Download Green`s functions for Maxwell`s equations: application to

Optical and Quantum Electronics 29 (1997) 199±216
Green's functions for Maxwell's equations:
application to spontaneous emission
F . W I J N A N D S , J . B . P E N D R Y , F . J . G A R C I A - V I D A L ,
P. M. BELL
Condensed Matter Theory Group, The Blackett Laboratory, Imperial College,
Prince Consort Road, London SW7 2BZ, UK
P. J. ROBERTS
Defence Research Agency, St Andrew's Road, Great Malvern,
Worcs. WR14 3PS, UK
L . M A R T ÂI N M O R E N O
Instituto de Ciencia de Materiales (Sede B), Universidad AutoÂnoma de Madrid,
280049 Madrid, Spain
Received 2 January; accepted 10 June 1996
We present a new formalism for calculating the Green's function for Maxwell's equations. As our aim is to apply our formalism to light scattering at surfaces of arbitrary
materials, we derive the Green's function in a surface representation. The only requirement on the material is that it should have periodicity parallel to the surface. We
calculate this Green's function for light of a speci®c frequency and a speci®c incident
direction and distance with respect to the surface. The material properties entering the
Green's function are the re¯ection coef®cients for plane waves at the surface. Using the
close relationship between the Green's function and the density of states (DOS), we
apply our method to calculate the spontaneous emission rate as a function of the
distance to a material surface. The spontaneous emission rate can be calculated using
Fermi's Golden Rule, which can be expressed in terms of the DOS of the optical modes
available to the emitted photon. We present calculations for a ®nite slab of cylindrical
rods, embedded in air on a square lattice. It is shown that the enhancement or suppression of spontaneous emission strongly depends on the frequency of the light.
1. Introduction
The Green's function is an extremely useful concept in, for example, scattering problems
or problems in which the density of states (DOS) plays an important role [1]. Green's
functions for Maxwell's equations have been used, for example, for calculating modal ®eld
solutions in waveguides, regarding the waveguide as a perturbation of homogeneous space
Present address: R&D Devices, Hewlett-Packard Fiber-Optic Components Operation, White House Road,
Ipswich, Su€olk IP1 5PB, UK.
0306±8919
Ó 1997 Chapman & Hall
199
F. Wijnands et al.
[2]. In a recent paper, Martin et al. [3] used the Green's function as a propagator to
calculate the scattered ®eld, given an arbitrary incident ®eld, for arbitrary geometries.
Often one is interested in the DOS of optical modes. For instance, the spontaneous
emission rate is directly related to the DOS of optical modes available to the spontaneously emitted photon. The DOS, in turn, can be modi®ed by changing the environment
in which the emitting atom is put. The control of spontaneous emission is one of the major
motivations for studying photonic band structures, and in particular, photonic bandgap
structures [4]. These have been studied both theoretically (e.g. [5±7]) and experimentally
(e.g. [8]). In a photonic bandgap material, spontaneous emission is prohibited inside the
forbidden frequency band [4]. Of particular interest is modi®cation of spontaneous
emission in microcavities [9]. This has a large technological impact on threshold-reduced
lasers [4]. Another area in which the DOS is relevant, is dispersion forces. These are Van
der Waals forces originating from the vacuum ¯uctuations, or zero-point ¯uctuations, in
the quantized electromagnetic ®eld [10]. Dispersion forces are relevant in calculating, for
example, the force between the tip of an atomic force microscope and the surface of a
material when the distance is suciently large.
Because of this interest in the DOS in surface geometries, and because of the close
relationship between the DOS and the Green's function, it would be very useful to have a
Green's function formalism which is appropriate for surface geometries. In order to study
surfaces of fairly arbitrary materials theoretically, there is a strong need for a computational approach, because complex geometries cannot be treated analytically in general. We
have developed a formalism to calculate the Green's function in the presence of a surface.
As will be shown, the material properties entering the Green's function are the re¯ection
coecients for plane waves at the surface. These re¯ection coecients can be calculated
using a transfer matrix method, derived by Pendry et al. [11±13].
The paper is organized as follows. In Section 2 we set up the Green's function which is
appropriate for our surface problem in continuous space. In Section 3 we derive the
Green's function on a lattice, necessary for our computational approach. The DOS calculated with the Green's function, is compared in Section 4 with independent derivation of
the DOS using a `particle in a box' approach, for a metal surface.
In Section 5 we apply our method to spontaneous emission. We calculate vacuum
¯uctuations near a ®nite slab of cylindrical dielectric rods embedded in air on a square
lattice. Conclusions are given in Section 6.
2. The Green's function in continuous space
We start from the source-free Maxwell equations:
rHˆ
@D
;
@t
rEˆÿ
@B
@t
…1†
Assuming a time dependence exp…ÿixt†, and using B ˆ l0 lH; D ˆ e0 eE; the Maxwell
equation for the electric ®eld reads:
c2 ‰r…r E† ÿ r2 ŠE ˆ x2 E
…2†
with c2 ˆ 1=…l0 le0 e†: The Fourier-transform of the Maxwell equations is given by
ik E ˆ ‡ixB
200
…3†
Green's functions for Maxwell's equations
ik H ˆ ÿixD
…4†
whereas the Fourier-transform of Equation 2 reads:
c2 …k k ÿ kkT †E M0 E ˆ x2 E
…5†
2.1. Green's function in homogeneous space
First, we set up the Green's function in an in®nite homogeneous space of dielectric constant e: Start with the Green's function in k-representation (cf. Equation 5):
…x2 1 ÿ M0 †G0 …x2 † ˆ 1
…6†
Notice that since electric ®elds are vector ®elds, the Green's function is a tensor. It is called
a dyadic Green's function [14]. The causal Green's function in the k-representation is
de®ned as:
X jEj ihEj j
ÿ1
0
2
2
…7†
ˆ
G‡
0 …x † ˆ ‰x 1 ÿ M ‡ id1Š
2
2
j x ÿ xj ‡ id
with d P
> 0 in®nitesimally small. Here fjEj ig is a complete set of eigenvectors in k-space,
hence j jEj ihEj j ˆ 13 : The causal Green's function in real space reads:
2
0
‡
2
0
G‡
0 …x ; r; r † ˆ h rjG0 …x †jr i ˆ
X hrjEj ihEj jr0 i;
j
x2 ÿ x2j ‡ id
ˆ
X Ej …r†‰EjT …r0 †Š
j
x2 ÿ x2j ‡ id
…8†
Now de®ne normalized electric ®eld vectors for the three polarizations s; p; l; w.r.t. the
normal ^n ˆ ‰0; 0; 1ŠT :
^vrˆs ˆ
k^
n
;
jk ^
nj
which take the explicit form
2
3
k
14 y 5
^vrˆs ˆ
ÿkx ;
kk
0
^vrˆp ˆ
^vrˆp
k k ^n
;
jk k ^nj
2
3
kx
kz 4
ky 5 ;
ˆ
kk k ÿk 2 =k
k
^vrˆl ˆ
^vrˆl
z
k
jkj
2 3
k
14 x 5
ˆ
ky
k
kz
…9†
…10†
with k jkj and kk jkk j; kk ˆ …kx ; ky †: The corresponding real space eigensolutions are
Ek;r …r† ˆ ^vr Lÿ3=2 exp‰i…k r ÿ xk;r t†Š
We have normalized these eigensolutions such that
Z
‰Ek;r …r†ŠT Ek;r …r†d3 r ˆ 1
…r ˆ s; p; l†
…11†
…12†
In Fourier-space this corresponds to a normalization,
^vTr ^vr ˆ 1;
r ˆ s; p; l
In terms of normalized eigenvectors, the Green's function reads
(
)
