive index of n = 3 at λ = 623 nm corresponding to a complex permittivity of εm = −9.2504 + 1.9880i. Hence, the enhancement frequencies and magnitudes clearly depend on the metal considered. Furthermore, for a given metallic surface, the resonant wavelength is red shifted and the resonance is stronger, with the increase of the dielectric refractive index.

2.5 Field far from the source

In the region where the distance to the slit is much larger than one wavelength, the width of the transient SPP peak at x=0, as shown in Fig. 2a, is negligible. Hence, exp(ikspx)E1(ikspx) may be approximated by a delta function and the scattered field expressed in Eq. (9) is simply the first order cylindrical harmonic with the propagation constant kd:
Hs(x,y)hs2πkspyx2+y2H1(1)(kdx2+y2)
(15)
which differs from the basic cylindrical wave by the presence of the factor sin(θ), corresponding to a diffraction shadow for small θ. Along the y-axis, the field drops as 1/y1/2 as the typical behavior of the Hankel function, but not as an evanescent surface wave. In the diffraction shadow region close to the interface where x >> y, the field takes an approximate form:

Hs(x,y)hs2πkspyxH1(1)(kdx)
(16)

As the Hankel function drops in amplitude as 1/x1/2, Hs(x,y) will drop as 1/x3/2. This result is in agreement with the work of Nikitin et al. [20

20. A. Y. Nikitin, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martín-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. 11(12), 123020 (2009). [CrossRef]

] which evaluated the field in the diffraction shadow using the asymptotic steepest descent method. Similar result has also been obtained by Lalanne et al. [13

13. P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. 2(8), 551–556 (2006). [CrossRef]

] with numerical integration methods. Note that when the substrate is a perfect conductor and no SPP mode is present in the solution, the scattered field is a cylindrical wave with a damping rate of 1/x1/2 away from the slit. When the substrate is real metal, the scattered field behaves as a flattened cylindrical wave only for the distances x<λ/10 as shown in Fig. 4a, and is damped more rapidly as the wave propagates along the interface. Clearly, this behavior is different from the perfect conductor case and can therefore be linked to the real metal and to the pumping of energy to the SPP mode. This result is in agreement with the numerical evaluation by Lalanne et al. [13

13. P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. 2(8), 551–556 (2006). [CrossRef]

,14

14. P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep. 64(10), 453–469 (2009). [CrossRef]

] and the asymptotic steepest descent solution by Ung et al. [19

19. B. Ung and Y. Sheng, “Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals revisited,” Opt. Express 16(12), 9073–9086 (2008). [CrossRef] [PubMed]

]

3. Numerical solution

To support the analytical result, a Finite Difference Time Domain (FDTD) simulation of the field radiated from a punctual magnetic line source located on a metallo-dielectric interface of a perfect conductor (Fig. 6a
Fig. 6 FDTD simulations (OptiwaveTM) for real part of the magnetic intensity at λ = 500 nm from a punctual magnetic line source located on a metallo-dielectric interface with (a): Perfect electric conductor and (b): Silver.
) and silver (Fig. 6b) has been performed. Snapshots of the real part of the magnetic intensity field are shown. The mesh size was 0.01 μm and time step was 0.02238 fs, at the wavelength λ = 500 nm. Results for the perfect conductor, which does not support the SPP mode, show no surface wave and the scattered field in the half space above the surface takes the form of the zero order cylindrical harmonic. As discussed in Section 2.1, this result is expected because the zero order cylindrical harmonic corresponds to the 2D Green’s function of Helmholtz equation. The scattered field behaves as the Hankel function H0(1)(kdr) which has a damping rate along radial axis as a logarithmic drop in the region close to the source, and changes to 1/r for distance greater than a wavelength from the source. In the case of a silver slit, the SPP mode is clearly visible along the interface as a consequence of the real metal permittivity allowing the metal surface to support SPP modes. However, the scattered field in dielectric region takes the form of the first order cylindrical harmonic, sin(θ)H1(1)(kdr), as predicted by Eq. (6). The diffraction shadow is clearly present.

