The exact solution of self-consistent equations in the scanning near-field optic microscopy problem
The macroscopic approach that allows one to obtain an exact solution of the self-consistent equation of the Lippmann-Schwinger type is developed. The main idea of our method consist in usage of diagram technique for exact summation of the infinite series corresponding to the iteration procedure for...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
1999
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nasplib_isofts_kiev_ua-123456789-1198662025-02-23T18:00:48Z The exact solution of self-consistent equations in the scanning near-field optic microscopy problem Lozovski, V. Bozhevolnyi, S. The macroscopic approach that allows one to obtain an exact solution of the self-consistent equation of the Lippmann-Schwinger type is developed. The main idea of our method consist in usage of diagram technique for exact summation of the infinite series corresponding to the iteration procedure for solving the self-consistent integral equation. The method developed is applied to calculations of near-field optical images obtained in illumination mode. It is assumed that the system under consideration consists of an object illuminated by the field scattered by a small probe. This assumption allows us to consider multiple scattering between a (point-like) probe and an extended object as well as inside the object. The exact solution for the self-consistent field is then obtained in terms of effective susceptibility of the probe-object system. Application of our method to the description of orientation of molecular complexes at the surface is discussed. 1999 Article The exact solution of self-consistent equations in the scanning near-field optic microscopy problem / V. Lozovski, S. Bozhevolnyi // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 3. — С. 45-56. — Бібліогр.: 28 назв. — англ. 1560-8034 PACS: 42.65.K https://nasplib.isofts.kiev.ua/handle/123456789/119866 en Semiconductor Physics Quantum Electronics & Optoelectronics application/pdf Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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The macroscopic approach that allows one to obtain an exact solution of the self-consistent equation of the Lippmann-Schwinger type is developed. The main idea of our method consist in usage of diagram technique for exact summation of the infinite series corresponding to the iteration procedure for solving the self-consistent integral equation. The method developed is applied to calculations of near-field optical images obtained in illumination mode. It is assumed that the system under consideration consists of an object illuminated by the field scattered by a small probe. This assumption allows us to consider multiple scattering between a (point-like) probe and an extended object as well as inside the object. The exact solution for the self-consistent field is then obtained in terms of effective susceptibility of the probe-object system. Application of our method to the description of orientation of molecular complexes at the surface is discussed. |
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| author |
Lozovski, V. Bozhevolnyi, S. |
| spellingShingle |
Lozovski, V. Bozhevolnyi, S. The exact solution of self-consistent equations in the scanning near-field optic microscopy problem Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Lozovski, V. Bozhevolnyi, S. |
| author_sort |
Lozovski, V. |
| title |
The exact solution of self-consistent equations in the scanning near-field optic microscopy problem |
| title_short |
The exact solution of self-consistent equations in the scanning near-field optic microscopy problem |
| title_full |
The exact solution of self-consistent equations in the scanning near-field optic microscopy problem |
| title_fullStr |
The exact solution of self-consistent equations in the scanning near-field optic microscopy problem |
| title_full_unstemmed |
The exact solution of self-consistent equations in the scanning near-field optic microscopy problem |
| title_sort |
exact solution of self-consistent equations in the scanning near-field optic microscopy problem |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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1999 |
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https://nasplib.isofts.kiev.ua/handle/123456789/119866 |
| citation_txt |
The exact solution of self-consistent equations in the scanning near-field optic microscopy problem / V. Lozovski, S. Bozhevolnyi // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 3. — С. 45-56. — Бібліогр.: 28 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT lozovskiv theexactsolutionofselfconsistentequationsinthescanningnearfieldopticmicroscopyproblem AT bozhevolnyis theexactsolutionofselfconsistentequationsinthescanningnearfieldopticmicroscopyproblem AT lozovskiv exactsolutionofselfconsistentequationsinthescanningnearfieldopticmicroscopyproblem AT bozhevolnyis exactsolutionofselfconsistentequationsinthescanningnearfieldopticmicroscopyproblem |
| first_indexed |
2025-11-24T06:15:44Z |
| last_indexed |
2025-11-24T06:15:44Z |
| _version_ |
1849651305738403840 |
| fulltext |
45© 1999, Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Semiconductor Physics, Quantum Electronics & Optoelectronics. 1999. V. 2, N 3. P. 45-56.
1. Introduction
Experimental studies of nano-scale systems such as quan-
tum dots, molecular complexes at the surface, etc. have been
intensively carried out by numerous research groups in re-
cent years. A very informative tool for high spatial resolu-
tion optical measurements is scanning near-field optical
microscopy (SNOM). In SNOM, the modifications of elec-
tromagnetic radiation, which are caused by the interaction
of a SNOM probe with an object smaller than a wavelength
of light, are detected. The SNOM technique has been suc-
cessfully used to image with subwavelenth resolution vari-
ous surface structures exhibiting different optical contrast
mechanisms, such as real and imaginary parts of refractive
index, polarization and fluorescence [1]. Quite recently, the
first experimental SNOM images related to variations in
second-order susceptibilities, i.e. near-field optical images
showing nonlinear optical contrast, have been reported [2,3].
Most theoretical papers dealing with the self-consistent
problem in near-field optics are based on the consideration
of the following linear integral equation (usually called
Lippmann-Schwinger equation) [4-6]:
∫ ′′′−=
V
RERRGRdiRERE ),()(),,(),(),( 0
)0( ωωχωωµωω
rrtrrtrrrrr
. (1)
Here, ),,( ωRRG ′
rrt
is the Green dyadic of the space in which
the particle having the volume V and linear susceptibility
tensor )(ωχt is situated. One of the way for obtaining the
solution of Eq.(1) is discretization of Eq.(1) leading to the
(self-consistent) system of linear algebraic equations [7]:
.),()(),,(),(),(
1
0
)0( ∑
=
−=
N
i
iiaaa RERRGiRERE ωωχωωµωω
rrtrrtrrrr
(2)
In Eq.(2) it is implicitly assumed that the self-consistent field
is constant inside of each of N subvolumes. The approxi-
mation (2) is thereby practically equivalent to the point-like
(dipole) approximation. Numerous results were obtained by
using Eq.(2) [5, 6, 8-11].
