Coherence matrix of plasmonic beams

We consider monochromatic electromagnetic beams of surface plasmon-polaritons created at interfaces between dielectric media and metals. We theoretically study non-coherent superpositions of elementary surface waves and discuss their spectral degree of polarization, Stokes parameters, and the form of the spectral coherence matrix. We compare the polarization properties of the surface plasmonspolaritons as three-dimensional and two-dimensional fields concluding that the latter is superior. Physics of electromagnetic beams has gained considerable attention owing to recent advances in the study of the Airy beams, Bessel beams, Bessel-Gauss beams, etc.[1]-[4]. Any of these beams can be formed using coherent superposition of elementary waves, e.g., plane waves. When elementary waves are not coherent, the beams are characterized by the Stokes parameters and the degree of polarization. These parameters can be conveniently combined in the coherence matrix [5]. If an electromagnetic field has no direction of propagation, i.e. it is not a beam, the elementary waves propagate in various random directions in the three-dimensional space. Such three-dimensional fields (like thermal fields) are described by the generalized coherence matrix and degree of polarization [6]-[8]. These fields exactly are assumed to be excitation sources for the surface waves in the current investigation. Localized surface plasmon-polariton (SPP) beams are not very special objects. Similarly to optical beams, they can be generated as Bessel and Airy plasmonic beams [11], [12]. That is why we can expect that the noncoherent superposition of the surface waves should be put forth. In the previous works [9], [10] the spatial coherence and polarization properties of the SPP fields were discussed for some specific cases. We use the tensorial generalization of the coherence matrix for statistically stationary fields, which is called a light beam tensor Φ [13]-[15]. Hermitian conjugated tensor Φ (Φ + = Φ) in the coordinate-free notations, which ignore setting a specific coordinate system, is: , ) , ( * ) ( ) (     s s s E E r  (1) * E-mail: [email protected] where dyad b a is the direct (Kronecker) product of vectors a and b, E (s) is the electric field of an elementary wave, the asterisk denotes complex conjugate. When the elementary non-coherent plane waves propagate in the direction of unit vector n, the orthogonal condition E (s) n=0 brings us to the two-dimensional beam tensor Φ restricted by Φn=nΦ=0, where the contraction of tensor Φ with vector n is assumed. Then the tensor has two nonzero eigenvalues λ1 and λ2 and can be presented as , ) ( 2 * 1 1 2 1 2 I u u         (2) where n n 1 u u u u I        2 2 1 1 is the projector onto the plane perpendicular to vector n, u1 and u2 are the in-plane normalized eigenvectors such that |u1|=|u2|=1. The first and second terms in the right-hand side of equation (2) correspond to the completely polarized and unpolarized parts, respectively. In general, the field has no definite propagation direction and can be treated as three-dimensional one. It is sufficient for our purposes to consider light as a superposition of completely polarized light with the threedimensional complex vector u1 and three-dimensionally unpolarized field proportional to the 3D identity tensor 1 (more general description is presented in Refs. [6]-[8]). Thus one introduces , ) ( 2 * 1 1 2 1 3 1 u u         (3) Let us move to the SPPs now. When the boundary conditions for a plane wave incident onto the metaldielectric interface are satisfied, there appear electron oscillations in the metal and a surface electromagnetic wave coupled to these collective oscillations (Fig. 1). The wave is called SPP [16]. It is well localized at the interface and can be described in the dielectric medium as