T
X ^vs^vTs ‡ ^vp^vTp
^
^
v
v
l
2
0
l
‡
G‡
Lÿ3 exp…ik …r ÿ r0 ††
0 …x ; r; r † ˆ
x2 ÿ c2 k 2 ‡ id x2 ‡ id
k
…13†
…14†
201
F. Wijnands et al.
As we are interested in surface geometries, it is appropriate to distinguish between waves
vÿ
going in the positive …^v‡
r † and in the negative …^
r † z-direction. In particular, we are
interested in the response of the system at z ˆ g; given an excited plane wave at z0 ˆ 0
having wave vector component kk parallel to the surface. This response is described by the
Green's function in a surface representation. As this Green's function is de®ned in terms of
a given incident kk ; we have to sum over kz instead of k. The Green's function then reads
(
)
T
T
T
X ^v‡
v‡
v‡
v‡
^v‡
v‡
s ^
p^
s ‡^
p
‡
2
0
l
l ^
‡ 2
Lÿ1
…15†
lim G0 …x ; kk ; z ˆ ‡g; z ˆ 0† ˆ
2 ÿ c2 k 2 ‡ id
g!0‡
x
x
‡
id
kz
R
P
with g > 0: Replacing the summation by an integral,
k z ! …L=2p† dkz ; the Green's
function becomes
q
i
ÿi h ‡ ‡T
T
2
0
^vs ^vs ‡ ^v‡
v‡
; Kz ˆ …x=c†2 ÿ kk2 ‡ id
lim‡ G‡
p
p^
0 …x ; kk ; z ˆ ‡g; z ˆ 0† ˆ
2
g!0
2c Kz
…16†
2.2. Green's function in presence of a surface
Consider a surface at a distance d > 0 from the position at which the plane wave is excited.
This surface can be taken account of by modifying the boundary conditions for the
Green's function. We assume the slab to be periodic in the xy-plane with lattice K. This
lattice has a rectangular unit cell with lengths Lx ; Ly in x; y respectively. Denote the twodimensional reciprocal lattice vectors by g 2 K . Consider an incident plane wave going in
0
0
0
the ‡z direction ^v‡
r0 g0 (hence g ˆ 0), and polarization r …r ˆ s or p†. The scattered wave
‡
at z ˆ ‡g (with g ! 0 ) is a superposition of plane waves going in the ÿz direction
^vÿ
rg …g 2 K † times a re¯ection
coecient R
rg;r0 g0 times optical path terms to and from the
0
0
surface exp iKzg jd ÿ z j exp iKzg jd ÿ zj . The Green's function reads
i
ÿi h ‡ ‡T
T
^vsg0 ^vsg0 ‡ ^v‡
v‡
lim‡ G‡ …x2 ; kk ; z ˆ ‡g; z0 ˆ 0† ˆ 2
pg0
pg0 ^
g!0
2c Kzg0
‡
i T
X X ÿi h
ÿ
0
0
0
v‡
exp‰iK
dŠR
exp‰iK
dŠ^
v
zg
rg;r g
zg
r0 g 0
rg ^
2K 0
2c
zg
g r;r0
…17†
where we summed over g 2 K ; r; r0 2 fs; pg, and Kz^g is the z-component of the incident
wave vector (with g0 ˆ 0).
Notice that, as for the scalar case [1], the dyadic Green's function appears in pair terms:
the right term (with superscript `T') denotes the excited wave at z0 , whereas the left term
denotes the response of the system at z.
3. Green's function on a lattice
In our numerical approach, space is discretized to form a cubic mesh with unit mesh length
a (more generally we can treat orthorhombic meshes). Recall that the material at the
surface of which scattering occurs, has a rectangular unit cell length Lx ; Ly in x; y respectively. Consequently, the number of reciprocal lattice vectors is …Lx =a† …Ly =a†. In
order to determine the Green's function (Equation 17) numerically for a given slab of
material, the re¯ection coecients Rrg;r0 g0 can be calculated using a transfer matrix method
202
Green's functions for Maxwell's equations
derived by Pendry et al. [11±13]. In this method, Maxwell's equations (Equations 1) are
transformed into a transfer matrix set of equations, which is appropriate for a numerical
approach. The most important steps are the following. First the Maxwell equations
(Equations 1) are Fourier transformed. A crucial step is to approximate the wave vector in
Equation 3 as
kj …exp…ikj a† ÿ 1†=ia kjE
… j ˆ x; y; z†
…18†
while in Equation 4 we approximate
kj ÿ…exp…ÿikj a† ÿ 1†=ia kjH
… j ˆ x; y; z†
…19†
Subsequently the equations are inverse Fourier-transformed, leading to a set of equations
which can be cast in transfer matrix form
F …z ‡ Dz† ˆ T …z†F …z†
…20†
This transfer matrix T …z† is not Hermitian. Hence, we have to distinguish between right
‡ÿ
‡ÿ
eigenvectors ^vrrg and left eigenvectors ^vlrg . Another aspect to be mentioned is that the
eigenvectors of the transfer matrix are given by Ex ; Ey ; Hx ; Hy -components, whereas the
Green's function eigenvectors are Ex ; Ey ; Ez -components. Hence one must be careful in
using the re¯ection coecients obtained with the transfer matrix method, in the Green's
function. The interested reader is referred to Appendix 1 where we derive left and right
eigenvectors for both the transfer matrix and the Green's function, and the consequences
for the re¯ection coecients.