Another way of solving the self-consistent problem (1)
consists in using the iteration procedure, i.e., Born approxi-
mation of a finite (n-th) order. This procedure implies that
multiple scattering inside the system is treated taking into
account the scattering processes up to the n-th order. Thus,
the first Born approximation )()1( RE
rr
is obtained from
Eq.(1) with the incidence field )()0( RE
rr
been substituted
into the integrand. This approximation corresponds to the
regime of single scattering. To take into account secondary
scattered waves one should invoke the second Born approxi-
PACS: 42.65.K
The exact solution of self-consistent equations in the
scanning near-field optic microscopy problem
V. Lozovski1, S. Bozhevolnyi2
1 Institute of Semiconductor Physics NAS of Ukraine,
45, prospect Nauki, 252650 Kyiv, Ukraine;Phone/Fax: (+380) 44 2655530; E-mail:lozovski@mol.semicond.kiev.ua
2 Research Center COM, DTU, Bldg.345 east, DK-2800 Lyngby, Denmark;Fax: (+45)45936581;
Phone: (+45)45255763; E-mail: sib@com.dtu.dk
Abstract. The macroscopic approach that allows one to obtain an exact solution of the self-consistent
equation of the Lippmann-Schwinger type is developed. The main idea of our method consist in usage
of diagram technique for exact summation of the infinite series corresponding to the iteration procedure
for solving the self-consistent integral equation. The method developed is applied to calculations of
near-field optical images obtained in illumination mode. It is assumed that the system under considera-
tion consists of an object illuminated by the field scattered by a small probe. This assumption allows us
to consider multiple scattering between a (point-like) probe and an extended object as well as inside the
object. The exact solution for the self-consistent field is then obtained in terms of effective susceptibility
of the probe-object system. Application of our method to the description of orientation of molecular
complexes at the surface is discussed.
Keywords: near-field image, self-consistent equation
Paper received 14.09.99; revised manuscript received 28.09.99; accepted for publication 11.10.99.
V. Lozovski, S. Bozhevolnyi: The exact solution of self-consistent equations...
46 SQO, 2(3), 1999
mation, in which the field )()1( RE
rr
found in the first ap-
proximation is substituted in the integrand of Eq.(1). In or-
der to include multiple scattered waves of the n-th order,
one should recurrently carry out the above procedure n times.
It should be emphasized, that method of direct solution
of self-consistent problem by numerical solution of the set
of the linear algebraic equations (2) is more correct then the
method, in which the n-th order Born approximation is em-
ployed. In order to clarify this point let us rewrite Eq.(1) in
the operator form
EJEE
rtrr
⋅+= )0( , (3)
with J
t
denoting the integral operator in the right part of
Eq.(1). The exact solution can then be formally expressed
in the form
( ) )0(1
EJUE
rttr
⋅−=
−
, (4)
where U
t
is the unite operator. The n-th order Born series
expansion becomes
( ) )0()0(2)0()0( )( EJEJEJEE n rt
K
rtrtrr
++⋅+⋅+= . (5)
This expansion cannot describe the resonant interactions
occurring when the (resonant) condition ( ) 0det =− JU
tt
is
satisfied. In such a case, any finite order of the Born ap-
proximation would not allow one to obtain the exact solu-
tion including resonances. To obtain an exact solution of
Eq.(1) one has to accomplish exact summation of an infi-
nite series corresponding to the iteration procedure.
The exact solution of the self-consistent integral equa-
tion (1) has been reported for some configurations. For the
situation, when particle susceptibility is a factorizing func-
tion of spatial coordinates, the effective method for direct
solution of Eq.(1) was proposed in Ref.[12]. On the other
hand, the approach using the diagram technique was pro-
posed for the same model of particle susceptibility in calcu-
lations concerned with nano-scale electrodynamics [13].
In the present work we develop the method which is based
on the diagram technique for exact solution of a self-con-
sistent equation. This method consists in exact summation
of infinite series corresponding to the iteration procedure
constructed for the self-consistent Lippmann-Schwin-ger
Eq. (1). Note that the field scattered by a probe is included
in the iteration procedure.
The proposed technique is demonstrated for general case
of spatial dependent susceptibility of a particle. Numerical
calculations are provided for simple configuration contain-
ing a probe and a sample with spatially independent suscep-
tibilities. Furthermore, in calculations we used the approxi-
mation of point-like probe and object in the form of paral-
lelepiped.
2. Theory
Let us consider the object and a probe contained in the
medium the electro-dynamical properties of which are char-
acterized by the Green function ),,( ωRRGij ′
rr
. Let us sup-
pose that object which, in general, can be a multi-connected
body is described by non-local susceptibilities. Namely, the
effective current in the object is connected with an electri-
cal field by integral constitutive equation
∫ ′′′=
sV
j
s
iji RERRRdRJ )(),()( )( rrrrr
χ , (6)
Integration in Eq.(6) is made over volume of an object which
in general case can consist of a few parts (to be multi-con-
nected one). It needs to make the next remark concerning
the form of the constitutive Eq. (6). Namely, there exist in-
dications that both linear and non-linear optical suscepti-
bilities can be in general represented in factorizing form,
i.e., the kernels of the integral Eq. (6) can be separated on
sum of products of functions dependent only on one vari-
able R
r
or R′
r
[14]. In this connection it could mind that in
general, the solution of Lippmann-Schwinger equation can
be obtained by methods developed for separable suscepti-
bility model (e.g. by the method developed in [12]). But,
firstly, the difficulties connected with solution of both «new»
constitutive equation and the Lippman-Schwinger equation
remain. And, secondly, there is no possibility to obtain a
self-consistent solution of Lippmann-Schwinger equation in
the case when the susceptibility )()( ωχ s
ij is independent on
spatial coordinates. Thus, the constitutive equation in the
form of Eq.(6) will be used in this work during determina-
tion of exact solution of the Lippmann-Schwinger equation.
Then, total field in the arbitrary point of the system must
obey the equation
)(),(),()()( )(
0
)0( RERRRdRRRdiRERE l
s
j
VV
ijii
ss
′′′′′′′′ℜ′−= ∫∫
rrrrrrrrr
χωµ ,
(7)
in which, for brevity, the generalized Green function
),( RRij ′ℜ
rr
is introduced. This Green function is a photon
propagator describing the propagation of photon from point
R′
r
to point R
r
of the system.
