Download Radiation pressure cross sections and optical forces over negative

Radiation pressure cross sections and optical forces
over negative refractive index spherical particles
by ordinary Bessel beams
Leonardo A. Ambrosio* and Hugo E. Hernández-Figueroa
Department of Microwave and Optics, School of Electrical and Computer Engineering,
University of Campinas (Unicamp), Avenida Albert Einstein, 400, 13083-970 Campinas, São Paulo, Brazil
*Corresponding author: [email protected]
Received 4 April 2011; revised 15 June 2011; accepted 16 June 2011;
posted 16 June 2011 (Doc. ID 145030); published 27 July 2011
When impinged by an arbitrary laser beam, lossless and homogeneous negative refractive index (NRI)
spherical particles refract and reflect light in an unusual way, giving rise to different scattered and
internal fields when compared to their equivalent positive refractive index particles. In the generalized
Lorenz–Mie theory, the scattered fields are dependent upon the Mie scattering coefficients, whose values
must reflect the metamaterial behavior of an NRI scatterer, thus leading to new optical properties such
as force and torque. In this way, this work is devoted to the analysis of both radial and longitudinal optical
forces exerted on lossless and simple NRI particles by zero-order Bessel beams, revealing how the force
profiles are changed whenever the refractive index becomes negative. © 2011 Optical Society of
America
OCIS codes: 080.0080, 160.3918, 290.4020, 350.3618, 350.4855.
1. Introduction
One of the fundamental properties of optical tweezers
is the optical force exerted on a nano- or microsized
particle by an arbitrary laser beam. Its theoretical
determination is of significant practical importance,
because it allows a previous knowledge of the corresponding displacement of the particle toward or away
from some relative high energy concentration region
of the impinging beam. Since the first optical trapping
experiments performed by Ashkin et al. by the 1970s
and 1980s [1–8], a variety of incident beams have been
proposed for achieving an efficient and stable trap.
Optical forces from single or improved combinations
of Gaussian, Laguerre, Laguerre–Gaussian, and Bessel beams (BBs), among others, have been under intense investigation first because of their immense
biomedical interest for studying physiological mechanisms of molecules and biological organelles
0003-6935/11/224489-10$15.00/0
© 2011 Optical Society of America
[9–14] and second because of all the promising possibilities opened by optical tweezers in treating diseases such as cancer [15].
It is relatively easy to explain the momentum transfer from the photons of a laser beam to a dielectric
particle based on a ray optics picture [14,16]. In this
optical regime, an incident ray will be partially reflected and partially transmitted at the surface of the
scatterer according to Snell’s law, and, given the refractive index (nm ) of the immersion medium, it is
common knowledge that particles with refractive index np > nm are (except for special cases in which np is
much higher than nm , when repulsive axial, or scattering, prevails over gradient forces [17]) using simple
ray tracing, directed toward high-intensity regions of
the impinging beam, whereas np < nm would lead to a
repulsion phenomenon, the particle then tends to be
trapped only with special beams whose spatiallimited relative low-intensity regions provide points
of stable equilibrium, as is the case of multiringed
beams such as BBs [18–20]. The limiting situation np ¼ nm (or, equivalently, nrel ¼ np =nm ¼ 1)
1 August 2011 / Vol. 50, No. 22 / APPLIED OPTICS
4489
represents the matched condition, meaning that no
momentum transfer occurs.
When the diameter of the spherical particle is of
the order of or smaller than the wavelength of the
incident laser beam, however, geometric optics becomes out of its range of validity. There are, in fact,
several approaches or approximations to deal with
these situations, but maybe the most robust theory
relies on an expansion of the incident electromagnetic fields into a series of spherical harmonics,
called the Lorenz–Mie theory in the case of a plane
wave [21] or, for arbitrary laser beams, the generalized Lorenz–Mie theory (GLMT) [22,23]. The coefficients of these expansions for the scattered fields of
the GLMT are the Mie scattering coefficients, intrinsically related to the beam-shape coefficients (BSCs)
that describe the spatial field distribution of the incident beam [22,23]. Finding the BSCs may involve
time-consuming methods such as quadratures (exact
solutions with double or triple integrations) [24], finite series approaches without closed-form solutions
[25], or the localized approximation [26] and its improved version, the integral localized approximation
(ILA) [27], the last two proving along the years to be
an efficient and fast algorithm for both on- and offaxis Gaussian beams [28,29].