The discrete equivalent of the normalization condition (Equation 13) is
‰^vlrg ŠT ^vrg ˆ ‰^vlrg ŠT ^vrrg ˆ 1;
‡
‡
ÿ
ÿ
8r 2 fs; pg; 8g 2 K
2
…21†
2 2
On a lattice, the dispersion relation di€ers from x ˆ c k , and is given by [13]:
X
x2 l0 e0 e ˆ k E k H ˆ
2=a2 1 ÿ cos kj a
…22†
jˆx;y;z
Now we are in a position to formulate the Green's function on a lattice in the presence of a
single surface:
h ‡ ‡T
i
ÿia
‡
‡T
^vrsg0 ^vlsg0 ‡ ^vrpg0 ^vlpg0
lim‡ G‡ …x2 ; kk ; z ˆ ‡g; z0 ˆ 0† ˆ 2
g!0
2c sin…Kzg0 a†
‡
X X
g
r;r0
h
i T
ÿia
rÿ l‡
0 d†Rrg;r0 g0 exp…iKzg d†^
^vr0 g0
exp…iK
v
zg
rg
2c2 sin…Kzg0 a†
…23†
where r; r0 2 fs; pg; g runs over all reciprocal lattice vectors, and g0 ˆ 0. Note that due to
the discrete dispersion relation (Equation 22), the pole term in the continuous case
1=…2c2 Kzg0 † has to be replaced by a=…2c2 sin…Kzg0 a††. Notice that the continuous result is
recovered in the limit a ! 0.
It is vital to have a general `intrinsic' test for the correctness of our computer codes,
based upon Equation 23. As Pendry [12] pointed out, current conservation is sensitive to
the slightest error in computer codes. As we can calculate re¯ection and transmission
coecients for a ®nite slab of material, we can test current conservation. In Appendix 2 it
is shown how current conservation is tested both for the transfer matrix basis and for the
Green's function basis.
203
F. Wijnands et al.
4. DOS for Green's function compared with `particle in box' approach
To illustrate that our Green's function (Equation 17) has been properly normalized, we
now compare the DOS according to the Green's function, with the DOS calculated according to a `particle in a box' approach.
The DOS is related to the imaginary part of a Green's function [1]. For scalar ®elds, the
Green's function is also a scalar [1]. The DOS for tensor Green's functions is slightly more
complicated. We de®ne
DOS…x; kk ; z†jj ˆ
I‰G‡ …x; kk ; z; z†Šjj
ˆ 2x
I‰G‡ …x2 ; kk ; z; z†Šjj
ÿp
ÿp
where we have taken into account that the Green's function was de®ned in terms of x2
instead of x, hence the factor dx2 =dx.
We treat the case of TE or TM polarized waves in the presence of a metal surface.
Consider two metal plates, both with normal along z, at position z ˆ 0 and z ˆ L (later, we
shall let L ! 1). Owing to the boundary conditions, Ek ; H? should be continuous. The
general solution is then of the following form. We treat TE and TM modes separately.
· TE modes (s waves).
Without loss of generality, we may assume that ky ˆ 0.
2
3
0
E ˆ 4 N sin…kz z ‡ /† 5
0
…24†
Owing to the continuity of Ek , it follows that sin…kz z ‡ /† ˆ 0 for z ˆ 0, z ˆ L, hence
kz ˆ np=L …n 2 Z† and / ˆ 0. The normalization constant N is found from the condition
r
Z L
L
2
…25†
dzN E E ˆ 1 ) N ˆ 1=
2
0
Now let L ! 1. Then, as the electric ®eld is along y, we can de®ne
dn
L sin2 …kz z†
…x; kk ; z†yy ˆ dkz
p
L=2
so that the DOS along y is
…26†
dn
dn dkz
L sin2 …kz z† x
2x sin2 …kz z†
…x; kk ; z†yy ˆ
…x; kk ; z†yy ˆ
ˆ
…27†
dx
dkz dx
p L=2 c2 kz
pc2 kz
Let us compare this with the Green's function. Inserting the two polarization
eigenvectors (Equations 10) in the continuous expression for the Green's function
containing both polarizations (Equation 17), we obtain
ÿi
G‡ …x2 ; kk ; z; z† ˆ 2
2c kz
8 2
32
3T
2
32
3T
ky
ky
ky
ky
>
<1
6
76
7 Rss exp…2ikz z† 6
76
7
4 ÿkx 54 ÿkx 5 ‡
4 ÿkx 54 ÿkx 5
2
>
kk2
: kk
0
0
0
0
204
Green's functions for Maxwell's equations
32
3
2
32
3 9
kx kz
ÿkx kz
kx kz T
kx kz T >
=
1
4 ky kz 54 ky kz 5 ‡ Rpp exp…2ikz z† 4 ÿky kz 54 ky kz 5
‡
kk2 …x=c†2 ÿk 2
kk2 …x=c†2
;
ÿkk2
ÿkk2
ÿkk2 >
k
2
…28†
Here Rss and Rpp are the Fresnel re¯ection coecients, which are Rss ˆ ÿ1 and
Rpp ˆ ‡1 for a perfect metal. For TE waves
82 3 2 3 T
2 32 3T 9
0
0
0 =
0
<
ÿi
2
4
5
4
5
4
4
5
…x
;
k
;
z;
z†
ˆ
ÿ
exp…2ik
z†
1
…29†
1
G‡
15
1
z
k
TE
;
2c2 kz :
0
0
0
0
Therefore the Green's function takes the form
(
ÿi ‰1 ÿ cos…2k z†Š
‡
z
2
I GTE …x ; kk ; z; z† jj ˆ 2c2 kz
0
Hence the DOS in the yy-direction can be written as
2
2
I‰G‡
TE …x ; kk ; z; z†Šyy d…x †
ÿp
Clearly, Expressions 27 and 31 agree.
dx
ˆ
… j ˆ y†
… j ˆ x; z†
2x sin2 …kz z†
pc2 kz
· TM modes (p waves).
