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Green Function Lecture Notes
C. Van Vlack
November 11, 2010
1 Mathematical Basis for Green Functions
The Green Function (or Green’s Function depending on how you would like to
say it [23]) is very easy to understand physically. From Morse and Feshbach [11]:
“To obtain the field cause by a distributed source (or charge or heat generator
or whatever it is that causes the field) we calculate the effects of each elementary
portion of the source and add them all. If G (r, r′) is the field at the observers point
r caused by a unit point source at the source point r′ then the field at r caused by
a source distribution ρ (r′) is the integral of G (r, r′) ρ (r′) over the whole range of
r′ occupied by the source. The function G is called the Green Function. ”
It turns out that boundary conditions can be solved the same way. Again
from Morse and Feshbach[11]: “We compute the field at r for the boundary value
(or normal gradient, depending on whether Dirichlet of Neumann conditions are
pertinent) zero at every point on the surface except r′s (which is on the surface).
At r′s the boundary value has a delta function behaviour, so that its integral over
a small surface area near r′s is unity. The field at r (not on the boundary) we
can call G (r, r′s); then the general solution, for an arbitrary choice of boundary
1
2
values ψ′ (r′s) (or else gradients) is equal to the integral of G (r, r′s)ψ′ (r′s) over the
boundary surface. These functions G are also called Green Functions.”
To show mathematically what is explained above, Novotny has an excellent
description [12] which I take almost exactly from his book. Consider the following
general, inhomogeneous equation:
LA (r) = B (r) . (1)
L is a linear operator acting on the vectorfield A representing the unknown response
of the system. The vectorfield B is a known source function and makes the differen-
tial equation inhomogeneous. A well known theorem for linear differential equations
states that the general solution is equal to the sum of the complete homogeneous
solution (B = 0) and a particular inhomogeneous solution. Here, we assume that
the homogeneous solution (A0 ) is known. We thus need to solve for an arbitrary
particular solution. Usually it is difficult to find a solution of Eq. 1 and it is easier
to consider the special inhomogeneity δ (r− r′), which is zero everywhere, except
in the point r = r. Then, the linear equation reads as
LGi (r, r′) = niδ (r− r′) (i = x, y, z) , (2)
where ni denotes an arbitrary constant unit vector. In general, the vectorfield Gi is
dependent on the location r of the inhomogeneity δ (r− r′). Therefore, the vector
r has been included in the argument of Gi. The three equations given by Eq. 2 can
be written in closed form as
LG (r, r′) = Iδ (r− r′) , (3)
where the operator L acts on each column of G separately and I is the unit dyad.
The dyadic function G fulfilling Eq. 3 is known as the dyadic Green function.
3
In a next step, assume that Eq. 3 has been solved and that G is known.
Postmultiplying Eq. 3 with B (r) on both sides and integrating over the volume V
in which B 6= 0 gives∫V
LG (r, r′)B (r′) dV ′ =
∫V
B (r′) δ (r− r′) dV ′, (4)
The right hand side simply reduces to B (r) and with Eq. 1 it follows that
LA (r) =
∫V
LG (r, r′)B (r′) dV ′. (5)
If on the right hand side the operator L is taken out of the integral, the solution of
Eq. 1 can be expressed as
A (r) =
∫V
G (r, r′)B (r′) dV ′. (6)
Thus, the solution of the original equation can be found by integrating the product
of the dyadic Green function and the inhomogeneity B over the source volume V .
The assumption that the operators L and dV can be interchanged is not
strictly valid and special care must be applied if the integrand is not well behaved.
Most often G is singular at r = r′ and an infinitesimal exclusion volume surrounding
r = r′ has to be introduced [1, 25]. Depolarization of the principal volume must be
treated separately resulting in a term (L) depending on the geometrical shape of
the volume. Furthermore, in numerical schemes the principal volume has a finite
size giving rise to a second correction term designated by M. As long as we consider
field points outside of the source volume V , i.e., r 6∈ V we do not need to consider
these tricky issues.
4
2 Derivation of the Green’s Functions for the Elec-
tric Field
The following sections are taken almost exactly from Novotny’s book [12].
