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(σ_y ŷ + σ_z ẑ), where k _o = 2π/λ_o and σ_y ² + σ_z ² = ε. In general, ${\sigma }_z=\pm \sqrt{\epsilon -{\sigma }_y^2}$, with both plus and minus signs admissible. In each of the semi-infinite cladding media, however, only one value of σ_z 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.)

The E and H-fields of each plane-wave are related through the Maxwell equation ∇ × H = ∂D/∂t (where D = ε _o εE) as follows:

Here H _o is the (complex) amplitude of the magnetic field, $Z_o=\sqrt{\frac{{\mu }_o}{{\epsilon }_o}}\approx 377\Omega$ is the impedance of the free space, and $\omega =k_{o}c=\frac{k_o}{\sqrt{{\mu }_o{\epsilon }_o}}$ is the temporal frequency of the light wave. In what follows, the time-dependence factor exp(-iωt) shall be omitted from the equations.

We confine our attention to symmetric structures where both cladding media have the same dielectric constant ε ₁. In general, the modal fields are either odd or even with respect to the y-axis, allowing one to express the H-field profile of a given mode as follows (plus sign for even, minus sign for odd modes):

 

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 $z=\pm \frac{1}{2}w$ w requires that k_y = k _o σ_y 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 H_x along the x-axis, while the electric field has two components (E_y, E_z) 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).

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The corresponding E-field for each mode can be found from Eqs. (1). Continuity of H_x and E_y at the $z=\pm \frac{1}{2}w$ w boundaries yields

Substituting for H ₁ from Eq. (3a) into Eq. (3b), rearranging the terms, and expressing σ _z1 and σ _z2 in terms of σ_y, we find:

This transcendental equation in σ_y = σ ^(r) _y + i σ ⁽ⁱ⁾ _y is the characteristic equation of the waveguide depicted in Fig. 1. Each solution σ_y 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 σ_y 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 r_p 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 ${\sigma }_y=\sqrt{\frac{{\epsilon }_1{\epsilon }_2}{\left({\epsilon }_1+{\epsilon }_2\right)}}$, 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, ${\epsilon }_1\sqrt{{\epsilon }_2-{\sigma }_y^2}+{\epsilon }_2\sqrt{{\epsilon }_1-{\sigma }_y^2}=0$.

A trivial solution of Eq. (4), ${\sigma }_y=\sqrt{{\epsilon }_2}$, 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 ${\sigma }_y=\sqrt{{\epsilon }_1}$ 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 |σ_y|, we note that $\sqrt{{\epsilon }_1-{\sigma }_y^2}\approx \sqrt{{\epsilon }_2-{\sigma }_y^2}\approx i{\sigma }_y=-{\sigma }_y^{\left(i\right)}+i{\sigma }_y^{\left(r\right)}$. Since the imaginary part of the square root must always be positive, this answer should be multiplied by -1 if σ_y 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 |σ_y| thus becomes:

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 |σ_y|, 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 ₁ − ε 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 ${\sigma }_{\mathrm{spp}}=\sqrt{\frac{{\epsilon }_1{\epsilon }_2}{\left({\epsilon }_1+{\epsilon }_2\right)}}=\left(1.0265+i0.6217\times {10}^{-3}\right)$. 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 ${\sigma }_y=\sqrt{{\epsilon }_1}$, 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 σ_y in the propagation direction (phase velocity V_p = c/σ_y). 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 ε_metal ensures the continuity of D _⊥, and E _∥ can be made continuous by the proper choice of σ_y, namely, σ_y = σ_spp. 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 σ_y = σ_spp. (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.

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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 − ω_p ²/(ω ² +iγω). Here ω_p and γ are the conduction electrons’ plasma frequency and damping coefficient, respectively. The SPP’s characteristic function is thus written

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 ω_p = 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(ω_p)] = 0 at the plasma frequency ω_p, which corresponds to λ _o = 143 nm, whereas $\left[\epsilon \left(\frac{{\omega }_p}{\sqrt{2}}\right)\right]=-1$, corresponding to λ _o = 202 nm. (b) Plots of the real and imaginary parts of ${\sigma }_y\left(\omega \right)=\sqrt{\frac{\epsilon \left(\omega \right)}{\left[1+\epsilon \left(\omega \right)\right]}}$.