We assume again that ky ˆ 0. A general solution is of the form
2
3
2
3
kz M sin…kz z ‡ /†
0
1
4
5
0
H ˆ 4 M cos…kz z ‡ /† 5; E ˆ
ÿixe0 e
0
ikx M cos…kz z ‡ /†
…30†
…31†
…32†
The continuity of Ek implies that sin…kz z ‡ /† ˆ 0 for z ˆ 0, z ˆ L, hence kz ˆ np=L
and / ˆ 0. The normalization constant N M=…xe0 e†, is found from the condition
r
r
Z L
L
L
2
2
2
2
ˆ …x=c† =
…33†
dzN E E ˆ 1 ) N ˆ …kx ‡ kz †=
2
2
0
Again let L ! 1. Then
8
kz2
2
>
>
… j ˆ x†
>
2 sin …kz z†
>
<
…x=c†
dn
dn dkz
2x
0
… j ˆ y†
…34†
ˆ
…x; kk ; z†jj ˆ
>
dx
dkz dx pc2 kz >
2
>
k
>
x
:
cos2 …kz z† … j ˆ z†
…x=c†2
The Green's function takes the form
ÿi
1
2c2 kz k 2 …x=c†2
k
82
32
3T
2
32
3 9
ÿkx kz
k
k
k
k
kx kz T >
x z
>
< x z
=
6 0 76 0 7
6
76
7
4
54
5 ‡ exp…2ikz z†4 0 54 0 5
>
>
: ÿk 2
ÿkk2
ÿkk2
ÿkk2 ;
k
2
G‡
TM …x ; kk ; z; z† ˆ
…35†
leading to a DOS:
205
F. Wijnands et al.
8
kz2
2
>
>
… j ˆ x†
>
2 sin …kz z†
>
‡
2
<
…x=c†
2
I‰GTM …x ; kk ; z; z†Šjj d…x †
2x
ˆ 2 0
… j ˆ y†
ÿp
dx
pc kz >
2
>
>
k
>
x
:
cos2 …kz z† … j ˆ z†
…x=c†2
Comparing Equations 34 and 36, the two approaches give the same result.
…36†
5. Application: spontaneous emission in presence of a surface
In this section we apply our numerical technique to calculate spontaneous emission in the
presence of a surface. The spontaneous emission rate Sj for dipole transitions oriented in
the jth direction … j ˆ x; y; z† of an isolated two-level atom is given by Fermi's Golden Rule
[15]:
X 2p 2
hf jHint jiij d…x ÿ x0 †
…37†
Sj …x† ˆ
2
h
f with Hint the interaction Hamiltonian between atom and electromagnetic ®eld, x0 is the
frequency of the photon emitted due to an electronic transition from initial state jii to
some ®nal state jf i. We have to sum over all ®nal states of the electromagnetic ®eld for
x. The radiative lifetime 1=trad …x† of an atomic transition is then given by
which x0 ˆ P
1=trad …x† ˆ jˆx;y;z Sj …x†. Equation 37 can be rewritten as
X
2p
hf jE
^ j jii2 d…x ÿ x0 †
Sj …x† ˆ 2 l2 …38†
h
f
^ j the electric ®eld operator in second
with l the electric dipoleP
moment
of the atom and E
hf jE
^ j jii2 d…x ÿ x0 † represents the vacuum ¯uctuations. The
quantization.
The
term
f
P
sum f d…x ÿ x0 † can be replaced by an integral over all directions, summed over both
transverse polarizations for electromagnetic plane waves with frequency x.
These vacuum ¯uctuations can be expressed in terms of a classical DOS, each state
having an energy hx=2 for each transverse polarization [16]. We are interested in the
vacuum ¯uctuations in homogeneous space with dielectric constant e, at a distance z from
some surface with normal along z. The vacuum ¯uctuations in the j-direction, say Vj , are
given by
Z
dkk hx DOS…x; kk ; z†rad jj
Vj …x; z† ˆ
2 2
…2p†
1
…39†
DOS…x; kk ; z†
e
1 I‰G‡ …x; kk ; z; z†Š 2x I‰G‡ …x2 ; kk ; z; z†Š
ˆ
ˆ ÿp
ÿp
e
e
The reason that the radiation DOS, denoted DOSrad , entering the vacuum ¯uctuations is a
factor e smaller than the total DOS, is the following. If each mode carries an energy hx=2,
then we should in fact have normalized the eigenvectors such that
Z
…40†
e…r†‰Ek;r …r†ŠT Ek;r …r†d3 r ˆ 1 …r ˆ s; p; l†
DOS…x; kk ; z†rad ˆ
206
Green's functions for Maxwell's equations
This means that the normalized electric ®eld vectors in Equations 9 and 11 have to be
multiplied by a factor eÿ1=2 , with a consequence that the Green's function describing
spontaneous emission is a factor 1=e smaller. This result is consistent with recent results by
Lagendijk et al. [17] and Sprik et al. [18]. They point out that in fact only part of the total
DOS contributes to the Einstein coef®cient describing spontaneous emission. The total
DOS (equal to I‰G‡ …x; kk ; z; z†Š=…ÿp†† is divided into a radiation part and a material part.
Only the radiation part (the DOS in Equation 39), which scales as e1=2 , enters the Einstein
coef®cient. Hence the Green's function (Equation 23) corresponds to the total DOS. In
order to calculate spontaneous emission, we use Equation 39. Notice that eigenfunction
methods need a weighting function to solve this problem [19, 20]. An advantage of the
Green's function approach is that it naturally describes systems with losses. Such systems
are not easily describable in terms of wavefunctions.
In in®nite homogeneous space, the vacuum ¯uctuations can be readily calculated by
multiplying Equation 16 by …1=e†…
hx=2† and integrating over all directions kk
Vj …x† ˆ
1 hx3
hx 3
ˆ
e1=2 ;
e 6p2 c3 6p2 c30
j ˆ x; y; z
…41†
of light in free space. Hence the total vacuum
in accordance with [15].
P Here c0 is the speed
hx3 e1=2 =2p2 c30 .
¯uctuations V …x† ˆ jˆx;y; z Vj …x† ˆ We apply our numerical technique to calculate vacuum ¯uctuations in air as a function
of the distance to a ¯at surface of a ®nite slab. The slab consists of eight layers of
cylindrical rods of dielectric constant e ˆ 8:9, embedded in air on a square lattice. The rods
are invariant in the direction y, and the square lattice in the xz-plane has unit cell length
L ˆ 1:87 mm. The diameter of the rods is 0.74 mm (notice that these lengths can be scaled
down to optical length scales). The geometry of this structure is shown in Fig. 1. The
photonic band structure (the dispersion relation kz …x; kk †† has been studied experimentally
[21] and theoretically [11, 13, 21] for normal incidence, and for light polarized parallel
(along y) and perpendicular (along x) to the cylinders.
