frozen waves

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Analytical approach of ordinary frozen waves
for optical trapping and micromanipulation
Article in Applied Optics · April 2015
Impact Factor: 1.78 · DOI: 10.1364/AO.54.002584
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Leonardo André Ambrosio
University of São Paulo
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2. Analytical approach of ordinary frozen waves for
optical trapping and micromanipulation
Leonardo André Ambrosio1,
* and Michel Zamboni-Rached2
1
Department of Electrical and Computer Engineering, São Carlos School of Engineering, University of São Paulo,
400 Trabalhador São-carlense Ave., 13566-590 São Carlos, Brazil
2
Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Street,
Toronto, ON, Canada
*Corresponding author: leo@sc.usp.br
Received 7 October 2014; revised 11 February 2015; accepted 11 February 2015;
posted 11 February 2015 (Doc. ID 224275); published 23 March 2015
The optical properties of frozen waves (FWs) are theoretically and numerically investigated using the
generalized Lorenz-Mie theory (GLMT) together with integral localized approximation. These waves are
constructed from a suitable superposition of equal-frequency ordinary Bessel beams and are capable of
providing almost any desired longitudinal intensity profile along their optical axis, thus being of potential
interest in applications in which intensity localization may be used advantageously, such as in optical
trapping and micromanipulation systems. In addition, because FWs are composed of nondiffracting
beams, they are also capable of overcoming the diffraction effects for longer distances when compared
to conventional (ordinary) beams, e.g., Gaussian beams. Expressions for the beam-shape coefficients of
FWs are provided, and the GLMT is used to reconstruct their intensity profiles and to predict their
optical properties for possible biomedical optics purposes. © 2015 Optical Society of America
OCIS codes: (050.1940) Diffraction; (140.3460) Lasers; (170.4520) Optical confinement and manipu-
lation; (290.4020) Mie theory.
http://dx.doi.org/10.1364/AO.54.002584
1. Introduction
The use of nondiffracting beams in optical trapping
and micromanipulating systems is recent and relies
on the possibility of simultaneously trapping several
scatterers [1]. Single-beam experimental setups with
Bessel beams (BBs) for manipulating microsized
structures appeared in the literature during the first
years of the past decade, proving to be an efficient
tool for two-dimensional (2D) micromanipulation
[2–6]. Particles embedded on a host medium can
be mechanically displaced and rotated by exchanging
(linear or angular) momentum with zero-order or
higher-order nondiffracting beams, and several of
them, depending on their refractive index relative to
the external medium, can be simultaneously trapped
in one of the various bright or dark annular disks of
intensity, aligned in the radial or in the longitudinal
directions [1–7].
One of the advantages of using BBs in biomedical
optics is the self-reconstruction property of such
beams after passing through an obstacle due to
the temporally constant lateral energy provided
to the optical axis, which is intrinsically related to
the multiple trap property. Simple optical elements
or computer-generated holograms can be adapted
in optical tweezers systems, for example, to generate
arbitrary-order BBs and alternative schemes for
three-dimensional (3D) micromanipulation [2,6,8].
In a recent work, we provided paraxial theoretical
derivations of the beam-shape coefficients (BSCs) for
a single ordinary (zero-order) BB using integral local-
ized approximation (ILA) [9] in the framework of the
generalized Lorenz-Mie theory (GLMT) [10–15]. Ana-
lytical expressions for the BSCs gm
n;TM and gm
n;TE, which
1559-128X/15/102584-10$15.00/0
© 2015 Optical Society of America
2584 APPLIED OPTICS / Vol. 54, No. 10 / 1 April 2015

3. ultimately describe the geometrical shape of a laser
beam satisfying Maxwell’s equations, were intro-
duced basically considering potential biomedical op-
tics applications, viz., for optical force calculations
over microsized scatterers. Because the GLMT ap-
plies to any arbitrary optical regime, our method
may be expected to serve as a robust alternative to
those already available in the literature [16,17].
Because of the nondiffracting nature of BBs, a type
of wave has been proposed in which a suitable super-
position of BBs with different longitudinal (or, equiv-
alently, transverse) wave numbers can generate an
almost unlimited number of longitudinal intensity
profiles (LIPs), even in absorbing media [18–21].
