Download Simplified description of optical forces acting on a nanoparticle in

Zemanek et al.
Vol. 19, No. 5 / May 2002 / J. Opt. Soc. Am. A
1025
Simplified description of optical forces acting on a
nanoparticle in the Gaussian standing wave
Pavel Zemánek and Alexandr Jonáš
Institute of Scientific Instruments, Academy of Sciences of the Czech Republic, Královopolská 147, 612 64 Brno,
Czech Republic
Miroslav Liška
Brno University of Technology, Faculty of Mechanical Engineering, Technická 2, 616 69 Brno, Czech Republic
Received March 23, 2001; revised manuscript received October 16, 2001; accepted October 16, 2001
We study the axial force acting on dielectric spherical particles smaller than the trapping wavelength that are
placed in the Gaussian standing wave. We derive analytical formulas for immersed particles with relative
refractive indices close to unity and compare them with the numerical results obtained by generalized Lorenz–
Mie theory (GLMT). We show that the axial optical force depends periodically on the particle size and that
the equilibrium position of the particle alternates between the standing-wave antinodes and nodes. For certain particle sizes, gradient forces from the neighboring antinodes cancel each other and disable particle confinement. Using the GLMT we compare maximum axial trapping forces provided by the Gaussian standingwave trap (SWT) and single-beam trap (SBT) as a function of particle size, refractive index, and beam waist
size. We show that the SWT produces axial forces at least ten times stronger and permits particle confinement in a wider range of refractive indices and beam waists compared with those of the SBT. © 2002 Optical
Society of America
OCIS codes: 140.7010, 170.4520, 260.2110, 260.3160, 290.4020, 290.5850.
1. INTRODUCTION
Within the past two decades optical trapping has proved
to be an invaluable method for noncontact manipulation
of nano-objects1,2 and micro-objects,1,3 measurement of
extremely weak forces,4,5 and study of single molecule
properties6,7 and surface properties.8–11 The most frequently used trapping set-up is based on a single laser
beam tightly focused by an immersion microscope objective of high numerical aperture.1 This classical singlebeam trap (SBT) set-up has been gradually modified by
using interference of co-propagating laser beams,12 selfaligned dual beams,13 or optical fibers.14,15 Several laser
beams,16,17 diffractive optics,18 and time sharing of a
single beam16,19 were employed to create several optical
traps. The common characteristic of all the above methods is that the axial force exerted on the object is smaller
than the radial one. Recently it has been shown that the
optical trapping can also be achieved in the standing
wave created by interference of two counter-propagating
coherent beams.20 In this case the axial force is stronger
than the radial one because of strongly inhomogeneous
optical intensity distribution in the periodic structure of
the standing-wave nodes (minimums) and antinodes
(maximums). Therefore particle confinement is achieved
even in weakly focused or aberrated beams. An interesting problem that arises here is to study the properties of
this type of optical trap [standing-wave trap (SWT)] and
compare them with the classical SBT for different parameters of the trapped object (size, refractive index) and
trapping beam (waist size).
Theoretical description of optical forces is simple only
0740-3232/2002/051025-10$15.00
for very small particles—so-called Rayleigh particles—
whose radius fulfils a ⭐ ␭/20, where ␭ is the trapping
wavelength in the medium.21,22 Such a small particle behaves as an induced elementary dipole, and the optical
forces acting on it can be divided into two components,
gradient and scattering forces. The gradient force comes
from electrostatic interaction of a particle (dielectric) with
an inhomogeneous electric field, and the scattering force
results from the scattering of the incident beam by the
object.21,23
For particles larger than ␭/20, a more complex concept
involving the Maxwell stress tensor of the electromagnetic field surrounding the particle must generally be
used. This requires knowledge of the total field outside
the confined particle. The original theory based on
plane-wave scattering24 has been gradually modified so
that it can be applied to the case of a spherical or spheroidal particle placed into an arbitrary field
distribution.25–27 It is commonly referred to as the generalized Lorenz–Mie theory (GLMT).
A recently presented intuitive approach28 assumes that
the electric field creates such a large gradient force that
the contribution of the scattering force to the total optical
force is negligible. Consequently, only the gradient force
controls the behavior of an irradiated dielectric object.
The stress tensor contains only the terms depending on
the electric field, and the problem is then reduced to the
evaluation of the change in electrostatic energy after insertion of the dielectric into the inhomogeneous electrostatic field. Because the dielectric is assumed to be
weak, its placement into the field does not change the ini© 2002 Optical Society of America
1026
J. Opt. Soc. Am. A / Vol. 19, No. 5 / May 2002
Zemanek et al.
tial field distribution appreciably. This allows use of an
analytic expression for the optical force acting on spherical particles in an isotropic Gaussian field or on a cube in
axially symmetric fields without further restriction on the
particle size. This method will be called electrostatic approximation (ESA) in this paper.
The application of this approach is limited to cases
where the scattering force can be neglected. It has been
shown22 that if the Gaussian standing wave (GSW) is
used for the optical trapping, the scattering force acting
on the Rayleigh particle is proportional to the difference
in the energy fluxes of counterpropagating waves. If the
two waves have the same intensity, the scattering force is
zero. A similar trend can also be expected for larger particles. Thus the ESA framework is well suited to the
evaluation of the optical forces in the GSW.
A. General Description of the Optical Forces by Using
Electrostatic Approximation
Let us assume that a dielectric object is placed into an inhomogeneous electromagnetic field in a dielectric liquid.
