Post on 06-Jun-2016
Simulation of Near Field Optical Manipulator by Boundary
Element Method�Aperture-Probe Coated with Metal
Masahiro Tanaka and Kazuo Tanaka
Faculty of Engineering, Gifu University, Gifu, Japan 501-1193
SUMMARY
In this paper, the integral equations (guided-mode
extracted integral equations) of a near field optics CAD
proposed earlier are extended for a two-dimensional near
field optics circuit using an aperture probe coated with
metal. As a specific example, an aperture-probe manipula-
tor with a two-dimensional configuration is chosen. In
particular, a derivation of the boundary integral equation is
described in detail for the incident TM mode, in which a
simple extension of the previous results is not possible. The
validity and effectiveness of the proposed integral equations
are demonstrated by simulation using the boundary element
method. It is shown that an attractive force directed toward
the aperture is exerted on an infinitesimal dielectric cylinder
placed near the aperture. The characteristics of the aperture
probe coated with metal are compared in detail with those
of the uncoated probe reported earlier. © 2000 Scripta
Technica, Electron Comm Jpn Pt 2, 83(9): 32�41, 2000
Key words: Near field optics; optical probe ma-
nipulator; simulation; boundary integral equation; metal
coating; aperture probe.
1. Introduction
Recently, many researches of near field optical cir-
cuits such as optical probe manipulators and optical scan-
ning tunneling microscopes have been reported [1�3]. The
optical manipulator allows an infinitesimal object to be
captured and moved without contact, which will be useful
for the extraction of biological cells and molecules [1, 2].
Most of the theoretical research related to the optical
manipulator deals with the incidence of a focused laser
beam on a dielectric sphere. There has been very little
theoretical study of the optical probe manipulator. The
fundamental physical process of an optical probe manipu-
lator making use of near field optics can be considered as
the mutual interaction between a dielectric probe and an
infinitesimal object. The present authors have recognized
the similarity between the near field optical circuit and the
problem of scattering at an optical waveguide discontinuity.
By applying the boundary integral equation that they pro-
posed previously [14, 15] to the near field optical circuit,
the authors have demonstrated that the integral equation can
be used in the basic theory of CAD in near field optics and
simulation software [12, 13]. In these earlier publications,
the probe is assumed to consist of a dielectric slab core
without metallic coating.
It is possible to consider forming an aperture at the
tip of the probe by metallic coating, so that the light can be
focused into an even smaller space and more precise posi-
tional control of an infinitesimal object can be obtained.
This is because an attractive force is exerted on a dielectric
body placed near the aperture due to the evanescent wave
near the aperture. The method employing an aperture probe
has been used to improve the resolution of the optical
scanning tunneling microscope [1]. However, so far as the
authors are aware, there has been no precise analysis of an
optical manipulator involving rigorous equations based on
© 2000 Scripta Technica
Electronics and Communications in Japan, Part 2, Vol. 83, No. 9, 2000Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J82-C-I, No. 8, August 1999, pp. 468�476
Contract grant sponsor: Research Grant for New Generation Research
Laboratories and in part by a Scientific Research Grant from the Ministry
of Education (10135210).
32
a global model that takes account of multiple reflections
between the aperture and the dielectric body.
In this paper, the integral equations (guided-mode
extracted integral equations) proposed in Ref. 12 and 13 are
extended to a form applicable to near field optics simulation
using a metal-coated aperture probe. In particular, in the
case of the incident mode TM whose fundamental mode is
odd, the result cannot be obtained by a simple extension of
Refs. 12 and 13. The derivation process is described in
detail here. By using the derived integral equation, the
characteristics of the two-dimensional aperture-probe ma-
nipulator are determined and the validity and effectiveness
of the proposed integral equation are demonstrated. Also,
the effect of the metal coating on the characteristics of the
optical probe manipulator is studied by simulation.
In the optical domain, a metal film can in general be
treated as a dielectric material with a complex index of
refraction. However, in this paper, the metal coating is
treated as a perfect conductor for simplicity. In fact, in the
case of the microwave experiments in Refs. 10 and 11, the
metal coating can be treated as a perfect conductor. The time
factor exp (jZt) is assumed and is omitted from the follow-
ing descriptions.
