Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

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
...
However, in the case of Electromagnetism,
this technique is not suitable because of the presence of
the “static” modes which generate a “Dirac” singularity
in the Green’s function (see reference14 for investigations
on the singularity)
...


(37)

R3

R3

[From now, the brakets h·, ·i denotes the standard inner
product (36) for the solely electric fields
...
Then, the


square integrable
coefficients φp , He (z)−1 φq can be used to define “formally”
Ge (x, y; z) =

X


φp , He (z)−1 φq φp (x) ⊗ φq (y) ,

(38)

n,m

where the symbol ⊗ means that the tensor product is
considered
...

The functions φ and ψ in (36) and (37) can be also chosen
to approach the identity
...

Then, for a small enough, φx0 (x) = φa (x − x0 ) ≈ δ(x − x0 )
and φy0 (y) = φa (y − y0 ) ≈ δ(y − y0 ), and the coefficient

5



φx0 , He (z)−1 φy0 can approach the Green’s function:



φx0 , He (z)−1 φy0
Z
Z
dy φa (x − x0 )Ge (x, y; z)φa (y − y0 )
dx
=
R3

R3

≈ Ge (x0 , y0 ; z)
...
Thus the coefficient φx0 , He (z)−1 φy0 just
allows to address an approximation of the Green’s function Ge (x0 , y0 ; z)
...
First, it is clear that all the analytic properties of He (z)−1 and H(z, ξ)−1 are directly transposable
to coefficients hφ, He (z)−1 ψi and hφ, H(z, ξ)−1 ψi
...
Another important properties of the Green’s function is the behavior
for large frequency z
...


used to analyze the dielectric permittivity, the permeability or the optical index1,3
...
This version shows that the general
expression of the permittivity is a continuous superposition of elementary resonances given by the elastically
bound electron model
...
In particular, the continuous superposition of resonances in (10) describes a regime with
absorption, while the quantum mechanics model is reduced to a discrete superposition of resonances and thus
to the description of systems without absorption
...
The objective is to transpose all the properties
of the permittivity, and to make it possible to use all the
knowledge on permittivity ε(x, z) for the electromagnetic
Green’s function
...
Thus the following operator is considered:
R(z) = He (z)−1 − H0 (z)−1 ,

|z|→∞

It has to be noticed that this asymptotic behavior is for
the modulus |z| of the complex frequency which tends
to infinity
...
e
...
In this case, one can show that


(41)
z 2 He (z)−1 − H0 (z)−1

is bounded when ω → ∞
...

Hence it is found that


z 2 He (z)−1 − H0 (z)−1
≈

ω→∞

≤

zH0 (z)−1 [∂t χ](x, 0+ ) zHe (z)−1

(43)

[∂t χ](x, 0+ )

...

IV
...
It is generally

(44)

First, it is noticed that, as well as He (z)−1 and H0 (z)−1 ,
the adjoint operator of R(z)−1 is


R(z)−1

†

= R(−z)−1 ,

(45)

which is related to ε(z) = ε(−z)
...
It is stressed that this integral is well defined
since, thanks to (43), R(z) is bounded and decreases like
1/ω 2
...
The integral expression of X(t) is
independent of η thanks to the analytic nature of the operator under the integral
...
In addition, it can be checked that X(t) vanishes for negative
times
...
Since all the functions are

6
analytic, it is found that X(t) = 0 if t < 0
...

Next, it is always possible to write for Imz = η > 0
Z ∞
1
dt exp[izt]X(t)
...
Let ξ = ν + iζ be a complex
number with positive imaginary part such that Im(z) =
η > ζ > 0
...

