## Abstract

Rigorous electromagnetic theory is utilized to characterize the partial spatial coherence and partial polarization of a two-mode field consisting of the long-range and the short-range surface-plasmon polariton at a metallic nanofilm. By employing appropriate formulations for the spectral degrees of coherence and polarization, we examine the fundamental limits for these quantities associated with such a superposition field and explore how the degrees are influenced when the media, frequency, and slab thickness are varied. It is in particular shown that coherence lengths extending from subwavelength scales up to thousands of wavelengths are possible and their physical origins are elucidated. In addition, we demonstrate that for ultra-thin films the generally highly polarized two-mode field can be partially polarized in close vicinity of the polariton excitation region. The results could benefit cross-disciplinary electro-optical applications in which near-field interactions between plasmons and nanoparticles are exploited.

© 2015 Optical Society of America

## 1. Introduction

Plasmonics [1–3 ], based on the utilization of electromagnetic evanescent near fields coupled to collective electron oscillations at a metal-dielectric interface, known as surface-plasmon polaritons (SPPs) [4, 5], has in recent years attracted extensive research activity within the realms of nanophotonics [6–8 ]. Due to their unique features, such as strong enhancement and localization of light energy within subwavelength domains, SPPs occupy a pivotal position in a wide range of optoelectronic and nanophotonic applications. Besides the rapid technological progress, research concerning the fundamental nature of SPPs has very recently suggested the existence of entirely new SPP classes, both at single boundaries [9] and at metallic films [10, 11], which further broadens the possibilities for novel polaritonic devices.

While a single boundary allows only one plasmon mode, the SPPs supported by the individual surfaces of a thin metal film can couple and form two bounded modes with symmetric or antisymmetric field profiles and different dispersion characteristics [10, 12–14 ]. In particular, as the thickness of the film becomes sufficiently small, the symmetric mode may acquire a propagation length which is several hundreds of times greater than that of the single-interface SPP [15, 16], whereupon it is frequently called the ‘long-range surface-plasmon polariton’ (LRSPP) [17]. The antisymmetric mode, on the other hand, becomes increasingly confined as the slab thickness decreases, resulting in a much smaller surface propagation distance than those of the LRSPP and the respective single-interface SPP, and therefore the antisymmetric mode is termed the ‘short-range surface-plasmon polariton’ (SRSPP) [16]. Due to its unique capability of long-range guidance the LRSPP has over the years received considerably more attention and practical significance than the SRSPP. However, several phenomena have been reported in which the SRSPP plays a crucial role, including plasmonic focusing [18], stimulated-amplification supported SPP propagation [19], extraordinarily low transmission through nanopatterned films [20], and plasmon-waveguide sensing [21]. Methods which allow efficient excitation of the SRSPP, either simultaneously with [22] or without [23] the LRSPP, have also been presented.

The electromagnetic coherence, the partial polarization, and the spectrum of random evanescent near fields, with or without plasmon excitations, can differ substantially from those associated with the far field. It has been demonstrated that subwavelength coherence lengths, as well as polarization states not allowed by the traditional beam-field formalism, may occur in purely evanescent waves generated at an interface between two lossless dielectric media [24]. Likewise, the correlation length in a fluctuating thermal near field can be significantly shorter than the wavelength of light, or it may extend over several tens of wavelengths under SPP excitation [25, 26]. The broadband near field radiated by a thermal half-space source may become essentially quasi-monochromatic [27], and highly polarized [28], when surface plasmons are involved. Owing to SPP interactions, bodies in thermal equilibrium may emit spectrally dependent, directional beams of radiation if gratings are fabricated on their surface [29]. These results illustrate that depending on the specific circumstances, the presence of surface plasmons may significantly alter, not just the intensity and the spectrum, but also the spatial coherence and polarization characteristics of the electromagnetic field near the surface.

Motivated by the conclusion above, in this work we analyze the spatial coherence and polarization properties of a two-mode surface-plasmon polariton field consisting of the LRSPP and the SRSPP at a metallic nanofilm. These are the two modes supported by a thin metal slab, and the SRSPP has a pivotal role as regards the coherence and polarization modifications. More specifically, on the basis of exact formulations of SPP modes and appropriate spectral electromagnetic coherence theory, we investigate the fundamental ranges in which the degrees of spatial coherence and polarization may vary in practice in such a two-mode configuration, regardless of the excitation method. The degrees of coherence and polarization are the basic measures of any partially coherent and partially polarized field, but no studies employing the rigorous definitions of these quantities, which take into account the complete vector-valued nature of the electromagnetic field, have been carried out for SPPs. We introduce a local and a global coherence length and show that the coherence may range from subwavelength scales up to hundreds or thousands of wavelengths depending on the choice of the media, frequency, and slab thickness. Both coherence lengths, describing different features of the SPP field, as we explain later, can be measured by use of leakage radiation microscopy [30]. Physically, the coherence and polarization states influence the interaction of SPPs with the surrounding and collections of nanoparticles located in close proximity of the metal slab. For example, SPPs propagating on the surfaces of a metallic nanolayer can form a highly sensitive interferometric biosensor [31]. Likewise, a coherent plasmon field may excite a random set of molecules to radiate coherently from different emitters [32]. Our results further indicate that the two-mode SPP field is generally highly polarized, but can be partially polarized in and near the plasmon excitation region. Nanoparticle scattering is known to depend on the polarization properties of the field, to the extent that the field’s polarization state can be fully deduced from measurements of the scattered radiation [33].