With the advent of a new class of materials possessing negative refractive indices, originally as a theoretical hypothesis [30] and more recently as an
experimental reality [31,32], the scientific community
has verified that plenty of new devices and unimaginable applications can be foreseen with such “metamaterials” (for a review of this subject, see Refs. [33–35]),
such as optical cloaking and transparency [36,37] and
the perfect lens [38]. New classes of antennas and
absorbers, transmission lines, waveguides and resonators, and lenses can overcome the limits of current
microwave and optics technology, emerging as
potential engineering machinery for the near future
[33–35,39].
Recently, we have proposed that negative refractive index (NRI) particles could find a place in optical
tweezers experiments. Numerical verification of
their trapping properties in the case of Gaussian
beams in ray optics [40], an arbitrary optical regime
using the GLMT [41], and also for arbitrary-order
paraxial BBs in ray optics [42] has demonstrated
that actual trapping limits can also be overcome,
an example being the possibility of an NRI particle
being either attracted or repelled under the influence
of a TEM00 Gaussian beam [40,41], this attraction or
repulsion depending solely on the relative distance
between the center of the particle and the optical axis
of the beam. Similar work was done on magnetodielectric particles [43]. Biomedical optics research,
especially those involving optical trapping systems,
could greatly benefit from such particles in the near
future.
As a natural consequence of our previous works,
this paper is devoted to the analysis of lossless
and simple NRI spherical particles under the influ4490
APPLIED OPTICS / Vol. 50, No. 22 / 1 August 2011
ence of zero-order BBs, with no privilege to any optical regime. This implies the adoption of the GLMT
as our mathematical support for calculating the optical forces, imposing the ILA in the determination of
the Mie scattering coefficients and, consequently, of
the BSCs. Therefore, this work is organized as follows: in Section 2 we introduce the GLMT together
with the ILA, providing a general analytical formula
for the BSCs and emphasizing the fundamental differences imposed by the NRI nature on the Mie scattering coefficients. In Section 3 we show how axial
(longitudinal) force profiles are modified by the introduction of the np < 0 inequality, while in Section 4
the same is done for transverse (radial) forces. Because of the phenomenological importance, we paid
special attention to the condition nrel ¼ −1, where
no analogies can be found in the positive refractive
index (PRI) case at all, because this situation can still
represent the matched condition whenever the permeabilities of both the external medium and the
particle are the same in modulus. Finally, our conclusions are presented.
2. GLMT and ILA Applied to Ordinary Bessel Beams
Suppose a monochromatic zero-order BB, propagating along þz in a host medium (unity permeability
pffiffiffiffiffiffi
μ0 and refractive index nm ¼ εm , εm being its permittivity) with wavelength λ, impinges on a lossless
and simple spherical particle with permeability μp ¼
pffiffiffiffiffiffiffiffiffi
1 and refractive index np ¼ μp εp , εp being its permittivity. In the framework of the GLMT, all incident,
scattered, and internal electromagnetic fields can be
mathematically described as a superposition of spherical harmonics, the coefficients associated with the
scattered fields being the BSCs, described in terms of