2.1 Frequency Dependent
The derivation of the Green function for the electric field is most conveniently
accomplished by considering the time-harmonic vector potential A and the scalar
potential φ in an infinite and homogeneous space which is characterized by the
constants ε and µ. In this case, A and φ are defined by the relationships
E (r;ω) = iωA (r;ω)−∇φ (r;ω) (7)
H (r;ω) =1
µ0µ∇×A (r;ω) . (8)
We can insert these into the Maxwell equation,
∇×H (r;ω) = −iωD (r;ω) + j (r;ω) (9)
and obtain
∇×∇×A (r;ω) = µ0µj (r;ω)− iωµ0µε0ε [iωA (r;ω)−∇φ (r;ω)] , (10)
where we used D = ε0εE. The potentials A and φ are not uniquely defined by Eqs.
7-8. We are still free to define the value of ∇ ·A which we choose as the Lorentz
gauge,
∇ ·A (r;ω) = iωµ0µε0εφ (r;ω) . (11)
Using the identity, ∇ × ∇× = −∇2 + ∇∇· along with the Lorentz gauge we can
rewrite Eq. 10 as, [∇2 + k2
]A (r;ω) = −µ0µj (r;ω) (12)
2.1 Frequency Dependent 5
which is the inhomogeneous Helmholtz equation. It holds independently for each
component Ai of A. A similar equation can be derived for the scalar potential φ
[∇2 + k2
]φ (r;ω) = −ρ (r;ω)
ε0ε. (13)
Thus, we obtain four scalar Helmholtz equations of the form
[∇2 + k2
]f (r;ω) = −g (r;ω) . (14)
To derive the scalar Green’s function G0 (r, r′;ω) for the Helmholtz operator we
replace the source term g (r) by a single point source δ (r− r′) and obtain
[∇2 + k2
]G0 (r, r′;ω) = −δ (r− r′) . (15)
The coordinate r denotes the location of the field point, i.e. the point in which the
fields are to be evaluated, whereas the coordinate r designates the location of the
point source. Once we have determined G0 we can state the particular solution for
the vector potential in Eq. 12 as
A (r;ω) = µ0µ
∫V
G0 (r, r′;ω) j (r′;ω) dV ′. (16)
A similar equation holds for the scalar potential. Both solutions require the knowl-
edge of the Green function defined through Eq. 15. In free space, the only physical
solution of this equation is [7]
G0 (r, r′;ω) =e(±ik|r−r′|)
4π |r− r′|. (17)
The solution with the plus sign denotes a spherical wave that propagates out of the
origin whereas the solution with the minus sign is a wave that converges towards
the origin. In the following we only retain the outwards propagating wave. The
2.1 Frequency Dependent 6
scalar Green function can be introduced into Eq. 16 and the vector potential can be
calculated by integrating over the source volume V . Thus, we are in a position to
calculate the vector potential and scalar potential for any given current distribution
j and charge distribution ρ. Notice, that the Green function in Eq. 17 applies only to
a homogeneous three-dimensional space. The Green function of a two-dimensional
space or a half-space will have a different form.
So far we reduced the treatment of Green functions to the potentials A and
φ because it allows us to work with scalar equations. The formalism becomes more
involved when we consider the electric and magnetic fields. The reason for this is
that a source current in x-direction leads to an electric and magnetic field with x,
y, and z- components. This is different for the vector potential: a source current in
x gives only rise to a vector potential with a x component. Thus, in the case of the
electric and magnetic fields we need a Green function which relates all components
of the source with all components of the fields, or, in other words, the Green function
must be a tensor. This type of Green function is denoted as dyadic Green function
and has been introduced in the previous section. To determine the dyadic Green
function we start with the wave equation for the electric field
∇× µ−1∇× E (r;ω)− k20εE (r;ω) = iωµ0j (r;ω) (18)
which, in a homogeneous space, reads as
∇×∇× E (r;ω)− k2E (r;ω) = iωµµ0j (r;ω) (19)
We can define for each component of j a corresponding Green function which can
all be brought together to obtain the general definition of the dyadic Green function
for the electric field,
∇×∇×G (r, r′;ω)− k2G (r, r′;ω) = Iδ (r− r′) (20)
2.1 Frequency Dependent 7
The first column of the tensor G corresponds to the field due to a point source in
x-direction, the second column to the field due to a point source in y-direction, and
the third column is the field due to a point source in z-direction. Thus a dyadic
Green function is just a compact notation for three vectorial Green functions.