## 2. Long-range and short-range SPPs

We consider SPPs, represented as monochromatic realizations of electromagnetic plane waves of angular frequency *ω*, at two planar interfaces lying in the *xy* plane between linear, homogeneous, isotropic, non-magnetic, stationary, and spatially non-dispersive media. The geometry, as illustrated in Fig. 1, includes an absorptive metal slab (region |*z*| < *d*/2) characterized by a complex (*ω*-dependent) relative permittivity *ε*
_{r1} = *ε′*
_{r1} + *iε″*
_{r1} (the prime ′ and double prime ″ denote real and imaginary parts, respectively), with *ε′*
_{r1} < 0 and *ε″*
_{r1} > 0, surrounded on both sides (regions |*z*| ≥ *d*/2) by a lossless dielectric medium having a real and positive permittivity *ε*
_{r2}. We consider, as is customary, a system with two (transversally counter-propagating) waves within and one on each side of the film.

The spatial part of the *p*-polarized electric field outside the slab can be written as

*E*with

_{α}*α*∈ {1, 2} is a complex (

*ω*-dependent) field amplitude, $\left|{\mathbf{k}}_{\alpha}\right|={\left[{\mathbf{k}}_{\alpha}^{*}\cdot {\mathbf{k}}_{\alpha}\right]}^{1/2}$ (the asterisk denotes complex conjugation) is the magnitude of the complex (

*ω*-dependent) wave vector

**k**

*=*

_{α}*k*

_{αx}**ê**

*+*

_{x}*k*

_{αz}**ê**

*, and*

_{z}**ê**

*and*

_{x}**ê**

*are the unit vectors in the positive*

_{z}*x*and

*z*directions, respectively. The tangential wave-vector component is continuous across the boundaries, i.e.,

*k*=

_{αx}*k*=

_{x}*k′*+

_{x}*ik″*, whereas the normal component

_{x}*k*=

_{αz}*k′*+

_{αz}*ik″*satisfies ${k}_{\alpha z}={\left({k}_{0}^{2}{\epsilon}_{r\alpha}-{k}_{x}^{2}\right)}^{1/2}$ (

_{αz}*k*

_{0}is the free-space wave number). In this work we examine only such fields which outside the slab are non-radiative (

*k′*

_{2z}≤ 0) and bound to the slab interfaces (

*k″*

_{2z}> 0), and whose phase-propagation and amplitude-attenuation directions along the

*x*axis are positive (

*k′*,

_{x}*k″*> 0).

_{x}The superscript (±) in Eqs. (1) and (2), where the + (−) sign corresponds to the upper (lower) signs inside the parentheses, identifies two different kinds of modes,
${\mathbf{E}}_{\alpha}^{(+)}(\mathbf{r},\omega )$ and
${\mathbf{E}}_{\alpha}^{(-)}(\mathbf{r},\omega )$, representing electric fields whose normal components, or magnetic fields, are symmetric and antisymmetric with respect to *z* = 0, respectively. It follows from Maxwell’s equations and the boundary conditions that Eqs. (1) and (2) have to satisfy [4, 5, 7, 10, 11, 13, 14, 16]

*k*and generally require numerical methods to solve. It can be verified, however, that when

_{x}*d*→ 0, Eq. (3) has a solution for which [10, 11, 14, 16]

*d*→ 0.

It should be emphasized, though, that in fact both Eqs. (3) and (4) stand for an infinite number of modes for any chosen media, frequency, and film thickness [10, 11, 14]. All of these modes can be divided into two main classes: fundamental modes (FMs) and higher-order modes (HOMs). The former are the two familiar ones (one symmetric and one antisymmetric) which become degenerate and reduce to the respective single-interface SPP as the film thickness gets large, while the latter instead are infinite in number and have no correspondence at a single boundary. Traditionally only the FMs have acquired attention and physical importance, since normally they possess the largest surface propagation lengths (the symmetric FM evolves into the LRSPP as *d* becomes small), whereas the HOMs have been rejected due to their extremely small propagation range [14]. We remark, however, that the existence of HOMs which assume the function of long-range guidance (together with even enhanced field confinement), in circumstances where the propagation distances of the FMs and the single-interface SPP are negligible, has been suggested very recently [11]. Nevertheless, in this work we analyze the FMs and refer to [10, 11, 14] for a more elaborate discussion concerning the HOMs.

