chapter 1 scattering of electromagnetic surface waves on ... · incident surface wave at the edge...

Chapter 1

Scattering of electromagnetic surface waves onimperfectly conducting canonical bodies

Mikhail A. Lyalinov1 and Ning Yan Zhu2

An imperfectly conducting surface may support surface waves provided appropri-ate impedance boundary conditions (Leontovich conditions) are satisfied. Electro-magnetic surface waves propagate along an impedance surface and interact with itssingular points such as edges or conical vertices giving rise to the reflection andtransmission of such surface waves as well as to those diffracted into the space sur-rounding the canonical body. In this work we discuss a mathematical approach de-scribing some physical processes dealing with the diffraction of surface waves bycanonical singularities like wedges and cones. We develop amathematically jus-tified theory of such processes with the attention centred ondiffraction of a skewincident surface wave at the edge of an impedance wedge. Questions of excitation ofthe electromagnetic surface waves by a Hertzian dipole are also addressed as well asthe Geometrical Optics laws of reflection and transmission of a surface wave acrossthe edge of an impedance wedge, and possible type conversionof the transmittedsurface wave.

1.1 Introduction and survey of some known results

It is well known that in many situations imperfectly conducting surfaces may sup-port propagation of electromagnetic surface waves; see forinstance a recent tutorial[1]. From the theoretical point of view these waves are some special (asymptotic orexact) solutions of Maxwell’s equations satisfying appropriate boundary conditionsand exponentially vanishing as the observation point goes away from the surface.On the other hand, different technical problems encountered, e.g. in plasmonics orin the theory of antenna design, dealing with the scatteringof surface waves, requireefficient study of the corresponding wave processes. This study becomes much moredifficult when a surface wave interacts with different kindsof geometrical (edges,conical points) and (or) material (e.g. abrupt change of thesurface impedance) ir-

1Department of Mathematics and Mathematical Physics, Saint-Petersburg University, Universitetskayanab. 7/9, Saint Petersburg, 199034, Russia ([email protected], [email protected])2Institut für Hochfrequenztechnik, Universität Stuttgart, Pfaffenwaldring 47, D-70569 Stuttgart, Germany([email protected])

2 Advances of Mathematical Methods in Electromagnetics

✣

✾✒

✍

✮

❃

✍

✙✒

✻

③

incident wave

reflected

Σ

z+

z−

transmitted

Z

X

✙

✻

attributed wedge

1

Figure 1.1 Surface wave on a curved wedge

regularities on the supporting surface. Some approriate short survey on this subjectis given in this section. In this work we also discuss some of these problems and of-fer an adequate and mathematically justified theory of the corresponding scatteringphenomena.

Let us imagine that an electromagnetic surface wave propagates along a curvedsurfaceΣ (curved wedge, see Fig.1.1) on which the impedance-type boundary con-ditions are postulated. It is assumed that the surface may have an edge and, besides,the line of the edge is also the line of the jump of the surface impedance. The inci-dent surface wave approaches the edge, reflects from the edgeand transmits acrossit. In this process also the edge-diffracted wave arises. Provided the radii of curva-ture of the wedge’s faces are much greater than the wavelength, we can make use ofthe localisation principle. The reflection and transmission coefficients as well as thediffraction coefficient are then obviously specified by the local characteristics of thescattering surface and by the incident angle of the surface wave. So one may con-sider an attributed wedge with planar faces and their commonedge being tangentialto the primary ‘curved’ wedge at the point of diffraction.

There are some recent results describing interaction of incident surface waveswith geometrical (edges, tips) or (and) material irregularities of the surfaces with thecanonical shape (wedges, cones), see [2], [3], [4] (Chapter4), [5], [6]. It is also worthmentioning the works [7], [8], [9], [10], [11] and [12], where some other results anddifferent applications in plasmonics are considered.

In Reference 3

a recent advance in applications of the Sommerfeld-Malyuzhinets tech-nique to the problem of diffraction of a surface wave by an angular break ofa thin material slab is discussed. The solution is represented by Sommer-feld intergrals, which are then substituted into boundary conditions. Theunknown spectral functions satisfy coupled Malyuzhinets functional equa-tions. The latter are reduced to Fredholm integral equations of the secondkind, which are solved numerically. The scattering diagramof the cylindri-cal wave arising from the edge of the structure is computed.

Scattering of electromagnetic surface waves3

Reference 2 deals

with the scattering of an incident surface wave propagatingto the vertexof a circular impedance cone. Diffraction of the incident surface wave bythe vertex gives rise to the reflected surface wave as well to the spheri-cal wave from the vertex of the cone in the far-field approximation. Thestudy is based on the Sommerfeld integrals, Fourier transform and on theincomplete separation of the spherical variables for the problem at hand.The analytic formulae for the reflected surface wave and for the diffractioncoefficient of the spherical wave are obtained.

It is worth noting that some experimental results dealing with the diffraction of asurface wave by the conical tip are considred in the work [13]. Diffraction of aplane wave by a penetrable cone is discussed in [14]. The authors of the work [11]study surface waves on a conductive right-circular cone, however, their approach canhardly be justified from the mathematical point of view. Electromagnetic surfacewaves on the conical surface with Leontovich impedance boundary conditions areconsequently derived and discussed in Chapter 6 of [4].

We study the process of scattering of an incident surface wave at the edge of awedge. The corresponding recent extentions of the Sommerfeld-Malyuzhinets tech-nique given in [4] (Chapter 2), [15], [16] enable us to develop an analytical-numericalprocedure of calculations of the reflection (transmission)coefficients as well as thosefor the edge diffraction coefficients in the case of an incident surface wave. In theshort-wavelength approximation we may expect that such computed coefficients willsuccessfully serve in order to describe the scattering in the common case of a wedgewith curved faces. It is worth noting, however, that the problem of diffraction ofan electromagnetic plane wave, which is skew incident at theedge of an impedancewedge, can be also treated by use of the analytical-numerical technique based on thementioned extentions of the Sommerfeld-Malyuzhinets approach. The explicit (i.e.in quadratures) solution has been obtained only in some particular or degeneratedcases (see e.g. [17]–[18]).