As before, we can view the source current in Eq. 19 as a superposition of
point currents. Thus, if we know the Green function G we can state a particular
solution of Eq. 19 as
E (r;ω) = iωµ0µ
∫V
G (r, r′;ω) j (r′;ω) dV ′. (21)
However, this is a particular solution and we need to add any homogeneous solutions
E0 . Thus, the general solution turns out to be
E (r;ω) = E0 (r;ω) + iωµ0µ
∫V
G (r, r′;ω) j (r′;ω) dV ′. (22)
The corresponding magnetic field reads as
H (r;ω) = H0 (r;ω) +
∫V
[∇r ×G (r, r′;ω)] j (r′;ω) dV ′. (23)
These equations are denoted as volume integral equations. They are very important
since they form the basis for various formalisms such as the method of moments,
the Lippmann-Schwinger equation, or the coupled dipole method. We have limited
the validity of the volume integral equations to the space outside the source volume
V in order to avoid the apparent singularity of G at r = r′ . This limitation will
be relaxed in later.
Note that the Green functions used above are electric field Green functions
that describe the propagation of electric fields. A similar magnetic Green function
exists from solving equations involving magnetic field sources,
∇×∇×H (r;ω)− k2H (r;ω) = iωµµ0∇× j (r;ω) (24)
2.2 Time Dependent 8
which gives the magnetic field Green function via
∇×∇×Gm (r, r′;ω)− k2Gm (r;ω) = Iδ (r− r′) . (25)
This function looks exactly the same as Eq. 20 magnetic sources are handled
differently from electric sources and the magnetic field boundary conditions are
different from the electric field boundary conditions. However, it is possible to
calculate the magnetic field Green function in the exact same manner as the electric
field Green function by interchanging µ with ε and E with H [20].
Some common relations used in derivations involving Green Functions are [16],
G∗ij (r, r′;ω) = Gij (r, r′;−ω) (26)
Gji (r′, r;ω) = Gij (r, r′;ω) (27)
ImG (r, r′;ω) =
∫d3s([
G (r, s;ω)×←−∇s
]Imµ−1 (s;ω)
[−→∇s ×G
∗(s, r′;ω)
]+ G (r, s;ω) Imε (s;ω)G
∗(s, r′;ω)
) (28)
where the following relation has been used,
G (r, r′;ω)×←−∇ ′ = −
[−→∇ ′ ×G
T(r, r′;ω)
]T(29)
2.2 Time Dependent
It is also useful to consider the derivation of the time dependent Green function,
especially in the context of ultrafast phenomena. We can rewrite Eqs. 7-8 in the
time domain as,
E (r; t) = − ∂
∂tA (r; t)−∇φ (r; t) (30)
H (r; t) =1
µ0µ∇×A (r; t) . (31)
9
from which we find the time-dependent Helmholtz equation in the Lorentz gauge
(assuming a none dispersive medium),[∇2 − n2
c2
∂
∂t
]A (r; t) = −µ0µj (r; t) (32)
with a similar equation for the scalar potential φ. The definition of the scalar Green
function is now generalized to,[∇2 − n2
c2
∂
∂t
]G0 (r, r′; t, t′) = −δ (r− r′) δ (t− t′) . (33)
The point source is now defined with respect to space and time. The solution for
G0 is [7]
G0 (r, r′; t, t′) =δ(t′ −
[t∓ n
c|r− r′|
])4π |r− r′|
, (34)
where the minus sign is associated with the response at a time t later than t′.
Using G0 we can construct the time-dependent dyadic Green function G (r, r′; t, t′)
similar to the previous case. Since we will mostly work with time-independent Green
functions we avoid further details and refer the interested reader to specialized
books on electrodynamics. Working with time-dependent Green functions accounts
for arbitrary time behavior but it is very difficult to incorporate dispersion. Time
dependent processes in dispersive media are more conveniently solved using Fourier
transforms of monochromatic fields.
3 Green Function Applications
The most straightforward use case for Green function techniques is for using Eqs.