∫
∫
′′′′′′′′′′′′×
×′′′′−′=′ℜ
P
P
V
lj
p
lk
V
ilijij
RRGRRRd
RRGRdiRRGRR
),(),(
),(),(),(
)(
0
rrrrr
rrrrrrr
χ
ωµ
(8)
The first term in Eq.(8) describes the direct propagation of
photon from point R′
r
to point R
r
. The second term of Eq.(8)
describes a propagation of photon from point R′
r
to point
R
r
of the system by reradiation by the probe with volume VP
and susceptibility ),()( RRp
lk ′
rr
χ . Eq.(7) could be solved with
iteration method. The first step of an iteration procedure is
)()( )0()( RERE i
I
i
rr
= . (9)
The second step of an iteration procedure is
).(),(),()( )0()(
0
)( RERRRdRRRdiRE l
s
jl
VV
ij
II
i
ss
′′′′′′′′ℜ′−= ∫∫
rrrrrrrr
χωµ
(10)
The third step of an iteration procedure is
V. Lozovski, S. Bozhevolnyi: The exact solution of self-consistent equations...
47SQO, 2(3), 1999
)(),()(),( )()( RERRRdRERRRd A
VV
A rrrrtrrrrrtr
⋅′Φ′≅′′Φ′ ∫∫ .
It is followed by
0)(),(),(
),(),(),(
))(,(),(),(
)()(
)()(
0
)(
=′⋅
′′′′′′ℜ−
−′′′′′′′′ℜ′′′′′′′′×
×−′−
′′′′′′′
∫
∫∫∫
∫ ∫
RERRRdRR
RRRdRRRdRRRd
iRRgRRRdRRgRd
A
l
V
s
jlij
V
IVs
nl
IV
V
mn
s
jm
V
ij
V V
s
jlij
s
sss
s s
rrrrrr
rrrrrrrrr
rrrrrrrr
χ
χχ
ωµχ
(20)
Since a field )()( RE A
l ′
r
in the integrand in Eq.(20) is an
arbitrary function of variable R′
r
, to satisfy this equation it
needs to be
0),(
),(),())(,(
),(),(),(),(
)(
)(
0
)()(
=′′′×
×′′′′′ℜ′′′′′′′′−′−
−′′′′′′ℜ−′′′′′′
∫
∫∫
∫∫
s
ss
ss
V
IVs
nl
IV
V
mn
s
jm
V
ij
V
s
jlij
V
s
jlij
RRRd
RRRdRRRdiRRg
RRRdRRRRRdRRg
rrr
rrrrrrrr
rrrrrrrrrr
χ
χωµ
χχ
(21)
Introducing the designations
∫ ′′′′′=′
sV
s
jljl RRRdRM ),()( )( rrrr
χ , (22)
and
)(),(),()()( )(
0 RMRRRdRRRdiRW nl
V
mn
s
jm
V
jl
ss
′′′′′′′′ℜ′′′′′′′′−=′ ∫∫
rrrrrrrr
χωµ ,
(23)
one can obtain from Eq.(21)
[ ] 1
lnln )()()(),(),(
−′−′′′ℜ=′ RWRMRMRRRRg jnijil
rrrrrrr
. (24)
Then, from Eqs.(14) and (24) one can find
)(),(),()( )0()(
0 RERRRdRRgRdiRE j
V
s
lj
V
iliji
ss
rrrrrrrr
′′′′′′′′−= ∫∫ χωµδ .
(25)
This expression is an exact solution of the Lippmann-
Schwinger equation (7) for the system linear dimensions of
which are less then wavelength of external field )()0( REi
r
.
3. The effective susceptibility
The expression for the self-consistent field in the arbitrary
point of a system (Eq.(25)) can be rewritten in terms of ef-
).(),(),(
))(,(),()(
)0()(
0
)(
0
)(
IV
m
IVs
km
V
IV
V
lk
s
jl
VV
ij
III
i
RERRRdRRRd
iRRRdRRRdiRE
ss
ss
rrrrrrr
rrrrrrr
′′′′′′′′ℜ′′′×
×−′′′′′′ℜ′−=
∫∫
∫∫
χ
ωµχωµ
(11)
Thus, the exact solution of Eq.(7) can be written as a
infinite series of iteration procedure, which in the diagram
representation [15] has a form
⋅⋅⋅+→↔↔+→↔+→=)(REi
r
, (12)
where the next definitions are used
↔=′′′′′′ℜ′−
→=
∫ ∫
s sV V
s
jlij
i
RRRdRRRdi
RE
...),(),(
,)(
)(
0
)0(
rrrrrr
r
χωµ
(13)
The series (12) could be rewritten in the skeleton diagram
representation
→⇔+→=)(REi
r
, (14)
where operator
∫∫ ′′′′′′′−=⇔
ss V
s
jl
V
ij RRRdRRgRdi ),(),( )(
0
rrrrrr
χωµ (15)
containing an unknown function ),( RRgij ′
rr
is represented
as an infinite series
...+↔↔↔+↔↔+↔=⇔ (16)
This series by the next mathematical trick
{ }↔+↔↔+↔+↔=⇔
=⇔
44 344 21
K (17)
can be rewritten in the standard Dyson equation form [15,
16]
⇔↔+↔=⇔ . (18)
To find unknown function ),( RRgij ′
rr
let it acts by operator
(15) which is obeyed to equation (18) on an arbitrary long-
range field. As a result one can obtain the equation
.)(),(),()(
),(),()(),(
),()(),(),(
)()(
0
)()()(
)()(
∫ ∫
∫∫∫
∫∫ ∫
′′′′′′′′ℜ′′′−×
×′′′′′′′+′′′′′′′×
×′ℜ′=′′′′′′′′′
s s
sss
ss s
V
IVA
m
IVs
km
IV
V
lk
s
jl
VV
il
A
l
s
jl
V
V
ij
V
A
l
s
jl
V
ij
RERRRdRRRdi
RRRdRRgRdRERRRd
RRRdRERRRdRRgRd
rrrrrrr
rrrrrrrrrr
rrrrrrrrrr
χωµ
χχ
χ
(19)
Then one can use the fact that long-range arbitrary field
slowly changes on the distances of the order of linear di-
mensions of the system. It means that it is possible to
remove the field from the internal integrands in Eq.(19)