2π R

(51)

The inverse Fourier transform is used: for t > 0,
Z
ξR(ξ) − ξR(ξ)†
1
∂t X(t) = dν exp[−iξt]

...

dν exp[−iξt]
iz
2iπ
0
R
(53)
Next, by analogy with the function σ(x, ν) defined by
(11), the following quantity is considered:

1 
D(ξ) =
ξR(ξ) − ξR(−ξ) ,
2iπ

(54)

which is selfadjoint (and, contrary to σ(x, ν), it is not
positive since it is the difference of the one in the media
minus the one in vacuum)
...

z − ξ2 z
R
The integral over ν in (52) and above is independent of
the imaginary part ζ of ξ
...
Let D(ν) be
defined by
D(ν) = lim D(ν + iζ) ,
ζ↓0

(56)

it is left invariant when ν is changed in −ν [as well as in
D(ξ)]
...


These equations define Kramers-Kronig expressions for
the inverse Helmholtz operator
...
It
is stressed that the quantity ρ(x, y; ν) is closely related to
the local density of states which is just proportional to
the trace of ρ(x, x; ν) [modulo the density of states in vacuum]
...
Indeed, consider the
case of a closed cavity filled with a non dispersive and
non absorptive dielectric media
...
In the considered closed cavity, the
operator L has a discrete set of eigenfunctions |φn i and
(real) eigenvalues ωn2
L|φn i = ωn2 |φn i ,

(62)

which can be used to develop the inverse Helmholtz operator and the Green’s function
...
(64)
+
ρn (ξ) =
2π (ν − ωn )2 + ζ 2
(ν + ωn )2 + ζ 2
1
D(ξ)|φn i =
2iπ

7
When the imaginary part of ξ tends to zero, the eigenvalue ρn (ξ) becomes

1
lim ρn (ξ) = ρn (ν) = δ(ν − ωn ) + δ(ν + ωn ) ,
ζ↓0
2

and the operator D(ν) can be written
X
D(ν) =
ρn (ν)|φn ihφn |
...
Finally, This expression is replaced into the
equation (59) without the free part G0 (x, y; z):
Ge (x, y; z) = −

X
n

1
φn (x) ⊗ φn (y) ,
z 2 − ωn2

(68)

Hence the classical expansion6 of the Green’s function
is retrieved
...
This Kramers-Kronig expression (59) may pave the way to obtain an eigenmodes
expansion in the general case of frequency dispersive and
absorptive systems
...
ANALYTIC PROPERTIES IN NON DISPERSIVE
SYSTEMS

In non dispersive systems, the dielectric permittivity
is independent of the frequency
...

(69)
ε(x, ω0 ) = ε0 − dν 2
ω0 − ν 2
R
The Helmholtz operator for the corresponding non dispersive system is


Hd (z)E (x) = z 2 ε(x, ω0 )µ0 E(x) − ∂x ×∂x ×E(x)
...
The identification
of Hd (z) with the expression (33) yields
z[ε(x, ω0 ) − ε0 ] = ξ[ε(x, ξ) − ε0 ]
...

z

(73)

Finally, the imaginary part of ξ can be related to the one
of z:

n

The function under the integral in (59) is then
X
ρ(x, y; ν) =
ρn (ν)φn (x) ⊗ φn (y) ,

A sufficient condition to ensure this equation is

(71)

The permittivities are replaced by their Kramers-Kronig
expressions (69) and (12), which provides the following
condition:
Z
Z
σ(x, ν)
σ(x, ν)
z dν 2
dν
=

...

|z|2

(74)

In order to preserve the analytic properties of the inverse
Helmholtz operator, it is necessary to have Im(ξ) > 0
...
This condition is realized if the function
σ(x, ν) vanishes for frequency ν smaller than a frequency
ν0 > ω0
...

(75)
ν
− ω02
|ν|≥ν0
The function under the integral is strictely positive and
purely real
...

It has been shown that the analytic properties of the
inverse Helmoltz operator can be preserved in non dispersive systems if the permittivity is the one of transparent dielectric
...

