Comptes Rendus Physique

Benoit Deveaud

Polariton interactions in semiconductor microcavities

Alain Priou

Cyril Lupi

Tome 17 (2016) – No 9 – novembre 
Poster of the URSI-France 2015 scientific days – Paris, CNAM, 24 & 25 March 2015.
Affiche des journées scientifiques d’URSI-France – Paris, Cnam, 24 & 25 mars 2015.
DOSSIER
Probing matter with electromagnetic waves / Sonder la matière par les ondes électromagnétiques
URSI-France 2015 scientific days – Paris, CNAM, 24 & 25 March 2015
Coordinators / Coordinateurs : Alain Priou, Cyril Lupi
• Foreword
Alain Priou, Cyril Lupi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957
• Combined complex-source beam and spherical-multipole analysis for the electromagnetic probing of conical 
structures
Ludger Klinkenbusch, Hendrik Brüns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 960
• A variety of radars designed to explore the hidden structures and properties of the Solar System’s planets and 
bodies
Valérie Ciarletti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966
• Star formation and chemical complexity in the Orion nebula: A new view with the IRAM and ALMA 
interferometers
Alain Baudry, Nathalie Brouillet, Didier Despois . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976
• Comets at radio wavelengths
Jacques Crovisier, Dominique Bockelée-Morvan, Pierre Colom, Nicolas Biver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985
• Scattering properties of a stratified air/snow/sea ice medium. Small slope approximation
Richard Dusséaux, Saddek Afifi, Monique Dechambre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995
• A new technique to characterize foliage attenuation using passive radar in the L-band
Marc Lesturgie, Laetitia Thirion-Lefèvre, Stéphane Saillant, Philippe Dorey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003
Continued on the next page
Contents (continued)
• Microwave imaging of magnetohydrodynamic instabilities in fusion plasma
Roland Sabot, Didier Elbèze, Woochang Lee, Yoonbum Nam, Hyeon Park, Junsong Shen, Gunsu Yun, 
Minjun Choi, Jean-Claude Giacalone, Timothée Nicolas, Christine Bottereau, Frédéric Clairet, 
Philippe Lotte, Diego Molina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018
• Superposition of fiber Bragg and LPG gratings for embedded strain measurement
Romain Guyard, Dominique Leduc, Yann Lecieux, Cyril Lupi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027
Guide for authors: http://www.em-consulte.com/getinfoProduit/COMREN/instructionsAuteurs/COMREN.pdf
Electronic Submission: http://ees.elsevier.com/crphysique/


