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

[1] J. Van Bladel. Some remarks on green’s dyadic for infinite space.

IRE Transactions on Antennas and Propagation, 9:533, 1961. doi: 10.

1109/TAP.1961.1145064. URL http://resolver.scholarsportal.info/

resolve/00961973/v09i0006/563_srogdfis.

[2] P. C. Chaumet and M. Nieto-Vesperinas. Coupled dipole method determination

of the electromagnetic force on a particle over a flat dielectric substrate. Phys.

Rev. B, 61(20):14119–14127, May 2000. doi: 10.1103/PhysRevB.61.14119.

[3] Patrick C. Chaumet, Anne Sentenac, and Adel Rahmani. Coupled dipole

method for scatterers with large permittivity. Phys. Rev. E, 70(3):036606,

Sep 2004. doi: 10.1103/PhysRevE.70.036606.

[4] B.T. Draine. The discrete dipole approximation and its application to inter-

stellar graphite grains. Astrophys J., 333:848, 1988.

[5] H. T. Dung, L. Knll, and D-G. Welsch. Spontaneous decay in the presence of

dispersing and absorbing bodies: General theory and application to a spherical

cavity. Phys. Rev. A., 62:053804, 2000. doi: 10.1103/PhysRevA.62.053804.

[6] S. Hughes, H. Gotoh, and H. Kamada. Classical and quantum optical corre-

lation effects between single quantum dots: The role of the hopping photon.

Phys. Rev. B., 74:115334, 2006. doi: 10.1103/PhysRevB.74.115334.

[7] J. D. Jackson. Classical Electrodynamics. Wiley, 1975.

[8] Philip Trøst Kristensen, Jesper Mørk, Peter Lodahl, and Stephen Hughes.

REFERENCES 19

Decay dynamics of radiatively coupled quantum dots in photonic crystal slabs.

Submitted, -:–, 2010.

[9] Akhlesh Lakhtakia. Strong and weak forms of the method of moments and the

coupled dipole method for scattering of time-harmonic electromagnetic fields.

J. Mod. Phys. C, 3:583, 1992. doi: 10.1142/S0129183192000385.

[10] O. J. F. Martin and N. B. Piller. Electromagnetic scattering in polarizable

backgrounds. Phys. Rev. E, 58:3909, 1998. doi: 10.1103/PhysRevE.58.3909.

[11] Philip McCord Morse and Herman Feshbach. Methods of Theoretical Physics.

Mcgraw-Hill, 1953.

[12] L. Novotny and B. Hecht. Principles of Nano-Optics. Cambridge, 2006.

[13] M. Paulus, P. Gay-Balmaz, and O. J. F. Martin. Accurate and efficient com-

putation of the greens tensor for stratified media. Phys. Rev. E., 62:5797, 2000.

doi: 10.1103/PhysRevE.62.5797.

[14] Michael Paulus and Olivier J.F. Martin. A fully vectorial technique for scatter-

ing and propagation in three-dimensional stratified photonic structures. Optical

and Quantum Electronics, 33:315, 2001. doi: 10.1023/A:1010804414399.

[15] E. M. Purcell. Spontaneous emission probabilities at radio frequencies. Phys.

Rev., 69:681, 1946.

[16] C. Raabe and D-G Welsch. Casimir force acting on magnetodielectric bodies

embedded in media. Phys. Rev. A, 71:013814, 2005. doi: 10.1103/PhysRevA.

71.013814.

REFERENCES 20

[17] C. Raabe and D-G Welsch. Dispersive forces on bodies and atoms: A unified

approach. Phys. Rev. A, 73:063822, 2006. doi: 10.1103/PhysRevA.73.063822.

[18] C. Raabe and D-G Welsch. Unified approach to dispersion forces within

macroscopic qed. Hyperfine Interactions, 172:149, 2007. doi: 10.1007/

s10751-007-9533-4.

[19] V. S. C. Manga Rao and S. Hughes. Single quantum dot spontaneous

emission in a finite-size photonic crystal waveguide: Proposal for an effi-

cient on chip single photon gun. Phys. Rev. Lett., 99:193901, 2007. doi:

10.1103/PhysRevLett.99.193901.

[20] Alejandro W. Rodriguez, Alexander P. McCauley, John D. Joannopoulos, and

Steven G. Johnson. Casimir forces in the time domain: Theory. Phys. Rev. A,

80(1):012115, Jul 2009. doi: 10.1103/PhysRevA.80.012115.

[21] L. Knll S. Scheel and D.-G. Welsch. Spontaneous decay of an excited atom in

an absorbing dielectric. Phys Rev. A, 60:4094, 1999. doi: 10.1103/PhysRevA.

60.4094.

[22] M. S. Tomas. Green function for multilayers: Light scattering in planar cavities.

Phys. Rev. A., 51:2545, 1995. doi: 10.1103/PhysRevA.51.2545.

[23] M. C. M. Wright. Green function or green’s function? Nature Physics, 2:

646, 2006. doi: 10.1038/nphys411. URL http://www.nature.com/nphys/

journal/v2/n10/full/nphys411.html.

[24] M. Wubs, L. G. Suttorp, and A. Lagendijk. Multiple-scattering approach to

interatomic interactions and superradiance in inhomogeneous dielectrics. Phys.

Rev. A., 70:053823, 2004. doi: 10.1103/PhysRevA.70.053823.

REFERENCES 21

[25] A.D. Yaghjian. Electric dyadic green’s functions in the source region. Proc.

IEEE, 68:248, 1980.

[26] P. Yao and S. Hughes. Macroscopic entanglement and violation of bells in-

equalities between two spatially separated quantum dots in a planar photonic

crystal system. Opt. Express, 17:11505, 2009.

[27] P. Yao, V.S.C. Manga Rao, and S. Hughes. On-chip single photon sources

using planar photonic crystals and single quantum dots. Laser and Photonics

Review, 1:1, 2009. doi: 10.1002/lpor.200810081.