the radial components of the incident fields as [27]
Z 2π
Zm
n
^ r ðr; θ; ϕÞe−imϕ dϕ;
gm
¼
ð1Þ
G½H
n;TE
2πH 0 0
gm
n;TM ¼
Zm
n
2πE0
Z
0
2π
^ r ðr; θ; ϕÞe−imϕ dϕ;
G½E
ð2Þ
where the TE and TM subscripts stands for TE or TM
modes, respectively. H r and Er are the magnetic and
electric fields, written in a spherical coordinate system ðr; θ; ϕÞ whose origin is located at the center of
^ performs
the scatterer. The localization operator G
the transformation r → ðn þ 0:5Þ=k, where k is the
wavenumber, while Zm
n are normalization factors
that depend on the integers n and m, −n ≤ m ≤ n
and 1 ≤ n < ∞ (for further details, see [27] and references therein). For a linearly polarized ordinary
BB in the paraxial regime with its optical axis parallel to the þz axis and displaced a radial distance ρ0
from the center of the sphere, with an azimuth angle
ϕ0 relative to the x axis, H r and Er are written as [44]
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cos ϕ
E x ¼ E0 J 0 sin θa ðkrÞ2 sin2 θ þ ρ20 k2 − 2ðkrÞρ0 sin θ cosðϕ − ϕ0 Þ e−ikr cos θa cos θ sin θ
;
sin ϕ
r;
y
ð3Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sin ϕ
2
2
2 2
−ikr cos θa cos θ
sin θ
; ð4Þ
H x ¼ H 0 cos θa J 0 sin θa ðkrÞ sin θ þ ρ0 k − 2ðkrÞρ0 sin θ cosðϕ − ϕ0 Þ e
− cos ϕ
r;
y
E0 and H 0 being, respectively, the electric and magnetic field amplitudes, and θa is the associated axicon
angle of the beam. The paraxial field profiles in
Eqs. (3) and (4) are valid as long as sin θa =ð1 þ
cos θa Þ ≪ 1 [45]. A temporal factor expðiωtÞ is implicitly assumed. Notice that the terms in slashes (sin ϕ
or cos ϕ) are to be used with the corresponding polarizations of the BB. By imposing the radial fields
[Eqs. (3) and (4)] into Eqs. (1) and (2) and after some
algebra, one finds the following BSCs:
where Ψn are Ricatti–Bessel functions [46] and the
primes indicate a derivative with respect to the argument, α ¼ πd=λ is the size parameter, d ¼ 2a is the
diameter (a is the radius) of the particle, and
β ¼ nrel =α. The original TM and TE Mie scattering
coefficients for plane waves are denoted by an and
bn , respectively. Finally, the radiation pressure
cross sections are given in terms of Eqs. (6) and (7)
[47]:
8
>
cos ϕ0
2nðnþ1Þ
>
>
i ð2nþ1Þ J 1 ðsin θa ðn þ 1=2ÞÞJ 1 ðρ0 k sin θa Þ
;
>
>
sin ϕ0
>
>
>
>
1
>
1
−2i jmj−1
>
J jmj−1 ðsin θa ðn þ 1=2ÞÞJ jmj−1 ðρ0 k sin θa Þ
<
2 2nþ1
∓i
m
¼
g
1
>
x
>
>
×½cosðjmj
−
1Þϕ
∓i
sinðjmj
−
1Þϕ
þ
J jmjþ1 ðsin θa ðn þ 1=2ÞÞJ jmjþ1 ðρ0 k sin θa Þ
n;TM
0
0
>
>
i
y
>
>
>
>
>
>
: ×½cosðjmj þ 1Þϕ0 ∓i sinðjmj þ 1Þϕ0 ;
m¼0
;
m≠0
ð5Þ
Cpr;z
with equivalent expressions for the TE BSCs [44].
Once the BSCs are found, the Mie scattering coefficients are calculated in a straightforward way:
∞
λ2 X
1
ðA g0 g0
¼
Re
π n¼1
n þ 1 n n;TM nþ1;TM
þ Bn g0n;TE g0
nþ1;TE Þ
n
X
1
ðn þ m þ 1Þ!
m
þ
ðAn gm
n;TM gnþ1;TM
2
ðn
−
mÞ!
ðn
þ
1Þ
m¼1
m
am
n ¼ gn;TM an
¼
gm
n;TM
pffiffiffiffiffiffiffiffiffiffiffi 0
ε0 =εp ψ n ðαÞψ n ðβÞ
pffiffiffiffiffiffiffiffiffiffiffi
;
ψ n ðαÞψ n ðβÞ − ε0 =εp ψ n ðαÞψ n ðβÞ
ψ n ðαÞψ 0n ðβÞ −
−m
m
m
þ An g−m
n;TM gnþ1;TM þ Bn gn;TE gnþ1;TE
ð6Þ
−m
þ Bn g−m
n;TE gnþ1;TE Þ
2n þ 1 ðn þ mÞ!
m
Cn ðgm
n;TM gn;TE
n ðn þ 1Þ2 ðn − mÞ!
−m
− g−m
g
Þ
;
n;TM n;TE
ð8Þ
Cpr;x ¼ Re½C and Cpr;y ¼ Im½C;
ð9Þ
þm
pffiffiffiffiffiffiffiffiffiffiffi
ε0 =εp ψ n ðαÞψ 0n ðβÞ − ψ 0n ðαÞψ n ðβÞ
m
m
;
¼
g
b
¼
g
bm
n
n;TE n
n;TE pffiffiffiffiffiffiffiffiffiffiffi
ε0 =εp ψ n ðαÞψ n ðβÞ − ψ n ðαÞψ n ðβÞ
ð7Þ
2
where
1 August 2011 / Vol. 50, No. 22 / APPLIED OPTICS
4491
C¼
∞ λ2 X
ð2n þ 2Þ! nþ1
Fn
−
2π n¼1
ðn þ 1Þ2
n
X
ðn þ mÞ!