## 3. Electromagnetic coherence and polarization in the space-frequency domain

All the information of second-order spatial coherence of a random electromagnetic field is in the space-frequency domain included in the cross-spectral density matrices [34]. For a stationary (vector) field, the elements of the electric cross-spectral density matrix, at a pair of points **r**
_{1} and **r**
_{2} (and at frequency *ω*), can be expressed as a correlation function over an ensemble of strictly monochromatic field realizations [35], i.e.,

The strength of correlations that exist between the orthogonal components of the electric field at two points is described by the (electromagnetic) degree of coherence for vectorial fields [36], which is defined as [37]

_{F}is the Frobenius norm and tr denotes the trace. Physically, for beam-like fields,

*μ*(

**r**

_{1},

**r**

_{2},

*ω*) characterizes the modulation of the four Stokes parameters in Young’s interference experiment, i.e., not only the intensity variation but also the polarization-state modulation [38, 39], and is therefore a generalization of the traditional degree of coherence for scalar fields (which only involves the visibility of intensity fringes). Analogously to its scalar counterpart it is bounded as [36, 37] where the upper and lower limits, respectively, correspond to complete spatial correlation and lack of correlation between the Cartesian field components, while the intermediate values stand for partially coherent fields. In the following we will only be interested in the degree of coherence for vectorial fields (SPPs are vector valued), so henceforth the term degree of coherence is used to refer to the degree of coherence for vectorial fields defined in Eq. (8).

The degree of polarization is a measure of the correlations that prevail between the orthogonal electric-field components at a point. In general, an electric field is composed of three orthogonal components and the concept of the degree of polarization must take into account such three-dimensional fields with wave fronts of arbitrary form [40, 41]. Nevertheless, since the electric field of the SPP modes only has two components, the traditional two-dimensional (2D) treatment of the polarization state is formally sufficient in this analysis. In our case, however, the degree of polarization is defined in the plane parallel to the wave vector (*xz* plane, where the modes are generally elliptically polarized) and thus the situation differs entirely from that of beam-like fields for which the electric field is perpendicular to the wave vector.

Mathematically, the (2D) degree of polarization is defined as [34]

*P*(

**r**,

*ω*) = 0. The highest value,

*P*(

**r**,

*ω*) = 1, corresponds to the case when the field is completely polarized and this occurs if, and only if, the components are fully correlated. Any other value of

*P*(

**r**,

*ω*) represents partial polarization.

## 4. Degree of coherence in the LRSPP-SRSPP superposition

As explained in Sec. 2, the slab supports both the long-range and the short-range SPPs, which individually are fully spatially coherent and completely polarized. We therefore consider a superposition of these two modes, which allows for the possibility of partial coherence and partial polarization. The two modes can be excited, for example, by employing the well-known Otto [42] or Kretschmann [43] configurations based on attenuated total reflection, or, alternatively, by end-fire coupling [12, 14] (see also [22, 23]). We will not examine in this work the fields inside the slab any further, but focus on the electric field outside the film. So, in order to keep the notation simpler, henceforward the subscript 2 in the field amplitudes, wave vectors, etc., referring to medium 2 (surrounding) is dropped off. In this case, by defining the unit polarization vector **p̂** ≡ |**k**|^{−1}(−*k _{z}*

**ê**

*+*

_{x}*k*

_{x}**ê**

*), and according to Eq. (1), for the region above the slab the cross-spectral density matrix given by Eq. (6) reads as*

_{z}
Equation (12) is general since the mutual correlation between the two modes can be anything. In a practical situation the mode correlation depends on the excitation process. From now on, however, the LRSPP and SRSPP are taken to be mutually uncorrelated, i.e., 〈*E*
^{(+)*}
*E*
^{(−)}〉 = 〈*E*
^{(−)*}
*E*
^{(+)}〉 = 0. This choice is expected to give the fundamental lower boundary for the coherence degree of such a superposition, since any mutual correlation between the constituents leads to a more coherent field. Furthermore, for the same reason, the modes are considered to have equal intensities at **r**
_{1} = *x*
_{1}
**ê**
* _{x}* +(

*d*/2)

**ê**

*which we define as the excitation point. With these decisions, and by using Eq. (12), the degree of coherence given by Eq. (8) becomes*