In the direction normal to the surface along the y-axis and sinθ = 1 according to Eq. (6), the field behaves as H1(1)(kdy), which drops as 1/y1/2 for y larger than a wavelength, in agreement with our results presented in Section 2.5, and similarly to that in perfect conductor case, which drops as H0(1)(kdy). In fact, the difference between H0(1)(kdy) and H1(1)(kdy) is when the argument tends to zero where the damping rates are –ln(y) and 1/y, respectively. In the diffraction shadow, the SPP mode is dominant along the interface. The mismatch between the wavelength of the scattered field in the dielectric, k0, and that of the SPP, ksp, can be seen clearly from the distortion of the wave fronts, again in agreement with our rigorous closed solution.

4. Conclusion

Appendix A

The inverse Fourier transform Eq. (10) is expressed as

Hb(x,y)=12πeiεdk02k2yeikxdk
(A1)

Using Euler’s formula, we rewrite Eq. (A1) as

Hb(x,y)=12πeiεdk02k2y(cos(kx)+isin(kx))dk
(A2)

The first exponential function is even with respect to k, so that

Hb(x,y)=1π0cos(kx)ek2εdk02ydk
(A3)

In this form, the following identity (see [29

29. I. S. Gradshteyn and I. m. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

], Section 3.914) can be used to solve the integral:
0cos(bx)eβγ2+x2dx=βγβ2+b2K1(γβ2+b2)
(A4)
where K1 is the modified Bessel function of the second kind. Hence, Eq. (A3) becomes

Hb(x,y)=1πy(±ik0εd)x2+y2K1(±iεdk0x2+y2)
(A5)

The K1 function with an imaginary argument can be expressed in term of the Hankel function:
Hb(x,y)=i2kdyx2+y2H1(1)(kdx2+y2)
(A6)
where we replaced εdk0 by kd. The negative sign has been chosen in order to retrieve the Hankel function of the first kind representing waves out-coming from the source. We then obtain the result presented in Eq. (9).

Appendix B

Expressing Eq. (1) is separated as a Fourier transform of a product of two functions:
H(x,y)=12πH˜a(k)H˜b(k,y)eikxdk
(B1)
with

H˜a(k)=ωJm*11εdεdk02k2+1εmεmk02k2
(B2)
H˜b(k,y)=eiεdk02k2y
(B3)

The total magnetic field can thus be obtained using the convolution theorem:

H(x,y)=Ha(x)Hb(x,y)
(B4)

Since Ha(x) and Hb(x,y) are given by respectively Eqs. (4) and (7), the convolution can be written explicitly:
H(x,y)=hspy(xx')2+y2H1(1)(kd(xx')2+y2)eikspx'dx'+hsy(xx')2+y2H1(1)(kd(xx')2+y2)eikspx'E1(ikspx')dx'
(B5)
where the convolution has been distributed on the sum of Ha and the following constants have been defined:

hsp=ωkdJm*(εdεm)322(εd2+εm2)
(B6)
hs=ωkdJm*εdεm4π(εd+εm)
(B7)

The following form is also valid by virtue of the commutability of the convolution:

H(x,y)=hspy(x')2+y2H1(1)(kd(x')2+y2)eiksp(xx')dx'+hsy(x')2+y2H1(1)(kd(x')2+y2)eiksp(xx')E1(iksp(xx'))dx'
(B8)

The first term is labeled Hsp(x,y) for the SPP mode and the second Hs(x,y) for the scattered field such that

H(x,y)=Hsp(x,y)+Hs(x,y)
(B9)

Appendix C

The first term of the convolution given by Eq. (B5) can be solved analytically:

Hsp(x,y)=hspy(x')2+y2H1(1)(kd(x')2+y2)eiksp(xx')dx'
(C1)

We use Euler formula to rewrite Eq. (C1):