2. The Boundary Integral Equation
Let us consider a near field optical circuit which
consists of a dielectric cylinder and an aperture probe
coated with a perfect conductor except at the probe tip and
which satisfies the single mode condition as shown in Fig.
1(a). A two-dimensional problem is considered in this
paper, so that the structure is uniform in the z direction. As
the incident wave, the TE or TM mode incident from the
back of the probe, namely, x �f, is assumed. Since the
treatment of the case of the incident TE mode can easily be
extended from the results in Refs. 12 and 13, only the
incident TM mode is described in detail here.
The basic equation is the two-dimensional Helmholtz
equation with indices of refraction ni�i 1, 2, 3� in each
region, as shown in Fig. 1. Inside the probe with an index
of refraction of n2, the Helmholtz equation can be written
as
where H�x, y� H�x� indicates the z component of the
magnetic field, and k0 is the wave number in free space. Let
us consider Green�s function G2(x|xc) in free space filled
with an index of refraction n2 associated with Eq. (1).
G2(x|xc) satisfies Eq. (2) and is given specifically by Eq. (3):
where H0�2��x� is the zeroth-order Hankel function of the
second kind. When Green�s theorem is applied to Eqs. (1)
and (2) and the boundary condition wH�x� /wn 0 on the
perfect conductor is used, the following boundary integral
equation can be derived:
Fig. 1. (a) Optical manipulator using metal-coated
aperture probe. (b) Integral boundaries.
(2) (4)
(1)
(3)
33
Here the boundary C is that inside the probe other than the
aperture boundary Ct in Fig. 1(b) and C = C1 + C2 + C3 + C4,
and w /wn indicates differentiation in the normal direction.
Since integral equation (4) contains an infinite length
boundary C3 + C4 where guided modes exist, the boundary
element method cannot be applied directly. Hence, the
total magnetic field along the infinite length boundary C3
+ C4 that can be considered as the boundary of the dielec-
tric waveguide with a uniform width is now separated into
the following components:
where R indicates the reflection coefficient while Hr�1��x�
indicates the guided mode functions for the incident wave (�)
and the reflected wave (+). HC�x� is the quantity obtained by
subtracting the incident and reflected modes from the total
magnetic field and is called the disturbed field [12�16]. The
total magnetic field on the boundary Ct � C1 � C2, on the
perfect electric conductor boundary C5 + C6 outside the probe,
and on the boundary CP of the dielectric cylinder, is expressed
as identical to the disturbed field:
When Eqs. (5) and (6) are substituted into Eq. (4) we obtain
where
In the derivation of Eq. (7), the following relationship to be
satisfied by the guided mode Hr�1��x� is used [12�16]:
As shown in Fig. 1(b), the boundary C10 is a virtual
boundary inside the probe between the tip and the
waveguide with uniform width. In Eq. (7), the reflection
coefficient R is contained as an unknown in addition to
the magnetic field HC(x). Hence, if the observation point
x is inside the probe far from the origin of coordinates,
the condition to be satisfied by the field is considered, so
that the reflection coefficient R is expressed by HC(x)
[13]. When the observation point x is far from the origin,
Green�s function can be expressed as follows in terms of
an asymptotic expansion in the cylindrical coordinate
system in Fig. 1(a):
As in Refs. 13 and 14, Eq. (10) is substituted into Eq. (7)
and both sides of the resultant expression are divided by
A2�r�. When the direction T = S of the observation point in
the probe is substituted, we obtain
where
In such a structure as the one in Fig. 1(a), Hr�1�(x) becomes
an odd mode with its field distribution asymmetric with
respect to the x axis if the incident mode is the fundamental
TM mode. It is then seen that ur�1��S� 0, by Eq. (14), so
that Eq. (13) does not contain R. Hence, it is not possible
to obtain a relationship between the reflection coefficient
R and HC(x).
Note that Eq. (13) has undetermined form with respect to
the variable T. Partial differentiation with respect to T is added
to the above procedure. Hence, Eq. (10) is substituted into Eq.