The idea is to take advantage of the long-range
propagation of BBs (when compared to conventional
diffracting beams) to design a pre-chosen arbitrary
LIP for specific purposes, such as in free-space optics,
atom guides, optical microlithography, and optical or
acoustical bistouries. These waves have been called
frozen waves (FWs) because they provide localized
wave fields with high transverse localization and
static intensity envelopes whose longitudinal inten-
sity patterns can be chosen a priori. Recently, FWs
were experimentally produced for the first time by
using computer-generated holograms and a spatial
light modulator [22].
Because of their intrinsic nondiffracting nature
and their high degree of freedom for the specification
of predetermined LIPs, they surely must be viewed
as promising alternative laser beams for optical trap-
ping experiments, and, therefore, it would be of inter-
est to delineate their basic properties, such as optical
forces (or, equivalently, radiation pressure cross sec-
tions) and torques.
This work is, therefore, devoted to the theoretical
derivation of the BSCs that correctly describes para-
xial FWs in the framework of the GLMT, thus
allowing the study of their optical properties when-
ever they are used as laser beams for optical trapping
and micromanipulation of particles, and is organized
as follows: Section 2 presents the GLMT for (single)
ordinary paraxial BBs using ILA, based on a pre-
vious work of the authors and assuming both linear
and circular polarizations [9]. In Section 3, we out-
line the theory of FWs and use the results of Section 2
for evaluating the BSCs for this class of laser beams.
In Section 4 we provide some examples of prechosen
LIPs, all providing good agreement with previous
works [18,19]. Then, in Section 5, we use the results
and numerical values of Sections 3 and 4 to calculate
longitudinal optical forces exerted by FWs over
homogeneous spherical particles with arbitrary ra-
tios between their radii and the wavelength and with
arbitrary (both positive and negative) real refractive
indices.
2. ILA Description of Ordinary Bessel Beams
ILA is an alternative to the more time-consuming
quadrature schemes [16] and finite series ap-
proaches [17] for the computation of the BSCs of
arbitrary laser beams in the framework of the GLMT,
which is an extension of the Lorenz-Mie theory for
the scattering of arbitrary waves by a spherical par-
ticle. It avoids the numerical difficulties arising in
the calculation of the BSCs when quadratures are
employed by eliminating the oscillatory behavior of
the integrands, while still maintaining a flexible
character (only the kernel is to be modified in both
quadratures and in ILA when the nature of the inci-
dent beam is changed, in contrast with finite series)
[23–25]. The principle of localization of van de Hulst
[26], which is deduced from the asymptotic behavior
of Bessel functions of order (n 1∕2), n representing
the corresponding order, was adopted from the plane-
wave situation (Lorenz-Mie theory) and successfully
applied for Gaussian beams, laser sheets, top-hat
beams, and, more recently, zero-order paraxial BBs.
(For a review on the GLMT, its different techniques
for determining the BSCs, and its applications, see
[12,13,27] and references therein.)
To start, let us now briefly resume ILA by consid-
ering a linearly polarized (along u or v) paraxial BB
propagating along positive w [9,14,15]. From sym-
metry considerations of the BSCs, circular polariza-
tion can be implemented [28–30]. Notice that the
origin Ouvw is displaced from Oxyz (which will be
considered as the reference for the center of the
spherical particle) by x0; y0; z0 and that the axes
u; v; w are parallel to the axes x; y; z, respectively.
Thus, a paraxial ordinary BB with linear polariza-
tion in x or y is described, using cylindrical coordi-
nates relative to the xyz system, as [9]
E ρ; ϕ; z
ˆx
ˆy
E0J0 χ e−ikz z−z0 ; (1)
where χ kρ ρ2 ρ2
0 − 2ρρ0 cos ϕ − ϕ0
q
, E0 is the
field amplitude, J0 · is the zero-order Bessel func-
tion, kρ (kz) is the transverse (longitudinal) wave
number, ρ x2
y2 1∕2
, ρ0 x2
0 y2
0
1∕2
, and
ϕ0 arctan y0∕x0 . A factor exp iωt is omitted
for simplicity. For a BB, kρ and kz are intrinsically
related to the axicon angle θa, that is, kρ k sin θa
and kz k cos θa (k nref 2π∕λ, nref being the re-
fractive index and λ the wavelength in vacuum),
thus forming the well-known k cone in space. For
the paraxial regime, one must ensure β
sin θa∕ 1 cos θa ≪ 1 [5]. Contrary to [9], we keep
the factor exp ikzz0 , because the longitudinal dis-
placement of the beam relative to the center of
the scatterer will now be important in the case
of FWs. The magnetic field H can be readily ob-
tained from Maxwell’s equations by assuming loss-
less media with no sources.
