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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 1 — Jan. 8, 2007
  • pp: 183–197
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Surface plasmon polaritons on metallic surfaces

A. R. Zakharian, J. V. Moloney, and M. Mansuripur  »View Author Affiliations


Optics Express, Vol. 15, Issue 1, pp. 183-197 (2007)
http://dx.doi.org/10.1364/OE.15.000183


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Abstract

A unified theoretical study of surface plasmon polaritons on flat metallic surfaces and interfaces is undertaken to clarify the nature of these electromagnetic waves, conditions under which they are launched, and the restrictions imposed by Maxwell’s equations that ultimately determine the strength of the excited plasmons. Finite Difference Time Domain computer simulations are used to provide a clear picture of the electromagnetic field distribution and the energy flow profile in a specific situation. The examined case involves the launching of plasmonic waves on the entrance facet of a metallic host perforated by a subwavelength slit, and the (simultaneous) excitation of the slit’s guided mode.

© 2007 Optical Society of America

1. Introduction

Recent advances in nano-fabrication have enabled a host of nano-photonic experiments involving subwavelength metallic structures.1-4 This flurry of activity has, in turn, reawakened interest in surface plasmon polaritons (SPPs) and inspired theoretical research in this area.5-8 Although the fundamental properties of SPPs have been known for nearly five decades,9,10 there remain certain subtle issues that could benefit from further critical analysis.

Metallic films or slabs surrounded by dielectrics, and dielectric-filled slits in metallic hosts, have similarities and differences that can be exploited for a better understanding of both systems. For instance, in addition to the well-known SPP modes, which usually have a long propagation range, Maxwell’s equations admit an infinite number of other solutions for both structures. The latter modes, which typically decay a short distance beyond their point of origination, may not be glamorous, but they play an important role in matching the boundary conditions at launch and, consequently, in determining the strength of the excited SPPs.

It is a goal of the present paper to provide a theoretical framework for the study of SPPs along with the other, “short-range” modes of metallo-dielectric interfaces. Another goal is to use numerical simulations to verify the detailed structure of long-range SPPs. Along the way, we present field distribution profiles and energy flow patterns aimed at promoting a physical understanding of SPP generation and propagation in ways that mathematical equations alone cannot convey. Thus, beginning with Maxwell’s equations, we determine the electromagnetic eigen-modes confined to flat metallo-dielectric interfaces. The behavior of these modes will be examined through computer simulations that show the excitation of SPPs in certain practical situations. Our numerical computations are based on the Finite Difference Time Domain (FDTD) method.11

Section 2 presents a general formulation of the problem that applies to a metallic slab surrounded by dielectrics, and also to a narrow slit within a metallic host. The subjects of discussion in Section 3 are the role of polarization in SPP formation, the wavelength dependence of SPP, and SPP’s group velocity. Section 4 describes the guiding of light through a subwavelength slit in a metallic host. Here, in addition to the guided mode (a bona fide SPP in its own right), we examine the nature of the SPPs launched at the entrance facet of the slit, and the subsequent lateral propagation of these waves away from the slit. Continuity of the electromagnetic fields at the entrance facet requires a continuum of surface modes, which are investigated in subsection 4.3.

2. General Formulation

With reference to Fig. 1, in a homogeneous medium of dielectric constant ε the propagation vector is k = k o(σy ŷ + σz ẑ), where k o = 2π/λo and σy 2 + σz 2 = ε. In general, σz=±εσy2, with both plus and minus signs admissible. In each of the semi-infinite cladding media, however, only one value of σz is allowed, corresponding to the solution that approaches zero when z → ±∞. This is why σ z1 of the upper cladding in Fig. 1 is chosen to have a plus sign, whereas that of the lower cladding has a minus sign. (σ z1, σ z2 have positive imaginary parts.)

Hxyzt=Hoexp{i[ko(σyy±σzz)ωt]}
(1a)
Eyyzt=(Zoσzε)Hxyzt
(1b)
Ezyzt=(Zoσyε)Hxyzt
(1c)

Here H o is the (complex) amplitude of the magnetic field, Zo=μoεo377Ω is the impedance of the free space, and ω=koc=koμoεo is the temporal frequency of the light wave. In what follows, the time-dependence factor exp(-iωt) shall be omitted from the equations.

