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Analytic properties of the electromagnetic Green’s function
Boris Gralak1, a)
CNRS, Aix-Marseille Universit´e, Centrale Marseille, Institut Fresnel, UMR 7249, 13013 Marseille,
France

arXiv:1512
...
The first frequency results from the frequency
decomposition of the electromagnetic field while the second frequency is associated with the dispersion of the
dielectric permittivity
...
Such analytic properties are
also extended to complex wavevectors
...
In addition, these Kramers-Kronig expressions are
shown to correspond to the classical eigengenmodes expansion of the Green’s function established in simple
situations
...

I
...
It is defined from the inverse of the Helmholtz operator2, which provides the electric field radiated by a current source
...
Then, the electromagnetic Green’s function
can be defined from the inverse of the Helmholtz operator by
Z


−1
dy Ge (x, y; z) · S(y) ,
(2)
He (z) S (x) =
R3

where S(x) is proportional to a current source density
...

It is well-known that the Green’s function Ge (x, y; z) is
an analytic function in the upper half space of complex
frequencies z
...
Notice that the frequency
dependence of the electromagnetic Green’s function has
two different origins in Maxwell’s equations: the first one
is the consequence of the frequency decomposition of the
time derivative of the fields in Maxwell’s equations, and
the second one is the frequency dispersion which results
in the frequency dependence of the permittivity ε(x, z)
...
In particular, this

a) boris
...
fr

is exploited in section V to provide a rigorous proof of
the analyticity and causality in the non-dispersive case
...

The analytic properties of the electromagnetic Green’s
function can be used to compute Sommerfeld integrals
and time-dependent electromagnetic fields4 , for instance
defining analytic continuation in the plane of complex
frequencies
...
For instance,
new Kramers-Kronig relations have been established in
reference5 for the reflection and transmission coefficients
in non-normal incidence
...
Arguments are provided to interpret these expressions as generalizations to dispersive
and absorptive systems of the well-known eigenmodes
expansion6 established for simple closed cavity (without
dispersion and absorption)
...


GENERALIZED HELMHOLTZ OPERATOR

A
...
Let E (x, t), H (x, t) and P(x, t) be respectively the
time-dependent electric, magnetic and polarization fields
...
In addition, the
electric field is related to the polarization through the

2
constitutive equation
Z
P (x, t) =

t
−∞

ds χ(x, t − s)E (x, s) ,

(4)

where χ(x, t) is the electric susceptibility
...

ε(x, ω) − ε0 =

(5)

(6)

0

Here, according to the causality principle, it has been
used that the susceptibility χ(x, t) vanishes for negative
times, i
...
χ(x, t) = 0 if t < 0
...
Consequently, t is
always positive in the integral above, and the permittivity remains well-defined if the real frequency is replaced
by the complex frequency z = ω + iη with positive imaginary part Im(z) = η > 0
...
It follows that
the permittivity ε(x, z) is an analytic function in the half
plane of complex frequencies z with positive imaginary
part, which will be denominated by “upper half plane”
from now on
...

The Helmholtz equation is directly deduced from the
set of equations (5), where the ω-dependence of the fields
has been omitted:


He (ω)E (x) = ω 2 ε(x, ω)µ0 E(x) − ∂x ×∂x ×E(x)
(9)
= −iωµ0 J(x)
...
It can be shown rigorously that the inverse of He (z) exists and is analytic
in this domain Imz > 0 using the the auxiliary field
formalism7
...
The inverse
[z − K]−1 is then well-defined for all complex number z
with Im(z) > 0, and is moreover an analytic function
of z
...

Since the projector on electric fields is z-independent, the
inverse of the Helmholtz operator has the same analytic
properties as the inverse [z − K]−1
...


B
...
The key point is the generalized expression of
Kramers-Kronig relations7,10 for the permittivity:
ε(x, z) = ε0 −

Z

dν

R

σ(x, ν)
,
z2 − ν2

(10)

where
σ(x, ν) = Im

ν[ε(x, ν) − ε0 ]
≥ 0
...
Notice that
it has been assumed that only passive media are considered
...
At the microscopic scale, this function corresponds to the oscillator strength11 which must
be positive
...
Indeed, using that σ(x, ν) = σ(x, −ν), the
expression (10) can be written as
z[ε(x, z) − ε0 ] = −

Z

dν

σ(x, ν)
,
z−ν

(12)

Z

dν

σ(x, ν)
≥ 0
...
It can be calculated using the Kramers-Kronig re-

3
lation (10) or (12):
(∂t χ)(x, t) =
=

1
2π
1
2π

Z

Γη

Z

dz exp[−izt] (−iz) [ε(x, z) − ε0 ]

dν σ(x, ν)

Z

dz (i)

Γ

R

exp[−izt]

...
It is retrieved that the susceptibility χ(x, t)
vanishes for t < 0 and, for t > 0:
Z
dν σ(x, ν) cos[νt]
...


(16)

Also, it can be checked that [∂t χ](x, t) is continuous of t
in the general case (except at t = 0), and that [∂t χ](x, t)
is bounded by [∂t χ](x, 0)
...
The function
[∂t χ](x, t) being bounded and continuous (except at t =
0), the second derivative [∂t2 χ](x, t) can be defined for all
t 6= 0
...