and use that
V. Lozovski, S. Bozhevolnyi: The exact solution of self-consistent equations...
48 SQO, 2(3), 1999
fective susceptibility of a system. Namely, using avowed
form of the function ),( RRgij ′
rr
it can be obtained from
Eq.(25)
)(),(),()( )0(
0 RERRRdRRRdiRE j
V V
ljiliji
s s
rrrrrrrr
′′′Χ′′′ℜ′−= ∫ ∫ωµδ ,
(26)
where a tensor
[ ] ),()()(),( )(11 RRRMRWRR s
kmnlknkllm ′′′′′−=′′′Χ
−− rrrrrr
χδ (27)
plays a role of effective susceptibility of the system. As it
takes after previous consideration, the value
)(),()( )0()( RERRRdRJ j
V
lj
eff
l
s
rrrrr
∫ ′′′Χ′′=′ (28)
is a current inside an object generated by external long-
range field. It means that tensor ),( RRlm ′′′Χ
rr
is a nonlocal
susceptibility, calculated with the account of all scattering
processes in the system.
The pole part of this tensor defines the resonances in the
system caused by multi-scattering processes into the sys-
tem. So-called configuration resonances [5] which depends
on mutual situation between different parts of a system are
described by poles of tensor ),( RRlm ′′′Χ
rr
, too. These
resonances are depended on the dimension and shape of an
object. The condition of resonance interaction is determined
by usual manner
[ ] 0)()(det 1 =′′− − RMRW nlknkl
rr
δ . (29)
The real and imaginary parts of an effective susceptibil-
ity must satisfy the Kramers-Kronig relations [17], which
express the causality principle. The real and imaginary parts
of ),( RRlm ′′′Χ
rr
must obey, moreover, optical theorem
conditions [18-20], which express the energy conservation
law. Both these fundamental principles can be considered
as an examination requirements for effective susceptibility
calculated in the framework of determined model.
4. The separable susceptibility
There exist a special case of a nonlocal susceptibility -
factorizing susceptibility. Namely, it can be often supposed
that susceptibility of a nano-object has a form [12, 14, 21]
)()(),(
0
RjRj
ia
RR jiij ′−=′
rrrr
ωµ
χ (30)
with
∑ −++
−=
nm nm
nm
EEi
ff
a
,
0 )( νω
µ
h
. (31)
Where )(Rji
r
and )(Rji ′
r
are the transition current densi-
ties between one-electron eigenstates m and n calculated at
the space points R
r
and R′
r
, respectively. The quantities
mf and nf are Fermi-Dirac distribution factors giving the
probabilities that the states m and n are occupied in thermal
equilibrium [12, 21].
For simplicity used was the two-level model suscepti-
bility of an object containing only transitions between n =
0 and m = 1 states. Then, using Eqs (22) and (23) one ob-
tains
)0(
0
)()( ljjl Rj
ia
RM γ
ωµ
′−=′
rr
, (32)
and
)0(
0
)()( ljjl aNRj
ia
RW γ
ωµ
′−=′
rr
, (33)
where
∫=
sV
ll RjRd )()0( rr
γ . (34)
In according to Ref.[12], let us introduce a parameter
)(),()( RjRRRdRjRdN n
V
mnm
V ss
′′′′′′′′ℜ′′′′′′′= ∫∫
rrrrrr
, (35)
then, substituting Eqs.(32)-(35) into Eq.(24), one obtains
( ) [ ] 11)0()0( 1)()(),(),( −−
−′′′ℜ=′ aNRjRjRRRRg nlnjijil γγ
rrrrrr
. (36)
Using the next equality
( ) jlnlnj RjRj δγγ =′′
−1)0()0( )()(
rr
, (37)
from Eq.(36) we can derive an expression
[ ] 11),(),( −−′ℜ=′ aNRRRRg ijil
rrrr
. (38)
At last, substitution of Eq.(38) into expression for self-
consistent field (27) gives the expression for a local field of
the system characterized by nonlocal susceptibility (30).
Namely,
)()(
1
1
)()( )0()0(
pjjiiji RERF
aN
aRRE
rrrr
−
−= γδ , (39)
where expression in the brackets plays the role of so-called
local field factor [12], and
∫ ′′ℜ′=
sV
lili RjRRRdRF )(),()(
rrrrr
. (40)
As it is easily seen, Eqs (39) and (40) are the expressions
obtained earlier (See, e.g., Ref.[12]) with other method.
4. The system consisting of a small probe and
an object characterized by susceptibility
independent on spatial coordinates
Let us consider the system consisting of an object with linear
dimensions less then wavelength of an external field and
small probe which could be considered as a point-like par-
ticle. Let us assume that probe is small as compared with
characteristic wavelength of a field. It means that for
arbitrary long-range field the equation for reradiated field
by a probe can be written in the form (Cf. Eq.(8))
V. Lozovski, S. Bozhevolnyi: The exact solution of self-consistent equations...
49SQO, 2(3), 1999
)(),(
)(),(),()(
)(
)()(
0
)(
p
A
jpij
V
A
j
p
lj
V
il
R
i
RERRG
RERRRdRRGRdiRE
p p
rrr
rrrrrrrr
≅
≅′′′′′′′′′−= ∫ ∫ χωµ
(41)
where
),(),(),( )(
0 RRRdRRGRdiRRG p
lj
VV
ilpij
pp
′′′′′′′−= ∫∫
rrrrrrrr
χωµ .