We propose an efficient approach based on the reduction of theproblem to asystem of functional Malyuzhinets equations and their transformation to the integralones completing the study by the calculation of the asymptotics of the far field fromthe corresponding Sommerfeld representation. Moreover, we exploit the plane waveexpansion of the field from a Hertzian dipole located close toone of the impedancefaces of the wedge. Such a dipole induces surface waves propagating to the edgeof the wedge. We consequently develop the theory of the reflection and transmis-sion of such a wave as well as of the space edge wave. The expressions for thereflection/transmission and diffraction coefficients are also addressed.

1.1.1 Electromagnetic surface waves on impedance surfacesWe describe some traditional approaches dealing with analytical or asymptotic con-structions of the electromagnetic surface waves. In the next two sections we makeuse of [19] (see also [20], [21] and [22]) and describe the results in a form which isconvenient for the rest of this chapter.


1.1.1.1 Electromagnetic surface waves supported by planarimpedance surfaces

In this section we present some known elementary results on construction of exactsolutions of Maxwell’s equations. These solutions are localised in some vicinity ofthe supporting surfaceΣ.

Consider solutions of the time-harmonic Maxwell’s equations

curlE = iωµc

Z0 H ,

curl(Z0H) = − iωεc

E +Z0J0,(1.1)

whereJ0 is a current (J0 = 0, µ ,ε, the relative permeability and permittivity of themedium, are assumed to be constant in this section),c denotes the speed of light infree space andZ0 is the intrinsic impedance of free space. An electromagnetic waveis

Z0H (q1,q2,n, t) = Z0H(q1,q2,n)e−iωt = Z0H0e−iωt+i(k1q

1+k2q2+iγn) ,

E (q1,q2,n, t) = E(q1,q2,n)e−iωt = E0e−iωt+i(k1q1+k2q

2+iγn) ,

γ > 0, ω > 0,

(1.2)

q1,q2,q3 = n are the Cartesian coordinates,q3 = n> 0 is a half spaceΩ of R3 (seeFig 1.1), where the solution (1.2) of the homogeneous equations (1.1) withJ0 = 0 issought. The solution (1.2) has the form of a plane wave with unknown parametersω,k1,k2,γ in the phase and with constant amplitudesE0,Z0H0. The wave field de-scribed by (1.2) exponentially vanishes asn→ +∞. We look for the solution in theform (1.2) which satisfies Leontovich impedance boundary conditions

E− (n,E)n = η n× (Z0 H) (1.3)on the surfaceΣ defined by the equationq3 = 0, η is the (with respect toZ0) nor-malised surface impedance.

The complex wave vectorK is taken such thatK = (k,0, iγ)T (T means trans-pose), i.e. the Cartesian coordinate system is oriented so thatk1 = k, k2 = 0, k is thewave number. Making use of Maxwell’s equations, we arrive at

K×E0 =ωµc

Z0 H0 ,

−K×Z0H0 =ωεc

E0 .(1.4)

From the boundary condition (1.3) we find that

E01 = −η Z0H02, E02 = η Z0H01.

The linear system of equations can be considered inC6 as that for determinationof the eigenvectorsE0 = (E01,E02,E03)T, H0 = (H01,H02,H03)T corresponding tothe eigenvalueω. We multiply the first equation in (1.4) byE0 and take into account


thatK×E0 is orthogonal toE0 then(E0,H0)= 0. In view of the boundary conditionson Σ we haveE01H01+E02H02 = η−1(E01E02/Z0−E02E01/Z0) = 0. As a result,

E03H03 = 0.

We consider only two cases: 1.E03 = 0, H03 6= 0 and 2.E03 6= 0, H03 = 0becauseE03 = 0, H03 = 0 lead to a trivial solutionE0 = H0 = 0 . In case 1, wewrite the equations in (1.4) in the coordinate form

AH01 = AH02 = 0

BH02 = −kH02 = 0

BH01 = kH03

ωµc

H03 = kη H01

withA= ω

µc+ iγη , B= ω

εηc

+ iγ .

From the latter system it is obvious thatH02 = 0 andA= 0 so thatω = − iγηcµ .Becauseω andγ are positive, we conclude that

η = i|η |.

We construct a non-trivial solution so that from the last twoequations of the systemthe expression forω is determined in an explicit form by means of

cηµω

k2 = B= ωεηc

+ iγ

and iγ =− µωcη . As a result, we arrive at

ω =c|η |k

√

εµ(|η |2+µ/ε). (1.5)

It is worth saying that the phase and group velocities of sucha wave coincide

vph =

(

c|η |√

εµ(|η |2+µ/ε),0,0

)T

,

vg =(

∂ω∂k

,0,0

)T

=

(

c|η |√

εµ(|η |2+µ/ε),0,0

)T

.

After elementary calculations we find

E0 =C∗(0,ηk,0)T, Z0 H0 =C∗(

k,0,iωc(ε |η |+µ/|η |)

)T

,


whereC∗ is an arbitrary complex constant. The constructed solutioncan be naturallycalledmagneticsurface wave.

In a similar manner we can construct anelectric surface wave asη = −i|η |.Thus we have

vph = vg ==

(

c√

εµ(|η |2ε/µ +1),0,0

)T

,

E0 =C∗(

k,0,iωc(ε |η |+µ/|η |)

)T

, Z0 H0 =C∗(0,−k/η ,0)T ,

γ =εω|η |

c, ω =

ck√

εµ(|η |2ε/µ +1).

A propagating surface wave is localised near a surface with apurely imaginary sur-face impedance. The sign of the imaginary part of the surfaceimpedanceη specifiesthe type of the propagating surface wave, i.e.electric or magnetic. In some sit-uations, e.g. for an axially anisotropic surface impedance(see [4], Chapter 2), thesurface waves of both types can be excited simultaneously.

Remark 1: It should be mentioned, however, that sometimes in applications theimpedances with small positive real parts are also considered for some surfaces withabsorption. In this case, the corresponding formal solution should be appropriatelyinterpreted. Traditionally this interpretation also means that such a surface waverapidly attenuates in the direction of propagation.