If the sources of the field are kept fixed, the change in total energy of the field due to the insertion of the object can
be written as the difference between the energy of the
field with the inserted medium and the initial energy
with the liquid dielectric29:
1
2
冕
共 ED ⫺ E0 D0 ⫹ HB ⫺ H0 B0 兲 dV,
(1)
V
where E (D) and E0 (D0 ) are the electric field vector (electric displacement) and H (B) and H0 (B0 ) are the magnetic field vector (magnetic induction) after and before insertion of the object, respectively. The integration is over
the whole space occupied by the field. During observation of the object we cannot follow the rapid optical frequencies; therefore, only the time-averaged values are accessible. It can be shown (see Appendix A) that if the
object and the liquid are nonmagnetic, the time-averaged
change in the field energy can be rewritten as
1
具 ⌬W 共 r, t 兲 典 T ⫽ ⫺ ⑀ 2 ␣
2
冕
V1
具 E共 r, t 兲 E0 共 r, t 兲 典 T dV,
I0 共 r兲 ⬅ 具 S共 r, t 兲 典 T
⬅ 具 E0 共 r, t 兲 ⫻ H0 共 r, t 兲 典 T ,
⯝ z0 具 n 2 ⑀ 0 c 兩 E0 共 r, t 兲 兩 2 典 T
⫽ z0 n 2 ⑀ 0 c 兩 E 0 共 r兲 兩 2 /2 ⫽ z0 I 0 共 r兲 .
(4)
Here z0 is the unit vector in the direction of the light
propagation and c is the speed of light. The equation for
the energy change consequently takes the final form
␣ n2
⌬W 共 r兲 ⬅ 具 ⌬W 共 r, t 兲 典 T ⫽ ⫺
2 c
2. ELECTROSTATIC APPROXIMATION
APPLIED TO A SPHERICAL DIELECTRIC
PARTICLE PLACED IN THE
GAUSSIAN STANDING WAVE
⌬W ⫽
Assuming that the incident beam is polarized mainly in
the transverse direction (paraxial beam), we obtain, using
the relation between the optical intensity and the field
vectors,
冕
I 0 共 r兲 dV.
(5)
V1
The time-averaged force can be expressed by the divergence theorem
F共 r兲 ⬅ ⫺ⵜ 具 ⌬W 共 r, t 兲 典 T
⫽
␣ n2
2 c
冕
V1
ⵜI 0 共 r兲 dV ⫽
␣ n2
2 c
冕
nI 0 共 r兲 dS,
(6)
S1
where n is the outward unit normal to the surface element dS.
B. Analytical Formulas for a Spherical Dielectric
Particle Placed in the Gaussian Standing Wave
Classical optical trapping requires a focused laser beam
with a spot diameter comparable to the trapping wavelength. To achieve this, high-quality immersion objectives are usually used. In this case the beam goes
through many dielectric interfaces (optical elements inside the objective, immersion oil layer, coverslip, and water layer), and the actual field distribution can differ considerably from the theoretical model, usually represented
by a Gaussian beam.30 However, since we want to
present here only the basic properties of the SWT, we
(2)
where ␣ ⫽ ⑀ 1 / ⑀ 2 ⫺ 1, ⑀ i ⫽ n i 2 ⑀ 0 is the permittivity of the
particle (i ⫽ 1), the liquid (i ⫽ 2), and the vacuum (i
⫽ 0). Symbols n 1 and n 2 represent the refractive indices of the particle and the liquid, respectively. The integration is now performed over the volume V 1 of the particle. If the particle is not strongly polarized, i.e., n 1
⯝ n 2 , the perturbed electric field E can be approximated
by the unperturbed one E0 and we obtain
1
具 ⌬W 共 r, t 兲 典 T ⫽ ⫺ ⑀ 2 ␣
2
冕
V1
具 兩 E0 共 r, t 兲 兩 典 T dV.
2
(3)
Fig. 1. Orientation of axes in the standing-wave apparatus.
The z axis follows the direction of the reflected wave. Positive z 0
means that the beam waist of radius w 0 is located in the reflected
wave.
Zemanek et al.
Vol. 19, No. 5 / May 2002 / J. Opt. Soc. Am. A
have chosen a moderately focused Gaussian beam to
reach a compromise among the exact description, speed of
calculation, and availability of analytical results.
Let us assume that the GSW is created by interference
of an incident wave and a wave reflected at a dielectric
surface (see Fig. 1), and let us omit any multiple scattering between the object and the surface. Let the orientation of the z axis be parallel to the direction of the reflected beam. Therefore the ‘‘trapping force,’’ which acts
against the incident wave, has positive sign in this coordinate system. The incident and the reflected Gaussian
beams can be written as22
E i 共 r B , z B 兲 ⫽ E 00
⫺
i kr B 2
2 Ri
E r 共 r B , z B 兲 ⫽ E 00␳
⫺
冉 冊 冋
冉 冊册
冉 冊 冋
冉 冊 册
r B2
exp ⫺ 2 exp ik 共 z B ⫹ z 0 兲
wi
wi
w0
⫺ i arctan
zB ⫹ z0
,
zR
r B2
exp ⫺ 2 exp ⫺ik 共 z B ⫺ z 0 兲
wr
wr
w0
i kr B 2
2 Rr
zB ⫺ z0
⫹ i arctan
冋
共 zB ⫾ z0兲2
w i/r 共 z B 兲 ⫽ w 0 1 ⫹
z R2
冋
R i/r 共 z B 兲 ⫽ ⫿共 z B ⫾ z 0 兲 1 ⫹
册
1/2
,
z R2
共zB ⫾ z0兲
2
册
,
(8)
and z R ⫽ kw 0 2 /2 is the Rayleigh length. Under these assumptions we can write for the total intensity distribution
I GSW共 r B , z B 兲 ⫽
n 2⑀ 0c
2
⫽ I 00
兩 E i共 r B , z B 兲 ⫹ E r共 r B , z B 兲兩 2
w 02
w i2
2
2
⫻ exp关 ⫺共 r B 2 /w r 2 兲兴 cos ␾ GSW
w 02
w r2
1
Ri
⫺
zB ⫹ z0
zR
zB ⫺ z0
zR
冊
冊
1
Rr
冊
⫹ ␺.