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
34
(7) and both sides are divided by A2�r�. Further, after taking
the partial derivative of both sides with respect to the
variable T, the direction T = S of the observation point within
the probe is substituted, yielding the following equation [15, 16]:
where
Since there is no radiation field in the probe, the left-hand
side of Eq. (15) can be set to zero if the observation point (r,
S) in the probe is sufficiently far from the origin. Hence, the
reflection coefficient R can be written in terms of HC(x) and
wHC�x� /wn as
When Eq. (18) is substituted into Eq. (7),
where
Equation (19) is the boundary integral equation of the
TM mode for the aperture probe coated with metal pro-
posed in this paper. In Eq. (19), the field HC(x) obtained by
subtracting the guide modes from the total field and its
derivative are the unknowns. Hence, the field on the bound-
ary C3 + C4 given by Eq. (19) is assumed to be zero far from
the origin. Therefore, the infinite length boundary C3 + C4
can be treated as of finite length. The boundary element
method can then be applied directly to Eq. (19).
The conventional boundary integral equations can be
derived by making the observation point approach each
boundary from the external space with an index of refrac-
tion n1 as shown in Fig. 1(a), or the cylindrical boundary
from the interior of the dielectric cylinder with an index of
refraction n3. When Eq. (6) is substituted into these expres-
sions , we obtain
Although Eq. (22) contains an infinite length boundary
C5 + C6 on the exterior surface of the metal coating,
HC(x) at a location sufficiently far from the aperture is
assumed to become zero, since the source is located only
at the aperture on the probe tip. Therefore, C5 + C6 is
considered to be of finite length, like the boundary C3 +
C4.
When the boundary element method is applied to
integral equations (19), (22), and (23), HC(x) and
wHC�x� /wn on all boundaries can be obtained numerically.
Once HC(x) and wHC�x� /wn on all boundaries are found, the
reflection coefficient R can be derived from Eq. (18), mak-
ing it possible to find the magnetic field at any point from
the integral expression.
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
35
3. Force Exerted on Dielectric Cylinder
Once the electromagnetic field distributions near the
dielectric cylinder are obtained, the force exerted on the
dielectric cylinder can be derived by the divergence of
Maxwell�s stress tensor or the Lorentz force [17, 18]. Their
time averages are as follows:
(i) Divergence of Maxwell�s stress tensor
(ii) Lorentz force
where <> indicates time average, S is the cross-sectional
area containing the dielectric cylinder, and C is its bound-
ary. T�e� and T�m� are the electric and magnetic Maxwell�s
stress tensors, and U and J are the polarization electric
charge density and the polarization current density. Note
that Eqs. (25) and (26) are equivalent. By comparing their
values, the validity of the method and the numerical accu-
racy of the results can be confirmed.
4. Simulations
Simulations were performed using integral equations
(19), (22), and (23) for the TM mode and the equations
obtained as extensions of the equations in Refs. 12 and 13
for the TE mode. The parameters used in the simulations
are as follows.
Index of refraction external n1 = 1.0
probe n2 = 1.5
dielectric
cylinder
n3 = 1.5
Probe width k0d = 2.4
Radius of curvature of probe
tip
k0t = 0.3
Probe tip angle D = 120°
Aperture width on tip k0w = 1.2
Metal coating thickness k0b = 0.3
Dielectric cylinder radius k0a = 0.5
Expansion functions for
boundary element method
quadratic
functions
Testing functions for bound-
ary element method
delta function
Table 1 shows the power reflection coefficient �*R� normal-
ized to the incident power, the normalized scattered power
�*S�, and the total power �*TOTAL�. Here, k0Lx and k0Ly are
the distances between the probe tip and the dielectric cyl-
inder. It is found from Table 1 that the simulation results
satisfy the energy conservation law with an error of less than
1%. Tables 2(a) and 2(b) give the normalized values Qx and
Qy of the forces derived from Eqs. (25) and (26). The
relationships of Qx and Qy to the x and y components of the
force, Fx and Fy, are given below for the TE and TM modes.
(i) TE mode
(24)
(25)
(26)
Table 1. Reflected power, radiated power, and their total power
(27)
36
(ii) TM mode
where E0 and H0 are the amplitudes of the incident mode.