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Plots of ε(ω) and σ_y(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 ω_p, where Re[ε] ≈ 0, and $\frac{{\omega }_p}{\sqrt{2}}$, where Re[ε] ≈ -1, the latter being a singularity of σ_y(ω). Surface plasmon polaritons exist below $\frac{{\omega }_p}{\sqrt{2}}$, where Im[σ_y(ω)] ≈ 0 and Re[σ_y(ω)]>1. Between $\frac{{\omega }_p}{\sqrt{2}}$ and ω_p, in the so-called SPP bandgap,⁸ the large values of Im[σ_y(ω)] preclude the possibility of long-range surface waves. Above ω_p, the dielectric function ε(ω) is essentially real and positive; the material, therefore, is transparent, with a refractive index $n\left(\omega \right)=\sqrt{\epsilon \left(\omega \right)}<1$. Under these circumstances, Re[σ_y(ω)] < 1 represents incidence at Brewster’s angle θ_B, where ${\sigma }_y\left(\omega \right)=\sin {\theta }_B=\frac{n}{\sqrt{\left(1+n^2\right)}}$.

Suppose at y = 0 the SPP’s time-dependence is f(y = 0, t) = ∫F(ω − ω _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:

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

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

 

Fig. 4. Group velocity V_g(ω), normalized by the speed of light c, in the frequency range (a) below $\frac{{\omega }_p}{\sqrt{2}}$ and (b) above ω_p. 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.

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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 V_g(ω _o) = c/[ωσ _y ^(r)(ω)]′, where the derivative is evaluated at ω = ω _o. Figure 4 shows plots of V_g(ω) as well as the distortion coefficients of Eq. (9) in the frequency range (a) below $\frac{{\omega }_p}{\sqrt{2}}$ and (b) above ω_p. The group velocity diminishes as $\omega \to \frac{{\omega }_p}{\sqrt{2}}$ from below, but pulse attenuation and distortion in this neighborhood become substantial.

4. Slit in metallic host

In a previous publication¹² we investigated transmission of light through subwavelength slits in metallic slabs, and discussed the role of slab thickness and slit width, among other factors. Here we confine our attention to slits in semi-infinitely thick slabs, to avoid complications arising from the guided mode’s multiple reflections at the slit’s entrance and exit facets.

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 ${\sigma }_y^{(-)}=\sqrt{{\epsilon }_2}=1.0$, 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 (λ_o = 650 nm, ε ₁ = -19.6224 + 0.443i, ε ₂ = 1.0; guided mode in red).

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The modes associated with large σ _y ⁽ⁱ⁾ decay rapidly along the y-axis; unlike the fundamental solution, none of the modes having large σ_y ⁽ⁱ⁾ can be considered “guided.” However, when an external beam arrives at the entrance facet of the slit, such rapidly-decaying modes are needed to match the boundary conditions. In the system depicted in Fig. 5, for example, the Gaussian beam, arriving from the free-space at the semi-infinite slit, excites the guided mode, but it also excites surface modes that, together with the guided mode and the free-space modes reflected toward the source, help to satisfy Maxwell’s boundary conditions in the y = 0 plane. (In a recent publication devoted to periodic slit arrays,¹³ we presented a complete modal analysis and showed the utility of lossy modes in determining the optical properties of such arrays. A similar analysis for single slits requires not only the discrete modes listed in Table 3, but also the continuum modes to be described in Section 4.3.)

 

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 (E_y, E_z) 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.

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4.1. Effect of the slit width

In Table 4 we list the first few values of σ_y 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 w ≪ λ _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 σ_y associated with this guided mode decrease. As w increases, an odd mode appears which, at first, has a fairly large loss factor, but σ_y ⁽ⁱ⁾ continues to decrease until, at w ∼ λ _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 λ_o/Re[σ _y ⁽⁺⁾] = 0.65/1.1225 = 0.579μm; see Table 4.