Hsp(x,y)=hspyeikspx1x'2+y2H1(1)(kdx'2+y2)(cos(kspx')isin(kspx'))dx'
(C2)

All the functions of x’ in the integrant are even, except sin(kspx’), which is odd. We also separate the Hankel function in its Bessel components:

Hsp(x,y)=2hspyeikspx[0J1(kdx'2+y2)x'2+y2cos(kspx')dx'+i0Y1(kdx'2+y2)x'2+y2cos(kspx')dx']
(C3)

The following identities (see [23

23. H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452(7188), 728–731 (2008). [CrossRef] [PubMed]

], Section 6.726) can be used to solve the two integrals:

0(x2+b2)ν2Jν(ax2+b2)cos(cx)dx=π2aνbν+12(a2c2)ν214Jν12(ba2c2)
(C4)
0(x2+b2)ν2Yν(ax2+b2)cos(cx)dx=π2aνbν+12(a2c2)ν214Yν12(ba2c2)
(C5)

Using Eqs. (C4) and (C5), Eq. (C3) becomes

Hsp(x,y)=2hspyeikspx[π2(kd2ksp2)14kdyJ12(±ykd2ksp2)+iπ2(kd2ksp2)14kdyY12(±ykd2ksp2)]
(C6)

Simplifying the equation and combining the two Bessel Functions in a Hankel function gives

Hsp(x,y)=hspeikspx2π(kd2ksp2)14kdyH12(1)(±ykd2ks2)
(C7)

The Hankel function can be expressed in an analytical form with the following identity (see [29

29. I. S. Gradshteyn and I. m. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

], Section 8.469):

H12(1)(z)=2πzeizi
(C8)

Hsp(x,y)=hsp2ikdeikspxe±iykd2ksp2
(C9)

Selecting the minus sign to satisfy the radiation condition gives the result presented in Eq. (C4).

Appendix D

To obtain a solution for the scattered field in the near-field region, which is when the distance to the slit is smaller than one wavelength, we take the convolution integral for Hs(x,y) in the form shown in Eq. (B8):

H(x,y)=hsy(x')2+y2H1(1)(kd(x')2+y2)eiksp(xx')E1(iksp(xx'))dx'
(D1)

For a small arguments z, the Hankel function can be approximated as
H1(1)(z)2iπz
(D2)
which is valid for 0<z<<21/2 and z = kd[(x-x’)2 + y2]1/2where x’ is smaller than a wavelength for E1(ikspx)≠0. The exponential integral can be expressed as
E1(z)=γln(z)+n=1(1)n+1znnn!
(D3)
where γ is the Euler constant. Hence, in the region with the distances to the slit smaller than the wavelength, Eq. (D1) can be approximated as

Hs(x,y)2hsyiπkdeikspx1(x')2+y2eikspx'[γln(iksp(xx'))+n=1(1)n+1(iksp(xx'))nnn!]dx'
(D4)

Equation (D4) can be solved using complex contour integration. The poles of the integrant are at x’iy. Closing the contour with a half-circular path in the upper half space encloses the positive pole. The integral is then computed with Cauchy’s theorem trough evaluation of the residue around the pole x’=iy, resulting in the following solution satisfying the radiation condition:
Hs(x,y)2hsikdeiksp(xiy)[γln(iksp(xiy))+n=1(1)n+1(iksp(xiy))nnn!]
(D5)
which according to Eq. (D3) can be expressed in terms of a new exponential integral function:

Hs(x,y)2hsikdeiksp(xiy)E1(iksp(xiy))
(D6)

References and links

1.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]

2.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, Berlin, 2007).

3.

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(6668), 667–669 (1998). [CrossRef]

4.

L. Matrin-Moreno, F. J. Gracia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through sub-wavelength hole arrays,” Phys. Rev. Lett. 86, 1112–1117 (2001).

5.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004). [CrossRef] [PubMed]

6.