Hxyz={H1exp(ikoσyy)exp[ikoσz1(z12w)];z+12wH2exp(ikoσyy)[exp(ikoσz2z)±exp(ikoσz2z)];z12w±H1exp(ikoσyy)exp[ikoσz1(z+12w)];z12w
(2)
Fig. 1. Slab of thickness w and dielectric constant ε 2, sandwiched between two homogeneous, semi-infinite media of dielectric constant ε 1. An electromagnetic mode of the structure consists of two (generally inhomogeneous) plane-waves within the slab and a single (inhomogeneous) plane-wave in each of the surrounding media. Continuity of the fields at z=±12w w requires that ky = k o σy be the same for all these plane-waves. Although the polarization state of the mode can, in general, be either TE or TM, only TM modes are considered in this paper. The magnetic field, therefore, has a single component Hx along the x-axis, while the electric field has two components (Ey, Ez) in the yz-plane. Throughout the paper λ o = 650 nm and the metallic medium is silver, having ε = -19.6224 + 0.443i (corresponding to n + i k = 0.05 + 4.43i).

The corresponding E-field for each mode can be found from Eqs. (1). Continuity of Hx and Ey at the z=±12w w boundaries yields

H2[exp(ikoσz2w2)±exp(ikoσz2w2)]=H1
(3a)
ZoH2(σz2ε2)[exp(ikoσz2w2)exp(ikoσz2w2)]=ZoH1(σz1ε1)
(3b)

Substituting for H 1 from Eq. (3a) into Eq. (3b), rearranging the terms, and expressing σ z1 and σ z2 in terms of σy, we find:

ε1ε2σy2ε2ε1σy2ε1ε2σy2+ε2ε1σy2exp(ikoε2σy2w)=±1
(4)

This transcendental equation in σy = σ (r) y + i σ (i) y is the characteristic equation of the waveguide depicted in Fig. 1. Each solution σy of Eq. (4) corresponds to a particular mode of the waveguide; when the plus (minus) sign is used on the right-hand side of Eq.(49), the solution represents an even (odd) mode. Since we are presently interested in modes that propagate from left to right in Fig. 1, the imaginary part of σy must be non-negative (i.e., (i) y ≥ 0), otherwise the mode will grow exponentially as y → ∞. Also, when computing the complex square roots in Eq. (4), one must always choose the root which has a positive imaginary part.

Note that the coefficient multiplying the complex exponential on the left-hand side of Eq. (4) is the Fresnel reflection coefficient rp for a p-polarized (TM) plane-wave at the interface between media of dielectric constants ε 1 and ε 2. The Fresnel coefficient has a singularity (pole) at σy=ε1ε2(ε1+ε2), where its denominator vanishes. The function on the left-hand side of Eq. (4) thus varies rapidly in the vicinity of this pole, where some of the solutions of the equation are to be found. In particular, when w → ∞, the complex exponential approaches zero and the pole itself becomes a solution. This can be seen most readily with reference to Eqs. (3); by allowing exp(+i k o σ z2 w/2) → 0 and substituting for H 1 from Eq. (3a) into Eq. (3b), we find σ z2/ε 2 = -σ z1/ε 1, namely, ε1ε2σy2+ε2ε1σy2=0.

A trivial solution of Eq. (4), σy=ε2, can be readily substituted in the equation and verified to be an odd solution. This, however, leads to σ z2 = 0, which, when plugged into Eqs. (2, 3), reveals the electromagnetic field to be identically zero everywhere.

In the limit when w → 0, the complex exponential in Eq. (4) approaches unity, and σy=ε1 becomes an even solution. This corresponds to σ z1 = 0, and yields a single plane-wave that propagates along the y-axis throughout the entire space; the plane-wave will be homogeneous if ε 1 happens to be real and positive, otherwise it will be inhomogeneous.

In searching for solutions of Eq. (4) that possess a very large |σy|, we note that ε1σy2ε2σy2iσy=σy(i)+iσy(r). Since the imaginary part of the square root must always be positive, this answer should be multiplied by -1 if σy happens to be in the second quadrant of the complex plane, that is, it must be written as -(σ (r) y/|σ (r) y|)σ (i) y + i|σ (r) y|. The simplified form of Eq. (4) in the limit of large |σy| thus becomes:

exp[kowσy(i)]exp[ikowσy(r)σy(r)σy(r)]±(ε1+ε2)(ε1ε2)
(5)

Since the magnitude of the left-hand side of Eq. (5) is below unity, for a solution to exist, the magnitude of the right-hand side must also be ≤ 1. This is possible when the angle between ε 1 and ε 2 in the complex plane happens to be greater than 90°. Thus, in the limit of large |σy|, the value of |σ (r) y| is fixed by |(ε 1 + ε 2)/(ε 1 − ε2)|. Subsequently, the choice of an even or odd solution (i.e., ± sign on the right-hand-side of Eq. (5)), the phase of (ε 1 + ε 2)/(ε 1ε 2), and a ± sign choice for σ (r) y determine the value of σ (i) y. An infinite number of such solutions for σ (i) y exist, as the left-hand-side of Eq. (5) is repeated whenever σ (i) y increases by λ o/w.

