1. Photonic nanojet tethers axially trapped microspheres
Murat Muradoglu∗
and Tuck Wah Ng
Laboratory for Optics & Applied Mechanics, Monash University, Clayton VIC 3800, Australia
(Dated: September 1, 2015)
Microspheres that approach each other axially in a focused Gaussian laser beam are shown here
to have overall trapping behaviour influenced by photonic nanojets (PNJs), through simulations
based on the Debye series analysis developed within the Generalized Lorenz-Mie Theory (GLMT)
formulation. Investigations with a single microsphere indicate that PNJs are produced most effec-
tively when held below the focal point of the focused beam. In investigations with two microspheres
that approach each other, scattering forces initially tended to cause them to try to leave the trap,
but are drawn together via a strong enough connecting PNJ tether that develops between them
when they have differentiated and specific refractive indices. It is found that the tethering of three
microspheres is possible when the microsphere closest to the light source maintains a differentiated
and specific refractive index to the microsphere next to it, but this condition is not needed for the
following pair of microspheres.
Since the seminal work of Ashkin demonstrating that
three dimensional trapping of dielectric particles was pos-
sible using a single, highly focused laser beam [1], now
popularly known as optical tweezing, there has been a
wide range of applications devised based on the princi-
ple [2–6]. Through it, a considerable degree of insight has
been gained on understanding micro-rheological mechan-
ics [7–11]. In investigations involving dielectric rather
than absorbing particles, light that interacts with one
particle is able to interact with a second particle that is
in close proximity. This gives rise to inter-particle forces
that are mediated by light, which lead to the effect often
described as optical binding [12]. Unlike single particles,
two or more optically bound particles can be expected to
possess higher degrees of complexity in their interactions
[12–14].
Initial experiments on optical binding applied elon-
gated (in the transverse sense) light beams in order
to accommodate the trapping of multiple particles [12].
Counter-propagating light beams have also been shown
to be able to achieve this in the longitudinal sense [13, 14].
The modus operandi in the latter lies with getting the
scattering and gradient forces arising from each beam to
be canceled and enhanced respectively, thereby creating
an extended space to constrain the particles. Yet even
when a single focused Gaussian beam is used, two mi-
crospheres that approach each other axially can possess
unexpected physical outcomes [15, 16]. In fact, one of the
known procedures to ensure that a laser beam is aligned
is to trap two similar sized microspheres axially, such that
only of them should be visible.
Dielectric cylinders and spheres, having a radius typ-
ically less than 10µm, are able to focus light beyond
the diffraction limit by acting as some sort of super-
lens [17, 18]. This manifestation, called Photonic Nano-
jets (PNJs) due to its similarity to jets encountered in
fluid mechanics, corresponds to the specific spatial region
∗ Corresponding author: murat.neyzen@gmail.com
within the external focal waist located near the shadow
side of the light wave that is diffracted and refracted
simulatenously by the dielectric particle. This effect has
been important in the development of applications such
as high spatial resolution sensors [19–21] and information
recording devices [22].
In combination, PNJs and optical tweezers offer the
capacity to advance a number of applications such as
nanosensing [23]. Generating a PNJ within an optical
tweezer is known to be difficult since the focal point is
required to be near the surface of the microsphere to ex-
cite a PNJ [24, 25]. This is contrary to the stable equilib-
rium trapping configuration of an optical tweezer, which,
over-time, results in the microsphere being centered close
to the focal point of the laser. Recently, PNJ excitation
within an optical tweezer has been shown to be feasible
when two counter-propagating beams are used, whereby
one beam is responsible for trapping the microsphere and
the other is used to genereate the PNJ [26].
In multiple particle trapping, most efforts have been
expended to account for the case where two microspheres
approached a focused laser beam laterally [27] or in the
optical binding sense mentioned earlier. In such in-
stances, any PNJ that develops will likely not have influ-
ence on the overall trapping behaviour. The illumination
of a chain of axially affixed microspheres with the same
refractive indices by a plane wave has revealed whisper-
ing gallery mode (WGM) excitations [28]. More recently,
it has been shown that the longitudinal dimension of the
PNJ can be reduced if the microspheres are differenti-
ated, leading to improved resolution without the need
for high refractive index contrasts [29].