exp关 ⫺共 2r B 2 /w r 2 兲兴 ,
(10)
Here, I 00 is the on-axis intensity at the position of the
beam waist and is related to the total power of the incident Gaussian beam by I 00 ⫽ 2P/ ␲ w 0 2 .
Because we assume that the dielectric object has
spherical shape, it is useful to rewrite radial and axial cylindrical coordinates (r B , z B ) referenced to the beam by
using spherical coordinates (r, ␹ , ␾ ) referenced to the
sphere center, which is shifted from the center of the cylindrical coordinate system by r s laterally and z s axially:
r B 2 ⫽ r 2 sin2 ␹ ⫹ rs2 ⫹ 2rrs sin ␹ cos ␾ ), z B ⫽z s ⫹ r cos ␹.
In this case the final forms of the energy change [Eq. (5)]
and the axial force [Eq. (6)] are
冕冕
a
0
␲
I GSW共 r B , z B 兲 r 2 sin ␹ d␹ dr,
0
(11)
F z共 r s , z s 兲 ⫽
␣ n2
2 c
2␲
冕
␲
I GSW共 r B , z B 兲 a 2 sin ␹ cos ␹ d␹ .
0
(12)
These integrals can be expressed analytically only if one
of the following approximations is employed.
1. a Ⰶ w 0 , a Ⰶ ␭ (Rayleigh Particle)
The particle is so small that the intensity can be considered constant over the integration volume, and therefore
it can be taken out of the integral in Eqs. (11) and (12).
Consequently, we obtain
2 n2
⌬W 共 r s , z s 兲 ⫽ ⫺
␣ ␲ a 3 I GSW共 r s , z s 兲 ,
3 c
F共 rs , zs兲 ⫽
2 n2
3 c
␣ ␲ a 3 ⵜI GSW共 r s , z s 兲 .
␣ ⫽ m 2 ⫺ 1 ⯝ 3 关共 m 2 ⫺ 1 兲 / 共 m 2 ⫹ 2 兲兴 ,
exp关 ⫺共 r B /w i 兲兴
⫹ 2 ␳ I 00
w iw r
⫹ ␳ 2 I 00
冉
冉
冉
(13)
(14)
Equation (14) is identical to the Harada et al. result for
the gradient force in a paraxial beam21 if we use the substitution
exp关 ⫺共 2r B 2 /w i 2 兲兴
w 02
2
⫺ arctan
(7)
where E 00 is the electric field amplitude at the beam
waist position, k is the wave number in the medium, ␳ is
the reflectivity of the surface, ␺ is the phase shift after reflection [the Fresnel reflection coefficient has the form
r m ⫽ ␳ exp(⫺i␺)], and z 0 is the distance of the beam waist
from the surface (mirror). Positive (negative) z 0 corresponds to beam waist created in the reflected (incident)
wave (see Fig. 1). Symbol w 0 is the beam waist size and
w i and w r are the widths of the incident and the reflected
beams at a given z B , respectively. R i and R r are the radii of the wave-front curvature for the incident and the reflected waves, respectively,
kr B 2
⫺ arctan
␣ n2
⌬W 共 r s , z s 兲 ⫽ ⫺
2␲
2 c
⫺i ␺ ,
zR
␾ GSW共 r B , z B 兲 ⫽ 2kz B ⫺
1027
(9)
(15)
which, however, is correct only for m ⯝ 1. The difference
between the left-hand and right-hand sides of Eq. (15) is
7% and 14.7% for m ⫽ 1.1 and m ⫽ 1.2, respectively.
Therefore the validity of ESA within the Rayleigh approximation can be extended into a region of higher refractive index by substituting Eq. (15) into Eq. (14).
1028
J. Opt. Soc. Am. A / Vol. 19, No. 5 / May 2002
Zemanek et al.
2. a Ⰶ w 0 , a ⯝ ␭ (Intermediate Particle in Moderately
Focused Beams)
This assumption means that over the integration volume
the intensity does not change materially in the transverse
direction, and therefore purely transverse-coordinatedependent terms can be taken out of the integral in Eqs.