1
nþmþ1 m
F
þ
−
F mþ1
n
2
ðn
−
mÞ!
n−mþ1 n
ðn
þ
1Þ
m¼1
2n þ 1
−mþ1
m
−m
ðCn gm−1
n;TM gn;TE − Cn gn;TM gnþ1;TE
n2
−mþ1
m−1 m
−m
þ Cn gn;TE gn;TM − Cn gn;TE gnþ1;TM Þ ;
þ
ð10Þ
m−1 m
m−1 m
Fm
n ¼ An gn;TM gnþ1;TM þ Bn gn;TE gnþ1;TE
−mþ1
−mþ1
−m
þ An g−m
nþ1;TM gn;TM þ Bn gnþ1;TE gn;TE ;
ð11Þ
An ¼ an þ anþ1 − 2an anþ1
Bn ¼ bn þ bnþ1 − 2bn bnþ1
Cn ¼ −iðan þ bnþ1 − 2an bnþ1 Þ:
ð12Þ
As expected for the matched condition, the Mie
scattering coefficients [Eqs. (6) and (7)] are zero
whenever β ¼ α (nrel ¼ 1), regardless of the spatial
confinement of the laser beam. Both axial and radial
forces (or, equivalently, both axial and radial radiation pressure cross sections) are also zero, and no
movement of the particle can be observed due to
the momentum transfer.
However, if we consider the particle to be of NRI
nature with β ¼ −α (nrel ¼ −1), even though we still
have matched conditions in a ray optics picture (no
wave is reflected at the interface, i.e., the wave fully
penetrates and passes through the particle), am
n and
will
no
longer
be
zero,
as
already
expected
bm
n
[30,38,41,48,49]. Mathematically, this happens because the Ricatti–Bessel functions ψ n ðβÞ and their
derivatives have analytic continuation equations
with respect to the argument of the form ψ n ð−jβjÞ ¼
−ð−1Þn ψ n ðjβjÞ and ψ 0n ð−jβjÞ ¼ ð−1Þn ψ 0n ðjβjÞ so that, for
the NRI scattering process, Eqs. (6) and (7) could be
equivalently written as
m
am
n;NRI ¼ gn;TM an;NRI
¼
gm
n;TM
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0
ε0 =jεp jψ n ðαÞψ n ðjβjÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
;
ψ n ðαÞψ n ðjβjÞ þ ε0 =jεp jψ n ðαÞψ n ðjβjÞ
ψ n ðαÞψ 0n ðjβjÞ þ
ð13Þ
m
bm
n;NRI ¼ gn;TE bn;NRI
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ε0 =jεp jψ n ðαÞψ 0n ðjβjÞ þ ψ 0n ðαÞψ n ðjβjÞ
m
;
¼ gn;TE pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ε0 =jεp jψ n ðαÞψ n ðjβjÞ þ ψ n ðαÞψ n ðjβjÞ
ð14Þ
gm
n;TM
4492
thus reinforcing the fact that the numerators of
Eqs. (6) and (7) do not go to zero in the NRI matched
situation. In fact, we can express the NRI Mie scattering coefficients as a sum of two terms, the first representing the conventional expressions (6) and (7)
as if the particle were a PRI sphere and the second
representing a pure metamaterial term that adds to
the previous term in such a way as to establish the
final scattered and internal fields for an NRI sphere
[47]. When nrel ¼ −1, only the pure metamaterial
term contributes to the scattered and internal fields
and to the optical forces, also with new resonance effects [40,41,50]. Physically, although no reflection occurs at the surface of the scatterer, there may be a
significant energy redistribution, as the vector energy flux inside the NRI particle is not at the same
direction as the one in the host medium, the same
being valid for the wave vector [30]. In fact, in the
ray optics regime, we interpret this as the inversion
of Snell’s law [30].
Finally, similar relations for the Ricatti–Bessel
functions could be introduced in Eqs. (6) and (7) when
the refractive index changes from positive to purely
imaginary, which happens when the particle is a lossless plasmonic metamaterial, for example. Even relations concerning the Mie scattering coefficients
between plasmonic and NRI particles could be analyzed, but this is beyond the scope of this work, and
we shall not be concerned with these possibilities.
