-
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])
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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.
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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.
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4 Advances of Mathematical Methods in Electromagnetics
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
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Scattering of electromagnetic surface waves5
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
,
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6 Advances of Mathematical Methods in Electromagnetics
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.
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Scattering of electromagnetic surface waves7
✶✿
✿
✿✻
✯
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)
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8 Advances of Mathematical Methods in Electromagnetics
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].
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Scattering of electromagnetic surface waves9
✇
✻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.
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10 Advances of Mathematical Methods in Electromagnetics
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.
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Scattering of electromagnetic surface waves11
✻
✙
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.
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12 Advances of Mathematical Methods in Electromagnetics
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.
-
Scattering of electromagnetic surface waves13
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.
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14 Advances of Mathematical Methods in Electromagnetics
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.
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Scattering of electromagnetic surface waves15
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)
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16 Advances of Mathematical Methods in Electromagnetics
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+)
-
Scattering of electromagnetic surface waves17
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
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18 Advances of Mathematical Methods in Electromagnetics
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
-
Scattering of electromagnetic surface waves19
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′).
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20 Advances of Mathematical Methods in Electromagnetics
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
-
Scattering of electromagnetic surface waves21
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.
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