Because interesting e€ects are due to strong suppression or enhancement of spontaneous
emission, we must choose the frequency suitably. We expect the modi®cation in spontaneous emission to be largest if re¯ection coecients at the surface have a large modulus
(in order to get large interference e€ects between incident and scattered waves). In absorption-free materials, if the material would have a complete photonic bandgap (i.e. a
frequency band in which the transmittance is zero irrespective of the incident direction and
polarization), then we would expect the re¯ection coecients to have a large modulus. Our
structure does not have a complete photonic bandgap. In Fig. 2 we show for s- and ppolarization the photonic band structure (Figs 2a, c) and the transmission coecient (Figs
2b, d) for waves incident at 30 o€ the normal, with parallel wave vector kk ˆ …k=2; 0†. The
number of mesh points per unit cell is Nx ˆ Nz ˆ 11, Ny ˆ 1. The transmission coecient is
the transmitted current in the z-direction, given an incident wave carrying unity current in
the z-direction (cf. Appendices 1 and 2). Comparing Fig. 2 with Fig. 5 of [13], we can see
that there is no complete bandgap. However, we can choose x to lie in a bandgap for a
large range of incident directions. We shall compare x ˆ 5:0 10ÿ5 eV which is well
below the bandgaps, and x ˆ 2:7 10ÿ4 eV which is in a bandgap for a wide range of
angles, especially for s-polarized incident waves.
First we test the minimal number of mesh points needed to obtain convergence in the
calculations. In Fig. 3 we plot ÿI‰G‡ …x2 ; kk ˆ 0; z; z†Šjj , calculated using Equation 23 for
207
F. Wijnands et al.
Figure 1 Geometry of the structure, described in the text. The structure consists of cylindrical rods on a
square lattice in the xz-plane with unit cell length L ˆ 1:87 mm. The rods have diameter d ˆ 0:74 mm and
dielectric constant e ˆ 8:9, are invariant in the y-direction and are embedded in air. A ®nite slab of eight layers
of rods in the z-direction is considered, the structure is in®nitely long in the x -direction.
x ˆ 2:7 10ÿ4 eV, as a function of surface distance, in units of the free space wavelength,
Z ˆ z=k0 . The calculations were performed for normal incidence, both for y-polarized light
( j ˆ y, Fig. 3a) and for x-polarized light ( j ˆ x, Fig. 3b), and we varied the number of
mesh points, Nx ˆ Nz ˆ 18, 30, 50, 70. In all calculations Ny ˆ 1. In order to avoid stability
problems for large Nz , the unit cell was subdivided into smaller parts, consisting of typically ten or less layers. For each part, we calculated re¯ection and transmission coecients. Re¯ection and transmission coecients of the di€erent parts were then combined to
give the corresponding coecients of the whole unit cell using a multiple re¯ection formula [12].
From Fig. 3 it becomes clear that the denser the mesh, the more the results converge.
The reason is the basic approximation (18), (19) of our transfer matrix approach. The
smaller the value of k0 a (recall that a ˆ L=Nx ), the more accurate these approximations
are. The di€erent calculations in Fig. 3 are labelled by their value of k0 a. Comparing Fig.
3a and Fig. 3b, it can be seen that for light polarized parallel to the cylinders, the convergence is much faster as a function of the mesh size than for light polarized perpendicular to the cylinders. The reason is that for the electric ®eld parallel to the cylinders, no
discontinuity occurs at the interface. In order to model the discontinuity at the interface
for the x-polarized ®eld, many more mesh points are needed than would be the case
without a discontinuity. In order to calculate vacuum ¯uctuations for this structure, we
can conclude from Fig. 3 that Nx ˆ Nz ˆ 30, is sucient. Here we make a few remarks.
208
Green's functions for Maxwell's equations
Figure 2 Photonic band structure kz L as a function of x for s- (a) and p-waves (c) and transmission
coef®cients
ÿ
for s- (b) and p-waves (d) for the structure described in the text, for incident parallel wave vector
kk ˆ 12 ; 0 jk j. The number of mesh points per unit cell Nx ˆ Nz ˆ 11, Ny ˆ 1. Here L is the unit cell length in
the xz-direction.
209
F. Wijnands et al.
Figure 3 Minus the imaginary part of Green's function ÿI‰G ‡ …x2 ;kk ˆ 0; z; z†Š as a function of the surface
distance Z ˆ z=k0 for the cylindrical structure described in the text. The calculation was done for normal
incidence, both for y - (a) and x- (b) polarized light, with x ˆ 2:7 10ÿ4 eV, and number of mesh points
Nx ˆ Nz ˆ 18; 30; 50; 70. In all calculations, Ny ˆ 1.
First, if Nx ˆ Nz ˆ 30 is sucient for x ˆ 2:7 10ÿ4 eV, then this number will certainly
be sucient for x ˆ 5:0 10ÿ5 eV, with regard to the approximations (18) and (19).
Secondly, the results in Fig. 2 have not fully converged because that calculation was
performed with Nx ˆ Nz ˆ 11. Nevertheless, Fig. 2 still gives a strong indication which
range of x is interesting for our vacuum ¯uctuations problem. In Fig. 4 total vacuum
¯uctuations in the x-, y- and z-direction are shown for x ˆ 5:0 10ÿ5 eV (Fig. 4a) and
x ˆ 2:7 10ÿ4 eV (Fig. 4b), in units of the free space value hx3 =…6p2 c30 †, as a function of
surface distance Z ˆ z=k0 . The number of mesh points per unit cell, Nx ˆ Nz ˆ 30,
Ny ˆ 1 …k0 a ˆ 0:0158†.
From Fig. 4 we see that the vacuum ¯uctuations (and therefore the spontaneous
emission rate (cf. Equation 38) in the x-direction, Vx ¯uctuate much more around their free
space value (which is normalized to one) for x ˆ 2:7 10ÿ4 eV than for x ˆ 5 10ÿ5 eV.
Near the surface, Vx is more suppressed for x ˆ 2:7 10ÿ4 eV than for x ˆ 5 10ÿ5 eV.
