1. Introduction

The generation of surface waves and optical transmission around and through subwavelength structures has been the object of an avalanche of publications over the past decade, and reviews of some of this work have started to appear [1, 2, 3, 4, 5]. Most studies have approached the subject either by emphasizing periodic array properties (delocalized optical Bloch states and energy bands) [2, 5] or have couched their analysis in terms of physical optics (diffraction, reflection, transmission) [6, 7, 8]. The interpretation of numerical solutions to the Maxwell equations, governing the electromagnetic response of a structured metal film to incident radiation, has also provided insight into charge, current, near- and far-field distributions in and around subwavelength apertures [9, 10, 11]. In a similar vein electromagnetic analysis of “prototypical” subwavelength objects (spheres, rods, squares, etc.) has led to the development of a plasmonic circuit theory of lumped elements, guiding the engineering design of functional devices [12].

This electromagnetic point of view has been recently applied to the problem of light transmission through subwavelength slits milled into thin metal films [13]. This study concluded that currents, induced within the metal skin depth by a time-varying standing wave at the surface, produce oscillating charge dipoles at the slit edges, and that these dipoles are primarily responsible for transmission through the slit as well as the launch of surface waves at the dielectricmetal interface. The overall layout of the situation considered here and in Ref. [13] is shown in Fig. 1, and the disposition of fields and structured metal surface is shown in Fig. 2. Panel (A) of Fig. 2 shows a numerical simulation of the standing wave above a silver metal film in which two slits have been milled. The B-field is adjacent to metal surface. Panel (B) is an enlarged diagram of the interface region, showing the optical half-cycle in which the B-field points along the positive y direction. The purpose of the present study is to explore the consequences of a petahertz driving field incident on the surface for the phase and magnitude of the conduction current in the metal, the displacement current and fields present in the slit gap, and the transmission of light through the slits separated by a distance close to the surface plasmon polariton wave length.

 

Fig. 1. Basic layout showing metal layer with two slits of length l and width d separated by distance p. Dimensions are those typical of experiments, for example, in Refs. [14], [15] with l~10-20 µm, d~50-100 nm, p~λ_spp. Metal layer thickness ~200–300 nm with skin depth δ⋍25 nm. Plane waves incident normal to the top surface are polarized in TM mode, penetrate to depth δ and reflect from the surface.

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2. Complex permittivity and conductivity

We start with Maxwell’s equations in a real metal conductor with no free charges present.

where E,B are the electric and magnetic induction fields respectively; µ,ε the permeability and permittivity of the material; and the conduction current J _c is given by J _c=σE with σ the material conductivity. Equations (1), (2) can be decoupled by first applying the curl operation to the Faraday-Maxwell equation,

$\nabla \times \left(\nabla \times \mathbf{E}\right)=\nabla (\nabla ·\mathbf{E})-{\nabla }^2\mathbf{E}=\nabla \times \left(-\frac{\partial \mathbf{B}}{\partial t}\right)=-\frac{\partial \left(\nabla \times \mathbf{B}\right)}{\partial t}$

$=-\mu \epsilon -\frac{{\partial }^2\mathbf{E}}{\partial t^2}-\mu \sigma \frac{\partial \mathbf{E}}{\partial t}$

which together with ∇·E=0 yields

 

Fig. 2. Incident light on a structured metal surface. Panel (A) shows a numerical simulation of the standing wave field set up by the incident and reflected light in TM mode as indicated in Fig. 1. Dark blue rectangle with two slits is a silver layer. Above the surface and in the slit gaps B-field amplitude is color coded with red maximum and blue minimum. Note that the B-field predominates near the surface. Panel (B) is an enlarged schematic of the interface region, showing the optical half-cycle in which the B-field points in the +y direction, the skin depth δ, the distance p between slits of width d, and the induced charges at the top edges of the slits.

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Similarly applying the curl to the Ampère-Maxwell equation results in

Equations (3), (4) are wave equations describing the propagation of electromagnetic waves in the metal. A standard approach in the analysis of these waves is to assume plane-wave solutions of the form

where the subscript m indicates a field in the metal, Ẽ_0m, B̃_0m denote complex amplitudes, and k̃ _m is the complex propagation parameter in the metal. Equations (5), (6) can be considered plane-wave basis functions from which wave solutions of arbitrary form propagating in the metal can be constructed. The amplitude distribution of these plane wave components is called the “angular spectrum” [16] of the field. Substituting these solutions back into Eqs. (3), (4) results in an expression for the propagation parameter in terms of the permeability, complex permittivity, and complex conductivity of the metal.

where the complex quantities are defined as

Substituting and gathering real and imaginary terms,

and writing the real and imaginary parts of the permittivity in terms of the corresponding dielectric constants, ε _r=ε ₀ ε ^′ and ε _i=ε ₀ ε ^″ we have

where the standard relations µ ₀ ε ₀=1/c ² and k ₀=ω/c have been used.