(17)
Since the derivative [∂t χ](x, t) corresponds to the microscopic currents, it is related to the impulsion of the
charges: it is then reasonable to assume that its variations are bounded because of the inertia (charges have
a mass), otherwise an infinite power is required
...

Thus, from now on, it is assumed that the second derivative [∂t2 χ](x, t) is bounded for t > 0
...

ω→∞

C
...

0
0
Let h·, ·i be the standard inner product in the Hilbert
space of square integrable electromagnetic fields:
Z





dx ε0 E1 (x) · E2 (x) + µ0 H1 (x) · H2 (x)
...
Using that the curl is selfadjoint, the following relationship is obtained
Z





Im F , M0 (z)F = Im(z)
dx ε0 |E(x)|2 + µ0 |H(x)|2
R3




= Im(z) F , F
...

R3

The combination of the two equations leads to





 
 F , M0 (z) + V(x, z) F  ≥ Im(z) F , F ,

(25)
(26)

which implies that the inverse [M0 (z) + V(x, z)]−1 is welldefined for Im(z) > 0 and bounded by α−1 = 1/Im(z)
...
The analyticity property can be also shown using the first resolvent
formula12
...

(20)
, S(x) =
H(x)
0

(19)

where, using expression (12) for V(x, z),


Z
σ(x, ν)
1+m 0

...
The identity (27)
implies

−1
M(z)−1 = M(z0 )−1 1 − (z0 − z)AM(z0 )−1
n
X
p o


...
It is bounded by
w
w
wH(z, ξ)−1 w ≤

1

...


p

(29)
The last series converges in norm provided |z − z0 | is
smaller than the inverse of the nom of [AM(z0 )−1 ]
...

In the last step, the Helmholtz operator is retrieved using the projector on electric fields P, defined by PF (x) =
E(x)
...
According to (9),
the inverse Helmholtz operator is given by E(x) =
−izµ0 He (z)−1 J(x), and the comparison with the equation above provides
He (z)−1 =


−1
1
P M0 (z) + V(x, z) P ,
zµ0 ε0

(31)

This expression shows that all the properties of the inverse [M0 (z) + V(x, z)]−1 are directly transposable to the
inverse Helmholtz operator
...

|z|ε0 µ0 Im(z)

(32)

It is stressed that the bound α = Im(z) in (26) is governed by the imaginary part of z in M0 (z) only, and thus
is independent of the complex number z in V(x, z)
...
As a result, it is obtained that the inverse

−1
M0 (z) + V(x, ξ)
≤ [ Im(z) ]−1 ,
(33)

exists and is analytic with respect to both complex frequencies z and ξ as soon as Im(z) > 0 and Im(ξ) > 0
...
This property
can be transposed to the inverse of a generalized version
of the Helmholtz operator
...


THE ELECTROMAGNETIC GREEN’S FUNCTION

The electromagnetic Green’s function can be introduced from the inverse Helmholtz operator as shown
by equation (2)
...
Indeed, the existence
of the Green’s function is usually the consequence of the
compact or Hilbert-Schmidt nature of the corresponding
operator12
...

For instance, similar property has been used in5 to
derive new Kramers-Kronig relations for the reflection
and transmission coefficients (via the Green’s function) in
the case of multilayered stacks illuminated with incident
angle θ 6= 0
...


VII
...
These properties are strongly related to
the causality principle and the passivity requirement in
frequency dispersive and absorptive media
...
Hence the general inverse
Helmholtz operator has been shown to be analytic in the
domain Im(z) − c |k ′′ | > 0 of complex frequencies z and
complex wavevectors k = k′ + ik′′ (sections II and VI)
...
This additional frequency
has been then exploited to retrieve that causal systems
with non dispersive permittivity must have purely real dielectric constant taking values above the vacuum permittivity ε0 (section V)
...
Such KramersKronig expressions can be considered as an extension
of the well-known eigenmodes expansion of the Green’s
function in the case of a closed cavity filled with non
dispersive and non absorptive media6
...
Indeed, in that case, it is
enough to add the permeability µ(x, z) in the expression
of the matrix V(x, z)


z[ε(x, z) − ε0 ]
0
,
(82)
V(x, z) =
0
z[µ(x, z) − µ0 ]

and then to apply the arguments proposed in this paper
...

The results established in this paper may be used
to calculate time-dependent electromagnetic fields4 and
to establish rigorous eigenmodes expansion in dispersive
and absorptive systems
...

REFERENCES
1 J
...
Jackson, Classical Electrodynamics, 3rd ed
...


9
2 M
...
Simon, Scattering theory, Vol
...
3) (Academic Press, 1979)
...
D
...
M
...
P
...

8), 2nd ed
...
C
...

4 B
...
Maystre, “Negative index materials and timeharmonic electromagnetic field,” C
...
Physique 13, 786 (2012)
...
Gralak, M
...
Zerrad, and C
...
Opt
...
Am
...

6 R
...
Caz´
e, D
...
Peragut, V
...
Pierrat, and Y
...

7 A
...
Rev
...

8 A
...
Moroz, and J
...
Combes, “Band structure of absorptive photonic crystals,” J
...
A: Math
...
33, 6223
(2000)
...
Gralak and A

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