3. Axial Forces
In this section, we shall use the GLMT to calculate
the radiation pressure cross section along the direction of propagation of the BB by means of Eq. (8) with
the aid of Eq. (5) and the NRI versions of the Mie
scattering coefficients [Eqs. (13) and (14)]. To do
so, a Fortran code was developed and is available
upon request.
Let us consider again the condition nrel ¼ −1,
where Cpr;z ¼ 0 for any PRI particle. This is possible
if we choose, for example, μ0 ¼ 1 and ε0 ¼ 1 for the
external medium and μp ¼ −1 and εp ¼ −1 for the
scatterer. For all subsequent simulations, the parameters μ0 ¼ 1 and μp ¼ −1 are assumed, the permittivities being the only responsible for the change in
magnitude of the relative refractive index. Besides,
suppose our þz-propagating BB is x linearly polarized with λ ¼ 1064 nm (θa ¼ 0:0141, with a spot of
Δρ ¼ 21:81 μm on a host medium with nm ¼ 1:33)
and is shifted along x (ρ0 ¼ x0 ; ϕ0 ¼ 0). This simplifies the BSCs expressions in Eq. (5), which now
read as
8
2nðnþ1Þ
>
>
>
< i ð2nþ1ÞJ 1 ðsin θa ðn þ 1=2ÞÞJ 1 ðρ0 k sin θa Þ;
¼ 1 −2i jmj−1 ½J jmj−1 ðsin θa ðn þ 1=2ÞÞJ jmj−1 ðρ0 k sin θa Þ
2 2nþ1
>
>
>
: þJ jmjþ1 ðsin θa ðn þ 1=2ÞÞJ jmjþ1 ðρ0 k sin θa Þ;
APPLIED OPTICS / Vol. 50, No. 22 / 1 August 2011
m¼0
:
m≠0
ð15Þ
Figure 1 shows Cpr;z as a function of ρ0 ¼ x0 for several sizes of the NRI particle (a=λ ¼ 0:01, 0.1, 1, 5, 10,
and 20) calculated by imposing Eq. (15) on Eq. (8) and
using Eq. (12). One sees that due to the NRI nature of
the scatterer, the energy redistribution provided by
the index contrast (from þ1 to −1 when the wave penetrates the surface of the sphere and from −1 to þ1
when the wave exits it) changes the direction of the
incident photons, thus imposing the axial forces observed. The NRI matched condition can still provide
momentum transfer from the photons of the laser
beam to the particle, and the associated axial optical
force will be stronger whenever the particle gets close
to a bright-intensity region of the beam, where more
photons hit it. Naturally, as the particle gets smaller,
the axial forces become weaker. For a=λ ¼ 10 and 20,
the radius of the particle is such that, even when it is
centered at some dark annular region of the beam,
axial forces are still present due to the momentum
transfer from the lateral bright regions. As expected
for a single BB, all optical forces are repulsive (positive Cpr;z means axial forces along þz), and no threedimensional trap can be achieved.
We may compare the optical force profile behavior
for an NRI particle with those commonly observed for
PRI particles by plotting Cpr;z as a function of both
nrel and x0 , as can be seen in Fig. 2 for a=λ ¼ 0:01,
and 20, where a dashed line emphasizes the jnrel j ¼
1 condition. Looking at the amplitude of Cpr;z for refractive indices of equal magnitude, Fig. 2 immediately reveals that, in general, repulsive forces are
stronger for NRI particles. Besides, Fig. 2(a) shows
a significant dissimilitude between NRI and PRI axial forces for a=λ ¼ 0:01, mainly because of the discrepant relative amplitude observed around np ≈ −1:88
(nrel ≈ −1:41), which occurs due to a resonance of
a1;NRI at this point, as illustrated in Fig. 3 (there
are other resonances in an;NRI for n > 1, but their contributions to the Cpr;z profile is insignificant). It is
worthwhile to see that, for this radius and at this
nrel , the maximum peak amplitude at x0 ¼ 0 is
approximately 3:20 × 104 times greater than the
maximum peak observed for all nrel > 0 (Cpr;z ≈
4:26 × 10−20 m2 for np ¼ 3:95, whereas Cpr;z ≈ 1:36×
10−15 m2 for np ¼ −1:88). In the case of a=λ ¼ 20,
the magnitude differences are smoothed, but higher
axial repulsive forces still happens for NRI particles
when compared to a PRI with nrel;PRI ¼ jnrel;NRI j. Similar conclusions follow from Fig. 4, where Cpr;z is
represented for a=λ ¼ 0:1 and 10.
