On solutions of Maxwell’s equations with dipole sources over a thin conducting film
Abstract
We derive and interpret solutions of timeharmonic Maxwell’s equations with a vertical and a horizontal electric dipole near a planar, thin conducting film, e.g. graphene sheet, lying between two unbounded isotropic and nonmagnetic media. Exact expressions for all field components are extracted in terms of rapidly convergent series of known transcendental functions when the ambient media have equal permittivities and both the dipole and observation point lie on the plane of the film. These solutions are simplified for all distances from the source when the film surface resistivity is large in magnitude compared to the intrinsic impedance of the ambient space. The formulas reveal the analytical structure of two types of waves that can possibly be excited by the dipoles and propagate on the film. One of these waves is intimately related to the surface plasmonpolariton of transversemagnetic (TM) polarization of plane waves.
I Introduction
In the last decade, rapid advances have been made in the design and fabrication of twodimensional materials Torresbook which can be used to manipulate light at small scales. Zhangbook The highly active field of plasmonics focuses on the interaction of electromagnetic radiation at the mid and nearinfrared spectrum with the collective motion of electrons in conducting interfaces and nanostructures. Maierbook A goal is to generate electromagnetic waves that propagate with relatively small energy loss close to the interface between a conducting material, e.g. graphene, and a dielectric. Vakil11 ; Ju11 ; Cheng13 ; Hanson08 ; Hanson11 ; Lovat15 In plasmonics such lateral waves should decay fast enough away from the interface; while, on the other hand, they should attenuate slowly enough in the direction of propagation along the interface. A wave that has attracted much attention in this context is the surface plasmonpolariton, Raetherbook ; Maierbook ; Bludov13 with a variety of reported applications Zhang12 ; Huidobro10 including invisibility cloaking, Renger10 photovoltaics Green12 and nanolithography. Luo04
Most recently, direct experimental evidence was provided for generating surface plasmons by placing a receiving resonant antenna near a graphene sheet. Gonzalez14 Motivated by this advance, our goal with this paper is to analytically study the generation of surface plasmons by currentcarrying sources via a solvable model for a fundamental setting. To this end, we formulate and solve exactly boundary value problems for the timeharmonic Maxwell equations in the presence of vertical and horizontal electric (Hertzian) dipoles near an isotropic and homogeneous conducting sheet between two isotropic and nonmagnetic unbounded media. Maierbook ; Cheng13
The underlying theme, wave propagation near boundaries, has been the subject of important studies for over a century; Sommerfeld1899 ; Sommerfeld1909 ; Zenneck1907 ; vanderPol31 ; Fok33 ; Norton36 ; Banos ; Wait ; KOW see particularly the systematic and extensive treatment of Ref. KOW, . In these works, approximation techniques are developed for radiowave propagation; these have offered valuable insights into the lateral electromagnetic waves traveling near the boundaries between media of very different indices of refraction. KOW In the frequency band of plasmonics, Zhangbook however, additional considerations have emerged because of properties of novel twodimensional materials used in microscale applications. For instance, a thin layer of graphene has a complex surface conductivity, with a positive or negative imaginary part depending on frequency and doping, and introduces a jump in the tangential component of the magnetic field across the interface. The analytical consequences of this discontinuity are largely unexplored. Hanson08 ; Hanson11 ; Nikitin11 This view suggests that the associated lateral electromagnetic waves be studied in detail in the near and midinfrared spectrum.
In this paper we analytically address aspects of the following question. What is the structure of the waves generated by dipole sources on a thin conducting film? Our goal is to single out fundamental attributes of the field which are intimately related to the film by analyzing a minimal solvable model. Our tasks can be summarized as follows.

We explicitly represent all field components in terms of onedimensional FourierBessel (Sommerfeldtype) integrals, in the spirit of Ref. KOW, . Our derivations, focusing on the electromagnetic field itself, differ from the use of the (nonphysical) Hertzian potential invoked, e.g. in Ref. Hanson08, , by which the field components are derived via successive differentiations.