Indeed, it can be checked that the causality principle is
preserved for the solution E (x, t) of Maxwell’s equation
...
Then, after the frequency
decomposition, the current source
Z ∞
J(x, z) =
dt exp[izt] J (x, t)
(76)
0

is an analytic function in the upper half plane of the
complex frequencies z
...
For negative times t, the integral
can be computed by closing the line Γ by a semi circle
with infinite radius in the upper half plane
...
Hence,
the causality principle is preserved
...

v=p
ε 0 µ0
ε(x, ω0 )µ0

(78)

8
Notice that the expression (75) obtained for the permittivity, and derived from equation (71), is just a sufficient condition which might be too strong
...


VI
...
The
starting point is the expression of Maxwell’s equations
(21) introduced in section II C
...
The resulting free operator
introduced in (21) becomes M0 (k, z) while the potential
V(x, z) is left invariant
...
Let k′ and
k′′ be the real and imaginary parts of the wavevector:
k = k′ + ik′′
...
(79)

The eigenvalues of this matrix are
λ0 = Im(z) ,

λ± = Im(z) ± ck ′′ ,

(80)

where c is the light velocity in vacuum (78), and (k ′′ )2 =
k′′ · k′′
...


(81)

Since the imaginary part of the potential V(x, z) is also
positive (25), the inverse [M0 (k, z) + V(x, z)] is welldefined and analytic in the domain Im(z) − c |k ′′ | > 0
of complex frequencies z and wavevectors k
...

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
...
Math
...

10 Y
...
Guenneau, and B
...
Rev
...

11 C
...
Diu, and F
...
II (Hermann, 1973)
...
Reed and B
...
, Vol
...

1) (Academic Press, 1980)
...
Reed and B
...
(Methods of
modern mathematical physics; V
...

14 C
...
Gu´
erin, B
...
Tip, “Singularity of the dyadic
greens function for heterogeneous dielectrics,” Phys
...
E 75,
056601 (2007)
...

As a first step, this asymptotic behavior is established
for the inverse of the free Helmholtz operator given by


H0 (z)E (x) = z 2 ε0 µ0 E(x) − ∂x ×∂x ×E(x)
...
The expression of H0 (z) in the Fourier space is


b 0 (z) = z 2 ε0 µ0 − k 2 1 − kk/k 2 ,
H
(A3)
where kk is the tensor product of k with k, and k 2 = k·k
...
(A5)
2
z 2 ε 0 µ0 − k

Let φ and ψ be square integrable fields, then the following
coefficients are considered:
Z



−1
ˆ
...
(A8)
1 − kk/k 2 ψ(k)
2
z 2 ε 0 µ0 − k

In order to obtain the property (40) for the free resolvent,
it is enough to show that the number I(z) tends to zero
for large |z|
...
Writting the complex number as z = |z| exp[iθ],
it is obtained that |z 2 ε0 µ0 −k 2 | ≥ k 2 sin2 θ
...
e
...
Next, the number |z| is chosen large enough to have
|z|2 ε0 µ0 > K 2 , and the second part is bounded by
Z




K2
ˆ ψ(k)
ˆ 
...
Thus it is concluded that





lim z 2 ε0 µ0 φ, H0 (z)−1 ψ = φ, ψ
...
The second resolvent
identity is used to express the difference
He (z)−1 −H0 (z)−1 = −H0 (z)−1 z 2 µ0 [ε(x, z)−ε0 ]He (z)−1
...


(A15)

It is well-defined for all z since zHe (z)−1 is bounded by
[ ε0 µ0 Im(z) ]−1 and, from (12), z[ε(x, z) − ε0] is bounded

10
by [∂t χ](x, 0+ )/Im(z)
...

|z|ε0 Im(z)

(A16)

Finally, the preceeding equations (A12–A16) imply





C(z) = z 2 ε0 µ0  φ, H0 (z)−1 Ψ(z) 
≈

|z|→∞

−→

|z|→∞

+



 φ, Ψ(z)  ≤ [∂t χ](x, 0 ) kφk kψk
|z|ε0 Im(z)

0
...




---