For the vacuum ¯uctuations in the direction of the cylindrical axis, Vy , the ¯uctuations
around the free space value are much more pronounced for x ˆ 2:7 10ÿ4 eV than for
x ˆ 5 10ÿ5 eV, especially close to the surface. For Vz , the behaviour for
x ˆ 2:7 10ÿ4 eV and for x ˆ 5 10ÿ5 eV is much more similar. We believe that the
reason that the effects are much more pronounced for the two transverse directions than
for the longitudinal direction, is that for x ˆ 2:7 10ÿ4 eV the re¯ection coef®cients have
a much larger modulus than for x ˆ 5:0 10ÿ5 eV for a wide range of incident angles,
210
Green's functions for Maxwell's equations
Figure 4 Total vacuum ¯uctuations in units of free space value h x3 =…6p2 c03 † in x ; y - and z-direction as a
function of the surface distance Z ˆ z=k0 for the cylindrical structure described in the text, for frequencies
x ˆ 5:0 10ÿ5 eV (a) and x ˆ 2:7 10ÿ4 eV (b), and number of mesh points Nx ˆ Nz ˆ 30, Ny ˆ 1.
especially for s-polarized waves (cf. Figs 2b, d). As the s-polarized waves have no electric
®eld component along z, this would explain why the difference between the two energies is
much less pronounced for Vz than for Vx and Vy . Notice that for large Z, the vacuum
¯uctuations can be seen to approach their free space value.
6. Conclusions
We have presented a new Green's function formalism, appropriate for studying scattering
problems in surface geometries. We have shown that for the continuous case, the DOS
calculated with our Green's function, is exactly the same as the DOS according to a
`particle in a box' approach. We have applied our lattice Green's function technique to
calculate vacuum ¯uctuations in the presence of the surface of a ®nite slab consisting of
cylindrical dielectric rods embedded in air on a square lattice. The results have shown that
the enhancement or suppression of spontaneous emission strongly depends on the frequency of the light. The accuracy of our method depends on the approximation of the
wave vectors in the transfer matrix. The smaller the mesh, the more accurate the calculation will be, at the expense of computer time.
Acknowledgements
F.W. thanks A. Lagendijk for interesting discussions, and for showing two manuscripts
prior to publication. F.J.G.-V. gratefully acknowledges ®nancial support from Spain's
Ministerio de Educacion y Ciencia. L.M.M. is supported by the CYCIT under contract
no. MAT94-0058-C02-01.
211
F. Wijnands et al.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
E. N. ECONOMOU, Green's Functions in Quantum Physics (Springer, Berlin, 1983).
E. W. KOLK, N. G. BAKEN and H. BLOK, IEEE Trans. Microwave Theory Tech.
Tech. 38 (1990) 78.
O. J. F. MARTIN, C. GIRARD and A. DEREUX, Phys. Rev. Lett. 74 (1995) 526.
E. YABLONOVITCH, T. J. GMITTER and R. BHAT, Phys. Rev. Lett. 61 (1988) 2546.
K. M. LEUNG and Y. F. LIU, Phys. Rev. Lett. 65 (1990) 2646.
Z. ZHANG and S. SATPATHY, Phys. Rev. Lett. 65 (1990) 2650.
K. M. HO, C. T. CHAN and C. M. SOUKOULIS, Phys. Rev. Lett. 65 (1990) 3152.
E. YABLONOVITCH, T. J. GMITTER and K. M. LEUNG, Phys. Rev. Lett. 67 (1991) 2295.
È RK, Coherence, Ampli®cation and Quantum E€ects in
Y. YAMAMOTO, S. MACHIDA, K. IGETA, and G. BJO
BJORK
10.
11.
12.
13.
14.
15.
16.
17.
18.
J. ISRAELACHVILI, Intermolecular and Surface Forces,
Forces, 2nd edn (Academic Press, London, 1991).
J. B. PENDRY, and A. MACKINNON, Phys. Rev. Lett. 69 (1992) 2772.
J. B. PENDRY, J. Mod. Opt. 41 (1994) 209.
P. M. BELL, J. B. PENDRY, L. MARTIN
MARTIÂN MORENO and A. J. WARD, Comp. Phys. Comm. 85 (1995) 306.
R. E. COLLIN, Field Theory of Guided Waves (IEEE Press, New York, 1991).
R. LOUDON, The Quantum Theory of Light (Clarendon Press, Oxford, 1983).
E. SNOEKS, A. LAGENDIJK and A. POLMAN, Phys. Rev. Lett. 74 (1995) 2459.
A. LAGENDIJK and B. A. VAN TIGGELEN, Phys. Rep. 270 (1996) 143.
R. SPRIK, A. LAGENDIJK and B. A. VAN TIGGELEN, in Photonic Band Gap Materials,
Materials, edited by
Semiconductor Lasers (Wiley, New York, 1991) 561.
C. M. Soukoulis (Kluwer, Dordrecht, 1996) p. 679.
19. H. KHOSRAVI and R. LOUDON, Proc. R. Soc. Lond. A 433 (1991) 337.
20. H. KHOSRAVI and R. LOUDON, Proc. R. Soc. Lond. A 436 (1992) 373.
21. W. M. ROBERTSON, G. ARJAVALINGAM, R. D. MEADE K. D. BROMMER, A. M. RAPPE and J. D. JOANNOPOULOS, Phys. Rev. Lett. 68 (1992) 2023.
Appendix 1
The central problem in this appendix is: suppose we know re¯ection and transmission
coecients through a ®nite slab of material on a plane wave basis (in our case the transfer
matrix basis), then how does the value of these coecients change when going to another
basis (in our case the Green's function basis).