Equations (11), (12) can be separated to find expressions for the real and imaginary propagation parameters, k _{m 1},k _{m 2}

The complex propagation parameter k̃_m can also be expressed in terms of a complex index of refraction, ñ.

with the usual relation between refractive index and dielectric constant, ñ²=ε ₁+iε ₂. Again equating real and imaginary terms results in expression for n ₁,n ₂ in terms of ε ₁,ε ₂

$n_1=\sqrt{\frac{{\epsilon }_1}{2}}\sqrt{1\pm \sqrt{1+{\left(\frac{{\epsilon }_2}{{\epsilon }_1}\right)}^4}}$

$n_2=\sqrt{\frac{{\epsilon }_1}{2}}\sqrt{-1\pm \sqrt{1+{\left(\frac{{\epsilon }_2}{{\epsilon }_1}\right)}^4}}$

and consequently

Comparing Eqs. (16), (17) to (13), (14) shows that

Equations (18), (19) separate the metal dielectric constant into two parts: first, ε ^′ and ε ^″, representing the dispersive and absorptive response of the metal excluding the conduction electrons and second, σ_i,r/ε ₀ ω, expressing the conducting electrons contribution to the dielectric constant. The next step is to model the conducting electron motion by a driven, damped harmonic oscillator.

3. Damped harmonic oscillator model for conduction electrons

We posit that the equation of motion of the conduction electrons is governed by harmonic acceleration issuing from an electromagnetic field propagating in the metal and polarized along the x direction.

$m_e\frac{d^{2}x}{d{t}^2}+m_e\Gamma \frac{dx}{dt}=eE(x,t)=e{E}_{0m}{e}^{-i\omega t}$

where m _e is the electron mass, Γ a phenomenological damping constant, and eE ₀ m ^{e -iωt} the driving force of an E-M field in the metal with amplitude E _0m and frequency ω. Then, dropping the common harmonic time factor, the solutions for position, velocity and acceleration are

$x=-\frac{1}{m_e\left({\omega }^2+i\Gamma \omega \right)}e{E}_{0m}\frac{dx}{dt}=\frac{i\omega }{m_e\left({\omega }^2+i\Gamma \omega \right)}e{E}_{0m}\frac{d^{2}x}{d{t}^2}=\frac{{\omega }^2}{m_e\left({\omega }^2+i\Gamma \omega \right)}e{E}_{0m}$

The conduction current density is then given by

$J_c=e{N}_e\frac{dx}{dt}=e{N}_e\frac{\omega }{m_e\left(\Gamma \omega -i{\omega }^2\right)}e{E}_{0m}$

The electron density N _e is related to the bulk plasmon frequency ω _p by

and therefore the magnitude of the conduction current density can be written

$J_c=\frac{{\omega }_p^2}{\left(\Gamma -i\omega \right)}{\epsilon }_0{E}_{0m}$

From the standard constitutive relations

and so the expression for the complex conductivity within the damped harmonic oscillator model can be written,

Compare the damping rate Γ to the bulk plasmon frequency ω _p. From Eq. (20) we can determine ω _p if we know the conduction electron density of the metal. A typical “good” metal such as silver exhibits an electron density N _e⋍5.85×10²⁸m^-3 [17], and therefore ω _p⋍1.4×10¹⁶ s^-1. The damping rate Γ due to radiative loss and phonon collisions is typically ~10¹⁴ s^-1 [18]. Therefore Γ≪ω _p, and in the visible and near-infrared spectral region (ω⋍10¹⁵ s^-1), Eq. (22) can be written

This last expression for the conductivity shows that at relatively low frequencies (RF, microwave, etc.) the real term dominates, and the conductivity is “ohmic”. As the frequency increases into the optical range, however, the electrons can no longer follow the driving field in phase, and the imaginary term dominates. At ω~10¹⁵ s^-1, the real term is only about 5% of the imaginary term. We can therefore, within the harmonic oscillator model, write Eqs. (18), (19) as

