Field Correlations in Surface Plasmon Speckle

In this work fluctuations in the electric field of surface plasmon polaritons undergoing random scattering on a rough metallic surface are considered. A rigorous closed form analytic expression is derived describing second order correlations in the resulting plasmon speckle pattern assuming statistically stationary and isotropic roughness. Partially coherent planar Schell-model source fields can also be described within the developed framework. Behaviour of the three-dimensional degree of cross polarisation and spectral degree of coherence is also discussed. Expressions derived take full account of dissipation in the metal with non-universal behaviour exhibited within the correlation length of the surface and source fields.


S1 Derivation of field correlation matrix
Within the electrostatic approximation the electric field of a surface plasmon polariton (SPP) propagating on a surface defined by the surface profile ζ (ρ), where ρ denotes a position vector in the x-y plane, is given by E(r) = −∇φ (r). The scalar potential φ is a solution to Laplace's equation ∇ 2 φ = 0 subject to continuity constraints on φ and its normal derivative on the supporting interface ζ (ρ) 1 . The general form of the associated vector Helmholtz equation for the SPP field E in a source free region can be written in the form where µ(r) and ε(r) are the spatial distributions of the magnetic permeability and electric permittivity respectively and ω is the angular frequency of the wave (assumed monochromatic with an exp[−iωt] time dependence). We henceforth shall consider only non-magnetic media for which µ(r) = µ 0 for all r. The electric permittivity distribution describing an arbitrary surface can be written in the form where H(z) is the Heaviside function. We also define the permittivity difference δ ε(r) = ε(r) − ε f (r) relative to the permittivity distribution for a flat surface ε f (r) = ε 1 + (ε 2 − ε 1 )H(−z).
The roughness can thus be modelled as an equivalent source current J(r) = −iωδ ε(r)E(r). The general solution for the scattered SPP field can be written directly by employing the Green's tensor G(r 1 , r 2 ) describing elastic scattering of SPPs on a flat surface (defined below). If the underlying surface profile is a random field, then so too is the electric field E. Random fields can be partially characterised by their second order moments and we thus consider the two-point field correlation matrix W(r 1 , r 2 ) = δ E * (r 1 )δ E T (r 2 ) here, where δ E(r) = E(r) − E(r) . Upon averaging Eq. (S4) we find the mean field E(r) satisfies the equation Making a Born approximation, the driving field appearing in J is replaced by the incident field E 0 which is the solution of Eq. (S1) with ε → ε f , such that δ ε(r)E(r) → δ ε(r)E 0 (r) = δ ε(r) E 0 (r) = 0 since δ ε(r) = 0 by definition. Hence subtracting Eqs. (S4) and (S5) we have which has the solution where we now introduce subscripts on the position vectors to denote different variables. Using Eq. (S7) it follows that where W J (r 1 , r 2 ) = J * (r 1 )J T (r 2 ) is the two point correlation matrix for the current J(r) = −iωδ ε(r)E 0 (r).

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Further using Eqs. (S12), (S16) and (S20) this expression reduces to The Dirac delta functions allow the integration of z 3 , z 4 and κ 2 to be easily performed, yielding where we have also dropped the unneeded subscript on κ 1 and used the definition of w J (−κ) in the final equality.
The SPP contribution to the total Green's tensor for a single planar interface is given by Eq. (21) of 2 . Specifically assuming an interface at z = 0 separates a dielectric medium (z > 0) with permittivity ε 1 and a metallic medium (z < 0) of complex permittivity ε 2 (Im[ε 2 ] > 0) the SPP Green's tensor can be shown to take the form are unit vectors in the radial, azimuthal and longitudinal directions respectively, we can use Eq. (S29) to give

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where letting Z = α * z 1 + αz 2 The matrix factor can be evaluated and takes the form At this point we make the further restriction that w jk (Q) = w jk (Q), i.e. the two-point current correlation matrix is a function of the separation of the two points only and not their relative direction. For this case we have where using Eq. (S37), letting P = P[cos θ , sin θ , 0] T and splitting the exponential term we can write Recalling the integration identity 4 2π 0 cos nβ sin nβ e −ia cos(β −γ) dβ = 2π(−i) n cos nγ sin nγ J n (a).
the angular integration over β in Eq. (S46) can be performed yielding Noting that these expressions are independent of φ we can similarly evaluate the integral over φ yielding Now defining the auxiliary matrix and hence Eq. (S53) can however be simplified further by considering the assumption w jk (Q) = w jk (Q) further. In particular, we require this angular independence to hold regardless of the two points r 1 and r 2 under consideration, i.e. for general Q. Satisfying this requirement, necessitates that w J must be invariant under rotations around the z axis, and therefore w 11 = w 22 , w 12 = −w 21 , w 13 = w 31 = w 23 = w 32 = 0. Rewriting Eq. (S53) in the form where, recalling m 11 = m 22 , and δ jk is the Kronecker delta, we find the simplified form for the correlation matrix as Our final task thus remains to find an expression for V 0 (we can also find expressions for V n without great effort and thus we keep things general for the sake of interest). In particular we must evaluate the integrals which make up the individual terms in V n viz. are the Hankel functions of the first and second kind of order m. This substitution however introduces a pole at κ = 0 that must be excluded, such that we consider Now letting κ = e −iπ κ and using the reflection formulae H The two integrals can be combined into a single integral according to where PV denotes the principal value, Res denotes the complex residue, the integration contour C 0 is shown in Figure S1 and g(κ) is an analytic function in the upper half plane given by To examine the residue at κ = 0 we can consider the small κ behaviour of g(κ) using the small argument expansions 5 where Γ(n) is the Gamma function. Specifically which tends to zero for m − n ≥ 3 cases considered in this work.