In the following sections we investigate the solutions of Eq. (4) under different circumstances, and present computer simulation results that elucidate the physical meaning of these solutions.

3. Metallic slab in free space

Consider the case of ε 1 = 1.0, ε 2 = -19.6224 + 0.443i (silver at λ o = 650 nm). Fixing the slab’s thickness at w = 50 nm and searching the complex-plane for solutions of Eq. (4) yields the first few values of σ (±) y = σ (r) y + iσ(i) y listed in Table 1; the ± superscripts identify the even and odd modes, respectively. Only solutions having non-negative values of σ (i) y are considered so that, as y → +∞, the corresponding modes will decay to zero. Although we will be concerned mainly with the top two (fundamental) solutions in Table 1, there exists an infinite number of solutions with large values of σ (i) y. The latter are generally needed to match the boundary conditions upon launching a SPP, but, due to their rapid decay along the y-axis, the modes with large σ (i) y do not appear to have any practical significance otherwise.

Table 1. First few solutions of Eq. (4) for a 50 nm-thick silver slab (λ o = 650 nm, ε1 = 1.0, ε2 = -19.6224 + 0.443i)

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As the slab thickness w increases, the fundamental solutions (highlighted in Table 1) approach each other, reaching the common value of σspp=ε1ε2(ε1+ε2)=(1.0265+i0.6217×103). Reducing the slab thickness causes the fundamental solutions to move apart (and also further away from σ spp). As w→ 0, the even solution approaches σy=ε1, while the odd solution acquires a large σ y (r) and a fairly large σ y (i),. Table 2 lists the fundamental solutions of Eq. (4) for a range of values of w.

Table 2. Fundamental modes of silver slabs of differing thickness (λ o = 650 nm, ε 1 = 1.0, ε 2 = -19.6224 + 0.443i)

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3.1. Polarization dependence of SPP

A mathematical analysis similar to that of Section 2 reveals that TE-polarized electromagnetic waves cannot support SPPs at metal-dielectric interfaces. The following argument proves the same point by appealing to the underlying physics of surface plasmons. For the sake of simplicity we consider a thick metal plate in vacuum; see Fig. 2. A SPP consists of two inhomogeneous plane-waves (one in the free space, the other in the metal), both having the same σy in the propagation direction (phase velocity Vp = c/σy). The diagram in Fig. 2(a) represents a true SPP, with the E-fields originating on positive (surface) charges and terminating on negative ones. If the continuity of H at the surface is assumed, then a negative εmetal ensures the continuity of D , and E can be made continuous by the proper choice of σy, namely, σy = σspp. In contrast, the diagram in Fig. 2(b) presents a physical impossibility: absence of magnetic charges in nature means that the H-field must be divergence-free everywhere and, in particular, at the metal-vacuum interface; however, since H will now have opposite directions above and below the surface, it cannot satisfy the requisite boundary condition. This is the reason why SPPs must, of necessity, be TM-polarized.

Fig. 2. (a) The SPP’s E-fields originate on positive charges and terminate on negative ones. Continuity of H and a negative ε metal ensure the continuity of D⊥, while E becomes continuous when σy = σspp. (b) A physical impossibility, since the divergence-free nature of the H-field requires H to have opposite directions above and below the surface, thus prohibiting the continuity of H at the boundary.

3.2. Dependence of SPP on wavelength: group velocity

We examine the case of resonant oscillations at the flat interface between the free space and a semi-infinite metallic medium whose dielectric constant, according to the Drude model, is σ 2(ω) = 1 − ωp 2/(ω 2 +iγω). Here ωp and γ are the conduction electrons’ plasma frequency and damping coefficient, respectively. The SPP’s characteristic function is thus written

σy(ω)=ε2(1+ε2)=[(ω2ωp2)+iγω][(2ω2ωp2)+2iγω].
(6)

For concreteness, we consider a fictitious material whose dielectric constant at λ o = 650 nm (corresponding to ω = 0.29 × 1016 rad/s) is ε = -19.6224 + 0.443i. The known value of ε at this frequency then yields ωp = 1.317 × 1016 rad/s and γ = 0.623 × 1014 s-1.

