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
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, , 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.)
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, is the impedance of the free space, and 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):
The corresponding E-field for each mode can be found from Eqs. (1). Continuity of Hx and Ey at the w boundaries yields
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 , 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, .
A trivial solution of Eq. (4), , 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 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 . 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:
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 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.
As the slab thickness w increases, the fundamental solutions (highlighted in Table 1) approach each other, reaching the common value of . 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 , 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.
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.
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
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.
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 , where Re[ε] ≈ -1, the latter being a singularity of σy(ω). Surface plasmon polaritons exist below , where Im[σy(ω)] ≈ 0 and Re[σy(ω)]>1. Between 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 . Under these circumstances, Re[σy(ω)] < 1 represents incidence at Brewster’s angle θB, where .
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 ωσy(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,
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 and (b) above ωp. The group velocity diminishes as 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 , corresponding to an odd mode, yields zero-amplitude fields in the slit and its surroundings; see Eqs. (2,3).]
The modes associated with large σ y (i) decay rapidly along the y-axis; unlike the fundamental solution, none of the modes having large σy (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.)
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 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 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 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).
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.
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.
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 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.
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 Hx and Ey at the boundaries yields:
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 σz 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.
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 () 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.
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. ) 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.
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.
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.
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