_{z}**r**≡

**r**

_{2}−

**r**

_{1}is the separation between the observation and the excitation points,

**k**

^{(+)}and

**k**

^{(−)}, and thereby one can verify that

*κ*given by Eq. (15) satisfies where the lower and upper limits correspond to

*d*→ 0 and

*d*→ ∞, respectively. We note that while

**k**

^{(+)′}·

**k**

^{(+)″}=

**k**

^{(−)′}·

**k**

^{(−)″}= 0 [10], Eqs. (16) and (17) imply that generally Δ

**k′**·Δ

**k″**≠ 0 (cf. Sec. 4.1 below).

#### 4.1. Spatial behavior

The cosine and the hyperbolic cosine in Eq. (14) indicate, respectively, that generally the degree of coherence both oscillates and decays [nevertheless, there exists one particular direction for which *μ*(Δ**r**, *ω*) is exclusively oscillating and another one for which *μ*(Δ**r**, *ω*) is solely decaying, namely, perpendicularly to Δ**k″** and perpendicularly to Δ**k′**, respectively]. The oscillation originates from the fact that at certain periodic distances the two-mode surface-plasmon polariton field is electromagnetically similar [44] to the total field at the excitation point (we neglect the decay of the two modes for the moment). For these positions the field shows a high degree of coherence [however, as the modes have different polarization states, *μ*(Δ**r**, *ω*) < 1] even though the two modes are mutually uncorrelated and hence do not interfere. Thus the situation in this context is completely different from the conventional (wave-field) beating of two modes excited coherently. The effect is instead akin to the customary oscillatory behavior of the (scalar) degree of coherence for two uncorrelated modes in a gas laser [45]. The decaying behavior of the degree of coherence, which arises from the decay of the two modes, implies that, as long as Δ**r** ⊥̸ Δ**k″**, for a given slab thickness

*μ*

_{∞}, is valid regardless of the material parameters, the frequency of the field, or the thickness of the slab. This is a consequence of the different decay rates of the two modes: for a sufficiently large |Δ

**r**|, the mode with the lower decay rate (LRSPP) dominates the superposition and the mode with a higher decay rate (SRSPP) can be neglected. Thus, at large distances, the field can practically be considered as a single LRSPP. For a small |Δ

**r**|, on the other hand, the amplitudes of both modes are strong, whereupon the lower value in Eq. (19) depends on

*ε*

_{r1},

*ε*

_{r2},

*ω*, and

*d*according to Eqs. (3), (4), and (15). Furthermore, the lower value is in fact the maximum for the degree of coherence (cf. Sec. 4.3).

As an example, Fig. 2 illustrates the spatial dependence of *μ*(Δ**r**, *ω*) given in Eq. (14) for Ag films of two different thicknesses in vacuum. We observe from Fig. 2 that, as discussed above, the degree of coherence shows an oscillatory behavior, and both the amplitude and the period become smaller as *d* decreases. In addition *μ*(Δ**r**, *ω*) decays, starting from its maximum at |Δ**r**| = 0, towards the constant value *μ*
_{∞} when |Δ**r**| increases, in accordance with Eq. (19), and the decay becomes stronger as the film gets thinner. Further one notices the differences along the *x* and *z* axes: there is no oscillation of *μ*(Δ**r**, *ω*) perpendicular to the interface, in contrast to the situation along the surface, and the decay is notably stronger (around 30 times) in the *z* direction than in the *x* direction. The reason for the absence of the oscillatory behavior perpendicularly to the slab is that Δ*k″*
_{2z} ≫ Δ*k′*
_{2z}, making the degree of coherence ‘overdamped’, and the decay of *μ*(Δ**r**, *ω*) in the *z* direction is much stronger than along the *x* axis because Δ*k″*
_{2z} ≫ Δ*k″ _{x}* [cf. Eqs. (14) and (17)]. To illustrate further the directions in which

*μ*(Δ

**r**,

*ω*) in Fig. 2 is purely oscillating and purely decaying, as pointed out in the paragraph above, we show in Fig. 3 the corresponding density plots.

#### 4.2. Slab-thickness dependence

By keeping Δ**r** fixed (0 < |Δ**r**| < ∞), it can be verified from Eqs. (14) and (18) that

*d*decreases towards zero, whereupon the field contains only the LRSPP for any |

**r**| > 0 [however, for |Δ

**r**| = 0 the SRSPP is always present, no matter how small the slab thickness becomes, and therefore, according to Eqs. (18) and (19), in this specific case $\mu (\mathrm{\Delta}\mathbf{r},\omega )\to \sqrt{3}/2$ when

*d*→ 0].

Figure 4 illustrates the *d*-dependent behavior of *μ*(Δ**r**, *ω*) given in Eq. (14) along the *x* and *z* axes for a Ag slab in vacuum. Figure 4 confirms the features given by Eq. (20): for a given |Δ**r**|, the degree of coherence starts from 1 when *d* is large and approaches
$1/\sqrt{2}$ when the slab thickness goes to zero. Furthermore, the larger |Δ**r**| is the sooner
$\mu (\mathrm{\Delta}\mathbf{r},\omega )\to 1/\sqrt{2}$ as *d* decreases. This behavior has its origin in the significantly higher decay rate of the SRSPP than of the LRSPP at small film thicknesses. For a sufficiently small |Δ**r**|, on the other hand, the amplitude of the SRSPP is strong even though *d* is small and hence also the SRSPP will contribute to the field’s coherence properties at that |Δ**r**|. Figure 4 further shows that along the *z* axis the degree of coherence decays monotonically from 1 to
$1/\sqrt{2}$ as *d* decreases, implying that a thinner slab results in a less coherent field for a given Δ*z*, while in the *x* direction *μ*(Δ**r**, *ω*) oscillates within a particular Δ*x*-dependent slab-thickness region before reaching
$1/\sqrt{2}$. When Δ*x* decreases the oscillation region shifts towards smaller film thicknesses and the separation between two nearby extrema (which is not constant for a fixed Δ*x*) becomes smaller.