(11) and (12). As the particle size is comparable to the
wavelength, there is an intensity change in the axial direction that cannot be neglected over the integration volume. If all the terms connected with Eq. (9) are transformed into the spherical coordinates (see Appendix B)
and the assumption a Ⰶ w 0 is applied, we obtain the following expressions:
⌬W 共 r s , z s 兲 ⫽ ⫺␣
n2
c
⫹ ␳2
⫹
P
冋冉
1
wr
2
1
w i2
exp关 ⫺共 2r s 2 /w r 2 兲兴
␳
3
exp关 ⫺共 2r s 2 /w i 2 兲兴
k w iw r
冊
4
3
Fig. 3. Behavior of two polystyrene spheres of slightly different
radii (a ⫽ 0.3␭, 0.35␭) placed symmetrically with respect to the
standing-wave node. The following parameters were used for
the force calculation: w 0 ⫽ ␭, ␳ ⫽ 1, ␺ ⫽ 3 ␲ /2, z 0 ⫽ 0 ␮ m, m
⫽ 1.95, P ⫽ 1 W, ␭ vac ⫽ 1064 nm.
a3
exp关 ⫺r s 2 共 1/w i 2 ⫹ 1/w r 2 兲兴
册
⫻ 共 sin 2ka ⫺ 2ka cos 2ka 兲 cos ␾ s ,
(16)
2n 2 ␣
P
F z共 r s , z s 兲 ⫽ ⫺
␳ exp关 ⫺r s 2 共 1/w i 2 ⫹ 1/w r 2 兲兴
2
c k w iw r
⫻ 共 sin 2ka ⫺ 2ka cos 2ka 兲 sin ␾ s ,
(17)
where ␾ s ⫽ ␾ GSW(r s , z s ). This approximation enables
us to perform the integration analytically even for off-axis
positions of the sphere center (i.e., r s ⫽ 0).
According to Eq. (17) the sign of the force is determined
by the product of two terms. The first one, sin ␾s , comes
from the spatial intensity profile described by the interference term in Eq. (9), and it determines the positions of
zero, minimum, and maximum optical forces along the optical axis. The second term, sin 2ka ⫺ 2ka cos 2ka, may
be set equal to G a and depends on the ratio of the particle
size and the trapping wavelength in the medium. Its
plot is shown in Fig. 2. Extremes of function G a can be
found for a extr /␭ ⫽ M/4, M ⫽ 1, 2,.... The zero value of
G a is obtained for the following values of the sphere radius: a zero /␭ ⫽ 0, 0.3576, 0.6148, 0.8677, 1.1194,.... If a
sphere of this particular size is placed into the GSW, the
change in the field energy does not depend on the sphere
position in the GSW. Consequently, the axial optical
force acting on the sphere is zero for all particle positions
in the GSW, and obviously this particle cannot be trapped.
Fig. 2. Profile of the size-dependent term G a in the analytical
form of the axial force acting on a dielectric sphere in the GSW.
The combined behavior of both terms has the effect
that the center of a particle of radius less than 0.3576␭
moves toward the place of maximum optical intensity,
that is, GSW antinode (2kz s ⫹ ␺ ⫽ 2 ␲ N, N ⫽ 1, 2,...),
while the center of a particle of radius between 0.3576
⬍ a/␭ ⬍ 0.6148 is pushed to the intensity minimum,
that is, GSW node (2kz s ⫹ ␺ ⫽ ␲ ⫹ 2 ␲ N, N
⫽ 1, 2,...), and so forth. An intuitive rule for this tendency can be stated: The sphere in the GSW moves so
that it covers the maximum number of GSW antinodes.
This effect is illustrated in Fig. 3, where two pairs of
polystyrene spheres of slightly different radii (a
⫽ 0.3␭, 0.35␭) are placed symmetrically with respect to
the node (the sphere centers are depicted by ⫹ and ⫻ and
their surfaces by continuous and dashed circles, respectively). The upper plots show the position of the spheres
with respect to the GSW nodes (lighter regions) and antinodes (darker regions). The lower plots show the axial
force acting on these objects for various distances of the
objects from the mirror. The arrows in the top plots show
the direction of the total optical force acting on the
sphere. The solid small circles depict the stable equilibrium position of the sphere, which is called the trap position. The optical trap position is shifted by ⬃␭/4 from antinode to node if the smaller particle is exchanged with
the larger one.
3. a ⯝ w 0 , a Ⰶ z R
Let us consider the slightly more general case of a sphere
that is so large that the intensity changes in the lateral
direction cannot be neglected over the integration volume.
To obtain analytical results we assume that the beam
widths w i (z) and w r (z) do not change with respect to the
particle size (see Appendix B) and that the particle is situated on the beam axis. Improved results with respect to
the previous case can be expected only in a narrow range
of particle sizes. Following Eqs. (11) and (12) the energy
change and the axial force can be expressed as
Zemanek et al.
Vol. 19, No. 5 / May 2002 / J. Opt. Soc. Am. A
⌬W 共 0, z s 兲
⫽ ⫺␣
P a共 1 ⫹ ␳2兲 ⫺
c
⫹ ␳2
⫹
冋
n2
wr
冑2
␳
daw
冉
冉 冊册
再
C 3/2
冑2
daw
exp共 i ␾ s 兲关 exp共 i2ka 兲 daw共 X ⫹ Y 兲
F z 共 0, z s 兲
n2
c
P
wi
sin共 2ka 兲共 cos ␾ s ⫺ Z W sin ␾ s 兲
⫹ exp共 ⫺i2ka 兲 daw共 ⫺X ⫹ Y 兲兴
⫽ 4␣
冑2a
wr
w iw r k共 1 ⫹ Z W2 兲
⫹ Re
wi
冑2a
2W 2
2W 3
冋 冉 冊
␳
w iw r
冠
冎 冊册
,
(18)
W2
⫺
sin 共 2ka 兲
1 ⫹ Z W2
再
ikW 3
C 3/2
exp共 i ␾ s 兲关 exp共 i2ka 兲 daw共 X ⫹ Y 兲
⫹ exp共 ⫺i2ka 兲 daw共 ⫺X ⫹ Y 兲兴
冎冊
lapped beam waists to get the initial field components of
the standing wave. Moreover, we assumed that the
spherical object was located on the beam axis so we could
employ the radial symmetry of the problem and simplify
the calculation.33 We wrote the modified code ourselves,
but we do not present a detailed mathematical description, because the method is well described in the
literature.26,27,31,33
In this study we neglected any electrostatic interactions between the surface and the particle as well as multiple scattering of the incident beam. Furthermore, we
assumed that the beam waist was placed on the surface
with reflectivity equal to 100%. The axial positions z s of
the sphere center, where we calculated the axial forces,
satisfy the inequality a ⭐ z s ⭐ (a ⫹ ␭).
