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).

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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