#### 4.3. Maxima and minima

The oscillation of *μ*(Δ**r**, *ω*) implies that the field may show a high or a low degree of coherence at certain locations. Equation (14) indicates that such local maxima and minima can be found for points where Δ**k′** ·Δ**r** = 2*mπ* and Δ**k′** ·Δ**r** = (2*m*+1)*π*, respectively, with *m* being an integer. The collection of these specific points, which we denote as
$\mathrm{\Delta}{\mathbf{r}}_{\text{max}}^{(m)}$ and
$\mathrm{\Delta}{\mathbf{r}}_{\text{min}}^{(m)}$, form an infinite number of straight lines in the *xz* plane (cf. Fig. 2) which are separated by *π*/|Δ**k′**| [cf. Eq. (24) in Sec. 5]. The respective extrema,
${\mu}_{\text{max}}^{(m)}$ and
${\mu}_{\text{min}}^{(m)}$, generally depend on *m* and the direction of
$\mathrm{\Delta}{\mathbf{r}}_{\text{max}}^{(m)}$ and
$\mathrm{\Delta}{\mathbf{r}}_{\text{min}}^{(m)}$. In particular, the case Δ**r** ⊥ Δ**k″**, whereby cosh(Δ**k″** · Δ**r**) = 1, is the most interesting, because, according to Eq. (14), it gives the (*m*-independent) global extrema,

*μ*(Δ

**r**,

*ω*) is restricted.

Equation (21) shows that ${\mu}_{\text{max}}^{2}+{\mu}_{\text{min}}^{2}=1$, indicating that an increase of one is accompanied with a decrease of the other. Furthermore, from Eqs. (18) and (21) one finds that the global extrema are bounded as

*μ*

_{max}(

*μ*

_{min}) correspond to

*d*→ 0 (

*d*→ ∞) and

*d*→ ∞ (

*d*→ 0), respectively. Equation (22) particularly demonstrates that, irrespective of the chosen materials, frequency, and slab thickness, there are always regions for which the total field displays a quite high or a rather low degree of coherence.

In Fig. 5 we have plotted the *d*-dependent behavior of *μ*
_{max} and *μ*
_{min} for a Ag slab at different wavelengths (left panel) and in various surroundings (right panel). Concerning the right panel in Fig. 5, we see that, practically, neither *μ*
_{max} nor *μ*
_{min} is affected at all when *ε*
_{r2} is varied. Neither is *μ*
_{max} in the left panel affected significantly as *λ*
_{0} is altered, whereas *μ*
_{min} slightly increases when the wavelength is reduced. Furthermore, in both panels the maxima are close to unity for *d* ≳ 50 nm and smoothly decrease towards
$\sqrt{3}/2$ when *d* → 0, while the minima approach 1/2 as the film thickness goes to zero and 0 for *d* → ∞, in accordance with Eq. (22).

## 5. Global and local coherence length

The definition of the degree of coherence in Eq. (8) enables us to introduce, in analogy with the traditional scalar theory, a coherence length, as a distance between **r**
_{1} and **r**
_{2} over which *μ*(**r**
_{1}, **r**
_{2}, *ω*) drops from its maximum value at **r**
_{1} = **r**
_{2} to a particular number (chosen appropriately for each situation). It was shown in the previous section that *μ*(Δ**r**, *ω*) generally decays towards
${\mu}_{\infty}=1/\sqrt{2}$ when |Δ**r**| increases, making it natural to define a coherence length with the ‘particular number’ to be close to *μ*
_{∞}. Therefore we define such a length, *L*
_{coh}, which we call the global coherence length, as the distance from the excitation point after which *μ*(Δ**r**, *ω*) differs from *μ*
_{∞} at most by a small value *ξ*, i.e.,

*μ*(Δ

**r**,

*ω*) does not essentially change anymore. We recall from Sec. 4 that the degree of coherence generally oscillates, starting from

*μ*

_{max}that according to Eq. (22) is close to unity. The quantity

*L*

_{coh}therefore marks the distance up to which

*μ*(Δ

**r**,

*ω*) fluctuates, physically meaning that within it there are regions in which the electromagnetic SPP field at the two points may be highly correlated and regions where the field may be rather uncorrelated.