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88(5), 057403–057407 (2002). [CrossRef] [PubMed]

7.

H. J. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express 12(16), 3629–3651 (2004). [CrossRef] [PubMed]

8.

B. Ung and Y. Sheng, “Interference of surface waves in a metallic nanoslit,” Opt. Express 15(3), 1182–1190 (2007). [CrossRef] [PubMed]

9.

M. W. Kowarz, “Homogeneous and evanescent contributions in scalar near-field diffraction,” Appl. Opt. 34(17), 3055–3063 (1995). [CrossRef] [PubMed]

10.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. Hooft, D. Lenstra, and E. R. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94(5), 053901 (2005). [CrossRef] [PubMed]

11.

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 model,” Nat. Phys. 2(4), 262–267 (2006). [CrossRef]

12.

G. Gay, O. Alloschery, J. Weiner, H. J. Lezec, C. O’Dwyer, M. Sukharev, and T. Seideman, “Surface quality and surface waves on subwavelength-structured silver films,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(1), 016612 (2007). [CrossRef] [PubMed]

13.

P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. 2(8), 551–556 (2006). [CrossRef]

14.

P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep. 64(10), 453–469 (2009). [CrossRef]

15.

A. N. Sommerfeld, “Propagation of waves in wireless telegraphy,” Ann. Phys. (Leipzig) 28, 665–737 (1909).

16.

J. Zenneck, “Propagation of plane EM waves along a plane conducting surface,” Ann. Phys. (Leipzig) 23, 846–866 (1907).

17.

K. A. Norton, “The propagation of radio waves over the surface of the earth and in the upper atmosphere,” Proc. IRE 24(10), 1367–1387 (1936). [CrossRef]

18.

R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: Some early and late 20th-century controversies,” IEEE Antennas Propagat. Mag. 46(2), 64–79 (2004). [CrossRef]

19.

B. Ung and Y. Sheng, “Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals revisited,” Opt. Express 16(12), 9073–9086 (2008). [CrossRef] [PubMed]

20.

A. Y. Nikitin, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martín-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. 11(12), 123020 (2009). [CrossRef]

21.

Y. Gravel and Y. Sheng, “Rigorous solution for the transient Surface Plasmon Polariton launched by subwavelength slit scattering,” Opt. Express 16(26), 21903–21913 (2008). [CrossRef] [PubMed]

22.

G. Lévêque, O. J. F. Martin, and J. Weiner, “Transient behaviour of surface plasmon polaritons scattered at a sub-wavelength groove,” Phys. Rev. B 76(15), 155418 (2007). [CrossRef]

23.

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452(7188), 728–731 (2008). [CrossRef] [PubMed]

24.

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(6668), 667–669 (1998). [CrossRef]

25.

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(7), 1399–1402 (2005). [CrossRef] [PubMed]

26.

R. W. P. King, M. Owens, and T. T. Wu, Lateral Electromagnetic Waves: Theory and Applications to Communications, Geophysical Exploration and Remote Sensing (Springer-Verlag, New York, 1992).

27.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

28.

L. Chen, J. T. Robinson, and M. Lipson, “Role of radiation and surface plasmon polaritons in the optical interactions between a nano-slit and a nano-groove on a metal surface,” Opt. Express 14(26), 12629–12636 (2006). [CrossRef] [PubMed]

29.

I. S. Gradshteyn and I. m. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(260.2110) Physical optics : Electromagnetic optics
(260.3910) Physical optics : Metal optics
(310.6628) Thin films : Subwavelength structures, nanostructures

ToC Category:
Diffraction and Gratings

History
Original Manuscript: October 28, 2011
Revised Manuscript: December 14, 2011
Manuscript Accepted: December 18, 2011
Published: January 17, 2012

Citation
Yann Gravel and Yunlong Sheng, "Rigorous solution for optical diffraction of a sub-wavelength real-metal slit," Opt. Express 20, 2149-2162 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2149


