A schematic illustration of the system under study is shown in Fig. 1. Stationary, polychromatic light incident from a dielectric medium in a Kretschmann configuration creates a surface plasmon polariton field **E**_{p} that propagates along the metal-air interface towards a nanostripe N. Most of **E**_{p} is reflected at N as a backward-propagating plasmon field **E**_{r}, while a small part is scattered into the half-space above the metal in the form of an effectively freely propagating electric field **E**_{s}. A detector D in the far zone of the nanostripe N measures the spectrum of the scattered light, from which the spectral amplitudes of **E**_{p} and thereby all the spatiotemporal coherence (and polarization) properties of the superposition surface plasmon field can be inferred.

### Theory

#### Surface plasmon polaritons

In the absence of the nanostripe the metal film on a dielectric medium is homogeneous and characterized by a complex relative permittivity *ε*_{r}(*ω*), accounting for dispersion and absorption. A *p*-polarized (TM), statistically stationary, tailored illumination excites a polychromatic SPP field on the metal-air interface (see Fig. 1). The film thickness *h* is large enough so that coupling among the plasmon modes at the two metal surfaces can be neglected. The spatial electric part of the SPP field in air, at point **r**=(*x*,*z*), frequency *ω*, and propagating in the positive *x* direction, may then be expressed as [19, 20]

$$ \mathbf{E}_{\mathrm{p}}(\mathbf{r},\omega) = E(\omega)\hat{\mathbf{p}}(\omega) \mathrm{e}^{\mathrm{i}\mathbf{k}(\omega)\cdot\mathbf{r}}, $$

(1)

where *E*(*ω*) is the complex field amplitude at the origin (**r**=0), and

$$\begin{array}{*{20}l} \mathbf{k}(\omega) &= k_{x}(\omega)\hat{\mathbf{e}}_{x} + k_{z}(\omega)\hat{\mathbf{e}}_{z}, \end{array} $$

(2)

$$\begin{array}{*{20}l} \hat{\mathbf{p}}(\omega) &= \left[\mathbf{k}_{\mathrm{p}}(\omega)\times\hat{\mathbf{e}}_{y}\right]/ \left|\mathbf{k}_{\mathrm{p}}(\omega)\right| = p_{x}(\omega)\hat{\mathbf{e}}_{x} + p_{z}(\omega)\hat{\mathbf{e}}_{z}, \end{array} $$

(3)

are the wave vector and the unit-normalized polarization vector, respectively, and \(\hat {\mathbf {e}}_{x}\), \(\hat {\mathbf {e}}_{y}\), and \(\hat {\mathbf {e}}_{z}\) are the Cartesian unit vectors. The wave-vector components in Eq. (2) read as [1, 2]

$$\begin{array}{*{20}l} k_{x}(\omega) &= \frac{\omega}{c}\sqrt{\frac{\epsilon_{\mathrm{r}}(\omega)} {\epsilon_{\mathrm{r}}(\omega)+1}}, \end{array} $$

(4)

$$\begin{array}{*{20}l} k_{z}(\omega) &= \frac{\omega}{c}\sqrt{\frac{1}{\epsilon_{\mathrm{r}}(\omega)+1}}, \end{array} $$

(5)

where *c* is the speed of light in free space.

When the nanostripe N is present, it serves as a barrier from which the SPP reflects back and may, to a good accuracy, be represented as

$$ \mathbf{E}_{\mathrm{r}}(\mathbf{r},\omega) = E_{\mathrm{r}}(\omega)\hat{\mathbf{p}}_{\mathrm{r}}(\omega) \mathrm{e}^{\mathrm{i}\mathbf{k}_{\mathrm{r}}(\omega) \cdot (\mathbf{r}-d\hat{\mathbf{e}}_{x})}, $$