In the following section we consider generalisation of the solutions obtainedonto the case of a curved supporting surface, inhomogeneousimpedance and elec-tromagnetic constants of the medium in the short-wavelength approximation.

1.1.1.2 Electromagnetic surface waves on a curved surface withvarying surface impedance in an inhomogeneous medium

Consider a regularly curved surfaceΣ in R3 with its parametric equationrs= rs(q1,q2),Fig. 1.2. In a vicinity ofΣ we make use of the coordinatesq1,q2,q3 = n, wheren ismeasured along the normaln to Σ pointing into the domain with the medium havingthe electromagnetic properties described byε ,µ which depend on the coordinatesq1,q2,q3 = n. The surface impedanceη in the boundary conditions

E1 = −η Z0H2, E2 = η Z0H1 on Σ

may also depend on the coordinatesq1,q2.It is assumed that the wavelengthl of a harmonic wave is much less than the

characteristic radiusR0 of curvature ofΣ as well as of characteristic length of thevariations ofε ,µ andη so thatkR0 ≫ 1, k= 2π/l ∼ ω/c. In what follows we implythat the coordinates are normalised byR0 and we useqi in place ofqi/R0 with thenotionk for the large parameter (k≫ 1) in place ofkR0.


✶✿

✿

✿✻

✯

z+

Σ

❪

q1

~nq2

surface rays

1

Figure 1.2 Surface wave on a curved surface

The asymptotic (ray) solution of Maxwell’s equations (1.1)satisfying the bound-ary conditions and(E0,Z0H0)→ 0 asν = kn→+∞ is found in the form

Z0H (q1,q2,n, t) = e−iωt+ikΘ(q

1,q2)Z0H0(q1,q2,ν)(

1+ O

(

1k

))

E (q1,q2,n, t) = e−iωt+ikΘ(q1,q2) E0(q1,q2,ν)

(

1 + O

(

1k

))

,

(1.6)

whereν = knand

Z0H0 = e−νλ (q1,q2)A0(q1,q2)h0,

E0 = e−νλ (q1,q2)A0(q1,q2)e0,

(1.7)

with

h0 = ∇Θ+niωc(ε |η |+µ/|η |), e0 = η n×∇Θ, λ =

µ|η |

for the magnetic surface wave,

e0 = ∇Θ+niωc(ε |η |+µ/|η |), h0 = −η−1n×∇Θ, λ = |η |ε

for the electric surface wave.In the expressions (1.6) the surface eikonal solves the equation

cs|∇Θ| = c, (1.8)where|∇Θ|=

√

gi j ΘiΘ j , gi j are the coefficients of the first quadratic form ofΣ, gi jis the inverse to the matrixgi j . It is worth mentioning that in (1.8) we imply that

cs = cms , c

ms =

c|η |√

εµ(|η |2+µ/ε)for magnetic wave and

cs = ces , c

es =

c√

εµ(|η |2ε/µ +1)


for the electric surface wave. The eikonal equation (1.8) issolved by use of the classi-cal method of characteristics for the Hamilton-Jacoby equation [22] which specifiesthe dependence of the ray coordinatesa,sonΣ with thoseq1,q2; q1 = q1(a,s), q2 =q2(a,s) with non-degenerate JacobianJ (spreading).

The complex amplitudeA0(q1,q2) is given by the expression

A0(q1,q2) = |A0(q1,q2)|eiV0(q

1,q2)

with

|A0(q1,q2)|=C(a)√

λE(e0,h0)J

, E(e0,h0) =ε |e0|2+µ |h0|2

8π.

ThegeometricalphaseV0(q1,q2) takes the form

V0 =V0|s=0 +s∫

0

ds

(

14λ 2

∂ε∂n |n=0|e0|2+

∂ µ∂n |n=0|h0|2

ε0|e0|2+µ0|h0|2+

c2λ 2(ε0|e0|2+µ0|h0|2)

(2MRe(∇Θ× e0 ·h0)+2Re(B∇Θ×h0 · e0)+

Re[∇Θ×Bh0 · e0 − ∇Θ×Be0 ·h0] )) ,

whereM is the mean curvature,B is a tensor defined by(Be) j = g jkbkiei , bki are thecontravariant components of the second quadratic form ofΣ. Remark thatC(a) andV0|s=0 are specified from some ‘initial’ data for the surface wave.Remark 2: In the latter integral for the geometrical phase the summands in theintegrand describe the influence of different geometrical and material charateristicsof the surface and of the medium on the propagating surface wave. It is worth men-tioning that the geometrical phase frequently arises in theasymptotic analysis of thewave phenomena and in the quantum theory.

1.1.2 Electromagnetic surface waves on a right circular conicalsurface

Let us consider, in this section, the spherical coordinate system (r,θ ,ϕ) connectedwith the axisX3 (Fig 1.3) of the cone having the origin at the vertex of the cone. Letη = sinζu be the surface impedance with Imζu < 0, η−1 = sinζv. For the conicalsurface we use the equationθ = θ1. We notice that under the sufficient conditions

θ −θ1−Reζu−gd(Imζu)> 0, Imζu < 0, Reζu = 0,

where gd(x) stands for the Gudermann function, from the integral representation ofthe wave field ([4], Chapter 6) the expressions for electromagnetic surface wave canbe derived. In these conditions the electrical type surfacewaves are excited providedthe incident electromagnetic plane wave interacts with theconical surface [4].