Note: Although the chosen value of ε at λ o = 650 nm is that of silver, its value at other wavelengths deviates from silver’s, the reason being that the Drude model ignores the contribution of bound electrons to the optical properties of the material. Dionne et al.8 have used the experimentally-determined ε(ω) in their calculations, and pointed out the consequences for SPP’s dispersion relation; for our purposes here, however, the Drude model should suffice.

Fig. 3. (a) Real and imaginary parts of ε(ω) for a metal that obeys the Drude model in the frequency range from near IR to UV; the inset is a close-up of the right tail of the curves. Re[ε(ωp)] = 0 at the plasma frequency ωp, which corresponds to λ o = 143 nm, whereas [ε(ωp2)]=1, corresponding to λ o = 202 nm. (b) Plots of the real and imaginary parts of σy(ω)=ε(ω)[1+ε(ω)].

Plots of ε(ω) and σy(ω) in a frequency range extending from the near infrared to the ultraviolet are shown in Fig. 3. The two critical frequencies specifically marked on these plots are ωp, where Re[ε] ≈ 0, and ωp2, where Re[ε] ≈ -1, the latter being a singularity of σy(ω). Surface plasmon polaritons exist below ωp2, where Im[σy(ω)] ≈ 0 and Re[σy(ω)]>1. Between ωp2 and ωp, in the so-called SPP bandgap,8 the large values of Im[σy(ω)] preclude the possibility of long-range surface waves. Above ωp, the dielectric function ε(ω) is essentially real and positive; the material, therefore, is transparent, with a refractive index n(ω)=ε(ω)<1. Under these circumstances, Re[σy(ω)] < 1 represents incidence at Brewster’s angle θB, where σy(ω)=sinθB=n(1+n2).

Suppose at y = 0 the SPP’s time-dependence is f(y = 0, t) = ∫F(ωω o)exp(-iωt)dω, where f(∙) is expressed as the Fourier transform of a narrowband function F(∙) centered at ω o. At a later point y = y o, each Fourier component of f(∙) acquires a y-dependent term as follows:

fyot=F(ωωo)exp{iω[σy(ω)yoct]}dω.
(7)

Expanding ωσy(ω) in a Taylor series around ω=ω o and retaining the first few terms yields,

ωσy(ω)ωoσy(ωo)+[ωσy(ω)](ωωo)+12[ωσy(ω)]′′(ωωo)2.
(8)

Here the first and second derivatives with respect to ω are evaluated at ω o. Consequently,

fyotexp{ko[σy(i)(ωo)iσy(r)(ωo)+i[ωσy(r)]]yo}
×exp{[[ωσy(i)](ωωo)+12[ωσy(i)]′′(ωωo)212i[ωσy(r)]′′(ωωo)2]yoc}F(ωωo)
×exp{[[ωσy(r)]yoct]}.
(9)
Fig. 4. Group velocity Vg(ω), normalized by the speed of light c, in the frequency range (a) below ωp2 and (b) above ωp. Also shown are the coefficients [ωσ y (i)(ω)]′, [ωσ y (r)(ω)]″, [ωσ y (i)(ω)]″, which appear in Eq. (9) and cause pulse distortion during propagation.

In Eq. (9), the first exponential function causes attenuation and phase shift in proportion to the propagation distance y o, the second exponential term distorts the waveform’s spectrum (albeit insignificantly, if the spectrum is narrow and/or y o is small), and the last term causes a delay in the arrival of the pulse at y o. The group velocity is thus obtained from the last term as Vg(ω o) = c/[ωσ y (r)(ω)]′, where the derivative is evaluated at ω = ω o. Figure 4 shows plots of Vg(ω) as well as the distortion coefficients of Eq. (9) in the frequency range (a) below ωp2 and (b) above ωp. The group velocity diminishes as ωωp2 from below, but pulse attenuation and distortion in this neighborhood become substantial.