(42)
It means that the generalized Green function describing
the electrodynamical properties of the media in which an
object situated have to consist of two parts
),(),(),(),( RRGRRGRRGRR pljpilijij ′⋅+′=′ℜ
rrrrrrrr
, (43)
This Green function is a photon propagator describing
the propagation of photon from point R′
r
to point R
r
of the
system. The first term of Eq.(43) describes the direct
propagation of a photon from point R′
r
to point R
r
. The
second term describes propagation of a photon from point
R′
r
to point R
r
, which is re-radiated by a point-like probe
situated at point pR
r
. In the case of independent on spatial
coordinate susceptibility of an object and a small probe,
Eq.(26) can be rewritten as
)()()(),()()( )0(
0 pj
V
mjlmiliji RERIRRRRdiRIRE
s
rrrrrrrr
′′Χ′ℜ′−= ∫ωµ ,
(44)
with effective susceptibility of a system
[ ] 11)( )()()(
−− −=Χ RDR ji
s
ijij
rr
χ , (45)
and «incoming field» tensor
),()( pijijij RRGRI
rrr
+= δ . (46)
In Eq.(45) the designation
∫ ′ℜ′−=
sV
ijij RRRdiRD ),()( 0
rrrr
ωµ (47)
is used for the self-energy part. It should be emphasized
that because generalized Green function is in the integrand
of Eq.(47), self-energy part describes all scattering proc-
esses including scattering of the field between a probe and
an object.
These equations will be used for numerical calculations
of near-field image of an object in the next part of the work.
5. Numerical calculations
To illustrate the developed method we calculated the image
of near-field for the system consisting of rectangular dielec-
tric parallelepiped and small spherical probe. For simplic-
ity, it is supposed that susceptibilities of an object and a
probe are independent on spatial coordinates. It means that
a probe is rather small for the approximation (40) to be valid.
Calculation was made for a model shown in Fig.1. It is sup-
posed that a small probe is scanned along the scanning plane.
A probe re-radiates the external field acting on the system.
Due to action of both external and re-radiated fields on the
object the currents are generated inside the object. These
currents cause radiation, which is detected as a near-field
image. The dependence of intensity of near-field image
on a probe position in the scanning plane was calculated.
Cartesian coordinate system was chosen with origin situated
at the center of an object.
In recent years the interest to nano-technologies based
not only on solid state materials but on molecular technolo-
gies is permanently increased [22, 23]. In this connection
the problem of molecular or molecular complexes orienta-
tion at the surface becomes important. There is a big num-
ber of the papers devoted both to theoretical and experi-
mental studies of the problem of orientation of molecules
and molecular clusters at the surface [24,25]. On the other
hand, the SNOM technique allows to obtain different imag-
es of different oriented nanostructures [26]. These facts al-
lows us to propose this new method of calculation of near-
field image to problem of termining orientation of nano-
objects with respect to axes of external field polarization.
These reasons were taken into account when three types of
objects are used in the calculations: an object of the 1-st
type is characterized by dimensions 40x200x20 nm3, with a
short side along Oz axis. An object of the 2-nd type is char-
acterized by dimensions 400x200x20 nm3 , with a short side
along Oz axis. And object of the 3-d type is characterized
by dimensions 200x200x20 nm3, with a short side along Oz
axis. Radius of a probe is much less then wavelength
λ<<0r , namely r0 = 4 nm, λ = 800 nm. Due to these
circumstances the near-field and middle-filed parts of the
sa m p le
z
x
y
p rob e
in c iden t
f ie ld
d etec to r
Fig.1. The scheme of SNOM experiment for which the calculations were
made in the work.
V. Lozovski, S. Bozhevolnyi: The exact solution of self-consistent equations...
50 SQO, 2(3), 1999
photon propagator were used in numerical calculations [5,
12]. Namely,
( ) d
dd
Rci
RR
dd
dd
eee
R
c
R
ic
U
R
c
R
ic
RRG
ω
ωω
ωωπ
⋅
−+
+⋅
+−=′
rr
trrt
22
2
2
32
2
3
33
4
1
),(
(48)
with U
t
unit dyadic, RRRd ′−=
rrr
, RRR ′−=
rr
and RReR /
rr
= .
Since the external field supposed as monochromatic one,
dielectric constants of both probe and object were supposed
to be constant and equal to εpr = 2.25 and ε =3 respectively.
Calculated was the value, which characterizes the in-
tensity of a light forming the near-field image in far zone. It
is clear that this intensity is equal to
2
)(),(
4
RERRGJ
sV
FF
E ′′−= ∫ −
rrrrt
π
χ
(49)
with ),( RRG FF ′−
rrt
far-field part of the Green dyadic. Since
the distance at which the near-field image is detected, Dr,
is much more than a linear dimension of an object, one can
write
J =
2
0 ),(
4
RRG FF
E
rrt
−−
π
χ
2
)(RERd
Vs
rrr
′′∫ (50)
with R0 coordinate of the center of an object. So, due to the
distance 0RRDr
rr
−= that is constant in the defined ex-
periment, the intensity of light which forms the near-field
image can be written as
2
)(RERdConstJ
sV
′′⋅= ∫
rrr
, (51)
This value is calculated in the present work. In Eq.(51) )(RE
rr
is the self-consistent field calculated in according to Eq.(44).
In the numerical calculations the problem of divergence
of integrals at 0=′− RR
rr
arises. For solution this problem
it is necessary to use the approach developed in the works
of van Bladel [26] and Yaghjian [27] in which the idea of
exclusive volume was discussed. This approach consist in
the formal procedure
00
0
)(
)(),(lim)(
ωµ
ωµ
δδ i
RJL
RdRJRRGiRE
VVs
rrt
rrrrrtrr ⋅−′′⋅′−= ∫ −→
,
(52)
where L
t
is a source dyadic which depends solely on the ge-
ometry of a «principal volume», δV , which excludes the sin-
gularity of G
t
and becomes infinitesimally small in the limit
as its maximum chord length δ approaches to zero [27]. The
dyadic L
t
for a rectangular parallelepiped was calculated in
[28] and is equal to ( )zzzyyyxxx eeeeee
rrrrrr
Ω+Ω+Ω
π4
1
with
-0.50
-0.25
0
0.25
0.50 -0.50
-0.25
0
0.25
0.50
0.54
0.57
0.60
0.63
E
E
x
x
(0
)
X , 8 00 n mx Y , 8 00 n mx
E
E
y
x
(0
)
X , 8 00 n mx Y , 8 00 n mx
-0.50
0.50 -0.50
-0.04
0
0.04
0.08
E
E
x
z
(0
)
X , 8 00 n mx Y, 8 00 n mx
-0.50 -0.50
-0.50
-0.10
-0.05
0
0.05
0.10
Fig.2. The 3d-field distribution at the probe scanning along scanning plane
situated at the distance of 30 nm from an object that is rectangular bulk
with dimensions (200x200x20 nm3).
a
b
c
V. Lozovski, S. Bozhevolnyi: The exact solution of self-consistent equations...
51SQO, 2(3), 1999
Fig.3. The gray scale map representation of near-field image calculations for stick-like object (80x400x20 nm3) when the distance between a scanned
plane and an object is equal to 30 nm. (a) - both external and detected fields are polarized along Ox axes; (b) - both external and detected fields are
polarized along Oy axes; (c) - the external field is polarized along Ox axes and detected field is polarized along Oy axes; (d) - the external field is
polarized along Oy axes and detected field is polarized along Ox axes.