✇

✻X3

☛θ1

O

incident plane wave

✢✎ ❲

❯✾

surface waves

Figure 1.3 Surface wave on a right circular impedance cone

After some calculations, in the leading approximation we obtain

Eswr (kr,θ ,ϕ) = ikTu(ϕ,θ) sin2 (θ1−θ +ζu)exp[ikr cos(θ1−θ +ζu)]

(−ikr)1/2+λu

×[

1+O

(

1kr

)]

,

Eswθ (kr,θ ,ϕ) =−ikTu(ϕ,θ)sin(θ1−θ +ζu)cos(θ1−θ+ζu)

× exp[ikr cos(θ1−θ+ζu)](−ikr)1/2+λu

[

1+O

(

1kr

)]

,

Hswϕ (kr,θ ,ϕ) = −ikTu(ϕ,θ) sin(θ1−θ +ζu)exp[ikr cos(θ1−θ +ζu)]

(−ikr)1/2+λu

×[

1+O

(

1kr

)]

,

(1.9)

with

Tu(ϕ,θ) =C0,u(ϕ)

π i

(

e2π iµ −1)

Γ(µ)[−sin(θ1−θ +ζ )]µ

√

sinθ1sinθ

.

with

C0u(ϕ) =∞

∑n=−∞

ine−inϕC0u(n) .

andλu =−(1/2)cotθ1 tanζu, µ = 1+λu. A procedure of derivation of the constantsC0u(n) in the excitation coefficients of the surface waves is described in Chapter 6 of[4].

In accordance with the introduced terminology it is naturalto call the wave (1.9)the electrical-type surface wave, which is excited, in particular, provided Imζu <0, Imζv > 0. Contrary to this case, provided Imζv < 0, Imζu > 0, the magnetic-typesurface wave propagates along the impedance cone.


From (1.9) we obtain some simple physical conclusions. The surface wave prop-agates with the velocityc/cosh(|ζu|), that is, slower than the spherical wave fromthe vertex having the velocityc. On the conical surface its amplitude decreases asO(1/(kr)1/2+λu). The factor(kr)−λu written in the form exp[−λu log(kr)] enables usto interpret it as being responsible for the geometrical phaseΦg(kr) = iλu log(kr).For the surface waves on an impedance cone the geometrical phase is specified bythe mean curvature of the circular cone, which also follows from [19] and [21].

1.2 Excitation of an electromagnetic surface wave by a dipolelocated near a plane impedance surface

In this section we briefly describe a way in order to find an expression for the surfacewave and its excitation coefficient, provided that this waveis generated by the currentJ0 = Pδ (x− x0)δ (y− y0)δ (z) in (1.1) of a Hertzian dipole located near a planesurfacex = 0 with impedanceη+, see Fig. 1.4. The dipole is arbitrarily oriented,P = (Px,Py,Pz)T. The cylindrical coordinates (r,ϕ,z) are introduced forx ≥ 0, Fig.1.4,x0 = r0cosϕ0, y0 = r sinϕ0, z0 = 0. As discussed in [4], Chapter 3, [5], in orderto asymptotically evaluate the integrals representing thesolution we deformed thecontours, took into account crossed polar singularities and applied steepest descent(or other) techniques. However, provided, say,ϕ0∼Φ, some additional contributionsto the far field come into being. In this case, the point sourceis close to the wedge’sfaceϕ = Φ then the factor exp(−ikr0cos[Φ−ϕ0− ζ+(β )]) in (3.53) of [4] is notsmall even, ifkr0 ≫ 1, and the corresponding primary surface wave is excited by thedipole.

This integral representation is determined from the doubleintegral (see 3.42 in[4]) representing the reflected wave from the impedance plane surface with the dipolebeing close to it. As is well known the surface wave generatedby the Hertzian dipoleover the surface with the impedance sinζ+ is due to the contribution of a pole of thereflection coefficientR

+(α,β ) which is captured in the process of deformation of the

contour into the SD path. These polar singularities are due to zeros of denominatorsD±(−α,β ) = [sin(−α±Φ)±sinθ±(β )][sin(−α±Φ)±sinχ±(β )] of the reflectioncoefficientR

+(α,β ). We consider the polar singularity

α∗(β ) = Φ+ζ+(β )

(with ζ+ = {θ+(β ),χ+(β )}).We assume thatϕ0 ∼ Φ and eitherζ+ = θ+ with Imθ+ < 0 or ζ+ = χ+ with

Imχ+ < 0 recalling thatζ± = {θ±,χ±}. Notice that by definition (see [4], Sect.3.3.2)

sinζ+(β ) =z+

sinβ,

wherez±= {η±,(η±)−1}. We assumeη±= i Im(η±) purely imaginary,| Im(η±)|=|η±| which implies that surface waves can actually propagate along the impedancesurface provided that either Im(η±)< 0 or Im(η±)−1 < 0.


✻

✙

ss Z

X

Y

~PO

η+

✿❄

θ0

surface wave

dipole

1

Figure 1.4 Surface wave from a dipole over an impedance plane

The expression for the surface wave on an impedance surface excited by a dipoletakes on the form (see also Sect. 3.5.1 in [4] )

[

Z0HswzEswz

]

z+= − k

3

4π

∫

Γ(π/2)

dβ resα∗(β )R+(α,β ) · U0(α∗(β ),β ) sin2β ×

eik{zcosβ+sinβ [r0 cos(α∗(β )−ϕ0)−r coscos(α∗(β )−2Φ+ϕ)]} ,

α∗(β ) = Φ+ζ+(β ). Here the reflection coefficientR+(α,β ) is explicitly given by

(3.40) from [4],

U0(α,β ) = [U10(α,β ), U20(α,β )]T ,

U10(α,β ) = sinαPx−cosαPy,

U20(α,β ) = cosβ (cosαPx+sinαPy)+sinβPz.

(1.10)

After the change of the variableτ = cosβ we find[

Z0HswzEswz

]

z+= − k

3

4πeik{−r0 sin(Φ−ϕ0)z

+−r sin(ϕ−Φ)z+}

∫

R′

dτ F p+(τ)eikρ0

{

zτ/ρ0+√

1+|z+|2−τ2}

,

withρ0 = r0cos[Φ−ϕ0]− r cos[Φ−ϕ] > 0

andF

p+(τ)|τ=cosβ = − sinβ resα∗(β )R

+(α,β ) ·U0(α∗(β ),β ) .

The contourR′ goes along the real axis comprising the branch points±√

1+ |z+|2from below(+) and above(−), the branch cuts are conducted from±

√

1+ |z+|2 to±∞ and

√

1+ |z+|2− τ2 > 0 asτ = 0.