If we consider the metal response, excluding the conduction electrons, to behave essentially as a lossless dielectric material, we can estimate ε ^′ and ε ^″ as 1 and 0, respectively. This assumption is equivalent to the Drude model for metals. Taking ω=2.0×10¹⁵ s^-1 (λ ₀=942 nm) as a specific example, we calculate ε ₁=-47 and ε ₂=2.4. We can compare these values for the dispersive and absorptive parts of the dielectric constant to measured optical constants for silver at λ ₀=942 nm. Johnson and Christy [19] report the complex index of refraction in the metal as ñ=η+iκ, and from interpolation of their data we find η=0.04 and κ=6.964 at λ ₀=942 nm. Converting these data to the equivalent dielectric constants yields ε ₁=-48 and ε ₂=0.6. We see that the large, negative real value for ε ₁ can be interpreted as arising from the in-quadrature component of the complex conductivity. With η=0.04 the measured data results in an absorptive dielectric constant about one fourth the value calculated from the assumed damping constant Γ=10¹⁴ s^-1. However, from another commonly cited data set [20] we find η=0.2, κ=6.7 with corresponding dielectric constants ε ₁=-45,ε ₂=2.6. Dielectric constants derived from the harmonic oscillator model are in good agreement with both the dispersive and absorptive terms from this second data set. Table 1 summarizes the comparison between measured data and the harmonic oscillator model(HOM) at ω=2.0×10¹⁵ s^-1, while

Table 1. Comparison of dielectric constants for silver derived from two frequently cited data sets and the damped harmonic oscillator model (HOM), ω=2.0×10¹⁵ s^-1. The real and imaginary terms of the index of refraction η,κ are equivalent to n₁,n₂ used here.

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Fig. 3 plots the real and imaginary permittivities from published data and compares them to the HOM in the petahertz range of frequencies. It follows from Eqs. (21), (23) that the conduction current is in quadrature with the E-field propagating in the metal. In order to determine the phase of the current density with respect to the incident E-M field we have to examine the continuity conditions at the surface.

 

Fig. 3. Left and right panels show the real and imaginary parts respectively of the permittivity for Ag metal over the petahertz frequency range. Plotted blue triangles are from [19], red circles, [20]. Black curve is the harmonic oscillator model (HOM), Eq. 24, with ω _p=1.4×10¹⁶ rad s^-1 and Γ=1×10¹⁴ rad s^-1. The model agrees well with data over the range 1≤ω≤6×10¹⁵ rad s^-1.

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4. Continuity conditions

4.1. Amplitude and phase relations

Figure 4 shows the orientation of the E- and B-fields incident and reflected normal to the metal surface.

Where Ẽ₀, B̃₀ denote complex amplitudes, Ẽ₀=E ₀ e ^iφ _E, B̃₀=B ₀ e ^iφ _B and the subscripts I,R,T indicate incident, reflected and transmitted fields, respectively. The complex propagation parameter of the wave in the metal is k̃_m and k̃_m/ω is the phase velocity of propagation there. As in Eq. (8) we have

and from Eq. (26) the relative phase of the transmitted B-field to the transmitted E-Field is given by

The relation between the complex refractive index and the propagation parameter in the metal is

Eliminating Ẽ_0R from Eqs. (25), (26)

Writing out the phases explicitly and using Eqs. (28), (29) we have

$\frac{E_{0T}}{E_{0I}}{e}^{i\left({\phi }_{\mathrm{ET}}-{\phi }_{\mathrm{EI}}\right)}\to \frac{2k_0}{\mid k_m\mid }{e}^{-i\phi }=\frac{2k_0}{\mid k_m\mid }{e}^{-i\left({\phi }_{\mathrm{BT}}-{\phi }_{\mathrm{ET}}\right)}$

Now from Table I we know that, |ε ₁|≫|ε ₂|; from Eqs. (16), (17), (29) |k _{m 2}|≫|k _{m 1}|; |n ₂|≫ |n ₁|. Then from Eqs. (27), (30)

To a quite good approximation the transmitted B-field lags the transmitted E-field by π/2 (Eq. 28), and the transmitted E-field leads the incident E-field by π/2 (Eq. 31). Furthermore from Eqs. (28), (31), and the relation E _0I=cB _0I we find

The amplitude of the transmitted B-field at the boundary is about twice the incident B-field and the two B-fields are in phase.