Although the adopted simplifications (CGB, absence of
spherical aberrations, diffraction, and multiple-scattering
events) may seem drastic, this model provides at the least
a correct qualitative description of the behavior of dielectric spheres in the GSW34 and an acceptable speed of calculation.
B. Validity of the Electrostatic Approximation
⫻ 共 sin ␾ s ⫹ Z W cos ␾ s 兲
⫹ Re
1029
,
(19)
where X, Y, W, C, Z W , and ␾ s are defined in Appendix B.
3. GENERALIZED LORENZ–MIE THEORY
APPLIED ON A SPHERE PLACED IN
THE GAUSSIAN STANDING WAVE AND
THE SINGLE BEAM
A. Calculation of Optical Forces by Using Generalized
Lorenz–Mie Theory
The GLMT uses the scattering procedure presented first
by Mie24 who derived expressions for the field distribution
outside a spherical object of arbitrary size placed into a
plane wave. This original method was generalized so
that it permits a mathematical description of the forces
acting on spherical and oblate objects placed in a general
electromagnetic field.25,27,31 For the description of a focused beam, a modification to the fundamental Gaussian
beam was introduced [so called fifth-order corrected
Gaussiam beam (CGB)].26,32 It uses a field expansion in
the beam size parameter s ⫽ 1/kw 0 to the fifth order,
achieves better agreement with the wave equation, and
therefore provides more precise calculation of the optical
forces. Because we consider the standing wave created
by the interference of two counterpropagating focused laser beams, we have easily adapted the GLMT formalism
to this case. Instead of a single CGB we summed field
components of two counterpropagating CGBs with over-
Validity of the ESA is limited mainly by the assumptions
that the object does not change the field distribution and
that the incident beam is polarized only in the transverse
direction [see Eq. (4)]. Therefore, it is desirable to study
the differences between the ESA and a more precise
method (GLMT) as a function of object size, refractive index, and Gaussian beam waist size.
Rough comparison of the GLMT and the ESA methods
revealed that the ESA methods are applicable only if the
object is smaller than 0.25␭. For larger spheres, only
very small relative refractive indices must be used.
Since the maximum trapping force is nonzero for particles
of applicable sizes, we can compare the methods by using
maximum relative error of the ESA trapping force defined
by the formula
Fig. 4. Maximum relative error of the ESA methods (in percent)
as a function of the sphere radius and relative refractive index
for four beam waist sizes (w 0 ⫽0.75, 1, 1.25, 2.5␭) and the following parameters: ␳ ⫽ 1, ␺ ⫽ 3 ␲ /2, ␭ vac ⫽ 1064 nm, z 0 ⫽ 0 ␮ m.
1030
⌬F z ⫽
J. Opt. Soc. Am. A / Vol. 19, No. 5 / May 2002
Zemanek et al.
max(兩 F apr共 a ⭐ z s ⭐ a ⫹ ␭ 兲 ⫺ F GLMT共 a ⭐ z s ⭐ a ⫹ ␭ 兲 兩 )
F max
,
(20)
where F max ⫽ max(F GLMT(a ⭐ z s ⭐ a ⫹ ␭)) and F apr is
the force value calculated using a particular approximation ‘‘apr.’’ Relative errors are summarized in Fig. 4 for
various beam waist sizes as a function of the sphere radius and the relative refractive index. The numbers on
the contour curves mark the relative error in percent.
We compared the GLMT with three ESA methods labeled
ESA-int, ESA-daw, and ESA-anal calculated by using
Eqs. (12), (B8), and (17), respectively. ESA-int is the
most precise method, giving 5% ⭐ ⌬F z ⭐ 22% for w 0
⫽ 0.75␭ and 0.5% ⭐ ⌬F z ⭐ 14% for w 0 ⫽ 2.5␭. The
ESA-daw method gives more precise results than the
ESA-anal one for intermediate sphere radii (0.1 ⭐ a/␭
⭐ 0.23) and smaller m, but negligible improvement is
found for greater m. The relative error of the ESA-daw
method satisfies 10% ⭐ ⌬F z ⭐ 34% for w 0 ⫽ 0.75␭ and
0.5%⌬F z ⭐ 16% for w 0 ⫽ 2.5␭. We see that the wider
the beam waist size, the better the coincidence with the
GLMT, but no significant improvement was found for w 0
⬎ 2.5␭.