The stationary point of the latter integral, askρ0 ≫ 1,

τp =z

ρ0

√

1+ |z+|2√

1+(z/ρ0)2

solves the equation

z= [r0cos(Φ−ϕ0)− r cos(Φ−ϕ)]τp

√

1+ |z+|2− τ2p,

and the asymptotic expression for the leading term is written as

[

Z0HswzEswz

]

z+=

k3ei3π/4

4π2

√

2πkρ0

ρ3/40 (1+ |z+|2)1/4(z2+ρ20)3/4

Fp+(τp)× (1.11)

×eik√

1+|z+|2√

z2+ρ20 eik{−r0 sin(Φ−ϕ0)z+−r sin(Φ−ϕ)z+}×

(

1+O

(

1kρ0

))

remarking thatr0sin(Φ−ϕ0)+ r sin(Φ−ϕ)≥ 0.It is convenient to introduce the angleβ0 by the equality

cosβ0 =z

√

ρ20 +z2< 1.

We assume thatr0cos(Φ−ϕ0) ≫ r cos(Φ−ϕ) and also define the angle of ‘inci-dence’θ0 by

cosθ0(β0) =√

1+ |z+|2cosβ0.

As a result, the expression (1.11) can be also written in the form

[

Z0HswzEswz

]

z+= A (r0,ϕ0,θ0)eik[zcosθ0− r sin(Φ−ϕ)z

+− r cos(Φ−ϕ)√

sin2 θ0+|z+|2]

with the complex amplitude

A (r0,ϕ0,θ0) =k3ei3π/4

4π2

√

2πkρ0

ρ3/40 (1+ |z+|2)1/4(z2+ρ20)3/4

Fp+(cosθ0)×

e−ik[ r0 sin(Φ−ϕ0)z++ r0 cos(Φ−ϕ0)

√sin2 θ0+|z+|2] .

It is worth remarking that in our assumptions we haveρ0 ≈ r0cos(Φ−ϕ0) and

ρ0(z2+ρ20)1/2

= sinβ0.


As a result, the complex amplitudeA (r0,ϕ0,θ0) of the excited surface wave is spec-ified by r0,ϕ0,θ0,z+, i.e. by the position and orientation of the Hertzian dipole. It isalso useful to introduce the complex angle

ϕ0(θ0) = Φ−ζ+(θ0),

where

sinζ+(θ0) =z+

sinθ0.

The surface wave excited by the dipole is then written as[

Z0HswzEswz

]

ζ+= A eik[zcosθ0− r sinθ0 cos(ϕ0(θ0)−ϕ)] (1.12)

with sinθ0cos(ϕ0(θ0)−ϕ)= sin(Φ−ϕ)z+ + cos(Φ−ϕ)√

sin2 θ0+ |z+|2. The wave(1.12) can be interpreted as a (with respect toOZ) skew incident plane wave with theangles(θ0,ϕ0(θ0)) (actuallyϕ0 is complex) specifying the direction of incidence. Inthe next section we make use of this simple observation and study the problem ofscattering of a surface wave (1.12) by an impedance wedge (see also [15], [16]).

1.3 Scattering of a skew incident surface wave by the edge on animpedance wedge

The impedance wedge under study is most conveniently described in a cylindricalcoordinate system(r,ϕ,z), with its edge coinciding with thez-axis and its upper andlower faces being the half-planesϕ =±Φ (Fig. 1.5). The boundary conditions to bemet by the electromagnetic field components turn out from (1.3)

Ez(r,±Φ,z) =±Z0η±Hr(r,±Φ,z), Er(r,±Φ,z) =∓Z0η±Hz(r,±Φ,z).

Now assume that the upper face of the wedge is purely inductive, i.e. η+ =−i |Imη+|. Let an E-mode electromagnetic surface wave of the type (1.12), but witha constant amplitudeU0, move on the upper face of the wedge towards its edge underthe angles(ϑ0,ϕ0)

[

Z0H inczEincz

]

=U0ei[k′′z−k′r cos(ϕ−ϕ0)], k′ = ksinϑ0, k′′ = kcosϑ0.

with U0 = [U10 U20]T = [−Z0H0sinβ0 −η+Z0H0cosβ0]T andβ0 being the angle

subtended by the wedge’s edge and the direction of propagation. Taking this direc-tion as the positiveq1-axis of the Cartesian coordinates(q1,q2,q3) for the upper faceof the wedge (see Sect. 1.1.1.1), then the magnetic field of the incident surface wavein that coordinate system reads

Z0Hinc = (0,Z0H0,0)T eik(n+q1−q3η+), n+ =

√

1− (η+)2.


Being the ratio of the speed of light to that of the surface wave, n+ is termed therefractive index of the surface wave supported by the upper face of the wedge. Therespective electric field can be given as in Sect. 1.1.1.1.

The incident angles(ϑ0,ϕ0) depend uponβ0 andη+ (via n+)

ϑ0 = arccos(n+ cosβ0), ϕ0 = Φ−ϑ+, ϑ± = arcsin(η±/sinϑ0). (1.13)In this section,ϑ0 is assumed at first to be real, limitingβ0 to |cosβ0| < 1/n+. Thelast subsection tackles the case with|cosβ0|> 1/n+.

Figure 1.5 Diffraction of a surface wave at an impedance wedge

Next we make use of an early work [15]. To this end, we considerthe z-components of the total field

[Z0Hz(r,ϕ;z) Ez(r,ϕ;z)]T =U(r,ϕ)exp(

ik′′z)

. (1.14)

U(r,ϕ) = [U1(r,ϕ)U2(r,ϕ)]T solves the two-dimensional Helmholtz equation in freespace outside the wedge

[

1r

∂∂ r

(

r∂∂ r

)

+1r2

∂ 2

∂ϕ2+(k′)2

]

U(r,ϕ) = 0,

and satisfies the respective conditions on the faces of the wedge

Ii∂U(r,ϕ)

kr∂ϕ

∣

∣

∣

∣

ϕ=±Φ=∓sin2 ϑ0A

±U(r,±Φ)+cosϑ0B

i∂U(r,±Φ)k∂ r

, (1.15)

with

I =

[

1 00 1

]

, A±=

[

η± 00 1/η±

]

, B =

[

0 −11 0

]

.