BSCs for circularly polarized BBs could easily be
constructed from Eq. (15) and its equivalent
pffiffiffi TE
m;circular
m
m
BSCs by writing gn;TE
¼ ðg
ign;TM Þ= 2 and
pffiffiffi n;TE
m;circular
m
m
gn;TM
¼ ðgn;TM ∓ign;TE Þ= 2, where the upper (lower) signs stand for right-hand (left-hand) polarization. No significant differences were observed
between linear and circular polarizations for the
same parameters used in Figs. 2(b) and 2(d).
Finally, we point out that the resonance character
of the Mie scattering coefficients is also reflected on
the axial forces if one considers an NRI or PRI particle with a fixed nrel , but with a varying radius. This
is well known for PRI particles in the framework of
Fig. 1. (Color online) Cpr;z (solid blue curve) as a function of ρ0 ¼ x0 for a=λ ¼ ðaÞ 0:01, (b) 0.1, (c) 1, (d) 5, (e) 10, and (f) 20. The NRI particle
and the host medium are matched (nrel ¼ −1, nm ¼ 1:33). The axial force is always repulsive (Cpr;z is positive), and weaker radiation pressure cross sections act on smaller particles, the profile resembling that of the BB intensity (dashed red curve), as expected. In all cases, a
three-dimensional trap is impossible.
1 August 2011 / Vol. 50, No. 22 / APPLIED OPTICS
4493
Fig. 2. (Color online) Cpr;z as a function of both x0 and np . (a) , (b) 3D views for a=λ ¼ 0:01 with 0 < np < 4 and −4 < np < 0, respectively.
(c), (d) Equivalent to (a), (b) for a=λ ¼ 20. Dashed lines are placed at np ¼ 1:33 (nrel ¼ 1), where Cpr;z ¼ 0 for PRI particles.
Fig. 3. (Color online) Mie scattering coefficients a1 and b1 for a=λ ¼ 0:01, λ ¼ 1064 nm. (a), (b) PRI particle, evidencing the matched condition nrel ¼ 1, where scattered fields are zero and no optical forces are exerted. (c), (d) NRI particle, where a resonance of a1 at nrel ≈ −1:41
increases the optical forces by many orders of magnitude.
4494
APPLIED OPTICS / Vol. 50, No. 22 / 1 August 2011
Fig. 4. (Color online) Same as Fig. 2 but for a=λ ¼ 0:1 [(a), (b)] and 10 [(c), (d)].
the GLMT [51]. Obviously, we expect that resonance
effects would also be present in the NRI case, but at
other specific ratios of a=λ due to the new denominators in Eqs. (13) and (14), as already pointed out.
4. Radial Forces
When a BB hits a PRI scatterer, radial forces tend to
push the particle toward either high- or low-intensity
regions of the multiringed transverse structure of the
Fig. 5. (Color online) Cpr;x (solid blue curve) as a function of ρ0 ¼ x0 for a=λ ¼ ðaÞ 0:01, (b) 0.1, (c) 1, (d) 5, (e) 10, and (f) 20. The NRI particle
and the host medium are matched (nrel ¼ −1, nm ¼ 1:33), and positive values represent an attractive force toward the optical axis. The
Bessel intensity profile is represented by a dashed (red) curve. For (a)–(e), stable equilibrium occurs at high-intensity regions of the beam.
In (f), the particle will be under stable equilibrium only when located at one of the low-intensity rings.
1 August 2011 / Vol. 50, No. 22 / APPLIED OPTICS
4495
Fig. 6. (Color online) Cpr;x as a function of both x0 and np . (a), (b) 3D views for a=λ ¼ 0:01 with 0 < np < 4 and −4 < np < 0, respectively. (c),
(d) Equivalent to (a), (b) for a=λ ¼ 20. The dashed lines at jnp j ¼ 1:33 (jnrel j ¼ 1) in (b), (d) correspond to the cut views (a), (f) of Fig. 5, when
Cpr;x ¼ 0 for PRI particles.
beam, the direction of displacement being determined mainly by the well-known inequalities nrel >
1 or 0 < nrel < 1, respectively. However, as already
discussed for an NRI spherical scatterer in ray optics
[42] and as pointed out in Section 2, the relation
jnrel j ¼ 1 does not seem to play any significant rule
or impose any special constraints over the optical
forces in this case.