-0.2 0 0.2
-1.0-1.0-1.0
-0.5-0.5-0.5
000
0.50.50.5
1.01.01.0
19.69 -- 21.00
18.38 -- 19.69
17.06 -- 18.38
15.75 -- 17.06
14.44 -- 15.75
13.13 -- 14.44
11.81 -- 13.13
10.50 -- 11.81
9.188 -- 10.50
7.875 -- 9.188
6.563 -- 7.875
5.250 -- 6.563
3.938 -- 5.250
2.625 -- 3.938
1.313 -- 2.625
0 -- 1.313
X , 8 00 n mx
Y
,
80
0
nm
x
Y
,
80
0
nm
x
Y
,
80
0
nm
x
-0.2 0 0.2
-1.0
-0.5
0
0.5
1.0
X , 8 00 n mx
Y
,
80
0
nm
x
0.3469 -- 0.3700
0.3238 -- 0.3469
0.3006 -- 0.3238
0.2775 -- 0.3006
0.2544 -- 0.2775
0.2313 -- 0.2544
0.2081 -- 0.2313
0.1850 -- 0.2081
0.1619 -- 0.1850
0.1388 -- 0.1619
0.1156 -- 0.1388
0.0925 -- 0.1156
0.0694 -- 0.0925
0.0463 -- 0.0694
0.02313 -- 0.0463
0 -- 0.02313
-0.2 0 0.2
-1.0-1.0-1.0
-0.5-0.5-0.5
000
0.50.50.5
1.01.01.0
X , 8 00 n mx
Y
,
80
0
nm
x
Y
,
80
0
nm
x
Y
,
80
0
nm
x
0.0722 -- 0.0770
0.0674 -- 0.0722
0.0626 -- 0.0674
0.0578 -- 0.0626
0.0529 -- 0.0578
0.0481 -- 0.0529
0.0433 -- 0.0481
0.0385 -- 0.0433
0.0337 -- 0.0385
0.02888 -- 0.0337
0.02406 -- 0.02888
0.01925 -- 0.02406
0.01444 -- 0.01925
0.00963 -- 0.01444
0.00481 -- 0.00963
0 -- 0.00481
-0.2 0 0.2
-1.0-1.0-1.0
-0.5-0.5-0.5
000
0.50.50.5
1.01.01.0
X , 8 00 n mx
Y
,
80
0
nm
x
Y
,
80
0
nm
x
Y
,
80
0
nm
x
2.250 -- 2.400
2.100 -- 2.250
1.950 -- 2.100
1.800 -- 1.950
1.650 -- 1.800
1.500 -- 1.650
1.350 -- 1.500
1.200 -- 1.350
1.050 -- 1.200
0.9000 -- 1.050
0.7500 -- 0.9000
0.6000 -- 0.7500
0.4500 -- 0.6000
0.3000 -- 0.4500
0.1500 -- 0.3000
0 -- 0.1500
zyx ΩΩΩ , , twice the solid angle subtended by a side
perpendicular to the x, y, z direction, respectively
( )π4=Ω+Ω+Ω zyx [27].
The near-field images were calculated for different val-
ues of distance l between a scanning plane and an object.
The main attention is paid to the influence of mutual
orientations of an object and external field polarization on
formation of near-field object image.
The first series of calculations was made for stick-like
object (the 1-type object). This model calculation can give
the notion about a possible near-field images of stick-like
polymers or molecular complexes. The results of these cal-
culations are shown in Figs 2-5. It needs to be pointed that
increasing of the distance between scanning plane and an
object leads not only to some smearing out the image and
decreasing of intensity of image. In the case of external light
polarization along a long axes of an object the near-field
image can have rather complicated structure (see, Fig.5) and
very similar to near-field image given by linear chain of a di-
poles.
a b
c d
V. Lozovski, S. Bozhevolnyi: The exact solution of self-consistent equations...
52 SQO, 2(3), 1999
The results of calculations of a near-field image of rec-
tangular bulk 2-nd type object are shown in Figs 6 and 7.
The results of the same calculations for rectangular bulk
3-d type object are shown in Figs.8 and 9. All these results
allows to hope that polarimetric measurements of near-field
images of an objects having rectangular form of an projec-
tion on the scanning plane, can give a possibility for deter-
mining orientation of an object with respective to external
field polarization direction. It needs to be emphasized that
near-field images of cross-excited fields (e.g., external field
is polarized along Ox axes and detected field is polarized
along Oy axes and vice versa) have a specific structure con-
sisting of four bright smears at the corners of an object pro-
jection on the scanning plane. This characteristic behavior
of near-field images could be observed in all calculated ex-
amples.