It is found from Tables 2(a) and 2(b) that the forces derived
from Maxwell�s stress tensor agree well with those derived
from the Lorentz force. Figures 2(a) and 2(b) show the
distributions of |wEC/wn| on the perfectly conducting
boundaries C3 inside the probe and C5 outside the probe
when the incident mode is TE and the dielectric cylinder is
located at k0Lx = 0.5 and k0Ly = 0.0. Figures 3(a) and 3(b)
show the distributions of |HC| on the perfectly conducting
boundaries C3 and C5 inside and outside the probe when the
incident mode is TM. In these figures, the horizontal axis
k0l represents the distance from the virtual boundary C10
along the boundaries C3 and C5 as shown in the figures. The
vertical axis expresses the absolute value of each field. Also,
the width of the elements (discretization width) k0' used
for discretization on the boundaries C3 and C5 is 0.05 (solid
lines), 0.1 (broken lines), and 0.5 (dotted lines). The dis-
cretization width on other boundaries is constant, k0' =
0.01.
From Figs. 2(a) and 3(a), it is found that the field on
the boundary inside the probe decreases exponentially as
the observation point moves away from the virtual bound-
ary C10, becoming constant at a point a certain distance
away. Since the probe satisfies the single mode condition,
HC(x) defined in Eq. (5) is considered to express the higher-
order modes. It is quite natural that wEC/wn of the TE mode
and HC of the TM mode decay exponentially. Therefore, the
constant values in the regions where k0l is greater than 2 in
Figs. 2(a) and 3(a) are considered to be numerical errors. It
is also found that these errors can be reduced by making the
discretization width k0' smaller. However, it proves diffi-
cult to reduce these errors further even if k0' is made
(28)
Table 2. (a) Comparison between numerical values calculated by Maxwell�s stress tensor and those by Lorentz�s force
(TE-mode) (b) Comparison between numerical values calculated by Maxwell�s stress
tensor and those by Lorentz�s force (TM-mode)
37
smaller than 0.05. Therefore, the residual errors are attrib-
uted to roundoff error in the computer.
From Figs. 2(b) and 3(b), it is found that the fields on
the perfectly conducting boundary outside the probe also
decay as one moves away from the virtual boundary C10.
Also, these fields depend little on the discretization width
k0'. From the above findings, it is considered that the
numerical efficiency is good if the boundary C3 + C4 is
discretized finely while the boundary C5 + C6 is discretized
rather coarsely.
Figures 4(a) and 4(b) show the directional distribu-
tions of the force exerted on a single dielectric cylinder
placed near the aperture on the probe tip. If the center of the
dielectric cylinder is placed at the origin of the vector, the
magnitude of the force relative to the direction of the force
exerted on the cylinder is expressed as a vector. Figures 4(a)
and 4(b) are for TE mode incidence and the TM mode
incidence, respectively. In each case, it is found that the
single dielectric cylinder is attracted to the probe tip. The
attractive force in the y direction (transverse to the probe)
is stronger in the case of the TE mode than in the case of
the TM mode. Hence, the dielectric cylinder is more
strongly attracted to the probe tip. Also, as the distance
between the dielectric cylinder and the probe is increased,
the force exerted on the dielectric cylinder decreases
quickly; thus a force is exerted only on a dielectric cylinder
near the probe tip.
Fig. 2. (a) |wEC /wn| on boundary C3 (TE-mode). (b)
|wEC /wn| on boundary C5 (TE-mode).
Fig. 3. (a) |HC| on boundary C3 (TM-mode). (b) |HC| on
boundary C5 (TM-mode).
38
The results obtained in this paper are now compared
with those in the case of a probe without metal coating [12,
13]. For noncoated and coated aperture probes with the
same characteristics as above, numerical examples of the
force exerted on the dielectric cylinder placed on the x axis
in Fig. 1(a) are shown in Fig. 5. In this case, the incident
mode is TE and the size of the cylinder is k0a = 0.5 and 0.2.
The horizontal axis represents the distance k0Lx between the
probe tip and the center of the cylinder. The vertical axis
represents the normalized magnitude Qx in the X direction
defined by Eq. (27). Positive values indicate a repulsive
force (in the direction opposite to the probe) and negative
values indicate an attractive force. Due to symmetry, no
force exists in the y direction.