 

Fig. 4. (A) Orientation of E-M fields propagating along +z. (B) Orientation of E-M fields propagating along -z. (C) Orientation of E-M fields for normal reflection at x-y plane of incidence. Applying the continuity conditions upon reflection at the surface reverses the direction of the E-field and leaves the B-field orientation unchanged, consistent with the Poynting vector condition S=1/µ ₀ (E×B).

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Eliminating Ẽ_0T between Eqs. (25), (26) results in

${\tilde{E}}_{0R}=\left(\frac{1-\tilde{n}}{1+\tilde{n}}\right){\tilde{E}}_{0I}\mathrm{and}\underset{\tilde{n}\gg 1}{\lim }{\tilde{E}}_{0R}\to {-\tilde{E}}_{0I}$

and, as expected, the reflected wave is π out of phase with the incident wave.

To summarize the amplitude and phase relations at the dielectric/metal boundary,

• The B-field in the metal lags the E-field in the metal by π/2.

• The E-field incident at the surface lags the E-field in the metal by π/2 and the amplitude is attenuated by a factor of 2k ₀/|k _m|.

• The amplitude of the B-field in the metal is about twice the incident amplitude and the phase is continuous across the boundary.

• The reflected E-field is π out of phase with respect to the incident E-field.

It follows from Eqs. (21), (23), (31) that in the petahertz frequency domain the conduction current J _c is in phase with the incident E-M field.

4.2. Comparison of field amplitudes and energy densities at the interface

In order to assess the relative importance of the E- and B-fields at the interface we investigate field amplitudes and energy densities on the dielectric and metal sides. It is a standard textbook result [21] that the components of an electromagnetic field parallel to a material interface are continuous through it. From the continuity conditions Eqs (30), (32) the relative amplitudes of the B- and E-fields at the interface are given by

where the superscript (i) indicates the field amplitudes at the interface. The E-field at the interface is attenuated from the incident E-field by n ₂, and from Table 1 we see that at the example frequency ω=2×10¹⁵ s^-1 the B-field is greater than the E-field by a factor of 6.7.

On the dielectric side of the interface the field energy density averaged over an optical cycle <u>_d is given by

where we have assumed the vacuum dielectric. Substituting from Eq. (33) and using the relation B ₀=E ₀/c,

Thus on the dielectric side of the interface the energy density of the standing-wave field is about twice the incident field, and the E- and B-fields contribute 2% and 98% respectively.

In contrast to the dielectric side, the field energy density penetrating into an electrically and magnetically dispersive and absorptive metal, averaged over an optical cycle is [22]

${<u>}_m=\left[\frac{{\epsilon }_0}{4}\left({\epsilon }_1+\frac{2\omega {\epsilon }_2}{{\Gamma }_e}\right){\mid E_{0T}\mid }^2+\frac{{\mu }_0}{4}\left({\mu }_1+\frac{2\omega {\mu }_2}{{\Gamma }_h}\right){\mid H_{0T}\mid }^2\right]e^{-2k_{2m}z}$

where H _0T=B _0T/µ ₀ is the magnetic field amplitude in the metal, µ ₁,µ ₂ are the real and imaginary terms of the magnetic permeability, and Γ_h is the magnetic absorptive damping constant. We take the common case of metals with nondispersive, nonabsorptive magnetic properties and write

${<u>}_m=\left[\frac{{\epsilon }_0}{4}\left({\epsilon }_1+\frac{2\omega {\epsilon }_2}{\Gamma }\right){\mid E_{0T}\mid }^2+\frac{{\mu }_0}{4}{\mid H_0\mid }^2\right]e^{-2k_{2m}z}$

Using the damped harmonic oscillator to model the frequency dependence ε(ω),

$\epsilon \left(\omega \right)={\epsilon }_1+i{\epsilon }_2\mathrm{with}{\epsilon }_1=1-\frac{{\omega }_p^2}{{\omega }^2},{\epsilon }_2=\frac{\Gamma {\omega }_p^2}{{\omega }^3}$

the energy density in the metal is

${<u>}_m=\left[\frac{{\epsilon }_0}{4}\left(1+\frac{{\omega }_p^2}{{\omega }^2}\right){\mid E_{0T}\mid }^2+\frac{1}{4{\mu }_0}{\mid B_{0T}\mid }^2\right]e^{-2k_{2m}z}$

and from the continuity relations

${\mid E_{0T}\mid }^2\simeq \frac{4}{n_2^2}{\mid E_{0I}\mid }^2\mathrm{and}{\mid B_{0T}\mid }^2={\mid B_{0I}^{\left(i\right)}\mid }^2\simeq {\mid 2B_{0I}\mid }^2\simeq \frac{4}{c^2}{\mid E_{0I}\mid }^2$

The energy density can be written as the product of the incident energy density and the sum of the separate contributions from the E- and B-fields in the metal.