We also checked whether intuitive substitution of ␣ by
3(m 2 ⫺ 1)/(m 2 ⫹ 2) in Eqs. (12), (B8), and (17), according to Eq. (15), improves the coincidence between the
GLMT and the ESA methods. We denote these methods
ESAn-int, ESAn-daw, and ESAn-anal, and results are
shown in Fig. 5. We can see that precision increased on
average by more than three times, especially for higher
refractive indices and intermediate particles (0.1␭ ⭐ a
⭐ 0.22␭). The relative error is smaller than 5% for the
majority of ESAn-int results. Contrary to the case with
ESA-daw, the ESAn-daw method slightly increases the
precision in a narrow region of higher m for small w 0 and
smaller m for wider w 0 . Together with the ESAn-anal
method the relative error with the ESAn-daw method is
smaller than 10% for wider beam waist sizes. No simple
Fig. 6. Contour plot of maximum SWT axial trapping force (in
piconewtons) with respect to the z axis as a function of the relative refractive index, particle size, and beam waist radius calculated by using the GLMT. Dotted–dashed and dashed lines represent the loci of maximum and minimum values of the
maximum trapping force with respect to the particle size and
relative refractive index. The shaded areas represent the nontrapping regions. The following parameters were used: P
⫽ 1 W, ␳ ⫽ ⫽ 1, ␺ ⫽ 3 ␲ /2, ␭ vac ⫽ 1064 nm, z 0 ⫽ 0 ␮ m.
tendency with respect to the beam waist size was found,
but there is a region of minimum error that moves from
the area of higher m for smaller w 0 toward the area of
smaller m for higher w 0 . The steep increase of ⌬F z for
larger a and m in Figs. 4 and 5 is caused by the fact that
F max , which appears in the denominator of Eq. (20), decreases here.
Fig. 5. Maximum relative error of the ESA methods (in percent)
if the ␣ ⫽ m 2 ⫺ 1 term is replaced by 3(m 2 ⫺ 1)/(m 2 ⫹ 2) as a
function of the sphere radius and relative refractive index for
four beam waist sizes (w 0 ⫽0.75, 1, 1.25, 2.5␭) and the following
parameters: ␳ ⫽ 1, ␺ ⫽ 3 ␲ /2, ␭ vac ⫽ 1064 nm, z 0 ⫽ 0 ␮ m.
C. Comparative Study of the Maximum Trapping Force
in the Standing-Wave Trap and the Single-Beam
Trap
Figure 6 presents the contour plots of the maximum axial
trapping force acting on the spherical particles placed
Zemanek et al.
Vol. 19, No. 5 / May 2002 / J. Opt. Soc. Am. A
1031
into the GSW for various values of particle size, relative
refractive index, and beam waist size. The numbers represent the levels of constant forces in piconewtons, and
the shaded regions correspond to the negative maximum
trapping force. For these parametric configurations the
particle is accelerated toward the surface without a
chance of its confinement in the GSW (nontrapping regions). This is caused by the gradient forces from the
neighboring GSW antinodes, which pull the object in opposite directions and thus cancel each other. Consequently, the weaker gradient force due to the focused
beam envelope dominates and accelerates the particle towards the beam waist placed on the mirror. The larger
the sphere, the wider is this nontrapping region. We also
see that this region becomes narrower with increasing
beam waist since the gradient of the Gaussian envelope of
the intensity becomes smaller.
Nontrapping predictions of the ESA models fit the
GLMT results only for m ⯝ 1. Figure 6 shows that the
difference between the consequent force-maximizing or
-minimizing sphere radii is smaller than ESA predictions
(equal to ␭/4) and still decreases with increasing value of
the relative refractive index.
Figure 7 shows the maximum trapping force in the SBT
for comparison. We see that the wider the beam waist,
the lower must be the relative refractive indices used to
confine the object in the SBT. Direct comparison with
the SWT forces (Fig. 6) acting on larger particles (outside
the nontrapping regions) reveals that the SWT provides
trapping forces that are at least one order of magnitude
stronger. For smaller particles and wider beam waists
this disproportion becomes even more pronounced.
4. SUMMARY AND CONCLUSIONS
In the work described in this paper the trapping properties of the Gaussian standing wave (GSW) were studied
theoretically by using electrostatic approximation (ESA)
and generalized Lorenz–Mie theory (GLMT). On the basis of the ESA method, analytical formulas for optical
forces acting on dielectric particles with relative refractive indices close to unity were derived. The validity of
the ESA and its analytical formulas was tested by comparison with the numerical results obtained from the
GLMT. Despite the drastic simplification, it turned out
that the ESA methods are convenient for fast estimation
of axial forces acting on particles of radii a ⭐ 0.25␭ and
relative refractive indices m ⭐ 1.21 placed into the GSW
of beam waist w 0 ⭓ 0.75␭. If the term ␣ ⫽ (m 2 ⫺ 1) is
replaced by the term 3(m 2 ⫺ 1)/(m 2 ⫹ 2) taken from the
Rayleigh scattering theory, the relative error of the ESA
method with respect to the GLMT decreases below 16%.
The ESA model straightforwardly explains why spheres
of radius smaller than ⬃0.3␭ are trapped in the antinodes
and spheres of radius larger than ⬃0.3␭, but smaller than
⬃0.55␭ are confined in the nodes.
We found that the SWT, in contrast to the SBT, permits
confinement of particles of high refractive indices even in
moderately focused beams and could therefore be employed for easier optical confinement of such particles in
air. This conclusion, however, is valid only for particle
Fig. 7. Contour plot of maximum SBT axial trapping force (in
piconewtons) with respect to the z axis as a function of relative
refractive index, particle size, and beam waist radius calculated
by using the GLMT. The dotted–dashed curves represent the
loci of maximum and minimum vales of the maximum trapping
force with respect to particle size and relative refractive index.
The dotted curves denote the borders of the trapping regions, and
the shaded areas represent the nontrapping regions. The inset
plots in the bottom two plots magnify the regions of interest with
the same horizontal scale. The following parameters were used:
P ⫽ 1 W, ␭ vac ⫽ 1064 nm.
sizes and refractive indices outside the nontrapping regions, where the effect of the GSW is not minimized.