Owing to the oscillations the degree of coherence shows local maxima and minima at certain positions which we in Sec. 4.3 labeled as
$\mathrm{\Delta}{\mathbf{r}}_{\text{max}}^{(m)}$ and
$\mathrm{\Delta}{\mathbf{r}}_{\text{min}}^{(m)}$, respectively. Thus it is reasonable to further introduce an additional, local coherence length, *l*
_{coh}, as the shortest distance between two such nearby points, i.e.,

*L*

_{coh}<

*l*

_{coh}. Equation (24) indicates that

*l*

_{coh}→ 0 for

*d*→ 0 and

*l*

_{coh}→ ∞ as

*d*→ ∞. Further Eq. (24) shows that

*l*

_{coh}is independent of

*m*and thus, for any chosen media, frequency, and slab thickness, the distances between all the nearby extrema are the same. Nevertheless, as was already pointed out in Sec. 4.3, the values of the extrema themselves generally depend on

*m*. The global extrema,

*μ*

_{max}and

*μ*

_{min}given by Eq. (21), are the most interesting ones, and therefore we refer specifically to the distance between

*μ*

_{max}and

*μ*

_{min}when addressing

*l*

_{coh}. Thus the local coherence length gives a rough estimation for the domain at the excitation region in which the field is electromagnetically coherent (meaning that all field components are fully correlated). Such a field interacts with nearby nanoscatterers in a coherent manner as in [32].

We next analyze the behavior of *L*
_{coh} and *l*
_{coh} along the *x* and *z* axes separately. When discussing the surface propagation length and the penetration depth into the surrounding of the field, *l _{x}* and

*l*, respectively, we refer to those of the LRSPP, i.e.,

_{z}*l*≡ 1/

_{x}*k*

_{x}^{(+)″}and

*l*≡ 1/

_{z}*k*

_{z}^{(+)″}.

#### 5.1. Along the slab

Let us first consider the local coherence length, *l*
_{coh,x} = *π*/|Δ*k′ _{x}*|, whose behavior along a Ag layer in vacuum is illustrated in Fig. 6 as a function of

*d*for some optical wavelengths. We find from Fig. 6 that increasing

*λ*

_{0}results in a larger

*l*

_{coh,x}in terms of the wavelength (left panel), but a smaller one with respect to the propagation distance (right panel), especially when

*λ*

_{0}approaches the near-ultraviolet regime. One further notices that, for slab thicknesses where

*l*

_{coh,x}≈

*l*, the coherence length can be several tens of

_{x}*λ*

_{0}at larger wavelengths, while for very thin films, where

*l*

_{coh,x}is only a fraction of the propagation range, a smaller wavelength may lead to subwavelength coherence lengths. Figure 7, on the other hand, illustrates the

*d*-dependent behavior of

*l*

_{coh,x}at

*λ*

_{0}= 632.8 nm along a Ag slab in different surroundings, from which we observe that a higher

*ε*

_{r2}results in a smaller coherence length compared to the wavelength in the surrounding medium ( ${\lambda}_{2}={\lambda}_{0}\sqrt{{\epsilon}_{r2}}$), but a larger one with respect to

*l*.

_{x}
Figures 8 and 9 depict the global coherence length introduced in Eq. (23) when *ξ* = 0.01 for the same fields considered in Figs. 6 and 7, from which one observes that the *λ*
_{0}- and *ε*
_{r2}-dependent behavior of *L*
_{coh,x} is somewhat similar to that of *l*
_{coh,x}. However, the order of magnitude of *L*
_{coh,x} differs from that of *l*
_{coh,x}. Particularly one finds that for thicker slabs the global coherence length can be many times larger than the actual propagation distance of the field, even several hundreds or thousands of wavelengths, indicating that *μ*(Δ**r**, *ω*) may oscillate over the field’s whole propagation regime. On the other hand, for very thin films *L*
_{coh,x} may be just a fraction of *l _{x}*, which signifies that, apart from a small |Δ

**r**| with respect to

*l*, the degree of coherence is practically constant along the surface of the slab. The small abrupt jumps in Figs. 8 and 9 arise from the formal definition of

_{x}*L*

_{coh}in Eq. (23).

#### 5.2. Perpendicular to the slab

In this case Eq. (24) for *l*
_{coh} is indeed still valid (*l*
_{coh,z} = *π*/Δ*k′ _{z}*), but it turns out that

*l*

_{coh,z}>

*L*

_{coh,z}(data not shown) due to the ‘overdamped’ behavior of

*μ*(Δ

**r**,

*ω*) in the

*z*direction (cf. Fig. 2), whereupon there is no meaning, at least in a practical sense, to investigate the local coherence perpendicularly to the slab. Therefore we consider only the global coherence length along the

*z*axis.

Figures 10 and 11 illustrate the *d*-dependent transverse behavior of *L*
_{coh,z} defined by Eq. (23) for *ξ* = 0.01 above a Ag layer for selected vacuum wavelengths and surroundings, respectively. We see from Figs. 10 and 11 that varying the free-space wavelength or the surrounding medium has the same effect on the global coherence length in the *z* direction as along the *x* axis (cf. Figs. 8 and 9): increasing *λ*
_{0} (*ε*
_{r2}) gives a larger (smaller) *L*
_{coh,z} with respect to the wavelength, but a smaller (larger) one in terms of *l _{z}*. One also finds that, unless the film gets very thin, the global coherence length is (much) larger than the actual penetration depth of the field. Consequently, since this feature implies that the degree of coherence decays slowly and monotonically [

*L*

_{coh,z}>

*l*

_{coh,z}and thus there is no oscillation of

*μ*(Δ

**r**,

*ω*) in the

*z*direction, cf. Fig. 2], the field is highly coherent perpendicularly to the slab, even for quite small slab thicknesses. Nonetheless, as

*d*becomes very small we see that

*L*

_{coh,z}<

*l*and

_{z}*L*

_{coh,z}<

*λ*

_{2}, indicating that

*μ*(Δ

**r**,

*ω*) can change notably at subwavelength distances when moving away from the surface of an ultra-thin film.