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References

  1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424(6950), 824–830 (2003). [CrossRef] [PubMed]
  2. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, Berlin, 2007).
  3. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998). [CrossRef]
  4. L. Matrin-Moreno, F. J. Gracia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through sub-wavelength hole arrays,” Phys. Rev. Lett.86, 1112–1117 (2001).
  5. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305(5685), 847–848 (2004). [CrossRef] [PubMed]
  6. Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett.88(5), 057403–057407 (2002). [CrossRef] [PubMed]
  7. H. J. Lezec and T. Thio, “Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays,” Opt. Express12(16), 3629–3651 (2004). [CrossRef] [PubMed]
  8. B. Ung and Y. Sheng, “Interference of surface waves in a metallic nanoslit,” Opt. Express15(3), 1182–1190 (2007). [CrossRef] [PubMed]
  9. M. W. Kowarz, “Homogeneous and evanescent contributions in scalar near-field diffraction,” Appl. Opt.34(17), 3055–3063 (1995). [CrossRef] [PubMed]
  10. H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. Hooft, D. Lenstra, and E. R. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett.94(5), 053901 (2005). [CrossRef] [PubMed]
  11. 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 model,” Nat. Phys.2(4), 262–267 (2006). [CrossRef]
  12. G. Gay, O. Alloschery, J. Weiner, H. J. Lezec, C. O’Dwyer, M. Sukharev, and T. Seideman, “Surface quality and surface waves on subwavelength-structured silver films,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.75(1), 016612 (2007). [CrossRef] [PubMed]
  13. P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys.2(8), 551–556 (2006). [CrossRef]
  14. P. Lalanne, J. P. Hugonin, H. T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep.64(10), 453–469 (2009). [CrossRef]
  15. A. N. Sommerfeld, “Propagation of waves in wireless telegraphy,” Ann. Phys. (Leipzig)28, 665–737 (1909).
  16. J. Zenneck, “Propagation of plane EM waves along a plane conducting surface,” Ann. Phys. (Leipzig)23, 846–866 (1907).
  17. K. A. Norton, “The propagation of radio waves over the surface of the earth and in the upper atmosphere,” Proc. IRE24(10), 1367–1387 (1936). [CrossRef]
  18. R. E. Collin, “Hertzian dipole radiating over a lossy earth or sea: Some early and late 20th-century controversies,” IEEE Antennas Propagat. Mag.46(2), 64–79 (2004). [CrossRef]
  19. B. Ung and Y. Sheng, “Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals revisited,” Opt. Express16(12), 9073–9086 (2008). [CrossRef] [PubMed]
  20. A. Y. Nikitin, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martín-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys.11(12), 123020 (2009). [CrossRef]
  21. Y. Gravel and Y. Sheng, “Rigorous solution for the transient Surface Plasmon Polariton launched by subwavelength slit scattering,” Opt. Express16(26), 21903–21913 (2008). [CrossRef] [PubMed]
  22. G. Lévêque, O. J. F. Martin, and J. Weiner, “Transient behaviour of surface plasmon polaritons scattered at a sub-wavelength groove,” Phys. Rev. B76(15), 155418 (2007). [CrossRef]
  23. H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature452(7188), 728–731 (2008). [CrossRef] [PubMed]
  24. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998). [CrossRef]
  25. 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(7), 1399–1402 (2005). [CrossRef] [PubMed]
  26. R. W. P. King, M. Owens, and T. T. Wu, Lateral Electromagnetic Waves: Theory and Applications to Communications, Geophysical Exploration and Remote Sensing (Springer-Verlag, New York, 1992).
  27. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).
  28. L. Chen, J. T. Robinson, and M. Lipson, “Role of radiation and surface plasmon polaritons in the optical interactions between a nano-slit and a nano-groove on a metal surface,” Opt. Express14(26), 12629–12636 (2006). [CrossRef] [PubMed]
  29. I. S. Gradshteyn and I. m. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

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