By a generalized SchwingerFeynman representation for a class of integrals, Margetis01 we compute all field components via fast convergent series of known functions such as the Fresnel integrals, when both the dipole and observation point lie on the plane of the film and the ambient media have equal permittivities. Our model is thus simplified, yet without obscuring the goal to analytically understand the role of the interface.

In accord with applications in plasmonics, Cheng13 we further simplify the exact solutions when the surface resistivity (inverse of conductivity) of the thin layer is much larger in magnitude than the intrinsic impedance of the ambient space. Then, a few terms are retained in the series expansions for the fields yielding simple approximate formulas for all distances from the source.
Our approach is based on systematically solving Maxwell’s equations in the spirit of Refs. KOW, ; Margetis01, . Thus, we avoid any apriori planewave approximations. We recognize a specific type of lateral wave as intimately related to the surface plasmonpolariton of transversemagnetic (TM) polarization of plane waves Maierbook ; Bludov13 via the contribution of a certain pole in the complex plane of the dual (Fourier) variable; see also the treatments of Refs. Hanson08, ; Hanson11, ; Nikitin11, . For a horizontal dipole on a thin film in free space, when contributions related to TM polarization in principle may coexist with contributions of transverseelectric (TE) polarization, our analysis reveals that the TM surface plasmonpolariton, when present, is accompanied by a wave expressed by Fresnel integrals. KOW
The analysis presented here, with focus on explicit, physically transparent expressions for the electromagnetic field in terms of known functions, differs in methodology from previous studies of waves in similar settings. Bludov13 ; Hanson08 ; Hanson11 ; Nikitin11 For example, in Ref. Bludov13, , the authors review dispersion relations for plane waves in plasmonics for a variety of experimentally relevant geometries, without discussing effects of point sources. In Ref. Hanson08, , exact integrals are formulated for the Hertzian potential produced by dipoles in the presence of a graphene sheet; the electromagnetic field is then computed by numerical evaluation of integrals. In Ref. Hanson11, , a similar task is carried out more extensively, with numerical evidence that the surface plasmonpolariton of TM polarization, recognized as a discrete spectral contribution, may dominate wave propagation under certain conditions on the surface conductivity. In Ref. Nikitin11, , the authors numerically describe the field produced by dipoles near a graphene sheet, distinguishing a “core region”, where the electric field can be much larger than its values in free space, from an “outer region”, where the field approaches its values in free space.
Our work expands previous numerical approaches Hanson08 ; Hanson11 ; Nikitin11 in the following sense. By focusing on a simple yet nontrivial model with a conducting thin film, we are able to derive closedform expressions which explicitly separate the primary field of the dipole, produced in the absence of the layer, from the scattered field which is sensitive to the film conductivity, for all distances from the point source. This approach singles out analytic aspects of the wave produced by the point source that are intimately connected to the surface conductivity of the film, thus showing how the surface plasmon related to TM polarization can dominate propagation in cases of physical and practical interest. Our analysis lacks generality, since we restrict attention to the case where the source and the observation point both lie on the plane of the thin film; nonetheless, we view our treatment as a step necessary for tackling the problem of radiation by a realistic currentcarrying source (rather than an incident plane wave) placed on the material surface. Gonzalez14 The character of the wave produced by the source of course becomes important at short distances or high enough frequencies. Our work aims at illuminating the complicated structure of the field in a simple nontrivial setting.
The present work illustrates analytic aspects of the tensor (or, dyadic) Green function for the geometry of a thin conducting film at fixed frequency. Naturally, the response to any imposed currentcarrying source can then be derived by superposition. This task lies beyond our present scope. Furthermore, we do not pursue numerical computations of the fields. The explicit computation by asymptotic methods KOW of field components when the observation point or the dipole is away from the layer is left for future work.
The remainder of this paper is organized as follows. In Section II, we describe the boundary value problem for Maxwell’s equations. Section III focuses on the derivation of FourierBessel representations, known as the Hankel transform, Lebedevbook for the electromagnetic field. In Section IV, we evaluate exactly the requisite integrals when the dipole and the observation point are on the film, which is placed in a homogeneous space. Section V discusses the field resulting from our computations for distances from the source that are comparable to the wavelength in free space. Section VI concludes our paper with an outline of open problems.