## 6. Degree of polarization in the LRSPP-SRSPP superposition

We now turn our attention to the polarization properties and consider, as before, the LRSPP-SRSPP superposition above the slab. For such a two-mode field, according to Eqs. (12) and (16), the spectral polarization matrix in Eq. (7) reads as

*E*

^{(+)*}

*E*

^{(−)}〉 = 0 and rewrite

**r**→

**r**

_{1}+

**r**where

**r**

_{1}is the excitation point at which the intensities of the two modes are equal. On the basis of Eq. (25), in this case the degree of polarization in Eq. (10) becomes

#### 6.1. Spatial behavior

Unlike the degree of coherence in Eq. (14), one notices from Eq. (26) that the degree of polarization does not display an oscillatory term, but is characterized only by a hyperbolic cosine. Excluding the particular direction along which *P*(**r**, *ω*) is constant, i.e., the one that is perpendicularly to Δ**k″**, for a given slab thickness Eq. (26) indicates that

*P*(

**r**,

*ω*). Consequently, since 1/2 ≤

*κ*≤ 1 [cf. Eq. (18)], we conclude that the degree of polarization for the LRSPP-SRSPP superposition is always bounded as The physical origin behind the (

*ε*

_{r1}-,

*ε*

_{r2}-,

*ω*-, and

*d*-independent) maximum, representing a fully polarized field, is by now apparent: when |

**r**| becomes large enough, the SRSPP vanishes and the field can practically be considered as a single LRSPP for which

*P*(

**r**,

*ω*) = 1 regardless of the chosen media, frequency, film thickness, and position [cf. the behavior of

*μ*(Δ

**r**,

*ω*) in Eq. (19)]. By a similar reasoning, i.e., as the contribution of the SRSPP to the degree of polarization cannot be neglected when |

**r**| → 0, the minimum, corresponding to partial polarization, depends on

*ε*

_{r1},

*ε*

_{r2},

*ω*, and

*d*, according to Eqs. (3), (4), and (15).

Figure 12 illustrates the spatial behavior of *P*(**r**, *ω*) given in Eq. (26) for Ag films of two different thicknesses in vacuum at *λ*
_{0} = 632.8 nm. It is seen from Fig. 12 that for *d* = 50 nm (left panel) the degree of polarization is about unity everywhere [even for the minimum at |**r**| = 0 one has *P*(**r**, *ω*) = 0.993], corresponding to a fully polarized field. However, when the slab thickness is reduced to 20 nm (right panel), we observe that the degree of polarization slightly decreases [*P*(**r**, *ω*) ≈ 0.9] within a small region (Δ*x* ≲ 3*λ*
_{0} and Δ*z* ≲ 0.15*λ*
_{0}), suggesting that the field can be partially polarized for ultra-thin films within, or close to, subwavelength distances from the excitation point.

#### 6.2. Slab-thickness dependence

By choosing an arbitrary **r**, such that |**r**| > 0 (and **r** ⊥̸ Δ**k″**), Eqs. (15) and (26) imply that

*μ*(Δ

**r**,

*ω*) in Eq. (20), the upper value of

*P*(

**r**,

*ω*) in Eq. (29) follows from the fact that, for a sufficiently large film thickness, the wave vectors for the constituents of the two-mode field become degenerate. The reason for the lower value in Eq. (29) is somewhat similar to that of

*μ*(Δ

**r**,

*ω*) found in Eq. (20), namely, for any |

**r**| > 0, no matter how small, the field contains only the LRSPP because the decay rate of the SRSPP diverges as the slab thickness goes to zero [if |

**r**| = 0, on the other hand, the SRSPP cannot be neglected and in that case Eqs. (18) and (27) imply that $P(\mathbf{r},\omega )\to 1/\sqrt{2}$ when

*d*→ 0].

Equations (26) and (29) together indicate that for a fixed |**r**| > 0 there exists a specific film thickness for which the degree of polarization is at its lowest. Furthermore, the results in Sec. 6.1 allude that the field gets partially polarized inside a particular (small) region as *d* is very small. Therefore, in Fig. 13 we demonstrate the behavior of *P*(**r**, *ω*) given in Eq. (26) along the *x* and *z* axes for ultra-thin Ag slabs in vacuum when *λ*
_{0} = 632.8 nm. Figure 13 shows that, indeed, within a particular region, the superposition becomes less polarized when the film thickness is reduced, but at the same time also the regime in which the field is partially polarized gets smaller. Nevertheless, the conclusion is clear: for a sufficiently thin slab, the degree of polarization of the two-mode field can fluctuate at subwavelength distances from the excitation point even if it is fully polarized farther away.