Foreword

Ludger Klinkenbusch

Hendrik Brüns

C. R. Physique 17 (2016) 960–965Contents lists available at ScienceDirect
Comptes Rendus Physique
www.sciencedirect.com
Probing matter with electromagnetic waves / Sonder la matière par les ondes électromagnétiques
Combined complex-source beam and spherical-multipole 
analysis for the electromagnetic probing of conical structures
Combinaison de faisceaux de sources complexes et décomposition en modes 
sphériques pour le sondage électromagnétique des structures coniques
Ludger Klinkenbusch ∗, Hendrik Brüns
Kiel University, Kaiserstr. 2, 24143 Kiel, Germany
a r t i c l e i n f o a b s t r a c t
Article history:
Available online 12 August 2016
Keywords:
Complex-source beam
Spherical-multipole analysis
Conical structures
Green’s functions
Radiation condition
Mots-clés :
Faisceau source complexe
Analyse multipôle sphérique
Structure conique
Fonction de Green
Condition de radiation
The paper addresses the combination of the spherical-multipole analysis in sphero-conal 
coordinates with a uniform complex-source beam (CSB) in order to analyze the scattering 
of a localized electromagnetic plane wave by any desired part of a perfectly conducting 
elliptic cone. The concept of uniform CSB is introduced and rigorously applied to the 
diffraction by a semi-infinite elliptic cone. The analysis takes into account the fact that 
the incident CSB does not satisfy the radiation condition. A new modal form of the Green’s 
function for the elliptic cone is derived based on the principle that there is no energy loss 
to infinity. The numerical evaluation includes the scattered far fields of a CSB incident on 
the corner of a plane angular sector with different opening angles.
© 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
r é s u m é
Cet article présente la combinaison de l’analyse de multipôles sphériques en coordonnées 
sphéro-coniques avec une faisceau de source complexe (Complex Source Beam, CSB) dans 
le but d’analyser la diffusion localisée par un cône elliptique parfaitement conducteur 
d’une onde plane électromagnétique. Le concept de CSB est introduit au travers de la 
diffraction par un cône elliptique semi-infini. L’analyse prend en compte le fait que l’onde 
CSB incidente ne satisfait pas les conditions de radiation. Un nouveau modèle de la fonction 
de Green pour un cône elliptique est développé en faisant l’hypothèse qu’il n’y a pas de 
pertes d’énergie à l’infini. Le modèle numérique inclut la diffusion en champ lointain d’une 
source CSB sur le coin d’un secteur angulaire avec différents angles d’ouverture.
© 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
* Tel.: +49 431 8806252; fax: +49 431 8806253.
E-mail address: lbk@tf.uni-kiel.de (L. Klinkenbusch).http://dx.doi.org/10.1016/j.crhy.2016.07.018
1631-0705/© 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
L. Klinkenbusch, H. Brüns / C. R. Physique 17 (2016) 960–965 9611. Introduction
Multipole analysis and complex-source beams (CSB) are important techniques in field theory. As examples for applica-
tions, multipole expansions are in use to solve and compactly characterize scalar, electrodynamic, and elastodynamic fields 
in the presence of canonical structures while complex-source beams are perfectly suited to elegantly produce analytic beam-
like fields. With a combination of both methods, it is possible to analytically investigate the scattering and diffraction of 
a beam-like electromagnetic wave by canonical structures. Particularly, the elliptic cone which is described as a coordinate 
surface in sphero-conal coordinates contains several interesting special cases, including the half plane, the quarter plane, 
and the circular cone. The corresponding electromagnetic boundary-value problem is solved via the dyadic Green’s func-