${<u>}_m=\underset{\mathrm{incident}}{\underbrace{\frac{1}{2}{\epsilon }_0{\mid E_{0I}\mid }^2}}\left[\left(1+\frac{{\omega }_p^2}{{\omega }^2}\right)\frac{2}{n_2^2}+2\right]e^{-2k_{2m}z}$

and using the values in Table I we find that at z=0

${<u>}_m=\frac{1}{2}{\epsilon }_0{\mid E_{0I}\mid }^2\left[2.2+2\right]$

Thus on the metal side of the interface the energy density of the E- and B-fields contribute about equally. It is worth noting that while the field amplitudes are continuous through the interface, the field energy densities are not. The energy density jumps by about a factor of 2 from the dielectric to the metal side of the interface.

5. E-M fields in the presence of a slit array

5.1. Induced E-M fields with no slits

Reference [13] discussed the phase of the oscillating charge dipole and surface wave generated at a slit or groove in the metal with the aim of explaining the π-out-of-phase relation between the incident and surface waves. In that study, however, the resistivity 𝓡, the inverse of the conductivity σ, was assumed to be real. At optical frequencies, however, Eq. (23) shows that the imaginary term of the complex conductivity predominates, and therefore the amplitudes and phase of the fields formed in the metal and in slits must be reexamined. In Section 2 we posited a damped harmonic plane wave solution to the Helmholtz equation in the metal as the point of departure and worked outward through the continuity conditions at the metal-dielectric interface to determine amplitude and phase relations between fields in the metal and incident fields in the dielectric. In this section, following Ref. [13], we start from the dielectric side and posit a plane wave incident on the interface, setting up a standing wave there; and, working inward through the continuity conditions, determine the fields set up within the skin depth of the metal. From this point of view the E-field and conduction current in the metal arise from induction by the standing-wave B-field present at the surface. Figure 5 shows the standing wave with the B-field adjacent to the surface. The B-field at the surface, with approximately twice the amplitude of the incident wave, attenuates exponentially in +z direction into the metal to a level characterized by the penetration depth (20–30 nm at optical frequencies). The magnetic field is time-harmonic, and the integral form of the Faraday-Maxwell law (Eq. 1) can be used to calculate the E-field in the metal.

The right hand side of Eq. (36) represents the integral of the time derivative of the magnetic flux penetrating the surface A ₁ shown in Fig. 5. This time-dependent magnetic flux induces an electric field in the skin depth represented by the integral of the curl expression on the left hand side of Eq. (36). By virtue of Stokes’ theorem this integral over an area can be converted into an integral over the circuit C ₁ also shown in Fig. 5. The integrals over both sides of Eq. (36) can be easily carried out. The B-field time derivative is

$-\frac{\partial {\mathbf{B}}_{\mathbf{T}}^{\left(\mathbf{0}\right)}}{\partial t}=i\omega {\mathbf{B}}_T^{\left(0\right)}{e}^{-i\omega t}$

and because the B-field amplitude is assumed constant along x (plane wave) and exponentially attenuated along z, we can write ${\mathbf{B}}_T^{\left(0\right)}(x,z)={\mathbf{B}}_T^{\left(0\right)}{e}^{-k_{m{2}^z}}$; and from the continuity conditions B ⁽⁰⁾ _T⋍2B _I where B _I is the incident B-field. Integrating over A ₁ on the right hand side and C ₁ over the left hand side of Eq. (36) yields,

This result for E ⁽¹⁾ _T is consistent with the field obtained from the continuity conditions (Eq. 30). The conduction current in the metal, the product of the electric field and the complex conductivity,

is also consistent with the continuity conditions at the surface; the conduction current in the metal is in phase with the incident field. The B-field in the metal B ⁽¹⁾ _T can be obtained by substituting E ⁽¹⁾ _T into Eq. (2) and integrating over an appropriate area and contour or more directly from Eq. (28).