4. Slit in metallic host

Listed in Table 3 are the first few solutions of Eq. (4) for a 100 nm-wide empty slit (ε 2 = 1.0) in a silver host at λ o = 650 nm. For small values of σ y (i) the transcendental equation yields only one solution, highlighted in Table 3, which corresponds to an even, low-loss, guided mode along the length of the slit. [The trivial solution σy()=ε2=1.0, corresponding to an odd mode, yields zero-amplitude fields in the slit and its surroundings; see Eqs. (2,3).]

Table 3. First few solutions of Eq. (4) for a 100 nm-wide slit in a silver host (λo = 650 nm, ε 1 = -19.6224 + 0.443i, ε 2 = 1.0; guided mode in red).

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Fig. 5. A Gaussian beam of wavelength λ o is focused at the entrance facet of a semi-infinite slit (width = w, dielectric constant = ε 2) in a metallic host having dielectric constant ε 1. (In this 2-dimensional system the beam is focused through a cylindrical lens; its shape, therefore, does not vary along the x-axis.) The beam is linearly polarized, having H-field along x and E-field components (Ey, Ez) confined to the plane of incidence. In our FDTD simulations, the incident beam (λ o = 650 nm, amplitude FWHM = 4 μm) is sourced at y = -10 nm, just before the entrance facet of the slit.

4.1. Effect of the slit width

In Table 4 we list the first few values of σy for TM modes in slits of varying width; the host medium is silver, the slit is empty, and the incident wavelength is λ o = 650 nm. For wλ o there exists only one guided mode (which turns out to be even). With an increasing slit-width, both the real and imaginary parts of σy associated with this guided mode decrease. As w increases, an odd mode appears which, at first, has a fairly large loss factor, but σy (i) continues to decrease until, at wλ o/2, this odd mode becomes guided as well. Further increases in w bring more and more guided modes into the system; at w = 980 nm, for example, there exists a total of four guided modes (two odd and two even).

Table 4. First few TM modes of slits of differing width in a silver host (λ o = 650 nm, ε 1 = -19.6224 + 0.443i, ε 2 = 1.0; guided modes in red)

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4.2. Launching a guided mode

In Figs. 6–9 we present FDTD simulation results for a sub-wavelength, semi-infinite slit (w = 200nm, ε 2 = 1.0) in a silver host. A Gaussian beam (λ o = 650 nm, FWHM = 4.0 μm) is normally incident at the entrance facet of the slit, as in Fig. 5. The guided mode is seen in Fig. 6 to propagate along the y-axis, and a certain amount of light is reflected back toward the source. (The incident beam having been removed, only reflected and transmitted beams appear in these pictures.) The guided mode’s period along the length of the slit in Fig. 6 is 0.58μm, in agreement with λo/Re[σ y (+)] = 0.65/1.1225 = 0.579μm; see Table 4.

Fig. 6. Left to right: Instantaneous field profiles Hx, Ev, Ez show a guided mode propagating down the slit (w = 200 nm, ε 2 = 1.0) and a reflected beam propagating backward in the incidence space. Long-range SPPs excited on the front facet of the metallic host (both above and below the slit) are clearly visible in the Hx and Ey plots. (The incident beam, sourced at Δy = 10 nm before the entrance facet, is largely absent from these plots, hence the absence of interference fringes between the incident and reflected beams.)

An intriguing feature of these results is the pair of SPPs launched at the slit edges and seen to propagate away from the slit on the front facet of the metallic host. In the Poynting vector plots of Fig. 7, the observed energy flow in the ±z directions is consistent with the lateral propagation of surface plasmons away from the slit. A fit to the H-field profile of the upward-traveling SPP (see Fig. 8) yields λspp = 634 nm and |Hx(y = 0,z)| ∼ exp(-0.00585 z), consistent with the presence of a lone SPP on either side of the slit.

Fig. 7. Left to right: Poynting vector components Sy, Sz, and magnitude |S| in and around the 200 nm-wide slit simulated in Fig. 6. The incident beam having been removed, only reflected and transmitted beams appear in these pictures.
Fig. 8. Profiles of Hx(z) at Δy = 10 nm beneath the metal surface. |Hx| (green) indicates the field strength, while Re[Hx] (black) reveals phase variations across the surface. (a) Beyond |z| ∼ 5 μm the field amplitude |Hx| becomes nearly constant, indicating the presence of a low-loss SPP propagating away from the slit. Oscillations of |Hx| in the immediate vicinity of the slit are caused by interference between the small fraction of the Gaussian beam that penetrates the metallic surface and the SPP launched at the slit. (b) Subtracting from the total Hx at Δy = 10 nm, an estimated profile of the Gaussian beam (obtained by setting w = 0 in a separate simulation), reveals the dominance of SPP at |z| > 1 μm