tion, which consists of bilinear products of spherical Bessel functions, periodic, and non-periodic Lamé functions. With an 
incident CSB, it is possible to probe any geometrical detail of the structure and thus to isolate the corresponding electromag-
netic response. Particularly, the tip and edge diffraction coefficients derived from these solutions can be applied to further 
complete asymptotic high-frequency methods like the Geometrical Theory of Diffraction (GTD) or the Uniform Theory of 
Diffraction (UTD) [1].
The paper starts with brief summaries of the spherical-multipole analysis in sphero-conal coordinates and the complex-
source beam technique. Then the concept of the uniform CSB is introduced and thoroughly applied to the diffraction by a 
semi-infinite elliptic cone. The analysis is similar to the derivation of the classical Green’s function of the cone, but takes 
into account the fact that the uniform CSB does not satisfy the radiation condition. The numerical evaluation includes the 
scattered far fields of a uniform CSB incident on the corner of a plane angular sector with different opening angles.
2. Spherical-multipole analysis in sphero-conal coordinates
Sphero-conal (conical) coordinates r, ϑ, ϕ are related to Cartesian coordinates by
x = r sin ϑ cosϕ (1)
y = r
√
1 − k2 cos2 ϑ sinϕ (2)
z = r cosϑ
√
1 − k′ 2 sin2 ϕ (3)
They belong to those famous eleven three-dimensional coordinate systems where the Helmholtz equation is fully separa-
ble [2]. k and k′ are positive real-valued parameters which satisfy k2 + k′ 2 = 1. The coordinate surface ϑ = ϑ0 generally 
describes a semi-infinite elliptic cone where the degree of ellipticity is defined by the choice of k. In the case where k = 1, 
the sphero-conal coordinates turn into regular spherical coordinates, and the elliptic cone becomes a right-circular cone. In 
the case where ϑ0 = 0 or ϑ0 = π , the elliptic cone degenerates to a plane angular sector in the yz-plane symmetrically 
around the +z and −z axis, respectively, with an opening angle defined by 2 arccos(k). We denote the normalized metric 
scaling coefficients by
sϑ = 1
r
∣∣∣∣ ∂r∂ϑ
∣∣∣∣ =
√
k2 sin2 ϑ + k′ 2 cos2 ϕ
1 − k2 cos2 ϑ (4)
sϕ = 1
r
∣∣∣∣ ∂r∂ϕ
∣∣∣∣ =
√
k2 sin2 ϑ + k′ 2 cos2 ϕ
1 − k′ 2 sin2 ϕ (5)
The scalar homogeneous Helmholtz equation reads in sphero-conal coordinates
σ (r) + κ2σ (r) = 0
1
r2
∂
∂r
(
r2
∂σ (r)
∂r
)
+ 1
r2
(r × ∇)2 σ (r) + κ2σ (r) = 0 (6)
where the angular part of the -operator is related to the square of the angular-momentum operator known from quantum 
mechanics and defined by [3]:
(r × ∇)2 = sϑ
sϕ
[
∂
∂ϑ
(
sϕ
sϑ
∂
∂ϑ
)
+ ∂
∂ϕ
(
sϑ
sϕ
∂
∂ϕ
)]
(7)
The method of separation of variables yields an elementary solution of (6) according to
σ (r,ϑ,ϕ) = zσ (κr)
σ (ϑ)σ (ϕ) (8)
The separation constants are denoted by σ and λ. zσ is a spherical Bessel function, while 
σ and σ represent non-periodic 
and periodic Lamé functions, respectively. The products of non-periodic and periodic Lamé functions are referred to as Lamé 
products Yσ (ϑ, ϕ) = 
σ (ϑ)σ (ϕ). They satisfy the eigenvalue equation
962 L. Klinkenbusch, H. Brüns / C. R. Physique 17 (2016) 960–965Fig. 1. Eigenvalue curves λ(σ ) and discrete Dirichlet eigenvalues σs(×) for ϑ0 = 180◦ and k2 = 0.5 (quarter plane).
(r × ∇)2 Yσ (ϑ,ϕ) + σ(σ + 1)Yσ (ϑ,ϕ) = 0 (9)
It can be shown [3] that for any valid solution of the boundary value problem at hand, the possible discrete values of σ and 
λ have to lie on the so-called eigenvalue curves, as exemplarily represented in Fig. 1. We are interested in those eigenvalues 
σs satisfying the Dirichlet condition (index s for acoustically soft) or the Neumann condition (σh; index h for acoustically 
hard) at ϑ0 according to