 

Fig. 6. Left to right: Instantaneous field profiles H_x, E_v, E_z 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 H_x and E_y 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.)

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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 λ_spp = 634 nm and |H_x(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 S_y, S_z, 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.

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Fig. 8. Profiles of H_x(z) at Δy = 10 nm beneath the metal surface. |H_x| (green) indicates the field strength, while Re[H_x] (black) reveals phase variations across the surface. (a) Beyond |z| ∼ 5 μm the field amplitude |H_x| becomes nearly constant, indicating the presence of a low-loss SPP propagating away from the slit. Oscillations of |H_x| 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 H_x 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

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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 V_g = 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 V_gτ= 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 w ∼ λ _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[H_x(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.

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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 E_y ($z=\frac{1}{2}w+6.2\mu m,y=-20\mathrm{nm}$) is taken as the SPP’s strength. The incident Gaussian beam is described in the caption to Fig. 5.

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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):

The corresponding E-field components may be found from Eqs. (1). Continuity of H_x and E_y at the $z=\pm \frac{1}{2}w$ boundaries yields:

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

 

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 $z=\pm \frac{1}{2}w$ boundaries requires that σ_y 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 (H_x, E_y, E_z) are considered in this paper.

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Plots of H ₁′/H ₁ versus σ _z1 (a positive, real-valued parameter) for λ _o = 650 nm, w = 100 nm, ε ₁ = -19.6224 + 0.443i, ε ₂ = 1.0, are shown in Figs. 12(a, b) for even and odd modes, respectively. Note that, for most values of σ _z1, the ratio H ₁′/H ₁ is essentially -1, with resonances occurring at σ _z1 intervals ∼ λ _o/w = 6.5. These resonances appear to be linked to the higher roots of Eq. (4), some of which are listed in Table 3. For instance, the first even resonance at σ _z1 = 4.65 is related to the pair of roots σ _y ⁽⁺⁾ = (0.0764 + 6.4243i) and σ _y ^(-) = (-0.0763 + 6.4193i), whose corresponding σ_z values in the metallic region are (-4.653 + 0.058i) and (4.648 + 0.153i). The exact nature of this relationship is not yet clear.

 

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.

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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 ($\frac{1}{2}w<z<\infty$) in the system of Fig. 5.

Figure 13 shows a typical plasmonic function f(z) along with its spatial Fourier spectrum F(σ_z). Note that F(σ_z) 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(σ_z) 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.

The surface-mode continuum described above, together with the discrete set of slit modes obtained in Section 2, form a complete set, which can be used to match the boundary conditions at the front facet of a slit aperture in order to determine the strength of all excited modes, including guided mode(s) and SPP modes bound to the entrance facet. A modal analysis of the single slit aperture (similar to that conducted for a periodic slit array in Ref. [13]) is beyond the scope of the present paper, but will be taken up in a future publication.

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, (σ_spp = 1.1 + 0.03i), rapidly drops to zero when z goes negative. (b) The Fourier transform F(σ_z) of f(z) has spatial frequency content confined to the vicinity of σ_z = Re[σ_spp]/λ ^o. (c, d) The function g(z) and its Fourier transform G(σ_z) obtained by flipping F(σ_z) to the negative side of the σ_z-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.

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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.

Sometimes what amounts to a short-range or lossy mode along one axis, turns out to exhibit long-range behavior along a different axis. This is the case, for instance, with the pair of SPPs launched at the entrance facet of a slit waveguide and analyzed in some detail in Section 4: Whereas the SPP pair (bound to the entrance facet) is genuinely long-range along the z-axis, it decays rapidly in the y-direction; nonetheless, its existence and properties are critical factors in determining the strength of the slit’s guided mode that is launched at the same entrance facet but travels subsequently down the slit along the y-axis. It has been an objective of the present paper to identify and characterize all short-range, lossy modes of metallo-dielectric interfaces, in anticipation of the role that such modes necessarily play in any comprehensive analysis of plasmonic devices. We have presented a modal analysis of periodic arrays of slits in a recent publication;¹³ the extension of this analysis to the case of single slits (with the help of the lossy modes identified in the present paper) will be the subject of a forthcoming paper.