(6)

where *E*_{r}(*ω*) is the complex amplitude of the reflected SPP at the nanostripe [ **r**=(*d*,0)]. Further, from the properties of SPPs it follows that [22, 23]

$$\begin{array}{*{20}l} \mathbf{k}_{\mathrm{r}}(\omega) &= -k_{x}(\omega)\hat{\mathbf{e}}_{x} + k_{z}(\omega)\hat{\mathbf{e}}_{z}, \end{array} $$

(7)

$$\begin{array}{*{20}l} \hat{\mathbf{p}}_{\mathrm{r}}(\omega) &= -p_{x}(\omega)\hat{\mathbf{e}}_{x} + p_{z}(\omega)\hat{\mathbf{e}}_{z}. \end{array} $$

(8)

Considering the SPP propagation at *z*=0, we may estimate

$$\begin{array}{*{20}l} E_{\mathrm{r}}(\omega) &= r(\omega) E(\omega) \mathrm{e}^{\mathrm{i}k_{x}(\omega) d}, \end{array} $$

(9)

$$\begin{array}{*{20}l} r(\omega) &= \frac{1 - n(\omega)}{1 + n(\omega)}, \end{array} $$

(10)

where *r*(*ω*) is the field reflection coefficient. In it \(n(\omega) = \sqrt {\epsilon _{\mathrm {r}}(\omega)}\) is the (complex) refractive index of the metal nanostripe [24]. The coefficient *r*(*ω*) corresponds to reflection of a normally incident plane wave, as is typically the case with SPPs [22].

From Eqs. (1) and (6) we now readily find for the total SPP field between the excitation point and the nanostripe N the expression

$$\begin{array}{*{20}l} \mathbf{E}_{\text{SPP}}(\mathbf{r},\omega) = \: &E(\omega) \mathrm{e}^{\mathrm{i}k_{z}(\omega)z} \left[ \hat{\mathbf{p}}(\omega) \mathrm{e}^{\mathrm{i}k_{x}(\omega)x} \right. \\ &\left. + \: r(\omega) \mathrm{e}^{\mathrm{i}2k_{x}(\omega)d} \hat{\mathbf{p}}_{\mathrm{r}}(\omega) \mathrm{e}^{-\mathrm{i}k_{x}(\omega)x} \right], \end{array} $$

(11)

or explicitly in the component form

$$ {\begin{aligned} E_{\text{SPP}x}(\mathbf{r},\omega) &\,=\, E(\omega) \mathrm{e}^{\mathrm{i}k_{z}(\omega)z} p_{x}(\omega) \left[ \mathrm{e}^{\mathrm{i}k_{x}(\omega)x} \,-\, r(\omega) \mathrm{e}^{\mathrm{i}2k_{x}(\omega)d} \mathrm{e}^{-\mathrm{i}k_{x}(\omega)x} \right], \end{aligned}} $$

(12)

$$ {\begin{aligned} E_{\text{SPP}z}(\mathbf{r},\omega) &\,=\, E(\omega) \mathrm{e}^{\mathrm{i}k_{z}(\omega)z} p_{z}(\omega) \left[ \mathrm{e}^{\mathrm{i}k_{x}(\omega)x} \,+\, r(\omega) \mathrm{e}^{\mathrm{i}2k_{x}(\omega)d} \mathrm{e}^{-\mathrm{i}k_{x}(\omega)x} \right]. \end{aligned}} $$

(13)

The relative sizes of these components depend on the metal and the frequency, which determine the SPP wave vector **k**(*ω*) and polarization vector \(\hat {\mathbf {p}}(\omega)\).