σs(ϑ0) = 0 ;
d
σh
dϑ
(ϑ)
∣∣∣∣
ϑ=ϑ0
= 0 (10)
respectively. The corresponding discrete pairs of eigenvalues (στ , λ) with τ ∈ {s, h} are found by a numerically performed 
search on the aforementioned eigenvalue curves. Note that the Lamé products satisfying (10) form an orthogonal system of 
functions on the domain 0 ≤ ϑ ≤ ϑ0; 0 ≤ ϕ ≤ 2π .
For the solution of the electromagnetic problem, the same spherical-multipole technique as in ordinary spherical coordi-
nates applies. As has been shown in [3] outside of a perfectly electrically conducting (PEC) elliptic cone in a homogeneous, 
isotropic, and linear medium characterized by permittivity ε and permeability μ, the time-harmonic electromagnetic field 
at a time factor e+jωt is represented by the spherical-multipole expansion
E(r) =
∑
σs
aσs Nσs(r) +
Z
j
∑
σh
bσh Mσh(r)
H(r) = j
Z
∑
σs
aσs Mσs(r) +
∑
σh
bσh Nσh(r)
(11)
Here, Z = √μ/ε is the intrinsic impedance of the medium, while the expansion functions, which are also known as vector 
spherical-multipole functions, are related to the elementary solutions of the scalar homogeneous Helmholtz equation by
Mστ (r) = (r × ∇)στ (r) ; Nστ (r) =
1
κ
∇ × Mστ (r) (τ ∈ s,h) (12)
where κ = ω√με is the wave number of the medium. The expansion coefficients in (11) are referred to as the spherical-
multipole amplitudes aσs and bσh . They contain the entire information of the total electromagnetic field for a specific source, 
e.g., a Hertzian dipole.
3. Uniform complex-source beam
By assigning a complex-valued location rc = rw − j b, a point source field (e.g., due to a Hertzian dipole) is turned into 
a complex-source beam (CSB) [4,5]. In a paraxial approximation, that CSB represents a Gaussian beam. In contrast to a 
Gaussian beam, the CSB is an exact solution of Maxwell’s equations. The location of the beam’s waist is defined by the 
real part rw, while the direction of beam propagation and the focus (Rayleigh) length is given by the imaginary part b. If 
the original point-source field satisfies the radiation condition (i.e. only outwardly traveling waves are allowed in the far 
field), that condition is preserved for a complex source location, i.e. all inwardly traveling field parts (towards the waist) 
of the resulting CSB are exponentially damped and virtually not relevant. On the other hand, if the location of a point 
sink—with only inwardly traveling waves—is defined as complex-valued, the CSB parts traveling outwardly (from the waist) 
are exponentially damped. Consequently, by simply adding these two fields, a uniform CSB (a complete CSB) is achieved, 
which includes an analytic local plane wave at the waist [6].
L. Klinkenbusch, H. Brüns / C. R. Physique 17 (2016) 960–965 963Fig. 2. Path of integration in the complex η-plane for solving (17).
4. Green’s function of an elliptic cone illuminated by a uniform complex-source beam
As mentioned above, a uniform CSB does not satisfy the radiation condition. Hence the Green’s function for the elliptic 
cone which usually takes into account that condition has to be re-deduced if the cone is illuminated by a uniform CSB. 
For the sake of simplicity and because the electromagnetic case is a straightforward extension of the scalar case, we will 
demonstrate the derivation for a scalar Green’s function satisfying
Gs(r, rc) + k2Gs(r, rc) = δs(r − rc) (13)
subject to the Dirichlet (soft) boundary condition
Gs(r, rc)|ϑ=ϑ0 = 0 (14)
First we express the three-dimensional δ distribution by using the corresponding completeness relation according to (see 
[3] p. 566 for a derivation):
δs(r − rc) = 2
π
∞∫
0
η2
∑
σs,
jσs(ηr) jσs(ηrc)dηYσs(ϑ,ϕ)Y
∗
σs
(ϑc,ϕc) (15)
where jσs represents the spherical Bessel function of the first kind of order σs , and the asterisk denotes the complex 
conjugation. Next we expand the Green’s function by means of the complete bilinear eigenfunction expansion
Gs(r, rc) = 2
π
∞∫
0
η2
∑
σs
aσ (η) jσs(ηr) jσs(ηrc)dηYσs(ϑ,ϕ)Y
∗
σs
(ϑc,ϕc) (16)
We insert (15) and (16) into (13) and use the eigenvalue equation (9) as well as the orthogonality properties of the Lamé 
products on the unit sphere and of the spherical Bessel functions on the interval [0, ∞) to obtain
Gs(r, rc) = 2
π
∞∫
0
η2
κ2 − η2
∑
σs
jσs(ηr) jσs(ηrc)dηYσs(ϑ,ϕ)Y
∗
σs
(ϑc,ϕc) (17)
The integral in (17) can be solved similarly to the approach in [7] using residue calculus in the complex η-plane. However, 
in opposite to [7], where the poles of (17) at η = ±κ have to be properly excluded from the path of integration to satisfy 
the radiation condition, in the present case of a uniform CSB we have to take in account that there is no energy loss to 
infinity. Consequently, as shown in Fig. 2, the path of integration has to stay on the real axis to account for that conservation 
of energy for any η.
For a complex-valued source location rc = rw − j b where rw and b are real and b > 0, we finally derive the modal 
expansion of the Green’s function as
Gs(r, rc) = 2jk