APPENDIX A: TIME-AVERAGED CHANGE
IN FIELD ENERGY
If object and immersion media are linear (i.e., D ⫽ ⑀ E,
B ⫽ ␮ H) and both media are nonmagnetic ( ␮ ⫽ ␮ 0 ), Eq.
(1) can be rewritten in the form
⌬W ⫽
1
2
冕
⫹
共 ED0 ⫺ DE0 兲 dV
V
1
2
冕
V
共 E ⫹ E0 兲共 D ⫺ D0 兲 dV.
(A1)
1032
J. Opt. Soc. Am. A / Vol. 19, No. 5 / May 2002
Zemanek et al.
We get for the first integral in Eq. (A1) 兰 V (ED0 ⫺ DE0 )
⫽ ⫺兰 V 1 ( ⑀ 2 ⫺ ⑀ 1 )EE0 dV. The integration is performed
over only the volume of the inserted dielectric V 1 , because outside the inserted object the integrand vanishes
as ( ⑀ 1 ⫺ ⑀ 1 )EE0 ⫽ 0.
If we employ the Maxwell equations and introduce potentials A, ␾ as E ⫽ ⫺ⵜ ␾ ⫺ ⳵ A/ ⳵ t, the second integral in
Eq. (A1) can be rewritten as
1
2
冕
共 E ⫹ E0 兲共 D ⫺ D0 兲 dV
V
1
⫽
2
1
⫽
2
冕冉
冕冋
冕
⫺ⵜ ␾ 0 ⫺
V
⳵ A0
⳵t
⫺ⵜ 共 ␾ 0 ⫹ ␾ 兲 ⫺
⳵t
冊
册
⳵ 共 A0 ⫹ A兲
⳵t
V
1
⫽⫺
2
⳵A
⫺ ⵜ␾ ⫺
ⵜ ␾ T 共 D ⫺ D0 兲 dV ⫺
V
冕
1
2
V
共 D ⫺ D0 兲 dV,
共 D ⫺ D0 兲 dV,
⳵ AT
⳵t
2
共 D ⫺ D0 兲 dV,
where new total potentials ␾ T ⫽ ␾ 0 ⫹ ␾ and AT ⫽ A0
⫹ A were defined.
Since we assume harmonic fields, the second term in
Eq. (A2) can be rewritten as
冕
⳵ 共 AT0 sin ␻ t 兲
⳵t
V
1
⫽⫺
2
冕
共 D ⫺ D0 兲 0 sin ␻ t dV
␻ AT0 共 D ⫺ D0 兲 0 cos ␻ t sin ␻ t dV. (A3)
冓冕
V
⳵ AT
⳵t
共 D ⫺ D0 兲 dV
冔
⫽ 0.
1
⫺
2
冕
1
ⵜ ␾ T 共 D ⫺ D0 兲 dV ⫽ ⫺
2
V
⫹
冕
2
w i/r 2
1
ⵜ 关 ␾ T 共 D ⫺ D0 兲兴 dV ⫽ ⫺
2
V
冕
⫽2
⯝2
冉
r s2
⫹2
w i/r 2
r s2
w i/r 2
冋 冉
冋 冉
⯝ w 02 1 ⫹
S
(A6)
2 R i/r
⫽ ⫿
r s r sin ␹ cos ␾
w i/r 2
r 2 sin2 ␹
⫹
w i/r 2
z s ⫾ z 0 ⫹ r cos ␹
zR
zs ⫾ z0
zR
冊册
冊册
2
2
,
(B2)
z s ⫾ z 0 ⫹ r cos ␹
k
2 z R 关 1 ⫹ 共 z s ⫾ z 0 ⫹ r cos ␰ 兲 2 /z R 2 兴
2
zs ⫾ z0
⯝⫿
,
z R w i/r 2
kr B 2
2
冉
1
Ri
冊
(B1)
w i/r 2 ⫽ w 0 2 1 ⫹
k 1
关 ␾ T 共 D ⫺ D0 兲兴 ndS.
(A7)
,
␾ T ⵜ 共 D ⫺ D0 兲 dV.
V
The second integral in Eq. (A5) is equal to zero if we make
use of the fact that the sources of the field are unchanged
during the placement of the dielectric particles, i.e., ⵜD
⫽ ⵜD0 ⫽ ␳ , where ␳ is the free-charge density. With
the Gauss theorem, the first integral in Eq. (A5) can be
rewritten as the integration of the normal component of
the product ␾ T (D ⫺ D0 ) over a surface embedding the object:
冕
r B2
T
(A5)
1
⫺
2
2
ⵜ 关 ␾ T 共 D ⫺ D0 兲兴 dV
冕
共 ⑀ 2 ⫺ ⑀ 1 兲 具 EE0 典 T dV.
1. a™w 0
Let us express the principal terms in Eq. (9) in spherical
coordinates and simplify them as
V
1
Vi
APPENDIX B: INTEGRATION IN THE
GAUSSIAN STANDING WAVE IN
SPHERICAL COORDINATES
(A4)
The first term in Eq. (A2) can be rewritten with use of integration by parts, as
冕
Because of the time averaging, the equation takes the
same form as in the electrostatic case.