ILA can be applied to Eq. (1) by (i) using a spherical
coordinate system r; θ; φ centered at Oxyz; (ii) finding
the corresponding radial fields Er and Hr; (iii) apply-
ing to Er and Hr an operator ˆG that performs the
transformations z → 0, kr → n 0.5 and introduc-
ing prefactors Zm
n , n, and m being integers related
1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS 2585

4. to the associated Legendre functions Pm
n · ; and
(iv) performing an integration over angular coordi-
nate ϕ. Details are found in [9]. Thus, using the pro-
cedure above and introducing the factor exp ikzz0 ,
the BSCs gm
n;TM and gm
n;TE for arbitrary ϕ0 and
z0 can be easily obtained, after some simple
algebra, as
g0
n;TMfx
yg i
2n n 1
2n 1
J1 ϖ J1 ξ
cos ϕ0
sin ϕ0
exp ikzz0 ;
(2)
gm≠0
n;TMfx
yg
1
2
−2i
2n 1
jmj−1 1
∓i
Jjmj−1 ϖ Jjmj−1 ξ
× cos jmj − 1 ϕ0∓i sin jmj − 1 ϕ0
1
i
Jjmj 1 ϖ Jjmj 1 ξ
× cos jmj 1 ϕ0∓i sin jmj 1 ϕ0
× exp ikzz0 ; (3)
g0
n;TEfx
yg i
2n n 1
2n 1
J1 ϖ J1 ξ
sin ϕ0
− cos ϕ0
exp ikzz0 ;
(4)
gm≠0
n;TEfx
yg
1
2
−2i
2n 1
jmj−1 ∓i
−1
Jjmj−1 ϖ Jjmj−1 ξ
× cos jmj − 1 ϕ0∓i sin jmj − 1 ϕ0
i
−1
Jjmj 1 ϖ Jjmj 1 ξ
× cos jmj 1 ϕ0∓i sin jmj 1 ϕ0
× exp ikzz0 ; (5)
where ϖ sin θa n 1∕2 , ξ ρ0k sin θa, and, as in
Eq. (1), the terms inside the slashes are related to the
corresponding x or y polarization. Notice that the
factor exp ikzz0 now appears explicitly. The set of
Eqs. (2)–(5) may be further simplified if we consider
particular polarizations and displacements such as
ϕ0 0 or π∕2 for x or y polarization [9]. For the
on-axis case (ρ0 0), when the longitudinal Poynting
vector does not depend on ϕ (axisymmetric case),
the GLMT requires nonzero BSCs to occur only for
m 1 or m −1, with the additional constraint
g1
n;TM g−1
n;TM ig1
n;TE −ig−1
n;TE. These BSCs reduce
to the plane wave case when θa 0, as expected,
because J0 ϖ → 1 [28]. Numerical values for gm
n;TM,
together with some examples of ordinary BB
descriptions, can be found elsewhere and will not
be reproduced here [9].
3. Extension to Frozen Waves
Initially, the method of the FWs was developed for
the ideal case where infinity-energy BBs, all with
the same frequency but different longitudinal wave
numbers, were superposed in order to achieve some
prechosen LIP. This, however, would require that
our BBs be generated by infinite apertures, thus
making the method unrealizable in practice
[18,19]. But situations may exist in which trun-
cated versions of FWs (using transmitters with fi-
nite apertures) can furnish the same LIP as its
ideal counterpart, with low error [20–22]. This is
possible because of a careful compromise between
the radius R of the aperture (which, by the way,
limits the maximum distance L up to which the
BBs can regenerate themselves) and the number
of lateral lobes of each BB for a radial distance
R (thus imposing the self-reconstruction capacity
of this BB [31,32]).
