1. Introduction

Over the past several years optical transmission through apertures and the generation of surface waves from subwavelength structures has been the subject of intensive study and lively debate. From these researches has gradually emerged a clearer understanding of the physics underlying some of these phenomena. The existence of transient “leaky” surface waves and long-lived surface plasmon polariton waves (SPPs), originating from the interaction between incident TM-polarized electromagnetic waves and one-dimensional (1-D) subwavelength slits and grooves at a metal-dielectric interface has now been definitely established through numerical simulation and direct measurement [1, 2]. In periodic array structures the resonance excitation of delocalized, Bloch-state SPPs has been proposed [3] to explain peaks in light transmission observed when the wavelength of light is tuned near integer multiples of the array pitch. Other theoretical studies have emphasized the role of wave interference rather than resonant excitation and have predicted transmission minima [4, 5] where Bloch-state resonances predict maxima. Recent quantitative measurements [6] in 1-D slit arrays confirm that the transmission indeed reaches a minimum when the array pitch is equal to the SPP wavelength. These results are consistent with the interpretation that destructive interference between the incident wave and the SPP is responsible for the extinction. But if this interpretation is correct, it implies a definite phase relation between the incident wave and the scattered surface wave; specifically that these two waves are π out of phase.

The phase of the surface wave has been examined from several perspectives. Comparing them requires care because various studies have emphasized either the incident and scattered H-field (a scalar quantity in TM polarization) or one of the components of the E-field, parallel and perpendicular to the surface plane. Figure 1 shows the coordinate system and field components of the incident and scattered fields appropriate to the discussion here. Lezec and Thio [7] proposed a composite diffracted evanescent wave (CDEW) model, one signature of which is a phase shift of π/2 between the incident and E_x component of the electric field diffracted on the surface. Subsequent finite-element field simulations [8] showed a phase shift of π between the incident and SPP scattered wave magnetic field H_y provided that the scattering slit was sub-wavelength in width. This finding is not inconsistent with the CDEW prediction since, as will be shown in what follows, the H_y and E_x components themselves are π/2 out of phase on the surface. A Green’s tensor study [9] concluded that with the incident wave impinging normal to the surface the scattered surface wave E_z is π out of phase with the incident E_x and H_y fields, again consistent with the simulations of Ref. [8], since the E_z and H_y surface wave can be shown to be in phase. Although these simulations, numerical calculations, and models lend credence to the idea that the suppression of transmission in slit arrays is due to antiphase destructive interference between the incident propagating and surface evanescent waves, they do not offer a physical explanation for this phase relation. Using field continuity conditions and a simple application of Faraday’s law of induction, the next two sections establish the phase relations first between the electric and magnetic field components of the surface wave and then between the magnetic fields of the incident wave and the surface wave.

 

Fig. 1. Orientation of orthogonal coordinate axes is shown along with a light wave TM polarized (H field aligned along the 0_y axis). Incident and scattered E-field components are confined to the x-z plane. A subwavelength slit milled at the interface between the metal slab (m) and a dielectric (d) is also aligned along the y- axis.

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2. Field relations

This section begins by following the development in the appendix of Raether [12]. In a source-free environment and in rationalized MKS units Faraday’s Law and Ampère’s Law are expressed as

with the constitutive relations

where ε ₀, μ ₀ are the permittivity and permeability of free space, and ε,μ are the dielectric constant and the “specific magnetic inductive capacity” [11]. In nonmagnetic media μ is essentially unity. The vector fields D, E are the displacement field and electric field, respectively, and H, B are the the magnetic field and magnetic induction, respectively. In TM polarization, with the magnetic field parallel to the y axis, plane-wave solutions to Eqs 1,2 are written

where $E_0=\sqrt{{\mid E_x\mid }^2+{\mid E_y\mid }^2}$ and $E_0=Z_0{H}_0\sqrt{\frac{\mu }{\epsilon }}$ with $Z_0=\sqrt{\frac{{\mu }_0}{{\epsilon }_0}}$ , the impedance of free space. At the dielectric-metal boundary, field continuity conditions require,

where H^d, E^d denote field components in the dielectric (z>0); H^m, E^m of the metal (z<0); and ε_d,ε_m are the dielectric constants in the dielectric and metal, respectively.

From Eq. 2 in the dielectric half-space,

$\frac{{\partial H}_y^d}{\partial z}=\frac{{\partial D}_x^d}{\partial t}={\epsilon }_d\frac{{\partial E}_x^d}{\partial t}$

$-\frac{{\partial H}_y^d}{\partial x}=\frac{{\partial D}_z^d}{\partial t}={\epsilon }_d\frac{{\partial E}_z^d}{\partial t}$

and substituting from Eqs 4,5,6,

Similarly in the metal half-space

From condition 7 and Eqs 10, 12 at the dielectric-metal interface

and in order that Eqs 14, 15 be satisfied simultaneously

which expresses the relation between the wave propagation parameters perpendicular to the surface on each side of the interface. Similarly from Eqs. 7,11,13,

showing that the propagation parameter parallel to the surface is continuous across the boundary. Furthermore, conservation of wave energy on both sides of the boundary requires

Using Eq. 16 to express the ratio k^d_z/k^m_z in terms of ε_d and ε_m, and dividing Eq. 18 by Eq. 19 to eliminate the k_z components results in the well-known expression for k_x at the boundary,

Further use of Eqs. 16, 18, 19 results in

and since the metal dielectric constant is always negative,

Then from Eqs. 10 and 11 and at the surface

Equations 23 and 24 show that at the surface E_x and H_y are in quadrature and in phase with E_z, respectively.