Notation and terminology. The time dependence is assumed throughout, where is the angular frequency. is the set of reals, is the set of integers, and boldface symbols denote vectors in . We write () to mean that is bounded by a nonzero constant (approaches zero) in a prescribed limit. implies in a prescribed limit. () denotes the real (imaginary) part of complex . The term “sheet” has a twofold meaning as either a material thin film or, as a “Riemann sheet”, a particular branch of a multiplevalue function of a complex variable; and the terms “top Riemann sheet” and “first Riemann sheet” are used interchangeably. We use the terms “TM polarization” and “TE polarization” in the context of waves produced by dipoles to indicate the presence of certain denominators, denoted by and in the main text, respectively, in the Fourier representations of the corresponding fields; each denominator appears in the reflection coefficient for the TM or TEpolarized plane wave incident upon the thin film. The terms “surface plasmonpolariton” Maierbook and “surface plasmon” are used interchangeably.
Ii Boundary value problem
In this section, we formulate the boundary value problem for Maxwell’s equations. The current density of the vertical unit electric dipole, shown in Fig. 1, is
(1) 
where is the distance of the dipole from the layer, , is the unit vector along the axis, and denotes the Dirac mass at point . For the horizontal unit electric dipole, shown in Fig. 2, the current density reads
(2) 
The film has infinitesimal thickness and scalar surface conductivity , which is in principle complex and dependent. Hanson08 The film lies in the plane which separates region 1, the upper half space with wave number , from region 2, the lower half space with wave number (Figs. 1 and 2). We assume that and (), i.e., a lossy medium , including the case with . Note that where is the magnetic permeability of free space, since the media are assumed nonmagnetic, and is the complex permittivity of medium ; in practically appealing situations, this has a small imaginary part.
The timeharmonic Maxwell equations for the field () in region dictate that
(3a)  
(3b) 
By Gauss’ law, and if ; however, these equations are not independent from (3) in the timeharmonic case and, therefore, are not utilized here. Equations (3) are supplemented with boundary conditions for the tangential components, viz., Bludov13 ; Hanson08
(4a)  
(4b) 
where denotes the (continuous) tangential electric field at . Notably, condition (4b) expresses the physical property that the thin conducting film amounts to an effective surface current of density at ; this is viewed as a free current density for the field outside the film. The boundary conditions for the normal components of are redundant for the derivation of a solution; see Appendix A. In addition to (4), we impose the Sommerfeld radiation condition, viz., Muller
(5) 
uniformly in if , for each scalar component () of the vectorvalued field (); is the position vector in Cartesian coordinates. Equations (3)–(5) with (1) or (2) constitute the desired boundary value problem.
Iii Fourier representation of solution
In this section, we derive onedimensional integral representations for for a vertical and a horizontal electric dipole. The starting point is the Fourier transform with respect to of (3) and (4). Accordingly, let
(6) 
where and is the Fourier transform of in , assuming that the integral converges in an appropriate sense. Consequently, by (3) the transformed variables obey
(7a)  
and  
(7b) 
where or is the Fourier transform of or , respectively; . Equations (7) are complemented with the transformation of boundary conditions (4) and radiation condition (5).
iii.1 Vertical dipole
Consider Fig. 1. In this case, by symmetry we have for . The remaining field components can be expressed in terms of , as shown below. Equations (7) combined yield the differential equation KOW
where
In compliance with radiation condition (5), we write
(8) 
where and are integration constants to be determined. In view of (7), the remaining field components are given in terms of by the relations KOW
(9) 
(10) 
To determine and we resort to conditions (4), by which , , , and at . In fact, we need only apply the first and third (or, second and fourth) of these conditions since we have set ab initio; the other two conditions are then satisfied. Thus, we obtain
where the factor is defined as
(11) 
which is associated with the reflection of TMpolarized plane waves from the thin layer (the subscript “m” stands for “magnetic”, implying TM polarization); and the corresponding denominator, , is Maierbook ; Hanson08
(12) 
The twodimensional Fourier integrals for then follow from (6) with (8)–(10).