22 and 23 and adding a scatterer into the system. This can be in free space where
scatterers are added to examine spectra [10], to calculate optical forces between
10
objects via stress tensor techniques [2] or to examine defects in systems [14]. How-
ever the Green function, which is an entirely classical quantity (there have been
no Hamiltonian’s thus far), can be related to a number quantities which are useful
in QED systems. I present here, without derivation, a number of quantities: The
homogeneous space density of states
ρ (ω) =6ω
πc2Im[Ghom (r, r;ω)
]=ω2√ε
π2c3
(35)
as the imaginary parts of the diagonal elements of the homogeneous Green function
for r = r′ is,
Im[Ghom (r, r;ω)
]=ω√ε
6πc. (36)
The local density of states is then given by,
ρloc (r, ω) =6ω
πc2Im[G (r, r;ω)
]. (37)
The Green function can be decomposed into two quantities, the homogeneous part
(or direct part), Ghom, and the scattered part (or indirect part), Gscatt. It is worth
noting that there are other decompositions that are useful as well, such as lon-
gitudinal (or local)/transverse (or propagating), or into the modes of the system
(relevant for structures with well defined modes such as photonic crystal cavities).
Decomposing into the homogeneous and scattered parts allow us to write the local
density of states as,
ρloc (r, ω) =ω2√ε
π2c3+
6ω
πc2Im[Gscatt (r, r;ω)
]. (38)
11
The scattered part of the Green function is well behaved at r = r′. These quantities
are related to the Purcell factor, PF (r;ω), via [15],
PF (r;ω) =ρloc (r, ω)
ρ (r, ω)
=Im[G (r, r;ω)
]Im[Ghom (r, r;ω)
]=1 +
Im[Gscatt (r, r;ω)
]Im[Ghom (r, r;ω)
](39)
Related to both the Purcell factor and the local density of states [? ] is the decay
rate of an emitter with dipole moment d (please check the units),
Γ (r;ω) = Γ0 +2ω2d · Im
[Gscatt (r, r;ω)
]· d
h̄ε0c2(40)
and the lamb shift (frequency shift) of the emitter,
∆ω (r;ω) =ω2d · Re
[Gscatt (r, r;ω)
]· d
h̄ε0c2(41)
It is also possible to calculate the spectrum, S (R;ω), that a detector at R would
see from a quantum emitter at rd, [24] [8], and the Casimir-Polder force on an atom
or on a larger body [17, 18].
4 Volume Integral Equations
The following section is just an overview, however it is covered very thoroughly in
the last chapter of Novotny [12]. The first issue we must deal with when considering
Eqs. 22 and 23 is how to handle the divergent homogeneous Green function at
r = r′. This is done by first dividing the total Green function as described above
into homogeneous and scattered parts,
E (r;ω) = E0 (r;ω) + iωµ0µ
∫V
(Ghom (r, r′;ω) + Gscatt (r, r′;ω)
)j (r′;ω) dV ′. (42)
12
To exclude the singularity we introduce a principal volume Vδ, which allows us to
rewrite Eq. 42 as
E (r;ω) =E0 (r;ω) + iωµ0µ
∫V
Gscatt (r, r′;ω) j (r′;ω) dV ′
+iωµ0µ
∫V−Vδ
Ghom (r, r′;ω) j (r′;ω) dV ′ + iωµ0µ
∫Vδ
Ghom (r, r′;ω) j (r′;ω) dV ′
=E0 (r;ω) + iωµ0µ
∫V
Gscatt (r, r′;ω) j (r′;ω) dV ′
+iωµ0µ
∫V−Vδ
Ghom (r, r′;ω) j (r′;ω) dV ′ +L · j (r′;ω)
iωεB (r;ω)
(43)
A similar expression can be found for the magnetic field H. In the limit as the max-
imum chord length δ approaches zero, the exclusion volume Vδ becomes infinitely
small. The source dyadic L accounts for the depolarization of the excluded volume
Vδ and turns out to entirely depend on the geometry of the principal volume [25]
L =1
4π
∫Sδ
n (r′) (r′ − r)
|r− r′|3dS ′. (44)
The limit as δ → 0 is omitted in the expression for L because the surface integral
depends only on the geometry of Vδ . For cubic or spherical principal volumes the
source dyadic turns out to be L = (1/3)I . As Yaghjian points out, the value of the
volume integral also varies with the geometry of the principal volume and in just
the right way to keep the sum of the volume and surface integral contributions a
unique value [25].