Figure 9 shows the SPP excited on the upper side of the 200 nm-wide slit detaching itself from the slit and moving away; the incident beam in this case is a τ= 20 fs Gaussian pulse. For this simulation the assumed functional form of ε(ω) was the same as that used in subsection 3.2 (i.e., Drude model). The computed group velocity Vg = 0.925c is very close to the theoretical value of 0.9217c (see Fig. 4), and the Gaussian profile of the SPP along the z-axis (FWHM = 5.495μm) is consistent with the value of Vgτ= 5.53μm.

Figure 10 shows the computed strength of the launched SPP as a function of the slit width w. For very small w, the SPP on either side of the slit is weak; this is expected, of course, considering that no SPP can be launched in the absence of a slit. As the slit widens, the SPP becomes strong at first, but weakens again and reaches a minimum when wλ o. We expect the SPP strength to eventually stabilize with further increases in w, as optical interference between the two metal blocks forming the slit’s walls should decline as the blocks move apart.

Fig. 9. The SPPs of Fig. 8 detach from the slit and travel away when the incident beam is a short, Gaussian pulse (FWHM width τ= 20 fs). Shown are profiles of Re[Hx(z)] on one side of the slit at Δy = 20 nm before the metal surface at t = 60, 80, 100 fs after the start of the pulse.
Fig. 10. Strength of the SPP launched on the front facet of the metallic host as function of the slit width (λ o = 650 nm, semi-infinite silver host). The magnitude of Ey (z=12w+6.2μm,y=20nm) is taken as the SPP’s strength. The incident Gaussian beam is described in the caption to Fig. 5.

Returning to the functional form of the SPPs launched along the z-axis at the slit’s edges, it should not come as a surprise that this mode is exactly the same as that of a thick slab described in Section 3 and listed in Table 2 under w = ∞; after all, the interface between the free-space and a semi-infinite metallic medium can sustain one and only one confined mode. What is perhaps surprising here is that none of the slit modes computed from Eq. (4) – and, for the specific case of w = 100 nm, listed in Table 3 – resemble the front facet SPPs shown, for instance, in Fig. 8 (w = 200 nm in this case). While the short-range/lossy modes of Table 3 (or their counterparts associated with other values of the slit-width w) exhibit long-range and rapid oscillations along the z-axis, their spatial frequencies (σ z) generally differ from the observed SPP frequency at the front facet. This observation leads one to suspect the existence of additional modes, i.e., modes that lie beyond the solution space of Eq. (4), at the entrance facet of the waveguide depicted in Fig. 5. In the following subsection we investigate the surface modes of such slit waveguides, uncover the existence of a continuum of modes at the front facet, and show consistency with the above FDTD simulations.

4.3. Surface modes of the slit waveguide

In this section we address the question of additional modes of the system of Fig. 1 that fall outside the solutions of Eq. (4). Whereas the solutions of Eq. (4) produce a discrete set of modes for the single slit system, the modes discussed in the present section form a continuum. Both sets of modes are needed to satisfy the boundary conditions at the entrance facet of a slit aperture, and, therefore, to provide a full description of the interaction between an incident beam and the single-slit system.

With reference to Fig. 11, we introduce the possibility of constructing additional modes that contain both incoming and outgoing waves in the cladding region. Such incoming waves (from z = ±∞) are physically admissible so long as the z-component of their k-vector is real (i.e., σ z1 is real-valued). In general, the field profiles are either odd or even around the y-axis, allowing one to express the H-field of a given mode as follows (± for even and odd modes):

Hxyz={exp(ikoσyy){H1exp[ikoσz1(z12w)]+H1exp[ikoσz1(z12w)]};z+12wH2exp(ikoσyy)[exp(ikoσz2z)±exp(ikoσz2z)];z12w±exp(ikoσyy){H1exp[ikoσz1(z+12w)]+H1exp[ikoσz1(z+12w)]};z12w
(10)

The corresponding E-field components may be found from Eqs. (1). Continuity of Hx and Ey at the z=±12w boundaries yields:

H2[exp(ikoσz2w2)±exp(ikoσz2w2)]=H1+H1
(11a)
ZoH2(σz2ε2)[exp(ikoσz2w2)exp(ikoσz2w2)]=Zo(σz1ε1)(H1H1)
(11b)