In the case of a noncoated probe, an attractive force
is exerted on a small dielectric cylinder [where k0a = 0.2
(Non-Coating) in Fig. 5]. On the other hand, a repulsive
force (in the direction opposite to the probe) is exerted on
a large dielectric cylinder [where k0a = 0.5 (Non-Coating)
in Fig. 5]. In the present analysis of an aperture probe, an
attractive force is exerted in both cases. Assuming the above
parameters, the radius of the dielectric cylinder is varied
from k0a = 0.2 to 1.0. No example is found in which the
force on the dielectric cylinder is repulsive.
The apparent reason for this result is that in the case
of the aperture probe, the radiated wave from the probe,
which exerts a repulsive force on a dielectric cylinder with
a large radius, is suppressed. It is found from Table 1 that
most of the incident energy is returned to the probe as a
reflection mode, so that the energy radiated outward from
the probe as a scattered wave is small. In the case without
a coating, most of the incident mode is radiated outside the
probe as the scattered wave [12, 13].
In the case of an aperture probe in Fig. 5, the magni-
tude of the attractive force is smaller than in the case of a
noncoated probe. In the aperture probe in Fig. 1(a), the
incident mode is cut off at the tapered section of the probe
and hence the magnitude of the electric field is reduced at
the aperture. It is necessary to make the taper as short as
possible in order to increase the force exerted on the dielec-
tric cylinder.
From Figs. 4(a), 4(b), and 5, it is found that the region
over which an attractive force is exerted can be localized in
the vicinity of the probe tip by using an aperture probe with
metal coating.
The above results are predictable if the difference
between the metallic waveguide and the dielectric
waveguide and the distribution of the evanescent wave near
the aperture are taken into account. This prediction is con-
firmed by accurate numerical results based on a global
model that takes account of multiple reflection between the
aperture and the dielectric body.
Since no specific value of the force is found in Figs.
4(a) and 4(b), the magnitude exerted on an actual dielectric
cylinder is obtained for a laser with the following parame-
ters:
Wavelength 1.06 Pm
Intensity 150 mW
Radius of spot 5 Pm
The radius of the dielectric cylinder is a = 84 nm and
the time average of the Poynting vector is 1.91 u 109 W/m2.
The x component of the normalized force exerted on the
dielectric cylinder at k0Lx = 0.0 and k0Ly = 0.0 [with the
Fig. 4. (a) Force on the dielectric pillar (TE-mode). (b)
Force on the dielectric pillar (TM-mode).
39
coordinates of the center at (0.5, 0.0)] in Figs. 4(a) and 4(b)
is |Qx| = 0.039 for the TE mode and |Qx| = 0.065 for the TM
mode.
When the above parameters are applied, the x com-
ponent of the actual force exerted on the dielectric cylinder
is Fx = 4.2 u 10�8 N/m for the TE mode and Fx = 7.0 u 10�8
N/m for the TM mode. If the dielectric cylinder is made of
glass, the above values of the force are larger than 5.2 u
10�10 N/m, the gravitational force on the dielectric cylinder.
For both the TE and TM modes, the y component is |Qy| =
0.0. The y component of the actual force is also Fx = 0.0.
5. Conclusions
The integral equation (guided-mode extracted inte-
gral equation) proposed in Refs. 12 and 13 for near field
optics CAD is extended for the near field optical circuits
using an aperture probe. In particular, the case of TM modes
for which the fundamental mode is odd symmetric is de-
scribed in detail. A simulation by the boundary element
method is performed for the aperture-probe manipulator
and the validity and effectiveness of the proposed method
are demonstrated. By means of the simulation, the charac-
teristics of an aperture probe coated with a perfect conduc-
tor and of the uncoated probe discussed earlier are com-
pared.
Acknowledgments. This work was supported in
part by a Research Grant for New Generation Research
Laboratories and in part by a Scientific Research Grant
from the Ministry of Education (10135210).
REFERENCES
1. Pohl DW, Courjon D (editors). Near field optics.
Kluwer Academic; 1993.
2. Ohtsu G. Motion control of single atoms by optics. J
Precis Eng Soc 1992;53:410�414.