The situation as depicted in Fig. (1), wherein two mi-
crospheres are located axially in a single focused Gaus-
sian laser beam, forms the basis of creating a chain of
multiple microspheres that can tether to each other via a
mechanism that is unlike optical binding. Consequently,
this can give rise to a totally different optical means to fa-
cilitate micro-rheology studies. In this work, we develop
the numerical simulation needed with specific attention
paid to the generation of PNJs in order to provide in-

2. 2
FIG. 1. Ray-like decomposition of the scattering process is
obtained by the Debye series, such that when p > 0, the
scattered field has been internally reflected (p − 1) times and
when p = 0, the scattered field is caused by a combination of
diffraction and surface reflection.
sights into the dynamical light-matter interactions that
can be expected to emerge. To facilitate the investiga-
tion, the microspheres are restricted to having the same
radius R = 1.5µm and the x-polarized Gaussian beam is
focused to have a numerical aperture of NA=0.98. The
presentation of results is done here in the order of one,
two, and three microspheres being used.
I. LIGHT-MATTER INTERACTION MODEL
The ability to model the optical trapping of a single
microsphere has been widely investigated [30–33]. In tra-
ditional models of optical tweezing, such as in the work of
Ashkin [33], the particle size was assumed to be typically
a few micrometers. Hence it was possible to decompose
the optical force in terms of two contributions: (i) the
scattering force and (ii) the gradient force, which loosely
correlate to the forces in the direction of propagation and
that towards the highest gradient intensity, respectively.
The exposition of light matter interaction issues as
presented by Gouesbet et. al [34], particularly prob-
lem 23, implies that the Generalized Lorenz-Mie The-
ory (GLMT) [35] will necessarily be useful in arriving at
insights into the scattering mechanisms that lead to op-
tical trapping when the PNJ effect is considered. To this
end, the Debye series analysis, originally introduced to
account for the case of light scattering by circular cylin-
ders [36], offer the capacity to decompose the interaction
of an incident beam with a sphere into an infinite series
of interactions with the sphere boundary [37]. Consider a
physical system where N spherical particles with radius
Rj and refractive index ηj are bound within a physical
medium, such as water which has a refractive index of
η = 1.33, with their centers located at dj = (xj, zj). The
suspended particles are then illuminated with a focused
laser beam whose electric and magnetic fields can be de-
scribed as a Gaussian beam. From classical Mie theory,
where the incident field upon the particle is a plane wave,
the application of the boundary conditions at the surface
of the jth
sphere yields the scattering coefficients a
(j)
n and
b
(j)
n , given explicitly as
a(j)
n =
Ψn(αj)Ψn(βj) − MjΨn(αj)Ψn(βj)
ξn(αj)Ψn(βj) − Mjξn(αj)Ψn(βj)
b(j)
n =
MjΨn(αj)Ψn(βj) − Ψn(αj)Ψn(βj)
Mjξn(αj)Ψn(βj) − ξn(αj)Ψn(βj)
(1)
where Ψn and ξn are the Riccati-Bessel functions con-
structed from the first spherical Bessel function and first
spherical Hankel function, respectively. The parameters
αj and βj correspond to the size parameters of the jth
sphere where αj = 2πRj, βj = ηjαj/η and Mj = βj/αj.
The arbitrary incident electromagnetic beam focused at
the origin (j = 0) can be described by
E
(j=0)
beam (r) =
∞
n=1
n
m=−n
gm,j=0
n,TM M(3)
mn(αr)
+ gm,j=0
n,TE N(3)
mn(αr)
(2)
where M
(k)
mn and N
(k)
mn vector spherical wavefunctions are
constructed with spherical Hankel functions of the first
and second kind, (k = 1, 2), corresponding to incoming
and outgoing waves or they can be constructed with a
spherical Bessel function of the first kind (k = 3) which
represents a regularized wave without a singularity at the
origin. The beam intercepted by the jth
particle can be
obtained by applying a translation and rotation matrix
onto gm,j=0
n,TM and gm,j=0
n,TE to obtain gm,j
n,TM and gm,j
n,TE [35].
The plane-wave derived scattering coefficients in Eq (1)
can be related to the generalized scattering coefficients,
p
(j)
nm = a
(j)
n gm,j
n,TM, q
(j)
nm = b
(j)
n gm,j
n,TE. The scattered field of
the jth
particle,
E(j)
sca(r) =
∞
n=1
n
m=−n
p(j)
nmM(k=1)
mn (αr)
+ q(j)
nmN(k=1)
nm (αr),
(3)
can be calculated for points exterior to the sphere j. Sim-
ilarly, the internal fields E
(j)
int(r) can be calculated by re-
placing α → β and the VSWF type with k = 3. To
put the representation in a more compact form, a com-
bined index l can be introduced to represent the indices
n and m as l = n(n + 1) + m, where n = {1, 2, ..., Nmax}
and −n ≤ m ≤ n, such that Lmax = Nmax (Nmax + 2).
In the case of spheres, this combined index permits the
expression of the coefficients as vectors, so that we may