In line with the Meixner edge condition,U(r,ϕ) remains finite asr → 0

U(r,ϕ) =[

C1+O(rδ ) C2+O(r

δ )]T

, δ > 0,

C1,2 being constant. Furthermore, it is subject to the radiationconditions, given mostconcisely for the spectra ofU(r,ϕ) in Sect. 1.3.1.


1.3.1 Integral Equations for the SpectraAs usual,U(r,ϕ) is expressed in terms of the Sommerfeld integrals:

U(r,ϕ) =1

2π i

∫

γf (α +ϕ)e−ik

′r cosαdα, (1.16)

whereγ denotes the Sommerfeld double-loop andf (α) = [ f1(α) f2(α)]T the spectrato be determined. The radiation condition demands thatf (α)−U0/(α −ϕ0) beregular in the strip|Reα| ≤ Φ, whereϕ0 is defined in (1.13).

Inserting (1.16) into the boundary condition (1.15) and inverting the Sommer-feld integrals, we get a system of equations for the spectra.For example, the equationfor f1(α) reads

f1(α +2Φ)−b+2 (α)b−2 (α

f1(α −2Φ) = q1(α) f1(α), (1.17)

with the coefficientsb+2 (α),b−2 (α) andq1(α) given in Sect. 1.5.

On use off1(α) =F0(α)F1(α), the above functional equation can be simplifiedto

F1(α +2Φ)+F1(α −2Φ) = Q1(α)F1(−α), (1.18)with Q1(α) = q1(α)F0(−α)/F0(α +2Φ) and the auxiliary functionF0(α) given inclosed form in Sect. 1.5.

By making use of the S-integrals and taking into account the edge and radiationconditions, an integral equivalent of (1.18) in the strip|Reα| ≤ 2Φ for Φ > π/2reads

F1(α) =νU10/F0(ϕ0)sinν(α −ϕ0)

+A+1 e−iνα +A−1 e

iνα

− i8Φ

∫ +i∞

−i∞

Q1(−t)F1(t)cosν(α + t)

dt, ν =π

4Φ. (1.19)

The constantsA±1 are fixed by deleting non-physical poles

f1(±Φ−π/2) = b±1 (∓Φ−π/2) f1(±Φ+π/2), (1.20)The coefficientsb±1 (α) are given in Sect. 1.5.

Relation (1.19), together with (1.20), amounts to an integral equation forF1(α)on the imaginary axis of the complexα-plane. These values can be obtained by solv-ing numerically the integral equation and then extrapolated into the strip|Reα| ≤ 2Φon use of (1.19). In a similar manner, the second spectrumf2(α) can be deduced. In-serting them into the Sommerfeld integrals (1.16) leads to an exact solution, althoughnot in an explicit form, to the problem under study.

1.3.2 Far-Field ExpansionDeforming the path of integrationγ in (1.16),U(r,ϕ) can be rewritten as

U(r,ϕ) =Ugo(r,ϕ)+Usw(r,ϕ)+Ud(r,ϕ). (1.21)


The geometrical-optical part is related to the incident surface wave according to

Ugo(r,ϕ) = H(ϕ −Φ+π −gd(Imϕ0))U0e−ik

′r cos(ϕ−ϕ0),

where H(·) stands for the Heaviside uni-step function.Of particular importance are the surface waves

Usw(r,ϕ) =

4

∑ℓ=1

H(Aℓ)Rℓe−ik′r cosαℓ , (1.22)

where the poles and residues off (α +ϕ) related to surface waves are given in Table1.1. And gd(x) = arctan(sinhx) stands for the Gudermann function.

Table 1.1 Surface-wave poles and residues off (α +ϕ)

ℓ Poles αℓ Residues Rℓ Arguments Aℓ

1 π +Φ+ϑ+−ϕ R+ϑ+ · f (Φ−π −ϑ+) −Φ+ϕ −Reϑ+−gd(Imϑ+)2 −π −Φ−ϑ−−ϕ R+ϑ− · f (−Φ+π +ϑ−) −Φ−ϕ −Reϑ−−gd(Imϑ−)3 π +Φ+χ+−ϕ R+χ+ · f (Φ−π −χ+) −Φ+ϕ −Reχ+−gd(Im χ+)4 −π −Φ−χ−−ϕ R+χ− · f (−Φ+π +χ−) −Φ−ϕ −Reχ−−gd(Im χ−)

For largek′r, the diffracted partUd(r,ϕ) is given by

Ud(r,ϕ)∼ Q(ϕ)eik′r/

√r (1.23)

with the non-uniform diffraction coefficient (scattering diagram)

Q(ϕ) =[

f (ϕ −π)− f (ϕ +π)]√

i/(2πk′). (1.24)

A uniform expression for the diffracted field can be given in asimilar way as in [15].Obviously, for the edge-diffracted rays the components of the wave vector are

Kz = kcosϑ0, Kr = ksinϑ0,

implying that in the far field, these waves are located on a cone whose axis is theedge of the wedge and whose interior semivertex angle isϑ0. Therefore, the lawof edge-diffraction excited by an incident surface wave at the edge of an impedancewedge is given by the first of (1.13) rewritten as

sinκd = n+ sinκ+, κd = π/2−ϑ0, κ+ = π/2−β0. (1.25)The similarity of the above relation to the Fresnel law of refraction for wave transmis-sion through an interface between two different media is dueto the edge-diffractedrays being excited by a slower incident surface wave. As a consequence, there existsa critical angle corresponding toκd = π/2 (henceϑ0 = 0)

κ+c = arcsin(1/n+)


beyond which, that is forκ+ > κ+c the ‘edge-diffracted’ rays propagate along thez-axis, being localised in a neighbourhood of the edge; see Subsect. 1.3.4.