3. Visscher K, Brakenhoff GJ. Theoretical study of op-
tically induced forces on spherical particles in a sin-
gle beam trap 1: Rayleigh scatterers. Optik
1992;89:174�180.
4. Visscher K, Brakenhoff GJ. Theoretical study of op-
tically induced forces on spherical particles in a sin-
gle beam trap II: Mie scatterers. Optik
1992;90:57�60.
Fig. 5. Comparison of force on a dielectric cylinder by metal-coated aperture probe with that by noncoating probe.
40
5. Greene WM, Spjut RE, Bar-Ziv E, Sarofim AF, Long-
well JP. Photophoresis of irradiated spheres: Absorp-
tion centers. J Opt Soc Am B 1985;2:998�1004.
6. Dusel PW, Kerker M, Cooke DD. Distribution of
absorption centers within irradiated spheres. J Opt
Soc Am 1979;69:55�59.
7. Barton JP, Alexander DR, Schaub SA. Internal and
near-surface electromagnetic fields for a spherical
particle irradiated by a focused laser beam. J Appl
Phys 1988;64:1632�1639.
8. Gauthier RC, Wallace S. Optical levitation of
spheres: Analytical development and numerical com-
putations of the force equations. J Opt Soc Am B
1995;12:1608�1686.
9. Li J, Agu M. On attractive force of evanescent elec-
tromagnetic field on dielectric slab. Record of Insti-
tute of Electrical Engineers of Japan, EMT-95-67, p
41�49, 1995.
10. Ash EA, Nicholls G. Super-resolution aperture scan-
ning microscope. Nature 1972;237:510�512.
11. Sukeno J, Takahashi N, Kitano M, Ogura H. Meas-
urement of near field by microwaves. Near Field
Optics Study Group, 2nd Research Meeting Proceed-
ings, p 64�69, 1994.
12. Tanaka M, Tanaka K. Boundary integral equation for
near field optics CAD. Trans IEICE 1996;J79-C-
I:101�108.
13. Tanaka M, Tanaka K. Boundary integral equations for
CAD and simulations of near-field optics: Two-di-
mensional optical manipulator. J Opt Soc Am A
1996;15:101�108.
14. Tanaka K, Kojima M. New boundary integral equa-
tions for computer-aided design of dielectric
waveguide circuits. J Opt Soc Am A 1989;6:667�
674.
15. Tanaka K, Nakahara M. New boundary integral equa-
tions for CAD of waveguide circuits: Guided-mode
extracted integral equations. IEEE Trans Microwave
Theory Tech 1992;40:1647�1654.
16. Katayama K, Tanaka K. Boundary integral equation
for electron wave circuit CAD (general expression).
Research Record, Institute of Electrical Engineers of
Japan, EMT-94-71, p 51�60, 1994.
17. Van Bladel J. Relativity and engineering. Springer-
Verlag; 1984.
18. Imai K. Let us consider electromagnetics. Science;
1990.
19. Almaas E, Brevik I. Radiation forces on a microme-
ter-sized sphere in an evanescent field. J Opt Soc Am
B 1995;12:2429�2438.
AUTHORS (from left to right)
Masahiro Tanaka (member) graduated from the Department of Electronic and Computer Engineering, Gifu University,
in 1992, and completed the M.S. program in 1994. He then became a research associate in the Department of Management
Information, Tokoha Gakuen Hamamatsu University. From 1997 to 1998 he was a visiting scholar at the University of Arizona.
Presently, he is a research associate in the Department of Information Science, Gifu University. He has been engaged in research
on optical guided wave circuits and near field optical CAD.
Kazuo Tanaka (member) graduated from the Department of Communication Engineering, Osaka University, in 1970,
and completed the doctoral program in 1975. He then became a research associate in the Department of Electrical Engineering,
Gifu University. In 1994, he was a visiting scholar at the University of Toronto. He has been engaged in research on relativistic
electromagnetic field theory, surface acoustic waves, radiation image engineering, optical circuits, and near field optical CAD.
He is now a professor in the Department of Information Science, Gifu University. He received a Uchida Paper Award from the
Medical Image Information Society in 1985. He is a member of the Information Processing Society, IEEE, the Japan Physical
Society, and the Medical Image Information Society.
41