∑
σs
jσs(κr) jσs(κrc)Yσs(ϑ,ϕ)Y
∗
σs
(ϑc,ϕc) (18)
Note that there is no case distinction |r| ≷ |rc| as necessary for the Green’s function satisfying the radiation condition. 
Moreover, the expansion (18) is equally convergent for |r| = |rc| as well. The acoustic field at r due to a point source Q
located at rc in the presence of a acoustically soft elliptic cone can then be represented by (r) = Gs(r, rc)Q .
The extension to the electromagnetic case, i.e. to a dyadic Green’s function can be performed as usual [8]. For a Hertzian 
dipole cel located at rc, the electromagnetic field in the presence of an elliptic cone can finally be represented by the 
spherical-multipole expansion (11) with multipole amplitudes
aσs = −κ2 Z
1
N∗σs(rc) · cel ; bσh = − jκ2
1
M∗σh(rc) · cel (19)σs (σs + 1) σh (σh + 1)
964 L. Klinkenbusch, H. Brüns / C. R. Physique 17 (2016) 960–965Fig. 3. Total field of a uniform CSB incident on the tip of a circular cone. The waist is located directly at the tip. The Rayleigh length is b = 10  ( =
wavelength). The angle of incidence is θ = 10◦ of the vertical axis. The polarizations are as indicated.
Fig. 4. Scattered far field of a uniform CSB incident on the tip of a circular cone. The waist is located directly at the tip. The Rayleigh length is b = 10 
( = wavelength). The angle of incidence is θ = 10◦ of the vertical axis. The polarizations are as indicated.
Fig. 5. Uniform CSB incident on the corner of a plane angular sector.
Note that we have to insert spherical Bessel functions of the first kind jστ (τ ∈ {s, h}) in all occurring vector spherical-
multipole functions in case of a complex-source location rc.
5. Numerical results
First we have applied the described theory to the case of a uniform CSB which is incident directly towards the tip of a 
circular cone, i.e. an elliptic cone with k = 1. As has been shown in [9,10], in this case only the radial component of rc , that 
is rc has to be chosen complex-valued while ϑc and ϕc remain real-valued. Fig. 3 shows the total near field for a CSB which 
is incident from θ = 10◦ off the z-axis for the two indicated polarizations, whereas Fig. 4 represents the corresponding 
scattered far fields. We have obtained the scattered field by simply subtracting the incident from the total field. As a second 
example we have chosen a uniform CSB incident directly towards the corner of a plane angular sector described by ϑ0 = π , 
see Fig. 5. Again, we have chosen the waist to be located at the corner for achieving a plane-wave front at the area of 
interaction with the sector. Fig. 6 finally shows the scattered fields for different opening angles of the sector. We clearly 
observe that the interaction between the edges is dramatically increasing for smaller opening angles. Consequently, not just 
the area which is illuminated by the CSB contributes to the scattered field. Moreover, the characteristic structure of the 
scattered field extremely changes as a function of the opening angle.
Finally, it is important to note that all multipole expansions evaluated for these results show strong convergence. This 
is in contrast to the case of a full plane-wave incidence as treated in [11], where the obtained multipole series were 
non-convergent and special summation techniques had to be applied to come to meaningful results. The computational 
resources for evaluating the fields are in the range of CPU seconds on a standard personal computer.
L. Klinkenbusch, H. Brüns / C. R. Physique 17 (2016) 960–965 965Fig. 6. Scattered field of a uniform CSB incident on the corner of a plane angular sector for different opening angles: Left upper 90◦, right upper 45◦ , left 
lower 25◦ , right lower 5◦ . The waist is located directly at the corner of the sector. The Rayleigh length is b = 10 . The electric field of the CSB is polarized 
in the y-direction, see Fig. 5.
Acknowledgements
This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under KL815/10-1&2.
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Combined complex-source beam and spherical-multipole analysis for the electromagnetic probing of conical structures

Richard Dusséaux

Saddek Afifi

Monique Dechambre

Scattering properties of a stratified air/snow/sea ice medium. Small slope approximation

Artis Svilans

Martin Leijnse

Heiner Linke

Experiments on the thermoelectric properties of quantum dots

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Polariton interactions in semiconductor microcavities

Foreword

Combined complex-source beam and spherical-multipole analysis for the electromagnetic probing of conical structures

Scattering properties of a stratified air/snow/sea ice medium. Small slope approximation

Experiments on the thermoelectric properties of quantum dots

Editorial Board

Dossier Sommaire

Silver and oxygen: Transition from clusters to nanoparticles

Editorial Board

Editorial Board

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