Regardless of how FWs are generated, as a first
approximation we can assume that a longitudinal
(ρ 0) intensity profile jΨ ρ 0; z j2 jF z j2 has
been previously specified inside the chosen interval
0 ≤ z ≤ L (or, as assumed hereafter, −L∕2 ≤ z ≤ L∕2)
and that the associated FW propagating along z is
given by the following scalar solution to the wave
equation [19]:
Ψ ρ; z
XN
q −N
AqJ0 kρqρ exp −ikzqz : (6)
In Eq. (6), kρq (kzq) are the transverse (longitudinal)
wave numbers of each BB with complex ampli-
tude Aq:
kzq Q
2π
L
q 0 ≤ Q
2π
L
q ≤
nref ω
c
kρq
nref ω
c
2
− k2
zq
1∕2
Aq
1
L
Z L∕2
−L∕2
F z exp i2πqz∕L dz; (7)
where ω is the angular frequency, and Q > 0 is a
constant value [limited according to Eq. (7)] chosen
according to the given experimental situation and
the desired degree of transverse field localization.
Notice that the constraints on Q ensure propagat-
ing waves (no evanescent waves) in the positive z
direction. The integrand for Aq differs from pre-
vious work in the sign of the exponential because
of our temporal convention exp iωt [19]. Now, if
one displaces the beam by a rectangular distance
x0; y0; z0 , in analogy with the previous section,
Eq. (6) can be recast in the form
Ψ ρ; ϕ; z
XN
q −N
AqJ0 χq e−ikzq z−z0 ; (8)
2586 APPLIED OPTICS / Vol. 54, No. 10 / 1 April 2015

5. where χq kρq ρ2 ρ2
0 − 2ρρ0 cos ϕ − ϕ0
1∕2. One
can assume Eq. (8) to represent our new transverse
(x or y) electric field component, the determination
of the BSCs being similar to that of a single BB.
In fact, because FWs are linear superposition of
equal-frequency BBs, gm FW
n;TM and gm FW
n;TE will also in-
corporate this linear condition. From Eqs. (2)–(5),
one finds
gm 0 FW
n;TMfx
yg i
2n n 1
2n 1
×
XN
q −N
AqJ1 ϖq J1 ξq
cos ϕ0
sin ϕ0
exp ikzqz0 ;
(9)
gm≠0 FW
n;TMfx
yg
1
2
−2i
2n 1
jmj−1 XN
q −N
Aq
1
∓i
Jjmj−1 ϖq Jjmj−1 ξq
× cos jmj − 1 ϕ0∓i sin jmj − 1 ϕ0
1
i
Jjmj 1 ϖq Jjmj 1 ξq
× cos jmj 1 ϕ0∓i sin jmj 1 ϕ0 exp ikzqz0 ;
(10)
gm 0 FW
n;TEfx
yg i
2n n 1
2n 1
×
XN
q −N
AqJ1 ϖq J1 ξq
sin ϕ0
− cos ϕ0
× exp ikzqz0 ; (11)
gm≠0 FW
n;TEfx
yg
1
2
−2i
2n 1
jmj−1 XN
q −N
Aq
×
∓i
−1
Jjmj−1 ϖq Jjmj−1 ξq
cos jmj − 1 ϕ0∓i sin jmj − 1 ϕ0
i
−1
Jjmj 1 ϖq Jjmj 1 ξq
× cos jmj 1 ϕ0∓i sin jmj 1 ϕ0
× exp ikzqz0 ; (12)
where ϖq sin θaq n 1∕2 , ξq ρ0k sin θaq, and
sin θaq 1 −
kzq
k
2 1∕2
: (13)
Obviously, only an approximation of the desired
LIP is observed from Eqs. (9)–(12) because the
sums are necessarily truncated (q ≤ Nmax). We em-
phasize that all the copropagating BBs that com-
pose our FW have the same frequency (the wave
number k is fixed) and are capable of maintaining
their nondiffracting properties up to a longitudinal
distance Z ≈ R∕ tan θa when generated by a finite
aperture of radius R. According to Eq. (13), there-
fore, L < Zmin, where Zmin is the field depth of the
BB with the smallest longitudinal wave num-
ber kz q −N Q − 2πN∕L.
Let us again look at the on-axis case but now for a
FW described in the GLMT by means of Eqs. (9)–(12).