As the surface waves given in (1.22) are excited by the incident surface wave,they can be regarded as the reflection and refraction of the latter at the edge of animpedance wedge.

1.3.3 Reflection and refraction of an incident surface wave attheedge of an impedance wedge

In close connection with the reflection and refraction of waves are their respectiveangles. The explicitly given polesαℓ (see Table 1.1) are very useful in this respect.On the upper face, the angle of incidence with respect to the normal to the edge isκ+ = π/2−β0, and the corresponding components of the wave vector are

Kz = kcosϑ0, Kr =−ksinϑ0√

1− (η+/sinϑ0)2.

For the excited surface wave on the upper face, we get from (1.22) that

Kz = kcosϑ0, Kr = ksinϑ0√

1− (η+/sinϑ0)2,

implying that the angle of reflectionκ+r equals that of incidenceκ+, namely

κ+r = κ+. (1.26)

Similarly, for the surface wave excited on the lower face of the wedge, the re-spective components of the wave vector are

Kz = kcosϑ0, Kr = ksinϑ0√

1− (z−/sinϑ0)2,

wherez± = {η±,1/η±} with Imz± < 0. Hence, the angle of refraction is definedaccording to

sinκ− =Kz

√

K2z +K2r=

cosϑ0n−

.

Here,n− =√

1− (z−)2 stands for the refractive index for the surface wave supportedby the lower face of the wedge.

On use of (1.13), we arrive at a formula

n+ sinκ+ = n− sinκ−, (1.27)

which reminds us of the Fresnel law of refraction for wave transmission through asmooth interface between two different media. In spite of this similarity, we shouldnot forget that in the present case withκ+ < κ+c , one more wave is born at the edge,namely the diffracted space-waveU

dgiven in (1.23).

Another aspect of the study concerns the type of reflected andrefracted surfacewaves. As can be expected, the reflected surface wave is of thesame type as thatof the incident surface wave; in this example also an E-type wave. The type of


the refracted part is determined by the electrical propertyof the lower face of thewedge, as implied in the above study of the law of refraction:if the lower face is alsoinductive with Imη− < 0, the refracted surface wave is of the same E-type; if thelower face is but capacitive (Imη− > 0), then the refracted part is of the other type,here the H-type. Obviously, such a conversion of the type of electromagnetic surfacewaves happens only under skew incidence, as shown by the explicit expressions of

R±ϑ±,χ±

R±ϑ± =±

2tanϑ±

sinϑ±−sinχ±[

−(csc2 ϑ0sinϑ±−sinχ±) ±cotϑ0cscϑ0cosϑ±∓cotϑ0cscϑ0cosϑ± −cot2 ϑ0sinϑ±

]

,

R±χ± =±

2tanχ±

sinχ±−sinϑ±[

−cot2 ϑ0sinχ± ±cotϑ0cscϑ0cosχ±∓cotϑ0cscϑ0cosχ± −(csc2 ϑ0sinχ±−sinϑ±)

]

.

Figure 1.6 Diffraction of an E-type surface wave at an impedance wedge with aninductive upper face and a capacitive lower face

Now let us look at a numerical example shown in Fig. 1.6. This figure displays|Z0Hz| of the different wave ingredients as well as the total field excited by an in-cident electromagnetic surface wave of the E-type by an impedance wedge with aninductive upper face and a capacitive lower face. Hence, thesurface wave propa-gating away from the edge along the lower face of the wedge is of the H-type. Itis noted that the reflection and refraction coefficients are calculated numerically, by


solving at first the Fredholm integral equations of the second kind and then insertingthe spectraf (α) in the residuesRℓ, as shown in Table 1.1.

1.3.4 Beyond the critical angle of edge diffractionNow consider incidence of a surface wave under the conditionκ+c < κ+ < π/2 with

κ+c = arcsin(1/n+), κ+ = π/2−β0.

In line with (1.13),ϑ0 becomes purely imaginary with

ϑ0 = iarccosh(n+ sinκ+). (1.28)

Therefore we get

ϑ+ = −π2− iarccosh

(

|η+|√

(n+ sinκ+)2−1

)

,

ϕ0 = Φ−ϑ+.

The Sommerfeld integral (1.16) still expressesU(r,ϕ). But for the sake of con-vergence, the contour of integration runs now along(π + i∞,π + iδ ]∪ [π + iδ ,−π +iδ ]∪ [−π + iδ ,−π + i∞) and its mirror image with respect to the origin of the com-plex α-plane. The positive constantδ is chosen in such a way that the two loops ofintegration contain no singularities of the spectra.

Even without determining the spectra, a formal asymptotic analysis of the Som-merfeld integral (1.16) affords useful insights into the related wave phenomena. Un-der this circumstance, the saddle points remain atα =∓π with the steepest-descentpaths (SDP) given by

SDP(−π) : (−π − i∞,−π + i∞),SDP(π) : (π + i∞,π − i∞).

As a result, the formulae derived in Sect. 1.3.2 remain valid, except that theGudermann functions appearing in the argumentsAℓ in Table 1.1 are set to zero.

Remark 3: In case of a complex-valued k′, the Gudermann functiongd(x) is to bereplaced by its generalisationGd(x,argk′) given by (see for example [6])

Gd(x,y) = arctansinhx cosy

1+coshx siny,

with the two special cases used in this section

Gd(x,y= 0) = gd(x), Gd(x,y= π/2) = 0.

Hence, the steepest-descent paths for the Sommerfeld integral (1.16) in the generalcase are

SDPargk′(∓π) : Reα =∓π −Gd(Imα,argk′).


According to (1.23), the ‘edge-diffracted’ componentUd(r,ϕ) decreases expo-

nentially away from the edge withr, leading to

[

Z0Hdz (r,ϕ;z) Edz (r,ϕ;z

]T=U

d(r,ϕ)eikzcosϑ0 ∼ Q(ϕ)e

k(izcosϑ0−r sinh|ϑ0|)√

r,

being a wave clung to and propagating along the edge. Furthermore, the larger theangle of incidenceκ+, the stronger is the concentration of this wave ingredient tothe edge; see the relationship betweenϑ0 andκ+ (1.28).