Dividing Eq. (11a) by Eq. (11b), rearranging the terms, and expressing σ z2 in terms of σ z1 (remembering that σ y 2 + σ z1 2 = ε 1 and σ y 2 + σ z2 2 = ε 2), we arrive at

H1H1=1±[(ε2σz1ε1σz12+ε2ε1)(ε2σz1+ε1σz12+ε2ε1)]exp(ik0σz12+ε2ε1w)[(ε2σz1ε1σz12+ε2ε1)(ε2σz1+ε1σz12+ε2ε1)]±exp(ik0σz12+ε2ε1w)
(12)
Fig. 11. Same as Fig. 1, except for the existence of counter-propagating waves in the semi-infinite cladding regions above and below the slab waveguide. An electromagnetic mode of the structure thus consists of six (generally inhomogeneous) plane-waves – two within each of the three media. Continuity of the fields at the z=±12w boundaries requires that σy be the same for all these plane-waves. We impose the further restriction that σ z1 be real-valued, as any imaginary component of σ z1 causes the beams in the cladding region to grow indefinitely when z → ±∞. Only TM modes having field components (Hx, Ey, Ez) are considered in this paper.

Fig. 12. Real and imaginary parts of H 1′/H 1 as functions of σ z1. (a) Even modes corresponding to the plus sign in Eq. (12). (b) Odd modes corresponding to the minus sign.

With regard to the plasmonic waves excited just beneath the metallic surface (see Fig. 8), the question arises as to the role played by surface modes of negative spatial frequency, i.e., the terms in Eq. (10) whose coefficients are denoted by H 1′. In what follows we shall focus our attention on the plasmonic wave excited on the upper half of the z-axis, namely, the SPP running along the positive z-axis (12w<z<) in the system of Fig. 5.

Figure 13 shows a typical plasmonic function f(z) along with its spatial Fourier spectrum F(σz). Note that F(σz) contains primarily positive frequencies located in the neighborhood of the SPP frequency. For every positive σ z1 in Eq. (10), however, there exists a corresponding negative frequency with the same amplitude (albeit multiplied by -1). Therefore, the negative spatial frequencies associated with the f(z) SPP (beneath the metallic surface) have the spectrum G(σz) of Fig. 13(d) and the spatial profile g(z) of Fig. 13(c). Note that g(z) is essentially zero on the positive z-axis, its main contribution to the plasmonic wave being in the immediate neighborhood of the origin at z = 0, where it overlaps with the initial part of f(z); see Fig. 13(e). The overlap will be even less significant if the SPP happens to have a sharper rising edge at z = 0. The conclusion is that the presence of both positive and negative spatial frequencies within the continuum of surface modes described by Eq. (10) is not incompatible with the existence of unidirectional plasmons at the entrance facet of the slit.

5. Concluding remarks

We have analyzed the modes of metallic surfaces, including those that are linked across the narrow gap of a subwavelength slit in a metallic host. Maxwell’s equations admit many solutions for electromagnetic fields that can be considered localized at and around metallic surfaces (or, in general, confined to the vicinity of metallo-dielectric interfaces). However, only a handful of such solutions extend far enough beyond their point of origination to be considered useful for practical applications. The odd and even waves that propagate along the surfaces of metallic films, and the guided modes of slit waveguides are examples of such long-range surface plasmon polaritons. The remaining solutions – properly classified as short-range or lossy modes – should not be ignored, however, as they participate in the matching of the boundary conditions wherever a long-range SPP is launched, or whenever an existing SPP crosses the boundary from one environment into another.

Fig. 13. (a) The function f(z), representing an arbitrary plasmonic wave on the positive z-axis (λ o = 0.65μm, (σspp = 1.1 + 0.03i), rapidly drops to zero when z goes negative. (b) The Fourier transform F(σz) of f(z) has spatial frequency content confined to the vicinity of σz = Re[σspp]/λ o. (c, d) The function g(z) and its Fourier transform G(σz) obtained by flipping F(σz) to the negative side of the σz-axis, then multiplying it by -1. (e) Combining the positive- and negative-frequency spectra of (b) and (d) yields the function f(z) + g(z), shown here on the positive side of the z-axis only. The main effect of adding negative-frequency terms to the plasmonic function f(z) is seen to be in the neighborhood of the origin; the inset is a close-up of |f(z) + g(z)| around z = 0.