 

Fig. 5. Panel (A): Front view of incident plane wave setting up a standing wave at the surface of a smooth, featureless metal slab. Red amplitude indicates B-field maxima, blue shows E-field maxima. Panel (B) Diagram of the field metal surface shown in panel (A) for an optical half-cycle with E _I pointing along +x and B _I pointing along +y. The skin depth in z is indicated as δ, and p is some arbitrary length along x. The area A ₁=pδ and contour C ₁ are used for the integration of Eq. (36). The conduction current induced in the metal, J _c, is in phase with the incident E-field.

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In the last term on the right the expression for σ_i from harmonic oscillator model Eq. (23) has been substituted.

5.2. E-M fields with a slit array

Introduction of discontinuities in the metal surface such as slits, grooves, and holes unleashes a wide spectrum of scattered waves in the space above, below and along the metal surface. A careful and quantitative analysis of the Bloch modes populated in a one-dimensional periodic array of slits in a silver slab has been carried out by Xie et al. [23]. In general many Bloch modes of the periodic array within the metal and within the slits must be populated to match the incident and reflected modes at the surface. Therefore, for an arbitrary slit periodicity, the description of transmission through the array does not lend itself to a simple physical model. However, in the special case when the separation between the slits approaches an integer number of λ_spp wave lengths, Xie et al. [23] have shown that only the normally reflected propagating mode and the surface plasmon polariton (spp) modes, evanescent along z but guided and propagating along the surface, remain. At these special slit positions two counterpropagating spps form a standing wave at the surface, superposing with the standing wave arising from the incident and reflected propagating plane wave. The spatial distributions of these two standing waves are shown in Fig. 6. Conduction currents J _p,J _spp, appear in the metal skin by Faraday induction from these B-field standing waves present at the surface. The conduction currents charge the slit edges similar to a conventional parallel plate capacitor, and the displacement current, J _slit, in the slit gap gives rise to an E-field and B-field, oscillating in quadrature with the incident plane wave. Consider first the slit fields induced by the plane standing wave.

5.2.1. Fields generated in one slit

From the condition of current density continuity across the slit gap, ∇·J=0,

and using Eqs. (38), (23) we can easily find the E-field in the slit induced by the plane wave,

We see that this slit field is in quadrature with the incident E-field. The dielectric constant in the gap εslit for subwavelength slit widths d will be somewhat greater than unity, determined by field continuity conditions for this metal-insulator-metal geometry along x. We can now calculate the B-field in the slit by integrating the Ampère-Maxwell equation (Eq. 2) over the appropriate contour and area, C ₂,A ₂ and using Eq. (40).

${\oint }_{C2}\frac{{\mathbf{B}}_{\mathrm{slit}}^p}{{\mu }_0}·d{\mathbf{s}}_2={\oint }_{A2}{\mathbf{J}}_{\mathrm{slit}}·{\mathbf{n}}_{A2}d{A}_2$

$={\oint }_{A2}{\mathbf{J}}_p·{\mathbf{n}}_{A2}d{A}_2={\oint }_{A2}\left({\epsilon }_0{\epsilon }_{\mathrm{slit}}\frac{\partial {\mathbf{E}}_{\mathrm{slit}}}{\partial t}\right)·{\mathbf{n}}_{A2}d{A}_2$

so that

where $n_{\mathrm{slit}}=\sqrt{{\epsilon }_{\mathrm{slit}}}$. We see that this B-field in the slit is also in quadrature with the incident E-M field, and therefore these fields can be associated with the propagating mode in the slit gap. Consider now the fields on the surface and in the slits induced by the spp standing wave.