-0.2 0 0.2
-1.0
-0.5
0
0.5
1.0
X , 8 00 n mx
Y
,
80
0
nm
x
1.781 -- 1.900
1.662 -- 1.781
1.544 -- 1.662
1.425 -- 1.544
1.306 -- 1.425
1.188 -- 1.306
1.069 -- 1.188
0.9500 -- 1.069
0.8312 -- 0.9500
0.7125 -- 0.8312
0.5938 -- 0.7125
0.4750 -- 0.5938
0.3562 -- 0.4750
0.2375 -- 0.3562
0.1187 -- 0.2375
0 -- 0.1187
-0.2 0 0.2
-1.0
-0.5
0
0.5
1.0
X , 8 00 n mx
Y
,
80
0
nm
x
0.002250 -- 0.002400
0.002100 -- 0.002250
0.001950 -- 0.002100
0.001800 -- 0.001950
0.001650 -- 0.001800
0.001500 -- 0.001650
0.001350 -- 0.001500
0.001200 -- 0.001350
0.001050 -- 0.001200
9E-4 -- 0.001050
7.5E-4 -- 9E-4
6E-4 -- 7.5E-4
4.5E-4 -- 6E-4
3E-4 -- 4.5E-4
1.5E-4 -- 3E-4
0 -- 1.5E-4
0.00328 -- 0.00350
0.003063 -- 0.00328
0.002844 -- 0.003063
0.002625 -- 0.002844
0.002406 -- 0.002625
0.002188 -- 0.002406
0.001969 -- 0.002188
0.001750 -- 0.001969
0.001531 -- 0.001750
0.001313 -- 0.001531
0.001094 -- 0.001313
8.75E-4 -- 0.001094
6.563E-4 -- 8.75E-4
4.375E-4 -- 6.563E-4
2.188E-4 -- 4.375E-4
0 -- 2.188E-4
-0.2-0.2-0.2 000 0.20.20.2
X , 8 00 n mxX , 8 00 n mxX , 8 00 n mx
-1.0-1.0-1.0-1.0-1.0
-0.5-0.5-0.5-0.5-0.5
00000
0.50.50.50.50.5
1.01.01.01.01.0
Y
,
80
0
nm
x
Y
,
80
0
nm
x
Y
,
80
0
nm
x
Y
,
80
0
nm
x
Y
,
80
0
nm
x
Fig.4. The same as in Fig.2, but calculated for distance between a scanned plane and an object l = 90 nm .
-0.2 0 0.2
X , 8 00 n mx
-1.0-1.0-1.0-1.0-1.0
-0.5-0.5-0.5-0.5-0.5
00000
0.50.50.50.50.5
1.01.01.01.01.0
Y
,
80
0
nm
x
Y
,
80
0
nm
x
Y
,
80
0
nm
x
Y
,
80
0
nm
x
Y
,
80
0
nm
x
0.0938 -- 0.1000
0.0875 -- 0.0938
0.0813 -- 0.0875
0.0750 -- 0.0813
0.0688 -- 0.0750
0.0625 -- 0.0688
0.0563 -- 0.0625
0.0500 -- 0.0563
0.0438 -- 0.0500
0.0375 -- 0.0438
0.03125 -- 0.0375
0.02500 -- 0.03125
0.01875 -- 0.02500
0.01250 -- 0.01875
0.00625 -- 0.01250
0 -- 0.00625
c d
a b
V. Lozovski, S. Bozhevolnyi: The exact solution of self-consistent equations...
53SQO, 2(3), 1999
-0.5 0 0.5
X , 8 00 n mxX , 8 00 n mx
-1.0-1.0-1.0
-0.5-0.5-0.5
00000
0.5
1.0
Y
,
80
0
nm
x
2.016 -- 2.150
1.881 -- 2.016
1.747 -- 1.881
1.612 -- 1.747
1.478 -- 1.612
1.344 -- 1.478
1.209 -- 1.344
1.075 -- 1.209
0.9406 -- 1.075
0.8062 -- 0.9406
0.6719 -- 0.8062
0.5375 -- 0.6719
0.4031 -- 0.5375
0.2687 -- 0.4031
0.1344 -- 0.2687
0 -- 0.1344
-0.50 -0.25 0 0.25 0.50
X , 8 00 n mx
-0.50
-0.25
0000
0.25
0.50
Y
,
80
0
nm
x
0.4594 -- 0.4900
0.4288 -- 0.4594
0.3981 -- 0.4288
0.3675 -- 0.3981
0.3369 -- 0.3675
0.3063 -- 0.3369
0.2756 -- 0.3063
0.2450 -- 0.2756
0.2144 -- 0.2450
0.1838 -- 0.2144
0.1531 -- 0.1838
0.1225 -- 0.1531
0.0919 -- 0.1225
0.0613 -- 0.0919
0.03063 -- 0.0613
0 -- 0.03063
-0.5 0 0.5
X , 8 00 n mx
-1.0-1.0-1.0
-0.5-0.5-0.5
00000
0.5
1.0
Y
,
80
0
nm
x
0.3094 -- 0.3300
0.2888 -- 0.3094
0.2681 -- 0.2888
0.2475 -- 0.2681
0.2269 -- 0.2475
0.2063 -- 0.2269
0.1856 -- 0.2063
0.1650 -- 0.1856
0.1444 -- 0.1650
0.1238 -- 0.1444
0.1031 -- 0.1238
0.0825 -- 0.1031
0.0619 -- 0.0825
0.0413 -- 0.0619
0.02063 -- 0.0413
0 -- 0.02063
-0.5 0 0.5
X , 8 00 n mx
-1.0-1.0-1.0
-0.5-0.5-0.5
00000
0.5
1.0
Y
,
80
0
nm
x
0.8906 -- 0.9500
0.8313 -- 0.8906
0.7719 -- 0.8313
0.7125 -- 0.7719
0.6531 -- 0.7125
0.5938 -- 0.6531
0.5344 -- 0.5938
0.4750 -- 0.5344
0.4156 -- 0.4750
0.3563 -- 0.4156
0.2969 -- 0.3563
0.2375 -- 0.2969
0.1781 -- 0.2375
0.1188 -- 0.1781
0.0594 -- 0.1188
0 -- 0.0594
a b
c d
Fig.5. The gray scale map representation of near-field image calculations an rectangular object (200x400x20 nm3) when the distance between a scanned
plane and an object is equal to 30 nm. (a) - both external and detected fields are polarized along Ox axes; (b) - both external and detected fields are
polarized along Oy axes; (c) - the external field is polarized along Ox axes and detected field is polarized along Oy axes; (d) - the external field is
polarized along Oy axes and detected field is polarized along Ox axes.