Due to the smooth behavior of Ghom at r 6= r′, we can split the integral N
into subvolumes ∆Vn. The integral in the expression for r 6∈ ∆VN can be approxi-
mated by ∆VnGhom (r, rn;ω). This approximation cannot be applied for r ∈ ∆Vn
because of the strong variation of Ghom near r = r′. Instead, the volume inte-
gral has to be carried out explicitly for a given geometry of the principal volume
13
Vδ. Since Gscatt is well behaved for all r, the integration of Gscatt can be replaced
by ∆VnGscatt (r, rn;ω) everywhere. For later convenience, the remaining volume
integral shall be denoted as
M = limδ→0
∫∆Vn−Vδ
Ghom (r, r′;ω) dV ′. (45)
After inserting Eqs. 44, and 45 into Eq. 43 and evaluating the field E at the
positions rk = rn, the following N vector equations are obtained (dropping the ω
argument to ease the notation):
E (rk) = E0 (rk) +iωµ
ε0c2
[M (rk)−
c2L (rk)
ω2εB (rk)+ ∆VkGscatt (rk, rk)
]j (rk)
+iωµ
ε0c2
N∑n=1,n6=k
G (rk, rn) j (rn) ∆Vn k = 1, ..., N.
(46)
These N equations are the basis for both method of moments (MOM) and the
coupled dipole method (CDM). It can be shown that M (rn) approaches zero as
the subvolume ∆Vn is reduced arbitrarily. Therefore, in the limit ∆Vn → 0 the
contribution of M (rn) can be ignored. The dyadic L (rn) on the other hand, does
not vanish in the limit ∆Vn → 0 This dyadic accounts for self-depolarization and
its incorporation is absolutely necessary in a self-consistent formalism.
Since Eq. 46 considers both M and L, the equation represents a so called
strong form. The weak form is obtained, if M is ignored and only L is considered.
According to Lakhtakia [9], only comparisons between strong forms or weak forms
are appropriate. A comparison between the strong form of MOM and the weak
form of CDM will show the same inconsistency as a comparison between the strong
and weak form of the same method.
It is possible to write M explicitly for a spherical exclusion volume of size ak,
M (rk) =2
3k2 (rk)[[1− ik (rk) ak] exp (ik (rk) ak)− 1] I. (47)
14
Note that in addition to the strong and weak forms, Draine [4] and Chaumet [2] use
this approximation to leading order in the imaginary part to obtain the radiative
reaction force on nanoparticles in optical tweezing. Hughes et. al. also use this form
extensively as it is the leading term when writing ImG (r, r′ = r;ω) [26, 6, 19, 27].
It is also possible to exactly calculate M for a cubic exclusion volume [3].
Typically, we do not introduce current sources into a system. We usually
introduce polarization sources or dipole sources. If we introduce a source of per-
mittivity ε (r) with a different permittivity than the background εB (r), this can be
related to the current via,
j (r;ω) = −iωε0 [ε (r)− εB (r)]E (r;ω) (48)
allowing us to write Eq. 22 in terms of the change in permittivity, ∆ε (r;ω) =
ε (r;ω)− εB (r;ω), as,
E (r;ω) = E0 (r;ω) +ω2µ
c2
∫V
G (r, r′;ω) ∆ε (r;ω)E (r′;ω) dV ′. (49)
and Eq. 46 as,
E (rk) = E0 (rk) +ω2µ
c2
[M (rk)−
c2L (rk)
ω2εB (rk)+ ∆VkGscatt (rk, rk)
]∆ε (rk)E (rk)
+ω2µ
c2
N∑n=1,n6=k
G (rk, rn) ∆ε (rn)E (rn) ∆Vn k = 1, ..., N.