A1.1. Basis and re¯ection coef®cients for transfer matrix
In the transfer matrix method, eigenvectors are constructed as follows. First, for given
incident parallel wavevector kk , with each reciprocal lattice vector g we associate a total
parallel wave vector given by kk ‡ g. Then kz follows from dispersion relation (22) in the
main text. Next, we construct vectors kEg and kH
g according to Equations 18 and 19 of the
main text, respectively. De®ne polarization states
i
0
E
H
n;
H
ˆ
k
k
n
Esg ˆ kH
g
sg
g
ax2 e0 l0 g
i
ia
H0pg ˆ
Hpg ˆ kEg n; Epg ˆ kH
kEg n
axe0
e g
where we introduced the reduced magnetic ®eld H0 …‡i=axe0 †H. Next, de®ne normalized six-component s- and p-polarization vectors:
"
^vr6;sg ˆ Nsg
212
Esg
H0sg
#
2
ˆ N sg 4
kH
g n
3
5
…i=ax2 e0 l0 †kEg kH
g n
Green's functions for Maxwell's equations
^vr6;pg
ˆ Npg
Epg
H0pg
"
ˆ Npg
#
ia kH kE n
g
e g
kEg n
…A1:1†
The Nsg , Npg are normalization factors such that the electromagnetic wave carries unity
current in the z-direction, provided the wave is not evanescent (cf. Appendix 2). As only
four of the six electromagnetic components are independent, we choose as right eigenvectors of the transfer matrix the Ex ; Ey ; Hx0 ; Hy0 -components of the six vectors:
h
iT
^vr4;rg ˆ …Ex †^vr ; …Ey †^vr ; …Hx0 †^vr ; …Hy0 †^vr
6;rg
6;rg
The left eigenvectors are de®ned as
2
3
0 ÿ1
0
0
61
0
0
0 7
6
a2 x 2 7
6
7 r
0
0
0
7
^vl4;sg ˆ 6
6
c20 e 7^v4;pg ;
6
7
2
2
4
5
ax
0
0 ÿ 2
0
c0 e
6;rg
2
^vl4;pg
0
61
6
6
0
ˆ6
6
6
4
0
6;rg
ÿ1
0
0
0
0
0
0
ÿ
a2 x 2
c20 e
3
0
0 7
a2 x 2 7
7 r
7
c20 e 7^v4;sg
7
5
0
Finally the left eigenvectors are normalized:
‰^vl4;rg ŠT ^vr4;rg ˆ 1
…A1:2†
The re¯ection and transmission coecients denote the following. Consider an incident
‡
right eigenvector ^vr4;r0 g0 travelling in the ‡z-direction. This wave enters a slab of material.
Eventually, we end up with a transmitted ®eld at the far end and a re¯ected ®eld at the
incident surface of the slab. The re¯ection coecient Rrg;r0 g0 …^v4 † denotes the re¯ected ®eld,
ÿ
projected upon the left eigenvector ^vl4;rg . But due to Equation A1.2, this projected ®eld
ÿ
corresponds to a re¯ected wave R1;rg;r0 g0 …^v4 †^vr4;rg , moving in the ÿz-direction. The transmission coecient Trg;r0 g0 …^v4 † denotes the transmitted ®eld, projected upon the left
‡
‡
eigenvector ^vl4;rg . This projected ®eld corresponds to a transmitted wave Trg;r0 g0 …^v4 †^vr4;rg ,
moving in the ‡z-direction. This agrees with our intuitive interpretation of the re¯ection
and transmission coecient.
A1.2. Basis and re¯ection coef®cients for Green's function
The electric ®eld right eigenvectors can be chosen to be just the E-components of the sixcomponent eigenvectors, de®ned in Equation A1.1. The right and left eigenvectors are
then de®ned as
‡ÿ
^vr3;sg ˆ Nsg ‰kH
g
‡ÿ
^vr3;pg ˆ Npg
‡ÿ
^vl3;sg
‡ÿ
^vl3;pg
‡ÿ
nŠ
ia H‡ÿ
‡ÿ
‰k
…kEg n†Š
eref g
ˆ
‡ÿ
Lsg kEg
ˆ
‡ÿ
Lpg kEg
‡ÿ
‰kH
g
‡ÿ
…kH
g
‡ÿ
…kEg
…A1:3†
n†Š
n†
Here Lsg , Lpg are chosen such that the normalization condition (21) of the main text, is
satis®ed. It can be checked that with this de®nition of the left eigenvectors, the orthogonality relations are ful®lled.
213
F. Wijnands et al.
The choice (A1.3) is not the most general choice, however. If we would multiply every
right eigenvector by some complex number a, and the corresponding left eigenvector by
1=a (with regard to Equation 21 in the main text), that would be a valid choice as well.
Hence the most general choice for our normalized right and left eigenvectors is
‡ÿ
H
^vr3;sg ˆ a‡ÿ
sg Nsg ‰kg
‡ÿ
^vr3;pg ˆ a‡ÿ
pg Npg
1
‡ÿ
^vl3;sg ˆ
a‡ÿ
sg
1
‡ÿ
^vl3;pg ˆ
a‡ÿ
pg
‡ÿ
nŠ
ia H‡ÿ
‡ÿ
‰k
…kEg n†Š
eref g
Lsg kEg
‡ÿ
‰kH
g
‡ÿ
…KH
g
Lpg kEg
‡ÿ
…kEg
‡ÿ
‡ÿ
n†Š
…A1:4†
n†
The most appropriate choice, however, is the choice (A1.3)
a‡
rg ˆ 1;
aÿ
rg ˆ 1;
8r; 8g
The reason is that for this choice, the right eigenvectors of the transfer matrix and of the
Green's function both refer to exactly the same electromagnetic wave. In that case the
re¯ection coecients, obtained with the transfer matrix method, can be directly inserted in
the Green's function. If the electromagnetic waves corresponding to four-component and
three-component eigenvectors are scaled with respect to each other (as is the case for the
general choice (A1.4), then the re¯ection and transmission coecients have to be renormalized when going from one basis to the other basis, as follows.
A1.3. Transformation of re¯ection and transmission coef®cients for
change of basis
Consider waves travelling in the ‡z-direction, incident on a slab of material. Suppose we
know re¯ection and transmission coecients Rrg;r0 g0 …^v4 † and Trg;r0 g0 …^v4 † on the transfer
matrix four-component basis. Suppose the basis set used for the Green's function is in the
‡ÿ
class (A1.4), so the electromagnetic wave associated to ^vr3;rj gj is a factor a‡ÿ
rj gj larger than the
‡ÿ
r
electromagnetic wave associated to ^v4;rj gj .
A1.3.1. Re¯ection coef®cients
The problem is to ®nd the re¯ection matrix coecient Rrg;r0 g0 …^v3 † corresponding to a
ÿ
‡
re¯ected wave Rrg;r0 g0 …^v3 †^vr3;rg , given an incident right eigenvector ^vr3;r0 g0 .
It is convenient to treat the renormalization of the incident and the scattered waves
separately.
‡
If the incident electromagnetic wave associated with the right eigenvector ^vr3;r0 g0 is a
‡
vr4;r0 g0 , then the amplitude
factor a‡
r0 g0 larger than the electromagnetic wave associated with ^
‡
of the total scattered ®eld will be a factor ar0 g0 larger as well. In particular, the re¯ection
coecient will be a factor a‡
r0 g0 larger.