#### Spatiotemporal coherence

We view the SPP field in Eq. (11) as a realization of a statistically stationary ensemble and compute the cross-spectral density matrix as [19, 20]

$$ \mathbf{W}(\mathbf{r}_{1},\mathbf{r}_{2},\omega) = \left\langle \mathbf{E}_{\text{SPP}}^{*}(\mathbf{r}_{1},\omega) \mathbf{E}_{\text{SPP}}^{\mathrm{T}}(\mathbf{r}_{2},\omega) \right\rangle, $$

(14)

where the asterisk and superscript T denote complex conjugation and matrix transpose, respectively, and the angle brackets stand for ensemble averaging. Since *E*(*ω*) is the only random quantity, the polychromatic SPP field clearly is fully coherent in the space-frequency domain. Its spatiotemporal coherence is obtained from the generalized Wiener-Khintchine theorem [25]

$$ \boldsymbol{\Gamma}(\mathbf{r}_{1},\mathbf{r}_{2},\tau) = \int_{0}^{\infty} \mathbf{W}(\mathbf{r}_{1},\mathbf{r}_{2},\omega) \mathrm{e}^{-\mathrm{i}\omega\tau} \mathrm{d}\omega, $$

(15)

in which *Γ*(**r**_{1},**r**_{2},*τ*) is the mutual coherence matrix and *τ* is a time difference. Expression (15), with **E**_{SPP}(**r**,*ω*) given by Eq. (11), is valid everywhere between the SPP creation point and the nanostripe N, for statistically stationary excitations of any spectral distribution.

#### Scattered far field

Interaction of **E**_{p} with the nanostripe N produces, besides the reflected SPP **E**_{r} of Eq. (6), also a field scattered into the half-space *z*>0. If the nanostripe is sufficiently small, it behaves much like a point scatterer [26, 27] (albeit in a 2D space). In the far-zone the scattered electric field is normal to the propagation direction and has spherical wave fronts. Within a good approximation, we may thus write for the scattered far-field amplitude the expression [28]

$$ E_{\mathrm{s}}(s,\phi,\omega) = r(\phi,\omega) E(d,\omega) \frac{\mathrm{e}^{\mathrm{i}(\omega/c)s}}{\sqrt{s}}, $$

(16)

where *r*(*ϕ*,*ω*) is the scattering coefficient with *ϕ* representing the angle between the *x* axis and the scattering direction. Further, *E*(*d*,*ω*) is the amplitude of the forward-going SPP at the location of the nanostripe N and *s* is the distance between N and the observation point. Denoting the spectral intensities at detector D and nanostripe N by *I*_{s}(*ϕ*,*ω*)=〈|*E*_{s}(*s*,*ϕ*,*ω*)|^{2}〉 and *I*(*d*,*ω*)=〈|*E*(*d*,*ω*)|^{2}〉, respectively, it follows from Eq. (16) that

$$ {sI}_{\mathrm{s}}(\phi,\omega) = |r(\phi,\omega)|^{2} I(d,\omega), $$

(17)

implying that the scattered far-zone intensity multiplied by distance *s* is a constant that, in general, depends on the scattering direction and the frequency.

### Simulation

The simulations of vectorial SPP fields on the metallic surface and their scattering from the nanostripe are performed in 2D by utilizing in-house numerical codes based on the Fourier modal method [29], as well as COMSOL Multiphysics software that employs the finite element method. The SPP excitation in the Kretschmann geometry takes place by means of a perfectly phase-matched focused beam at each frequency. In all our simulations the two different computational methods lead to substantially similar results.

We illustrate in Fig. 2 the salient features of the simulations with a monochromatic excitation. As in all subsequent analyses, the metal layer thickness *h*=65 nm and the nanostripe side length *w*=80 nm. The material in both the layer and the nanostripe is gold (Au), while the dielectric prism consists of glass (SiO_{2}). Figure 2a demonstrates clearly the existence of a standing SPP wave pattern left of the nanostripe N (since the plasmon survival length is large compared to the separation *d*), and Fig. 2b shows a series of cone-like radiation lobes (due to the square shape of the scatterer) emerging from N with their strength decreasing as the scattering angle increases. We note that this latter figure is plotted starting from slightly above the metal surface so that the standing waves are not, for the most part, visible in it anymore.