As said, the only nonzero coefficients are those with
m 1. This is easily seen by imposing ρ0 0. Be-
cause ξq ρ0k sin θaq 0 and Jv 0 0 for v ≠ 0 (v
integer), Eqs. (9)–(12) for x-polarized BBs simplify,
for example, to
8
>>><
>>>:
gm≠1 FW
n;TMfx
yg gm≠1 FW
n;TEfx
yg 0
gm 1 FW
n;TMfx
yg
1
2
PN
q −N AqJ0 ϖq exp ikzqz0
gm 1 FW
n;TMfx
yg
∓i
2
PN
q −N AqJ0 ϖq exp ikzqz0
; 14
and, as expected, Eq. (14) represents a summation
of the BSCs given by Eq. (5), the contribution of
each ordinary BB to the LIP being weighted by
complex amplitudes Aq. Normalization factors can
be introduced into Eqs. (9)–(12) and Eq. (14) to en-
sure that BSCs for FWs reduce to the plane-wave
case when θa → 0 and z0 0. These factors would
explicitly take into account the fact that all BBs
are generated by the same (and not by distinct)
power sources.
Finally, the correct description of the prechosen
LIP, jF z j2
, using Eq. (6), depends, as we will see
in the next section, on the number (2N 1) of BBs
or, equivalently, on N Nmax, the choice of which
being limited to real values of the transverse wave
numbers kρq (kzq < k). This, according to Eq. (7), also
depends on the longitudinal distance L up to which
the desired LIP is to be reproduced and on the
parameter Q. For example, for Q Q0nrelω∕c
0.99nrelω∕c, Nmax 0.01 L∕λm , where λm is the
wavelength in the propagating medium. Thus, for
a FW generated by zero-order BBs with λm
λ∕nrel 1064 nm ∕1.33 (nrel 1.33 being approxi-
mately the refractive index of water, and λ
1064 nm the wavelength usually adopted in optical
tweezers systems), Nmax ∼ 18 for L 2 mm (thus,
the LIP of interest is within jzj < 1 mm). This en-
sures propagating waves only, and sin θa is real.
For the above values, one can verify that the aperture
radius R necessary to generate such LIP must
satisfy R ≥ 3.76 × 10−4
m.
1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS 2587

6. 4. FWs Generated by the Generalized Lorenz-Mie
Theory
The representation of a laser beam in the framework
of the GLMT allows one to calculate optical forces
and torques, radiation pressure cross sections, scat-
tering amplitudes, incident and scattered electro-
magnetic fields, etc., only by the knowledge of the
Mie scattering coefficients and the BSCs gm
n;TM and
gm
n;TE. In this section, we shall consider a few exam-
ples of applications of the analytical solutions shown
in Section 3 for reproducing the intensity profile of
some FWs using the GLMT formulation. We assume
that all BSCs have been normalized so that the
plane-wave case can be recovered.
A. First Example
In this first example, let us suppose that a constant
LIP is desired over the range −Zmax ≤ z ≤ Zmax by
superposing on-axis BBs with λ 1064 nm in water
(nrel 1.33). For such, we chose L 10−3
m, Zmax
0.1L, and Q 0.95nrelω∕c Q0 0.95 , which implies
Nmax 15 (that is, our FW will be generated by
2Nmax 1 31 BBs) [19]. All BSCs with jmj ≠ 1
are zero. According to the previous section, determin-
ing g1 FW
n;TM automatically allows us to find g1 FW
n;TE ,
g−1 FW
n;TM , and g−1 FW
n;TE . BSCs g1 FW
n;TM (up to n 400)
can be viewed in Fig. 1. Notice that, as n increases,
the amplitude of the BSCs and, consequently, their
contribution to the generated field, decreases. It
should be emphasized that we have chosen to
normalize the gm FW
n;TM s over g1 FW;OA
n;TM so that their
maximum possible value (on-axis case, OA) is 0.50
(as seen in Fig. 1). This ensures that the BSCs for
ordinary FWs will reduce to those expected for plane
waves in the limit θaq→0 [27].
The intensity profile associated with this FW can
be evaluated in the GLMT by using expressions for
the electromagnetic fields available in the literature
[27]. Figure 2 shows 2D and 3D plots of both the
expected FW generated by using the method avail-
able in [19] [Figs. 2(a) and 2(b)] and that obtained
by the GLMT with BSCs up to n 400 [Figs. 2(c)
and 2(d)]. Good agreement is achieved, as expected.