5.2.2. Fields generated by an spp standing wave in a slit array

When the slits are separated by λ_spp, the counterpropagating spps launched at the slit sites form a standing wave between them, the E-field of which is illustrated in Fig. 6 (B). This E-field “loop,” a half sine wave originating at the positive charge at the right slit and terminating on the negative charge on the left slit, oscillates at the incident frequency and in phase with the dipoles at the slits. The standing wave spp B-field, aligned along y and supported by J _spp running along x in the metal, must oscillate in quadrature with this E-field and therefore in quadrature with the dipoles at the slits. But Eqs. (41), (42) show that the dipoles themselves oscillate in quadrature with the incident E-M field; and therefore, the standing wave spp B-field is π out of phase with the incident E-M field. The standing wave spp B-field opposes the standing wave B-field at the surface. A Green’s tensor analysis [24] of a line dipole oriented just above a metal surface confirms the quadrature phase relation between the slit dipoles and the incident field. The induced current in the metal from the standing wave spp J _spp therefore opposes the induced current from the plane standing wave J _pw. The two opposing conduction currents result in a net conduction current J _net(x)=J ^max _spp cos[(k _spp/2)(x-λ _spp/2)]-J _pw. If J ^max _spp>J _pw, then, as x moves away from the midpoint and J _spp(x) decreases from its maximum value, there will be two null points where the two B-fields cancel and J _net=0. Using results from Table 1 of Ref. [23] it is easy to show that these null points for that study are near x=λ _spp/4 and x=3λ _spp/4. Points of null current are points of maximum charge. The overall effect of establishing the standing wave spp is to shift the charge centers from the slit edges to the one-quarter and three-quarter points along the surface between the slits. The slit corners transform from dipole sources of transmission to sites of maximum E-and B-field in phase with the reflected wave. At petahertz frequencies the physical slits present almost negligible capacitive impedance to the shifted field. The analysis of Ref. [23] confirms that the critical slit separation of λ _spp results in a near-extinction of the transmission mode through the slits and a corresponding maximum in reflection mode. This transmission extinction at slit separations of an integral number of λ _spp has been confirmed by precise measurements [15]. Numerical field simulations of Ref. [11] (see especially the right-hand column of Fig. 3 of that study) are consistent with the conclusion that at the spp standing wave condition the charge centers shift away from the slit sites to the quarter points between the slits.

 

Fig. 6. Panel (A): Standing plane wave above the two-slit structure, showing transmission through the slits at some arbitrary slit separation. Red (blue) color shows maximum B-field (E-field) amplitudes. Note that propagating fields within the slits set up Fabry-Perot-like cavities, the characteristics of which depend on the slit width, metal film thickness, and metal permittivity. The simulation shown here is for 100 nm slit in a silver film of thickness ⋍350 nm. Panel (B): A diagram of the area between the slits separated by λ _spp, showing the two superimposed standing-wave B-field amplitudes, one from the incident plane wave, the other from the spp standing wave. The E-field of the spp standing wave is indicated as a thick black arrow (E _spp) pointing from positive to negative charge at the two slit edges. The induced current density in the metal from the plane standing wave is denoted J _pw and from the spp standing wave J _spp.

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6. Conclusions

We have shown here how the complex dielectric constant of a metal can be separated into two parts: one that depends primarily on the conduction electrons and the other that depends only on the core. By describing the response of the conduction electrons to an external driving field with a damped harmonic oscillator model we have identified a separate frequency dependence for the real and imaginary parts of the complex conductivity. The low-frequency real part of the conductivity corresponds to the familiar ohmic response in which the electron flow in the metal is controlled by collisional diffusion. However, at petahertz frequencies (visible and near-infrared) the imaginary term dominates, corresponding to ballistic electron motion. Application of the E-M field continuity conditions through the metal surface shows that the conduction current density within the metal penetration depth is in phase with the external driving field. We have also shown that although the field amplitudes and energy density on the dielectric side of the interface are dominated by the B-field, on the metal side, due to dispersion of the material, the electric and magnetic field components contribute nearly equally to the energy density. The energy density is discontinuous through the boundary. Integration of the Faraday-Maxwell and Ampère-Maxwell equations, starting from the standing-wave B-field at the dielectric side of the interface, determines E- and B-fields within the metal as well as the displacement current density, E- and B-fields in the slit gap. The fields in the gap oscillate in phase with the corner charge dipoles and in quadrature with the incident E-M field. For arrays of slits, a periodicity of λ _spp sets up an spp standing wave and results in near-extinction of transmission through the slits. There exists now a significant body of evidence supporting the view that surface wave response to optical excitation of subwavelength metallic structures can be modeled by oscillating charge dipoles at the dielectric-metallic interface [25], [24]. The launch of surface waves originating from these structures is often couched in the language of diffraction [25] or more generally “scattering,” [24]. Here we take the view that the principal origin of the dipole charging current is the illuminated planar surface in the vicinity of the metal discontinuity and that this current is induced by standing-wave, time-harmonic E-M fields at the surface. If this view is correct, the “effective source” of E-M fields in the subwavelength opening is the oscillating dipole induced at the aperture entrance; and direct illumination of the structure aperture by the incident plane wave contributes little to optical transmission through the opening. These ideas must be confirmed and quantified by careful experiments.