On the lower face of the wedge, the Brewster angleζ− is given by

ζ− =

−π2 − iarccosh|z−|√

(n+ sinκ+)2−1, for κ+c < κ+ < κ+t.r.,

−arcsin |z−|√

(n+ sinκ+)2−1, for κ+t.r. < κ+ < π2 ,

in case of|z−|< |z+| withκ+t.r. = arcsin

n−

n+

and

ζ− =−π2− iarccosh |z

−|√

(n+ sinκ+)2−1, for κ+c < κ+ <

π2

otherwise. (See Sect. 1.5 for the second Brewster angles of the upper and lower faceof the wedge.)

It is worth taking a close look at the case|z−| < |z+|. For κ+c < κ+ < κ+t.r., thelaw of the refraction for surface waves at an edge (1.27) holds good. If the angleof incidenceκ+ exceedsκ+t.r., the ‘transmitted’ surface wave at the lower face of thewedge behaves in line with (1.22):

ek{izcosϑ0−[r cos(Φ+ϕ)sinh|ϑ0|cosζ−+r sin(Φ+ϕ)|z−|]},

again clung to and moving along the edge! It implies that the incident surface waveis completely reflected at the edge, and the very angleκ+t.r. is called the angle of totalreflection for an incident surface wave.

The geometrical properties of an incident surface wave at the edge of an impedancewedge, like the laws of reflection (1.26) and refraction (1.27) and possible total re-flection, have been known for scalar waves since 1965 [8]. Theconversion of surfacewaves of the E-type to those of the H-type and vice versa at theedge of an impedancewedge, however, seems to be unique for vectorial waves such as electromagneticwaves studied in this chapter.

1.4 Conclusion

In this chapter we made use of the concept of the Leontovich (impedance) bound-ary conditions and discussed excitation and propagation ofsurface waves as well as


their interaction with some canonical singularities of thesurface like edges or con-ical points. Although the validity of the impedance boundary conditions fails in aclose neighbourhood of the singular points, nevertheless these conditions are widelyapplicable in practice and the corresponding applicationsgive reliable and accurateresults.

An adequate use of the mathematical methods enabled us to give a motivateddescription of the wave phenomena arising in the process of propagation and scat-tering of the surface waves supported by impedance surfaces. In this way we couldefficiently describe scattering of an incident surface waveat the edge, calculate am-plitude and phases of the reflected, transmitted and diffracted waves. In a simplemanner the Geometrical Optics laws of reflection and transmission of the surfacewave at the edge of the wedge are also deduced as well as some analysis of the typeconversion of the transmitted surface wave is given.

These results have been obtained for the angle of incidence of the surface wavewhich is less than the first critical angle define above. However, the study of thereflection and transmission coefficient for the other anglesof incidence of the surfacewave is being carried out and its results will be given elsewhere.

1.5 Appendix

To make this Chapter self-sustained, several functions used in Sect. 1.3 are explicitlygiven below.

b±1 (α) =−2cot2 ϑ0sin2(α ±Φ)− [sin(α ±Φ)∓sinϑ±][sin(α ±Φ)±sinχ±]

[sin(α ±Φ)±sinϑ±][sin(α ±Φ)±sinχ±] ,

b±2 (α) =2cotϑ0cscϑ0sin(α ±Φ)cos(α ±Φ)

[sin(α ±Φ)±sinϑ±][sin(α ±Φ)±sinχ±] ,

q1(α) = b+1 (α)−b+2 (α)b−2 (α)

b−1 (α).

F0(α) =Ψ0(α)

sin[ν(α −Φ−π/2)]sin[ν(α +Φ+π/2)] , ν =π

4Φ,

Ψ0(α) =χΦ(α +Φ−π)χΦ(α +Φ)χΦ(α +Φ+π/2)χΦ(α −Φ+π)χΦ(α −Φ)χΦ(α −Φ−π/2)

×χΦ(α +Φ−π/2)χΦ(α −Φ− χ−+π)χΦ(α −Φ+ χ−)

χΦ(α −Φ+π/2)χΦ(α +Φ+ χ+−π)χΦ(α +Φ− χ+)

×χΦ(α −Φ−ϑ−+π)χΦ(α −Φ+ϑ−)

χΦ(α +Φ+ϑ+−π)χΦ(α +Φ−ϑ+),

whereχΦ(α) stands for a special function introduced by Bobrovnikov andis definedby the first-order functional difference equation [23]

χΦ(α +2Φ) = cos(α/2)χΦ(α −2Φ).

22 Book

More on this special function, and especially its efficient computation, can be foundfor instance in [4].

1.5.1 Brewster anglesThe second Brewster angle for the upper faceχ+ may be either complex-valued orpurely real, depending upon|η+| andκ+. In case of|η+|> 1, we have

χ+ =

π2+ iarccosh

1

|η+|√

(n+ sinκ+)2−1, for κ+c < κ+ < arcsin 1|η+| ,

arcsin1

|η+|√

(n+ sinκ+)2−1, for arcsin 1|η+| < κ

+ < π2 .

In case of|η+|< 1, there is

χ+ =π2+ iarccosh

1

|η+|√

(n+ sinκ+)2−1

for κ+c < κ+ < π/2.The second Brewster angle of the lower faceζ̃− takes the form

ζ̃− =

π2 + iarccosh

1|z−|

√(n+ sinκ+)2−1

, for κ+c < κ+ < arcsin√

1+1/|z−|21+|z+|2 ,

arcsin 1|z−|

√(n+ sinκ+)2−1

, for arcsin

√

1+1/|z−|21+|z+|2 < κ

+ < π2 ,

in case of 1/|z−|< |z+| and

ζ̃− =π2+ iarccosh

1

|z−|√

(n+ sinκ+)2−1, for κ+c < κ+ <

π2

otherwise.

1.6 Acknowledgements

One of the authors (MAL) was supported in part by the grant of the Russian ScienceFoundation, RSCF 17-11-01126.

References

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