V
Therefore the time-averaged value over the field period is
equal to zero:
1
⫺
2
1
具 ⌬W E 典 T ⫽ ⫺
(A2)
1
⫺
2
If the radius R of the surface goes to infinity, the area increases as R 2 but the product ␾ T (D ⫺ D0 ) for electrostatic fields decreases as R ⫺3 . However, this relationship is not valid for the radiation fields. To eliminate this
integral, we apply Wolf ’s arguments35 even though they
cannot be regarded as the rigorous ones. We assume
that the radiation field does not exist at all times but instead is produced by some source at time t 0 . At any time
t ⬎ t 0 , the field fills the region of space reaching to c(t
⫺ t 0 ) from the source (c is the velocity of the light). If
the integration surface is far enough from the source, the
field does not reach it within t ⫺ t 0 , and the integral can
be omitted. Therefore we can conclude that the timeaveraged energy change caused by the insertion of the dielectric object into the electric field is described by
⫺
1
Rr
冊
⯝ ⫺r s 2
(B3)
冉
zs ⫹ z0
z Rw i
2
⫹
zs ⫺ z0
z Rw r2
冊
. (B4)
Integration of noninterference terms in Eqs. (11) and (12)
is equal to zero, or these terms can be taken in front of the
integral. The rewritten interference term, with the substitution ␰ ⫽ cos ␹ and ␾ s ⫽ ␾ (r a , z s ), gives for the force
Zemanek et al.
Vol. 19, No. 5 / May 2002 / J. Opt. Soc. Am. A
␣ n2
F z共 r s , z n 兲 ⫽
2 c
I 004 ␲␳
⫺ r s 2 /w r 2 兲
n2
⫽␣
c
I 00␲␳
w 02
w iw r
冕
1
⫺1
W ⫽ w i 共 z s 兲 w r 共 z s 兲 / 关 w i 2 共 z s 兲 ⫹ w r 2 共 z s 兲兴 1/2 ,
a 2 exp共 ⫺r s 2 /w i 2
X ⫽ ⫺kW/ 冑C,
Y ⫽ a 冑C/W,
cos共 2ka ␰ ⫹ ␾ s 兲 ␰ d␰ dr,
C ⫽ 1 ⫺ iZ W ,
w 02
exp共 ⫺r s 2 /w i 2
k 2w iw r
Z W ⫽ W 2 /R 0 2 ,
1/R 0 2 ⫽ 共 z s ⫹ z 0 兲 / 共 z R w i 2 兲 ⫹ 共 z s ⫹ z 0 兲 / 共 z R w r 2 兲 .
⫺ r s 2 /w r 2 兲共 2ak cos 2ak
⫺ sin 2ak 兲 sin ␾ s .
(B5)
If the error function is transformed into the Dawson integral by using
Equation (16) can be obtained in similar manner.
2. a™z R , a¶w 0
If the sphere is placed on axis (r s ⫽ 0) and is so large
that the lateral variation of the intensity cannot be neglected, but the condition a Ⰶ z R is satisfied, the approximations in Eqs. (B2) and (B3) can be used again, but instead of relations (B1) and (B4) the following expressions
have to be employed:
2
kr B 2
2
冉
1
Ri
⫺
1
Rr
冊
r B2
w t/r 2
⫽2
r 2 sin2 ␹
w i/r 2
⯝ ⫺r 2 sin2 ␹
冉
,
(B6)
zs ⫹ z0
z Rw i
2
⫹
zs ⫺ z0
z Rw r
2
冊
␣n2
c
I 00␳ 2 ␲ a 2
冋
w 02
w iw r
冕
(B7)
⫽
␣n2
c
I 00␳ 2 ␲ a
冕
w 02
2
w iw r
a2
␣n2
R0
2
册
冠
Re exp共 i ␾ s 兲
c
w iw r
再
kW 3 冑␲
2C 3/2
Author Pavel Zemánek may be reached by e-mail at
[email protected].
5.
7.
exp共 i ␾ s ⫺ X 2 ⫺ Y 2 兲
⫻ 关 erf共 iX ⫹ iY 兲 ⫹ erf共 ⫺iX ⫹ iY 兲兴
4.
6.
W2
⫺
sin 共 2ak 兲
1 ⫹ Z W2
⫻ 共 sin ␾ s ⫹ Z W cos ␾ s 兲
⫹ Re
The authors thank J. Komrska for fruitful discussion and
comments and the Grant Agency of the Czech Republic for
financial support (grants 202/99/0959, 101/98/P106 and
101/00/0974).
3.
⫺1
冠
exp共 t 2 ⫺ x 2 兲 dt,
Eq. (19) can be obtained. Moreover, the expression for the
energy change needs integration by parts of the noninterference and the interference terms, and Eq. (18) can be
obtained in this way.
2.
冡
w 02
x
0
1.
共 1 ⫺ ␰ 2 兲 ␰ d␰ ,
exp兵 关 ⫺a 2 C 共1 ⫺ ␰ 2 兲/W 2 兴 ⫹ 2iak ␰ 其 ␰ d␰ ,
I 00␳ 2 ␲
冕
exp共 x 2 兲 daw共 x 兲 ,
REFERENCES
(B8)
⫽
冑␲
exp关 ⫺a 2 共 1 ⫺ ␰ 2 兲 /W 2 兴
1
⫻
daw共 x 兲 ⫽
2i
1
⫺1
⫻ cos ␾ s ⫹ 2ak ␰ ⫹
erf共 ix 兲 ⫽
ACKNOWLEDGMENTS
.
The axial force can be obtained by using Eq. (12). It
can be shown that the integral of the noninterference
terms is equal to zero, and the interference term gives,
with the substitution ␰ ⫽ cos ␹,
F z 共 0, z s 兲 ⫽
1033
8.
冎冡
9.
, (B9)
where erf and Re denote the error function and the real
part, respectively, and
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