One clearly sees that lateral energy, provided by the
lateral lobules of the 31 BBs, constantly feeds the
longitudinal axis close to the region of interest
−0.1L ≤ z ≤ 0.1L in order to reproduce the LIP.
Notice that the constant LIP as shown in Fig. 2
could, in principle, be significantly improved by using
lower values for Q0 (or, for instance, by choosing a
larger L). As long as decreasing the value of Q0 in-
creases Nmax, the LIP is expected to get closer to that
of the ideal case (constant). Although this would be
desired for practical purposes, one may not accom-
plish it without having to introduce nonparaxial
BBs [the condition sin θaq∕ 1 cos θaq ≪ 1 is no
longer valid for all BBs, specially for those with
q < 0]. This extension to nonparaxial BBs is cur-
rently under investigation and will be considered
in a future work.
B. Second Example
We now consider a LIP in the range −Zmax ≤ z ≤
Zmax that exhibits an exponential growth of the
form exp 5z∕L . Again we impose on-axis BBs with
λ 1064 nm in water. The following parameters
have been chosen: L 2 × 10−3
m, Zmax 0.10L,
Q 0.99nrelω∕c Q0 0.99), and Nmax 25 [i.e., 51
BBs for the superpositions in Eqs. (9)–(12)].
Figure 3 is equivalent to Fig. 1. Notice again that,
as n increases, the amplitude of the BSCs and, con-
sequently, their contribution to the generated field,
increases. The imaginary values (red squares) have
been multiplied by 10 for visualization purposes.
Figure 4 shows the 3D and 2D patterns for
jΨ ρ; z j2
, revealing the exponential growth of the in-
tensity along the optical axis as a consequence of a
constant lateral energy feeding.
The results presented in Figs. 2 and 4 of this sec-
tion can be compared with their long-range versions
[18–21]. Whereas it is relatively easy to fulfill the
paraxial requirement of small axicon angles for
long-range FWs, designing FWs for optical trapping
and micromanipulation using a scalar theory may
possess serious limitations, mainly because L must
be of the order of millimeters, which significantly
limits the number of BBs with small axicon angles
when their wavelengths are of the order of a
micrometer.
5. Applications in Optical Trapping and
Micromanipulation
We saw in the previous sections that paraxial FWs
can be accurately described through knowledge of
the BSCs and their introduction in the GLMT
formulation. The general off-axis BSCs given by
Eqs. (9)–(12) or their on-axis simplified versions,
Eq. (14), may be readily implemented not only in
any GLMT algorithm but also in similar methods
by introducing some prefactors [27,33,34].
All above considerations lead us to consider
FWs as potential laser beams for biomedical optics
Fig. 1. BSCs g1
n;TM (real part as blue triangles, imaginary part as
red squares) using ILA for an on-axis FW that provides an ideal
constant LIP. As expected, g1
n;TM → 0 as n increases, their contri-
bution to jF z j2 being insignificant for n > 400.
2588 APPLIED OPTICS / Vol. 54, No. 10 / 1 April 2015

7. purposes, e.g., the optical micromanipulation or trap-
ping of viruses, bacteria, biological organelles, etc.,
using laser beams [3,35–37]. In addition, due to
the spatial localization of their optical field, FWs
could also serve, for instance, as alternative laser
beams for optical bistouries.
According to Zamboni–Rached et al., practical FWs
could be generated by an array of annular disks, using
light space modulators or even by means of computer
holograms [18–22]. They would take advantage of the
nondiffracting and self-reconstruction properties
of BBs with an additional degree of freedom associ-
ated with its longitudinal field localization potential-
ities, thus providing an effective 3D trap for optical
tweezers systems. As the advantages of multiple trap-
ping and manipulation of particles using BBs are well
known, it would certainly be desired, even if only
theoretically and numerically, to study the trapping
properties of FWs such as optical forces or, equiva-
lently, radiation pressure cross sections.
The analytical expressions developed in the pre-
vious sections were numerically implemented on a
GLMT Fortran code, the evaluation of the BSCs
being based on ILA. Longitudinal and transverse ra-
diation pressure cross sections over homogeneous
spherical particles can then be calculated based on
the available literature [38,39]. These cross sections
are linear with respect to the longitudinal and
Fig. 3. BSCs g1
n;TM (real part as blue triangles and imaginary part
in red squares) necessary to generate the growing exponential LIP.