Given a total re¯ected ®eld, say W, suppose this ®eld is projected upon the left eigenÿ
vector ^vl4;rg , to give projection value R1;rg;r0 g0 …^v4 †. This corresponds to a wave
ÿ
ÿ
R1;rg;r0 g0 …^v4 †^vr4;rg . If we project upon the left eigenvector ^vl3;r0 g0 ; with projection value
ÿ
R1;rg;r0 g0 …^v3 †; this corresponds to a wave R1;rg;r0 g0 …^v3 †^vr3;rg : But as we assumed a ®xed re¯ected ®eld W; it should hold that
214
Green's functions for Maxwell's equations
ÿ
ÿ
Rrg;r0 g0 …^v4 †^vr4;rg ˆ Rrg;r0 g0 …^v3 †^vr3;rg
ÿ
ÿ
Hence, projecting upon ^vl3;r0 g0 instead of ^vl4;r0 g0 ; gives a factor 1=aÿ
rg for the re¯ection
coecient.
Combining these two arguments, Rrg;r0 g0 …^v4 † and Rrg;r0 g0 …^v3 †; are related by
Rrg;r0 g0 …^v3 † ˆ
a‡
r0 g 0
aÿ
rg
Rrg;r0 g0 …^v4 †
…A1:5†
Trg;r0 g0 …^v4 †
…A1:6†
A1.3.2. Transmission coef®cients
Trg;r0 g0 …^v3 † ˆ
a‡
r0 g 0
a‡
rg
Hence the most convenient choice for the Green's function basis is the choice (A1.3): all
ÿ
constants a‡
rg ˆ 1; arg ˆ 1; 8r; 8g. Then the re¯ection and transmission coecients, determined on the transfer matrix basis, do not have to be renormalized when using these
re¯ection coecients in the Green's function.
Appendix 2. Current conservation
A very useful general test for the correctness of our computer codes is current conservation. Pendry [12] has formulated the current in the z-direction on a lattice. For the transfer
matrix four-component eigenvectors, a current conservation test has been described by
Bell et al. [13]. After recapitulating these results, we derive the current conservation formula for the three-component electric ®eld eigenvectors used in the Green's function.
A2.1. Current conservation on a four-component basis
Suppose space is discretized on a cubic mesh with unit mesh vectors a; b; c; The current in
the z-direction is given by [12]:
1
J …r ‡ c; r† ˆ ‰Ex …r ‡ c†Hy …r† ÿ Ey …r ‡ c†Hx …r†Š
a
For complex ®elds, and using reduced magnetic ®elds, the current used in the computer
codes is
Hy0 …r†
H 0 …r†
ÿ fexp…ikz a†Ey …r†g x
J 0 …r ‡ c; r† ˆ R fexp…ikz a†Ex …r†g
…A2:1†
i
i
We de®ne a normalized current J^0 …r ‡ c; r; ^vr4;rg † being the current in the z-direction ac‡ÿ
cording to Equation A2.1, for a three-component basis vector ^vr3;rg . The eigenvectors are
‡ÿ
0
r
normalized such that J^ …r ‡ c; r; ^v4;rg † ˆ 1. The current conservation test is then to check
‡
for each incident right eigenvector vr4;r0 g0 , the relation
o
Xn
‡
ÿ
jR1;rg;r0 g0 …^v4 †j2 J^0 …r ‡ c; r; ^vr4;rg †
J^0 …r ‡ c; r; ^vr4;r0 g0 † ‡
‡ÿ
ÿ
Xn
rg
rg
o
‡
jT2;rg;r0 g0 …^v4 †j2 J^0 …r ‡ c; r; ^vr4;rg † ˆ 0
…A2:2†
and analogously for incident waves travelling in the ÿz-direction. Here `1', `2' refer to the
lower and upper surface, respectively, of the slab of material.
215
F. Wijnands et al.
A2.2. Current conservation on a three-component basis
The expressions just obtained can be rewritten in terms of three-component electric ®eld
eigenvectors, by expressing Hx ; Hy in Ex ; Ey ; Ez [12]. The current in the z-direction (Equation A2.1) then takes the form
‰exp…ikz a† ÿ 1ŠEx …r† ÿ ‰exp…ikx a† ÿ 1ŠEz …r†
J 0 …r ‡ c; r† ˆ R fexp…ikz a†Ex …r†g
i
‰exp…iky a† ÿ 1ŠEz …r† ÿ ‰exp…ikz a† ÿ 1ŠEy …r†
…A2:3†
ÿ exp…ikz a†Ey …r†
i
Assuming that current conservation on the four-component basis is valid, we check
whether the same is true on the three-component basis in the class (A1.4). We shall use the
transformation rules for re¯ection and transmission coecients, given by Equations A1.5
and A1.6.
Suppose there is current conservation for a given incident normalized four-component
‡ÿ
eigenvector, as expressed in Equation A2.2. De®ne a normalized current J^0 …r ‡ c; r; ^vr3;rg †
being the current in the z-direction according to Equation A2.3, for a three-component
‡ÿ
basis vector ^vr3;rg . The normalized currents in the z-direction for the two basis sets are
related as
‡ÿ
J^0 …r ‡ c; r; ^vr4;rg † ˆ
1
2
ja‡ÿ
rg j
‡ÿ
J^0 …r ‡ c; r; ^vr3;rg †
The current expression (A2.2) can be re-expressed in terms of normalized currents
‡ÿ
J^0 …r ‡ c; r; ^vr3;rg † using the renormalization properties of the re¯ection and transmission
coecients given in Appendix 1. For example, for incident waves travelling in the ‡zdirection, the result is
1
2
ja‡
r0 g 0 j
‡
‡
J^0 …r ‡ c; r; ^vr3;r0 g0 †
(
2
X jaÿ
rg j
rg
ÿ
2
ja‡
r0 g 0 j
(
2
X ja‡
rg j
rg
2
ja‡
r0 g 0 j
2
jR1;rg;r0 g0 …^v3 †j
jT2;rg;r0 g0 …^v3 †j2
in other words
‡
J^ 0 …r ‡ c; r; ^vr3;r0 g0 † ‡
ÿ
Xn
rg
Xn
rg
1
2
jaÿ
rg j
1
2
ja‡
rg j
)
^0
J …r ‡
ÿ
c; r; ^vr3;rg †
)
J^0 …r ‡ c; r; ^vr3;rg †
‡
ˆ0
jR1;rg;r0 g0 …^v3 †j2 J^0 …r ‡ c; r; ^vr3;rg †
ÿ
o
o
‡
jT1;rg;r0 g0 …^v3 †j2 J^0 …r ‡ c; r; ^vr3;rg † ˆ 0
Therefore, if current in the z-direction is conserved on the four-component basis, it is also
conserved on the three-component basis, as it should.
216