Our FDTD simulations have verified the validity of our simple theoretical analysis, but they have also provided a physical picture of field distributions and energy flow patterns in realistic systems that are generally inaccessible to simple mathematical analysis. We have shown, for example, that direct illumination of the entrance facet of a slit in a metallic host, while exciting the slit’s guided mode, also launches a pair of relatively strong SPPs at the entrance facet; the SPP’s then propagate away from the slit with a group velocity that is in excellent agreement with the theoretically predicted value.

Acknowledgements

We are grateful to John Weiner and Pavel Polynkin for many helpful discussions. This work has been supported by the AFOSR contracts F49620-03-1-0194, FA9550-04-1-0213, FA9550-04-1-0355 awarded by the Joint Technology Office.

References

1.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 39,667–69 (1998). [CrossRef]

2.

R. D. Averitt, S. L. Westcott, and N. J. Halas, “Ultrafast electron dynamics in gold nanoshells,” Phys. Rev. B 58,R10203–R10206 (1998). [CrossRef]

3.

J. J. Mock, S. J. Oldenburg, D. R. Smith, D. A. Schultz, and S. Schultz, “Composite Plasmon Resonant Nanowires,” Nano Letters 2,465–69 (2002). [CrossRef]

4.

G. Gay, O. Alloschery, B. V. de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nano-structured surfaces and the composite diffracted evanescent wave model,” Nature Phys. 264,262–67 (2006). [CrossRef]

5.

H. F. Ghaemi, T. Thio, and D. E. Grupp, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58,6779–82 (1998). [CrossRef]

6.

F. J. Garcia-Vidal and H. J. Lezec, et al, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90,213901(4) (2003). [CrossRef]

7.

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A 2,48–51 (2000). [CrossRef]

8.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72,075405 (2005). [CrossRef]

9.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Springer-Verlag, Berlin, 1986.

10.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33,5186–5201 (1986). [CrossRef]

11.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd edition, Artech House, 2000.

12.

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]

13.

Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through periodic arrays of sub-wavelength slits in metallic hosts,” Opt. Express 14,6400–6413 (2006). [CrossRef] [PubMed]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(260.3910) Physical optics : Metal optics
(310.2790) Thin films : Guided waves

ToC Category:
Optics at Surfaces

History
Original Manuscript: September 14, 2006
Manuscript Accepted: December 19, 2006
Published: January 8, 2007

Citation
Armis R. Zakharian, Jerome V. Moloney, and Masud Mansuripur, "Surface plasmon polaritons on metallic surfaces," Opt. Express 15, 183-197 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-1-183


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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 subwavelength hole arrays," Nature 39, 667-69 (1998). [CrossRef]
  2. R. D. Averitt, S. L. Westcott, and N. J. Halas, "Ultrafast electron dynamics in gold nanoshells," Phys. Rev. B 58, R10203-R10206 (1998). [CrossRef]
  3. J. J. Mock, S. J. Oldenburg, D. R. Smith, D. A. Schultz, and S. Schultz, "Composite Plasmon Resonant Nanowires," Nano Lett. 2,465 -69 (2002). [CrossRef]
  4. G. Gay, O. Alloschery, B. V. de Lesegno, C. O'Dwyer, J. Weiner, and H. J. Lezec, "The optical response of nano-structured surfaces and the composite diffracted evanescent wave model," Nat. Phys. 264, 262 - 67 (2006). [CrossRef]
  5. H. F. Ghaemi, T. Thio, and D. E. Grupp, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998). [CrossRef]
  6. F. J. Garcia-Vidal, H. J. Lezec,  et al, "Multiple paths to enhance optical transmission through a single subwavelength slit," Phys. Rev. Lett. 90, 213901(4) (2003). [CrossRef]
  7. Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, "One-mode model and Airy-like formulae for one-dimensional metallic gratings," J. Opt. A, Pure Appl. Opt.  2, 48-51 (2000). [CrossRef]
  8. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," Phys. Rev. B 72, 075405 (2005). [CrossRef]
  9. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer-Verlag, Berlin, 1986).
  10. J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986). [CrossRef]
  11. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed., (Artech House, 2000).
  12. 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]
  13. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, "Transmission of light through periodic arrays of sub-wavelength slits in metallic hosts," Opt. Express 14, 6400-6413 (2006). [CrossRef] [PubMed]

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