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With a view to extending the integration contour C 0 we consider the behaviour of J n (κQ)H (1) m (κP) for |κ| → ∞. The asymptotic formulae Splitting κ = κ r + iκ i into its real and imaginary parts, we see that the two position dependent exponents scale as e iκ(P±Q) ∼ e −κ i (P±Q) , both of which decay for κ i > 0, |κ| → ∞ if P > Q. Assuming this latter condition to hold true, the integration contour C 0 in Eq. (S63) can be extended to include the semicircular contour C ∞ (as shown in Figure S1) where the radius of C ∞ is taken to infinity, since the value of the integrand, g(κ), is zero along this contour. Accordingly the integration path is now a closed loop and noting Jordan's lemma, the integral can be evaluated using the residual theorem. The function g(κ) has four simple poles at κ = ±k SPP and ±k * SPP , only two of which lie within the integration contour namely κ 1 = k SPP and κ 2 = −k * SPP . Accordingly Evaluating the limits gives L nm (P, Q) = l nm (P, Q, k SPP ) + l nm (P, Q, −k * SPP ) where

S2 Degree of polarisation
In this section we evaluate the 2D and 3D degrees of polarisation (DOP) at P = 0 defined by 6,7 and respectively where · · · F denotes the Frobenius norm. We thus must first evaluate W(0, z 1 , z 2 ). Using Eqs. (S60) and (S74) at P = 0 we have l nm (0, Q, κ) = Λ nm (κ)H where W 0 is a constant found by evaluating the integral. Substituting this expression into Eq. (S76) directly gives .
For the 2D case describing correlations between in-plane field components we consider the reduced correlation matrix yielding D 2D = 0, whilst to describe the 2D DOP between in and out-of-plane field components we consider

S3 Degree of coherence
The degree of coherence of a 3D field is defined as µ(P, where r 1 = 1 and r 2 = −Γ. Hence Eq. (S87) gives a general expression for the DOC which is dependent on the surface and source correlation functions through w 11 = w 22 and w 33 , however, at this point we make a further approximation. Specifically, we split κ 1 = k SPP into its real and imaginary parts viz. κ 1 = κ 1r + iκ 1i and note that for typical SPPs κ 1i κ 1r . We wish to approximate the integral terms in 8/10 Eq. (S87) and thus we expand the arguments of the Bessel and Hankel functions as κ 1r Q + iκ 1i Q. For κ 1i Q 1 (or equivalently Q 2L where L = 1/(2Im[k SPP ]) is the SPP attenuation length) we can perform a series expansion about κ 1i = 0 to first order. Doing this yields an expression of the same form as Eq. (S87) albeit with the replacements J 0 (κ p Q) → J 0 (κ rp Q) and similarly for the H 0 (κ p Q) terms, where κ r2 = −κ r1 . Using the reflection formulae for the Bessel and Hankel functions 5 we have J 0 (κ 2r Q) = J 0 (κ 1r Q), H (1) Upon making a number of further approximations we can simplify Eq. (S87) for two cases, namely that of (i) a loss-free metal such that ε 2 is purely real and negative, and (ii) a narrow correlation function w jk . We shall consider both cases now in turn.

S3.2 Narrow correlation function
Typically the correlation functions w 11 (Q) = w 22 (Q) and w 33 (Q) fall-off to negligible values as Q increases such that we can define a characteristic length Q 0 above which w 11 (Q) ≈ 0. When considering points P such that P > Q 0 we can note that the second term in both Eq. (S87) is negligible such that µ(P, z 1 , z 2 ) = e −κ 1 Z ∑ 2 p=1 r p H (1) 0 (κ p P) P 0 Q W (Q)J 0 (κ p Q)dQ Since, however, for Q > Q 0 we have assumed that the w 11 (Q) ≈ w 33 (Q) ≈ 0, the integration limits in Eq. (S89) can both be replaced by It is worthwhile to note at this point that the DOC for points separated by a distance greater than the characteristic correlation length Q 0 is described by a form that is independent of the precise nature of the source and surface correlation functions. This 9/10 statement is however only true for P Q 0 , meaning when considering more closely spaced points the DOC between these positions exhibits non-universal behaviour i.e. the precise functional form of µ depends on w J . Nevertheless, for very narrow correlation functions (i.e. Q 0 → 0) the DOC is given by Eq. (S98) for all P = 0. By construction we note that µ(0, 0, 0) = 1. Importantly, Eq. (S98) does not hold in the large loss limit as can be seen by studying the limiting behaviour as P → 0. In particular which does not equal unity for k 1i > 0.