#### 6.3. Minimum

Finally, we examine how the choice of *λ*
_{0} and *ε*
_{r2} affects the degree of polarization of the superposition. Since
$1/\sqrt{\kappa}$ is the fundamental lower boundary for the polarization degree of the two-mode field [cf. Eq. (28)], labeled as *P*
_{min}, we focus specifically on this minimum value.

Figure 14 illustrates the *d*-dependent behavior of *P*
_{min} above a Ag layer for selected vacuum wavelengths and surroundings, from which one finds that decreasing *λ*
_{0} (left panel) reduces the degree of polarization, while varying *ε*
_{r2} (right panel) has a negligible effect on *P*
_{min}.

## 7. Conclusions

Within a rigorous electromagnetic framework, we have investigated the spatial coherence and polarization characteristics of a superposition field that consists of the long-range and the short-range surface-plasmon polaritons at a thin metal slab. The two polariton modes can be excited in a number of different ways and their mutual correlation depends on the details of the actual generation arrangement. The SRSPP, which has attracted little attention and practical importance as compared to the LRSPP, plays a crucial role in this study as it can be viewed to enable the creation of partial coherence and partial polarization in the two-mode SPP field. The aim of the work was to examine the fundamental limits that the degrees of electromagnetic coherence and polarization associated with such a LRSPP-SRSPP superposition field can assume. To this end, the two modes were taken uncorrelated and of equal intensity at their excitation point. Furthermore, we have explored how the two degrees vary within their extremal values when the media, frequency, and film thickness are changed.

Regarding coherence, we specifically analyzed the (spectral) degree of spatial coherence of the two-mode superposition field above the metal slab. Due to the vectorial nature of SPPs, for which the traditional degree of coherence for scalar fields is inadequate, an appropriate degree of coherence valid for vector fields was employed. It was shown that the superposition field is highly coherent and assumes the maximum value for the degree of coherence at the point where the two modes are generated. Nevertheless, away from the excitation region, the coherence degree oscillates and sufficiently far the field generally becomes partially coherent with an essentially constant degree of coherence. Because of this behavior, we introduced a global coherence length as the distance from the excitation point beyond which the degree of coherence does not essentially change anymore. Physically this means that within the global coherence length there are locations at which the field may be highly correlated or relatively uncorrelated with respect to the field at the excitation point. It was demonstrated that for thicker slabs the global coherence length can extend over ranges many times larger than the actual propagation distance or penetration depth of the field, from tens to even thousands of wavelengths, while thinner films may result in subwavelength global coherence lengths. The oscillatory behavior, particularly along the surface of the slab, led us further to introduce a local coherence length, as the distance from the excitation point to the first minimum (which is also the global minimum) of the degree of coherence. Hence the local coherence length serves as a rough measure for the regime in which the two-mode field is practically electromagnetically coherent. Similarly to the global coherence length, it was shown that the local coherence length can be greater than the field’s propagation range when the film thickness is large, or alternatively, just a fraction of it if the slab is very thin. Furthermore, we demonstrated that increasing the frequency and/or the permittivity of the surrounding medium results in larger global as well as local coherence lengths with respect to the propagation distance and the penetration depth, but smaller ones in terms of the wavelength.

Concerning polarization, we investigated the (spectral) degree of polarization of the two-mode field above the slab. Even though the usual two-dimensional treatment is adequate for describing the polarization state of such a superposition, the degree of polarization is here defined in the plane of incidence where the two modes are elliptically polarized, and thus the situation differs substantially from that for ordinary beam-like wave fields. As might be expected, since the two modes are *p* polarized and excited in the same plane, we found that the superposition field is generally highly polarized. Nevertheless, the analysis suggested that for ultra-thin films the field can be partially polarized and the degree of polarization may fluctuate within subwavelength distances from the excitation region. Fully polarized and partially polarized electromagnetic fields interact with nanoparticles differently, and such scattering may be used to determine the state of polarization of the field. Finally we investigated the influence of the frequency and the surrounding on the polarization and found that reducing the frequency increases the degree of polarization of the field, while varying the permittivity of the surrounding has a negligible effect on it.

The analysis in this work on the LRSPP-SRSPP superposition can be regarded quite generic, since it explores the fundamental coherence and polarization limits of such a two-mode field, irrespective of the excitation technique. Thus our results could particularly find use in various interdisciplinary plasmonic and nanophotonic applications, including optoelectronic thin-film waveguiding and near-field interactions, where controlled partial spatial coherence is desirable. Our results also indicate the possibility for achieving and tailoring genuine subwavelength polarization effects by utilizing such a configuration.

## Acknowledgments

This work was partially funded by the Academy of Finland (projects 268480 and 268705). A. Norrman is thankful for support from the Jenny and Antti Wihuri Foundation.

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