To observe the attractive/repulsive character of the
radial forces, consider again our BB with θa ¼ 0:0141
and λ ¼ 1064 nm. This wavelength is chosen because
of its common use in optical trapping systems. For
comparison, Fig. 5 is the equivalent of Fig. 1 (nrel ¼
−1, or εp ¼ −εm ) for the radial radiation pressure
forces Cpr;x along x (the direction of both displacement and polarization of the beam). Notice that,
Fig. 7. (Color online) Same as Fig. 6 but for a=λ ¼ 0:1 [(a), (b)] and 10 [(c), (d)]. The dashed lines at jnp j ¼ 1:33 (jnrel j ¼ 1) in (b), (d)
correspond to the cut views (b), (e) of Fig. 5.
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for a=λ ≪ 1, radial forces will displace the particle toward high-intensity regions of the beam. Furthermore, for a=λ ¼ 20, the radial forces will pull the
particle into some of the low-intensity rings, depending on the relative distance between the center of the
particle and the optical axis. Three-dimensional
views of Cpr;x with the same parameters of Fig. 2
and 4 are shown in Figs. 6 and 7, respectively.
In Figs. 5(e) and 5(f), the radius of the particle is
much greater than the wavelength of the BB, and
ray optics considerations can be used to explain
the inversion of the radial optical forces observed
[42], when the stable equilibrium points are shifted
from ρ0 ¼ 25:02, 51.42, 79.16, 107:24 μm, and so on
(the mean radius of each dark ring of the beam) to
ρ0 ¼ 0, 34.43, 63.26, 91:81 μm, and so on (the mean
radius of each bright ring of the beam). By considering each single ray with its own longitudinal axis [40]
and nrel ¼ −1, the momentum transfer will induce a
repulsive force on the particle whenever the incident
angle is greater or equal 45°, and a ray optics diagram can be drawn similarly to Fig. 1 of Ref. [41]
for focused beams.
As expected, because of the new and different resonance points of the Mie scattering coefficients for
NRI scatterers, the intensity profile of the radial radiation pressure cross section Cpr;x , as one immediately sees from observing Figs. 6 and 7, is also
changed accordingly when compared to those of PRI
particles. Besides, there is not a single and simple
rule for previously knowing the attraction/repulsion
character of the optical force, as is the nrel ¼ 1 condition for dielectric PRI optical trapping [16,40,41].
This makes NRI optical trapping much more interesting and with particular optical properties for
every set of parameters concerning both the incident
beam (wavenumber and shape) and the spherical
particle (geometry and electromagnetic susceptibility and permittivity).
5. Conclusions
In this paper, we have shown how radiation pressure
cross sections or, equivalently, optical forces, are
modified when we replace a PRI by an NRI spherical
particle under the influence of a BB. The multiringed
structure of this type of laser beams allows a simultaneous study of multiple organelles and biological
structures for biomedical optics purposes.
The introduction of metamaterial particles as
auxiliary nano- or microstructures in biomedical research, for cancer treatment or in studying physiological, mechanical, or structural characteristics of
bacteria, red blood cells, and fungi, has revolutionized biomedical optics and helped in paving the
way into the era of artificial manufactured nanorobots and nanomotors. We believe that NRI metamaterials can also be implemented in optical trapping or
bistoury systems as auxiliary particles, with new
trapping properties.
The fundamental differences in the optical forces
observed when the refractive index of the particle
becomes negative lies on the fact that the Mie scattering coefficients have new resonance effects, which
are, in fact, incorporated into any optical property
that depends on them, such as optical torques and
scattering fields. Our mathematical formulation
can be extended to plasmonic metamaterials by introducing analogous analytic continuation equations
for the Ricatti–Bessel functions in order to rewrite
the PRI Mie scattering coefficients. This would also
allow comparisons between NRI and plasmonic
structures with emphasis on their pure metamaterial contributions to the optical trapping properties.
The fabrication of such three-dimensional NRI
structures, however, is still a challenge for current
researchers, but it is expected that such materials
will have significant impact for biomedical optics.
The authors wish to thank FAPESP—the Fundação
de Amparo à Pesquisa do Estado de São Paulo—under
contracts 2009/54494-9 (Ambrosio’s postdoctorate
grant) and 2005/51689-2 (CePOF, Optics and Photonics Research Center), for supporting this work.
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