(50)
This equation is the central equation of the method of moments. Note that we can
similarly write this in terms of a polarizability. This is given classically as,
αk = ∆Vk∆ε (rk) ε0
[I−
[ω2
c2M (rk)−
L (rk)
εB (rk)
]∆ε (rk)
]−1
(51)
4.1 Dyson Equation 15
which reduces to the Clausius-Mossotti form for the polarizability when M = 0,
αk = ∆VkεB (rk) ε0∆ε (rk)
ε (rk) + 2εB (rk). (52)
Inserting Eq. 51 into Eq. 50 allows us to write the CDM in terms of polarizabilities,
E (rk) = E0 (rk) +ω2µ
c2∆VkGscatt (rk, rk)αkEexc (rk)
+ω2µ
c2
N∑n=1,n6=k
G (rk, rn)αnEexc (rn) ∆Vn k = 1, ..., N.(53)
Here, Eexc is the field that excites the system. This is the central equation of the
coupled dipole method. I have only briefly described the relation between the two,
however full derivations and explanations are in Novotny [12]. It is also possible to
use quantum mechanical polarizabilities in 53 where
αQM (r;ω) =|d|2
h̄
2ωrω2 − ω2
r − iωγ, (54)
and the alpha that is used is,
αk =
[I− ω2
c2ε0
αQM ImG (rk, rk)
]−1
αQM (55)
4.1 Dyson Equation
The Dyson equation is very similar to Eq. 49. In Eq. 49, we start with an
unperturbed system that can be described by the Green function G0 that has an
initial electric field distribution of E0. After the perturbation ∆ε is introduced we
are able to find the new electric field E,
E (r;ω) = E0 (r;ω) +ω2µ
c2
∫V
G0 (r, r′;ω) ∆ε (r;ω)E (r′;ω) dV ′. (56)
4.1 Dyson Equation 16
We can similarly calculate the new electric field via,
E (r;ω) = E0 (r;ω) +ω2µ
c2
∫V
G (r, r′;ω) ∆ε (r;ω)E0 (r′;ω) dV ′. (57)
where now the Green function G includes the perturbation. The Green function is
calculated via the Dyson equation, which is derived as follows; Consider the Green
function for the initial system which satisfies,
∇×∇×G0 (r, r′;ω)− ω2
c2εB (r;ω)G0 (r, r′;ω) = Iδ (r− r′) (58)
The Green function that includes the perturbation satisfies
∇×∇×G (r, r′;ω)− ω2
c2(εB (r;ω) + ∆ε (r;ω))G (r, r′;ω) = Iδ (r− r′) . (59)
If we invert Eq. 58, we obtain
G0 (r, r′;ω) =
(∇×∇×−ω
2
c2εB (r;ω)
)−1
Iδ (r− r′) (60)
and we can perform a similar inversion on Eq. 59 to obtain,
G (r, r′;ω)−(∇×∇×−ω
2
c2εB (r;ω)
)−1ω2
c2∆εG (r, r′;ω)
=
(∇×∇×−ω
2
c2εB (r;ω)
)−1
Iδ (r− r′) .
(61)
and inserting Eq. 60 twice gives the Dyson equation,
G (r, r′;ω) = G0 (r, r′;ω) +ω2
c2
∫G0 (r, s;ω) ∆ε (r;ω)G (s, r′;ω) . (62)
Note I have been very sloppy with my notation and dropping integrals but the
above procedure can be followed with more rigour to the same result.
17
A Appendix
A.1 Homogeneous Green Function
Given the wavevector k = ωn/c, the refractive index n, and R is the absolute value
of the separation R = r− r′, the homogeneous Green function is given by,
G (r, r′;ω) =exp (ikR)
4πR
[(1 +
ikR− 1
k2R2
)I +
(3− 3ikR− k2R2
k2R2
)RR
R2
]. (63)
This can be written out explicitly as,
G =
Gxx Gxy Gxz
Gxy Gyy Gyz
Gxz Gyz Gzz
G =
exp (ikR)
4πR
((1 +
ikR− 1
k2R2
)I
+
(3− 3ikR− k2R2
k2R4
)(x− x′)2 (x− x′) (y − y′) (x− x′) (z − z′)
(y − y′) (x− x′) (y − y′)2 (y − y′) (z − z′)
(z − z′) (x− x′) (z − z′) (y − y′) (z − z′)2
(64)
The tensor ∇×G is given by,
∇×G (r, r′;ω) =exp (ikR)
4πR
k(R× I
)R
(i− 1
kR
)(65)
The calculation of the Green function in more complicated geometries have also
been calculated. For instance, planar interfaces [12], planar multilayers [13, 22],
and for spherical multilayers [21, 5]
REFERENCES 18
References
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