In comparison with Fig. 1, one sees that a higher number of BSCs
are necessary to adequately represent the desired jΨ ρ; z j2 in the
framework of the GLMT. The imaginary values are scaled by a
factor of 10.
Fig. 2. (a) 3D and (b) 2D views of jΨ ρ; z j2 generated by the technique developed by Zamboni–Rached et al. [18], using L 10−3 m,
Q0 0.95, Nmax 15, Zmax 0.1L. All on-axis BBs have λ 1064 nm and are supposed to generate a constant LIP (along ρ 0) within
the range −Zmax ≤ z ≤ Zmax. (c) and (d) Same as (a) and (b) but using the method developed in Section 3.
1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS 2589

8. transverse optical forces exerted over these scatter-
ers, respectively, so that the results for the former
can be automatically extended to the latter [27].
Let us refer to the case of the FW with constant
LIP (first example, Section 4) and suppose that it im-
pinges on a spherical dielectric particle with radius
a 17.5 μm. The longitudinal radiation pressure
cross section (Cpr;z) for this scatterer is sketched in
Fig. 5 for four positive relative refractive indices
(nrel 0.95, 1.005, 1.01, and 1.20) between the par-
ticle and the water. It is clear that, depending on
nrel, this type of FW could provide a 3D trap because
possible points of stable equilibrium can be observed
for z0 ≈ −37 μm (the two zero force points repre-
sented by points P and Q for nrel 1.005 and 1.01,
respectively). As z0 changes, scattering forces tend
to push the particle back to its stable equilibrium
point. Finally, Fig. 6 shows plots of Cpr;z for negative
refractive index particles (simultaneous negative
permeability μ and permittivity ε with μ −1.0),
and, as noticed by Ambrosio and Hernández-
Figueroa [40–45], achieving a 3D trap with nrel < 0
may be much more challenging.
When we replace the previous FW with the grow-
ing exponential one, the results for Cpr;z again reveal
possible positions of stable equilibrium for a 3D trap.
These points occur at z0 ≈ −105 μm and are denoted,
respectively, as P0 and Q0 in Fig. 7. Figure 8 corre-
sponds to Fig. 6 when nrel < 0, and again we observe
restrictions in achieving an efficient 3D trap for the
values chosen.
Fig. 5. Longitudinal radiation pressure cross section Cpr;z for the
constant LIP of the previous sections over a dielectric spherical
particle with radius a 17.5 μm. Four different nrel nwater
1.33 are shown, and the center of the scatterer is fixed at
z 0. Possible points of stable equilibrium for nrel 1.005 and
1.01 are indicated as P and Q, respectively. For visualization pur-
poses, slopes nrel 1.005 and 1.01 are scaled by factors of 10 and 5,
respectively.
Fig. 4. jΨ ρ; z j2 for the growing exponential profile.
2590 APPLIED OPTICS / Vol. 54, No. 10 / 1 April 2015

9. Incidentally, neither a constant nor an exponential
LIP is capable of providing multiple axial traps, at
least with the results predicted by ILA in the
framework of the GLMT and using the geometrical
and electromagnetic parameters considered. The flex-
ibility in designing virtually any desired jF z j,
however, can be used advantageously to specify any
arbitrary-shape intensity pattern along ρ 0. As
an example, consider a LIP given by the two inverted
parabola
F z
8
<
:
−4 z−l1 z−l2
l1−l2
2 ; for l1 ≤ z ≤ l2
−4 2
p z−l3 z−l4
l3−l4
2 for l3 ≤ z ≤ l4
0 elsewhere;
(15)
where l1 1.5L∕10 − Δz, l2 1.5L∕10 Δz, l3
−1.5L∕10 − Δz, and l4 −1.5L∕10 Δz, with Δz
L∕70. When Eq. (15) is inserted in Eq. (7) and the first
n 300 jmj 1 BSCs [Eqs. (9)–(12)] are evaluated,
the (normalized) resulting jΨ ρ; z j becomes that
shown in Fig. 9 for L 10−3 m, Zmax 0.4L, and Q
0.988nrelω∕c (Nmax 15). This is an adaptation of
Fig. 2 from [19] and, in comparison with the previous
examples, a computationally time-consuming pattern
in the GLMT due to the fact that the BSCs (jmj 1)
decay very slowly with increasing n. Lateral energy
feeding the LIP is immediately observed in this figure
(notice that Fig. 11 represents jΨ ρ; z j and not
jΨ ρ; z j2
as in Figs. 3 and 6).