V. Lozovski, S. Bozhevolnyi: The exact solution of self-consistent equations...
54 SQO, 2(3), 1999
0.8438 -- 0.9000
0.7875 -- 0.8438
0.7313 -- 0.7875
0.6750 -- 0.7313
0.6188 -- 0.6750
0.5625 -- 0.6188
0.5062 -- 0.5625
0.4500 -- 0.5062
0.3937 -- 0.4500
0.3375 -- 0.3937
0.2813 -- 0.3375
0.2250 -- 0.2813
0.1688 -- 0.2250
0.1125 -- 0.1688
0.0563 -- 0.1125
0 -- 0.0563
-1.0-1.0-1.0-1.0-1.0
-0.5-0.5-0.5-0.5-0.5
00000
0.50.50.50.50.5
1.01.01.01.01.0
Y
,
80
0
nm
x
-0.5 0 0.5
X , 8 00 n mx
-1.0
-1.5
-0.5
00000
1.5
0.5
1.0
Y
,
80
0
nm
x
-0.5-1.0 0 0.5 1.0
X , 8 00 n mx
0.0741 -- 0.0790
0.0691 -- 0.0741
0.0642 -- 0.0691
0.0593 -- 0.0642
0.0543 -- 0.0593
0.0494 -- 0.0543
0.0444 -- 0.0494
0.0395 -- 0.0444
0.0346 -- 0.0395
0.02963 -- 0.0346
0.02469 -- 0.02963
0.01975 -- 0.02469
0.01481 -- 0.01975
0.00988 -- 0.01481
0.00494 -- 0.00988
0 -- 0.00494
-1.0-1.0-1.0-1.0-1.0
-0.5-0.5-0.5-0.5-0.5
00000
0.50.50.50.50.5
1.01.01.01.01.0
Y
,
80
0
nm
x
-0.5 0 0.5
X , 8 00 n mx
0.02063 -- 0.02200
0.01925 -- 0.02063
0.01788 -- 0.01925
0.01650 -- 0.01788
0.01513 -- 0.01650
0.01375 -- 0.01513
0.01238 -- 0.01375
0.01100 -- 0.01238
0.00963 -- 0.01100
0.00825 -- 0.00963
0.00688 -- 0.00825
0.00550 -- 0.00688
0.00413 -- 0.00550
0.002750 -- 0.00413
0.001375 -- 0.002750
0 -- 0.001375
-1.0-1.0-1.0-1.0-1.0
-0.5-0.5-0.5-0.5-0.5
0
0.5
1.0
Y
,
80
0
nm
x
-0.5 0 0.5
X , 8 00 n mx
0.0633 -- 0.0675
0.0591 -- 0.0633
0.0548 -- 0.0591
0.0506 -- 0.0548
0.0464 -- 0.0506
0.0422 -- 0.0464
0.0380 -- 0.0422
0.0338 -- 0.0380
0.02953 -- 0.0338
0.02531 -- 0.02953
0.02109 -- 0.02531
0.01688 -- 0.02109
0.01266 -- 0.01688
0.00844 -- 0.01266
0.00422 -- 0.00844
0 -- 0.00422
a b
c d
Fig.6. The same as in Fig.4, but calculated for distance between a scanned plane and an object l = 90nm.
V. Lozovski, S. Bozhevolnyi: The exact solution of self-consistent equations...
55SQO, 2(3), 1999
a b
c d
Fig.7. The gray scale map (a), (c) representation and 3d intencity distribution (b), (d) of near-field image of an rectangular object (200x200x20 nm3)
when the distance between a scanned plane and an object is equal to 30 nm. (a), (b) - both external and detected fields are polarized along Ox axes; (c),
(d) - the external field is polarized along Oy axes and detected field is polarized along Ox axes.
Y
,
80
0
nm
x
-0.2
-0.4
0
0.4
0.2
-0.2-0.4 0 0.2 0.4
X , 8 00 n mx
0
0
0
X , 80 0 nmx
Y, 80 0 nmx
0.25
0.20
0.15
0.10
0.05
0.5
-0.5
-0.5
0.5
a b
Fig.8a,b. The same as in Fig.6, but calculated for a distance between a scanned plane and an object l = 90 nm.
X , 8 00 n mx
Y , 8 00 n mx
0.75
-0.4
0.50
-0.2
0.25
0
0.2
0.4
-0.4
0.4
-0.2
0.2
0
0
X , 8 00 n mx
Y , 8 00 n mx
0.75
0.50
-0.5
0.25
0
0.5
0
0.5
1.0
-0.5
-1.0
0
-0.50 -0.25 0
0
0.25 0.50
-0.50
-0.25
0.25
0.50
0.4500
--
0.4800
0.4200
--
0.4500
0.3900
--
0.4200
0.3600
--
0.3900
0.3300
--
0.3600
0.3000
--
0.3300
0.2700
--
0.3000
0.2400
--
0.2700
0.2100
--
0.2400
0.1800
--
0.2100
0.1500
--
0.1800
0.1200
--
0.1500
0.0900
--
0.1200
0.0600
--
0.0900
0.03000 -- 0.0600
0 -- 0.03000
X , 8 00 n mx
Y
,
8
00
n
m
x
-0.50
-0.25
0
0.25
0.50
X , 8 00 n mx
Y
,
8
00
n
m
x
-0.4 -0.2 0 0.2 0.4
0.5156
--
0.5500
0.4813
--
0.5156
0.4469
--
0.4813
0.4125
--
0.4469
0.3781
--
0.4125
0.3438
--
0.3781
0.3094
--
0.3438
0.2750
--
0.3094
0.2406
--
0.2750
0.2063
--
0.2406
0.1719
--
0.2063
0.1375
--
0.1719
0.1031
--
0.1375
0.0688
--
0.1031
0.0344 -- 0.0688
0 -- 0.0344
V. Lozovski, S. Bozhevolnyi: The exact solution of self-consistent equations...
56 SQO, 2(3), 1999
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Y
,
80
0
nm
x
-0.2
-0.4
0
0.4
0.2
-0.2-0.4 0 0.2 0.4
X , 8 00 n mx
c d
Fig.8c,d. The same as in Fig.6, but calculated for a distance between a scanned plane and an object l = 90 nm.
0
0
X , 80 0 nmx
Y, 80 0 nmx
0.045
0.030
0.015
0
0.5
-0.5
-0.5
0.5
|