To reduce representation (6) to onedimensional integrals, we resort to the cylindrical coordinates where and (), following Ref. KOW, . The cylindrical components of the field () include and . Accordingly, let with where and ; thus, and . By direct integration in , we find
as expected by symmetry because of the dipole orientation. For the remaining components, we invoke the known formula BatemanII
(13) 
for and , where is the Bessel function of th order, and .
Consequently, after some algebra, we find the following integral representations.
For (region ) with (,
(14a)  
(14b) 
(15) 
and for (region ),
(16a)  
(16b) 
(17) 
In the above, if and if ; and, by (11) and (12), and are functions of with
(18a)  
In accord with the Sommerfeld radiation condition, the top (physical) Riemann sheet in the foursheeted Riemann surface for the field components is fixed by imposition of  
(18b) 
for each ; thus, in this Riemann sheet. We note in passing that in the limit where and , integrals (14)–(17) approach expressions that are divergent in the conventional sense yet become meaningful as finite in the sense of Abel. Hardy ; Margetis01 This physically transparent interpretation permeates Section IV.
iii.2 Horizontal dipole
Next, we focus on the geometry of Fig. 2. In this case, all cylindrical field components are in principle nonzero. By (7), and satisfy KOW
where and is defined in (18), with admissible solutions
(19)  
(20) 
consistent with radiation condition (5). Note that here because, in view of condition (4b), the tangential component of the field is not continuous across , in contrast to the formulation of Ref. KOW, . The integration constants , , and are determined through boundary conditions (4). By transformed Maxwell equations (7), the remaining field components can be expressed in terms of and as
(21) 
(22) 
We proceed to compute the fields. By (4a) we impose at ; thus,
We henceforth set for ease of notation. By (4b), we enforce at which yields
Now introduce . Thus, it suffices to determine the integration constants and .
By (4), the continuity condition and the jump condition at lead to the system
After some algebra, the determinant for this system is found to be
where is defined by (12), and
(23) 
corresponds to TE polarization. Hanson08 It follows that
Accordingly, all field components are now obtained via (19)–(22) by use of Fourier integral (6). By representing the resulting twodimensional integrals in cylindrical coordinates and using (13), KOW we obtain the following expressions.
For (region 1):
(24a)  
(24b)  
(24c) 
(25a)  
(25b)  
(25c) 
and for (region ),
(26a)  
(26b)  
(26c) 
(27a)  
(27b)  
(27c) 
In the above, the factor
(28) 
is associated with the reflection of plane waves with TE polarization from a thin layer (the subscript “e” stands for “electric”, implying TE polarization); also, recall (11), (12) and (23) for , and , respectively. Similarly to the verticaldipole case (Section III.1), the top Riemann sheet is defined by , as in (18b).
iii.3 Poles in Riemann surface
We now discuss the role of singularities in evaluating integrals (14)–(17) and (24)–(27). The singularities of the integrands include: branch points at because of the multiplevalued of (18a); and simple poles identified with zeros of and by (12) and (23). The Riemann surface associated with each integrand consists of four Riemann sheets if ; the physically relevant (top) one is fixed by . The location of the poles in this Riemann surface is closely related to the complex surface conductivity, . Hanson08
iii.3.1 Denominator
By (12), the zeros of satisfy
(29) 
Equation (29) is identified with the dispersion relation for the surface plasmon in the context of TM polarization for plane waves Maierbook if is viewed as the component of the vector wave number tangential to the interface. For nearly lossless ambient media (), roots of (29) lie in the first Riemann sheet if
(30) 
For a graphene layer, this is consistent with the Kubo formula in the farinfrared regime. Hanson08 ; Falkovsky07 ; Gan12 ; Cheng13
Consider the special case with (Section IV) in which the Riemann surface for the fields has two sheets. Then, and (29) reduces to , which is solved in the first Riemann sheet if . Equation (29) has two solutions, , where
(31) 
the branch of the square root is chosen so that . These are simple poles of the corresponding integrands for the fields. In the “nonretarded frequency regime”, Cheng13 one imposes by which