Cpr;z curves can be appreciated in Figs. 10 and 11,
again for the same nrel as before. Notice that two
points of stable equilibrium along z are readily ob-
served in Fig. 10, which means that this FW allows
for multiple axial trapping and, therefore, simultane-
ous micromanipulation of organic particles.
The results presented in this paper are limited by
the discrete superposition of zero-order BBs, all with
the same frequency, and this does not give us any
control over the transverse intensity profile as the
FW propagates along z. However, this difficulty
can be overcome by using higher-order BBs, which,
as a consequence, will shift jF z j along ρ and create
an transverse annular ring of intensity whose inten-
sity profile along z can also be designed at will [21].
Fig. 7. Cpr;z for the exponential LIP of the previous sections over a
dielectric spherical particle with radius a 17.5 μm. Four
different nrel nwater 1.33 are shown, and the center of the scat-
terer is fixed at z 0. Possible points of stable equilibrium for
nrel 1.005 and 1.01 are indicated as P0 and Q0, respectively.
For visualization purposes, the slopes nrel 1.005 and 0.950 are
scaled by factors of 10.
Fig. 8. Similar to Fig. 6 but for a growing exponential LIP.
Fig. 9. 3D plot of jΨ ρ; z j for the two inverted parabola given by
Eq. (15). Multiple trapping along the optical axis can be obtained
using FWs.
Fig. 6. Cpr;z for the constant LIP of Fig. 2 assuming a 17.5 μm
and considering negative values for the refractive index of the
particle. In contrast with Fig. 5, Cpr;z is always repulsive (pushing
the scatterer away from the laser source) and no 3D trap would be
possible for such parameters.
1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS 2591

10. Concerning the experimental generation of the FW
beams, we can cite the use of computer-generated
holograms (CGH) optically reconstructed by spatial
light modulators (SLMs), as it was made in [22,46].
More specifically, once the analytical solution of a FW
beam is known, we can use the information about its
amplitudes and phases at the initial plane to define
the complex transmittance hologram function and
perform the amplitude CGH.
Distinct values of L, Q0, and Nmax can be adopted
to create some particular FW. The efficient genera-
tion of the desired beam, however, depends on the
resolution of the SLM. If the amplitudes and phases
of the signal used to create the CGH don’t undergo
significant changes within a spatial interval of the
order of the SLM spatial resolution, then the gener-
ation process occurs without further problems.
6. Conclusions
The BSCs that correctly describe FWs in the frame-
work of the GLMT have been derived under ILA.
Because these waves are composed of a suitable
superposition of equal-frequency BBs, their LIPs
can be constructed from a continuous supply of lat-
eral energy, a behavior typical of nondiffracting
beams. Furthermore, they can be modeled in order
to provide virtually any prechosen pattern, simulta-
neously preserving the nondiffracting character of its
constituents (BBs) and enabling an additional degree
of freedom due do the axial control over the intensity.
Because of this, they certainly deserve a more pro-
found analysis of their capabilities in both long- and
short-range applications (including practical imple-
mentations), with possible restrictions of the scalar
solution provided in this paper depending on the
LIP imposed for a specific situation. In a case where
there are axicon angles (for a given BB composing the
FW) for which the paraxial approximation must be
discarded, we may still find the BSCs using ILA but
now at a higher computational cost. In fact, this
happens because the azimuth symmetry is broken
in the nonparaxial regime, and the fields of vector
BBs are described in terms of not only a zero-order
but also of higher-order Bessel functions.
Because of their propagating properties, FWs could
be advantageously used as laser beams for optical
trapping and micromanipulation, and we have also
calculated the longitudinal radiation pressure cross
sections for some specific LIPs and scatterers, includ-
ing negative refractive index spherical particles.
These cross sections are directly related to the scat-
tering forces. The results showed that single FW laser
beams could allow a 3D manipulation of organic and
biological particles, with simultaneous traps along
the optical axis, when desired. An experimental
implementation of such beams may confirm our
predictions.
The authors wish to thank FAPESP (contract nos.
2014/04867-1 and 2013/26437-6) for supporting this
work.
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