3. Relative phase between the incident wave and the surface wave

It is often said that subwavelength structures “launch” surface waves without specifying the physical origins of the launching process. From a physical optics point of view, the “launch” mechanism is diffraction of the incident wave, but it is doubtful that standard Fraunhofer diffraction theory can be legitimately applied at the subwavelength scale, and even the more general Kirchoff diffraction theory invokes field boundary conditions in the immediate vicinity of the structure that are only approximate [13]. Furthermore, surface plasmon polaritons always involve a metallic surface with high conductivity, and therefore it is more natural to seek an explanation in terms of charges and currents on the surface and in the immediate vicinity of the structure. It has been recently shown[9] that a groove or slit “launch” site can be successfully modeled by replacing the structure with an oscillating charge dipole on the surface. The results of this model are in good agreement with simulations [2,14] and measurements [15] of both the “near zone” transient surface wave and the “far zone” surface plasmon polariton. It therefore appears that under the influence of the incident wave charges accumulate at the slit or groove edges, oscillating at the optical frequency. This charge accumulation itself is the result of currents induced at the surface and within the skin depth of the metal by the standing wave established by the incident and reflected plane waves at the surface. The amplitude of the H_y-field at the surface is twice the amplitude of the incident wave, while the amplitude of the E_x-field is nearly zero. The top panel of Fig. 2 shows a numerical simulation of the intensity of the magnetic field standing wave. The simulation is for a two-slit structure milled in a silver layer with 100 nm widths and incident free-space wavelength λ ₀=852 nm. The spatially standing wave H_y(z, t)=2H ₀cos(k ₀ z)exp(-iω ₀ t) oscillates temporally at the optical frequency ω ₀ and induces an emf within the skin depth of the metal according to the Faraday law of induction. The magnetic field intensity map clearly shows that the presence of the slits has little effect on the standing wave pattern, and therefore the structured surface can be considered essentially a plane, highly reflecting mirror. At z=0

and the magnetic flux is given by

where S=δp, is the cross sectional area defined by the distance between the slits p and the skin depth δ, normal to the flux lines (see Fig. 2). Then according to Faraday’s Law the emf, oriented along the x direction, is given by

The linear current density, oriented along x and flowing perpendicular to the slit axes aligned along y, is given by

where 𝓡 is the resistivity of the metal in the skin depth. Here 𝓡 is taken to be real since the imaginary term in the wavelength range of interest (near IR) is very small compared to the real term. The linear charge density accumulated along the edge of each slit during a time t is

 

Fig. 2. Top panel is a numerical simulation of an incident plane wave TM polarized and incident on a two-slit structure from the top. The color map indicates the amplitude of the magnetic field component (red highest amplitude, dark blue lowest amplitude) and shows the standing wave set up by the incident and reflected wave. Note that the standing H-field is maximum at the surface. Middle and lower panels show how the time-oscillating H-field induces current in the skin depth δ of the metal slit structure, resulting in charge accumulation at the slit edges with widths d. Middle panel: first optical half-cycle; lower panel: second optical half-cycle.

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Finally the oscillating dipole density 𝒫 along the slit of width d is

Comparison of Eq. 25 with Eq. 30 shows that the oscillating dipole at the slit is in antiphase with the incident field. The phase shift of π occurs through the sum of two steps: first, a phase shift of π/2 from the magnetic induction in the metal skin depth and a second phase shift of π/2 from the capacitive charge buildup at the slits. The induced emf within the skin depth cannot couple directly to the SPP wave because the propagation parameter k_x≃k ₀ associated with the magnetic field is always less than k _SPP. In contrast the oscillating E-field at the slit is not harmonic but characterized by a band of propagation parameters δk≃2π/d where d is the slit width. In practice the surface is driven by this “broad-band” k-mode excitation and couples efficiently to the k _SPP mode.

4. Conclusions

The lines of force of the E-field at the surface must converge at points of charge concentration, and therefore the E_z amplitude at the surface is correlated to and in phase with 𝒫(t). Thus the surface wave component E_z is in antiphase with the incident magnetic field H_y in agreement with the Green’s tensor analysis of Ref. [9]. Field continuity at the surface and Eq. 23 shows that E_x is in quadrature with E_z. This result is in accord with the diffraction analysis of Ref. [7]. Similarly Eq. 24 shows that for the wave propagating on the surface H ^SPP _y is in phase with E ^SPP _z. Therefore the incident and surface H_y fields must be in antiphase, again in accord with the simulations of Ref. [8] and the interpretation of the measurements in Ref. [6]. The present analysis therefore lends support to the interference model proposed in Ref. [6] to explain optical transmission minima through slit arrays when the pitch becomes equal to an integer number of SPP wavelengths. It is worthwhile noting that the π phase shift may not hold for surface waves launched by structures other than slits or grooves. In particular arrays of subwavelength holes, even if they are capacitively charged by induced currents, will drive the surface with cylindrical rather than plane waves as in the case of slits. The form of these waves are essentially half-integer Bessel functions, J _n+1/2(r) and their asymptotic form has an extra phase shift of π/2 that would have to be taken into account.